From 7443c78a4bf1834c1877fc3933d9741bc95be085 Mon Sep 17 00:00:00 2001 From: Thomas Tuerk Date: Sun, 11 Sep 2016 10:19:41 +0200 Subject: [PATCH 1/7] MSetIntervals: implement opertions more efficiently This commit provides for sets represented by interval more efficient implementations of - union - inter - diff - subset - choose --- MSetIntervals.v | 4395 +++++++++++++++++++++++++++++++++++------------ 1 file changed, 3315 insertions(+), 1080 deletions(-) diff --git a/MSetIntervals.v b/MSetIntervals.v index 03ec64e..930c8e9 100644 --- a/MSetIntervals.v +++ b/MSetIntervals.v @@ -19,6 +19,12 @@ * If not, see . *) +(* CHANGES + * + * 25-09-2016 Thomas Tuerk + * implement union, inter, diff, subset, choose efficiently + *) + (** * Weak sets implemented by interval lists This file contains an implementation of the weak set interface @@ -56,6 +62,44 @@ Proof. apply N2Z.is_nonneg. Qed. +Lemma Z_lt_add_r : forall (z : Z) (n : N), + (n <> 0)%N -> + z < z + Z.of_N n. +Proof. + move => z n H_neq_0. + suff : (z + Z.of_N 0 < z + Z.of_N n). { + rewrite Z.add_0_r //. + } + apply Z.add_lt_mono_l, N2Z.inj_lt. + by apply N.neq_0_lt_0. +Qed. + +Lemma Z_lt_le_add_r : forall y1 y2 c, + y1 < y2 -> + y1 <= y2 + Z.of_N c. +Proof. + intros y1 y2 c H. + apply Z.le_trans with (m := y2). { + by apply Z.lt_le_incl. + } { + apply Z_le_add_r. + } +Qed. + +Lemma Z_to_N_minus_neq_0 : forall (x y : Z), + y < x -> + Z.to_N (x - y) <> 0%N. +Proof. + intros x y H_y_lt. + apply N.neq_0_lt_0. + apply N2Z.inj_lt. + suff H : 0 < x - y. { + rewrite Z2N.id => //. + by apply Z.lt_le_incl. + } + by apply Z.lt_0_sub. +Qed. + Lemma add_add_sub_eq : forall (x y : Z), (x + (y - x) = y). Proof. intros x y. @@ -190,7 +234,7 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. Definition empty : t := nil. Definition is_empty (l : t) := match l with nil => true | _ => false end. - + (** Defining the list of elements, is much more tricky, especially, if it needs to be executable. *) Lemma acc_pred : forall n p, n = Npos p -> Acc N.lt n -> Acc N.lt (N.pred n). @@ -208,8 +252,6 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. | c => fun Heq => elementsZ_aux'' (x::acc) (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end (refl_equal _). - Extraction Inline elementsZ_aux''. - Definition elementsZ_aux' acc x c := elementsZ_aux'' acc x c (lt_wf_0 _). Fixpoint elementsZ_aux acc (s : t) : list Z := @@ -238,26 +280,83 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. Definition mem (x : elt) (s : t) := memZ (Enc.encode x) s. + (** Comparing intervals *) + Inductive interval_compare_result := + ICR_before + | ICR_before_touch + | ICR_overlap_before + | ICR_overlap_after + | ICR_equal + | ICR_subsume_1 + | ICR_subsume_2 + | ICR_after + | ICR_after_touch. + + Definition interval_compare (i1 i2 : (Z * N)) : interval_compare_result := + match (i1, i2) with ((y1, c1), (y2, c2)) => + let yc2 := (y2+Z.of_N c2) in + match (Z.compare yc2 y1) with + | Lt => ICR_after + | Eq => ICR_after_touch + | Gt => let yc1 := (y1+Z.of_N c1) in + match (Z.compare yc1 y2) with + | Lt => ICR_before + | Eq => ICR_before_touch + | Gt => (* y2 < y1+c1 /\ y1 < y2 + c2 *) + match (Z.compare y1 y2, Z.compare yc1 yc2) with + | (Lt, Lt) => ICR_overlap_before + | (Lt, _) => ICR_subsume_2 + | (Eq, Lt) => ICR_subsume_1 + | (Eq, Gt) => ICR_subsume_2 + | (Eq, Eq) => ICR_equal + | (Gt, Gt) => ICR_overlap_after + | (Gt, _) => ICR_subsume_1 + end + end + end + end. + + Definition interval_1_compare (y1 : Z) (i : (Z * N)) : interval_compare_result := + match i with (y2, c2) => + let yc2 := (y2+Z.of_N c2) in + match (Z.compare yc2 y1) with + | Lt => ICR_after + | Eq => ICR_after_touch + | Gt => match (Z.compare (Z.succ y1) y2) with + | Lt => ICR_before + | Eq => ICR_before_touch + | Gt => ICR_subsume_1 + end + end + end. + + (** Auxiliary functions for inserting at front and merging intervals *) + Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N := + (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))). + + Fixpoint insert_interval_begin (x : Z) (c : N) (l : t) := + match l with + | nil => (x,c)::nil + | (y, c')::l' => + match (Z.compare (x + Z.of_N c) y) with + | Lt => (x, c) :: l + | Eq => (x, (c+c')%N) :: l' + | Gt => insert_interval_begin x (merge_interval_size x c y c') l' + end + end. + (** adding an element needs to be defined carefully again in order to generate efficient code *) Fixpoint addZ_aux (acc : list (Z * N)) (x : Z) (s : t) := match s with | nil => List.rev' ((x, (1%N))::acc) | (y, c) :: l => - match (Z.compare (Z.succ x) y) with - | Lt => List.rev_append ((x, (1%N))::acc) s - | Eq => List.rev_append ((x, N.succ c)::acc) l - | Gt => match (Z.compare x (y+Z.of_N c)) with - | Lt => List.rev_append acc s - | Gt => addZ_aux ((y,c) :: acc) x l - | Eq => match l with - | nil => List.rev' ((y, N.succ c)::acc) - | (z, c') :: l' => if (Z.eqb z (Z.succ x)) then - List.rev_append ((y,N.succ (c+c')) :: acc) l' - else - List.rev_append ((y,N.succ c) :: acc) l - end - end + match (interval_1_compare x (y,c)) with + | ICR_before => List.rev_append ((x, (1%N))::acc) s + | ICR_before_touch => List.rev_append ((x, N.succ c)::acc) l + | ICR_after => addZ_aux ((y,c) :: acc) x l + | ICR_after_touch => List.rev_append acc (insert_interval_begin y (N.succ c) l) + | _ => List.rev_append ((y, c)::acc) l end end. @@ -279,7 +378,7 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. (** removing needs to be done with code extraction in mind again. *) - Definition removeZ_aux_insert_guarded (x : Z) (c : N) s := + Definition insert_intervalZ_guarded (x : Z) (c : N) s := if (N.eqb c 0) then s else (x, c) :: s. Fixpoint removeZ_aux (acc : list (Z * N)) (x : Z) (s : t) : t := @@ -288,9 +387,9 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. | (y, c) :: l => if (Z.ltb x y) then List.rev_append acc s else if (Z.ltb x (y+Z.of_N c)) then ( - List.rev_append (removeZ_aux_insert_guarded (Z.succ x) + List.rev_append (insert_intervalZ_guarded (Z.succ x) (Z.to_N ((y+Z.of_N c)- (Z.succ x))) - (removeZ_aux_insert_guarded y (Z.to_N (x-y)) acc)) l + (insert_intervalZ_guarded y (Z.to_N (x-y)) acc)) l ) else removeZ_aux ((y,c)::acc) x l end. @@ -300,30 +399,130 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. Definition remove_list (l : list elt) (s : t) : t := List.fold_left (fun s x => remove x s) l s. - (** all other operations are defined trivially (if not always efficiently) - in terms of already defined ones. In the future it might be worth implementing - some of them more efficiently. *) - Definition union (s1 s2 : t) := - add_list (elements s1) s2. + (** union *) + Fixpoint union_aux (s1 : t) := + fix aux (s2 : t) (acc : list (Z * N)) := + match (s1, s2) with + | (nil, _) => List.rev_append acc s2 + | (_, nil) => List.rev_append acc s1 + | ((y1, c1) :: l1, (y2, c2) :: l2) => + match (interval_compare (y1, c1) (y2,c2)) with + | ICR_before => union_aux l1 s2 ((y1, c1)::acc) + | ICR_before_touch => + union_aux l1 ( + insert_interval_begin y1 ((c1+c2)%N) l2) acc + | ICR_after => aux l2 ((y2, c2)::acc) + | ICR_after_touch => union_aux l1 ( + insert_interval_begin y2 ((c1+c2)%N) l2) acc + | ICR_overlap_before => + union_aux l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2) acc + | ICR_overlap_after => + union_aux l1 (insert_interval_begin y2 (merge_interval_size y2 c2 y1 c1) l2) acc + | ICR_equal => union_aux l1 s2 acc + | ICR_subsume_1 => union_aux l1 s2 acc + | ICR_subsume_2 => aux l2 acc + end + end. + + Definition union s1 s2 := union_aux s1 s2 nil. + + (** diff *) - Definition filter (f : elt -> bool) (s : t) : t := - from_elements (List.filter f (elements s)). - Definition inter (s1 s2 : t) : t := - filter (fun x => mem x s2) s1. + Fixpoint diff_aux (y2 : Z) (c2 : N) (acc : list (Z * N)) (s : t) : (list (Z * N) * t) := + match s with + | nil => (acc, nil) + | ((y1, c1) :: l1) => + match (interval_compare (y1, c1) (y2,c2)) with + | ICR_before => diff_aux y2 c2 ((y1, c1)::acc) l1 + | ICR_before_touch => diff_aux y2 c2 ((y1, c1)::acc) l1 + | ICR_after => (acc, s) + | ICR_after_touch => (acc, s) + | ICR_overlap_before => diff_aux y2 c2 ((y1, Z.to_N (y2 - y1))::acc) l1 + | ICR_overlap_after => (acc, (y2+Z.of_N c2, Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) :: l1) + | ICR_equal => (acc, l1) + | ICR_subsume_1 => diff_aux y2 c2 acc l1 + | ICR_subsume_2 => ((insert_intervalZ_guarded y1 + (Z.to_N (y2 - y1)) acc), + insert_intervalZ_guarded (y2+Z.of_N c2) (Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) l1) + end + end. - Definition diff (s1 s2 : t) : t := - remove_list (elements s2) s1. + Fixpoint diff_aux2 (acc : list (Z * N)) (s1 s2 : t) : (list (Z * N)) := + match (s1, s2) with + | (nil, _) => rev_append acc s1 + | (_, nil) => rev_append acc s1 + | (_, (y2, c2) :: l2) => + match diff_aux y2 c2 acc s1 with + (acc', s1') => diff_aux2 acc' s1' l2 + end + end. - Definition subset s s' := - List.forallb (fun x => mem x s') (elements s). + Definition diff s1 s2 := diff_aux2 nil s1 s2. + + (** subset *) + Fixpoint subset (s1 : t) := + fix aux (s2 : t) := + match (s1, s2) with + | (nil, _) => true + | (_ :: _, nil) => false + | ((y1, c1) :: l1, (y2, c2) :: l2) => + match (interval_compare (y1, c1) (y2,c2)) with + | ICR_before => false + | ICR_before_touch => false + | ICR_after => aux l2 + | ICR_after_touch => false + | ICR_overlap_before => false + | ICR_overlap_after => false + | ICR_equal => subset l1 l2 + | ICR_subsume_1 => subset l1 s2 + | ICR_subsume_2 => false + end + end. + (** equal *) Fixpoint equal (s s' : t) : bool := match s, s' with | nil, nil => true | ((x,cx)::xs), ((y,cy)::ys) => andb (Z.eqb x y) (andb (N.eqb cx cy) (equal xs ys)) | _, _ => false end. + (** inter *) + Fixpoint inter_aux (y2 : Z) (c2 : N) (acc : list (Z * N)) (s : t) : (list (Z * N) * t) := + match s with + | nil => (acc, nil) + | ((y1, c1) :: l1) => + match (interval_compare (y1, c1) (y2,c2)) with + | ICR_before => inter_aux y2 c2 acc l1 + | ICR_before_touch => inter_aux y2 c2 acc l1 + | ICR_after => (acc, s) + | ICR_after_touch => (acc, s) + | ICR_overlap_before => inter_aux y2 c2 ((y2, Z.to_N (y1 + Z.of_N c1 - y2))::acc) l1 + | ICR_overlap_after => ((y1, Z.to_N (y2 + Z.of_N c2 - y1))::acc, s) + | ICR_equal => ((y1,c1)::acc, l1) + | ICR_subsume_1 => inter_aux y2 c2 ((y1, c1)::acc) l1 + | ICR_subsume_2 => ((y2, c2)::acc, s) + end + end. + + Fixpoint inter_aux2 (acc : list (Z * N)) (s1 s2 : t) : (list (Z * N)) := + match (s1, s2) with + | (nil, _) => List.rev' acc + | (_, nil) => List.rev' acc + | (_, (y2, c2) :: l2) => + match inter_aux y2 c2 acc s1 with + (acc', s1') => inter_aux2 acc' s1' l2 + end + end. + + Definition inter s1 s2 := inter_aux2 nil s1 s2. + + + (** Simple wrappers *) + + Definition filter (f : elt -> bool) (s : t) : t := + from_elements (List.filter f (elements s)). + Definition fold {B : Type} (f : elt -> B -> B) (s : t) (i : B) : B := List.fold_left (flip f) (elements s) i. @@ -336,7 +535,6 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. Definition partition (f : elt -> bool) (s : t) : t * t := (filter f s, filter (fun x => negb (f x)) s). - Fixpoint cardinalN c (s : t) : N := match s with | nil => c | (_,cx)::xs => cardinalN (c + cx)%N xs @@ -345,15 +543,15 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s). Definition chooseZ (s : t) : option Z := - match List.rev' (elementsZ s) with + match s with | nil => None - | x :: _ => Some x + | (x, _) :: _ => Some x end. Definition choose (s : t) : option elt := - match elements s with - | nil => None - | e :: _ => Some e + match (chooseZ s) with + | None => None + | Some x => Some (Enc.decode x) end. End Ops. @@ -370,23 +568,23 @@ Module Raw (Enc : ElementEncode). Include (Ops Enc). (** *** Defining invariant [IsOk] *) + Definition is_encoded_elems_list (l : list Z) : Prop := + (forall x, List.In x l -> exists e, Enc.encode e = x). - Definition inf (x:Z) (l: t) := - match l with - | nil => true - | (y,_)::_ => Z.ltb x y - end. + Definition interval_list_elements_greater (x : Z) (l : t) : bool := + match l with + | nil => true + | (y, _)::_ => Z.ltb x y + end. - Fixpoint isok (l : t) := + Fixpoint interval_list_invariant (l : t) := match l with | nil => true - | (x, c) ::l => inf (x+(Z.of_N c)) l && negb (N.eqb c 0) && isok l + | (x, c) :: l' => + interval_list_elements_greater (x + (Z.of_N c)) l' && negb (N.eqb c 0) && interval_list_invariant l' end. - Definition is_encoded_elems_list (l : list Z) : Prop := - (forall x, List.In x l -> exists e, Enc.encode e = x). - - Definition IsOk s := (isok s = true /\ is_encoded_elems_list (elementsZ s)). + Definition IsOk s := (interval_list_invariant s = true /\ is_encoded_elems_list (elementsZ s)). (** *** Defining notations *) @@ -453,7 +651,6 @@ Module Raw (Enc : ElementEncode). } Qed. - Lemma elementsZ_single_succ_front : forall x c, elementsZ_single x (N.succ c) = x :: elementsZ_single (Z.succ x) c. @@ -471,32 +668,23 @@ Module Raw (Enc : ElementEncode). induction c as [| c' IH] using N.peano_ind. { intros y x. rewrite elementsZ_single_base Z.add_0_r /=. - split; first done. - move => [H_x_le H_y_lt]. - have := Z.le_lt_trans _ _ _ H_x_le H_y_lt. - apply Z.lt_irrefl. + omega. } { intros y x. rewrite elementsZ_single_succ in_app_iff IH /= N2Z.inj_succ Z.add_succ_r Z.lt_succ_r. split. { - move => []. { - move => [H_x_le H_y_le]. - split; first done. - by apply Z.lt_le_incl. - } - move => [] // <-. - split. { - apply Z_le_add_r. + move => [ | []] //. { + move => [H_x_le H_y_le]. + omega. } { - by apply Z.le_refl. + move => <-. + split. + - by apply Z_le_add_r. + - by apply Z.le_refl. } } { move => [H_x_le] H_y_lt. - apply Z.lt_eq_cases in H_y_lt as [H_y_lt | <-]. { - by left. - } { - by right; left. - } + omega. } } Qed. @@ -506,13 +694,8 @@ Module Raw (Enc : ElementEncode). (x = y). Proof. intros y x. - rewrite In_elementsZ_single /= Z.add_1_r Z.lt_succ_r. - split. { - move => [? ?]. by apply Z.le_antisymm. - } { - move => ->. - split; apply Z.le_refl. - } + rewrite In_elementsZ_single /= Z.add_1_r Z.lt_succ_r. + omega. Qed. Lemma length_elementsZ_single : forall cx x, @@ -625,6 +808,83 @@ Module Raw (Enc : ElementEncode). rewrite !rev_map_alt_def map_app rev_app_distr map_rev rev_involutive //. Qed. + Lemma elementsZ_app : forall (s1 s2 : t), elementsZ (s1 ++ s2) = + ((elementsZ s2) ++ (elementsZ s1)). + Proof. + induction s1 as [| [x1 c1] s1 IH1]. { + move => s2. + rewrite elementsZ_nil app_nil_r //. + } + move => s2. + rewrite -app_comm_cons !elementsZ_cons IH1 -app_assoc //. + Qed. + + Lemma InZ_nil : forall y, InZ y nil <-> False. + Proof. + intro y. + done. + Qed. + + Lemma InZ_cons : forall y x c s, InZ y (((x, c) :: s) : t) <-> + List.In y (elementsZ_single x c) \/ InZ y s. + Proof. + intros y x c s. + rewrite /InZ elementsZ_cons in_app_iff -in_rev. + firstorder. + Qed. + + Lemma InZ_app : forall s1 s2 y, + InZ y (s1 ++ s2) <-> InZ y s1 \/ InZ y s2. + Proof. + intros s1 s2 y. + rewrite /InZ elementsZ_app in_app_iff. + tauto. + Qed. + + Lemma InZ_rev : forall s y, + InZ y (List.rev s) <-> InZ y s. + Proof. + intros s x. + rewrite /InZ. + induction s as [| [y c] s' IH]; first done. + simpl. + rewrite elementsZ_app in_app_iff IH. + rewrite !elementsZ_cons !in_app_iff elementsZ_nil + -!in_rev /=. + tauto. + Qed. + + Lemma In_elementsZ_single_dec : forall y x c, + {List.In y (elementsZ_single x c)} + + {~ List.In y (elementsZ_single x c)}. + Proof. + intros y x c. + case (Z_le_dec x y); last first. { + right; rewrite In_elementsZ_single; tauto. + } + case (Z_lt_dec y (x + Z.of_N c)); last first. { + right; rewrite In_elementsZ_single; tauto. + } + left; rewrite In_elementsZ_single; tauto. + Qed. + + Lemma InZ_dec : forall y s, + {InZ y s} + {~InZ y s}. + Proof. + intros y. + induction s as [| [x c] s IH]. { + by right. + } + move : IH => [] IH. { + by left; rewrite InZ_cons; right. + } + case (In_elementsZ_single_dec y x c). { + by left; rewrite InZ_cons; left. + } { + by right; rewrite InZ_cons; tauto. + } + Qed. + Lemma In_elementsZ_single_hd : forall (c : N) x, (c <> 0)%N -> List.In x (elementsZ_single x c). Proof. intros c x H_c_neq. @@ -640,274 +900,248 @@ Module Raw (Enc : ElementEncode). Qed. - (** *** Alternative definition of addZ *) - - (** [addZ] is defined with efficient execution in mind. - We derive first an alternative definition that demonstrates - the intention better and is better suited for proofs. *) - Lemma addZ_ind : - forall (P : Z -> list (Z * N) -> Prop), - (* case s = nil *) - (forall (x : Z), P x nil) -> - - (* case s = [y, c] :: l, compare (x+1) y = Eq *) - (forall (x : Z) (l : list (Z * N)) (c : N), - P x ((x + 1, c) :: l)) -> - - (* case s = [y, c] :: l, compare (x+1) y = Lt *) - (forall (x : Z) (l : list (Z * N)) (y : Z) (c : N), - (x + 1 ?= y) = Lt -> - P x ((y, c) :: l)) -> - - (* case s = [y, c] :: l, compare (x+1) y = Gt, compare x (y+c) = Eq, - l = nil *) - (forall (y : Z) (c : N), - ((y + Z.of_N c) + 1 ?= y) = Gt -> - P (y + Z.of_N c) ((y, c) :: nil)) -> - - (* case s = (y, c) :: (z, c') :: l, compare (x+1) y = Gt, compare x (y+c) = Eq, - (z = x + 1) *) - (forall (l : list (Z * N)) (y : Z) (c c' : N), - ((y + Z.of_N c) + 1 ?= y) = Gt -> - (P (y+Z.of_N c) l) -> - P (y+Z.of_N c) ((y, c) :: (((y+Z.of_N c) + 1, c') :: l))) -> - - (* case s = (y, c) :: (z, c') :: l, compare (x+1) y = Gt, compare x (y+c) = Eq, - (z <> x + 1) *) - (forall (l : list (Z * N)) (y : Z) (c : N) (z : Z) (c' : N), - ((y + Z.of_N c) + 1 ?= y) = Gt -> - (z =? (y+Z.of_N c) + 1) = false -> - (P (y+Z.of_N c) ((y, c) :: (z, c') :: l))) -> - - - (* case s = (y, c) :: l, compare (x+1) y = Gt, compare x (y+c) = Lt *) - (forall (x : Z) (l : list (Z * N)) (y : Z) (c : N), - (x + 1 ?= y) = Gt -> - (x ?= y + Z.of_N c) = Lt -> - P x ((y, c) :: l)) -> - - (* case s = (y, c) :: l, compare (x+1) y = Gt, compare x (y+c) = Lt *) - (forall (x : Z)(l : list (Z * N)) (y : Z) (c : N), - (x + 1 ?= y) = Gt -> - (x ?= y + (Z.of_N c)) = Gt -> - (P x l) -> - P x ((y, c) :: l)) -> - - - forall (x : Z) (s : list (Z * N)), - P x s. - Proof. - intros P H1 H2 H3 H4 H5 H6 H7 H8. - move => x s. - remember (length s) as n. - revert s Heqn. - induction n as [n IH] using Wf_nat.lt_wf_ind. - case. { - move => _. apply H1. - } { - move => [y c] s' H_n. - simpl in H_n. - have IH_s': (P x s'). { - apply (IH (length s')) => //. - rewrite H_n //. - } - case_eq (x + 1 ?= y). { - move => /Z.compare_eq_iff <-. - apply H2. - } { - move => H_x1y_comp. - by apply H3. + (** *** comparing intervals *) + + Ltac Z_named_compare_cases H := match goal with + | [ |- context [Z.compare ?z1 ?z2] ] => + case_eq (Z.compare z1 z2); [move => /Z.compare_eq_iff | move => /Z.compare_lt_iff | move => /Z.compare_gt_iff]; move => H // + end. + + Ltac Z_compare_cases := let H := fresh "H" in Z_named_compare_cases H. + + Lemma interval_compare_elim : forall (y1 : Z) (c1 : N) (y2 : Z) (c2 : N), + match (interval_compare (y1, c1) (y2, c2)) with + | ICR_before => (y1 + Z.of_N c1) < y2 + | ICR_before_touch => (y1 + Z.of_N c1) = y2 + | ICR_after => (y2 + Z.of_N c2) < y1 + | ICR_after_touch => (y2 + Z.of_N c2) = y1 + | ICR_equal => (y1 = y2) /\ (c1 = c2) + | ICR_overlap_before => (y1 < y2) /\ (y2 < y1 + Z.of_N c1) /\ (y1 + Z.of_N c1 < y2 + Z.of_N c2) + | ICR_overlap_after => (y2 < y1) /\ (y1 < y2 + Z.of_N c2) /\ (y2 + Z.of_N c2 < y1 + Z.of_N c1) + | ICR_subsume_1 => (y2 <= y1) /\ (y1 + Z.of_N c1 <= y2 + Z.of_N c2) /\ (y2 < y1 \/ y1 + Z.of_N c1 < y2 + Z.of_N c2) + | ICR_subsume_2 => (y1 <= y2) /\ (y2 + Z.of_N c2 <= y1 + Z.of_N c1) /\ (y1 < y2 \/ y2 + Z.of_N c2 < y1 + Z.of_N c1) + end. + Proof. + intros y1 c1 y2 c2. + rewrite /interval_compare. + (repeat Z_compare_cases); subst; repeat split; + try (by apply Z.eq_le_incl); + try (by apply Z.lt_le_incl); + try (by left); try (by right). + + apply Z.add_reg_l in H2. + by apply N2Z.inj. + Qed. + + Lemma interval_compare_swap : forall (y1 : Z) (c1 : N) (y2 : Z) (c2 : N), + (c1 <> 0%N) \/ (c2 <> 0%N) -> + interval_compare (y2, c2) (y1, c1) = + match (interval_compare (y1, c1) (y2, c2)) with + | ICR_before => ICR_after + | ICR_before_touch => ICR_after_touch + | ICR_after => ICR_before + | ICR_after_touch => ICR_before_touch + | ICR_equal => ICR_equal + | ICR_overlap_before => ICR_overlap_after + | ICR_overlap_after => ICR_overlap_before + | ICR_subsume_1 => ICR_subsume_2 + | ICR_subsume_2 => ICR_subsume_1 + end. + Proof. + intros y1 c1 y2 c2 H_c12_neq_0. + rewrite /interval_compare. + move : (Z.compare_antisym y1 y2) => ->. + move : (Z.compare_antisym (y1 + Z.of_N c1) (y2 + Z.of_N c2)) => ->. + have H_suff : y1 + Z.of_N c1 <= y2 -> y2 + Z.of_N c2 <= y1 -> False. { + move => H1 H2. + case H_c12_neq_0 => H_c_neq_0. { + suff : (y1 + Z.of_N c1 <= y1). { + apply Z.nle_gt. + by apply Z_lt_add_r. + } + eapply Z.le_trans; eauto. + eapply Z.le_trans; eauto. + apply Z_le_add_r. } { - move => H_x1y_comp. - case_eq (x ?= y + Z.of_N c). { - move => H_x_eqb. - move : (H_x_eqb) => /Z.compare_eq_iff H_x_eq. - case_eq s'. { - move => _. - rewrite H_x_eq. - apply H4. - rewrite -H_x1y_comp H_x_eq //. - } { - move => [z c'] l H_s'. - have IH_l: (P x l). { - apply (IH (length l)) => //. - rewrite H_n H_s' /=. - apply Lt.lt_trans with (m := S (length l)) => //. - } - case_eq (z =? x + 1). { - move => /Z.eqb_eq ->. - rewrite H_x_eq. - apply H5. { - rewrite -H_x1y_comp H_x_eq //. - } { - rewrite -H_x_eq. - apply IH_l. - } - } { - move => H_z_neq. - rewrite H_x_eq. - eapply H6 => //; rewrite -H_x_eq //. - } - } - } { - by apply H7. - } { - move => H_x_gt. - apply H8 => //. + suff : (y2 + Z.of_N c2 <= y2). { + apply Z.nle_gt. + by apply Z_lt_add_r. } + eapply Z.le_trans; eauto. + eapply Z.le_trans; eauto. + apply Z_le_add_r. } } - Qed. - - Lemma addZ_aux_alt_def : forall x s acc, - addZ_aux acc x s = (List.rev acc) ++ addZ x s. - Proof. - move => ? ?. - eapply addZ_ind with (P := (fun x s => forall acc, addZ_aux acc x s = rev acc ++ addZ x s)); clear. { - intros x acc. - rewrite /addZ_aux /rev' -rev_alt //. + repeat Z_compare_cases. { + exfalso; apply H_suff. + - by rewrite H; apply Z.le_refl. + - by rewrite H0; apply Z.le_refl. } { - intros x l c acc. - rewrite /addZ /= Z.compare_refl rev_append_rev //. + exfalso; apply H_suff. + - by rewrite H; apply Z.le_refl. + - by apply Z.lt_le_incl. } { - intros x l y c H_y_lt acc. - rewrite /addZ /= H_y_lt rev_append_rev //. + exfalso; apply H_suff. + - by apply Z.lt_le_incl. + - by rewrite H0; apply Z.le_refl. } { - intros y c H_y_comp acc. - rewrite /addZ /= H_y_comp Z.compare_refl /rev' -rev_alt //. - } { - intros l y c c' H_y_comp IH acc. - rewrite /addZ /= H_y_comp Z.compare_refl Z.eqb_refl. - rewrite rev_append_rev //. - } { - intros l y c z c' H_y_comp H_z_comp acc. - rewrite /addZ /= H_y_comp Z.compare_refl H_z_comp. - rewrite rev_append_rev /= //. - } { - intros x l y c H_y_comp H_x_comp acc. - rewrite /addZ /= H_x_comp H_y_comp rev_append_rev //. - } { - intros x l y c H_y_comp H_x_comp IH acc. - rewrite /addZ /= H_x_comp H_y_comp. - rewrite !IH /= -app_assoc //. + exfalso; apply H_suff. + - by apply Z.lt_le_incl. + - by apply Z.lt_le_incl. } Qed. - - Lemma addZ_alt_def : forall x s, - addZ x s = - match s with - | nil => (x, 1%N)::nil - | (y, c) :: l => - match (Z.compare (x+1) y) with - | Lt => (x, 1%N)::s - | Eq => (x, (c+1)%N)::l - | Gt => match (Z.compare x (y+Z.of_N c)) with - | Lt => s - | Gt => (y,c) :: addZ x l - | Eq => match l with - | nil => (y, (c+1)%N)::nil - | (z, c') :: l' => if (Z.eqb z (x + 1)) then - (y, (c + c' + 1)%N) :: l' - else - (y,(c+1)%N) :: (z, c') :: l' - end - end - end + + Lemma interval_1_compare_alt_def : forall (y : Z) (i : (Z * N)), + interval_1_compare y i = match (interval_compare (y, (1%N)) i) with + | ICR_equal => ICR_subsume_1 + | ICR_subsume_1 => ICR_subsume_1 + | ICR_subsume_2 => ICR_subsume_1 + | r => r end. Proof. - intros x s. - rewrite /addZ Z.add_1_r. - case s => //. - move => [y c] s' /=. - case (Z.succ x ?= y). { - rewrite N.add_1_r //. - } { - reflexivity. - } { - case (x ?= y+ Z.of_N c) => //. { - rewrite !N.add_1_r. - case s' => //. - move => [z c'] s'' //. - rewrite !N.add_1_r //. - } { - rewrite addZ_aux_alt_def //. - } + move => y1 [y2 c2]. + rewrite /interval_1_compare /interval_compare. + replace (y1 + Z.of_N 1) with (Z.succ y1); last done. + repeat Z_compare_cases. { + contradict H1. + by apply Zle_not_lt, Zlt_succ_le. + } { + contradict H. + by apply Zle_not_lt, Zlt_succ_le. } Qed. - - - (** *** Alternative definition of removeZ *) - (** [removeZ] is defined with efficient execution in mind. - We derive first an alternative definition that demonstrates - the intention better and is better suited for proofs. *) - - Lemma removeZ_aux_alt_def : forall s x acc, - removeZ_aux acc x s = (List.rev acc) ++ removeZ x s. + Lemma interval_1_compare_elim : forall (y1 : Z) (y2 : Z) (c2 : N), + match (interval_1_compare y1 (y2, c2)) with + | ICR_before => Z.succ y1 < y2 + | ICR_before_touch => y2 = Z.succ y1 + | ICR_after => (y2 + Z.of_N c2) < y1 + | ICR_after_touch => (y2 + Z.of_N c2) = y1 + | ICR_equal => False + | ICR_overlap_before => False + | ICR_overlap_after => False + | ICR_subsume_1 => (c2 = 0%N) \/ ((y2 <= y1) /\ (y1 < y2 + Z.of_N c2)) + | ICR_subsume_2 => False + end. Proof. - induction s as [| [y c] s' IH]. { - intros x acc. - rewrite /removeZ /removeZ_aux /= app_nil_r /rev' -rev_alt //. + intros y1 y2 c2. + move : (interval_compare_elim y1 (1%N) y2 c2). + rewrite interval_1_compare_alt_def. + have H_succ: forall z, z + Z.of_N 1 = Z.succ z by done. + + case_eq (interval_compare (y1, 1%N) (y2, c2)) => H_comp; + rewrite ?H_succ ?Z.lt_succ_r ?Z.le_succ_l //. { + move => [H_lt] [H_le] _. + contradict H_lt. + by apply Zle_not_lt. + } { + move => [_] [H_lt] H_le. + contradict H_lt. + by apply Zle_not_lt. } { - intros x acc. - rewrite /removeZ /removeZ_aux -/removeZ_aux. - case (x [->] <-. + rewrite ?Z.lt_succ_r. + right. + split; apply Z.le_refl. + } { + tauto. + } { + case (N.zero_or_succ c2). { + move => -> _; by left. } { - case (x [c2'] ->. + rewrite !N2Z.inj_succ Z.add_succ_r -Z.succ_le_mono Z.le_succ_l. + move => [H_y1_le] [H_le_y1]. + suff -> : y1 = y2. { + move => [] H_pre; contradict H_pre. { + apply Z.lt_irrefl. + } { + apply Zle_not_lt, Z_le_add_r. + } } + apply Z.le_antisymm => //. + eapply Z.le_trans; last apply H_le_y1. + apply Z_le_add_r. } } Qed. - - Lemma removeZ_alt_def : forall x s, - removeZ x s = + + (** *** Alternative definition of addZ *) + Lemma addZ_aux_alt_def : forall x s acc, + addZ_aux acc x s = (List.rev acc) ++ addZ x s. + Proof. + intros y1 s. + unfold addZ. + induction s as [| [y2 c2] s' IH] => acc. { + rewrite /addZ_aux /addZ /= /rev' !rev_append_rev /= app_nil_r //. + } { + unfold addZ_aux. + case (interval_1_compare y1 (y2, c2)); fold addZ_aux; + rewrite ?rev_append_rev /= ?app_assoc_reverse //. + rewrite (IH ((y2,c2)::acc)) (IH ((y2,c2)::nil)). + rewrite /= app_assoc_reverse //. + } + Qed. + + Lemma addZ_alt_def : forall x s, + addZ x s = match s with - | nil => nil + | nil => (x, (1%N))::nil | (y, c) :: l => - if (Z.ltb x y) then s else - if (Z.ltb x (y+Z.of_N c)) then ( - (removeZ_aux_insert_guarded y (Z.to_N (x-y)) - (removeZ_aux_insert_guarded (Z.succ x) (Z.to_N ((y+Z.of_N c)- (Z.succ x))) l)) - ) else (y, c) :: removeZ x l + match (interval_1_compare x (y,c)) with + | ICR_before => (x, (1%N))::s + | ICR_before_touch => (x, N.succ c)::l + | ICR_after => (y,c)::(addZ x l) + | ICR_after_touch => insert_interval_begin y (N.succ c) l + | _ => (y, c)::l + end end. Proof. intros x s. - rewrite /removeZ /=. + rewrite /addZ. case s => //. - move => [y c] s' /=. - case(x //. - case(x //. - } { - rewrite removeZ_aux_alt_def //. - } + move => [y c] s'. + unfold addZ_aux. + case (interval_1_compare x (y, c)); fold addZ_aux; + rewrite ?rev_append_rev /= ?app_assoc_reverse //. + rewrite addZ_aux_alt_def //. Qed. - + + (** *** Auxiliary Lemmata about Invariant *) - Lemma inf_impl : forall x y s, - (y <= x) -> inf x s = true -> inf y s = true. + Lemma interval_list_elements_greater_cons : forall z x c s, + interval_list_elements_greater z ((x, c) :: s) = true <-> + (z < x). + Proof. + intros z x c s. + rewrite /=. + apply Z.ltb_lt. + Qed. + + Lemma interval_list_elements_greater_impl : forall x y s, + (y <= x) -> + interval_list_elements_greater x s = true -> + interval_list_elements_greater y s = true. Proof. intros x y s. case s => //. move => [z c] s'. - rewrite /inf. + rewrite /interval_list_elements_greater. move => H_y_leq /Z.ltb_lt H_x_lt. apply Z.ltb_lt. eapply Z.le_lt_trans; eauto. Qed. + Lemma interval_list_invariant_nil : interval_list_invariant nil = true. + Proof. + by []. + Qed. + Lemma Ok_nil : Ok nil <-> True. Proof. - rewrite /Ok /IsOk /isok /is_encoded_elems_list //. + rewrite /Ok /IsOk /interval_list_invariant /is_encoded_elems_list //. Qed. Lemma is_encoded_elems_list_app : forall l1 l2, @@ -934,10 +1168,12 @@ Module Raw (Enc : ElementEncode). ). Qed. - Lemma isok_cons : forall y c s', isok ((y, c) :: s') = true <-> - (inf (y+Z.of_N c) s' = true /\ ((c <> 0)%N) /\ isok s' = true). + Lemma interval_list_invariant_cons : forall y c s', + interval_list_invariant ((y, c) :: s') = true <-> + (interval_list_elements_greater (y+Z.of_N c) s' = true /\ + ((c <> 0)%N) /\ interval_list_invariant s' = true). Proof. - rewrite /isok -/isok. + rewrite /interval_list_invariant -/interval_list_invariant. intros y c s'. rewrite !Bool.andb_true_iff negb_true_iff. split. { @@ -947,148 +1183,318 @@ Module Raw (Enc : ElementEncode). } Qed. + Lemma interval_list_invariant_sing : forall x c, + interval_list_invariant ((x, c)::nil) = true <-> (c <> 0)%N. + Proof. + intros x c. + rewrite interval_list_invariant_cons. + split; tauto. + Qed. + Lemma Ok_cons : forall y c s', Ok ((y, c) :: s') <-> - (inf (y+Z.of_N c) s' = true /\ ((c <> 0)%N) /\ + (interval_list_elements_greater (y+Z.of_N c) s' = true /\ ((c <> 0)%N) /\ is_encoded_elems_list (elementsZ_single y c) /\ Ok s'). Proof. intros y c s'. - rewrite /Ok /IsOk isok_cons elementsZ_cons is_encoded_elems_list_app + rewrite /Ok /IsOk interval_list_invariant_cons elementsZ_cons is_encoded_elems_list_app is_encoded_elems_list_rev. tauto. Qed. Lemma Nin_elements_greater : forall s y, - inf y s = true -> - isok s = true -> - forall x, x <= y -> - ~(InZ x s). + interval_list_elements_greater y s = true -> + interval_list_invariant s = true -> + forall x, x <= y -> ~(InZ x s). Proof. induction s as [| [z c] s' IH]. { intros y _ _ x _. - done. + by simpl. } { - rewrite /inf. - move => y /Z.ltb_lt H_y_lt /isok_cons [H_inf'] [H_c] H_s' x H_x_le. - rewrite /InZ elementsZ_cons in_app_iff -!in_rev In_elementsZ_single. - move => []. { - eapply IH; eauto. - apply Z.le_trans with (m := z). { - eapply Z.lt_le_incl. - eapply Z.le_lt_trans; eauto. - } { - apply Z_le_add_r. - } - } { + move => y /interval_list_elements_greater_cons H_y_lt + /interval_list_invariant_cons [H_gr] [H_c] H_s' + x H_x_le. + rewrite InZ_cons In_elementsZ_single. + have H_x_lt : x < z by eapply Z.le_lt_trans; eauto. + + move => []. { move => [H_z_leq] _; contradict H_z_leq. - apply Z.nle_gt. - eapply Z.le_lt_trans; eauto. + by apply Z.nle_gt. + } { + eapply IH; eauto. + by apply Z_lt_le_add_r. } } Qed. - Lemma isok_inf_nin : + Lemma Nin_elements_greater_equal : forall x s, - isok s = true -> - inf x s = true -> + interval_list_elements_greater x s = true -> + interval_list_invariant s = true -> ~ (InZ x s). Proof. - move => x. - induction s as [| [y c] s' IH]. { - move => _ _. - rewrite /InZ elementsZ_nil //. - } { - rewrite isok_cons. - move => [H_inf] [_] H_ok_s' /Z.ltb_lt H_x_lt_y. - rewrite /InZ elementsZ_cons in_app_iff -in_rev In_elementsZ_single Z.le_ngt. - suff : inf x s' = true by tauto. - eapply inf_impl; last apply H_inf. - apply Z.le_trans with (m := y). { - by apply Z.lt_le_incl. - } { - by apply Z_le_add_r. - } - } + move => x s H_inv H_gr. + apply (Nin_elements_greater s x) => //. + apply Z.le_refl. Qed. - (** *** Properties of In and InZ *) + Lemma interval_list_elements_greater_alt_def : forall s y, + interval_list_invariant s = true -> - Lemma In_alt_def : forall x s, Ok s -> - (In x s <-> List.In x (elements s)). + (interval_list_elements_greater y s = true <-> + (forall x, x <= y -> ~(InZ x s))). Proof. - intros x s H_ok. - rewrite /In InA_alt /elements rev_map_alt_def. + intros s y H_inv. split. { - move => [y] [H_y_eq]. - rewrite -!in_rev !in_map_iff. - move => [x'] [H_y_eq'] H_x'_in. - suff H_x'_eq : (Enc.encode x = x'). { - exists x'. - split => //. - rewrite -H_x'_eq Enc.decode_encode_ok //. - } - have H_enc_list : is_encoded_elems_list (elementsZ s). { - move : H_ok. - rewrite /Ok /IsOk => [] [] //. - } - move : (H_enc_list _ H_x'_in) => [x''] H_x'_eq. - move : H_y_eq'. - rewrite -!H_x'_eq Enc.decode_encode_ok => H_y_eq'. - subst. - suff -> : Z.eq (Enc.encode x) (Enc.encode y) by done. - by rewrite Enc.encode_eq. + move => H_gr. + apply Nin_elements_greater => //. } { - move => H_enc_in. - exists x. - split => //. - apply Enc.E.eq_equiv. + move : H_inv. + case s as [| [x2 c] s'] => //. + rewrite interval_list_invariant_cons interval_list_elements_greater_cons. + move => [_] [H_c_neq] _ H. + apply Z.nle_gt => H_ge. + apply (H x2) => //. + rewrite InZ_cons; left. + apply In_elementsZ_single_hd => //. } Qed. - Lemma In_InZ : forall x s, Ok s -> - (In x s <-> InZ (Enc.encode x) s). + Lemma interval_list_elements_greater_alt2_def : forall s y, + interval_list_invariant s = true -> + + (interval_list_elements_greater y s = true <-> + (forall x, InZ x s -> y < x)). Proof. - intros x s H_ok. - rewrite /InZ In_alt_def /elements rev_map_alt_def -in_rev in_map_iff. - split; last first. { - exists (Enc.encode x). - by rewrite Enc.decode_encode_ok. - } - move => [y] [<-] H_y_in. - suff : exists z, (Enc.encode z = y). { - move => [z] H_y_eq. - move : H_y_in. - by rewrite -!H_y_eq Enc.decode_encode_ok. - } + intros s y H. + rewrite interval_list_elements_greater_alt_def //. + firstorder. + apply Z.nle_gt. + move => H_lt. + eapply H0; eauto. + Qed. + + Lemma interval_list_elements_greater_intro : forall s y, + interval_list_invariant s = true -> + (forall x, InZ x s -> y < x) -> + interval_list_elements_greater y s = true. + Proof. + intros s y H1 H2. + rewrite interval_list_elements_greater_alt2_def //. + Qed. + + Lemma interval_list_elements_greater_app_elim_1 : forall s1 s2 y, + interval_list_elements_greater y (s1 ++ s2) = true -> + interval_list_elements_greater y s1 = true. + Proof. + intros s1 s2 y. + case s1 => //. + Qed. + + Lemma interval_list_invariant_app_intro : forall s1 s2, + interval_list_invariant s1 = true -> + interval_list_invariant s2 = true -> + (forall (x1 x2 : Z), InZ x1 s1 -> InZ x2 s2 -> Z.succ x1 < x2) -> + interval_list_invariant (s1 ++ s2) = true. + Proof. + induction s1 as [| [y1 c1] s1' IH]. { + move => s2 _ //. + } { + move => s2. + rewrite -app_comm_cons !interval_list_invariant_cons. + move => [H_gr] [H_c1_neq] H_inv_s1' H_inv_s2 H_inz_s2. + split; last split. { + move : H_gr H_inz_s2. + case s1' as [| [y1' c1'] s1'']; last done. + move => _ H_inz_s2. + rewrite app_nil_l. + apply interval_list_elements_greater_intro => //. + move => x H_x_in_s2. + suff H_inz : InZ (Z.pred (y1 + Z.of_N c1)) ((y1, c1) :: nil). { + move : (H_inz_s2 _ _ H_inz H_x_in_s2). + by rewrite Z.succ_pred. + } + rewrite InZ_cons In_elementsZ_single -Z.lt_le_pred; left. + split. { + by apply Z_lt_add_r. + } { + apply Z.lt_pred_l. + } + } { + assumption. + } { + apply IH => //. + intros x1 x2 H_in_x1 H_in_x2. + apply H_inz_s2 => //. + rewrite InZ_cons; by right. + } + } + Qed. + + + Lemma interval_list_invariant_app_elim : forall s1 s2, + interval_list_invariant (s1 ++ s2) = true -> + interval_list_invariant s1 = true /\ + interval_list_invariant s2 = true /\ + (forall (x1 x2 : Z), InZ x1 s1 -> InZ x2 s2 -> Z.succ x1 < x2). + Proof. + move => s1 s2. + induction s1 as [| [y1 c1] s1' IH]; first done. + rewrite -app_comm_cons !interval_list_invariant_cons. + + move => [H_gr] [H_c1_neq_0] /IH [H_inv_s1'] [H_inv_s2] H_in_s1'_s2. + repeat split; try assumption. { + move : H_gr. + case s1'; first done. + move => [y2 c2] s1''. + rewrite interval_list_elements_greater_cons //. + } { + move => x1 x2. + rewrite InZ_cons In_elementsZ_single. + move => []; last by apply H_in_s1'_s2. + move => [] H_y1_le H_x1_lt H_x2_in. + move : H_gr. + rewrite interval_list_elements_greater_alt2_def; last first. { + by apply interval_list_invariant_app_intro. + } + move => H_in_s12'. + have : (y1 + Z.of_N c1 < x2). { + apply H_in_s12'. + rewrite InZ_app. + by right. + } + move => H_lt_x2. + apply Z.le_lt_trans with (m := y1 + Z.of_N c1) => //. + by apply Zlt_le_succ. + } + Qed. + + Lemma interval_list_invariant_app_iff : forall s1 s2, + interval_list_invariant (s1 ++ s2) = true <-> + (interval_list_invariant s1 = true /\ + interval_list_invariant s2 = true /\ + (forall (x1 x2 : Z), InZ x1 s1 -> InZ x2 s2 -> Z.succ x1 < x2)). + Proof. + intros s1 s2. + split. { + by apply interval_list_invariant_app_elim. + } { + move => [] H_inv_s1 []. + by apply interval_list_invariant_app_intro. + } + Qed. + + Lemma interval_list_invariant_snoc_intro : forall s1 y2 c2, + interval_list_invariant s1 = true -> + (c2 <> 0)%N -> + (forall x, InZ x s1 -> Z.succ x < y2) -> + interval_list_invariant (s1 ++ ((y2, c2)::nil)) = true. + Proof. + intros s1 y2 c2 H_inv_s1 H_c2_neq H_in_s1. + apply interval_list_invariant_app_intro => //. { + rewrite interval_list_invariant_cons; done. + } { + intros x1 x2 H_in_x1. + rewrite InZ_cons. + move => [] //. + rewrite In_elementsZ_single. + move => [H_y2_le] _. + eapply Z.lt_le_trans; eauto. + } + Qed. + + + (** *** Properties of In and InZ *) + + Lemma In_alt_def : forall x s, Ok s -> + (In x s <-> List.In x (elements s)). + Proof. + intros x s H_ok. + rewrite /In InA_alt /elements rev_map_alt_def. + split. { + move => [y] [H_y_eq]. + rewrite -!in_rev !in_map_iff. + move => [x'] [H_y_eq'] H_x'_in. + suff H_x'_eq : (Enc.encode x = x'). { + exists x'. + split => //. + rewrite -H_x'_eq Enc.decode_encode_ok //. + } + have H_enc_list : is_encoded_elems_list (elementsZ s). { + move : H_ok. + rewrite /Ok /IsOk => [] [] //. + } + move : (H_enc_list _ H_x'_in) => [x''] H_x'_eq. + move : H_y_eq'. + rewrite -!H_x'_eq Enc.decode_encode_ok => H_y_eq'. + subst. + suff -> : Z.eq (Enc.encode x) (Enc.encode y) by done. + by rewrite Enc.encode_eq. + } { + move => H_enc_in. + exists x. + split => //. + apply Enc.E.eq_equiv. + } + Qed. + + Lemma In_InZ : forall x s, Ok s -> + (In x s <-> InZ (Enc.encode x) s). + Proof. + intros x s H_ok. + rewrite /InZ In_alt_def /elements rev_map_alt_def -in_rev in_map_iff. + split; last first. { + exists (Enc.encode x). + by rewrite Enc.decode_encode_ok. + } + move => [y] [<-] H_y_in. + suff : exists z, (Enc.encode z = y). { + move => [z] H_y_eq. + move : H_y_in. + by rewrite -!H_y_eq Enc.decode_encode_ok. + } suff H_enc_list : is_encoded_elems_list (elementsZ s). { by apply H_enc_list. } apply H_ok. Qed. + Lemma InZ_In : forall x s, Ok s -> + (InZ x s -> In (Enc.decode x) s). + Proof. + intros x s H_ok. + rewrite In_InZ /InZ. + move : H_ok. + rewrite /Ok /IsOk /is_encoded_elems_list. + move => [_] H_enc. + move => H_in. + move : (H_enc _ H_in) => [e] H_x. + subst. + by rewrite Enc.decode_encode_ok. + Qed. + (** *** Membership specification *) Lemma memZ_spec : - forall (s : t) (x : Z) (Hs : Ok s), memZ x s = true <-> InZ x s. + forall (s : t) (x : Z) (Hs : Ok s), memZ x s = true <-> InZ x s. Proof. induction s as [| [y c] s' IH]. { intros x _. rewrite /InZ elementsZ_nil //. } { move => x /Ok_cons [H_inf] [H_c] [H_is_enc] H_s'. - rewrite /InZ /memZ elementsZ_cons -/mem. - rewrite /isok -/isok in_app_iff -!in_rev In_elementsZ_single. + rewrite /InZ /memZ elementsZ_cons -/memZ. + rewrite in_app_iff -!in_rev In_elementsZ_single. case_eq (x /Z.ltb_lt H_x_lt. split; first done. move => []. { move => H_x_in; contradict H_x_in. - eapply Nin_elements_greater; eauto; first apply H_s'. - apply Z.le_trans with (m := y). { - by apply Z.lt_le_incl. + apply Nin_elements_greater with (y := (y + Z.of_N c)) => //. { + apply H_s'. } { - apply Z_le_add_r. + apply Z_lt_le_add_r => //. } } { move => [H_y_le]; contradict H_y_le. @@ -1120,201 +1526,345 @@ Module Raw (Enc : ElementEncode). rewrite /mem memZ_spec In_InZ //. Qed. + Lemma merge_interval_size_neq_0 : forall x1 c1 x2 c2, + (c1 <> 0%N) -> + (merge_interval_size x1 c1 x2 c2 <> 0)%N. + Proof. + intros x1 c1 x2 c2. + rewrite /merge_interval_size !N.neq_0_lt_0 N.max_lt_iff. + by left. + Qed. - (** *** add specification *) - Lemma addZ_spec : - forall (s : t) (x y : Z) (Hs : Ok s), - InZ y (addZ x s) <-> Z.eq y x \/ InZ y s. - Proof. - intros s x y. - eapply addZ_ind with (P := (fun (x:Z) s => Ok s -> (InZ y (addZ x s) <-> Z.eq y x \/ InZ y s))); clear s x. { - move => x _. - rewrite /InZ addZ_alt_def elementsZ_cons elementsZ_nil app_nil_l -in_rev - In_elementsZ_single1 /=. - firstorder. - } { - intros x l c _. - rewrite /InZ addZ_alt_def Z.compare_refl !elementsZ_cons !in_app_iff -!in_rev N.add_1_r - Z.add_1_r !In_elementsZ_single N2Z.inj_succ Z.add_succ_l - Z.add_succ_r Z.le_succ_l Z.lt_succ_r. - split. { - move => []; first by tauto. - move => [] /Z.lt_eq_cases. - firstorder. + + (** *** insert if length not 0 *) + + Lemma interval_list_invariant_insert_intervalZ_guarded : forall x c s, + interval_list_invariant s = true -> + interval_list_elements_greater (x + Z.of_N c) s = true -> + interval_list_invariant (insert_intervalZ_guarded x c s) = true. + Proof. + intros x c s. + rewrite /insert_intervalZ_guarded. + case_eq (c =? 0)%N => //. + move => /N.eqb_neq. + rewrite interval_list_invariant_cons. + tauto. + Qed. + + Lemma interval_list_elements_greater_insert_intervalZ_guarded : forall x c y s, + interval_list_elements_greater y (insert_intervalZ_guarded x c s) = true <-> + (if (c =? 0)%N then (interval_list_elements_greater y s = true) else (y < x)). + Proof. + intros x c y s. + rewrite /insert_intervalZ_guarded. + case (c =? 0)%N => //. + rewrite /interval_list_elements_greater Z.ltb_lt //. + Qed. + + Lemma insert_intervalZ_guarded_app : forall x c s1 s2, + (insert_intervalZ_guarded x c s1) ++ s2 = + insert_intervalZ_guarded x c (s1 ++ s2). + Proof. + intros x c s1 s2. + rewrite /insert_intervalZ_guarded. + case (N.eqb c 0) => //. + Qed. + + Lemma insert_intervalZ_guarded_rev_nil_app : forall x c s, + rev (insert_intervalZ_guarded x c nil) ++ s = + insert_intervalZ_guarded x c s. + Proof. + intros x c s. + rewrite /insert_intervalZ_guarded. + case (N.eqb c 0) => //. + Qed. + +Lemma elementsZ_insert_intervalZ_guarded : forall x c s, + elementsZ (insert_intervalZ_guarded x c s) = elementsZ ((x, c) :: s). + Proof. + intros x c s. + rewrite /insert_intervalZ_guarded. + case_eq (c =? 0)%N => //. + move => /N.eqb_eq ->. + rewrite elementsZ_cons elementsZ_single_base /= app_nil_r //. + Qed. + + Lemma InZ_insert_intervalZ_guarded : forall y x c s, + InZ y (insert_intervalZ_guarded x c s) = InZ y ((x, c) :: s). + Proof. + intros y x c s. + rewrite /InZ elementsZ_insert_intervalZ_guarded //. + Qed. + + (** *** Merging intervals *) + + Lemma merge_interval_size_add : forall x c1 c2, + (merge_interval_size x c1 (x + Z.of_N c1) c2 = (c1 + c2))%N. + Proof. + intros x c1 c2. + rewrite /merge_interval_size. + replace (x + Z.of_N c1 + Z.of_N c2 - x) with + (Z.of_N c1 + Z.of_N c2) by omega. + rewrite -N2Z.inj_add N2Z.id. + apply N.max_r, N.le_add_r. + Qed. + + Lemma merge_interval_size_eq_max : forall y1 c1 y2 c2, + y1 <= y2 + Z.of_N c2 -> + y1 + Z.of_N (merge_interval_size y1 c1 y2 c2) = + Z.max (y1 + Z.of_N c1) (y2 + Z.of_N c2). + Proof. + intros y1 c1 y2 c2 H_y1_le. + rewrite /merge_interval_size N2Z.inj_max Z2N.id; last first. { + by apply Zle_minus_le_0. + } + rewrite -Z.add_max_distr_l. + replace (y1 + (y2 + Z.of_N c2 - y1)) with (y2 + Z.of_N c2) by omega. + done. + Qed. + + Lemma merge_interval_size_invariant : forall y1 c1 y2 c2 z s, + interval_list_invariant s = true -> + y1 + Z.of_N c1 <= y2 + Z.of_N c2 -> + y2 + Z.of_N c2 <= z -> + interval_list_elements_greater z s = true -> + (c1 <> 0)%N -> + interval_list_invariant ((y1, merge_interval_size y1 c1 y2 c2) :: s) = + true. + Proof. + intros y1 c1 y2 c2 z s H_inv H_le H_le_z H_gr H_c1_neq_0. + rewrite interval_list_invariant_cons. + split; last split. { + rewrite merge_interval_size_eq_max; last first. { + eapply Z.le_trans; last apply H_le. + apply Z_le_add_r. } { - move => [-> | ]. { - right; split. - - by apply Z.le_refl. - - by apply Z_le_add_r. - } { - move => []; first by tauto. - move => [H_x_lt H_y_le]. - right; split => //. - by apply Z.lt_le_incl. - } - } - } { - intros x l y0 c H_y_comp _. - rewrite /InZ addZ_alt_def H_y_comp elementsZ_cons in_app_iff - In_elementsZ_single1. - firstorder. - } { - move => y0 c H_y_comp H_ok. - rewrite addZ_alt_def H_y_comp Z.compare_refl. - rewrite /InZ !elementsZ_cons elementsZ_nil !app_nil_l N.add_1_r. - rewrite elementsZ_single_succ -!in_rev in_app_iff /=. - firstorder. - } { - move => l y0 c c' H_y_comp _ _. - rewrite addZ_alt_def H_y_comp Z.compare_refl Z.eqb_refl. - rewrite /InZ !elementsZ_cons !in_app_iff -!in_rev. - have -> : (c + c' + 1 = N.succ c + c')%N. { - rewrite -N.add_1_r. ring. + rewrite Z.max_r => //. + eapply interval_list_elements_greater_impl; first apply H_le_z. + done. } - rewrite Z.add_1_r elementsZ_single_add elementsZ_single_succ !in_app_iff - N2Z.inj_succ Z.add_succ_r. - firstorder. } { - intros l y0 c z c'. - move => H_y_comp H_z_comp _. - rewrite addZ_alt_def H_y_comp Z.compare_refl H_z_comp. - rewrite /InZ !elementsZ_cons N.add_1_r !in_app_iff -!in_rev. - rewrite elementsZ_single_succ in_app_iff /=. - firstorder. + apply merge_interval_size_neq_0. + assumption. } { - intros x l y0 c H_y_comp H_x_comp _. - rewrite addZ_alt_def H_y_comp H_x_comp. - split. { - by right. + assumption. + } + Qed. + + + Lemma In_merge_interval : forall x1 c1 x2 c2 y, + x1 <= x2 -> + x2 <= x1 + Z.of_N c1 -> ( + List.In y (elementsZ_single x1 (merge_interval_size x1 c1 x2 c2)) <-> + List.In y (elementsZ_single x1 c1) \/ List.In y (elementsZ_single x2 c2)). + Proof. + intros x1 c1 x2 c2 y H_x1_le H_x2_le. + rewrite !In_elementsZ_single merge_interval_size_eq_max; + last first. { + eapply Z.le_trans; eauto. + by apply Z_le_add_r. + } + rewrite Z.max_lt_iff. + split. { + move => [H_x_le] [] H_y_lt. { + by left. } { - move => [] // ->. - rewrite /InZ elementsZ_cons in_app_iff -in_rev In_elementsZ_single. - right. split. { - move : H_y_comp => /Z.compare_gt_iff. - rewrite Z.add_1_r Z.lt_succ_r //. + case_eq (Z.leb x2 y). { + move => /Z.leb_le H_y'_le. + by right. } { - move : H_x_comp => /Z.compare_lt_iff //. + move => /Z.leb_gt H_y_lt_x2. + left. + split => //. + eapply Z.lt_le_trans; eauto. } } } { - intros x l y0 c H_y_comp H_x_comp IH H_ok. - have H_ok' : Ok l. { - move : H_ok. - rewrite Ok_cons. + move => []. { tauto. + } { + move => [H_x2_le'] H_y_lt. + split. { + eapply Z.le_trans; eauto. + } { + by right. + } } - rewrite addZ_alt_def H_y_comp H_x_comp. - unfold InZ in IH. - rewrite /InZ !elementsZ_cons !in_app_iff -!in_rev (IH H_ok'). - firstorder. } Qed. - Lemma addZ_isok : forall s x, isok s = true -> isok (addZ x s) = true. - Proof. - intros s x. - have H_add_1_neq : (forall x, x + 1 <> 0)%N. { - move => x'. - rewrite N.eq_add_0. - have : (1 <> 0)%N by done. - tauto. - } - - eapply addZ_ind with (P := (fun (x : Z) (s : t) => isok s = true -> - isok (addZ x s) = true)); clear s x. { - move => x _ //. - } { - intros x l c. - rewrite addZ_alt_def Z.compare_refl. - rewrite !isok_cons. - move => [H_inf] [H_c_neq] H_ok_l. - split; last split => //. { - suff -> : (x + Z.of_N (c + 1) = x + 1 + Z.of_N c) by done. - rewrite N2Z.inj_add /=. - ring. - } - } { - intros x l y c H_y_comp. - rewrite addZ_alt_def H_y_comp. - rewrite !isok_cons => [[H_inf] [H_c_neq] H_ok_l]. - split; last split; last split; last split; try done. - by apply Z.ltb_lt. - } { - move => y c H_y_comp _. - rewrite addZ_alt_def H_y_comp Z.compare_refl isok_cons. - done. - } { - move => l y c c' H_y_comp IH. - rewrite addZ_alt_def H_y_comp Z.compare_refl Z.eqb_refl. - rewrite !isok_cons. - move => [H_inf1] [H_c_neq] [H_inf2] [H_c'_neq] H_ok_l. - split; last split => //. { - suff -> : (y + Z.of_N (c + c' + 1) = y + Z.of_N c + 1 + Z.of_N c') by assumption. - rewrite !N2Z.inj_add /=. - ring. - } + Lemma insert_interval_begin_spec : forall y s x c, + interval_list_invariant s = true -> + interval_list_elements_greater x s = true -> + (c <> 0)%N -> + ( + interval_list_invariant (insert_interval_begin x c s) = true /\ + (InZ y (insert_interval_begin x c s) <-> + (List.In y (elementsZ_single x c) \/ InZ y s))). + Proof. + intros y. + induction s as [| [y' c'] s' IH]. { + intros x c _ H_c_neq H_z_lt. + rewrite /insert_interval_begin InZ_cons interval_list_invariant_cons //. } { - intros l y c z c'. - move => H_y_comp H_z_comp. - rewrite addZ_alt_def H_y_comp Z.compare_refl H_z_comp. - rewrite isok_cons => [] [] H_inf [] H_c_neq H_ok_l. - rewrite isok_cons. - split; last split; try done. - move : H_inf. - rewrite /inf. - move : H_z_comp => /Z.eqb_neq H_z_neq /Z.ltb_lt H_z_lt. - apply Z.ltb_lt. + intros x c. + rewrite interval_list_invariant_cons + interval_list_elements_greater_cons. + move => [H_gr] [H_c'_neq_0] H_inv_s' H_x_lt H_c_neq_0. + unfold insert_interval_begin. + Z_named_compare_cases H_y'; fold insert_interval_begin. { + subst. + rewrite !InZ_cons elementsZ_single_add in_app_iff. + split; last tauto. + rewrite interval_list_invariant_cons N2Z.inj_add + Z.add_assoc N.eq_add_0. + tauto. + } { + rewrite !InZ_cons !interval_list_invariant_cons + interval_list_elements_greater_cons. + repeat split => //. + } { + set c'' := merge_interval_size x c y' c'. + have H_x_lt' : x < y' + Z.of_N c'. { + eapply Z.lt_le_trans with (m := y') => //. + by apply Z_le_add_r. + } - have -> : Z.of_N (c + 1) = Z.of_N c + 1. { - rewrite N2Z.inj_add //. - } + have H_pre : interval_list_elements_greater x s' = true. { + eapply interval_list_elements_greater_impl; eauto. + by apply Z.lt_le_incl. + } + have H_pre2 : c'' <> 0%N. { + by apply merge_interval_size_neq_0. + } + move : (IH x c'' H_inv_s' H_pre H_pre2) => {IH} {H_pre} {H_pre2} [->] ->. - suff : y + (Z.of_N c + 1) <= z. { - rewrite Z.lt_eq_cases. - move => [] // H_z_eq. - contradict H_z_neq. - rewrite -H_z_eq. - ring. + split; first reflexivity. + unfold c''; clear c''. + rewrite In_merge_interval. { + rewrite InZ_cons. + tauto. + } { + by apply Z.lt_le_incl. + } { + by apply Z.lt_le_incl. + } } - suff H_suff : (Z.pred (y + (Z.of_N c + 1)) = y + Z.of_N c). { - apply Z.lt_pred_le. - by rewrite H_suff. + } + Qed. + + + (** *** add specification *) + Lemma addZ_InZ : + forall (s : t) (x y : Z), + interval_list_invariant s = true -> + (InZ y (addZ x s) <-> x = y \/ InZ y s). + Proof. + move => s x y. + induction s as [| [z c] s' IH]. { + move => _. + rewrite /InZ addZ_alt_def + elementsZ_cons elementsZ_nil app_nil_l -in_rev + In_elementsZ_single1 /=. + firstorder. + } { + move => /interval_list_invariant_cons [H_greater] [H_c_neq_0] H_inv_c'. + move : (IH H_inv_c') => {IH} IH. + rewrite addZ_alt_def. + have H_succ : forall z, z + Z.of_N 1 = Z.succ z by done. + move : (interval_1_compare_elim x z c). + case (interval_1_compare x (z, c)); + rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. { + move => ->. + rewrite elementsZ_single_succ_front /=. + tauto. + } { + move => [] // H_x_in. + split; first tauto. + move => [] // <-. + left. + by rewrite In_elementsZ_single. + } { + rewrite IH. + tauto. + } { + move => H_x_eq. + have -> : (InZ y (insert_interval_begin z (N.succ c) s') <-> + List.In y (elementsZ_single z (N.succ c)) \/ InZ y s'). { + eapply insert_interval_begin_spec. { + by apply H_inv_c'. + } { + eapply interval_list_elements_greater_impl; eauto. + apply Z_le_add_r. + } { + by apply N.neq_succ_0. + } + } + rewrite -H_x_eq elementsZ_single_succ in_app_iff /=. + tauto. } - rewrite Z.add_1_r Z.add_succ_r Z.pred_succ //. - } { - intros x l y c H_y_comp H_x_comp. - rewrite addZ_alt_def H_y_comp H_x_comp //. - } { - intros x l y c H_y_comp H_x_comp IH. - rewrite addZ_alt_def H_y_comp H_x_comp. - rewrite !isok_cons => [] [] H_inf [] H_c_neq H_ok_l. - split; last split; try done. { - suff : exists x0 c0 l0, addZ x l = (x0, c0)::l0 /\ (y+Z.of_N c < x0). { - move => [x0] [c0] [l0] [->]. - rewrite /inf. - apply Z.ltb_lt. + } + Qed. + + Lemma addZ_invariant : forall s x, + interval_list_invariant s = true -> + interval_list_invariant (addZ x s) = true. + Proof. + move => s x. + induction s as [| [z c] s' IH]. { + move => _. + by simpl. + } { + move => /interval_list_invariant_cons [H_greater] [H_c_neq_0] + H_inv_c'. + move : (IH H_inv_c') => {IH} IH. + rewrite addZ_alt_def. + have H_succ : forall z, z + Z.of_N 1 = Z.succ z by done. + move : (interval_1_compare_elim x z c). + case_eq (interval_1_compare x (z, c)) => H_comp; + rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. { + move => H_z_gt. + rewrite interval_list_invariant_cons /= !andb_true_iff !H_succ. + repeat split => //. { + by apply Z.ltb_lt. + } { + apply negb_true_iff, N.eqb_neq => //. } - apply Z.compare_gt_iff in H_x_comp. - move : H_inf. - case l. { - exists x, 1%N, nil. - by rewrite addZ_alt_def. - } { - move => [z c'] l0 /Z.ltb_lt H_z_gt. - rewrite addZ_alt_def. - case (x + 1 ?= z); try by eauto. - case (x ?= z + Z.of_N c'); try by eauto. - case l0; try by eauto. - move => [z' c''] ?. - case (z' =? x + 1); by eauto. + } { + move => ?; subst. + rewrite /= !andb_true_iff. + repeat split => //. { + move : H_greater. + rewrite Z.add_succ_l -Z.add_succ_r N2Z.inj_succ //. + } { + apply negb_true_iff, N.eqb_neq => //. + apply N.neq_succ_0. } } { - apply IH => //. + move => [] // _. + rewrite interval_list_invariant_cons /=. + tauto. + } { + rewrite interval_list_invariant_cons. + move => H_lt_x. + repeat split => //. + apply interval_list_elements_greater_intro => //. + move => xx. + rewrite addZ_InZ => //. + move => [<- //|]. + apply interval_list_elements_greater_alt2_def => //. + } { + move => H_x_eq. + apply insert_interval_begin_spec => //. { + eapply interval_list_elements_greater_impl; eauto. + apply Z_le_add_r. + } { + by apply N.neq_succ_0. + } } } Qed. - - + Global Instance add_ok s x : forall `(Ok s), Ok (add x s). Proof. move => H_ok_s. @@ -1322,13 +1872,13 @@ Module Raw (Enc : ElementEncode). rewrite /Ok /IsOk /is_encoded_elems_list /add. move => [H_isok_s] H_pre. split. { - apply addZ_isok => //. + apply addZ_invariant => //. } { intros y. - move : (addZ_spec s (Enc.encode x) y H_ok_s). + move : (addZ_InZ s (Enc.encode x) y H_isok_s). rewrite /InZ => ->. move => []. { - move => ->. + move => <-. by exists x. } { move => /H_pre //. @@ -1343,351 +1893,1923 @@ Module Raw (Enc : ElementEncode). intros s x y Hs. have Hs' := (add_ok s x Hs). rewrite !In_InZ. - rewrite /add addZ_spec Enc.encode_eq //. + rewrite /add addZ_InZ. { + rewrite -Enc.encode_eq; firstorder. + } { + apply Hs. + } Qed. - (** *** remove specification *) + (** *** empty specification *) - Lemma isok_removeZ_aux_insert_guarded : forall x c s, - isok s = true -> inf (x + Z.of_N c) s = true -> - isok (removeZ_aux_insert_guarded x c s) = true. + Global Instance empty_ok : Ok empty. Proof. - intros x c s. - rewrite /removeZ_aux_insert_guarded. - case_eq (c =? 0)%N => //. - move => /N.eqb_neq. - rewrite isok_cons. - tauto. + rewrite /empty Ok_nil //. + Qed. + + Lemma empty_spec' : forall x, (In x empty <-> False). + rewrite /Empty /empty /In elements_nil. + intros a. + rewrite InA_nil //. Qed. - Lemma inf_removeZ_aux_insert_guarded : forall x c y s, - inf y (removeZ_aux_insert_guarded x c s) = true <-> - (if (c =? 0)%N then (inf y s = true) else (y < x)). + Lemma empty_spec : Empty empty. Proof. - intros x c y s. - rewrite /removeZ_aux_insert_guarded. - case (c =? 0)%N => //. - rewrite /inf Z.ltb_lt //. + rewrite /Empty => a. + rewrite empty_spec' //. + Qed. + + (** *** is_empty specification *) + + Lemma is_empty_spec : forall (s : t) (Hs : Ok s), is_empty s = true <-> Empty s. + Proof. + intros [ | [x c] s]. { + split => // _. + apply empty_spec. + } { + rewrite /= /Empty Ok_cons. + move => [_] [H_c_neq] [H_enc] _. + split => //. + move => H. + contradiction (H (Enc.decode x)) => {H}. + rewrite /In InA_alt elements_cons. + exists (Enc.decode x). + split; first by apply Enc.E.eq_equiv. + rewrite in_app_iff; left. + rewrite /elements_single in_map_iff. + exists x. + split => //. + apply In_elementsZ_single_hd => //. + } Qed. - Lemma removeZ_counter_pos_aux : forall y c x, - x < y + Z.of_N c -> - 0 <= y + Z.of_N c - Z.succ x. + (** *** singleton specification *) + + Global Instance singleton_ok x : Ok (singleton x). Proof. - intros y c x H_x_lt. - rewrite Z.le_0_sub Z.le_succ_l. - assumption. + rewrite singleton_alt_def. + apply add_ok. + apply empty_ok. Qed. - Lemma removeZ_isok : forall s x, isok s = true -> isok (removeZ x s) = true. + Lemma singleton_spec : forall x y : elt, In y (singleton x) <-> Enc.E.eq y x. Proof. - intros s x. - induction s as [| [y c] s' IH]. { - rewrite removeZ_alt_def //. - } { - rewrite removeZ_alt_def. - case_eq (x //. - move => /Z.ltb_ge H_y_le_x. - case_eq (x /Z.ltb_lt H_x_lt /isok_cons [H_inf] [H_c_neq] H_ok_s'. - apply isok_removeZ_aux_insert_guarded; first apply isok_removeZ_aux_insert_guarded. { - assumption. + intros x y. + rewrite singleton_alt_def. + rewrite (add_spec empty x y) /empty /In elements_nil InA_nil. + split. { + move => [] //. + } { + by left. + } + Qed. + + (** *** add_list specification *) + + Lemma add_list_ok : forall l s, Ok s -> Ok (add_list l s). + Proof. + induction l as [| x l' IH]. { + done. + } { + move => s H_s_ok /=. + apply IH. + by apply add_ok. + } + Qed. + + Lemma add_list_spec : forall x l s, Ok s -> + (In x (add_list l s) <-> (SetoidList.InA Enc.E.eq x l) \/ In x s). + Proof. + move => x. + induction l as [| y l' IH]. { + intros s H. + rewrite /= InA_nil. + tauto. + } { + move => s H_ok /=. + rewrite IH add_spec InA_cons. + tauto. + } + Qed. + + (** *** union specification *) + + Lemma union_aux_flatten_alt_def : forall (s1 s2 : t) acc, + union_aux s1 s2 acc = + match (s1, s2) with + | (nil, _) => List.rev_append acc s2 + | (_, nil) => List.rev_append acc s1 + | ((y1, c1) :: l1, (y2, c2) :: l2) => + match (interval_compare (y1, c1) (y2,c2)) with + | ICR_before => union_aux l1 s2 ((y1, c1)::acc) + | ICR_before_touch => + union_aux l1 ( + insert_interval_begin y1 ((c1+c2)%N) l2) acc + | ICR_after => union_aux s1 l2 ((y2, c2)::acc) + | ICR_after_touch => union_aux l1 ( + insert_interval_begin y2 ((c1+c2)%N) l2) acc + | ICR_overlap_before => + union_aux l1 ( + insert_interval_begin y1 + (merge_interval_size y1 c1 y2 c2) l2) acc + | ICR_overlap_after => + union_aux l1 ( + insert_interval_begin y2 + (merge_interval_size y2 c2 y1 c1) l2) acc + | ICR_equal => union_aux l1 s2 acc + | ICR_subsume_1 => union_aux l1 s2 acc + | ICR_subsume_2 => union_aux s1 l2 acc + end + end. + Proof. + intros s1 s2 acc. + case s1, s2 => //. + Qed. + + Lemma union_aux_alt_def : forall (s1 s2 : t) acc, + union_aux s1 s2 acc = + List.rev_append acc (union s1 s2). + Proof. + rewrite /union. + intros s1 s2 acc. + move : acc s2. + induction s1 as [| [y1 c1] l1 IH1]. { + intros acc s2. + rewrite !union_aux_flatten_alt_def. + rewrite !rev_append_rev //. + } + intros acc s2; move : acc. + induction s2 as [| [y2 c2] l2 IH2]; first by simpl. + move => acc. + rewrite !(union_aux_flatten_alt_def ((y1, c1)::l1) ((y2, c2)::l2)). + case (interval_compare (y1, c1) (y2, c2)); + rewrite ?(IH1 ((y1, c1) :: acc)) ?(IH1 ((y1, c1) :: nil)) + ?(IH2 ((y2, c2) :: acc)) ?(IH2 ((y2, c2) :: nil)) + ?(IH1 acc) //. + Qed. + + Lemma union_alt_def : forall (s1 s2 : t), + union s1 s2 = + match (s1, s2) with + | (nil, _) => s2 + | (_, nil) => s1 + | ((y1, c1) :: l1, (y2, c2) :: l2) => + match (interval_compare (y1, c1) (y2,c2)) with + | ICR_before => (y1, c1) :: (union l1 s2) + | ICR_before_touch => + union l1 (insert_interval_begin y1 ((c1+c2)%N) l2) + | ICR_after => (y2, c2) :: union s1 l2 + | ICR_after_touch => union l1 + (insert_interval_begin y2 ((c1+c2)%N) l2) + | ICR_overlap_before => + union l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2) + | ICR_overlap_after => + union l1 (insert_interval_begin y2 (merge_interval_size y2 c2 y1 c1) l2) + | ICR_equal => union l1 s2 + | ICR_subsume_1 => union l1 s2 + | ICR_subsume_2 => union s1 l2 + end + end. + Proof. + intros s1 s2. + rewrite /union union_aux_flatten_alt_def. + case s1 as [| [y1 c1] l1] => //. + case s2 as [| [y2 c2] l2] => //. + case (interval_compare (y1, c1) (y2, c2)); + rewrite union_aux_alt_def //. + Qed. + + Lemma union_InZ : + forall (s1 s2 : t), + interval_list_invariant s1 = true -> + interval_list_invariant s2 = true -> + forall y, (InZ y (union s1 s2) <-> InZ y s1 \/ InZ y s2). + Proof. + intro s1. + induction s1 as [| [y1 c1] l1 IH1]. { + intros s2 _ _ y. + rewrite union_alt_def /InZ /=. + tauto. + } { + induction s2 as [| [y2 c2] l2 IH2]. { + intros _ _ y. + rewrite union_alt_def /InZ /=. + tauto. + } { + move => H_inv_s1 H_inv_s2. + move : (H_inv_s1) (H_inv_s2). + rewrite !interval_list_invariant_cons. + move => [H_gr_l1] [H_c1_neq_0] H_inv_l1. + move => [H_gr_l2] [H_c2_neq_0] H_inv_l2. + move : (IH2 H_inv_s1 H_inv_l2) => {IH2} IH2. + have : forall s2 : t, + interval_list_invariant s2 = true -> + forall y : Z, InZ y (union l1 s2) <-> InZ y l1 \/ InZ y s2. { + intros. by apply IH1. + } + move => {IH1} IH1 y. + rewrite union_alt_def. + move : (interval_compare_elim y1 c1 y2 c2). + case (interval_compare (y1, c1) (y2, c2)). { + rewrite !InZ_cons IH1 // InZ_cons. + tauto. } { - rewrite Z2N.id; last by apply removeZ_counter_pos_aux. - rewrite add_add_sub_eq. - assumption. + move => H_y2_eq. + replace (c1 + c2)%N with (merge_interval_size y1 c1 y2 c2); + last first. { + rewrite -H_y2_eq merge_interval_size_add //. + } + set c'' := merge_interval_size y1 c1 y2 c2. + have [H_inv_insert H_InZ_insert] : + interval_list_invariant (insert_interval_begin y1 c'' l2) = true /\ + (InZ y (insert_interval_begin y1 c'' l2) <-> + List.In y (elementsZ_single y1 c'') \/ InZ y l2). { + apply insert_interval_begin_spec => //. { + eapply interval_list_elements_greater_impl; eauto. + rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add. + apply Z_le_add_r. + } { + by apply merge_interval_size_neq_0. + } + } + + rewrite IH1 => //. + rewrite H_InZ_insert !InZ_cons /c''. + rewrite -H_y2_eq In_merge_interval. { + tauto. + } { + apply Z_le_add_r. + } { + by apply Z.le_refl. + } } { - rewrite /removeZ_aux_insert_guarded Z2N.id; last first. { - by rewrite Z.le_0_sub. + move => [H_y1_lt] [H_y2_lt] H_y1_c1_lt. + set c'' := merge_interval_size y1 c1 y2 c2. + have [H_inv_insert H_InZ_insert] : + interval_list_invariant (insert_interval_begin y1 c'' l2) = true /\ + (InZ y (insert_interval_begin y1 c'' l2) <-> + List.In y (elementsZ_single y1 c'') \/ InZ y l2). { + apply insert_interval_begin_spec => //. { + eapply interval_list_elements_greater_impl; eauto. + apply Z_lt_le_add_r => //. + } { + by apply merge_interval_size_neq_0. + } } - rewrite add_add_sub_eq. - case (Z.to_N (y + Z.of_N c - Z.succ x) =? 0)%N. { - eapply inf_impl; eauto. + rewrite IH1 => //. + rewrite H_InZ_insert !InZ_cons /c''. + rewrite In_merge_interval. { + tauto. + } { by apply Z.lt_le_incl. } { - rewrite /inf. - apply Z.ltb_lt, Z.lt_succ_diag_r. + by apply Z.lt_le_incl. } - } - } { - move => /Z.ltb_ge H_yc_le_x. - rewrite !isok_cons. - move => [H_inf] [H_c_neq] H_ok_s'. - split; last split => //; last by apply IH. - move : H_inf. - case_eq s' => //. - move => [z c'] l' H_s'_eq. - rewrite removeZ_alt_def. - case (x /Z.ltb_lt H_lt_z. - rewrite inf_removeZ_aux_insert_guarded. - case (Z.to_N (x - z) =? 0)%N => //. - rewrite inf_removeZ_aux_insert_guarded. - case (Z.to_N (z + Z.of_N c' - Z.succ x) =? 0)%N. { - move : H_ok_s'. - rewrite H_s'_eq !isok_cons. - move => [H_inf] _. - eapply inf_impl; last apply H_inf. - apply Z.le_trans with (m := z). { + } { + move => [H_y2_lt] [H_y1_lt] H_y2_c_lt. + set c'' := merge_interval_size y2 c2 y1 c1. + have [H_inv_insert H_InZ_insert] : + interval_list_invariant (insert_interval_begin y2 c'' l2) = true /\ + (InZ y (insert_interval_begin y2 c'' l2) <-> + List.In y (elementsZ_single y2 c'') \/ InZ y l2). { + apply insert_interval_begin_spec => //. { + eapply interval_list_elements_greater_impl; eauto. + apply Z_le_add_r. + } { + by apply merge_interval_size_neq_0. + } + } + + rewrite IH1 => //. + rewrite H_InZ_insert !InZ_cons /c''. + rewrite In_merge_interval. { + tauto. + } { by apply Z.lt_le_incl. } { - by apply Z_le_add_r. + by apply Z.lt_le_incl. + } + } { + move => [? ?]; subst. + rewrite IH1 => //. + rewrite !InZ_cons. + tauto. + } { + move => [H_y2_le] [H_y1_c1_le] _. + rewrite IH1 => //. + rewrite !InZ_cons. + suff : (List.In y (elementsZ_single y1 c1) -> + List.In y (elementsZ_single y2 c2)). { + tauto. + } + rewrite !In_elementsZ_single. + move => [H_y1_le H_y_lt]. + split; omega. + } { + move => [H_y1_le] [H_y2_c2_le] _. + rewrite IH2. + rewrite !InZ_cons. + suff : (List.In y (elementsZ_single y2 c2) -> + List.In y (elementsZ_single y1 c1)). { + tauto. } + rewrite !In_elementsZ_single. + move => [H_y2_le H_y_lt]. + split; omega. + } { + rewrite !InZ_cons IH2 !InZ_cons. + tauto. } { - rewrite Z.lt_succ_r //. + move => H_y1_eq. + replace (c1 + c2)%N with (merge_interval_size y2 c2 y1 c1); + last first. { + rewrite -H_y1_eq merge_interval_size_add N.add_comm //. + } + set c'' := merge_interval_size y2 c2 y1 c1. + have [H_inv_insert H_InZ_insert] : + interval_list_invariant (insert_interval_begin y2 c'' l2) = true /\ + (InZ y (insert_interval_begin y2 c'' l2) <-> + List.In y (elementsZ_single y2 c'') \/ InZ y l2). { + apply insert_interval_begin_spec => //. { + eapply interval_list_elements_greater_impl; eauto. + apply Z_le_add_r. + } { + by apply merge_interval_size_neq_0. + } + } + + rewrite IH1 => //. + rewrite H_InZ_insert !InZ_cons /c''. + rewrite -H_y1_eq In_merge_interval. { + tauto. + } { + apply Z_le_add_r. + } { + by apply Z.le_refl. + } } } } Qed. - - Lemma elementsZ_removeZ_aux_insert_guarded : forall x c s, - elementsZ (removeZ_aux_insert_guarded x c s) = elementsZ ((x, c) :: s). - Proof. - intros x c s. - rewrite /removeZ_aux_insert_guarded. - case_eq (c =? 0)%N => //. - move => /N.eqb_eq ->. - rewrite elementsZ_cons elementsZ_single_base /= app_nil_r //. - Qed. - - Lemma removeZ_spec : - forall (s : t) (x y : Z) (Hs : isok s = true), - InZ y (removeZ x s) <-> InZ y s /\ ~Z.eq y x. + Lemma union_invariant : + forall (s1 s2 : t), + interval_list_invariant s1 = true -> + interval_list_invariant s2 = true -> + interval_list_invariant (union s1 s2) = true. Proof. - intros s x y Hs. - move : Hs x y. - induction s as [| [y c] s' IH] . { - move => _ x y. - rewrite /=. - firstorder. + intro s1. + induction s1 as [| [y1 c1] l1 IH1]. { + intros s2 _ H_inv_s2. + rewrite union_alt_def /InZ //. } { - intros H_ok x y0. - rewrite removeZ_alt_def. - case_eq (x /Z.ltb_lt H_x_lt. - split; last tauto. - move => H_in. - split => // H_y0_eq. - apply (isok_inf_nin y0 ((y, c)::s')) => //. - rewrite /inf H_y0_eq. - by apply Z.ltb_lt. + induction s2 as [| [y2 c2] l2 IH2]. { + intros H_inv_s1 _. + rewrite union_alt_def /InZ //. } { - move => /Z.ltb_ge H_y_le. - have H_ok' : isok s' = true. { - move : H_ok. - rewrite isok_cons; tauto. - } - move : (IH H_ok') => {IH} IH. - case_eq (x /Z.ltb_lt H_x_lt /=. - rewrite /InZ elementsZ_removeZ_aux_insert_guarded !elementsZ_cons. - rewrite elementsZ_removeZ_aux_insert_guarded elementsZ_cons !in_app_iff -!in_rev. - rewrite !In_elementsZ_single !Z2N.id; last first. { - by apply removeZ_counter_pos_aux. + move => H_inv_s1 H_inv_s2. + move : (H_inv_s1) (H_inv_s2). + rewrite !interval_list_invariant_cons. + move => [H_gr_l1] [H_c1_neq_0] H_inv_l1. + move => [H_gr_l2] [H_c2_neq_0] H_inv_l2. + move : (IH2 H_inv_s1 H_inv_l2) => {IH2} IH2. + have : forall s2 : t, + interval_list_invariant s2 = true -> + interval_list_invariant (union l1 s2) = true. { + intros. by apply IH1. + } + move => {IH1} IH1. + + rewrite union_alt_def. + move : (interval_compare_elim y1 c1 y2 c2). + case (interval_compare (y1, c1) (y2, c2)). { + move => H_lt_y2. + have H_inv' : interval_list_invariant (union l1 ((y2, c2) :: l2)) = true. { + by apply IH1. + } + + rewrite interval_list_invariant_cons. + repeat split => //. + apply interval_list_elements_greater_intro => // x. + rewrite union_InZ => //. + move => []. { + apply interval_list_elements_greater_alt2_def => //. } { - by rewrite Z.le_0_sub. + apply interval_list_elements_greater_alt2_def => //. + rewrite interval_list_elements_greater_cons //. } - rewrite !add_add_sub_eq Z.le_succ_l. - case_eq (y0 =? x). { - move => /Z.eqb_eq ->. - split; last first. { - move => [_] H_neq. - contradict H_neq => //. - } { - move => [[] |]. { - move => H_in. exfalso. - apply (isok_inf_nin x s') => //. - move : H_ok. - case s' => //. - move => [z c'] s''. - rewrite !isok_cons /inf. - move => [] /Z.ltb_lt H_lt_x' _. - apply Z.ltb_lt. - eapply Z.lt_trans; eauto. - } { - move => [] /Z.lt_irrefl //. - } { - move => [_] /Z.lt_irrefl //. - } - } + } { + move => H_y2_eq. + apply IH1. + apply insert_interval_begin_spec => //. { + eapply interval_list_elements_greater_impl; last apply H_gr_l2. + rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add. + apply Z_le_add_r. + } { + rewrite N.eq_add_0. + tauto. } - move => /Z.eqb_neq H_y0_neq. - split. { - move => H. - split; last by apply H_y0_neq. - move : H => [[] |]. { - by left. - } { - move => [H_x_lt'] H_y0_lt. - right; split => //. - apply Z.le_trans with (m := x) => //. - by apply Z.lt_le_incl. - } { - move => [H_y_le'] H_y0_lt. - right; split => //. - eapply Z.lt_trans; eauto. - } + } { + move => [H_y1_lt] _. + apply IH1. + apply insert_interval_begin_spec => //. { + eapply interval_list_elements_greater_impl; last apply H_gr_l2. + apply Z_lt_le_add_r => //. } { - move => [[]]; first by tauto. - move => [H_y_le'] H_y0_lt _. - case_eq (y0 /Z.ltb_lt H_y0_lt_x. - right. done. - } { - move => /Z.ltb_ge /Z.lt_eq_cases []. { - move => H_x_lt'. - left; right. - done. - } { - move => H_x_eq. - contradict H_y0_neq => //. - } - } - } + apply merge_interval_size_neq_0 => //. + } } { - move => /Z.ltb_ge H_yc_le. - unfold InZ in IH. - rewrite /InZ !elementsZ_cons !in_app_iff !IH. - split; last tauto. - move => []; first tauto. - rewrite -!in_rev In_elementsZ_single. - move => [H_y_le'] H_y0_lt. - split; first by right. - move => H_y0_eq. - move : H_y0_lt. - apply Z.nlt_ge. - by rewrite H_y0_eq. + move => [H_y2_lt] _. + apply IH1. + apply insert_interval_begin_spec => //. { + eapply interval_list_elements_greater_impl; last apply H_gr_l2. + apply Z_le_add_r => //. + } { + apply merge_interval_size_neq_0 => //. + } + } { + move => [? ?]; subst. + apply IH1 => //. + } { + move => _. + apply IH1 => //. + } { + move => _. + apply IH2 => //. + } { + move => H_lt_y1. + rewrite interval_list_invariant_cons => //. + repeat split => //. + apply interval_list_elements_greater_intro => // x. + rewrite union_InZ => //. + move => []. { + apply interval_list_elements_greater_alt2_def => //. + rewrite interval_list_elements_greater_cons //. + } { + apply interval_list_elements_greater_alt2_def => //. + } + } { + move => H_y1_eq. + apply IH1 => //. + apply insert_interval_begin_spec => //. { + eapply interval_list_elements_greater_impl; last apply H_gr_l2. + apply Z_le_add_r. + } { + rewrite N.eq_add_0. + tauto. + } } } } Qed. - - Global Instance remove_ok s x : forall `(Ok s), Ok (remove x s). + + Global Instance union_ok s1 s2 : forall `(Ok s1, Ok s2), Ok (union s1 s2). Proof. - rewrite /Ok /IsOk /remove. - move => [H_is_ok_s H_enc_s]. + move => H_ok_s1 H_ok_s2. + move : (H_ok_s1) (H_ok_s2). + rewrite /Ok /IsOk /is_encoded_elems_list /add. + move => [H_inv_s1] H_pre1. + move => [H_inv_s2] H_pre2. split. { - by apply removeZ_isok. + apply union_invariant => //. } { - rewrite /is_encoded_elems_list => y. - move : (removeZ_spec s (Enc.encode x) y H_is_ok_s). - rewrite /InZ => -> [] H_y_in _. - apply H_enc_s => //. + intros y. + move : (union_InZ s1 s2 H_inv_s1 H_inv_s2). + rewrite /InZ => ->. + move => []. { + apply H_pre1. + } { + apply H_pre2. + } } Qed. - Lemma remove_spec : - forall (s : t) (x y : elt) (Hs : Ok s), - In y (remove x s) <-> In y s /\ ~Enc.E.eq y x. + Lemma union_spec : + forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'), + In x (union s s') <-> In x s \/ In x s'. Proof. - intros s x y Hs. - have H_rs := (remove_ok s x Hs). - rewrite /remove !In_InZ. - rewrite removeZ_spec. { - rewrite Enc.encode_eq //. + intros s s' x H_ok H_ok'. + rewrite !In_InZ. + rewrite union_InZ => //. { + apply H_ok. } { - apply Hs. + apply H_ok'. } Qed. - (** *** empty specification *) - - Global Instance empty_ok : Ok empty. + (** *** inter specification *) + + Lemma inter_aux_alt_def : + forall (y2 : Z) (c2 : N) (s : t) acc, + inter_aux y2 c2 acc s = match inter_aux y2 c2 nil s with + (acc', s') => (acc' ++ acc, s') + end. Proof. - rewrite /empty Ok_nil //. - Qed. - - Lemma empty_spec' : forall x, (In x empty <-> False). - rewrite /Empty /empty /In elements_nil. - intros a. - rewrite InA_nil //. + intros y2 c2. + induction s as [| [y1 c1] s' IH] => acc. { + rewrite /inter_aux app_nil_l //. + } { + simpl. + case_eq (inter_aux y2 c2 nil s') => acc'' s'' H_eq. + case (interval_compare (y1, c1) (y2, c2)); + rewrite ?(IH acc) + ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: acc)) + ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: nil)) + ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: acc)) + ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: nil)) + ?(IH ((y1, c1) :: acc)) + ?(IH ((y1, c1) :: nil)) + + ?H_eq -?app_assoc -?app_comm_cons //. + } Qed. - Lemma empty_spec : Empty empty. + Lemma inter_aux_props : + forall (y2 : Z) (c2 : N) (s : t) acc, + interval_list_invariant (rev acc) = true -> + interval_list_invariant s = true -> + (forall x1 x2, InZ x1 acc -> InZ x2 s -> + List.In x2 (elementsZ_single y2 c2) -> + Z.succ x1 < x2) -> + (c2 <> 0%N) -> + match (inter_aux y2 c2 acc s) with (acc', s') => + (forall y, (InZ y acc' <-> + (InZ y acc \/ (InZ y s /\ (List.In y (elementsZ_single y2 c2)))))) /\ + (forall y, InZ y s' -> InZ y s) /\ + (forall y, InZ y s -> y2 + Z.of_N c2 < y -> InZ y s') /\ + interval_list_invariant (rev acc') = true /\ + interval_list_invariant s' = true + end. Proof. - rewrite /Empty => a. - rewrite empty_spec' //. - Qed. - - (** *** is_empty specification *) + intros y2 c2. + induction s as [| [y1 c1] s1' IH] => acc. { + rewrite /inter_aux. + move => H_inv_acc _ _ _. + split; last split; try done. + move => y. rewrite InZ_nil. + tauto. + } { + rewrite interval_list_invariant_cons. + move => H_inv_acc [H_gr_s1'] [H_c1_neq_0] H_inv_s1'. + move => H_in_acc_lt H_c2_neq_0. + + rewrite inter_aux_alt_def. + case_eq (inter_aux y2 c2 nil ((y1, c1) :: s1')). + move => acc' s' H_inter_aux_eq. + + set P1 := forall y : Z, + (InZ y acc' <-> + ((InZ y ((y1, c1) :: s1') /\ List.In y (elementsZ_single y2 c2)))). + set P2 := (forall y, + (InZ y s' -> InZ y ((y1, c1) :: s1')) /\ + (InZ y ((y1, c1) :: s1') -> + y2 + Z.of_N c2 < y -> InZ y s')). + set P3 := interval_list_invariant (rev acc') = true. + set P4 := interval_list_invariant s' = true. + + suff : (P1 /\ P2 /\ P3 /\ P4). { + move => [H_P1] [H_P2] [H_P3] H_P4. + split; last split; last split; last split. { + move => y. + move : (H_P1 y). + rewrite !InZ_app InZ_cons !In_elementsZ_single. + move => <-. + tauto. + } { + move => y H_y_in. + by apply H_P2. + } { + move => y H_y_in. + by apply H_P2. + } { + rewrite rev_app_distr. + apply interval_list_invariant_app_intro => //. + move => x1 x2. + rewrite !InZ_rev. + move => H_x1_in /H_P1 [H_x2_in1] H_x2_in2. + apply H_in_acc_lt => //. + } { + apply H_P4. + } + } - Lemma is_empty_spec : forall (s : t) (Hs : Ok s), is_empty s = true <-> Empty s. + move : (H_gr_s1'). + rewrite interval_list_elements_greater_alt2_def => // => H_gr_s1'_alt. + + have : forall (acc : list (Z * N)), + interval_list_invariant (rev acc) = true -> + (forall y, InZ y acc <-> ( + y1 <= y < y1 + Z.of_N c1 /\ + y2 <= y < y2 + Z.of_N c2)) -> + (y1 + Z.of_N c1 <= y2 + Z.of_N c2) -> + (inter_aux y2 c2 acc s1' = (acc', s')) -> + P1 /\ P2 /\ P3 /\ P4. { + + intros acc0 H_inv_acc0 H_in_acc0 H_y2c_lt H_inter_aux_eq0. + have H_in_acc0_lt : (forall x1 x2 : Z, + InZ x1 acc0 -> + InZ x2 s1' -> + List.In x2 (elementsZ_single y2 c2) -> + Z.succ x1 < x2). { + + intros x1 x2 H_x1_in_acc0 H_x2_in_s1' H_x2_in_yc2. + + suff H_yc1_lt_x2 : Z.succ x1 <= y1 + Z.of_N c1. { + apply Z.le_lt_trans with (m := y1 + Z.of_N c1) => //. + by apply H_gr_s1'_alt. + } + move : (H_x1_in_acc0). + rewrite H_in_acc0 Z.le_succ_l. + tauto. + } + + move : (IH acc0 H_inv_acc0 H_inv_s1' H_in_acc0_lt H_c2_neq_0). + rewrite H_inter_aux_eq0. + move => [H1] [H2] [H3] [H4] H5. + split; last split => //. { + move => y. + rewrite (H1 y). + rewrite InZ_cons !In_elementsZ_single + H_in_acc0. + tauto. + } { + move => y. + split. { + move => /H2. + rewrite InZ_cons. + by right. + } { + rewrite InZ_cons In_elementsZ_single. + move => []. { + move => [_] H_y_lt H_lt_y. + exfalso. + suff : (y < y) by apply Z.lt_irrefl. + apply Z.lt_le_trans with (m := y1 + Z.of_N c1) => //. + apply Z.le_trans with (m := y2 + Z.of_N c2) => //. + + by apply Z.lt_le_incl. + } { + apply H3. + } + } + } + } + move => {IH} IH. + + clear H_inv_acc H_in_acc_lt acc. + move : (interval_compare_elim y1 c1 y2 c2) H_inter_aux_eq. + unfold inter_aux. + case_eq (interval_compare (y1, c1) (y2, c2)) => H_comp; + fold inter_aux. { + move => H_lt_y2. + apply IH. { + done. + } { + move => x. + rewrite InZ_nil. + split => //. + omega. + } { + apply Z.le_trans with (m := y2). { + by apply Z.lt_le_incl. + } { + apply Z_le_add_r. + } + } + } { + move => H_eq_y2. + apply IH. { + done. + } { + move => x. + rewrite InZ_nil. + split => //. + omega. + } { + rewrite H_eq_y2. + apply Z_le_add_r. + } + } { + move => [H_y1_lt_y2] [H_y2_lt_yc1] H_yc1_lt_yc2. + apply IH. { + rewrite interval_list_invariant_sing. + by apply Z_to_N_minus_neq_0. + } { + move => x. + rewrite InZ_cons InZ_nil In_elementsZ_single Z2N.id; last omega. + replace (y1 + (y2 - y1)) with y2 by omega. + split; omega. + } { + by apply Z.lt_le_incl. + } + } { + rewrite /P1 /P2 /P3 /P4. + move => [H_y2_lt] [H_y1_lt] H_yc1_lt. + move => [] H_acc' H_s'. + clear IH P1 P2 P3 P4 H_comp. + subst s' acc'. + split; last split; last split. { + move => y. + have H_yc2_intro : y1 + Z.of_N (Z.to_N (y2 + Z.of_N c2 - y1)) = + y2 + Z.of_N c2. { + rewrite Z2N.id; omega. + } + + rewrite !InZ_cons !In_elementsZ_single InZ_nil H_yc2_intro. + split. { + move => [] //. + move => [H_y1_le] H_y_lt. + split; last by omega. + left. omega. + } { + move => [H_in_s] [H_y2_le] H_y_lt. + left. + split; last assumption. + move : H_in_s => []. { + tauto. + } { + move => /H_gr_s1'_alt H_lt_y. + apply Z.le_trans with (m := y1 + Z.of_N c1). { + by apply Z_le_add_r. + } { + by apply Z.lt_le_incl. + } + } + } + } { + move => y. + split; done. + } { + rewrite interval_list_invariant_sing. + by apply Z_to_N_minus_neq_0. + } { + by rewrite interval_list_invariant_cons. + } + } { + rewrite /P1 /P2 /P3 /P4. + move => [H_y12_eq] H_c12_eq [] H_acc' H_s'. + clear IH P1 P2 P3 P4 H_comp. + subst. + split; last split; last split. { + move => y. + rewrite !InZ_cons InZ_nil In_elementsZ_single. + split; last by tauto. { + move => [] //. + tauto. + } + } { + move => y. + rewrite InZ_cons In_elementsZ_single. + split; first by right. + move => [] //. + move => [_] H_y_lt H_lt_y. + exfalso. + suff : (y2 + Z.of_N c2 < y2 + Z.of_N c2) by + apply Z.lt_irrefl. + apply Z.lt_trans with (m := y) => //. + } { + rewrite interval_list_invariant_sing //. + } { + assumption. + } + } { + move => [H_y2_le_y1] [H_yc1_le_yc2] _. + apply IH. { + by rewrite interval_list_invariant_sing. + } { + move => y. + rewrite InZ_cons InZ_nil In_elementsZ_single. + split. { + move => [] //. + move => [H_y1_le] H_y_lt. + split; first done. + split; omega. + } { + move => [?] _. + by left. + } + } { + assumption. + } + } { + rewrite /P1 /P2 /P3 /P4. + move => [H_y1_le] [H_yc2_le] _. + move => [] H_acc' H_s'. + clear IH P1 P2 P3 P4 H_comp. + subst. + split; last split; last split. { + move => y. + rewrite !InZ_cons !In_elementsZ_single InZ_nil. + split. { + move => [] //. + move => [H_y2_le] H_y_lt. + split; last by omega. + left. omega. + } { + move => [H_in_s] [H_y2_le] H_y_lt. + by left. + } + } { + tauto. + } { + by rewrite interval_list_invariant_sing. + } { + by rewrite interval_list_invariant_cons. + } + } { + rewrite /P1 /P2 /P3 /P4. + move => H_yc2_lt [] H_acc' H_s'. + clear IH P1 P2 P3 P4 H_comp. + subst. + split; last split; last split. { + move => y. + rewrite InZ_cons !In_elementsZ_single InZ_nil. + split; first done. + move => [] []. { + move => [H_y1_le_y] H_y_lt_yc1. + move => [H_y2_le_y] H_y_lt_yc2. + omega. + } { + move => /H_gr_s1'_alt H_lt_y [_] H_y_lt. + suff : (y1 + Z.of_N c1 < y1). { + apply Z.nlt_ge. + apply Z_le_add_r. + } + omega. + } + } { + tauto. + } { + done. + } { + by rewrite interval_list_invariant_cons. + } + } { + rewrite /P1 /P2 /P3 /P4. + move => H_y1_eq [] H_acc' H_s'. + clear IH P1 P2 P3 P4 H_comp. + subst acc' s'. + split; last split; last split. { + move => y. + rewrite InZ_cons !In_elementsZ_single InZ_nil. + split; first done. + move => [] []. { + move => [H_y1_le_y] H_y_lt_yc1. + move => [H_y2_le_y] H_y_lt_yc2. + omega. + } { + move => /H_gr_s1'_alt H_lt_y [_] H_y_lt. + suff : (y1 + Z.of_N c1 < y1). { + apply Z.nlt_ge. + apply Z_le_add_r. + } + omega. + } + } { + tauto. + } { + done. + } { + by rewrite interval_list_invariant_cons. + } + } + } + Qed. + + Lemma inter_aux2_props : + forall (s2 s1 acc : t), + interval_list_invariant (rev acc) = true -> + interval_list_invariant s1 = true -> + interval_list_invariant s2 = true -> + (forall x1 x2, InZ x1 acc -> InZ x2 s1 -> InZ x2 s2 -> Z.succ x1 < x2) -> + ((forall y, (InZ y (inter_aux2 acc s1 s2) <-> + (InZ y acc) \/ ((InZ y s1) /\ InZ y s2))) /\ + (interval_list_invariant (inter_aux2 acc s1 s2) = true)). Proof. - intros [ | [x c] s]. { - split => // _. - apply empty_spec. + induction s2 as [| [y2 c2] s2' IH]. { + move => s1 acc. + move => H_inv_acc _ _ _. + unfold inter_aux2. + replace (match s1 with + | nil => rev' acc + | _ :: _ => rev' acc + end) with (rev' acc); last by case s1. + rewrite /rev' rev_append_rev app_nil_r. + split; last done. + move => y. + rewrite InZ_nil InZ_rev. + tauto. } { - rewrite /= /Empty Ok_cons. - move => [_] [H_c_neq] [H_enc] _. - split => //. - move => H. - contradiction (H (Enc.decode x)) => {H}. - rewrite /In InA_alt elements_cons. - exists (Enc.decode x). - split; first by apply Enc.E.eq_equiv. - rewrite in_app_iff; left. - rewrite /elements_single in_map_iff. - exists x. - split => //. - apply In_elementsZ_single_hd => //. + intros s1 acc H_inv_acc H_inv_s1. + rewrite interval_list_invariant_cons. + move => [H_gr_s2'] [H_c2_neq_0] H_inv_s2'. + move => H_acc_s12. + + move : H_gr_s2'. + rewrite interval_list_elements_greater_alt2_def //. + move => H_gr_s2'. + + rewrite /inter_aux2; fold inter_aux2. + case_eq s1. { + move => H_s1_eq. + split. { + move => y. + rewrite /rev' rev_append_rev app_nil_r InZ_nil + InZ_rev. + tauto. + } { + rewrite /rev' rev_append_rev app_nil_r //. + } + } { + move => [_ _] _ <-. + case_eq (inter_aux y2 c2 acc s1). + move => acc' s1' H_inter_aux_eq. + + have H_acc_s1_yc2 : forall x1 x2 : Z, + InZ x1 acc -> + InZ x2 s1 -> + List.In x2 (elementsZ_single y2 c2) -> + Z.succ x1 < x2. { + intros x1 x2 H_x1_in H_x2_in1 H_x2_in2. + apply H_acc_s12 => //. + rewrite InZ_cons; by left. + } + + move : (inter_aux_props y2 c2 s1 acc H_inv_acc H_inv_s1 H_acc_s1_yc2 H_c2_neq_0). + rewrite H_inter_aux_eq. + move => [H_in_acc'] [H_in_s1'_elim] [H_in_s1'_intro] + [H_inv_acc'] H_inv_s1'. + + have H_in_acc'_s2' : forall x1 x2 : Z, + InZ x1 acc' -> InZ x2 s1' -> InZ x2 s2' -> Z.succ x1 < x2. { + move => x1 x2 /H_in_acc'. + move => []. { + move => H_in_acc H_in_s1' H_in_s2'. + apply H_acc_s12 => //. { + by apply H_in_s1'_elim. + } { + rewrite InZ_cons; by right. + } + } { + rewrite In_elementsZ_single. + move => [H_in_s1] [_] H_x1_lt _. + move => /H_gr_s2' H_lt_x2. + apply Z.le_lt_trans with (m := y2 + Z.of_N c2) => //. + by apply Z.le_succ_l. + } + } + + move : (IH s1' acc' H_inv_acc' H_inv_s1' H_inv_s2' H_in_acc'_s2'). + move => [H_inZ_res] H_inv_res. + split; last assumption. + move => y. + rewrite H_inZ_res H_in_acc' InZ_cons + In_elementsZ_single. + split. { + move => []; first by tauto. + move => [H_y_in_s1' H_y_in_s2']. + right. + split; last by right. + by apply H_in_s1'_elim. + } { + move => []. { + move => H_y_in_acc. + by left; left. + } { + move => [H_y_in_s1]. + move => []. { + move => H_in_yc2. + by left; right. + } { + right. + split; last assumption. + apply H_in_s1'_intro => //. + by apply H_gr_s2'. + } + } + } + } + } + Qed. + + Lemma inter_InZ : + forall (s1 s2 : t), + interval_list_invariant s1 = true -> + interval_list_invariant s2 = true -> + forall y, (InZ y (inter s1 s2) <-> InZ y s1 /\ InZ y s2). + Proof. + intros s1 s2 H_inv_s1 H_inv_s2 y. + rewrite /inter. + + move : (inter_aux2_props s2 s1 nil). + move => [] //. + move => H_in_inter _. + rewrite H_in_inter InZ_nil. + tauto. + Qed. + + Lemma inter_invariant : + forall (s1 s2 : t), + interval_list_invariant s1 = true -> + interval_list_invariant s2 = true -> + interval_list_invariant (inter s1 s2) = true. + Proof. + intros s1 s2 H_inv_s1 H_inv_s2. + rewrite /inter. + + move : (inter_aux2_props s2 s1 nil). + move => [] //. + Qed. + + + Global Instance inter_ok s1 s2 : forall `(Ok s1, Ok s2), Ok (inter s1 s2). + Proof. + move => H_ok_s1 H_ok_s2. + move : (H_ok_s1) (H_ok_s2). + rewrite /Ok /IsOk /is_encoded_elems_list /add. + move => [H_inv_s1] H_pre1. + move => [H_inv_s2] H_pre2. + split. { + apply inter_invariant => //. + } { + intros y. + move : (inter_InZ s1 s2 H_inv_s1 H_inv_s2). + rewrite /InZ => ->. + move => []. + move => /H_pre1 //. + } + Qed. + + Lemma inter_spec : + forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'), + In x (inter s s') <-> In x s /\ In x s'. + Proof. + intros s s' x H_ok H_ok'. + rewrite !In_InZ. + rewrite inter_InZ => //. { + apply H_ok. + } { + apply H_ok'. + } + Qed. + + + (** *** filter specification *) + + Global Instance filter_ok s f : forall `(Ok s), Ok (filter f s). + Proof. + move => _. + rewrite /filter /from_elements. + apply add_list_ok, empty_ok. + Qed. + + Lemma filter_spec : + forall (s : t) (x : elt) (f : elt -> bool), + Proper (Enc.E.eq==>eq) f -> + (In x (filter f s) <-> In x s /\ f x = true). + Proof. + move => s x f H_prop. + rewrite /filter /from_elements add_list_spec empty_spec' /In !InA_alt. + setoid_rewrite filter_In. + split. { + move => [] //. + move => [y] [H_y_eq] [H_y_in] H_fy. + split; first by exists y. + rewrite -H_fy. + by f_equiv. + } { + move => [] [y] [H_y_eq] H_y_in H_fx. + left. + exists y. + split; last split; try assumption. + apply Logic.eq_sym. + rewrite -H_fx. + by f_equiv. + } + Qed. + + + (** *** diff specification *) + + Lemma diff_aux_alt_def : + forall (y2 : Z) (c2 : N) (s : t) acc, + diff_aux y2 c2 acc s = match diff_aux y2 c2 nil s with + (acc', s') => (acc' ++ acc, s') + end. + Proof. + intros y2 c2. + induction s as [| [y1 c1] acc' IH] => acc. { + rewrite /diff_aux app_nil_l //. + } { + simpl. + case_eq (diff_aux y2 c2 nil acc') => acc'' s'' H_eq. + case (interval_compare (y1, c1) (y2, c2)); + rewrite ?(IH ((y1, c1)::acc)) ?(IH ((y1, c1)::nil)) + ?(IH acc) ?(IH ((y1, Z.to_N (y2 - y1)) :: acc)) + ?(IH ((y1, Z.to_N (y2 - y1)) :: nil)) ?H_eq; + rewrite ?insert_intervalZ_guarded_app -?app_assoc -?app_comm_cons //. + } + Qed. + + + Lemma diff_aux_props : + forall (y2 : Z) (c2 : N) (s : t) acc, + interval_list_invariant (List.rev acc) = true -> + interval_list_invariant s = true -> + (forall x1 x2, InZ x1 acc -> InZ x2 s -> Z.succ x1 < x2) -> + (forall x, InZ x acc -> x < y2) -> + (c2 <> 0%N) -> + match (diff_aux y2 c2 acc s) with + (acc', s') => (forall y, InZ y (List.rev_append acc' s') <-> + InZ y (List.rev_append acc s) /\ ~(List.In y (elementsZ_single y2 c2))) /\ + (interval_list_invariant (List.rev_append acc' s') = true) /\ + (forall x, InZ x acc' -> x < y2 + Z.of_N c2) + end. + Proof. + intros y2 c2. + induction s as [| [y1 c1] s1' IH] => acc. { + rewrite /diff_aux -rev_alt. + move => H_inv_acc _ _ H_in_acc H_c2_neq. + split; last split. { + move => y; split; last by move => [] //. + rewrite InZ_rev. + move => H_in. split => //. + move => /In_elementsZ_single => [] [] /Z.nlt_ge H_neq. + contradict H_neq. + by apply H_in_acc. + } { + assumption. + } { + intros x H_in_acc'. + apply Z.lt_le_trans with (m := y2). { + by apply H_in_acc. + } { + by apply Z_le_add_r. + } + } + } { + rewrite interval_list_invariant_cons. + move => H_inv_acc [H_gr_s1'] [H_c1_neq_0] H_inv_s1'. + move => H_in_s1 H_in_acc H_c2_neq_0. + + rewrite diff_aux_alt_def. + case_eq (diff_aux y2 c2 nil ((y1, c1) :: s1')). + move => acc' s' H_diff_aux_eq. + + set P1 := forall y : Z, + (InZ y acc' \/ InZ y s') <-> + InZ y ((y1, c1) :: s1') /\ ~ List.In y (elementsZ_single y2 c2). + set P2 := interval_list_invariant (rev acc' ++ s') = true. + set P3 := forall x : Z, InZ x acc' -> (x < y2 + Z.of_N c2). + + suff : (P1 /\ P2 /\ P3). { + move => [H_P1] [H_P2] H_P3. + split; last split. { + move => y. + move : (H_P1 y). + rewrite !rev_append_rev rev_app_distr !InZ_app + !InZ_rev In_elementsZ_single. + suff : (InZ y acc -> ~ y2 <= y < y2 + Z.of_N c2). { + tauto. + } + move => /H_in_acc H_y_lt [H_y_ge] _. + contradict H_y_ge. + by apply Zlt_not_le. + } { + rewrite rev_append_rev rev_app_distr -app_assoc. + apply interval_list_invariant_app_intro => //. + move => x1 x2. + rewrite InZ_app !InZ_rev. + move => H_in_acc' H_x2_in_s'. + suff : (InZ x2 ((y1, c1)::s1')). { + by apply H_in_s1. + } + move : (H_P1 x2). + tauto. + } { + move => x. + rewrite InZ_app. + move => []. { + apply H_P3. + } { + move => /H_in_acc H_x_lt. + eapply Z.lt_trans; eauto. + by apply Z_lt_add_r. + } + } + } + + move : (H_gr_s1'). + rewrite interval_list_elements_greater_alt2_def => // => H_gr_s1'_alt. + + have : forall (acc : list (Z * N)), + interval_list_invariant (rev acc) = true -> + (forall x : Z, + InZ x acc <-> + ((y1 <= x < y1 + Z.of_N c1) /\ (x < y2))) -> + (y1 + Z.of_N c1 <= y2 + Z.of_N c2) -> + (diff_aux y2 c2 acc s1' = (acc', s')) -> + P1 /\ P2 /\ P3. { + + intros acc0 H_inv_acc0 H_in_acc0 H_c1_before H_diff_aux_eq0. + have H_in_s1' : (forall x1 x2 : Z, + InZ x1 acc0 -> InZ x2 s1' -> Z.succ x1 < x2). { + intros x1 x2 H_x1_in_acc0. + move => /H_gr_s1'_alt. + + eapply Z.le_lt_trans. + move : H_x1_in_acc0. + rewrite Z.le_succ_l H_in_acc0. + tauto. + } + have H_in_acc0' : (forall x : Z, InZ x acc0 -> x < y2). { + move => x. + rewrite H_in_acc0. + move => [_] //. + } + move : (IH acc0 H_inv_acc0 H_inv_s1' H_in_s1' H_in_acc0' H_c2_neq_0). + rewrite H_diff_aux_eq0 !rev_append_rev. + move => [H1] [H2] H3. + split; last split => //. { + move => y. + move : (H1 y). + rewrite !InZ_app !InZ_rev In_elementsZ_single. + move => ->. + rewrite InZ_cons In_elementsZ_single. + split. { + rewrite H_in_acc0 -(Z.nle_gt y2 y). + tauto. + } { + rewrite H_in_acc0 -(Z.nle_gt y2 y). + move => [] H_in H_nin_i2. + split; last by assumption. + move : H_in => [] H_in; last by right. + left. + omega. + } + } + } + move => {IH} IH. + + clear H_inv_acc H_in_s1 H_in_acc acc. + move : (interval_compare_elim y1 c1 y2 c2) H_diff_aux_eq. + unfold diff_aux. + case_eq (interval_compare (y1, c1) (y2, c2)) => H_comp; + fold diff_aux. { + move => H_lt_y2. + apply IH. { + by rewrite interval_list_invariant_sing. + } { + move => x. + rewrite InZ_cons In_elementsZ_single. + split; last by tauto. + move => []; last done. + move => [H_y1_le H_x_lt]. + split; first done. + eapply Z.lt_trans; eauto. + } { + apply Z.le_trans with (m := y2). + - by apply Z.lt_le_incl. + - by apply Z_le_add_r. + } + } { + move => H_eq_y2. + apply IH. { + by rewrite interval_list_invariant_sing. + } { + move => x. + rewrite InZ_cons In_elementsZ_single -H_eq_y2. + split; last by tauto. + move => []; last done. + move => []. done. + } { + rewrite H_eq_y2. + by apply Z_le_add_r. + } + } { + move => [H_y1_lt_y2] [H_y2_lt_yc1] H_yc1_lt_yc2. + apply IH. { + rewrite interval_list_invariant_sing. + by apply Z_to_N_minus_neq_0. + } { + move => x. + rewrite InZ_cons In_elementsZ_single Z2N.id; last omega. + replace (y1 + (y2 - y1)) with y2 by omega. + split; last tauto. + move => [] //. + move => [H_y1_le] H_x_lt. + repeat split => //. + apply Z.lt_trans with (m := y2) => //. + } { + by apply Z.lt_le_incl. + } + } { + rewrite /P1 /P2 /P3. + move => [H_y2_lt] [H_y1_lt] H_yc1_lt [H_acc'] H_s'. + clear IH P1 P2 P3 H_comp. + subst. + have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. { + rewrite Z2N.id; omega. + } + have H_nin_yc2 : forall y, + InZ y s1' -> ~ y2 <= y < y2 + Z.of_N c2. { + move => y /H_gr_s1'_alt H_lt_y. + move => [H_y2_le_y]. + apply Z.le_ngt, Z.lt_le_incl. + by apply Z.lt_trans with (m := y1 + Z.of_N c1). + } + split; last split. { + move => y. + rewrite !InZ_cons !In_elementsZ_single H_yc1_intro. + split. { + move => [] //. + move => []. { + move => [H_le_y] H_y_lt. + split. { + left; omega. + } { + move => [_]. + by apply Z.nlt_ge. + } + } { + move : (H_nin_yc2 y). tauto. + } + } { + move => [] []; last by right; right. + move => [H_y_ge] H_y_lt_yc1 H_nin_yc2'. + right; left. omega. + } + } { + rewrite interval_list_invariant_cons H_yc1_intro. + split => //. + split => //. + by apply Z_to_N_minus_neq_0. + } { + move => [] //. + } + } { + rewrite /P1 /P2 /P3. + move => [H_y12_eq] H_c12_eq [] H_acc' H_s'. + clear IH P1 P2 P3 H_comp. + subst. + split; last split. { + move => y. + rewrite InZ_cons In_elementsZ_single. + split; last by tauto. { + move => [] //. + move => H_y_in. + split; first by right. + move => [] _. + by apply Z.nlt_ge, Z.lt_le_incl, H_gr_s1'_alt. + } + } { + apply H_inv_s1'. + } { + move => x [] //. + } + } { + move => [H_y2_le_y1] [H_yc1_le_yc2] _. + apply IH. { + done. + } { + move => x. + split; first done. + omega. + } { + assumption. + } + } { + rewrite /P1 /P2 /P3. + move => [H_y1_le] [H_yc2_le_yc1] _ [] H_acc' H_s'. + clear IH P1 P2 P3 H_comp. + subst. + have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. { + rewrite Z2N.id; omega. + } + have H_y1_intro : y1 + Z.of_N (Z.to_N (y2 - y1)) = y2. { + rewrite Z2N.id; omega. + } + split; last split. { + move => y. + rewrite !InZ_insert_intervalZ_guarded + !InZ_cons !In_elementsZ_single + H_yc1_intro H_y1_intro InZ_nil. + split. { + rewrite -!or_assoc. + move => [[[]|]|] //. { + move => [H_y1_le_y] H_y_lt. + split. { + left. + split => //. + apply Z.lt_le_trans with (m := y2 + Z.of_N c2) => //. + apply Z.lt_trans with (m := y2) => //. + by apply Z_lt_add_r. + } { + move => [] /Z.le_ngt //. + } + } { + move => [H_y2c_le_y] H_y_lt_yc1. + split. { + left. + split => //. + apply Z.le_trans with (m := y2 + Z.of_N c2) => //. + apply Z.le_trans with (m := y2) => //. + apply Z_le_add_r. + } { + move => [] _ /Z.lt_nge //. + } + } { + move => H_y_in_s1'. + split; first by right. + suff H_suff : y2 + Z.of_N c2 <= y. { + move => [] _ /Z.lt_nge //. + } + apply Z.le_trans with (m := y1 + Z.of_N c1) => //. + apply Z.lt_le_incl. + by apply H_gr_s1'_alt. + } + } { + move => [] []; last by tauto. + move => [H_y1_le_y] H_y_lt H_neq_y2. + apply not_and in H_neq_y2; last by apply Z.le_decidable. + case H_neq_y2. { + move => /Z.nle_gt H_y_lt'. + left; left; done. + } { + move => /Z.nlt_ge H_le_y. + right; left; done. + } + } + } { + rewrite insert_intervalZ_guarded_rev_nil_app. + rewrite !interval_list_invariant_insert_intervalZ_guarded => //. { + rewrite H_yc1_intro => //. + } { + rewrite /insert_intervalZ_guarded. + case_eq ((Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2)) =? 0)%N). { + rewrite H_y1_intro. + move => /N.eqb_eq /N2Z.inj_iff. + rewrite Z2N.id; last first. { + by apply Z.le_0_sub. + } + move => /Zminus_eq H_yc1_eq. + eapply interval_list_elements_greater_impl; + last apply H_gr_s1'. + rewrite H_yc1_eq. + apply Z_le_add_r. + } { + move => _. + rewrite interval_list_elements_greater_cons + H_y1_intro. + by apply Z_lt_add_r. + } + } + } { + move => x. + rewrite InZ_insert_intervalZ_guarded InZ_cons In_elementsZ_single H_y1_intro InZ_nil. + move => [] //. + move => [_] H_x_lt. + apply Z.lt_le_trans with (m := y2) => //. + apply Z_le_add_r. + } + } { + rewrite /P1 /P2 /P3. + move => H_yc2_lt [] H_acc' H_s'. + clear IH P1 P2 P3 H_comp. + subst. + split; last split. { + move => y. + rewrite InZ_cons In_elementsZ_single. + split; last by tauto. { + move => [] //. + move => H_y_in. + split; first assumption. + rewrite In_elementsZ_single. + move => [] H_y2_le H_y_lt. + case H_y_in; first by omega. + move => /H_gr_s1'_alt H_lt_y. + suff : y2 + Z.of_N c2 < y. { + move => ?. omega. + } + apply Z.lt_trans with (m := y1 + Z.of_N c1) => //. + apply Z.lt_le_trans with (m := y1) => //. + apply Z_le_add_r. + } + } { + by rewrite interval_list_invariant_cons. + } { + done. + } + } { + rewrite /P1 /P2 /P3. + move => H_yc2_eq [] H_acc' H_s'. + clear IH P1 P2 P3 H_comp. + subst. + split; last split. { + move => y. + rewrite InZ_cons In_elementsZ_single. + split; last by tauto. { + move => [] //. + move => H_y_in. + split; first assumption. + rewrite In_elementsZ_single. + move => [] H_y2_le H_y_lt. + case H_y_in; first by omega. + move => /H_gr_s1'_alt H_lt_y. + suff : y2 + Z.of_N c2 < y. { + move => ?. omega. + } + apply Z.lt_trans with (m := (y2 + Z.of_N c2) + Z.of_N c1) => //. + by apply Z_lt_add_r. + } + } { + by rewrite interval_list_invariant_cons. + } { + done. + } + } + } + Qed. + + + Lemma diff_aux2_props : + forall (s2 s1 acc : t), + interval_list_invariant (rev_append acc s1) = true -> + interval_list_invariant s2 = true -> + (forall x1 x2, InZ x1 acc -> InZ x2 s2 -> Z.succ x1 < x2) -> + ((forall y, (InZ y (diff_aux2 acc s1 s2) <-> + ((InZ y acc) \/ (InZ y s1)) /\ ~InZ y s2)) /\ + (interval_list_invariant (diff_aux2 acc s1 s2) = true)). + Proof. + induction s2 as [| [y2 c2] s2' IH]. { + move => s1 acc H_inv_acc_s1 _ _. + rewrite /diff_aux2. + replace (match s1 with + | nil => rev_append acc s1 + | _ :: _ => rev_append acc s1 + end) with (rev_append acc s1); last by case s1. + split. { + move => y. + rewrite rev_append_rev InZ_app InZ_rev InZ_nil. + tauto. + } { + assumption. + } + } { + intros s1 acc H_inv_acc_s1. + rewrite interval_list_invariant_cons. + move => [H_gr_s2'] [H_c2_neq_0] H_inv_s2'. + move => H_acc_s2. + rewrite /diff_aux2; fold diff_aux2. + case_eq s1. { + move => H_s1_eq. + split. { + move => y. + rewrite rev_append_rev InZ_app InZ_nil InZ_rev. + split; last tauto. + move => [] // H_y_in. + split; first by left. + move => H_y_in'. + move : (H_acc_s2 _ _ H_y_in H_y_in'). + apply Z.nlt_succ_diag_l. + } { + move : H_inv_acc_s1. + by rewrite H_s1_eq. + } + } { + move => [_ _] _ <-. + case_eq (diff_aux y2 c2 acc s1). + move => acc' s1' H_diff_aux_eq. + + have H_acc_lt_y2 : (forall x : Z, InZ x acc -> x < y2). { + move => x H_x_in. + have H_y2_in: (InZ y2 ((y2,c2) :: s2')). { + rewrite InZ_cons. + left. + by apply In_elementsZ_single_hd. + } + move : (H_acc_s2 _ _ H_x_in H_y2_in). + apply Z.lt_trans, Z.lt_succ_diag_r. + } + + have [H_inv_acc [H_inv_s1 H_acc_s1]] : + interval_list_invariant (rev acc) = true /\ + interval_list_invariant s1 = true /\ + (forall x1 x2 : Z, + InZ x1 acc -> InZ x2 s1 -> Z.succ x1 < x2). { + + move : H_inv_acc_s1. + rewrite rev_append_rev. + move => /interval_list_invariant_app_elim. + move => [?] [?] H_x. + split; first assumption. + split; first assumption. + move => x1 x2 H_in_x1. + apply H_x. + by rewrite InZ_rev. + } + + move : (diff_aux_props y2 c2 s1 acc H_inv_acc H_inv_s1 H_acc_s1 H_acc_lt_y2 H_c2_neq_0). + rewrite !H_diff_aux_eq. + move => [H_inZ_res] [H_inv_res] H_inZ_acc'. + + have H_acc'_s2' : (forall x1 x2 : Z, + InZ x1 acc' -> InZ x2 s2' -> Z.succ x1 < x2). { + move => x1 x2 H_x1_in H_x2_in. + apply Z.le_lt_trans with (m := y2 + Z.of_N c2). { + apply Z.le_succ_l. + by apply H_inZ_acc'. + } { + move : H_gr_s2'. + rewrite interval_list_elements_greater_alt2_def //. + move => H_gr_s2'. + by apply H_gr_s2'. + } + } + + move : (IH s1' acc' H_inv_res H_inv_s2' H_acc'_s2'). + move => [] H_inZ_diff_res ->. + split; last done. + move => y. + rewrite H_inZ_diff_res. + move : (H_inZ_res y). + rewrite !rev_append_rev !InZ_app !InZ_rev InZ_cons. + move => ->. + tauto. + } + } + Qed. + + + Lemma diff_InZ : + forall (s1 s2 : t), + interval_list_invariant s1 = true -> + interval_list_invariant s2 = true -> + forall y, (InZ y (diff s1 s2) <-> InZ y s1 /\ ~InZ y s2). + Proof. + intros s1 s2 H_inv_s1 H_inv_s2 y. + rewrite /diff. + + move : (diff_aux2_props s2 s1 nil). + move => [] //. + move => H_in_diff _. + rewrite H_in_diff InZ_nil. + tauto. + Qed. + + Lemma diff_invariant : + forall (s1 s2 : t), + interval_list_invariant s1 = true -> + interval_list_invariant s2 = true -> + interval_list_invariant (diff s1 s2) = true. + Proof. + intros s1 s2 H_inv_s1 H_inv_s2. + rewrite /diff. + + move : (diff_aux2_props s2 s1 nil). + move => [] //. + Qed. + + + Global Instance diff_ok s1 s2 : forall `(Ok s1, Ok s2), Ok (diff s1 s2). + Proof. + move => H_ok_s1 H_ok_s2. + move : (H_ok_s1) (H_ok_s2). + rewrite /Ok /IsOk /is_encoded_elems_list /add. + move => [H_inv_s1] H_pre1. + move => [H_inv_s2] H_pre2. + split. { + apply diff_invariant => //. + } { + intros y. + move : (diff_InZ s1 s2 H_inv_s1 H_inv_s2). + rewrite /InZ => ->. + move => []. + move => /H_pre1 //. + } + Qed. + + Lemma diff_spec : + forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'), + In x (diff s s') <-> In x s /\ ~In x s'. + Proof. + intros s s' x H_ok H_ok'. + rewrite !In_InZ. + rewrite diff_InZ => //. { + apply H_ok. + } { + apply H_ok'. + } + Qed. + + + (** *** remove specification *) + + Lemma removeZ_alt_def : forall x s acc, + interval_list_invariant s = true -> + removeZ_aux acc x s = match diff_aux x (1%N) acc s with + (acc', s') => rev_append acc' s' + end. + Proof. + intros y2. + induction s as [| [y1 c1] s' IH]; first done. + + move => acc. + rewrite interval_list_invariant_cons /=. + move => [H_gr] [H_c1_neq_0] H_inv_s'. + move : (interval_1_compare_elim y2 y1 c1). + rewrite interval_1_compare_alt_def. + rewrite !(interval_compare_swap y1 c1 y2); last first. { + right. done. + } + move : (interval_compare_elim y1 c1 y2 (1%N)). + case_eq (interval_compare (y1, c1) (y2, (1%N))) => H_eq. { + move => H_lt_y2 _. + have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). { + by apply Z.nlt_ge, Z.lt_le_incl. + } + have H_y2_nlt : ~(y2 < y1). { + move => H_y2_y1. + apply H_yc2_nlt. + apply Z.lt_le_trans with (m := y1) => //. + apply Z_le_add_r. + } + move : (H_y2_nlt) (H_yc2_nlt) => /Z.ltb_nlt -> /Z.ltb_nlt ->. + rewrite IH //. + } { + move => H_lt_y2 _. + have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). { + apply Z.nlt_ge. + rewrite H_lt_y2. + apply Z.le_refl. + } + have H_y2_nlt : ~(y2 < y1). { + move => H_y2_y1. + apply H_yc2_nlt. + apply Z.lt_le_trans with (m := y1) => //. + apply Z_le_add_r. + } + move : (H_y2_nlt) (H_yc2_nlt) => /Z.ltb_nlt -> /Z.ltb_nlt ->. + rewrite IH //. + } { + done. + } { + done. + } { + move => [H_y1_eq] H_c1_eq. + move => [] // . + move => [H_lt_y2] H_y2_lt. + have H_y2_nlt : ~(y2 < y1). { + apply Z.nlt_ge => //. + } + move : (H_y2_nlt) (H_y2_lt) => /Z.ltb_nlt -> /Z.ltb_lt ->. + rewrite /insert_intervalZ_guarded. + + have -> : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. { + rewrite H_y1_eq H_c1_eq Z.add_1_r Z.sub_diag //. + } + + have -> : (Z.to_N (y2 - y1) =? 0 = true)%N. { + rewrite H_y1_eq Z.sub_diag //. + } + done. + } { + move => [H_y2_le] [H_yc1_le] _. + move => [] //. + move => [H_y1_le] H_y2_lt. + have H_y2_nlt : ~(y2 < y1). { + apply Z.nlt_ge => //. + } + move : (H_y2_nlt) (H_y2_lt) => /Z.ltb_nlt -> /Z.ltb_lt ->. + + have H_y1_eq : (y1 = y2) by omega. + have H_yc1_eq : (y1 + Z.of_N c1 = Z.succ y2). { + apply Z.le_antisymm. { + move : H_yc1_le. + rewrite Z.add_1_r //. + } { + by apply Z.le_succ_l. + } + } + + rewrite /insert_intervalZ_guarded. + have -> : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. { + rewrite H_yc1_eq Z.sub_diag //. + } + + have -> : (Z.to_N (y2 - y1) =? 0 = true)%N. { + rewrite H_y1_eq Z.sub_diag //. + } + + suff -> : diff_aux y2 (1%N) acc s' = (acc, s') by done. + + move : H_gr. + rewrite H_yc1_eq. + case s' as [| [y' c'] s'']. { + done. + } { + rewrite interval_list_elements_greater_cons /= + /interval_compare. + move => H_lt_y'. + have -> : y2 + Z.of_N 1 ?= y' = Lt. { + apply Z.compare_lt_iff. + by rewrite Z.add_1_r. + } + done. + } + } { + move => [H_y1_le] [H_yc2_le] _. + move => [] //. + move => [_] H_y2_lt. + have H_y2_nlt : ~(y2 < y1). { + apply Z.nlt_ge => //. + } + move : (H_y2_nlt) (H_y2_lt) => /Z.ltb_nlt -> /Z.ltb_lt ->. + rewrite !rev_append_rev /insert_intervalZ_guarded Z.add_1_r. + case ((Z.to_N (y2 - y1) =? 0)%N), (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0)%N. { + reflexivity. + } { + rewrite /= -!app_assoc //. + } { + reflexivity. + } { + rewrite /= -!app_assoc //. + } + } { + move => _ H_y2_lt'. + have H_y2_lt : (y2 < y1). { + apply Z.lt_trans with (m := Z.succ y2) => //. + apply Z.lt_succ_diag_r. + } + move : (H_y2_lt) => /Z.ltb_lt -> //. + } { + move => _ H_y1_eq. + have H_y2_lt : (y2 < y1). { + rewrite H_y1_eq. + apply Z.lt_succ_diag_r. + } + move : (H_y2_lt) => /Z.ltb_lt -> //. } Qed. - - (** *** singleton specification *) - - Global Instance singleton_ok x : Ok (singleton x). + Lemma removeZ_interval_list_invariant : forall s x, interval_list_invariant s = true -> interval_list_invariant (removeZ x s) = true. Proof. - rewrite singleton_alt_def. - apply add_ok. - apply empty_ok. + intros s x H_inv. + rewrite /removeZ removeZ_alt_def //. + move : (diff_aux_props x (1%N) s nil). + case_eq (diff_aux x 1%N nil s). + move => acc' s' H_eq [] //. + move => _ [] //. Qed. - Lemma singleton_spec : forall x y : elt, In y (singleton x) <-> Enc.E.eq y x. + Lemma removeZ_spec : + forall (s : t) (x y : Z) (Hs : interval_list_invariant s = true), + InZ y (removeZ x s) <-> InZ y s /\ ~Z.eq y x. Proof. - intros x y. - rewrite singleton_alt_def. - rewrite (add_spec empty x y) /empty /In elements_nil InA_nil. - split. { - move => [] //. - } { - by left. - } + intros s x y H_inv. + rewrite /removeZ removeZ_alt_def //. + move : (diff_aux_props x (1%N) s nil). + case_eq (diff_aux x 1%N nil s). + move => acc' s' H_eq [] //. + move => -> _. + rewrite rev_append_rev InZ_app InZ_rev InZ_nil + In_elementsZ_single1. + split; move => [H1 H2]; split => //; + move => H3; apply H2; by rewrite H3. Qed. - (** *** add_list specification *) - - Lemma add_list_ok : forall l s, Ok s -> Ok (add_list l s). + Global Instance remove_ok s x : forall `(Ok s), Ok (remove x s). Proof. - induction l as [| x l' IH]. { - done. + rewrite /Ok /interval_list_invariant /remove. + move => [H_is_ok_s H_enc_s]. + split. { + by apply removeZ_interval_list_invariant. } { - move => s H_s_ok /=. - apply IH. - by apply add_ok. + rewrite /is_encoded_elems_list => y. + move : (removeZ_spec s (Enc.encode x) y H_is_ok_s). + rewrite /InZ => -> [] H_y_in _. + apply H_enc_s => //. } Qed. - Lemma add_list_spec : forall x l s, Ok s -> - (In x (add_list l s) <-> (SetoidList.InA Enc.E.eq x l) \/ In x s). + Lemma remove_spec : + forall (s : t) (x y : elt) (Hs : Ok s), + In y (remove x s) <-> In y s /\ ~Enc.E.eq y x. Proof. - move => x. - induction l as [| y l' IH]. { - intros s H. - rewrite /= InA_nil. - tauto. + intros s x y Hs. + have H_rs := (remove_ok s x Hs). + rewrite /remove !In_InZ. + rewrite removeZ_spec. { + rewrite Enc.encode_eq //. } { - move => s H_ok /=. - rewrite IH add_spec InA_cons. - tauto. + apply Hs. } Qed. + (** *** remove_list specification *) @@ -1717,120 +3839,269 @@ Module Raw (Enc : ElementEncode). } Qed. - (** *** union specification *) - - Global Instance union_ok s s' : forall `(Ok s, Ok s'), Ok (union s s'). - Proof. - move => H_ok H_ok'. - rewrite /union. - by apply add_list_ok. - Qed. - - Lemma union_spec : - forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'), - In x (union s s') <-> In x s \/ In x s'. + + (** *** subset specification *) + + Lemma subset_flatten_alt_def : forall (s1 s2 : t), + subset s1 s2 = + match (s1, s2) with + | (nil, _) => true + | (_ :: _, nil) => false + | ((y1, c1) :: l1, (y2, c2) :: l2) => + match (interval_compare (y1, c1) (y2,c2)) with + | ICR_before => false + | ICR_before_touch => false + | ICR_after => subset s1 l2 + | ICR_after_touch => false + | ICR_overlap_before => false + | ICR_overlap_after => false + | ICR_equal => subset l1 l2 + | ICR_subsume_1 => subset l1 s2 + | ICR_subsume_2 => false + end + end. Proof. - intros s s' x H_ok H_ok'. - rewrite /union add_list_spec //. + intros s1 s2. + case s1, s2 => //. Qed. - - (** *** filter specification *) - - Global Instance filter_ok s f : forall `(Ok s), Ok (filter f s). + Lemma subset_props_aux : forall y1 c1 l1 y2 c2 l2, + (exists y, InZ y ((y1, c1) :: l1) /\ ~InZ y ((y2, c2) :: l2)) -> + (false = true <-> + (forall y : Z, + InZ y ((y1, c1) :: l1) -> InZ y ((y2, c2) :: l2))). Proof. - move => _. - rewrite /filter /from_elements. - apply add_list_ok, empty_ok. + intros y1 c1 l1 y2 c2 l2. + move => [y] [H_y_in H_y_nin]. + split; first done. + move => H. + contradict H_y_nin. + by apply H. Qed. - - Lemma filter_spec : - forall (s : t) (x : elt) (f : elt -> bool), - Proper (Enc.E.eq==>eq) f -> - (In x (filter f s) <-> In x s /\ f x = true). + + Lemma subset_props_aux_before : forall y1 c1 l1 y2 c2 l2, + (c1 <> 0%N) -> + interval_list_invariant ((y2, c2) :: l2) = true -> + (y1 < y2) -> + (false = true <-> + (forall y : Z, + InZ y ((y1, c1) :: l1) -> InZ y ((y2, c2) :: l2))). Proof. - move => s x f H_prop. - rewrite /filter /from_elements add_list_spec empty_spec' /In !InA_alt. - setoid_rewrite filter_In. + intros y1 c1 l1 y2 c2 l2. + rewrite interval_list_invariant_cons. + move => H_c1_neq_0 [H_gr] [H_inv_l2] H_c2_neq_0 H_y1_lt. + apply subset_props_aux. + exists y1. split. { - move => [] //. - move => [y] [H_y_eq] [H_y_in] H_fy. - split; first by exists y. - rewrite -H_fy. - by f_equiv. + rewrite InZ_cons. + left. + by apply In_elementsZ_single_hd. } { - move => [] [y] [H_y_eq] H_y_in H_fx. - left. - exists y. - split; last split; try assumption. - apply Logic.eq_sym. - rewrite -H_fx. - by f_equiv. + rewrite InZ_cons. + suff : ~ (List.In y1 (elementsZ_single y2 c2)) /\ ~ InZ y1 l2 by tauto. + split. { + rewrite In_elementsZ_single. + move => [] /Z.le_ngt //. + } { + eapply Nin_elements_greater; eauto. + apply Z.le_trans with (m := y2). { + by apply Z.lt_le_incl. + } { + apply Z_le_add_r. + } + } } Qed. - - (** *** inter specification *) - - Global Instance inter_ok s s' : forall `(Ok s, Ok s'), Ok (inter s s'). + + Lemma subset_props : forall s1 s2 : t, + interval_list_invariant s1 = true -> + interval_list_invariant s2 = true -> + (subset s1 s2 = true <-> + (forall y, InZ y s1 -> InZ y s2)). Proof. - intros H_ok H_ok'. - rewrite /inter. - by apply filter_ok. - Qed. + induction s1 as [| [y1 c1] l1 IH1]. { + move => s2 _ _. + rewrite subset_flatten_alt_def. + split; done. + } { + induction s2 as [| [y2 c2] l2 IH2]. { + rewrite interval_list_invariant_cons + subset_flatten_alt_def. + move => [_] [H_c1_neq_0] _ _. + split => //. + move => H; move : (H y1). + rewrite InZ_nil => {H} H. + contradict H. + rewrite InZ_cons; left. + by apply In_elementsZ_single_hd. + } { + move => H_inv_s1 H_inv_s2. + move : (H_inv_s1) (H_inv_s2). + rewrite !interval_list_invariant_cons. + move => [H_gr_l1] [H_c1_neq_0] H_inv_l1. + move => [H_gr_l2] [H_c2_neq_0] H_inv_l2. + move : (IH2 H_inv_s1 H_inv_l2) => {IH2} IH2. + have : forall s2 : t, + interval_list_invariant s2 = true -> + (subset l1 s2 = true <-> + (forall y : Z, InZ y l1 -> InZ y s2)). { + intros. by apply IH1. + } + move => {IH1} IH1. - Lemma inter_spec : - forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'), - In x (inter s s') <-> In x s /\ In x s'. - Proof. - intros s s' x Hs Hs'. - rewrite /inter. - setoid_rewrite filter_spec. { - by rewrite mem_spec. - } { - repeat red. - intros y y' H_y_y'_eq. - apply eq_true_iff_eq. - rewrite !mem_spec /In. - setoid_rewrite H_y_y'_eq. - done. + have H_yc2_nin : ~ InZ (y2 + Z.of_N c2) ((y2, c2) :: l2). { + rewrite !InZ_cons !In_elementsZ_single. + move => []. { + move => [_] /Z.lt_irrefl //. + } { + eapply Nin_elements_greater; eauto. + apply Z.le_refl. + } + } + + rewrite subset_flatten_alt_def. + move : (interval_compare_elim y1 c1 y2 c2). + case (interval_compare (y1, c1) (y2, c2)). { + move => H_lt_y2. + apply subset_props_aux_before => //. + apply Z.le_lt_trans with (m := y1 + Z.of_N c1) => //. + apply Z_le_add_r. + } { + move => H_y2_eq. + apply subset_props_aux_before => //. + rewrite -H_y2_eq. + by apply Z_lt_add_r. + } { + move => [H_y1_lt] _. + apply subset_props_aux_before => //. + } { + move => [H_y2_lt] [H_y1_lt] H_yc2_lt. + apply subset_props_aux. + exists (y2 + Z.of_N c2). + split => //. + rewrite !InZ_cons !In_elementsZ_single. + left. + split => //. + by apply Z.lt_le_incl. + } { + move => [H_y1_eq] H_c1_eq; subst. + rewrite IH1 => //. + split; move => H_pre y; move : (H_pre y) => {H_pre}; + rewrite !InZ_cons. { + tauto. + } { + move => H_pre H_y_in_l1. + suff : ~(List.In y (elementsZ_single y2 c2)). { + tauto. + } + move : H_gr_l1. + rewrite interval_list_elements_greater_alt2_def + // In_elementsZ_single. + move => H; move : (H y H_y_in_l1) => {H}. + move => /Z.lt_ngt H_neq [_] //. + } + } { + move => [H_y2_lt_y1] [H_yc1_le] _. + rewrite IH1. + split; move => H_pre y; move : (H_pre y) => {H_pre}; + rewrite !InZ_cons. { + move => H []; last apply H. move => {H}. + rewrite !In_elementsZ_single. + move => [H_y1_le] H_y_lt. + left. + omega. + } { + move => H_pre H_y_in_l1. + apply H_pre. + by right. + } { + assumption. + } + } { + move => [H_y1_le] [_] []. { + apply subset_props_aux_before => //. + } { + move => H_yc2_lt. + apply subset_props_aux. + exists (y2 + Z.of_N c2). + split => //. + rewrite !InZ_cons !In_elementsZ_single. + left. + split => //. + apply Z.le_trans with (m := y2) => //. + apply Z_le_add_r. + } + } { + move => H_yc2_lt_y1. + rewrite IH2. + split; move => H_pre y; move : (H_pre y) => {H_pre}; + rewrite !InZ_cons. { + tauto. + } { + rewrite !In_elementsZ_single. + move => H_pre H_y_in. + suff : ~(y2 <= y < y2 + Z.of_N c2). { + tauto. + } + move => [H_y2_le H_y_lt]. + move : H_y_in => []. { + move => [H_y1_le] H_y_lt'. + omega. + } { + eapply Nin_elements_greater; eauto. + apply Z.le_trans with (m := y1); last first. { + apply Z_le_add_r. + } + apply Z.lt_le_incl. + apply Z.lt_trans with (m := y2 + Z.of_N c2) => //. + } + } + } { + move => H_y1_eq. + apply subset_props_aux. + exists y1. + rewrite !InZ_cons. + split. { + left. + by apply In_elementsZ_single_hd. + } { + rewrite !In_elementsZ_single H_y1_eq. + move => []. { + move => [_] /Z.lt_irrefl //. + } { + eapply Nin_elements_greater; eauto. + rewrite H_y1_eq. + apply Z.le_refl. + } + } + } + } } - Qed. - - - (** *** diff specification *) - - Global Instance diff_ok s s' : forall `(Ok s, Ok s'), Ok (diff s s'). - Proof. - intros ? ?. - by apply remove_list_ok. - Qed. - - Lemma diff_spec : - forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'), - In x (diff s s') <-> In x s /\ ~In x s'. - Proof. - intros s s' x Hs Hs'. - rewrite /diff remove_list_spec. - tauto. - Qed. - - - (** *** subset specification *) - + Qed. + Lemma subset_spec : forall (s s' : t) (Hs : Ok s) (Hs' : Ok s'), subset s s' = true <-> Subset s s'. Proof. intros s s' Hs Hs'. - rewrite /subset forallb_forall /Subset. - split; ( - move => H x /In_alt_def H_x_in; - move : (H x H_x_in) => {H}; - rewrite mem_spec // - ). + move : (Hs) (Hs'). + rewrite /Ok /IsOk. + move => [H_inv_s H_enc_s] [H_inv_s' H_enc_s']. + rewrite (subset_props s s' H_inv_s H_inv_s'). + rewrite /Subset. + split. { + move => H_pre enc_y. + rewrite !In_InZ. + apply H_pre. + } { + move => H_pre y H_y_in. + move : (H_enc_s _ H_y_in) => [e] H_e. subst. + move : (H_pre e) H_y_in. + rewrite !In_InZ //. + } Qed. - + (** *** elements and elementsZ specification *) @@ -1879,7 +4150,7 @@ Module Raw (Enc : ElementEncode). apply NoDupA_nil. } { move => H_ok_s. - move : (H_ok_s) => /Ok_cons [H_inf] [H_c] [H_enc] H_s'. + move : (H_ok_s) => /Ok_cons [H_interval_list_elements_greater] [H_c] [H_enc] H_s'. rewrite elementsZ_cons. apply NoDupA_app. { apply Z.eq_equiv. @@ -1899,8 +4170,13 @@ Module Raw (Enc : ElementEncode). move : H_y_in'. rewrite -in_rev In_elementsZ_single /=. move => [H_x_le] H_y_lt. - eapply Nin_elements_greater; eauto; first apply H_s'. - by apply Z.lt_le_incl. + eapply (Nin_elements_greater s' (x + Z.of_N c)) => //. { + apply H_s'. + } { + apply Z.lt_le_incl, H_y_lt. + } { + apply H_y_in. + } } } Qed. @@ -1947,134 +4223,93 @@ Module Raw (Enc : ElementEncode). Qed. Lemma elementsZ_cons_le_start : forall x cx xs y cy ys, - isok ((x, cx) :: xs) = true -> - isok ((y, cy) :: ys) = true -> - (forall z, List.In z (elementsZ ((y, cy) :: ys)) -> - List.In z (elementsZ ((x, cx) :: xs))) -> + interval_list_invariant ((x, cx) :: xs) = true -> + interval_list_invariant ((y, cy) :: ys) = true -> + (forall z, InZ z ((y, cy) :: ys) -> + InZ z ((x, cx) :: xs)) -> (x <= y). Proof. intros x cx xs y cy ys. - rewrite !isok_cons. - move => [H_inf_xs] [H_cx] H_xs [H_inf_ys] [H_cy] H_ys H. + rewrite !interval_list_invariant_cons. + move => [H_interval_list_elements_greater_xs] [H_cx] H_xs [H_interval_list_elements_greater_ys] [H_cy] H_ys H_impl. + have H_lt_xs : (forall zz : Z, InZ zz xs -> (x + Z.of_N cx) < zz). { + apply interval_list_elements_greater_alt2_def => //. + } - case_eq (x <=? y). { - move => /Z.leb_le //. + move : (Z.lt_ge_cases y x) => [] // H_y_lt. + apply Z.le_trans with (m := x + Z.of_N cx). { + apply Z_le_add_r. } { - move => /Z.leb_gt H_y_lt. + apply Z.lt_le_incl, H_lt_xs. + have H_pre : InZ y ((y, cy) :: ys). { + rewrite InZ_cons. + left. + by apply In_elementsZ_single_hd. + } + move : (H_impl y H_pre). + rewrite InZ_cons => [] [] //. + rewrite In_elementsZ_single. + move => [H_x_le]. exfalso. - move : (H y). - have -> : (List.In y (elementsZ ((x, cx) :: xs)) <-> False). { - split => //. - rewrite elementsZ_cons in_app_iff -in_rev In_elementsZ_single. - move => []. { - suff : (~ InZ y xs). { - rewrite /InZ //. - } - eapply Nin_elements_greater; eauto. - eapply Z.le_trans with (m := x). { - by apply Z.lt_le_incl. - } { - apply Z_le_add_r. - } - } { - apply Z.lt_nge in H_y_lt. - tauto. - } - } - have -> : (List.In y (elementsZ ((y, cy) :: ys)) <-> True). { - split => // _. - rewrite elementsZ_cons in_app_iff -in_rev. - right. - apply In_elementsZ_single_hd => //. - } - tauto. + eapply Z.nlt_ge; first apply H_x_le. + done. } Qed. Lemma elementsZ_cons_le_end : forall x cx xs y cy ys, - isok ((x, cx) :: xs) = true -> - isok ((y, cy) :: ys) = true -> + interval_list_invariant ((x, cx) :: xs) = true -> + interval_list_invariant ((y, cy) :: ys) = true -> (x <= y + Z.of_N cy) -> - (forall z, List.In z (elementsZ ((x, cx) :: xs)) -> - List.In z (elementsZ ((y, cy) :: ys))) -> + (forall z, InZ z ((x, cx) :: xs) -> + InZ z ((y, cy) :: ys)) -> (x + Z.of_N cx <= y + Z.of_N cy). Proof. intros x cx xs y cy ys. - rewrite !isok_cons. - move => [H_inf_xs] [H_cx] H_xs [H_inf_ys] [H_cy] H_ys H_x_le H. + rewrite !interval_list_invariant_cons. + move => [H_interval_list_elements_greater_xs] [H_cx] H_xs [H_interval_list_elements_greater_ys] [H_cy] H_ys H_x_le H_impl. - case_eq (x + Z.of_N cx <=? y + Z.of_N cy). { - move => /Z.leb_le //. - } { - move => /Z.leb_gt H_y_lt. - exfalso. - move : (H (y + Z.of_N cy)). - have -> : ((List.In (y + Z.of_N cy)) (elementsZ ((y, cy) :: ys)) <-> False). { - split => //. - rewrite elementsZ_cons in_app_iff -in_rev In_elementsZ_single. - move => []. { - suff : (~ InZ (y + Z.of_N cy) ys). { - rewrite /InZ //. - } - eapply Nin_elements_greater; eauto. - apply Z.le_refl. - } { - move => [_] /Z.lt_irrefl //. - } - } - have -> : (List.In (y + Z.of_N cy) (elementsZ ((x, cx) :: xs)) <-> True). { - split => // _. - rewrite elementsZ_cons in_app_iff -in_rev. - right. - rewrite In_elementsZ_single => //. - } - tauto. + apply Z.nlt_ge => H_y_lt. + + have H_nin : ~(InZ (y + Z.of_N cy) ((y, cy) :: ys)). { + rewrite InZ_cons In_elementsZ_single. + move => []. { + move => [_]. + apply Z.lt_irrefl. + } { + eapply (Nin_elements_greater ys (y + Z.of_N cy)) => //. + apply Z.le_refl. + } } + apply H_nin, H_impl => {H_nin}. + + rewrite InZ_cons In_elementsZ_single. + by left. Qed. Lemma elementsZ_cons_equiv_hd : forall x cx xs y cy ys, - isok ((x, cx) :: xs) = true -> - isok ((y, cy) :: ys) = true -> - (forall z, List.In z (elementsZ ((x, cx) :: xs)) <-> - List.In z (elementsZ ((y, cy) :: ys))) -> + interval_list_invariant ((x, cx) :: xs) = true -> + interval_list_invariant ((y, cy) :: ys) = true -> + (forall z, InZ z ((x, cx) :: xs) <-> + InZ z ((y, cy) :: ys)) -> (x = y) /\ (cx = cy). Proof. - intros x cx xs y cy ys H_ok_xs H_ok_ys H. + intros x cx xs y cy ys H_ok_xs H_ok_ys H_equiv. have H_xy_eq : x = y. { - have H_x_le : (x <= y). { - eapply (elementsZ_cons_le_start); eauto. - intro z. - rewrite H => //. - } - have H_y_le : (y <= x). { - eapply (elementsZ_cons_le_start); eauto. - intro z. - rewrite H => //. - } - apply Z.le_antisymm => //. + apply Z.le_antisymm; ( + eapply (elementsZ_cons_le_start); eauto; + apply H_equiv + ). } - subst. - split => //. + subst. split => //. have : (y + Z.of_N cx = y + Z.of_N cy). { - have H_cx_le : (y + Z.of_N cx <= y + Z.of_N cy). { - eapply elementsZ_cons_le_end; eauto. { - apply Z_le_add_r. - } { - intro Z. - rewrite H => //. - } - } - have H_cy_le : (y + Z.of_N cy <= y + Z.of_N cx). { - eapply elementsZ_cons_le_end; eauto. { - apply Z_le_add_r. - } { - intro Z. - rewrite H => //. - } - } - apply Z.le_antisymm => //. + apply Z.le_antisymm; ( + eapply elementsZ_cons_le_end; eauto; [ + apply Z_le_add_r | + apply H_equiv + ] + ). } rewrite Z.add_cancel_l. apply N2Z.inj. @@ -2090,13 +4325,13 @@ Module Raw (Enc : ElementEncode). Proof. intros x cx y cy H_cx H_cy H. apply elementsZ_cons_equiv_hd with (xs := nil) (ys := nil). { - rewrite isok_cons //. + rewrite interval_list_invariant_cons //. } { - rewrite isok_cons //. + rewrite interval_list_invariant_cons //. } { intro z. move : (H z) => {H}. - rewrite !elementsZ_cons !in_app_iff !elementsZ_nil -!in_rev /= => -> //. + rewrite !InZ_cons => -> //. } Qed. @@ -2126,14 +4361,14 @@ Module Raw (Enc : ElementEncode). rewrite /InZ. move => H_in. - have [H_ok_xs H_inf_xs] : Ok xs /\ inf (x + Z.of_N cx) xs = true. { + have [H_ok_xs H_interval_list_elements_greater_xs] : Ok xs /\ interval_list_elements_greater (x + Z.of_N cx) xs = true. { move : H_ok_xxs => /Ok_cons; tauto. } - have [H_ok_ys H_inf_ys] : Ok ys /\ inf (y + Z.of_N cy) ys = true. { + have [H_ok_ys H_interval_list_elements_greater_ys] : Ok ys /\ interval_list_elements_greater (y + Z.of_N cy) ys = true. { move : H_ok_yys => /Ok_cons. tauto. } - have H_isok_xs : isok xs=true by apply H_ok_xs. - have H_isok_ys : isok ys=true by apply H_ok_ys. + have H_isok_xs : interval_list_invariant xs=true by apply H_ok_xs. + have H_isok_ys : interval_list_invariant ys=true by apply H_ok_ys. have [H_x_eq H_cx_eq] : (x = y) /\ (cx = cy). { eapply elementsZ_cons_equiv_hd. @@ -2148,8 +4383,8 @@ Module Raw (Enc : ElementEncode). move => z. move : (H_in z) => {H_in}. - move : (Nin_elements_greater _ _ H_inf_xs H_isok_xs z). - move : (Nin_elements_greater _ _ H_inf_ys H_isok_ys z). + move : (Nin_elements_greater _ _ H_interval_list_elements_greater_xs H_isok_xs z). + move : (Nin_elements_greater _ _ H_interval_list_elements_greater_ys H_isok_ys z). rewrite /InZ !elementsZ_cons !in_app_iff -!in_rev !In_elementsZ_single. move => H_nin_ys H_nin_xs H_equiv. split. { @@ -2218,40 +4453,53 @@ Module Raw (Enc : ElementEncode). (** *** choose specification *) - Lemma choose_alt_def : forall s, - choose s = match chooseZ s with - | None => None - | Some e => Some (Enc.decode e) - end. - Proof. + Definition chooseZ_spec1 : + forall (s : t) (x : Z), + interval_list_invariant s = true -> + chooseZ s = Some x -> InZ x s. + Proof. + intros s x. + case s as [| [x' c] s']. { + rewrite /chooseZ //. + } { + rewrite /chooseZ InZ_cons interval_list_invariant_cons. + move => [_] [H_c_neq] _ [->]. + left. + by apply In_elementsZ_single_hd. + } + Qed. + + Definition chooseZ_spec2 : + forall (s : t), + chooseZ s = None -> forall x, ~InZ x s. + Proof. intros s. - rewrite /choose /chooseZ /elements /rev' rev_append_rev app_nil_r - rev_map_alt_def -map_rev. - case (rev (elementsZ s)) => //. + case s as [| [x' c] s']; + rewrite /chooseZ //. + move => _ x //. Qed. Definition choose_spec1 : - forall (s : t) (x : elt), choose s = Some x -> In x s. + forall (s : t) (x : elt) (Hs : Ok s), choose s = Some x -> In x s. Proof. - intros s x. - rewrite /choose /In. - case (elements s) => //. - move => z l [<-]. - apply InA_cons_hd. - apply Enc.E.eq_equiv. + rewrite /choose. + move => s x H_ok. + case_eq (chooseZ s) => //. + move => z H_choose [<-]. + apply InZ_In => //. + apply chooseZ_spec1 => //. + apply H_ok. Qed. Definition choose_spec2 : forall s : t, choose s = None -> Empty s. Proof. - rewrite /choose /Empty /In. - intro s. - case (elements s) => //. - move => _ a. - rewrite InA_nil //. + rewrite /choose /chooseZ /Empty /In. + case s as [| [x' c] s'] => //. + move => _ e. + rewrite elements_nil InA_nil //. Qed. - Lemma chooseZ_min : forall (s : t) (x y : Z) (Hs : Ok s), chooseZ s = Some x -> InZ y s -> ~ Z.lt y x. @@ -2260,13 +4508,9 @@ Module Raw (Enc : ElementEncode). eapply (Nin_elements_greater s (Z.pred x)) => //; last apply H_in. { move : H_ok H_min. case s => //. - move => [z c] s'. - rewrite /chooseZ Ok_cons elementsZ_cons /rev' -rev_alt rev_app_distr - rev_involutive. - move => [_] [/N.neq_0_r] [c'] -> _. - rewrite elementsZ_single_succ_front /=. - move => [->]. - apply Z.ltb_lt, Z.lt_pred_l. + move => [z c] s' _ [<-]. + rewrite interval_list_elements_greater_cons. + apply Z.lt_pred_l. } { apply H_ok. } { @@ -2274,18 +4518,6 @@ Module Raw (Enc : ElementEncode). } Qed. - Lemma chooseZ_InZ : - forall (s : t) (x : Z), - chooseZ s = Some x -> InZ x s. - Proof. - intros s x. - rewrite /chooseZ /InZ /rev' -rev_alt in_rev. - case (rev (elementsZ s)) => //. - move => z l [<-] /=. - by left. - Qed. - - Lemma chooseZ_spec3: forall s s' x x', Ok s -> Ok s' -> chooseZ s = Some x -> chooseZ s' = Some x' -> Equal s s' -> x = x'. Proof. @@ -2295,6 +4527,7 @@ Module Raw (Enc : ElementEncode). rewrite H_s_eq => -> [] //. Qed. + (** *** fold specification *) Lemma fold_spec : @@ -2522,7 +4755,7 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E. Lemma elements_spec2w : NoDupA E.eq (elements s). Proof. exact (Raw.elements_spec2w _ _). Qed. Lemma choose_spec1 : choose s = Some x -> In x s. - Proof. exact (Raw.choose_spec1 _ _). Qed. + Proof. exact (Raw.choose_spec1 _ _ _). Qed. Lemma choose_spec2 : choose s = None -> Empty s. Proof. exact (Raw.choose_spec2 _). Qed. @@ -2607,3 +4840,5 @@ End ElementEncodeNat. Module MSetIntervalsNat <: WSetsOn NPeano.Nat := MSetIntervals ElementEncodeNat. + + From c7d98c561ab210d74c10417866af9a4453877127 Mon Sep 17 00:00:00 2001 From: Thomas Tuerk Date: Mon, 3 Oct 2016 23:46:11 +0200 Subject: [PATCH 2/7] MSetIntervals: implement opertions more efficiently, implement strong set interface This commit implements the full ordered set interface with interval sets. Moreover a decent folding operator for interval sets is defined. This is used to implement certain operations more efficiently: - fold - for_all - exists_ - filter - partition --- MSetIntervals.v | 1886 +++++++++++++++++++++++++++++++++++++---------- 1 file changed, 1500 insertions(+), 386 deletions(-) diff --git a/MSetIntervals.v b/MSetIntervals.v index 930c8e9..2095774 100644 --- a/MSetIntervals.v +++ b/MSetIntervals.v @@ -23,12 +23,14 @@ * * 25-09-2016 Thomas Tuerk * implement union, inter, diff, subset, choose efficiently + * 03-10-2016 Thomas Tuerk + * implement fold, exists_, for_all, filter, partition efficiently *) (** * Weak sets implemented by interval lists - This file contains an implementation of the weak set interface - [WSetsOn] which uses internally intervals of Z. This allows some + This file contains an implementation of the set interface + [SetsOn] which uses internally intervals of Z. This allows some large sets, which naturally map to intervals of integers to be represented very efficiently. @@ -186,7 +188,7 @@ Qed. provides encode and decode function. *) Module Type ElementEncode. - Declare Module E : DecidableType. + Declare Module E : OrderedType. Parameter encode : E.t -> Z. Parameter decode : Z -> E.t. @@ -202,6 +204,10 @@ Module Type ElementEncode. Axiom encode_eq : forall (e1 e2 : E.t), (Z.eq (encode e1) (encode e2)) <-> E.eq e1 e2. + (** Encoding is compatible with the order of elements. *) + Axiom encode_lt : forall (e1 e2 : E.t), + (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2. + End ElementEncode. @@ -226,7 +232,7 @@ End ElementEncode. the decode function when checking it. This allows for sets with other element types than Z. *) -Module Ops (Enc : ElementEncode) <: WOps Enc.E. +Module Ops (Enc : ElementEncode) <: Ops Enc.E. Definition elt := Enc.E.t. Definition t := list (Z * N). @@ -245,29 +251,34 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. apply N.lt_pred_l. discriminate. Defined. - - Fixpoint elementsZ_aux'' (acc : list Z) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : list Z := - match c as c0 return c = c0 -> list Z with - | N0 => fun _ => acc - | c => fun Heq => elementsZ_aux'' (x::acc) (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) + + Fixpoint fold_elementsZ_aux {A} (f : A -> Z -> option A) (acc : A) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : (bool * A) := + match c as c0 return c = c0 -> (bool * A) with + | N0 => fun _ => (false, acc) + | c => fun Heq => match (f acc x) with + | None => (true, acc) + | Some acc' => + fold_elementsZ_aux f acc' (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end end (refl_equal _). - Definition elementsZ_aux' acc x c := elementsZ_aux'' acc x c (lt_wf_0 _). + Definition fold_elementsZ_single {A} f (acc : A) x c := fold_elementsZ_aux f acc x c (lt_wf_0 _). - Fixpoint elementsZ_aux acc (s : t) : list Z := + Fixpoint fold_elementsZ {A} f (acc : A) (s : t) : (bool * A) := match s with - | nil => acc - | (x, c) :: s' => - elementsZ_aux (elementsZ_aux' acc x c) s' + | nil => (false, acc) + | (x, c) :: s' => + match fold_elementsZ_single f acc x c with + (false, acc') => fold_elementsZ f acc' s' + | (true, acc') => (true, acc') + end end. Definition elementsZ (s : t) : list Z := - elementsZ_aux nil s. + snd (fold_elementsZ (fun l x => Some (x :: l)) nil s). Definition elements (s : t) : list elt := rev_map Enc.decode (elementsZ s). - - + (** membership is easily defined *) Fixpoint memZ (x : Z) (s : t) := match s with @@ -330,6 +341,23 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. end end. + Fixpoint compare (s1 s2 : t) := + match (s1, s2) with + | (nil, nil) => Eq + | (nil, _ :: _) => Lt + | (_ :: _, nil) => Gt + | ((y1, c1)::s1', (y2, c2)::s2') => + match (Z.compare y1 y2) with + | Lt => Lt + | Gt => Gt + | Eq => match N.compare c1 c2 with + | Lt => Lt + | Gt => Gt + | Eq => compare s1' s2' + end + end + end. + (** Auxiliary functions for inserting at front and merging intervals *) Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N := (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))). @@ -363,8 +391,8 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. Definition addZ x s := addZ_aux nil x s. Definition add x s := addZ (Enc.encode x) s. - (** [add_list] simple extension to add many elements, which then allows to - define from_elements. *) + (** [add_list] is a simple extension to add many elements. + This is used to define the function [from_elements]. *) Definition add_list (l : list elt) (s : t) : t := List.fold_left (fun s x => add x s) l s. @@ -517,23 +545,104 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. Definition inter s1 s2 := inter_aux2 nil s1 s2. + + (** Partition and filter *) + + Definition partitionZ_fold_insert + (cur : option (Z * N)) (x : Z) := + match cur with + None => (x, 1%N) + | Some (y, c) => (y, N.succ c) + end. + + Definition partitionZ_fold_skip (acc : list (Z * N)) + (cur : option (Z * N)) : (list (Z * N)) := + match cur with + None => acc + | Some yc => yc::acc + end. - (** Simple wrappers *) + Definition partitionZ_fold_fun f st (x : Z) := + match st with ((acc_t, c_t), (acc_f, c_f)) => + if (f x) then + ((acc_t, Some (partitionZ_fold_insert c_t x)), + (partitionZ_fold_skip acc_f c_f, None)) + else + ((partitionZ_fold_skip acc_t c_t, None), + (acc_f, Some (partitionZ_fold_insert c_f x))) + end. + + Definition partitionZ_single_aux f st (x : Z) (c : N) := + snd (fold_elementsZ_single (fun st x => Some (partitionZ_fold_fun f st x)) st x c). + + Definition partitionZ_single f acc_t acc_f x c := + match partitionZ_single_aux f ((acc_t, None), (acc_f, None)) x c with + | ((acc_t, c_t), (acc_f, c_f)) => + (partitionZ_fold_skip acc_t c_t, + partitionZ_fold_skip acc_f c_f) + end. + + Fixpoint partitionZ_aux acc_t acc_f f s := + match s with + | nil => (List.rev acc_t, List.rev acc_f) + | (y, c) :: s' => + match partitionZ_single f acc_t acc_f y c with + | (acc_t', acc_f') => partitionZ_aux acc_t' acc_f' f s' + end + end. + + Definition partitionZ := partitionZ_aux nil nil. + + Definition partition (f : elt -> bool) : t -> (t * t) := + partitionZ (fun z => f (Enc.decode z)). + + + Definition filterZ_fold_fun f st (x : Z) := + match st with (acc_t, c_t) => + if (f x) then + (acc_t, Some (partitionZ_fold_insert c_t x)) + else + (partitionZ_fold_skip acc_t c_t, None) + end. + + Definition filterZ_single_aux f st (x : Z) (c : N) := + snd (fold_elementsZ_single (fun st x => Some (filterZ_fold_fun f st x)) st x c). + + Definition filterZ_single f acc x c := + match filterZ_single_aux f (acc, None) x c with + | (acc, c) => + (partitionZ_fold_skip acc c) + end. + + Fixpoint filterZ_aux acc f s := + match s with + | nil => (List.rev acc) + | (y, c) :: s' => + filterZ_aux (filterZ_single f acc y c) f s' + end. + + Definition filterZ := filterZ_aux nil. + + Definition filter (f : elt -> bool) : t -> t := + filterZ (fun z => f (Enc.decode z)). - Definition filter (f : elt -> bool) (s : t) : t := - from_elements (List.filter f (elements s)). + + (** Simple wrappers *) Definition fold {B : Type} (f : elt -> B -> B) (s : t) (i : B) : B := - List.fold_left (flip f) (elements s) i. + snd (fold_elementsZ (fun b z => Some (f (Enc.decode z) b)) i s). Definition for_all (f : elt -> bool) (s : t) : bool := - List.forallb f (elements s). + snd (fold_elementsZ (fun b z => + if b then + Some (f (Enc.decode z)) + else None) true s). Definition exists_ (f : elt -> bool) (s : t) : bool := - List.existsb f (elements s). - - Definition partition (f : elt -> bool) (s : t) : t * t := - (filter f s, filter (fun x => negb (f x)) s). + snd (fold_elementsZ (fun b z => + if b then + None + else Some (f (Enc.decode z))) false s). Fixpoint cardinalN c (s : t) : N := match s with | nil => c @@ -542,18 +651,33 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E. Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s). - Definition chooseZ (s : t) : option Z := + Definition min_eltZ (s : t) : option Z := match s with | nil => None | (x, _) :: _ => Some x end. - Definition choose (s : t) : option elt := - match (chooseZ s) with + Definition min_elt (s : t) : option elt := + match (min_eltZ s) with | None => None | Some x => Some (Enc.decode x) end. + Definition choose := min_elt. + + Fixpoint max_eltZ (s : t) : option Z := + match s with + | nil => None + | (x, c) :: nil => Some (Z.pred (x + Z.of_N c)) + | (x, _) :: s' => max_eltZ s' + end. + + Definition max_elt (s : t) : option elt := + match (max_eltZ s) with + | None => None + | Some x => Some (Enc.decode x) + end. + End Ops. @@ -710,85 +834,181 @@ Module Raw (Enc : ElementEncode). } Qed. - Lemma elementsZ_aux''_irrel : forall c acc x H1 H2, - elementsZ_aux'' acc x c H1 = elementsZ_aux'' acc x c H2. + Lemma fold_elementsZ_aux_irrel {A} : + forall f c (acc : A) x H1 H2, + fold_elementsZ_aux f acc x c H1 = + fold_elementsZ_aux f acc x c H2. Proof. - intro c. - induction c using (well_founded_ind lt_wf_0). + intros f c. + induction c as [c IH] using (well_founded_ind lt_wf_0). case_eq c. { - intros H_c acc x; case; intro; case; intro. + intros H_c acc x; case; intro H_H1; case; intro H_H2. reflexivity. } { - intros p H_c acc x; case; intro; case; intro. - simpl. - apply H. + intros p H_c acc x; case; intro H_H1; case; intro H_H2. + unfold fold_elementsZ_aux; fold (@fold_elementsZ_aux A). + case (f acc x) => // acc'. + apply IH. rewrite H_c. - rewrite N.pos_pred_spec. apply N.lt_pred_l. discriminate. } Qed. - Lemma elementsZ_aux'_pos : forall s x p, - elementsZ_aux' s x (N.pos p) = elementsZ_aux' (x::s) (Z.succ x) (Pos.pred_N p). + Lemma fold_elementsZ_single_pos {A} : forall f (acc : A) x p, + fold_elementsZ_single f acc x (N.pos p) = + match f acc x with + | Some acc' => + fold_elementsZ_single f acc' (Z.succ x) + (N.pred (N.pos p)) + | None => (true, acc) + end. Proof. - intros s x p. - unfold elementsZ_aux'. - unfold elementsZ_aux''. + intros f acc x p. + unfold fold_elementsZ_single. + unfold fold_elementsZ_aux. case: (lt_wf_0 _). + fold (@fold_elementsZ_aux A). intro. - apply elementsZ_aux''_irrel. + case (f acc x) => // acc'. + apply fold_elementsZ_aux_irrel. Qed. - Lemma elementsZ_aux'_zero : forall s x, - elementsZ_aux' s x (0%N) = s. + Lemma fold_elementsZ_single_zero {A} : forall f (acc : A) x, + fold_elementsZ_single f acc x (0%N) = (false, acc). Proof. - intros s x. - unfold elementsZ_aux'. - by case (lt_wf_0 (0%N)); intro. + intros f acc x. + unfold fold_elementsZ_single. + case (lt_wf_0 (0%N)); intro. + unfold fold_elementsZ_aux. + reflexivity. + Qed. + + + Lemma fold_elementsZ_single_succ {A} : forall f (acc : A) x c, + fold_elementsZ_single f acc x (N.succ c) = + match f acc x with + | Some acc' => + fold_elementsZ_single f acc' (Z.succ x) c + | None => (true, acc) + end. + Proof. + intros f acc x c. + case c. { + by rewrite fold_elementsZ_single_pos. + } { + intro p; simpl. + rewrite fold_elementsZ_single_pos. + case (f acc x) => // acc' /=. + by rewrite Pos.pred_N_succ. + } Qed. + Fixpoint fold_opt {A B} f (acc : A) (bs : list B) : (bool * A) := + match bs with + | nil => (false, acc) + | (b :: bs') => + match f acc b with + | Some acc' => fold_opt f acc' bs' + | None => (true, acc) + end + end. - Lemma elementsZ_aux'_succ : forall s x c, - elementsZ_aux' s x (N.succ c) = elementsZ_aux' (x::s) (Z.succ x) c. + Lemma fold_opt_list_cons : forall {A} (bs : list A) (acc : list A), + fold_opt (fun l x => Some (x :: l)) acc bs = + (false, List.rev bs ++ acc). Proof. - intros s x c. - case c. - - by rewrite elementsZ_aux'_pos. - - intro p; simpl. - rewrite elementsZ_aux'_pos. - by rewrite Pos.pred_N_succ. + induction bs as [| b bs' IH] => acc. { + by simpl. + } { + rewrite /= IH -app_assoc //. + } Qed. + Lemma fold_opt_app {A B} : forall f (acc : A) (l1 l2 : list B), + fold_opt f acc (l1 ++ l2) = + (let (ab, acc') := fold_opt f acc l1 in + if ab then (true, acc') else fold_opt f acc' l2). + Proof. + intros f acc l1 l2. + move : acc. + induction l1 as [| b l1' IH] => acc. { + rewrite app_nil_l //. + } { + rewrite /=. + case (f acc b); last done. + intro acc'. + rewrite IH //. + } + Qed. - Lemma elementsZ_single_intro : forall c s x, - elementsZ_aux' s x c = - (List.rev (elementsZ_single x c)) ++ s. + + Lemma fold_elementsZ_single_alt_def {A} : forall f c (acc : A) x, + fold_elementsZ_single f acc x c = + fold_opt f acc (elementsZ_single x c). Proof. + intro f. induction c as [| c' IH] using N.peano_ind. { - intros s x. - rewrite elementsZ_aux'_zero. - rewrite elementsZ_single_base //. + intros acc x. + rewrite fold_elementsZ_single_zero + elementsZ_single_base /fold_opt //. } { - intros s x. - rewrite elementsZ_aux'_succ elementsZ_single_succ_front IH. - rewrite -app_assoc /= //. + intros acc x. + rewrite fold_elementsZ_single_succ + elementsZ_single_succ_front /=. + case (f acc x); last reflexivity. + intro acc'. + apply IH. } Qed. + Lemma fold_elementsZ_nil {A} : forall f (acc : A), + fold_elementsZ f acc nil = (false, acc). + Proof. done. Qed. - Lemma elementsZ_aux_alt_def : forall s acc, - elementsZ_aux acc s = elementsZ s ++ acc. + Lemma fold_elementsZ_cons {A} : forall f (acc : A) y c s, + fold_elementsZ f acc ((y, c)::s) = + (let (ab, acc') := fold_elementsZ_single f acc y c in + if ab then (true, acc') else fold_elementsZ f acc' s). Proof. - induction s as [| [x c] s' IH]. { - move => acc. - rewrite /elementsZ_aux elementsZ_nil app_nil_l //. + intros f acc y c s. + done. + Qed. + + + Lemma fold_elementsZ_alt_def_aux : forall (s : t) base, + (snd (fold_elementsZ + (fun (l : list Z) (x : Z) => Some (x :: l)) base s)) = + elementsZ s ++ base. + Proof. + induction s as [| [y1 c1] s' IH] => base. { + rewrite elementsZ_nil /fold_elementsZ /fold_opt /snd + app_nil_l //. + } { + rewrite /elementsZ !fold_elementsZ_cons. + rewrite !fold_elementsZ_single_alt_def !fold_opt_list_cons. + rewrite !IH app_nil_r app_assoc //. + } + Qed. + + + Lemma fold_elementsZ_alt_def {A} : forall f s (acc : A), + fold_elementsZ f acc s = + fold_opt f acc (rev (elementsZ s)). + Proof. + intro f. + induction s as [| [y1 c1] s' IH] => acc. { + by simpl. } { - move => acc. - rewrite /elementsZ /elementsZ_aux -/elementsZ_aux !IH !elementsZ_single_intro. - rewrite -!app_assoc app_nil_l //. + rewrite /elementsZ !fold_elementsZ_cons. + rewrite !fold_elementsZ_single_alt_def + fold_opt_list_cons app_nil_r + fold_elementsZ_alt_def_aux rev_app_distr + rev_involutive fold_opt_app. + case (fold_opt f acc (elementsZ_single y1 c1)). + move => [] //. } Qed. @@ -796,8 +1016,12 @@ Module Raw (Enc : ElementEncode). ((elementsZ s) ++ (List.rev (elementsZ_single x c))). Proof. intros x c s. - rewrite /elementsZ /elementsZ_aux -/elementsZ_aux. - rewrite !elementsZ_aux_alt_def elementsZ_single_intro !app_nil_r //. + rewrite /elementsZ fold_elementsZ_cons + !fold_elementsZ_alt_def + fold_elementsZ_single_alt_def. + rewrite !fold_opt_list_cons. + rewrite !app_nil_r !rev_involutive /=. + rewrite fold_elementsZ_alt_def_aux //. Qed. Lemma elements_cons : forall x c s, elements (((x, c) :: s) : t) = @@ -899,7 +1123,7 @@ Module Raw (Enc : ElementEncode). } Qed. - + (** *** comparing intervals *) Ltac Z_named_compare_cases H := match goal with @@ -1405,6 +1629,16 @@ Module Raw (Enc : ElementEncode). (** *** Properties of In and InZ *) + Lemma encode_decode_eq : forall x s, Ok s -> InZ x s -> + (Enc.encode (Enc.decode x) = x). + Proof. + intros x s. + rewrite /Ok /IsOk /InZ. + move => [_] H_enc H_in_x. + move : (H_enc _ H_in_x) => [x'] <-. + rewrite Enc.decode_encode_ok //. + Qed. + Lemma In_alt_def : forall x s, Ok s -> (In x s <-> List.In x (elements s)). Proof. @@ -1412,7 +1646,7 @@ Module Raw (Enc : ElementEncode). rewrite /In InA_alt /elements rev_map_alt_def. split. { move => [y] [H_y_eq]. - rewrite -!in_rev !in_map_iff. + rewrite -!in_rev !in_map_iff. move => [x'] [H_y_eq'] H_x'_in. suff H_x'_eq : (Enc.encode x = x'). { exists x'. @@ -1472,7 +1706,7 @@ Module Raw (Enc : ElementEncode). by rewrite Enc.decode_encode_ok. Qed. - + (** *** Membership specification *) Lemma memZ_spec : @@ -2958,41 +3192,6 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, Qed. - (** *** filter specification *) - - Global Instance filter_ok s f : forall `(Ok s), Ok (filter f s). - Proof. - move => _. - rewrite /filter /from_elements. - apply add_list_ok, empty_ok. - Qed. - - Lemma filter_spec : - forall (s : t) (x : elt) (f : elt -> bool), - Proper (Enc.E.eq==>eq) f -> - (In x (filter f s) <-> In x s /\ f x = true). - Proof. - move => s x f H_prop. - rewrite /filter /from_elements add_list_spec empty_spec' /In !InA_alt. - setoid_rewrite filter_In. - split. { - move => [] //. - move => [y] [H_y_eq] [H_y_in] H_fy. - split; first by exists y. - rewrite -H_fy. - by f_equiv. - } { - move => [] [y] [H_y_eq] H_y_in H_fx. - left. - exists y. - split; last split; try assumption. - apply Logic.eq_sym. - rewrite -H_fx. - by f_equiv. - } - Qed. - - (** *** diff specification *) Lemma diff_aux_alt_def : @@ -4222,195 +4421,37 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, } Qed. - Lemma elementsZ_cons_le_start : forall x cx xs y cy ys, - interval_list_invariant ((x, cx) :: xs) = true -> - interval_list_invariant ((y, cy) :: ys) = true -> - (forall z, InZ z ((y, cy) :: ys) -> - InZ z ((x, cx) :: xs)) -> - (x <= y). - Proof. - intros x cx xs y cy ys. - rewrite !interval_list_invariant_cons. - move => [H_interval_list_elements_greater_xs] [H_cx] H_xs [H_interval_list_elements_greater_ys] [H_cy] H_ys H_impl. - have H_lt_xs : (forall zz : Z, InZ zz xs -> (x + Z.of_N cx) < zz). { - apply interval_list_elements_greater_alt2_def => //. - } - - move : (Z.lt_ge_cases y x) => [] // H_y_lt. - apply Z.le_trans with (m := x + Z.of_N cx). { - apply Z_le_add_r. - } { - apply Z.lt_le_incl, H_lt_xs. - have H_pre : InZ y ((y, cy) :: ys). { - rewrite InZ_cons. - left. - by apply In_elementsZ_single_hd. - } - move : (H_impl y H_pre). - rewrite InZ_cons => [] [] //. - rewrite In_elementsZ_single. - move => [H_x_le]. - exfalso. - eapply Z.nlt_ge; first apply H_x_le. - done. - } - Qed. - - Lemma elementsZ_cons_le_end : forall x cx xs y cy ys, - interval_list_invariant ((x, cx) :: xs) = true -> - interval_list_invariant ((y, cy) :: ys) = true -> - (x <= y + Z.of_N cy) -> - (forall z, InZ z ((x, cx) :: xs) -> - InZ z ((y, cy) :: ys)) -> - (x + Z.of_N cx <= y + Z.of_N cy). - Proof. - intros x cx xs y cy ys. - rewrite !interval_list_invariant_cons. - move => [H_interval_list_elements_greater_xs] [H_cx] H_xs [H_interval_list_elements_greater_ys] [H_cy] H_ys H_x_le H_impl. - - apply Z.nlt_ge => H_y_lt. - - have H_nin : ~(InZ (y + Z.of_N cy) ((y, cy) :: ys)). { - rewrite InZ_cons In_elementsZ_single. - move => []. { - move => [_]. - apply Z.lt_irrefl. - } { - eapply (Nin_elements_greater ys (y + Z.of_N cy)) => //. - apply Z.le_refl. - } - } - apply H_nin, H_impl => {H_nin}. - - rewrite InZ_cons In_elementsZ_single. - by left. - Qed. - - - Lemma elementsZ_cons_equiv_hd : forall x cx xs y cy ys, - interval_list_invariant ((x, cx) :: xs) = true -> - interval_list_invariant ((y, cy) :: ys) = true -> - (forall z, InZ z ((x, cx) :: xs) <-> - InZ z ((y, cy) :: ys)) -> - (x = y) /\ (cx = cy). - Proof. - intros x cx xs y cy ys H_ok_xs H_ok_ys H_equiv. - have H_xy_eq : x = y. { - apply Z.le_antisymm; ( - eapply (elementsZ_cons_le_start); eauto; - apply H_equiv - ). - } - subst. split => //. - - have : (y + Z.of_N cx = y + Z.of_N cy). { - apply Z.le_antisymm; ( - eapply elementsZ_cons_le_end; eauto; [ - apply Z_le_add_r | - apply H_equiv - ] - ). - } - rewrite Z.add_cancel_l. - apply N2Z.inj. - Qed. - - - Lemma elementsZ_single_equiv : forall x cx y cy, - (cx <> 0)%N -> - (cy <> 0)%N -> - (forall z, List.In z (elementsZ_single x cx) <-> - List.In z (elementsZ_single y cy)) -> - (x = y) /\ (cx = cy). - Proof. - intros x cx y cy H_cx H_cy H. - apply elementsZ_cons_equiv_hd with (xs := nil) (ys := nil). { - rewrite interval_list_invariant_cons //. - } { - rewrite interval_list_invariant_cons //. - } { - intro z. - move : (H z) => {H}. - rewrite !InZ_cons => -> //. - } - Qed. - Lemma equal_elementsZ : forall (s s' : t) {Hs : Ok s} {Hs' : Ok s'}, (forall x, (InZ x s <-> InZ x s')) -> (s = s'). Proof. - induction s as [| [x cx] xs IH]. { - move => [] //. - move => [y cy ys] _ /Ok_cons [_] [H_cy] _. - rewrite /InZ elementsZ_nil /= => H. - exfalso. - rewrite (H y) elementsZ_cons in_app_iff -in_rev. - right. - apply In_elementsZ_single_hd => //. - } { - move => []. { - move => /Ok_cons [_] [H_xc] _ _. - rewrite /InZ elementsZ_nil /= => H. - exfalso. - rewrite -(H x) elementsZ_cons in_app_iff -in_rev. - right. - apply In_elementsZ_single_hd => //. - } { - move => [y cy ys] H_ok_xxs H_ok_yys. - rewrite /InZ. - move => H_in. - - have [H_ok_xs H_interval_list_elements_greater_xs] : Ok xs /\ interval_list_elements_greater (x + Z.of_N cx) xs = true. { - move : H_ok_xxs => /Ok_cons; tauto. - } - have [H_ok_ys H_interval_list_elements_greater_ys] : Ok ys /\ interval_list_elements_greater (y + Z.of_N cy) ys = true. { - move : H_ok_yys => /Ok_cons. tauto. - } - have H_isok_xs : interval_list_invariant xs=true by apply H_ok_xs. - have H_isok_ys : interval_list_invariant ys=true by apply H_ok_ys. - - have [H_x_eq H_cx_eq] : (x = y) /\ (cx = cy). { - eapply elementsZ_cons_equiv_hd. - - by apply H_ok_xxs. - - by apply H_ok_yys. - - by apply H_in. - } - subst. - - suff -> : (xs = ys) by []. - suff H_suff : (forall x : Z, InZ x xs <-> InZ x ys) by apply IH. - - move => z. - move : (H_in z) => {H_in}. - move : (Nin_elements_greater _ _ H_interval_list_elements_greater_xs H_isok_xs z). - move : (Nin_elements_greater _ _ H_interval_list_elements_greater_ys H_isok_ys z). - rewrite /InZ !elementsZ_cons !in_app_iff -!in_rev !In_elementsZ_single. - move => H_nin_ys H_nin_xs H_equiv. - split. { - move => H_z_in. - have : List.In z (elementsZ ys) \/ y <= z < y + Z.of_N cy. { - rewrite -H_equiv. - by left. - } - move => [] //. - move => [_] /Z.lt_le_incl H_z_le. - contradict H_z_in. - by apply H_nin_xs. - } { - move => H_z_in. - have : List.In z (elementsZ xs) \/ y <= z < y + Z.of_N cy. { - rewrite H_equiv. - by left. - } - move => [] //. - move => [_] /Z.lt_le_incl H_z_le. - contradict H_z_in. - by apply H_nin_ys. - } - } + intros s s'. + move => H_ok_s H_ok_s' H_InZ_eq. + have [] : ((subset s s' = true) /\ (subset s' s = true)). { + rewrite !subset_spec /Subset. + split => x; rewrite !In_InZ H_InZ_eq //. } + have : interval_list_invariant s' = true by apply H_ok_s'. + have : interval_list_invariant s = true by apply H_ok_s. + clear H_ok_s H_ok_s' H_InZ_eq. + move : s s'. + induction s as [| [x1 c1] s1 IH]; + case s' as [| [x2 c2] s2] => //. + rewrite !interval_list_invariant_cons. + move => [H_gr_s1] [H_c1_neq_0] H_inv_s1. + move => [H_gr_s2] [H_c2_neq_0] H_inv_s2. + rewrite subset_flatten_alt_def + (subset_flatten_alt_def ((x2, c2)::s2)). + rewrite (interval_compare_swap x1 c1); last by left. + move : (interval_compare_elim x1 c1 x2 c2). + case (interval_compare (x1, c1) (x2, c2)) => //. + move => [->] -> H_sub_s12 H_sub_s21. + + suff -> : s1 = s2 by done. + by apply IH. Qed. + Lemma equal_spec : forall (s s' : t) {Hs : Ok s} {Hs' : Ok s'}, @@ -4450,76 +4491,408 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, } Qed. + (** *** compare *) + Definition lt (s1 s2 : t) : Prop := (compare s1 s2 = Lt). - (** *** choose specification *) + Lemma compare_eq_Eq : forall s1 s2, + (compare s1 s2 = Eq <-> equal s1 s2 = true). + Proof. + induction s1 as [| [y1 c1] s1' IH]; + case s2 as [| [y2 c2] s2'] => //. + rewrite /= !andb_true_iff -IH Z.eqb_eq N.eqb_eq. + move : (Z.compare_eq_iff y1 y2). + case (Z.compare y1 y2). { + move => H. + have -> : y1 = y2. by apply H. + clear H. - Definition chooseZ_spec1 : - forall (s : t) (x : Z), - interval_list_invariant s = true -> - chooseZ s = Some x -> InZ x s. - Proof. - intros s x. - case s as [| [x' c] s']. { - rewrite /chooseZ //. + move : (N.compare_eq_iff c1 c2). + case (N.compare c1 c2). { + move => H. + have -> : c1 = c2. by apply H. + tauto. + } { + move => H. + have H_neq : ~(c1 = c2). by rewrite -H => {H}. + tauto. + } { + move => H. + have H_neq : ~(c1 = c2). by rewrite -H => {H}. + tauto. + } } { - rewrite /chooseZ InZ_cons interval_list_invariant_cons. - move => [_] [H_c_neq] _ [->]. - left. - by apply In_elementsZ_single_hd. + move => H. + have H_neq : ~(y1 = y2). by rewrite -H => {H}. + tauto. + } { + move => H. + have H_neq : ~(y1 = y2). by rewrite -H => {H}. + tauto. } Qed. - Definition chooseZ_spec2 : - forall (s : t), - chooseZ s = None -> forall x, ~InZ x s. - Proof. + Lemma compare_eq_Lt_nil_l : forall s, + compare nil s = Lt <-> s <> nil. + Proof. intros s. - case s as [| [x' c] s']; - rewrite /chooseZ //. - move => _ x //. - Qed. - - Definition choose_spec1 : - forall (s : t) (x : elt) (Hs : Ok s), choose s = Some x -> In x s. - Proof. - rewrite /choose. - move => s x H_ok. - case_eq (chooseZ s) => //. - move => z H_choose [<-]. - apply InZ_In => //. - apply chooseZ_spec1 => //. - apply H_ok. + case s => //=. + split => //. Qed. - Definition choose_spec2 : - forall s : t, choose s = None -> Empty s. - Proof. - rewrite /choose /chooseZ /Empty /In. - case s as [| [x' c] s'] => //. - move => _ e. - rewrite elements_nil InA_nil //. + Lemma compare_eq_Lt_nil_r : forall s, + ~(compare s nil = Lt). + Proof. + intros s. + case s as [| [y1 c1] s'] => //=. Qed. - - Lemma chooseZ_min : - forall (s : t) (x y : Z) (Hs : Ok s), - chooseZ s = Some x -> InZ y s -> ~ Z.lt y x. + + Lemma compare_eq_Lt_cons : forall y1 y2 c1 c2 s1 s2, + compare ((y1, c1)::s1) ((y2, c2)::s2) = Lt <-> + (y1 < y2) \/ ((y1 = y2) /\ (c1 < c2)%N) \/ + ((y1 = y2) /\ (c1 = c2) /\ compare s1 s2 = Lt). Proof. - intros s x y H_ok H_min H_in H_y_lt_x. - eapply (Nin_elements_greater s (Z.pred x)) => //; last apply H_in. { - move : H_ok H_min. - case s => //. - move => [z c] s' _ [<-]. - rewrite interval_list_elements_greater_cons. - apply Z.lt_pred_l. + intros y1 y2 c1 c2 s1 s2. + rewrite /=. + case_eq (Z.compare y1 y2). { + move => /Z.compare_eq_iff ->. + case_eq (N.compare c1 c2). { + move => /N.compare_eq_iff ->. + split. { + move => H. + right; right. + done. + } { + move => [| []]. { + move => /Z.lt_irrefl //. + } { + move => [_] /N.lt_irrefl //. + } { + move => [_] [_] -> //. + } + } + } { + move => /N.compare_lt_iff H_c1_lt. + split => //. + move => _. + right; left. done. + } { + move => /N.compare_gt_iff H_c2_lt. + split => //. + move => [| []]. { + move => /Z.lt_irrefl //. + } { + move => [_] /N.lt_asymm //. + } { + move => [_] [] H_c1_eq. + contradict H_c2_lt. + subst c1. + by apply N.lt_irrefl. + } + } } { - apply H_ok. + move => /Z.compare_lt_iff. + tauto. } { - by apply Z.lt_le_pred. - } - Qed. + move => /Z.compare_gt_iff H_y2_lt. + split => //. + move => [| []]. { + move => /Z.lt_asymm //. + } { + move => [] H_y1_eq. + exfalso. omega. + } { + move => [] H_y1_eq. + exfalso. omega. + } + } + Qed. + + + Lemma compare_antisym: forall (s1 s2 : t), + (compare s1 s2) = CompOpp (compare s2 s1). + Proof. + induction s1 as [| [y1 c1] s1' IH]; + case s2 as [| [y2 c2] s2'] => //. + rewrite /= (Z.compare_antisym y1 y2) (N.compare_antisym c1 c2). + case (Z.compare y1 y2) => //=. + case (N.compare c1 c2) => //=. + Qed. - Lemma chooseZ_spec3: forall s s' x x', Ok s -> Ok s' -> - chooseZ s = Some x -> chooseZ s' = Some x' -> Equal s s' -> x = x'. + + Lemma compare_spec : forall s1 s2, + CompSpec eq lt s1 s2 (compare s1 s2). + Proof. + intros s1 s2. + rewrite /CompSpec /lt (compare_antisym s2 s1). + case_eq (compare s1 s2). { + rewrite compare_eq_Eq equal_alt_def => ->. + by apply CompEq. + } { + move => _. + by apply CompLt. + } { + move => _. + by apply CompGt. + } + Qed. + + Lemma lt_Irreflexive : Irreflexive lt. + Proof. + rewrite /Irreflexive /Reflexive /complement /lt. + intros x. + suff -> : compare x x = Eq by done. + rewrite compare_eq_Eq equal_alt_def //. + Qed. + + Lemma lt_Transitive : Transitive lt. + Proof. + rewrite /Transitive /lt. + induction x as [| [y1 c1] s1' IH]; + case y as [| [y2 c2] s2']; + case z as [| [y3 c3] s3'] => //. + + rewrite !compare_eq_Lt_cons. + move => [H_y1_lt | [[->] H_c1_lt | [->] [->] H_comp]] + [H_y2_lt | [[<-] H_c2_lt | [<-] [<-] H_comp']]. { + left. + by apply Z.lt_trans with (m := y2). + } { + by left. + } { + by left. + } { + by left. + } { + right; left. + split => //. + by apply N.lt_trans with (m := c2). + } { + by right; left. + } { + by left. + } { + by right; left. + } { + right; right. + split => //. + split => //. + by apply (IH s2'). + } + Qed. + + (** *** elements is sorted *) + + Lemma elementsZ_single_sorted : forall c x, + sort Z.lt (elementsZ_single x c). + Proof. + induction c as [| c' IH] using N.peano_ind. { + intro x. + rewrite elementsZ_single_base. + apply Sorted_nil. + } { + intro x. + rewrite elementsZ_single_succ_front. + apply Sorted_cons. { + apply IH. + } { + case (N.zero_or_succ c'). { + move => ->. + rewrite elementsZ_single_base //. + } { + move => [c''] ->. + rewrite elementsZ_single_succ_front. + constructor. + apply Z.lt_succ_diag_r. + } + } + } + Qed. + + Lemma elementsZ_sorted : forall s, + interval_list_invariant s = true -> + sort Z.lt (rev (elementsZ s)). + Proof. + induction s as [| [y c] s' IH]. { + move => _. + rewrite elementsZ_nil. + apply Sorted_nil. + } { + rewrite interval_list_invariant_cons elementsZ_cons + rev_app_distr rev_involutive. + move => [H_gr] [H_c_neq_0] H_inv_s'. + apply SortA_app with (eqA := Logic.eq). { + apply eq_equivalence. + } { + apply Z.lt_strorder. + } { + apply elementsZ_single_sorted. + } { + by apply IH. + } { + intros x1 x2. + move => /InA_alt [_] [<-] /In_elementsZ_single [_ H_x1_lt]. + move => /InA_alt [_] [<-]. + rewrite -In_rev => H_x2_in. + + apply Z.lt_trans with (m := (y + Z.of_N c)) => //. + eapply interval_list_elements_greater_alt2_def; + eauto. + } + } + Qed. + + + Lemma elements_sorted : forall s, + Ok s -> + sort Enc.E.lt (elements s). + Proof. + move => s [H_inv] H_enc. + rewrite /elements rev_map_alt_def -map_rev. + have : (forall x : Z, List.In x (rev (elementsZ s)) -> + exists e : Enc.E.t, Enc.encode e = x). { + move => x. + move : (H_enc x). + rewrite In_rev //. + } + move : (elementsZ_sorted s H_inv) => {H_enc}. + generalize (rev (elementsZ s)). + induction l as [| x xs IH]. { + rewrite /= => _ _. + apply Sorted_nil. + } { + move => H_sort H_enc. + apply Sorted_inv in H_sort as [H_sort H_hd_rel]. + simpl. + apply Sorted_cons. { + apply IH => //. + move => xx H_xx_in. + apply H_enc. + by apply in_cons. + } { + move : H_hd_rel H_enc. + case xs => //=. + move => x' xs' H_hd_rel H_enc. + apply HdRel_inv in H_hd_rel. + apply HdRel_cons. + rewrite -Enc.encode_lt. + have [y H_y] : (exists y, Enc.encode y = x). { + apply H_enc. by left. + } + have [y' H_y'] : (exists y', Enc.encode y' = x'). { + apply H_enc. by right; left. + } + move : H_hd_rel. + rewrite -!H_y -!H_y' !Enc.decode_encode_ok //. + } + } + Qed. + + + (** *** choose specification *) + + Definition min_eltZ_spec1 : + forall (s : t) (x : Z), + interval_list_invariant s = true -> + min_eltZ s = Some x -> InZ x s. + Proof. + intros s x. + case s as [| [x' c] s']. { + rewrite /min_eltZ //. + } { + rewrite /min_eltZ InZ_cons interval_list_invariant_cons. + move => [_] [H_c_neq] _ [->]. + left. + by apply In_elementsZ_single_hd. + } + Qed. + + Lemma min_eltZ_spec2 : + forall (s : t) (x y : Z) (Hs : Ok s), + min_eltZ s = Some x -> InZ y s -> ~ Z.lt y x. + Proof. + intros s x y H_ok H_min H_in H_y_lt_x. + eapply (Nin_elements_greater s (Z.pred x)) => //; last apply H_in. { + move : H_ok H_min. + case s => //. + move => [z c] s' _ [<-]. + rewrite interval_list_elements_greater_cons. + apply Z.lt_pred_l. + } { + apply H_ok. + } { + by apply Z.lt_le_pred. + } + Qed. + + Definition min_eltZ_spec3 : + forall (s : t), + min_eltZ s = None -> forall x, ~InZ x s. + Proof. + intros s. + case s as [| [x' c] s']; + rewrite /min_eltZ //. + move => _ x //. + Qed. + + Definition min_elt_spec1 : + forall (s : t) (x : elt) (Hs : Ok s), min_elt s = Some x -> In x s. + Proof. + rewrite /min_elt. + move => s x H_ok. + case_eq (min_eltZ s) => //. + move => z H_min_elt [<-]. + apply InZ_In => //. + apply min_eltZ_spec1 => //. + apply H_ok. + Qed. + + Definition min_elt_spec2 : + forall (s : t) (x y : elt) (Hs : Ok s), min_elt s = Some x -> In y s -> ~(Enc.E.lt y x). + Proof. + rewrite /min_elt. + move => s x y H_ok. + case_eq (min_eltZ s) => //. + move => z H_min_elt [<-]. + rewrite In_InZ => H_inZ. + have H_y_eq : y = Enc.decode (Enc.encode y). { + by rewrite Enc.decode_encode_ok. + } + rewrite H_y_eq -Enc.encode_lt. + apply (min_eltZ_spec2 _ _ _ H_ok); last first. { + by rewrite Enc.decode_encode_ok. + } + suff -> : Enc.encode (Enc.decode z) = z by assumption. + apply encode_decode_eq with (s := s) => //. + apply min_eltZ_spec1 => //. + apply H_ok. + Qed. + + Definition min_elt_spec3 : + forall s : t, min_elt s = None -> Empty s. + Proof. + rewrite /min_elt /min_eltZ /Empty /In. + case s as [| [x' c] s'] => //. + move => _ e. + rewrite elements_nil InA_nil //. + Qed. + + + Definition choose_spec1 : + forall (s : t) (x : elt) (Hs : Ok s), choose s = Some x -> In x s. + Proof. + rewrite /choose. + apply min_elt_spec1. + Qed. + + Definition choose_spec2 : + forall s : t, choose s = None -> Empty s. + Proof. + rewrite /choose. + apply min_elt_spec3. + Qed. + + Lemma choose_spec3: forall s s' x x', Ok s -> Ok s' -> + choose s = Some x -> choose s' = Some x' -> Equal s s' -> x = x'. Proof. intros s s' x x' Hs Hs' Hx Hx'. rewrite -equal_spec equal_alt_def => H_s_eq. @@ -4527,6 +4900,131 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, rewrite H_s_eq => -> [] //. Qed. + + Definition max_eltZ_spec1 : + forall (s : t) (x : Z), + interval_list_invariant s = true -> + max_eltZ s = Some x -> InZ x s. + Proof. + intros s x. + induction s as [| [x' c] s' IH]. { + rewrite /max_eltZ //. + } { + rewrite InZ_cons interval_list_invariant_cons /=. + move => [_] [H_c_neq]. + case s' as [| [y' c'] s'']. { + move => _ [<-]. + left. { + rewrite In_elementsZ_single. + split. { + rewrite -Z.lt_le_pred. + by apply Z_lt_add_r. + } { + apply Z.lt_pred_l. + } + } + } { + move => H_inv H_max_eq. + right. + by apply IH. + } + } + Qed. + + Lemma max_eltZ_spec2 : + forall (s : t) (x y : Z), + interval_list_invariant s = true -> + max_eltZ s = Some x -> InZ y s -> ~ Z.lt x y. + Proof. + induction s as [| [y c] s' IH]. { + done. + } { + move => x x'. + rewrite interval_list_invariant_cons. + move => [H_gr] [H_c_neq_0] H_inv_s'. + have H_gr' : (forall xx : Z, InZ xx (s') -> y + Z.of_N c < xx). { + apply interval_list_elements_greater_alt2_def => //. + } + + case s' as [| [y' c'] s'']. { + move => [<-]. + rewrite InZ_cons InZ_nil In_elementsZ_single. + omega. + } { + move => H_max_eq. + rewrite InZ_cons. + move => []; last by apply IH. + rewrite In_elementsZ_single. + move => [_] H_x'_lt H_lt_x'. + + have H_x_in : InZ x ((y', c')::s''). { + by apply max_eltZ_spec1. + } + + move : (H_gr' _ H_x_in). + apply Z.nlt_ge, Z.lt_le_incl. + by apply Z.lt_trans with (m := x'). + } + } + Qed. + + Lemma max_eltZ_eq_None : + forall (s : t), + max_eltZ s = None -> s = nil. + Proof. + induction s as [| [x' c] s' IH] => //. + move : IH. + case s' as [| [y' c'] s''] => //=. + move => H H_pre. + move : (H H_pre) => //. + Qed. + + + Definition max_eltZ_spec3 : + forall (s : t), + max_eltZ s = None -> forall x, ~InZ x s. + Proof. + move => s /max_eltZ_eq_None -> x /InZ_nil //. + Qed. + + Definition max_elt_spec1 : + forall (s : t) (x : elt) (Hs : Ok s), max_elt s = Some x -> In x s. + Proof. + rewrite /max_elt. + move => s x H_ok. + case_eq (max_eltZ s) => //. + move => z H_max_elt [<-]. + apply InZ_In => //. + apply max_eltZ_spec1 => //. + apply H_ok. + Qed. + + Definition max_elt_spec2 : + forall (s : t) (x y : elt) (Hs : Ok s), max_elt s = Some x -> In y s -> ~(Enc.E.lt x y). + Proof. + rewrite /max_elt. + move => s x y H_ok. + move : (H_ok) => [H_inv] _. + case_eq (max_eltZ s) => //. + move => z H_max_elt [<-]. + rewrite In_InZ => H_inZ. + rewrite -Enc.encode_lt. + apply (max_eltZ_spec2 _ _ _ H_inv) => //. + suff -> : Enc.encode (Enc.decode z) = z => //. + apply encode_decode_eq with (s := s) => //. + apply max_eltZ_spec1 => //. + Qed. + + Definition max_elt_spec3 : + forall s : t, max_elt s = None -> Empty s. + Proof. + intro s. + rewrite /max_elt /Empty. + case_eq (max_eltZ s) => //. + move => /max_eltZ_eq_None -> _ x. + rewrite /In elements_nil InA_nil //. + Qed. + (** *** fold specification *) @@ -4535,7 +5033,16 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, fold f s i = fold_left (flip f) (elements s) i. Proof. intros s A i f. - rewrite /fold //. + rewrite /fold fold_elementsZ_alt_def /elements + rev_map_alt_def -map_rev. + move : i. + generalize (rev (elementsZ s)). + induction l as [| x xs IH]. { + done. + } { + move => i. + rewrite /= IH //. + } Qed. @@ -4574,11 +5081,43 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, (for_all f s = true <-> For_all (fun x => f x = true) s). Proof. intros s f Hs H. - rewrite /for_all forallb_forall /For_all //. - split; ( - move => HH x; move : {HH} (HH x); - rewrite In_alt_def // - ). + rewrite /for_all /For_all /In fold_elementsZ_alt_def + /elements rev_map_alt_def -map_rev. + generalize (rev (elementsZ s)). + induction l as [| x xs IH]. { + split => // _ x /= /InA_nil //. + } { + rewrite /=. + case_eq (f (Enc.decode x)) => H_f_eq. { + rewrite IH. + split. { + move => HH x' /InA_cons []. { + by move => ->. + } { + apply HH. + } + } { + move => HH x' H_in. + apply HH. + apply InA_cons. + by right. + } + } { + split; move => HH. { + contradict HH. + case xs => //. + } { + exfalso. + have H_in: (InA Enc.E.eq (Enc.decode x) (Enc.decode x :: map Enc.decode xs)). { + apply InA_cons. + left. + apply Enc.E.eq_equiv. + } + move : (HH _ H_in). + rewrite H_f_eq => //. + } + } + } Qed. (** *** exists specification *) @@ -4589,46 +5128,526 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, (exists_ f s = true <-> Exists (fun x => f x = true) s). Proof. intros s f Hs H. - rewrite /exists_ /Exists existsb_exists //. - split; ( - move => [x] [HH1 HH2]; exists x; move : HH1 HH2; - rewrite In_alt_def // - ). + rewrite /exists_ /Exists /In fold_elementsZ_alt_def + /elements rev_map_alt_def -map_rev. + generalize (rev (elementsZ s)). + induction l as [| x xs IH]. { + split => //. + move => [x] /= [] /InA_nil //. + } { + rewrite /=. + case_eq (f (Enc.decode x)) => H_f_eq. { + split => _. { + exists (Enc.decode x). + split => //. + apply InA_cons. + left. + apply Enc.E.eq_equiv. + } { + case xs => //. + } + } { + rewrite IH. + split. { + move => [x0] [H_in] H_f_x0. + exists x0. + split => //. + apply InA_cons. + by right. + } { + move => [x0] [] /InA_cons H_in H_f_x0. + exists x0. + split => //. + move : H_in => [] // H_in. + contradict H_f_x0. + rewrite H_in H_f_eq //. + } + } + } + Qed. + + + (** *** filter specification *) + + Definition partitionZ_aux_invariant (x : Z) acc c := + interval_list_invariant (List.rev (partitionZ_fold_skip acc c)) = true /\ + match c with + None => (forall y', InZ y' acc -> Z.succ y' < x) + | Some (y, c') => (x = y + Z.of_N c') + end. + + Lemma partitionZ_aux_invariant_insert : forall x acc c, + partitionZ_aux_invariant x acc c -> + partitionZ_aux_invariant (Z.succ x) acc + (Some (partitionZ_fold_insert c x)). + Proof. + intros x acc c. + rewrite /partitionZ_fold_insert /partitionZ_aux_invariant + /partitionZ_fold_skip. + case c; last first. { + move => [H_inv] H_in. + rewrite /= interval_list_invariant_app_iff Z.add_1_r. + split; last done. + split; first done. + split; first done. + move => x1 x2. + rewrite InZ_rev InZ_cons InZ_nil In_elementsZ_single1. + move => H_x1_in [] // <-. + by apply H_in. + } { + move => [y c']. + rewrite /= !interval_list_invariant_app_iff + N2Z.inj_succ Z.add_succ_r . + rewrite !interval_list_invariant_cons !interval_list_invariant_nil. + move => [] [H_inv_acc] [] [] _ [H_c_neq_0] _ + H_in_c ->. + split; last done. + split; first done. + split. { + split; first done. + split; last done. + apply N.neq_succ_0. + } { + move => x1 x2. + rewrite InZ_cons InZ_nil In_elementsZ_single. + move => H_x1_in [] // [H_y_le] H_x2_lt. + apply Z.lt_le_trans with (m := y) => //. + apply H_in_c => //. + rewrite InZ_cons In_elementsZ_single. + left. + split. { + apply Z.le_refl. + } { + by apply Z_lt_add_r. + } + } + } + Qed. + + Lemma partitionZ_aux_invariant_skip : forall x acc c, + partitionZ_aux_invariant x acc c -> + partitionZ_aux_invariant (Z.succ x) (partitionZ_fold_skip acc c) None. + Proof. + intros x acc c. + rewrite /partitionZ_fold_skip /partitionZ_aux_invariant + /partitionZ_fold_skip. + case c; last first. { + move => [H_inv] H_in. + split; first done. + move => y' H_y'_in. + apply Z.lt_trans with (m := x). { + by apply H_in. + } { + apply Z.lt_succ_diag_r. + } + } { + move => [y c'] [H_inv] ->. + split => //. + move => y'. + rewrite InZ_cons In_elementsZ_single. + move => []. { + move => [_]. + rewrite -Z.succ_lt_mono //. + } { + move : H_inv. + rewrite /= !interval_list_invariant_app_iff interval_list_invariant_cons. + move => [_] [] [_] [H_c'_neq] _ H_pre H_y'_in. + apply Z.lt_trans with (m := y). { + apply H_pre. { + by rewrite InZ_rev. + } { + rewrite InZ_cons. + left. + by apply In_elementsZ_single_hd. + } + } + apply Z.lt_succ_r, Z_le_add_r. + } + } + Qed. + + Definition partitionZ_fold_current (c : option (Z * N)) := + match c with + None => nil + | Some yc => yc::nil + end. + + Lemma InZ_partitionZ_fold_current_Some : forall yc y, + InZ y (partitionZ_fold_current (Some yc)) <-> + InZ y (yc :: nil). + Proof. done. Qed. + + Lemma InZ_partitionZ_fold_insert : forall c x y l, + match c with + | Some (y, c') => x = y + Z.of_N c' + | None => True + end -> ( + InZ y (partitionZ_fold_insert c x :: l) <-> + ((x = y) \/ InZ y (partitionZ_fold_current c) \/ + InZ y l)). + Proof. + intros c x y l. + rewrite /partitionZ_fold_insert /partitionZ_fold_current + /partitionZ_fold_skip. + case c. { + move => [y' c'] ->. + rewrite !InZ_cons elementsZ_single_succ in_app_iff + InZ_nil /=. + tauto. + } { + rewrite InZ_cons InZ_nil In_elementsZ_single1. + tauto. + } + Qed. + + Lemma InZ_partitionZ_fold_skip : forall c acc y, + InZ y (partitionZ_fold_skip acc c) <-> + (InZ y (partitionZ_fold_current c) \/ InZ y acc). + Proof. + intros c acc y. + rewrite /partitionZ_fold_skip /partitionZ_fold_current + /partitionZ_fold_skip. + case c. { + move => [y' c']. + rewrite !InZ_cons InZ_nil /=. + tauto. + } { + rewrite InZ_nil. + tauto. + } + Qed. + + Lemma filterZ_single_aux_props : + forall f c x acc cur, + partitionZ_aux_invariant x acc cur -> + match (filterZ_single_aux f (acc, cur) x c) with + (acc', c') => + let r := partitionZ_fold_skip acc' c' in + interval_list_invariant (List.rev r) = true /\ + (forall y', InZ y' r <-> (InZ y' (partitionZ_fold_skip acc cur) \/ + (f y' = true /\ List.In y' (elementsZ_single x c)))) + + end. + Proof. + intro f. + induction c as [| c' IH] using N.peano_ind. { + intros x acc cur. + rewrite /partitionZ_aux_invariant. + move => [H_inv] _. + rewrite /filterZ_single_aux fold_elementsZ_single_zero /=. + tauto. + } + intros x acc cur H_inv. + have -> : filterZ_single_aux f (acc, cur) x (N.succ c') = + filterZ_single_aux f (filterZ_fold_fun f (acc, cur) x) (Z.succ x) c'. { + by rewrite /filterZ_single_aux fold_elementsZ_single_succ. + } + case_eq (filterZ_fold_fun f (acc, cur) x). + move => acc' cur' H_fold_eq. + + case_eq (filterZ_single_aux f (acc', cur') (Z.succ x) c'). + move => acc'' cur'' H_succ_eq. + + have H_inv' : partitionZ_aux_invariant (Z.succ x) acc' cur'. { + move : H_fold_eq H_inv. + rewrite /filterZ_fold_fun. + case (f x); move => [<-] <-. { + apply partitionZ_aux_invariant_insert. + } { + apply partitionZ_aux_invariant_skip. + } + } + + move : (IH (Z.succ x) acc' cur' H_inv') => {IH}. + rewrite H_succ_eq /=. + set r := partitionZ_fold_skip acc'' cur''. + move => [H_inv_r] H_in_r. + split; first assumption. + + move => y'. + move : H_fold_eq. + rewrite H_in_r /filterZ_fold_fun. + case_eq (f x) => H_fx [<-] <-. { + rewrite InZ_partitionZ_fold_skip InZ_partitionZ_fold_current_Some InZ_partitionZ_fold_skip elementsZ_single_succ_front. + rewrite InZ_partitionZ_fold_insert; last first. { + move : H_inv. + rewrite /partitionZ_aux_invariant => [[_]]. + case cur => //. + } + rewrite InZ_nil /=. + split; last by tauto. + move => []; last by tauto. + move => []; last by tauto. + move => []. { + move => <-. + tauto. + } { + tauto. + } + } { + rewrite InZ_partitionZ_fold_skip /partitionZ_fold_current InZ_partitionZ_fold_skip elementsZ_single_succ_front !InZ_nil /=. + split; first by tauto. + move => []; first by tauto. + move => [] H_fy' []. { + move => H_x_eq; subst y'. + contradict H_fy'. + by rewrite H_fx. + } { + tauto. + } + } + Qed. + + + Lemma filterZ_single_props : + forall f c x acc, + interval_list_invariant (rev acc) = true -> + (forall y' : Z, InZ y' acc -> Z.succ y' < x) -> + match (filterZ_single f acc x c) with + r => + interval_list_invariant (List.rev r) = true /\ + (forall y', InZ y' r <-> (InZ y' acc \/ + (f y' = true /\ List.In y' (elementsZ_single x c)))) + + end. + Proof. + intros f c x acc. + move => H_inv H_acc. + rewrite /filterZ_single. + + have H_inv' : partitionZ_aux_invariant x acc None. { + by rewrite /partitionZ_aux_invariant /=. + } + move : (filterZ_single_aux_props f c x acc None H_inv'). + case_eq (filterZ_single_aux f (acc, None) x c). + move => acc' cur' /= H_res. + tauto. + Qed. + + + Lemma filterZ_aux_props : + forall f s acc, + interval_list_invariant s = true -> + interval_list_invariant (rev acc) = true -> + (forall x1 x2 : Z, InZ x1 acc -> InZ x2 s -> Z.succ x1 < x2) -> + match (filterZ_aux acc f s) with + r => + interval_list_invariant r = true /\ + (forall y', InZ y' r <-> (InZ y' acc \/ + (f y' = true /\ InZ y' s))) + end. + Proof. + intro f. + induction s as [| [y c] s' IH]. { + intros acc. + move => _ H_inv _. + rewrite /filterZ_aux. + split; first assumption. + move => y'; rewrite InZ_rev InZ_nil; tauto. + } { + intros acc. + rewrite interval_list_invariant_cons. + move => [H_gr] [H_c_neq_0] H_inv_s' H_inv H_in_acc /=. + move : H_gr. + rewrite interval_list_elements_greater_alt2_def => // H_gr. + + have H_pre : (forall y' : Z, InZ y' acc -> Z.succ y' < y). { + move => x1 H_x1_in. + apply H_in_acc => //. + rewrite InZ_cons. + by left; apply In_elementsZ_single_hd. + } + + move : (filterZ_single_props f c y acc H_inv H_pre) => {H_pre}. + set acc' := filterZ_single f acc y c. + move => [H_inv'] H_in_acc'. + + have H_pre : (forall x1 x2 : Z, + InZ x1 acc' -> InZ x2 s' -> Z.succ x1 < x2). { + move => x1 x2. + rewrite H_in_acc' In_elementsZ_single. + move => []. { + move => H_x1_in H_x2_in. + apply H_in_acc => //. + rewrite InZ_cons. + by right. + } { + move => [_] [_] H_x1_lt H_x2_in. + apply Z.le_lt_trans with (m := y + Z.of_N c). + - by apply Z.le_succ_l. + - by apply H_gr. + } + } + move : (IH acc' H_inv_s' H_inv' H_pre) => {H_pre}. + + move => [H_inv_r] H_in_r. + split; first assumption. + move => y'. + rewrite H_in_r H_in_acc' InZ_cons. + tauto. + } + Qed. + + Lemma filterZ_props : + forall f s, + interval_list_invariant s = true -> + match (filterZ f s) with r => + interval_list_invariant r = true /\ + (forall y', InZ y' r <-> (f y' = true /\ InZ y' s)) + end. + Proof. + intros f s H_inv_s. + rewrite /filterZ. + + have H_pre_1 : interval_list_invariant (rev nil) = true by done. + have H_pre_2 : (forall x1 x2 : Z, InZ x1 nil -> InZ x2 s -> Z.succ x1 < x2) by done. + + move : (filterZ_aux_props f s nil H_inv_s H_pre_1 H_pre_2) => {H_pre_1} {H_pre_2}. + move => [H_inv'] H_in_r. + split; first assumption. + move => y'. + rewrite H_in_r InZ_nil. + tauto. + Qed. + + Global Instance filter_ok s f : forall `(Ok s), Ok (filter f s). + Proof. + move => [H_inv H_enc]. + rewrite /filter. + set f' := (fun z : Z => f (Enc.decode z)). + move : (filterZ_props f' s H_inv). + move => [H_inv'] H_in_r. + rewrite /Ok /IsOk /is_encoded_elems_list. + split; first assumption. + move => x /H_in_r [_] H_x_in. + by apply H_enc. + Qed. + + Lemma filter_spec : + forall (s : t) (x : elt) (f : elt -> bool), + Ok s -> + (In x (filter f s) <-> In x s /\ f x = true). + Proof. + move => s x f H_ok. + suff H_suff : + (forall x, (InZ x (filter f s)) <-> + InZ x s /\ f (Enc.decode x) = true). { + rewrite !In_alt_def /elements !rev_map_alt_def + -!in_rev !in_map_iff. + setoid_rewrite H_suff. + rewrite /InZ. + split. { + move => [y] [<-] [?] ?. + split => //. + by exists y. + } { + move => [] [y] [<-] ? ?. + by exists y. + } + } + rewrite /filter. + set f' := (fun z : Z => f (Enc.decode z)). + move : (H_ok) => [H_inv _]. + move : (filterZ_props f' s H_inv). + move => [H_inv'] H_in_r. + move => y; rewrite H_in_r; tauto. + Qed. + + + (** *** partition specification *) + + Lemma partitionZ_single_aux_alt_def : forall f c y acc_t c_t acc_f c_f, + partitionZ_single_aux f ((acc_t, c_t), (acc_f, c_f)) y c = + (filterZ_single_aux f (acc_t, c_t) y c, + filterZ_single_aux (fun x : Z => negb (f x)) (acc_f, c_f) y c). + Proof. + intros f. + rewrite /partitionZ_single_aux /filterZ_single_aux. + induction c as [| c' IH] using N.peano_ind. { + intros y acc_t c_t acc_f c_f. + rewrite !fold_elementsZ_single_zero //. } { + intros y acc_t c_t acc_f c_f. + rewrite !fold_elementsZ_single_succ. + case_eq (partitionZ_fold_fun f (acc_t, c_t, (acc_f, c_f)) y) => [] [acc_t' c_t'] [acc_f' c_f'] H_fold_eq. + rewrite IH => {IH}. + suff : (filterZ_fold_fun f (acc_t, c_t) y = (acc_t', c_t')) /\ + (filterZ_fold_fun (fun x : Z => negb (f x)) (acc_f, c_f) y = (acc_f', c_f')). { + move => [->] -> //. + } + move : H_fold_eq. + rewrite /partitionZ_fold_fun /filterZ_fold_fun. + case (f y); move => [<-] <- <- <- //. + } + Qed. + + Lemma partitionZ_aux_alt_def : forall f s acc_t acc_f, + partitionZ_aux acc_t acc_f f s = + (filterZ_aux acc_t f s, + filterZ_aux acc_f (fun x : Z => negb (f x)) s). + Proof. + intros f. + induction s as [| [y c] s' IH]. { + done. + } { + intros acc_t acc_f. + rewrite /= /partitionZ_single /filterZ_single + partitionZ_single_aux_alt_def. + case (filterZ_single_aux f (acc_t, None) y c) => acc_t' c_t'. + case (filterZ_single_aux (fun x : Z => negb (f x)) (acc_f, None) y c) => acc_f' c_f'. + rewrite IH //. + } + Qed. + + Lemma partitionZ_alt_def : forall f s, + partitionZ f s = (filterZ f s, + filterZ (fun x => negb (f x)) s). + Proof. + intros f s. + rewrite /partitionZ /filterZ + partitionZ_aux_alt_def //. + Qed. + + Lemma partition_alt_def : forall f s, + partition f s = (filter f s, + filter (fun x => negb (f x)) s). + Proof. + intros f s. + rewrite /partition /filter partitionZ_alt_def. + done. Qed. - (** *** partition specification *) Global Instance partition_ok1 s f : forall `(Ok s), Ok (fst (partition f s)). Proof. - intros H_ok. - rewrite /partition /fst. + move => H_ok. + rewrite partition_alt_def /fst. by apply filter_ok. - Qed. + Qed. Global Instance partition_ok2 s f : forall `(Ok s), Ok (snd (partition f s)). Proof. - intros H_ok. - rewrite /partition /snd. + move => H_ok. + rewrite partition_alt_def /snd. by apply filter_ok. Qed. Lemma partition_spec1 : forall (s : t) (f : elt -> bool), - Proper (Enc.E.eq==>eq) f -> Equal (fst (partition f s)) (filter f s). + Equal (fst (partition f s)) (filter f s). Proof. - intros s f H_ok. - rewrite /partition /fst /Equal //. + intros s f. + rewrite partition_alt_def /fst /Equal //. Qed. Lemma partition_spec2 : forall (s : t) (f : elt -> bool), - Proper (Enc.E.eq==>eq) f -> + Ok s -> Equal (snd (partition f s)) (filter (fun x => negb (f x)) s). Proof. - intros s f H_ok. - rewrite /partition /snd /Equal //. + intros s f. + rewrite partition_alt_def /snd /Equal //. Qed. - + End Raw. @@ -4637,7 +5656,7 @@ End Raw. We can now build the invariant into the set type to obtain an instantiation of module type [WSetsOn]. *) -Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E. +Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E. Module E := Enc.E. Module Raw := Raw Enc. @@ -4670,7 +5689,10 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E. Definition empty : t := Mkt Raw.empty. Definition is_empty (s : t) := Raw.is_empty s. Definition elements (s : t) : list elt := Raw.elements s. + Definition min_elt (s : t) : option elt := Raw.min_elt s. + Definition max_elt (s : t) : option elt := Raw.max_elt s. Definition choose (s : t) : option elt := Raw.choose s. + Definition compare (s1 s2 : t) : comparison := Raw.compare s1 s2. Definition fold {A : Type}(f : elt -> A -> A)(s : t) : A -> A := Raw.fold f s. Definition cardinal (s : t) := Raw.cardinal s. Definition filter (f : elt -> bool)(s : t) : t := Mkt (Raw.filter f s). @@ -4693,7 +5715,6 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E. Instance eq_equiv : Equivalence eq. Proof. firstorder. Qed. - Definition eq_dec : forall (s s':t), { eq s s' }+{ ~eq s s' }. Proof. intros (s,Hs) (s',Hs'). @@ -4702,7 +5723,39 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E. rewrite <- Raw.equal_spec; congruence. Defined. + Definition lt : t -> t -> Prop := Raw.lt. + + Instance lt_strorder : StrictOrder lt. + Proof. + unfold lt. + constructor. { + move : Raw.lt_Irreflexive. + rewrite /Irreflexive /complement /Reflexive. + move => H x. + apply H. + } { + move : Raw.lt_Transitive. + rewrite /Transitive. + move => H x y z. + apply H. + } + Qed. + Instance lt_compat : Proper (eq==>eq==>iff) lt. + Proof. + repeat red. + move => [x1 H_x1_ok] [y1 H_y1_ok] H_eq. + move => [x2 H_x2_ok] [y2 H_y2_ok]. + move : H_eq. + rewrite /eq /lt /Equal /In /=. + replace (forall a : elt, Raw.In a x1 <-> Raw.In a y1) with + (Raw.Equal x1 y1) by done. + replace (forall a : elt, Raw.In a x2 <-> Raw.In a y2) with + (Raw.Equal x2 y2) by done. + rewrite -!Raw.equal_spec !Raw.equal_alt_def. + move => -> -> //. + Qed. + Section Spec. Variable s s' : t. Variable x y : elt. @@ -4738,7 +5791,7 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E. Proof. exact (@Raw.cardinal_spec s). Qed. Lemma filter_spec : compatb f -> (In x (filter f s) <-> In x s /\ f x = true). - Proof. exact (@Raw.filter_spec _ _ _). Qed. + Proof. move => _; exact (@Raw.filter_spec _ _ _ _). Qed. Lemma for_all_spec : compatb f -> (for_all f s = true <-> For_all (fun x => f x = true) s). Proof. exact (@Raw.for_all_spec _ _ _). Qed. @@ -4746,19 +5799,60 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E. (exists_ f s = true <-> Exists (fun x => f x = true) s). Proof. exact (@Raw.exists_spec _ _ _). Qed. Lemma partition_spec1 : compatb f -> Equal (fst (partition f s)) (filter f s). - Proof. exact (@Raw.partition_spec1 _ _). Qed. + Proof. move => _; exact (@Raw.partition_spec1 _ _). Qed. Lemma partition_spec2 : compatb f -> Equal (snd (partition f s)) (filter (fun x => negb (f x)) s). - Proof. exact (@Raw.partition_spec2 _ _). Qed. + Proof. move => _; exact (@Raw.partition_spec2 _ _ _). Qed. Lemma elements_spec1 : InA E.eq x (elements s) <-> In x s. Proof. rewrite /In /Raw.In /elements //. Qed. Lemma elements_spec2w : NoDupA E.eq (elements s). Proof. exact (Raw.elements_spec2w _ _). Qed. + Lemma elements_spec2 : sort E.lt (elements s). + Proof. exact (Raw.elements_sorted _ _). Qed. Lemma choose_spec1 : choose s = Some x -> In x s. Proof. exact (Raw.choose_spec1 _ _ _). Qed. Lemma choose_spec2 : choose s = None -> Empty s. Proof. exact (Raw.choose_spec2 _). Qed. + Lemma choose_spec3 : choose s = Some x -> choose s' = Some y -> + Equal s s' -> E.eq x y. + Proof. + intros H1 H2 H3. + suff -> : x = y. { + apply E.eq_equiv. + } + move : H1 H2 H3. + exact (Raw.choose_spec3 _ _ _ _ _ _). + Qed. + Lemma min_elt_spec1 : choose s = Some x -> In x s. + Proof. exact (Raw.min_elt_spec1 _ _ _). Qed. + Lemma min_elt_spec2 : min_elt s = Some x -> In y s -> ~ E.lt y x. + Proof. exact (Raw.min_elt_spec2 _ _ _ _). Qed. + Lemma min_elt_spec3 : choose s = None -> Empty s. + Proof. exact (Raw.min_elt_spec3 _). Qed. + + Lemma max_elt_spec1 : max_elt s = Some x -> In x s. + Proof. exact (Raw.max_elt_spec1 _ _ _). Qed. + Lemma max_elt_spec2 : max_elt s = Some x -> In y s -> ~ E.lt x y. + Proof. exact (Raw.max_elt_spec2 _ _ _ _). Qed. + Lemma max_elt_spec3 : max_elt s = None -> Empty s. + Proof. exact (Raw.max_elt_spec3 _). Qed. + + Lemma compare_spec : CompSpec eq lt s s' (compare s s'). + Proof. + generalize s s'. + move => [s1 H_ok_s1] [s2 H_ok_s2]. + move : (Raw.compare_spec s1 s2). + rewrite /CompSpec /eq /Equal /In /lt /compare /=. + replace (forall a : elt, Raw.In a s1 <-> Raw.In a s2) with + (Raw.Equal s1 s2) by done. + suff H_eq : (Raw.Equal s1 s2) <-> (s1 = s2). { + move => [] H; constructor => //. + by rewrite H_eq. + } + rewrite -Raw.equal_spec Raw.equal_alt_def //. + Qed. + End Spec. End MSetIntervals. @@ -4785,10 +5879,14 @@ Module ElementEncodeZ <: ElementEncode. (Z.eq (encode e1) (encode e2)) <-> E.eq e1 e2. Proof. by []. Qed. + Lemma encode_lt : forall (e1 e2 : E.t), + (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2. + Proof. by []. Qed. + End ElementEncodeZ. -Module MSetIntervalsZ <: WSetsOn Z := MSetIntervals ElementEncodeZ. +Module MSetIntervalsZ <: SetsOn Z := MSetIntervals ElementEncodeZ. (** *** N *) @@ -4811,9 +5909,17 @@ Module ElementEncodeN <: ElementEncode. intros e1 e2. rewrite /encode /Z.eq N2Z.inj_iff /E.eq //. Qed. + + Lemma encode_lt : forall (e1 e2 : E.t), + (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2. + Proof. + intros e1 e2. + rewrite /encode -N2Z.inj_lt //. + Qed. + End ElementEncodeN. -Module MSetIntervalsN <: WSetsOn N := MSetIntervals ElementEncodeN. +Module MSetIntervalsN <: SetsOn N := MSetIntervals ElementEncodeN. (** *** nat *) @@ -4836,9 +5942,17 @@ Module ElementEncodeNat <: ElementEncode. intros e1 e2. rewrite /encode /Z.eq Nat2Z.inj_iff /E.eq //. Qed. + + Lemma encode_lt : forall (e1 e2 : E.t), + (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2. + Proof. + intros e1 e2. + rewrite /encode -Nat2Z.inj_lt //. + Qed. + End ElementEncodeNat. -Module MSetIntervalsNat <: WSetsOn NPeano.Nat := MSetIntervals ElementEncodeNat. +Module MSetIntervalsNat <: SetsOn NPeano.Nat := MSetIntervals ElementEncodeNat. From 5f8af2477e0203931ccc6ee7ac02db203c8d1de5 Mon Sep 17 00:00:00 2001 From: Thomas Tuerk Date: Mon, 3 Oct 2016 23:47:54 +0200 Subject: [PATCH 3/7] update coqdoc --- coqdoc/MSetIntervals.html | 1609 +++++++++++++++++++++++++++------- coqdoc/MSetListWithDups.html | 10 +- coqdoc/index.html | 462 ++++++++-- coqdoc/mset_doc.pdf | Bin 434865 -> 653870 bytes coqdoc/toc.html | 74 +- 5 files changed, 1696 insertions(+), 459 deletions(-) diff --git a/coqdoc/MSetIntervals.html b/coqdoc/MSetIntervals.html index e26f698..a963e7f 100644 --- a/coqdoc/MSetIntervals.html +++ b/coqdoc/MSetIntervals.html @@ -20,6 +20,8 @@

Library MSetIntervals

+
+
@@ -29,8 +31,8 @@

Library MSetIntervals

- This file contains an implementation of the weak set interface - WSetsOn which uses internally intervals of Z. This allows some + This file contains an implementation of the set interface + SetsOn which uses internally intervals of Z. This allows some large sets, which naturally map to intervals of integers to be represented very efficiently. @@ -69,6 +71,19 @@

Library MSetIntervals

Lemma Z_le_add_r : forall (z : Z) (n : N),
  z <= z + Z.of_N n.
+
+Lemma Z_lt_add_r : forall (z : Z) (n : N),
+  (n <> 0)%N ->
+  z < z + Z.of_N n.
+ +
+Lemma Z_lt_le_add_r : forall y1 y2 c,
+  y1 < y2 ->
+  y1 <= y2 + Z.of_N c.
+Lemma Z_to_N_minus_neq_0 : forall (x y : Z),
+    y < x ->
+    Z.to_N (x - y) <> 0%N.
+
Lemma add_add_sub_eq : forall (x y : Z), (x + (y - x) = y).
@@ -127,7 +142,7 @@

Library MSetIntervals


Module Type ElementEncode.
-  Declare Module E : DecidableType.
+  Declare Module E : OrderedType.

  Parameter encode : E.t -> Z.
@@ -156,6 +171,16 @@

Library MSetIntervals

  Axiom encode_eq : forall (e1 e2 : E.t),
    (Z.eq (encode e1) (encode e2)) <-> E.eq e1 e2.
+
+ + +
+Encoding is compatible with the order of elements. +
+
+  Axiom encode_lt : forall (e1 e2 : E.t),
+    (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2.
+
End ElementEncode.
@@ -198,7 +223,7 @@

Library MSetIntervals

-Module Ops (Enc : ElementEncode) <: WOps Enc.E.
+Module Ops (Enc : ElementEncode) <: Ops Enc.E.
  Definition elt := Enc.E.t.
  Definition t := list (Z * N).
@@ -225,29 +250,32 @@

Library MSetIntervals

  Lemma acc_pred : forall n p, n = Npos p -> Acc N.lt n -> Acc N.lt (N.pred n).

-  Fixpoint elementsZ_aux´´ (acc : list Z) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : list Z :=
-    match c as c0 return c = c0 -> list Z with
-    | N0 => fun _ => acc
-    | c => fun Heq => elementsZ_aux´´ (x::acc) (Z.succ x) (N.pred c) (acc_pred _ _ Heq H)
+  Fixpoint fold_elementsZ_aux {A} (f : A -> Z -> option A) (acc : A) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : (bool * A) :=
+    match c as c0 return c = c0 -> (bool * A) with
+    | N0 => fun _ => (false, acc)
+    | c => fun Heq => match (f acc x) with
+        | None => (true, acc)
+        | Some acc´ =>
+          fold_elementsZ_aux f acc´ (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end
    end (refl_equal _).

-  Extraction Inline elementsZ_aux´´.
+  Definition fold_elementsZ_single {A} f (acc : A) x c := fold_elementsZ_aux f acc x c (lt_wf_0 _).

-  Definition elementsZ_aux´ acc x c := elementsZ_aux´´ acc x c (lt_wf_0 _).
- -
-  Fixpoint elementsZ_aux acc (s : t) : list Z :=
+  Fixpoint fold_elementsZ {A} f (acc : A) (s : t) : (bool * A) :=
    match s with
-    | nil => acc
-    | (x, c) :: s´ =>
-        elementsZ_aux (elementsZ_aux´ acc x c) s´
+    | nil => (false, acc)
+    | (x, c) :: s´ =>
+      match fold_elementsZ_single f acc x c with
+          (false, acc´) => fold_elementsZ f acc´ s´
+        | (true, acc´) => (true, acc´)
+      end
    end.

  Definition elementsZ (s : t) : list Z :=
-    elementsZ_aux nil s.
+    snd (fold_elementsZ (fun l x => Some (x :: l)) nil s).

  Definition elements (s : t) : list elt :=
@@ -275,6 +303,104 @@

Library MSetIntervals


+
+Comparing intervals +
+
+  Inductive interval_compare_result :=
+      ICR_before
+    | ICR_before_touch
+    | ICR_overlap_before
+    | ICR_overlap_after
+    | ICR_equal
+    | ICR_subsume_1
+    | ICR_subsume_2
+    | ICR_after
+    | ICR_after_touch.
+ +
+  Definition interval_compare (i1 i2 : (Z * N)) : interval_compare_result :=
+    match (i1, i2) with ((y1, c1), (y2, c2)) =>
+      let yc2 := (y2+Z.of_N c2) in
+      match (Z.compare yc2 y1) with
+        | Lt => ICR_after
+        | Eq => ICR_after_touch
+        | Gt => let yc1 := (y1+Z.of_N c1) in
+                match (Z.compare yc1 y2) with
+                | Lt => ICR_before
+                | Eq => ICR_before_touch
+                | Gt =>
+                        match (Z.compare y1 y2, Z.compare yc1 yc2) with
+                        | (Lt, Lt) => ICR_overlap_before
+                        | (Lt, _) => ICR_subsume_2
+                        | (Eq, Lt) => ICR_subsume_1
+                        | (Eq, Gt) => ICR_subsume_2
+                        | (Eq, Eq) => ICR_equal
+                        | (Gt, Gt) => ICR_overlap_after
+                        | (Gt, _) => ICR_subsume_1
+                        end
+                end
+      end
+    end.
+ +
+  Definition interval_1_compare (y1 : Z) (i : (Z * N)) : interval_compare_result :=
+    match i with (y2, c2) =>
+      let yc2 := (y2+Z.of_N c2) in
+      match (Z.compare yc2 y1) with
+        | Lt => ICR_after
+        | Eq => ICR_after_touch
+        | Gt => match (Z.compare (Z.succ y1) y2) with
+                | Lt => ICR_before
+                | Eq => ICR_before_touch
+                | Gt => ICR_subsume_1
+                end
+      end
+    end.
+ +
+  Fixpoint compare (s1 s2 : t) :=
+    match (s1, s2) with
+      | (nil, nil) => Eq
+      | (nil, _ :: _) => Lt
+      | (_ :: _, nil) => Gt
+      | ((y1, c1)::s1´, (y2, c2)::s2´) =>
+        match (Z.compare y1 y2) with
+          | Lt => Lt
+          | Gt => Gt
+          | Eq => match N.compare c1 c2 with
+                    | Lt => Lt
+                    | Gt => Gt
+                    | Eq => compare s1´ s2´
+                  end
+        end
+    end.
+ +
+
+ +
+Auxiliary functions for inserting at front and merging intervals +
+
+  Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N :=
+    (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))).
+ +
+  Fixpoint insert_interval_begin (x : Z) (c : N) (l : t) :=
+    match l with
+    | nil => (x,c)::nil
+    | (y, c´)::l´ =>
+         match (Z.compare (x + Z.of_N c) y) with
+         | Lt => (x, c) :: l
+         | Eq => (x, (c+c´)%N) :: l´
+         | Gt => insert_interval_begin x (merge_interval_size x c y c´) l´
+         end
+    end.
+ +
+
+
adding an element needs to be defined carefully again in order to generate efficient code @@ -284,20 +410,12 @@

Library MSetIntervals

    match s with
    | nil => List.rev´ ((x, (1%N))::acc)
    | (y, c) :: l =>
-        match (Z.compare (Z.succ x) y) with
-        | Lt => List.rev_append ((x, (1%N))::acc) s
-        | Eq => List.rev_append ((x, N.succ c)::acc) l
-        | Gt => match (Z.compare x (y+Z.of_N c)) with
-                 | Lt => List.rev_append acc s
-                 | Gt => addZ_aux ((y,c) :: acc) x l
-                 | Eq => match l with
-                           | nil => List.rev´ ((y, N.succ c)::acc)
-                           | (z, c´) :: l´ => if (Z.eqb z (Z.succ x)) then
-                                List.rev_append ((y,N.succ (c+c´)) :: acc) l´
-                             else
-                                List.rev_append ((y,N.succ c) :: acc) l
-                         end
-                end
+        match (interval_1_compare x (y,c)) with
+          | ICR_before => List.rev_append ((x, (1%N))::acc) s
+          | ICR_before_touch => List.rev_append ((x, N.succ c)::acc) l
+          | ICR_after => addZ_aux ((y,c) :: acc) x l
+          | ICR_after_touch => List.rev_append acc (insert_interval_begin y (N.succ c) l)
+          | _ => List.rev_append ((y, c)::acc) l
        end
    end.
@@ -309,8 +427,8 @@

Library MSetIntervals

-add_list simple extension to add many elements, which then allows to - define from_elements. +add_list is a simple extension to add many elements. + This is used to define the function from_elements.
  Definition add_list (l : list elt) (s : t) : t :=
@@ -338,7 +456,7 @@

Library MSetIntervals

removing needs to be done with code extraction in mind again.
-  Definition removeZ_aux_insert_guarded (x : Z) (c : N) s :=
+  Definition insert_intervalZ_guarded (x : Z) (c : N) s :=
     if (N.eqb c 0) then s else (x, c) :: s.

@@ -348,9 +466,9 @@

Library MSetIntervals

    | (y, c) :: l =>
        if (Z.ltb x y) then List.rev_append acc s else
        if (Z.ltb x (y+Z.of_N c)) then (
-           List.rev_append (removeZ_aux_insert_guarded (Z.succ x)
+           List.rev_append (insert_intervalZ_guarded (Z.succ x)
              (Z.to_N ((y+Z.of_N c)- (Z.succ x)))
-             (removeZ_aux_insert_guarded y (Z.to_N (x-y)) acc)) l
+             (insert_intervalZ_guarded y (Z.to_N (x-y)) acc)) l
        ) else removeZ_aux ((y,c)::acc) x l
    end.
@@ -366,52 +484,281 @@

Library MSetIntervals

-all other operations are defined trivially (if not always efficiently) - in terms of already defined ones. In the future it might be worth implementing - some of them more efficiently. +union +
+
+  Fixpoint union_aux (s1 : t) :=
+    fix aux (s2 : t) (acc : list (Z * N)) :=
+    match (s1, s2) with
+    | (nil, _) => List.rev_append acc s2
+    | (_, nil) => List.rev_append acc s1
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before => union_aux l1 s2 ((y1, c1)::acc)
+          | ICR_before_touch =>
+              union_aux l1 (
+               insert_interval_begin y1 ((c1+c2)%N) l2) acc
+          | ICR_after => aux l2 ((y2, c2)::acc)
+          | ICR_after_touch => union_aux l1 (
+              insert_interval_begin y2 ((c1+c2)%N) l2) acc
+          | ICR_overlap_before =>
+              union_aux l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2) acc
+          | ICR_overlap_after =>
+              union_aux l1 (insert_interval_begin y2 (merge_interval_size y2 c2 y1 c1) l2) acc
+          | ICR_equal => union_aux l1 s2 acc
+          | ICR_subsume_1 => union_aux l1 s2 acc
+          | ICR_subsume_2 => aux l2 acc
+        end
+    end.
+ +
+  Definition union s1 s2 := union_aux s1 s2 nil.
+ +
+
+ +
+diff
-  Definition union (s1 s2 : t) :=
-    add_list (elements s1) s2.

-  Definition filter (f : elt -> bool) (s : t) : t :=
-    from_elements (List.filter f (elements s)).
+  Fixpoint diff_aux (y2 : Z) (c2 : N) (acc : list (Z * N)) (s : t) : (list (Z * N) * t) :=
+    match s with
+    | nil => (acc, nil)
+    | ((y1, c1) :: l1) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before => diff_aux y2 c2 ((y1, c1)::acc) l1
+          | ICR_before_touch => diff_aux y2 c2 ((y1, c1)::acc) l1
+          | ICR_after => (acc, s)
+          | ICR_after_touch => (acc, s)
+          | ICR_overlap_before => diff_aux y2 c2 ((y1, Z.to_N (y2 - y1))::acc) l1
+          | ICR_overlap_after => (acc, (y2+Z.of_N c2, Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) :: l1)
+          | ICR_equal => (acc, l1)
+          | ICR_subsume_1 => diff_aux y2 c2 acc l1
+          | ICR_subsume_2 => ((insert_intervalZ_guarded y1
+                (Z.to_N (y2 - y1)) acc),
+              insert_intervalZ_guarded (y2+Z.of_N c2) (Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) l1)
+        end
+    end.

-  Definition inter (s1 s2 : t) : t :=
-    filter (fun x => mem x s2) s1.
+  Fixpoint diff_aux2 (acc : list (Z * N)) (s1 s2 : t) : (list (Z * N)) :=
+    match (s1, s2) with
+    | (nil, _) => rev_append acc s1
+    | (_, nil) => rev_append acc s1
+    | (_, (y2, c2) :: l2) =>
+      match diff_aux y2 c2 acc s1 with
+        (acc´, s1´) => diff_aux2 acc´ s1´ l2
+      end
+    end.

-  Definition diff (s1 s2 : t) : t :=
-    remove_list (elements s2) s1.
+  Definition diff s1 s2 := diff_aux2 nil s1 s2.

-  Definition subset s s´ :=
-    List.forallb (fun x => mem x s´) (elements s).
+
+ +
+subset +
+
+  Fixpoint subset (s1 : t) :=
+    fix aux (s2 : t) :=
+    match (s1, s2) with
+    | (nil, _) => true
+    | (_ :: _, nil) => false
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before => false
+          | ICR_before_touch => false
+          | ICR_after => aux l2
+          | ICR_after_touch => false
+          | ICR_overlap_before => false
+          | ICR_overlap_after => false
+          | ICR_equal => subset l1 l2
+          | ICR_subsume_1 => subset l1 s2
+          | ICR_subsume_2 => false
+        end
+    end.

+
+ +
+equal +
+
  Fixpoint equal (s s´ : t) : bool := match s, s´ with
    | nil, nil => true
    | ((x,cx)::xs), ((y,cy)::ys) => andb (Z.eqb x y) (andb (N.eqb cx cy) (equal xs ys))
    | _, _ => false
  end.
+
+
+ +
+inter +
+
+  Fixpoint inter_aux (y2 : Z) (c2 : N) (acc : list (Z * N)) (s : t) : (list (Z * N) * t) :=
+    match s with
+    | nil => (acc, nil)
+    | ((y1, c1) :: l1) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before => inter_aux y2 c2 acc l1
+          | ICR_before_touch => inter_aux y2 c2 acc l1
+          | ICR_after => (acc, s)
+          | ICR_after_touch => (acc, s)
+          | ICR_overlap_before => inter_aux y2 c2 ((y2, Z.to_N (y1 + Z.of_N c1 - y2))::acc) l1
+          | ICR_overlap_after => ((y1, Z.to_N (y2 + Z.of_N c2 - y1))::acc, s)
+          | ICR_equal => ((y1,c1)::acc, l1)
+          | ICR_subsume_1 => inter_aux y2 c2 ((y1, c1)::acc) l1
+          | ICR_subsume_2 => ((y2, c2)::acc, s)
+        end
+    end.
+ +
+  Fixpoint inter_aux2 (acc : list (Z * N)) (s1 s2 : t) : (list (Z * N)) :=
+    match (s1, s2) with
+    | (nil, _) => List.rev´ acc
+    | (_, nil) => List.rev´ acc
+    | (_, (y2, c2) :: l2) =>
+      match inter_aux y2 c2 acc s1 with
+        (acc´, s1´) => inter_aux2 acc´ s1´ l2
+      end
+    end.
+ +
+  Definition inter s1 s2 := inter_aux2 nil s1 s2.
+ +
+
+ +
+Partition and filter +
+
+ +
+  Definition partitionZ_fold_insert
+             (cur : option (Z * N)) (x : Z) :=
+    match cur with
+       None => (x, 1%N)
+     | Some (y, c) => (y, N.succ c)
+    end.
+ +
+  Definition partitionZ_fold_skip (acc : list (Z * N))
+             (cur : option (Z * N)) : (list (Z * N)) :=
+    match cur with
+       None => acc
+     | Some yc => yc::acc
+    end.
+ +
+  Definition partitionZ_fold_fun f st (x : Z) :=
+    match st with ((acc_t, c_t), (acc_f, c_f)) =>
+      if (f x) then
+        ((acc_t, Some (partitionZ_fold_insert c_t x)),
+         (partitionZ_fold_skip acc_f c_f, None))
+      else
+        ((partitionZ_fold_skip acc_t c_t, None),
+         (acc_f, Some (partitionZ_fold_insert c_f x)))
+    end.
+ +
+  Definition partitionZ_single_aux f st (x : Z) (c : N) :=
+    snd (fold_elementsZ_single (fun st x => Some (partitionZ_fold_fun f st x)) st x c).
+ +
+  Definition partitionZ_single f acc_t acc_f x c :=
+    match partitionZ_single_aux f ((acc_t, None), (acc_f, None)) x c with
+    | ((acc_t, c_t), (acc_f, c_f)) =>
+        (partitionZ_fold_skip acc_t c_t,
+         partitionZ_fold_skip acc_f c_f)
+    end.
+ +
+  Fixpoint partitionZ_aux acc_t acc_f f s :=
+    match s with
+    | nil => (List.rev acc_t, List.rev acc_f)
+    | (y, c) :: s´ =>
+      match partitionZ_single f acc_t acc_f y c with
+      | (acc_t´, acc_f´) => partitionZ_aux acc_t´ acc_f´ f s´
+      end
+    end.
+ +
+  Definition partitionZ := partitionZ_aux nil nil.
+ +
+  Definition partition (f : elt -> bool) : t -> (t * t) :=
+    partitionZ (fun z => f (Enc.decode z)).
+ +
+  Definition filterZ_fold_fun f st (x : Z) :=
+    match st with (acc_t, c_t) =>
+      if (f x) then
+        (acc_t, Some (partitionZ_fold_insert c_t x))
+      else
+        (partitionZ_fold_skip acc_t c_t, None)
+    end.
+ +
+  Definition filterZ_single_aux f st (x : Z) (c : N) :=
+    snd (fold_elementsZ_single (fun st x => Some (filterZ_fold_fun f st x)) st x c).
+ +
+  Definition filterZ_single f acc x c :=
+    match filterZ_single_aux f (acc, None) x c with
+    | (acc, c) =>
+        (partitionZ_fold_skip acc c)
+    end.
+ +
+  Fixpoint filterZ_aux acc f s :=
+    match s with
+    | nil => (List.rev acc)
+    | (y, c) :: s´ =>
+      filterZ_aux (filterZ_single f acc y c) f s´
+    end.
+ +
+  Definition filterZ := filterZ_aux nil.
+ +
+  Definition filter (f : elt -> bool) : t -> t :=
+    filterZ (fun z => f (Enc.decode z)).
+ +
+ +
+
+ +
+Simple wrappers +
+
+
  Definition fold {B : Type} (f : elt -> B -> B) (s : t) (i : B) : B :=
-    List.fold_left (flip f) (elements s) i.
+    snd (fold_elementsZ (fun b z => Some (f (Enc.decode z) b)) i s).

  Definition for_all (f : elt -> bool) (s : t) : bool :=
-    List.forallb f (elements s).
+    snd (fold_elementsZ (fun b z =>
+      if b then
+        Some (f (Enc.decode z))
+      else None) true s).

  Definition exists_ (f : elt -> bool) (s : t) : bool :=
-    List.existsb f (elements s).
- -
-  Definition partition (f : elt -> bool) (s : t) : t * t :=
-    (filter f s, filter (fun x => negb (f x)) s).
+    snd (fold_elementsZ (fun b z =>
+      if b then
+        None
+      else Some (f (Enc.decode z))) false s).

  Fixpoint cardinalN c (s : t) : N := match s with
@@ -423,17 +770,35 @@

Library MSetIntervals

  Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s).

-  Definition chooseZ (s : t) : option Z :=
-    match List.rev´ (elementsZ s) with
+  Definition min_eltZ (s : t) : option Z :=
+    match s with
+    | nil => None
+    | (x, _) :: _ => Some x
+    end.
+ +
+  Definition min_elt (s : t) : option elt :=
+    match (min_eltZ s) with
+    | None => None
+    | Some x => Some (Enc.decode x)
+    end.
+ +
+  Definition choose := min_elt.
+ +
+  Fixpoint max_eltZ (s : t) : option Z :=
+    match s with
    | nil => None
-    | x :: _ => Some x
+    | (x, c) :: nil => Some (Z.pred (x + Z.of_N c))
+    | (x, _) :: s´ => max_eltZ s´
    end.

-  Definition choose (s : t) : option elt :=
-    match elements s with
-      | nil => None
-      | e :: _ => Some e
+  Definition max_elt (s : t) : option elt :=
+    match (max_eltZ s) with
+    | None => None
+    | Some x => Some (Enc.decode x)
    end.

@@ -466,27 +831,26 @@

Library MSetIntervals

+  Definition is_encoded_elems_list (l : list Z) : Prop :=
+    (forall x, List.In x l -> exists e, Enc.encode e = x).

-  Definition inf (x:Z) (l: t) :=
-   match l with
-   | nil => true
-   | (y,_)::_ => Z.ltb x y
-   end.
+  Definition interval_list_elements_greater (x : Z) (l : t) : bool :=
+    match l with
+      | nil => true
+      | (y, _)::_ => Z.ltb x y
+    end.

-  Fixpoint isok (l : t) :=
+  Fixpoint interval_list_invariant (l : t) :=
   match l with
   | nil => true
-   | (x, c) ::l => inf (x+(Z.of_N c)) l && negb (N.eqb c 0) && isok l
+   | (x, c) :: l´ =>
+       interval_list_elements_greater (x + (Z.of_N c)) l´ && negb (N.eqb c 0) && interval_list_invariant l´
   end.

-  Definition is_encoded_elems_list (l : list Z) : Prop :=
-    (forall x, List.In x l -> exists e, Enc.encode e = x).
- -
-  Definition IsOk s := (isok s = true /\ is_encoded_elems_list (elementsZ s)).
+  Definition IsOk s := (interval_list_invariant s = true /\ is_encoded_elems_list (elementsZ s)).

@@ -571,27 +935,73 @@

Library MSetIntervals

  Lemma In_elementsZ_single1 : forall y x,
    List.In y (elementsZ_single x (1%N)) <->
    (x = y).
+ +
  Lemma length_elementsZ_single : forall cx x,
    length (elementsZ_single x cx) = N.to_nat cx.
-  Lemma elementsZ_aux´´_irrel : forall c acc x H1 H2,
-      elementsZ_aux´´ acc x c H1 = elementsZ_aux´´ acc x c H2.
-  Lemma elementsZ_aux´_pos : forall s x p,
-      elementsZ_aux´ s x (N.pos p) = elementsZ_aux´ (x::s) (Z.succ x) (Pos.pred_N p).
+  Lemma fold_elementsZ_aux_irrel {A} :
+    forall f c (acc : A) x H1 H2,
+      fold_elementsZ_aux f acc x c H1 =
+      fold_elementsZ_aux f acc x c H2.
+  Lemma fold_elementsZ_single_pos {A} : forall f (acc : A) x p,
+    fold_elementsZ_single f acc x (N.pos p) =
+    match f acc x with
+    | Some acc´ =>
+        fold_elementsZ_single f acc´ (Z.succ x)
+         (N.pred (N.pos p))
+    | None => (true, acc)
+    end.

-  Lemma elementsZ_aux´_zero : forall s x,
-      elementsZ_aux´ s x (0%N) = s.
+  Lemma fold_elementsZ_single_zero {A} : forall f (acc : A) x,
+      fold_elementsZ_single f acc x (0%N) = (false, acc).

-  Lemma elementsZ_aux´_succ : forall s x c,
-      elementsZ_aux´ s x (N.succ c) = elementsZ_aux´ (x::s) (Z.succ x) c.
+  Lemma fold_elementsZ_single_succ {A} : forall f (acc : A) x c,
+    fold_elementsZ_single f acc x (N.succ c) =
+    match f acc x with
+      | Some acc´ =>
+          fold_elementsZ_single f acc´ (Z.succ x) c
+      | None => (true, acc)
+    end.
+  Fixpoint fold_opt {A B} f (acc : A) (bs : list B) : (bool * A) :=
+    match bs with
+      | nil => (false, acc)
+      | (b :: bs´) =>
+        match f acc b with
+        | Some acc´ => fold_opt f acc´ bs´
+        | None => (true, acc)
+        end
+    end.

-  Lemma elementsZ_single_intro : forall c s x,
-     elementsZ_aux´ s x c =
-     (List.rev (elementsZ_single x c)) ++ s.
-  Lemma elementsZ_aux_alt_def : forall s acc,
-    elementsZ_aux acc s = elementsZ s ++ acc.
+  Lemma fold_opt_list_cons : forall {A} (bs : list A) (acc : list A),
+    fold_opt (fun l x => Some (x :: l)) acc bs =
+    (false, List.rev bs ++ acc).
+  Lemma fold_opt_app {A B} : forall f (acc : A) (l1 l2 : list B),
+    fold_opt f acc (l1 ++ l2) =
+    (let (ab, acc´) := fold_opt f acc l1 in
+     if ab then (true, acc´) else fold_opt f acc´ l2).
+  Lemma fold_elementsZ_single_alt_def {A} : forall f c (acc : A) x,
+     fold_elementsZ_single f acc x c =
+     fold_opt f acc (elementsZ_single x c).
+  Lemma fold_elementsZ_nil {A} : forall f (acc : A),
+     fold_elementsZ f acc nil = (false, acc).
+ +
+  Lemma fold_elementsZ_cons {A} : forall f (acc : A) y c s,
+    fold_elementsZ f acc ((y, c)::s) =
+    (let (ab, acc´) := fold_elementsZ_single f acc y c in
+     if ab then (true, acc´) else fold_elementsZ f acc´ s).
+ +
+  Lemma fold_elementsZ_alt_def_aux : forall (s : t) base,
+    (snd (fold_elementsZ
+      (fun (l : list Z) (x : Z) => Some (x :: l)) base s)) =
+    elementsZ s ++ base.
+  Lemma fold_elementsZ_alt_def {A} : forall f s (acc : A),
+     fold_elementsZ f acc s =
+     fold_opt f acc (rev (elementsZ s)).
  Lemma elementsZ_cons : forall x c s, elementsZ (((x, c) :: s) : t) =
     ((elementsZ s) ++ (List.rev (elementsZ_single x c))).
@@ -600,115 +1010,118 @@

Library MSetIntervals

     ((elements_single x c) ++ elements s).

-  Lemma In_elementsZ_single_hd : forall (c : N) x, (c <> 0)%N -> List.In x (elementsZ_single x c).
+  Lemma elementsZ_app : forall (s1 s2 : t), elementsZ (s1 ++ s2) =
+     ((elementsZ s2) ++ (elementsZ s1)).
+ +
+  Lemma InZ_nil : forall y, InZ y nil <-> False.
+ +
+  Lemma InZ_cons : forall y x c s, InZ y (((x, c) :: s) : t) <->
+     List.In y (elementsZ_single x c) \/ InZ y s.
+ +
+  Lemma InZ_app : forall s1 s2 y,
+     InZ y (s1 ++ s2) <-> InZ y s1 \/ InZ y s2.
+ +
+  Lemma InZ_rev : forall s y,
+     InZ y (List.rev s) <-> InZ y s.
+ +
+  Lemma In_elementsZ_single_dec : forall y x c,
+    {List.In y (elementsZ_single x c)} +
+    {~ List.In y (elementsZ_single x c)}.
+ +
+  Lemma InZ_dec : forall y s,
+     {InZ y s} + {~InZ y s}.
+   Lemma In_elementsZ_single_hd : forall (c : N) x, (c <> 0)%N -> List.In x (elementsZ_single x c).
-

Alternative definition of addZ

+

comparing intervals

-
- - addZ is defined with efficient execution in mind. - We derive first an alternative definition that demonstrates - the intention better and is better suited for proofs.
-  Lemma addZ_ind :
-    forall (P : Z -> list (Z * N) -> Prop),
-       
-       (forall (x : Z), P x nil) ->
-
-       
-       (forall (x : Z) (l : list (Z * N)) (c : N),
-        P x ((x + 1, c) :: l)) ->
-
-       
-       (forall (x : Z) (l : list (Z * N)) (y : Z) (c : N),
-        (x + 1 ?= y) = Lt ->
-        P x ((y, c) :: l)) ->
-
-       
-       (forall (y : Z) (c : N),
-        ((y + Z.of_N c) + 1 ?= y) = Gt ->
-        P (y + Z.of_N c) ((y, c) :: nil)) ->
-
-       
-       (forall (l : list (Z * N)) (y : Z) (c c´ : N),
-        ((y + Z.of_N c) + 1 ?= y) = Gt ->
-        (P (y+Z.of_N c) l) ->
-        P (y+Z.of_N c) ((y, c) :: (((y+Z.of_N c) + 1, c´) :: l))) ->
-
-       
-       (forall (l : list (Z * N)) (y : Z) (c : N) (z : Z) (c´ : N),
-        ((y + Z.of_N c) + 1 ?= y) = Gt ->
-        (z =? (y+Z.of_N c) + 1) = false ->
-        (P (y+Z.of_N c) ((y, c) :: (z, c´) :: l))) ->
-
-
-       
-       (forall (x : Z) (l : list (Z * N)) (y : Z) (c : N),
-        (x + 1 ?= y) = Gt ->
-        (x ?= y + Z.of_N c) = Lt ->
-        P x ((y, c) :: l)) ->
-
-       
-       (forall (x : Z)(l : list (Z * N)) (y : Z) (c : N),
-        (x + 1 ?= y) = Gt ->
-        (x ?= y + (Z.of_N c)) = Gt ->
-        (P x l) ->
-        P x ((y, c) :: l)) ->
-
-
-       forall (x : Z) (s : list (Z * N)),
-       P x s.
-  Lemma addZ_aux_alt_def : forall x s acc,
-    addZ_aux acc x s = (List.rev acc) ++ addZ x s.
-  Lemma addZ_alt_def : forall x s,
-    addZ x s =
-    match s with
-    | nil => (x, 1%N)::nil
-    | (y, c) :: l =>
-        match (Z.compare (x+1) y) with
-        | Lt => (x, 1%N)::s
-        | Eq => (x, (c+1)%N)::l
-        | Gt => match (Z.compare x (y+Z.of_N c)) with
-                 | Lt => s
-                 | Gt => (y,c) :: addZ x l
-                 | Eq => match l with
-                           | nil => (y, (c+1)%N)::nil
-                           | (z, c´) :: l´ => if (Z.eqb z (x + 1)) then
-                                (y, (c + c´ + 1)%N) :: l´
-                             else
-                                (y,(c+1)%N) :: (z, c´) :: l´
-                         end
-                end
-        end
+  Ltac Z_named_compare_cases H := match goal with
+    | [ |- context [Z.compare ?z1 ?z2] ] =>
+      case_eq (Z.compare z1 z2); [move => /Z.compare_eq_iff | move => /Z.compare_lt_iff | move => /Z.compare_gt_iff]; move => H //
+  end.
+  Ltac Z_compare_cases := let H := fresh "H" in Z_named_compare_cases H.
+  Lemma interval_compare_elim : forall (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
+    match (interval_compare (y1, c1) (y2, c2)) with
+      | ICR_before => (y1 + Z.of_N c1) < y2
+      | ICR_before_touch => (y1 + Z.of_N c1) = y2
+      | ICR_after => (y2 + Z.of_N c2) < y1
+      | ICR_after_touch => (y2 + Z.of_N c2) = y1
+      | ICR_equal => (y1 = y2) /\ (c1 = c2)
+      | ICR_overlap_before => (y1 < y2) /\ (y2 < y1 + Z.of_N c1) /\ (y1 + Z.of_N c1 < y2 + Z.of_N c2)
+      | ICR_overlap_after => (y2 < y1) /\ (y1 < y2 + Z.of_N c2) /\ (y2 + Z.of_N c2 < y1 + Z.of_N c1)
+      | ICR_subsume_1 => (y2 <= y1) /\ (y1 + Z.of_N c1 <= y2 + Z.of_N c2) /\ (y2 < y1 \/ y1 + Z.of_N c1 < y2 + Z.of_N c2)
+      | ICR_subsume_2 => (y1 <= y2) /\ (y2 + Z.of_N c2 <= y1 + Z.of_N c1) /\ (y1 < y2 \/ y2 + Z.of_N c2 < y1 + Z.of_N c1)
+    end.
+ +
+  Lemma interval_compare_swap : forall (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
+    (c1 <> 0%N) \/ (c2 <> 0%N) ->
+    interval_compare (y2, c2) (y1, c1) =
+    match (interval_compare (y1, c1) (y2, c2)) with
+      | ICR_before => ICR_after
+      | ICR_before_touch => ICR_after_touch
+      | ICR_after => ICR_before
+      | ICR_after_touch => ICR_before_touch
+      | ICR_equal => ICR_equal
+      | ICR_overlap_before => ICR_overlap_after
+      | ICR_overlap_after => ICR_overlap_before
+      | ICR_subsume_1 => ICR_subsume_2
+      | ICR_subsume_2 => ICR_subsume_1
+    end.
+  Lemma interval_1_compare_alt_def : forall (y : Z) (i : (Z * N)),
+    interval_1_compare y i = match (interval_compare (y, (1%N)) i) with
+      | ICR_equal => ICR_subsume_1
+      | ICR_subsume_1 => ICR_subsume_1
+      | ICR_subsume_2 => ICR_subsume_1
+      | r => r
+    end.
+  Lemma interval_1_compare_elim : forall (y1 : Z) (y2 : Z) (c2 : N),
+    match (interval_1_compare y1 (y2, c2)) with
+      | ICR_before => Z.succ y1 < y2
+      | ICR_before_touch => y2 = Z.succ y1
+      | ICR_after => (y2 + Z.of_N c2) < y1
+      | ICR_after_touch => (y2 + Z.of_N c2) = y1
+      | ICR_equal => False
+      | ICR_overlap_before => False
+      | ICR_overlap_after => False
+      | ICR_subsume_1 => (c2 = 0%N) \/ ((y2 <= y1) /\ (y1 < y2 + Z.of_N c2))
+      | ICR_subsume_2 => False
    end.
-

Alternative definition of removeZ

+

Alternative definition of addZ

-
- - removeZ is defined with efficient execution in mind. - We derive first an alternative definition that demonstrates - the intention better and is better suited for proofs.
-  Lemma removeZ_aux_alt_def : forall s x acc,
-    removeZ_aux acc x s = (List.rev acc) ++ removeZ x s.
-  Lemma removeZ_alt_def : forall x s,
-    removeZ x s =
+  Lemma addZ_aux_alt_def : forall x s acc,
+    addZ_aux acc x s = (List.rev acc) ++ addZ x s.
+   Lemma addZ_alt_def : forall x s,
+    addZ x s =
    match s with
-    | nil => nil
+    | nil => (x, (1%N))::nil
    | (y, c) :: l =>
-        if (Z.ltb x y) then s else
-        if (Z.ltb x (y+Z.of_N c)) then (
-           (removeZ_aux_insert_guarded y (Z.to_N (x-y))
-             (removeZ_aux_insert_guarded (Z.succ x) (Z.to_N ((y+Z.of_N c)- (Z.succ x))) l))
-        ) else (y, c) :: removeZ x l
+        match (interval_1_compare x (y,c)) with
+          | ICR_before => (x, (1%N))::s
+          | ICR_before_touch => (x, N.succ c)::l
+          | ICR_after => (y,c)::(addZ x l)
+          | ICR_after_touch => insert_interval_begin y (N.succ c) l
+          | _ => (y, c)::l
+        end
    end.
+ +
+ +
@@ -716,8 +1129,20 @@

Library MSetIntervals

-  Lemma inf_impl : forall x y s,
-    (y <= x) -> inf x s = true -> inf y s = true.
+ +
+  Lemma interval_list_elements_greater_cons : forall z x c s,
+    interval_list_elements_greater z ((x, c) :: s) = true <->
+    (z < x).
+ +
+  Lemma interval_list_elements_greater_impl : forall x y s,
+    (y <= x) ->
+    interval_list_elements_greater x s = true ->
+    interval_list_elements_greater y s = true.
+ +
+  Lemma interval_list_invariant_nil : interval_list_invariant nil = true.

  Lemma Ok_nil : Ok nil <-> True.
@@ -733,23 +1158,73 @@

Library MSetIntervals

    is_encoded_elems_list l.

-  Lemma isok_cons : forall y c s´, isok ((y, c) :: s´) = true <->
-    (inf (y+Z.of_N c) s´ = true /\ ((c <> 0)%N) /\ isok s´ = true).
+  Lemma interval_list_invariant_cons : forall y c s´,
+    interval_list_invariant ((y, c) :: s´) = true <->
+    (interval_list_elements_greater (y+Z.of_N c) s´ = true /\
+      ((c <> 0)%N) /\ interval_list_invariant s´ = true).
+  Lemma interval_list_invariant_sing : forall x c,
+    interval_list_invariant ((x, c)::nil) = true <-> (c <> 0)%N.
+ +
  Lemma Ok_cons : forall y c s´, Ok ((y, c) :: s´) <->
-    (inf (y+Z.of_N c) s´ = true /\ ((c <> 0)%N) /\
+    (interval_list_elements_greater (y+Z.of_N c) s´ = true /\ ((c <> 0)%N) /\
     is_encoded_elems_list (elementsZ_single y c) /\ Ok s´).

  Lemma Nin_elements_greater : forall s y,
-     inf y s = true ->
-     isok s = true ->
-     forall x, x <= y ->
-     ~(InZ x s).
-  Lemma isok_inf_nin :
+     interval_list_elements_greater y s = true ->
+     interval_list_invariant s = true ->
+     forall x, x <= y -> ~(InZ x s).
+  Lemma Nin_elements_greater_equal :
     forall x s,
-       isok s = true ->
-       inf x s = true ->
+       interval_list_elements_greater x s = true ->
+       interval_list_invariant s = true ->
       ~ (InZ x s).
+ +
+  Lemma interval_list_elements_greater_alt_def : forall s y,
+     interval_list_invariant s = true ->
+
+     (interval_list_elements_greater y s = true <->
+      (forall x, x <= y -> ~(InZ x s))).
+  Lemma interval_list_elements_greater_alt2_def : forall s y,
+     interval_list_invariant s = true ->
+
+     (interval_list_elements_greater y s = true <->
+      (forall x, InZ x s -> y < x)).
+ +
+  Lemma interval_list_elements_greater_intro : forall s y,
+     interval_list_invariant s = true ->
+     (forall x, InZ x s -> y < x) ->
+     interval_list_elements_greater y s = true.
+ +
+  Lemma interval_list_elements_greater_app_elim_1 : forall s1 s2 y,
+    interval_list_elements_greater y (s1 ++ s2) = true ->
+    interval_list_elements_greater y s1 = true.
+ +
+  Lemma interval_list_invariant_app_intro : forall s1 s2,
+      interval_list_invariant s1 = true ->
+      interval_list_invariant s2 = true ->
+      (forall (x1 x2 : Z), InZ x1 s1 -> InZ x2 s2 -> Z.succ x1 < x2) ->
+      interval_list_invariant (s1 ++ s2) = true.
+  Lemma interval_list_invariant_app_elim : forall s1 s2,
+      interval_list_invariant (s1 ++ s2) = true ->
+      interval_list_invariant s1 = true /\
+      interval_list_invariant s2 = true /\
+      (forall (x1 x2 : Z), InZ x1 s1 -> InZ x2 s2 -> Z.succ x1 < x2).
+  Lemma interval_list_invariant_app_iff : forall s1 s2,
+      interval_list_invariant (s1 ++ s2) = true <->
+      (interval_list_invariant s1 = true /\
+      interval_list_invariant s2 = true /\
+      (forall (x1 x2 : Z), InZ x1 s1 -> InZ x2 s2 -> Z.succ x1 < x2)).
+  Lemma interval_list_invariant_snoc_intro : forall s1 y2 c2,
+      interval_list_invariant s1 = true ->
+      (c2 <> 0)%N ->
+      (forall x, InZ x s1 -> Z.succ x < y2) ->
+      interval_list_invariant (s1 ++ ((y2, c2)::nil)) = true.
@@ -757,11 +1232,21 @@

Library MSetIntervals

+  Lemma encode_decode_eq : forall x s, Ok s -> InZ x s ->
+    (Enc.encode (Enc.decode x) = x).
+ +
  Lemma In_alt_def : forall x s, Ok s ->
    (In x s <-> List.In x (elements s)).
  Lemma In_InZ : forall x s, Ok s ->
    (In x s <-> InZ (Enc.encode x) s).
+
+  Lemma InZ_In : forall x s, Ok s ->
+    (InZ x s -> In (Enc.decode x) s).
+ +
+
@@ -773,68 +1258,116 @@

Library MSetIntervals


  Lemma memZ_spec :
-   forall (s : t) (x : Z) (Hs : Ok s), memZ x s = true <-> InZ x s.
+    forall (s : t) (x : Z) (Hs : Ok s), memZ x s = true <-> InZ x s.
  Lemma mem_spec :
   forall (s : t) (x : elt) (Hs : Ok s), mem x s = true <-> In x s.
+
+  Lemma merge_interval_size_neq_0 : forall x1 c1 x2 c2,
+     (c1 <> 0%N) ->
+     (merge_interval_size x1 c1 x2 c2 <> 0)%N.
+
-

add specification

+

insert if length not 0

-  Lemma addZ_spec :
-   forall (s : t) (x y : Z) (Hs : Ok s),
-    InZ y (addZ x s) <-> Z.eq y x \/ InZ y s.
-  Lemma addZ_isok : forall s x, isok s = true -> isok (addZ x s) = true.
-   Global Instance add_ok s x : forall `(Ok s), Ok (add x s).
-   Lemma add_spec :
-   forall (s : t) (x y : elt) (Hs : Ok s),
-     In y (add x s) <-> Enc.E.eq y x \/ In y s.
+ +
+  Lemma interval_list_invariant_insert_intervalZ_guarded : forall x c s,
+    interval_list_invariant s = true ->
+    interval_list_elements_greater (x + Z.of_N c) s = true ->
+    interval_list_invariant (insert_intervalZ_guarded x c s) = true.
+ +
+  Lemma interval_list_elements_greater_insert_intervalZ_guarded : forall x c y s,
+    interval_list_elements_greater y (insert_intervalZ_guarded x c s) = true <->
+    (if (c =? 0)%N then (interval_list_elements_greater y s = true) else (y < x)).
+ +
+  Lemma insert_intervalZ_guarded_app : forall x c s1 s2,
+    (insert_intervalZ_guarded x c s1) ++ s2 =
+    insert_intervalZ_guarded x c (s1 ++ s2).
+ +
+  Lemma insert_intervalZ_guarded_rev_nil_app : forall x c s,
+    rev (insert_intervalZ_guarded x c nil) ++ s =
+    insert_intervalZ_guarded x c s.
+ +
+Lemma elementsZ_insert_intervalZ_guarded : forall x c s,
+    elementsZ (insert_intervalZ_guarded x c s) = elementsZ ((x, c) :: s).
+ +
+  Lemma InZ_insert_intervalZ_guarded : forall y x c s,
+    InZ y (insert_intervalZ_guarded x c s) = InZ y ((x, c) :: s).

-

remove specification

+

Merging intervals


-  Lemma isok_removeZ_aux_insert_guarded : forall x c s,
-    isok s = true -> inf (x + Z.of_N c) s = true ->
-    isok (removeZ_aux_insert_guarded x c s) = true.
+  Lemma merge_interval_size_add : forall x c1 c2,
+     (merge_interval_size x c1 (x + Z.of_N c1) c2 = (c1 + c2))%N.

-  Lemma inf_removeZ_aux_insert_guarded : forall x c y s,
-    inf y (removeZ_aux_insert_guarded x c s) = true <->
-    (if (c =? 0)%N then (inf y s = true) else (y < x)).
+  Lemma merge_interval_size_eq_max : forall y1 c1 y2 c2,
+     y1 <= y2 + Z.of_N c2 ->
+     y1 + Z.of_N (merge_interval_size y1 c1 y2 c2) =
+     Z.max (y1 + Z.of_N c1) (y2 + Z.of_N c2).

-  Lemma removeZ_counter_pos_aux : forall y c x,
-     x < y + Z.of_N c ->
-     0 <= y + Z.of_N c - Z.succ x.
+  Lemma merge_interval_size_invariant : forall y1 c1 y2 c2 z s,
+    interval_list_invariant s = true ->
+    y1 + Z.of_N c1 <= y2 + Z.of_N c2 ->
+    y2 + Z.of_N c2 <= z ->
+    interval_list_elements_greater z s = true ->
+    (c1 <> 0)%N ->
+    interval_list_invariant ((y1, merge_interval_size y1 c1 y2 c2) :: s) =
+   true.
+  Lemma In_merge_interval : forall x1 c1 x2 c2 y,
+    x1 <= x2 ->
+    x2 <= x1 + Z.of_N c1 -> (
+    List.In y (elementsZ_single x1 (merge_interval_size x1 c1 x2 c2)) <->
+    List.In y (elementsZ_single x1 c1) \/ List.In y (elementsZ_single x2 c2)).
+  Lemma insert_interval_begin_spec : forall y s x c,
+     interval_list_invariant s = true ->
+     interval_list_elements_greater x s = true ->
+     (c <> 0)%N ->
+      (
+     interval_list_invariant (insert_interval_begin x c s) = true /\
+     (InZ y (insert_interval_begin x c s) <->
+     (List.In y (elementsZ_single x c) \/ InZ y s))).
+
-
-  Lemma removeZ_isok : forall s x, isok s = true -> isok (removeZ x s) = true.
-  Lemma elementsZ_removeZ_aux_insert_guarded : forall x c s,
-    elementsZ (removeZ_aux_insert_guarded x c s) = elementsZ ((x, c) :: s).
+
+

add specification

-
-  Lemma removeZ_spec :
-   forall (s : t) (x y : Z) (Hs : isok s = true),
-    InZ y (removeZ x s) <-> InZ y s /\ ~Z.eq y x.
-  Global Instance remove_ok s x : forall `(Ok s), Ok (remove x s).
-  Lemma remove_spec :
+
+
+  Lemma addZ_InZ :
+   forall (s : t) (x y : Z),
+    interval_list_invariant s = true ->
+    (InZ y (addZ x s) <-> x = y \/ InZ y s).
+  Lemma addZ_invariant : forall s x,
+    interval_list_invariant s = true ->
+    interval_list_invariant (addZ x s) = true.
+   Global Instance add_ok s x : forall `(Ok s), Ok (add x s).
+   Lemma add_spec :
   forall (s : t) (x y : elt) (Hs : Ok s),
-    In y (remove x s) <-> In y s /\ ~Enc.E.eq y x.
+     In y (add x s) <-> Enc.E.eq y x \/ In y s.
-

empty specification

+

empty specification

@@ -850,7 +1383,7 @@

Library MSetIntervals

-

is_empty specification

+

is_empty specification

@@ -860,7 +1393,7 @@

Library MSetIntervals

-

singleton specification

+

singleton specification

@@ -871,7 +1404,7 @@

Library MSetIntervals

-

add_list specification

+

add_list specification

@@ -880,96 +1413,283 @@

Library MSetIntervals

     (In x (add_list l s) <-> (SetoidList.InA Enc.E.eq x l) \/ In x s).
-
-

remove_list specification

- -
-
-  Lemma remove_list_ok : forall l s, Ok s -> Ok (remove_list l s).
-  Lemma remove_list_spec : forall x l s, Ok s ->
-     (In x (remove_list l s) <-> ~(InA Enc.E.eq x l) /\ In x s).
-
-

union specification

-  Global Instance union_ok s s´ : forall `(Ok s, Ok s´), Ok (union s s´).
+  Lemma union_aux_flatten_alt_def : forall (s1 s2 : t) acc,
+    union_aux s1 s2 acc =
+    match (s1, s2) with
+    | (nil, _) => List.rev_append acc s2
+    | (_, nil) => List.rev_append acc s1
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before => union_aux l1 s2 ((y1, c1)::acc)
+          | ICR_before_touch =>
+              union_aux l1 (
+                insert_interval_begin y1 ((c1+c2)%N) l2) acc
+          | ICR_after => union_aux s1 l2 ((y2, c2)::acc)
+          | ICR_after_touch => union_aux l1 (
+              insert_interval_begin y2 ((c1+c2)%N) l2) acc
+          | ICR_overlap_before =>
+              union_aux l1 (
+                insert_interval_begin y1
+                  (merge_interval_size y1 c1 y2 c2) l2) acc
+          | ICR_overlap_after =>
+              union_aux l1 (
+                insert_interval_begin y2
+                  (merge_interval_size y2 c2 y1 c1) l2) acc
+          | ICR_equal => union_aux l1 s2 acc
+          | ICR_subsume_1 => union_aux l1 s2 acc
+          | ICR_subsume_2 => union_aux s1 l2 acc
+        end
+    end.

+  Lemma union_aux_alt_def : forall (s1 s2 : t) acc,
+    union_aux s1 s2 acc =
+    List.rev_append acc (union s1 s2).
+ +
+  Lemma union_alt_def : forall (s1 s2 : t),
+    union s1 s2 =
+    match (s1, s2) with
+    | (nil, _) => s2
+    | (_, nil) => s1
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before => (y1, c1) :: (union l1 s2)
+          | ICR_before_touch =>
+              union l1 (insert_interval_begin y1 ((c1+c2)%N) l2)
+          | ICR_after => (y2, c2) :: union s1 l2
+          | ICR_after_touch => union l1
+              (insert_interval_begin y2 ((c1+c2)%N) l2)
+          | ICR_overlap_before =>
+              union l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2)
+          | ICR_overlap_after =>
+              union l1 (insert_interval_begin y2 (merge_interval_size y2 c2 y1 c1) l2)
+          | ICR_equal => union l1 s2
+          | ICR_subsume_1 => union l1 s2
+          | ICR_subsume_2 => union s1 l2
+        end
+     end.
+ +
+  Lemma union_InZ :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    forall y, (InZ y (union s1 s2) <-> InZ y s1 \/ InZ y s2).
+  Lemma union_invariant :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    interval_list_invariant (union s1 s2) = true.
+  Global Instance union_ok s1 s2 : forall `(Ok s1, Ok s2), Ok (union s1 s2).
  Lemma union_spec :
   forall (s s´ : t) (x : elt) (Hs : Ok s) (Hs´ : Ok s´),
   In x (union s s´) <-> In x s \/ In x s´.
- -
-

filter specification

+

inter specification

- -
-  Global Instance filter_ok s f : forall `(Ok s), Ok (filter f s).
- -
-  Lemma filter_spec :
-   forall (s : t) (x : elt) (f : elt -> bool),
-   Proper (Enc.E.eq==>eq) f ->
-   (In x (filter f s) <-> In x s /\ f x = true).
+  Lemma inter_aux_alt_def :
+    forall (y2 : Z) (c2 : N) (s : t) acc,
+      inter_aux y2 c2 acc s = match inter_aux y2 c2 nil s with
+                               (acc´, s´) => (acc´ ++ acc, s´)
+                             end.
+  Lemma inter_aux_props :
+    forall (y2 : Z) (c2 : N) (s : t) acc,
+      interval_list_invariant (rev acc) = true ->
+      interval_list_invariant s = true ->
+      (forall x1 x2, InZ x1 acc -> InZ x2 s ->
+                     List.In x2 (elementsZ_single y2 c2) ->
+                     Z.succ x1 < x2) ->
+      (c2 <> 0%N) ->
+      match (inter_aux y2 c2 acc s) with (acc´, s´) =>
+        (forall y, (InZ y acc´ <->
+                   (InZ y acc \/ (InZ y s /\ (List.In y (elementsZ_single y2 c2)))))) /\
+        (forall y, InZ y s´ -> InZ y s) /\
+        (forall y, InZ y s -> y2 + Z.of_N c2 < y -> InZ y s´) /\
+        interval_list_invariant (rev acc´) = true /\
+        interval_list_invariant s´ = true
+      end.
+  Lemma inter_aux2_props :
+   forall (s2 s1 acc : t),
+    interval_list_invariant (rev acc) = true ->
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    (forall x1 x2, InZ x1 acc -> InZ x2 s1 -> InZ x2 s2 -> Z.succ x1 < x2) ->
+    ((forall y, (InZ y (inter_aux2 acc s1 s2) <->
+               (InZ y acc) \/ ((InZ y s1) /\ InZ y s2))) /\
+    (interval_list_invariant (inter_aux2 acc s1 s2) = true)).
+  Lemma inter_InZ :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    forall y, (InZ y (inter s1 s2) <-> InZ y s1 /\ InZ y s2).
+ +
+  Lemma inter_invariant :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    interval_list_invariant (inter s1 s2) = true.
+ +
+ +
+  Global Instance inter_ok s1 s2 : forall `(Ok s1, Ok s2), Ok (inter s1 s2).
+  Lemma inter_spec :
+   forall (s s´ : t) (x : elt) (Hs : Ok s) (Hs´ : Ok s´),
+   In x (inter s s´) <-> In x s /\ In x s´.
-

inter specification

+

diff specification

-  Global Instance inter_ok s s´ : forall `(Ok s, Ok s´), Ok (inter s s´).
- -
-  Lemma inter_spec :
+  Lemma diff_aux_alt_def :
+    forall (y2 : Z) (c2 : N) (s : t) acc,
+      diff_aux y2 c2 acc s = match diff_aux y2 c2 nil s with
+                               (acc´, s´) => (acc´ ++ acc, s´)
+                             end.
+  Lemma diff_aux_props :
+    forall (y2 : Z) (c2 : N) (s : t) acc,
+      interval_list_invariant (List.rev acc) = true ->
+      interval_list_invariant s = true ->
+      (forall x1 x2, InZ x1 acc -> InZ x2 s -> Z.succ x1 < x2) ->
+      (forall x, InZ x acc -> x < y2) ->
+      (c2 <> 0%N) ->
+      match (diff_aux y2 c2 acc s) with
+        (acc´, s´) => (forall y, InZ y (List.rev_append acc´ s´) <->
+                                 InZ y (List.rev_append acc s) /\ ~(List.In y (elementsZ_single y2 c2))) /\
+                      (interval_list_invariant (List.rev_append acc´ s´) = true) /\
+                      (forall x, InZ x acc´ -> x < y2 + Z.of_N c2)
+      end.
+  Lemma diff_aux2_props :
+   forall (s2 s1 acc : t),
+    interval_list_invariant (rev_append acc s1) = true ->
+    interval_list_invariant s2 = true ->
+    (forall x1 x2, InZ x1 acc -> InZ x2 s2 -> Z.succ x1 < x2) ->
+    ((forall y, (InZ y (diff_aux2 acc s1 s2) <->
+               ((InZ y acc) \/ (InZ y s1)) /\ ~InZ y s2)) /\
+    (interval_list_invariant (diff_aux2 acc s1 s2) = true)).
+  Lemma diff_InZ :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    forall y, (InZ y (diff s1 s2) <-> InZ y s1 /\ ~InZ y s2).
+ +
+  Lemma diff_invariant :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    interval_list_invariant (diff s1 s2) = true.
+ +
+ +
+  Global Instance diff_ok s1 s2 : forall `(Ok s1, Ok s2), Ok (diff s1 s2).
+  Lemma diff_spec :
   forall (s s´ : t) (x : elt) (Hs : Ok s) (Hs´ : Ok s´),
-   In x (inter s s´) <-> In x s /\ In x s´.
+   In x (diff s s´) <-> In x s /\ ~In x s´.
-

diff specification

+

remove specification

-  Global Instance diff_ok s s´ : forall `(Ok s, Ok s´), Ok (diff s s´).
+  Lemma removeZ_alt_def : forall x s acc,
+    interval_list_invariant s = true ->
+    removeZ_aux acc x s = match diff_aux x (1%N) acc s with
+                    (acc´, s´) => rev_append acc´ s´
+                  end.
+  Lemma removeZ_interval_list_invariant : forall s x, interval_list_invariant s = true -> interval_list_invariant (removeZ x s) = true.

-  Lemma diff_spec :
-   forall (s s´ : t) (x : elt) (Hs : Ok s) (Hs´ : Ok s´),
-   In x (diff s s´) <-> In x s /\ ~In x s´.
+  Lemma removeZ_spec :
+   forall (s : t) (x y : Z) (Hs : interval_list_invariant s = true),
+    InZ y (removeZ x s) <-> InZ y s /\ ~Z.eq y x.

+  Global Instance remove_ok s x : forall `(Ok s), Ok (remove x s).
+  Lemma remove_spec :
+   forall (s : t) (x y : elt) (Hs : Ok s),
+    In y (remove x s) <-> In y s /\ ~Enc.E.eq y x.
-

subset specification

+

remove_list specification

+  Lemma remove_list_ok : forall l s, Ok s -> Ok (remove_list l s).
+  Lemma remove_list_spec : forall x l s, Ok s ->
+     (In x (remove_list l s) <-> ~(InA Enc.E.eq x l) /\ In x s).
+
+ +
+

subset specification

+ +
+
+  Lemma subset_flatten_alt_def : forall (s1 s2 : t),
+    subset s1 s2 =
+    match (s1, s2) with
+    | (nil, _) => true
+    | (_ :: _, nil) => false
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before => false
+          | ICR_before_touch => false
+          | ICR_after => subset s1 l2
+          | ICR_after_touch => false
+          | ICR_overlap_before => false
+          | ICR_overlap_after => false
+          | ICR_equal => subset l1 l2
+          | ICR_subsume_1 => subset l1 s2
+          | ICR_subsume_2 => false
+        end
+    end.

+  Lemma subset_props_aux : forall y1 c1 l1 y2 c2 l2,
+    (exists y, InZ y ((y1, c1) :: l1) /\ ~InZ y ((y2, c2) :: l2)) ->
+    (false = true <->
+    (forall y : Z,
+        InZ y ((y1, c1) :: l1) -> InZ y ((y2, c2) :: l2))).
+ +
+  Lemma subset_props_aux_before : forall y1 c1 l1 y2 c2 l2,
+    (c1 <> 0%N) ->
+    interval_list_invariant ((y2, c2) :: l2) = true ->
+    (y1 < y2) ->
+    (false = true <->
+    (forall y : Z,
+        InZ y ((y1, c1) :: l1) -> InZ y ((y2, c2) :: l2))).
+  Lemma subset_props : forall s1 s2 : t,
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    (subset s1 s2 = true <->
+     (forall y, InZ y s1 -> InZ y s2)).
  Lemma subset_spec :
   forall (s s´ : t) (Hs : Ok s) (Hs´ : Ok s´),
   subset s s´ = true <-> Subset s s´.
- -
-

elements and elementsZ specification

+

elements and elementsZ specification

- -
  Lemma elements_spec1 : forall (s : t) (x : elt) (Hs : Ok s), List.In x (elements s) <-> In x s.

@@ -980,83 +1700,148 @@

Library MSetIntervals

-

equal specification

+

equal specification

  Lemma equal_alt_def : forall s1 s2,
    equal s1 s2 = true <-> (s1 = s2).
-  Lemma elementsZ_cons_le_start : forall x cx xs y cy ys,
-     isok ((x, cx) :: xs) = true ->
-     isok ((y, cy) :: ys) = true ->
-     (forall z, List.In z (elementsZ ((y, cy) :: ys)) ->
-                List.In z (elementsZ ((x, cx) :: xs))) ->
-     (x <= y).
-  Lemma elementsZ_cons_le_end : forall x cx xs y cy ys,
-     isok ((x, cx) :: xs) = true ->
-     isok ((y, cy) :: ys) = true ->
-     (x <= y + Z.of_N cy) ->
-     (forall z, List.In z (elementsZ ((x, cx) :: xs)) ->
-                List.In z (elementsZ ((y, cy) :: ys))) ->
-     (x + Z.of_N cx <= y + Z.of_N cy).
-  Lemma elementsZ_cons_equiv_hd : forall x cx xs y cy ys,
-     isok ((x, cx) :: xs) = true ->
-     isok ((y, cy) :: ys) = true ->
-     (forall z, List.In z (elementsZ ((x, cx) :: xs)) <->
-                List.In z (elementsZ ((y, cy) :: ys))) ->
-     (x = y) /\ (cx = cy).
- -
-  Lemma elementsZ_single_equiv : forall x cx y cy,
-     (cx <> 0)%N ->
-     (cy <> 0)%N ->
-     (forall z, List.In z (elementsZ_single x cx) <->
-                List.In z (elementsZ_single y cy)) ->
-     (x = y) /\ (cx = cy).
  Lemma equal_elementsZ :
    forall (s s´ : t) {Hs : Ok s} {Hs´ : Ok s´},
    (forall x, (InZ x s <-> InZ x s´)) -> (s = s´).
+ +
  Lemma equal_spec :
    forall (s s´ : t) {Hs : Ok s} {Hs´ : Ok s´},
    equal s s´ = true <-> Equal s s´.
-

choose specification

+

compare

-  Lemma choose_alt_def : forall s,
-    choose s = match chooseZ s with
-      | None => None
-      | Some e => Some (Enc.decode e)
-    end.
+  Definition lt (s1 s2 : t) : Prop := (compare s1 s2 = Lt).
+ +
+  Lemma compare_eq_Eq : forall s1 s2,
+    (compare s1 s2 = Eq <-> equal s1 s2 = true).
+  Lemma compare_eq_Lt_nil_l : forall s,
+    compare nil s = Lt <-> s <> nil.
+ +
+  Lemma compare_eq_Lt_nil_r : forall s,
+    ~(compare s nil = Lt).
+ +
+  Lemma compare_eq_Lt_cons : forall y1 y2 c1 c2 s1 s2,
+    compare ((y1, c1)::s1) ((y2, c2)::s2) = Lt <->
+    (y1 < y2) \/ ((y1 = y2) /\ (c1 < c2)%N) \/
+    ((y1 = y2) /\ (c1 = c2) /\ compare s1 s2 = Lt).
+  Lemma compare_antisym: forall (s1 s2 : t),
+    (compare s1 s2) = CompOpp (compare s2 s1).
+ +
+  Lemma compare_spec : forall s1 s2,
+    CompSpec eq lt s1 s2 (compare s1 s2).
+  Lemma lt_Irreflexive : Irreflexive lt.
+ +
+  Lemma lt_Transitive : Transitive lt.
+
+
+

elements is sorted

+ +
+
+  Lemma elementsZ_single_sorted : forall c x,
+    sort Z.lt (elementsZ_single x c).
+  Lemma elementsZ_sorted : forall s,
+    interval_list_invariant s = true ->
+    sort Z.lt (rev (elementsZ s)).
+  Lemma elements_sorted : forall s,
+    Ok s ->
+    sort Enc.E.lt (elements s).
+
+ +
+

choose specification

+ +
+
+  Definition min_eltZ_spec1 :
+    forall (s : t) (x : Z),
+      interval_list_invariant s = true ->
+      min_eltZ s = Some x -> InZ x s.
+   Lemma min_eltZ_spec2 :
+    forall (s : t) (x y : Z) (Hs : Ok s),
+    min_eltZ s = Some x -> InZ y s -> ~ Z.lt y x.
+  Definition min_eltZ_spec3 :
+    forall (s : t),
+      min_eltZ s = None -> forall x, ~InZ x s.
+ +
+  Definition min_elt_spec1 :
+    forall (s : t) (x : elt) (Hs : Ok s), min_elt s = Some x -> In x s.
+ +
+  Definition min_elt_spec2 :
+    forall (s : t) (x y : elt) (Hs : Ok s), min_elt s = Some x -> In y s -> ~(Enc.E.lt y x).
+ +
+  Definition min_elt_spec3 :
+    forall s : t, min_elt s = None -> Empty s.
+
  Definition choose_spec1 :
-    forall (s : t) (x : elt), choose s = Some x -> In x s.
+    forall (s : t) (x : elt) (Hs : Ok s), choose s = Some x -> In x s.

  Definition choose_spec2 :
    forall s : t, choose s = None -> Empty s.

-  Lemma chooseZ_min :
-    forall (s : t) (x y : Z) (Hs : Ok s),
-    chooseZ s = Some x -> InZ y s -> ~ Z.lt y x.
-  Lemma chooseZ_InZ :
+  Lemma choose_spec3: forall s s´ x x´, Ok s -> Ok s´ ->
+    choose s = Some x -> choose s´ = Some x´ -> Equal s s´ -> x = x´.
+ +
+ +
+  Definition max_eltZ_spec1 :
    forall (s : t) (x : Z),
-    chooseZ s = Some x -> InZ x s.
+      interval_list_invariant s = true ->
+      max_eltZ s = Some x -> InZ x s.
+   Lemma max_eltZ_spec2 :
+    forall (s : t) (x y : Z),
+    interval_list_invariant s = true ->
+    max_eltZ s = Some x -> InZ y s -> ~ Z.lt x y.
+  Lemma max_eltZ_eq_None :
+    forall (s : t),
+      max_eltZ s = None -> s = nil.
+ +
+  Definition max_eltZ_spec3 :
+    forall (s : t),
+      max_eltZ s = None -> forall x, ~InZ x s.
+ +
+  Definition max_elt_spec1 :
+    forall (s : t) (x : elt) (Hs : Ok s), max_elt s = Some x -> In x s.

-  Lemma chooseZ_spec3: forall s s´ x x´, Ok s -> Ok s´ ->
-   chooseZ s = Some x -> chooseZ s´ = Some x´ -> Equal s s´ -> x = x´.
+  Definition max_elt_spec2 :
+    forall (s : t) (x y : elt) (Hs : Ok s), max_elt s = Some x -> In y s -> ~(Enc.E.lt x y).
+ +
+  Definition max_elt_spec3 :
+    forall s : t, max_elt s = None -> Empty s.

-

fold specification

+

fold specification

@@ -1065,19 +1850,13 @@

Library MSetIntervals

  Lemma fold_spec :
   forall (s : t) (A : Type) (i : A) (f : elt -> A -> A),
   fold f s i = fold_left (flip f) (elements s) i.
- -
- -
-

cardinal specification

+

cardinal specification

- -
  Lemma cardinalN_spec : forall (s : t) (c : N),
    cardinalN c s = (c + N.of_nat (length (elements s)))%N.
  Lemma cardinal_spec :
@@ -1088,7 +1867,7 @@

Library MSetIntervals

-

for_all specification

+

for_all specification

@@ -1098,31 +1877,143 @@

Library MSetIntervals

   forall (s : t) (f : elt -> bool) (Hs : Ok s),
   Proper (Enc.E.eq==>eq) f ->
   (for_all f s = true <-> For_all (fun x => f x = true) s).
- -
-

exists specification

+

exists specification

- -
  Lemma exists_spec :
   forall (s : t) (f : elt -> bool) (Hs : Ok s),
   Proper (Enc.E.eq==>eq) f ->
   (exists_ f s = true <-> Exists (fun x => f x = true) s).
+
+ +
+

filter specification

+ +
+
+  Definition partitionZ_aux_invariant (x : Z) acc c :=
+    interval_list_invariant (List.rev (partitionZ_fold_skip acc c)) = true /\
+    match c with
+      None => (forall y´, InZ y´ acc -> Z.succ y´ < x)
+    | Some (y, c´) => (x = y + Z.of_N c´)
+    end.
+ +
+  Lemma partitionZ_aux_invariant_insert : forall x acc c,
+    partitionZ_aux_invariant x acc c ->
+    partitionZ_aux_invariant (Z.succ x) acc
+      (Some (partitionZ_fold_insert c x)).
+  Lemma partitionZ_aux_invariant_skip : forall x acc c,
+    partitionZ_aux_invariant x acc c ->
+    partitionZ_aux_invariant (Z.succ x) (partitionZ_fold_skip acc c) None.
+  Definition partitionZ_fold_current (c : option (Z * N)) :=
+    match c with
+       None => nil
+     | Some yc => yc::nil
+    end.
+ +
+  Lemma InZ_partitionZ_fold_current_Some : forall yc y,
+     InZ y (partitionZ_fold_current (Some yc)) <->
+     InZ y (yc :: nil).
+ +
+  Lemma InZ_partitionZ_fold_insert : forall c x y l,
+   match c with
+   | Some (y, c´) => x = y + Z.of_N c´
+   | None => True
+   end -> (
+   InZ y (partitionZ_fold_insert c x :: l) <->
+    ((x = y) \/ InZ y (partitionZ_fold_current c) \/
+       InZ y l)).
+  Lemma InZ_partitionZ_fold_skip : forall c acc y,
+    InZ y (partitionZ_fold_skip acc c) <->
+    (InZ y (partitionZ_fold_current c) \/ InZ y acc).
+  Lemma filterZ_single_aux_props :
+    forall f c x acc cur,
+      partitionZ_aux_invariant x acc cur ->
+      match (filterZ_single_aux f (acc, cur) x c) with
+        (acc´, c´) =>
+        let r := partitionZ_fold_skip acc´ c´ in
+        interval_list_invariant (List.rev r) = true /\
+        (forall y´, InZ y´ r <-> (InZ y´ (partitionZ_fold_skip acc cur) \/
+                                    (f y´ = true /\ List.In y´ (elementsZ_single x c))))
+
+      end.
+  Lemma filterZ_single_props :
+    forall f c x acc,
+      interval_list_invariant (rev acc) = true ->
+      (forall y´ : Z, InZ y´ acc -> Z.succ y´ < x) ->
+      match (filterZ_single f acc x c) with
+        r =>
+        interval_list_invariant (List.rev r) = true /\
+        (forall y´, InZ y´ r <-> (InZ y´ acc \/
+                                 (f y´ = true /\ List.In y´ (elementsZ_single x c))))
+
+      end.
+ +
+ +
+  Lemma filterZ_aux_props :
+    forall f s acc,
+      interval_list_invariant s = true ->
+      interval_list_invariant (rev acc) = true ->
+      (forall x1 x2 : Z, InZ x1 acc -> InZ x2 s -> Z.succ x1 < x2) ->
+      match (filterZ_aux acc f s) with
+        r =>
+        interval_list_invariant r = true /\
+        (forall y´, InZ y´ r <-> (InZ y´ acc \/
+                                 (f y´ = true /\ InZ y´ s)))
+      end.
+  Lemma filterZ_props :
+    forall f s,
+      interval_list_invariant s = true ->
+      match (filterZ f s) with r =>
+        interval_list_invariant r = true /\
+        (forall y´, InZ y´ r <-> (f y´ = true /\ InZ y´ s))
+      end.
+ +
+  Global Instance filter_ok s f : forall `(Ok s), Ok (filter f s).
+ +
+  Lemma filter_spec :
+   forall (s : t) (x : elt) (f : elt -> bool),
+   Ok s ->
+   (In x (filter f s) <-> In x s /\ f x = true).

-

partition specification

+

partition specification

+
+  Lemma partitionZ_single_aux_alt_def : forall f c y acc_t c_t acc_f c_f,
+    partitionZ_single_aux f ((acc_t, c_t), (acc_f, c_f)) y c =
+    (filterZ_single_aux f (acc_t, c_t) y c,
+     filterZ_single_aux (fun x : Z => negb (f x)) (acc_f, c_f) y c).
+  Lemma partitionZ_aux_alt_def : forall f s acc_t acc_f,
+   partitionZ_aux acc_t acc_f f s =
+   (filterZ_aux acc_t f s,
+    filterZ_aux acc_f (fun x : Z => negb (f x)) s).
+  Lemma partitionZ_alt_def : forall f s,
+    partitionZ f s = (filterZ f s,
+                      filterZ (fun x => negb (f x)) s).
+ +
+  Lemma partition_alt_def : forall f s,
+    partition f s = (filter f s,
+                     filter (fun x => negb (f x)) s).
+
  Global Instance partition_ok1 s f : forall `(Ok s), Ok (fst (partition f s)).
@@ -1132,12 +2023,12 @@

Library MSetIntervals


  Lemma partition_spec1 :
   forall (s : t) (f : elt -> bool),
-   Proper (Enc.E.eq==>eq) f -> Equal (fst (partition f s)) (filter f s).
+   Equal (fst (partition f s)) (filter f s).

  Lemma partition_spec2 :
   forall (s : t) (f : elt -> bool),
-   Proper (Enc.E.eq==>eq) f ->
+   Ok s ->
   Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).

@@ -1147,7 +2038,7 @@

Library MSetIntervals

-

Main Module

+

Main Module

@@ -1158,7 +2049,7 @@

Library MSetIntervals


-Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E.
+Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E.
  Module E := Enc.E.
  Module Raw := Raw Enc.
@@ -1191,7 +2082,10 @@

Library MSetIntervals

 Definition empty : t := Mkt Raw.empty.
 Definition is_empty (s : t) := Raw.is_empty s.
 Definition elements (s : t) : list elt := Raw.elements s.
+ Definition min_elt (s : t) : option elt := Raw.min_elt s.
+ Definition max_elt (s : t) : option elt := Raw.max_elt s.
 Definition choose (s : t) : option elt := Raw.choose s.
+ Definition compare (s1 s2 : t) : comparison := Raw.compare s1 s2.
 Definition fold {A : Type}(f : elt -> A -> A)(s : t) : A -> A := Raw.fold f s.
 Definition cardinal (s : t) := Raw.cardinal s.
 Definition filter (f : elt -> bool)(s : t) : t := Mkt (Raw.filter f s).
@@ -1212,6 +2106,13 @@

Library MSetIntervals


 Definition eq_dec : forall (s s´:t), { eq s s´ }+{ ~eq s s´ }.
+
+ Definition lt : t -> t -> Prop := Raw.lt.
+ +
+ Instance lt_strorder : StrictOrder lt.
+ Instance lt_compat : Proper (eq==>eq==>iff) lt.
+
 Section Spec.
  Variable s s´ : t.
@@ -1245,9 +2146,25 @@

Library MSetIntervals

      Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
  Lemma elements_spec1 : InA E.eq x (elements s) <-> In x s.
  Lemma elements_spec2w : NoDupA E.eq (elements s).
+   Lemma elements_spec2 : sort E.lt (elements s).
  Lemma choose_spec1 : choose s = Some x -> In x s.
  Lemma choose_spec2 : choose s = None -> Empty s.
+   Lemma choose_spec3 : choose s = Some x -> choose s´ = Some y ->
+    Equal s s´ -> E.eq x y.
+ +
+  Lemma min_elt_spec1 : choose s = Some x -> In x s.
+   Lemma min_elt_spec2 : min_elt s = Some x -> In y s -> ~ E.lt y x.
+   Lemma min_elt_spec3 : choose s = None -> Empty s.
+ +
+  Lemma max_elt_spec1 : max_elt s = Some x -> In x s.
+   Lemma max_elt_spec2 : max_elt s = Some x -> In y s -> ~ E.lt x y.
+   Lemma max_elt_spec3 : max_elt s = None -> Empty s.
+
+  Lemma compare_spec : CompSpec eq lt s s´ (compare s s´).
+
 End Spec.
@@ -1258,7 +2175,7 @@

Library MSetIntervals

-

Instantiations

+

Instantiations

@@ -1266,7 +2183,7 @@

Library MSetIntervals

It remains to provide instantiations for commonly used datatypes.
-

Z

+

Z

@@ -1287,17 +2204,21 @@

Library MSetIntervals

  Lemma encode_eq : forall (e1 e2 : E.t),
    (Z.eq (encode e1) (encode e2)) <-> E.eq e1 e2.
+
+  Lemma encode_lt : forall (e1 e2 : E.t),
+    (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2.
+
End ElementEncodeZ.

-Module MSetIntervalsZ <: WSetsOn Z := MSetIntervals ElementEncodeZ.
+Module MSetIntervalsZ <: SetsOn Z := MSetIntervals ElementEncodeZ.

-

N

+

N

@@ -1317,16 +2238,22 @@

Library MSetIntervals


  Lemma encode_eq : forall (e1 e2 : E.t),
    (Z.eq (encode e1) (encode e2)) <-> E.eq e1 e2.
- End ElementEncodeN.
+ +
+  Lemma encode_lt : forall (e1 e2 : E.t),
+    (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2.

-Module MSetIntervalsN <: WSetsOn N := MSetIntervals ElementEncodeN.
+End ElementEncodeN.
+ +
+Module MSetIntervalsN <: SetsOn N := MSetIntervals ElementEncodeN.

-

nat

+

nat

@@ -1344,10 +2271,16 @@

Library MSetIntervals


  Lemma encode_eq : forall (e1 e2 : E.t),
    (Z.eq (encode e1) (encode e2)) <-> E.eq e1 e2.
- End ElementEncodeNat.
+ +
+  Lemma encode_lt : forall (e1 e2 : E.t),
+    (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2.
+ +
+End ElementEncodeNat.

-Module MSetIntervalsNat <: WSetsOn NPeano.Nat := MSetIntervals ElementEncodeNat.
+Module MSetIntervalsNat <: SetsOn NPeano.Nat := MSetIntervals ElementEncodeNat.

diff --git a/coqdoc/MSetListWithDups.html b/coqdoc/MSetListWithDups.html index 2289b36..bda67bb 100644 --- a/coqdoc/MSetListWithDups.html +++ b/coqdoc/MSetListWithDups.html @@ -24,7 +24,7 @@

Library MSetListWithDups

-

Weak sets implemented as lists with duplicates

+

Weak sets implemented as lists with duplicates

@@ -55,7 +55,7 @@

Library MSetListWithDups

-

Removing duplicates from sorted lists

+

Removing duplicates from sorted lists

@@ -200,7 +200,7 @@

Library MSetListWithDups

-

Operations Module

+

Operations Module

@@ -328,7 +328,7 @@

Library MSetListWithDups

-

Main Module

+

Main Module

@@ -350,7 +350,7 @@

Library MSetListWithDups

-

Proofs of set operation specifications.

+

Proofs of set operation specifications.

Logical predicates
diff --git a/coqdoc/index.html b/coqdoc/index.html index 301d042..32a5c83 100644 --- a/coqdoc/index.html +++ b/coqdoc/index.html @@ -47,7 +47,7 @@ Z _ other -(451 entries) +(596 entries) Notation Index @@ -207,7 +207,7 @@ Z _ other -(168 entries) +(257 entries) Axiom Index @@ -239,7 +239,7 @@ Z _ other -(31 entries) +(32 entries) Constructor Index @@ -257,7 +257,7 @@ L M N -O +O P Q R @@ -271,7 +271,7 @@ Z _ other -(2 entries) +(11 entries) Projection Index @@ -321,7 +321,7 @@ L M N -O +O P Q R @@ -335,7 +335,7 @@ Z _ other -(1 entry) +(2 entries) Section Index @@ -399,7 +399,7 @@ Z _ other -(15 entries) +(17 entries) Abbreviation Index @@ -463,7 +463,7 @@ Z _ other -(156 entries) +(199 entries) Record Index @@ -510,22 +510,26 @@

Global Index

ElementEncodeNat.decode_encode_ok [lemma, in MSetIntervals]
ElementEncodeNat.E [module, in MSetIntervals]
ElementEncodeNat.encode [definition, in MSetIntervals]
+ElementEncodeNat.encode_lt [lemma, in MSetIntervals]
ElementEncodeNat.encode_eq [lemma, in MSetIntervals]
ElementEncodeN.decode [definition, in MSetIntervals]
ElementEncodeN.decode_encode_ok [lemma, in MSetIntervals]
ElementEncodeN.E [module, in MSetIntervals]
ElementEncodeN.encode [definition, in MSetIntervals]
+ElementEncodeN.encode_lt [lemma, in MSetIntervals]
ElementEncodeN.encode_eq [lemma, in MSetIntervals]
ElementEncodeZ [module, in MSetIntervals]
ElementEncodeZ.decode [definition, in MSetIntervals]
ElementEncodeZ.decode_encode_ok [lemma, in MSetIntervals]
ElementEncodeZ.E [module, in MSetIntervals]
ElementEncodeZ.encode [definition, in MSetIntervals]
+ElementEncodeZ.encode_lt [lemma, in MSetIntervals]
ElementEncodeZ.encode_eq [lemma, in MSetIntervals]
ElementEncode.decode [axiom, in MSetIntervals]
ElementEncode.decode_encode_ok [axiom, in MSetIntervals]
ElementEncode.E [module, in MSetIntervals]
ElementEncode.encode [axiom, in MSetIntervals]
+ElementEncode.encode_lt [axiom, in MSetIntervals]
ElementEncode.encode_eq [axiom, in MSetIntervals]


F

foldWithAbortGtLtSpecPred [definition, in MSetFoldWithAbort]
@@ -636,13 +640,17 @@

Global Index

MSetIntervals.cardinal [definition, in MSetIntervals]
MSetIntervals.cardinal_spec [lemma, in MSetIntervals]
MSetIntervals.choose [definition, in MSetIntervals]
+MSetIntervals.choose_spec3 [lemma, in MSetIntervals]
MSetIntervals.choose_spec2 [lemma, in MSetIntervals]
MSetIntervals.choose_spec1 [lemma, in MSetIntervals]
+MSetIntervals.compare [definition, in MSetIntervals]
+MSetIntervals.compare_spec [lemma, in MSetIntervals]
MSetIntervals.compatb [abbreviation, in MSetIntervals]
MSetIntervals.diff [definition, in MSetIntervals]
MSetIntervals.diff_spec [lemma, in MSetIntervals]
MSetIntervals.E [module, in MSetIntervals]
MSetIntervals.elements [definition, in MSetIntervals]
+MSetIntervals.elements_spec2 [lemma, in MSetIntervals]
MSetIntervals.elements_spec2w [lemma, in MSetIntervals]
MSetIntervals.elements_spec1 [lemma, in MSetIntervals]
MSetIntervals.elt [definition, in MSetIntervals]
@@ -672,8 +680,19 @@

Global Index

MSetIntervals.is_empty_spec [lemma, in MSetIntervals]
MSetIntervals.is_empty [definition, in MSetIntervals]
MSetIntervals.is_ok [projection, in MSetIntervals]
+MSetIntervals.lt [definition, in MSetIntervals]
+MSetIntervals.lt_compat [instance, in MSetIntervals]
+MSetIntervals.lt_strorder [instance, in MSetIntervals]
+MSetIntervals.max_elt_spec3 [lemma, in MSetIntervals]
+MSetIntervals.max_elt_spec2 [lemma, in MSetIntervals]
+MSetIntervals.max_elt_spec1 [lemma, in MSetIntervals]
+MSetIntervals.max_elt [definition, in MSetIntervals]
MSetIntervals.mem [definition, in MSetIntervals]
MSetIntervals.mem_spec [lemma, in MSetIntervals]
+MSetIntervals.min_elt_spec3 [lemma, in MSetIntervals]
+MSetIntervals.min_elt_spec2 [lemma, in MSetIntervals]
+MSetIntervals.min_elt_spec1 [lemma, in MSetIntervals]
+MSetIntervals.min_elt [definition, in MSetIntervals]
MSetIntervals.Mkt [constructor, in MSetIntervals]
MSetIntervals.partition [definition, in MSetIntervals]
MSetIntervals.partition_spec2 [lemma, in MSetIntervals]
@@ -716,15 +735,14 @@

Global Index

Ops.cardinalN [definition, in MSetIntervals]
Ops.choose [definition, in MSetIntervals]
Ops.choose [definition, in MSetListWithDups]
-Ops.chooseZ [definition, in MSetIntervals]
+Ops.compare [definition, in MSetIntervals]
Ops.diff [definition, in MSetIntervals]
Ops.diff [definition, in MSetListWithDups]
+Ops.diff_aux2 [definition, in MSetIntervals]
+Ops.diff_aux [definition, in MSetIntervals]
Ops.elements [definition, in MSetIntervals]
Ops.elements [definition, in MSetListWithDups]
Ops.elementsZ [definition, in MSetIntervals]
-Ops.elementsZ_aux [definition, in MSetIntervals]
-Ops.elementsZ_aux' [definition, in MSetIntervals]
-Ops.elementsZ_aux'' [definition, in MSetIntervals]
Ops.elements_dist [definition, in MSetListWithDups]
Ops.elt [definition, in MSetIntervals]
Ops.elt [definition, in MSetListWithDups]
@@ -736,28 +754,63 @@

Global Index

Ops.exists_ [definition, in MSetListWithDups]
Ops.filter [definition, in MSetIntervals]
Ops.filter [definition, in MSetListWithDups]
+Ops.filterZ [definition, in MSetIntervals]
+Ops.filterZ_aux [definition, in MSetIntervals]
+Ops.filterZ_single [definition, in MSetIntervals]
+Ops.filterZ_single_aux [definition, in MSetIntervals]
+Ops.filterZ_fold_fun [definition, in MSetIntervals]
Ops.fold [definition, in MSetIntervals]
Ops.fold [definition, in MSetListWithDups]
+Ops.fold_elementsZ [definition, in MSetIntervals]
+Ops.fold_elementsZ_single [definition, in MSetIntervals]
+Ops.fold_elementsZ_aux [definition, in MSetIntervals]
Ops.for_all [definition, in MSetIntervals]
Ops.for_all [definition, in MSetListWithDups]
Ops.from_elements [definition, in MSetIntervals]
+Ops.ICR_after_touch [constructor, in MSetIntervals]
+Ops.ICR_after [constructor, in MSetIntervals]
+Ops.ICR_subsume_2 [constructor, in MSetIntervals]
+Ops.ICR_subsume_1 [constructor, in MSetIntervals]
+Ops.ICR_equal [constructor, in MSetIntervals]
+Ops.ICR_overlap_after [constructor, in MSetIntervals]
+Ops.ICR_overlap_before [constructor, in MSetIntervals]
+Ops.ICR_before_touch [constructor, in MSetIntervals]
+Ops.ICR_before [constructor, in MSetIntervals]
+Ops.insert_intervalZ_guarded [definition, in MSetIntervals]
+Ops.insert_interval_begin [definition, in MSetIntervals]
Ops.inter [definition, in MSetIntervals]
Ops.inter [definition, in MSetListWithDups]
+Ops.interval_1_compare [definition, in MSetIntervals]
+Ops.interval_compare [definition, in MSetIntervals]
+Ops.interval_compare_result [inductive, in MSetIntervals]
+Ops.inter_aux2 [definition, in MSetIntervals]
+Ops.inter_aux [definition, in MSetIntervals]
Ops.is_empty [definition, in MSetIntervals]
Ops.is_empty [definition, in MSetListWithDups]
+Ops.max_elt [definition, in MSetIntervals]
+Ops.max_eltZ [definition, in MSetIntervals]
Ops.mem [definition, in MSetIntervals]
Ops.mem [definition, in MSetListWithDups]
Ops.memZ [definition, in MSetIntervals]
+Ops.merge_interval_size [definition, in MSetIntervals]
+Ops.min_elt [definition, in MSetIntervals]
+Ops.min_eltZ [definition, in MSetIntervals]
Ops.MX [module, in MSetListWithDups]
Ops.partition [definition, in MSetIntervals]
Ops.partition [definition, in MSetListWithDups]
+Ops.partitionZ [definition, in MSetIntervals]
+Ops.partitionZ_aux [definition, in MSetIntervals]
+Ops.partitionZ_single [definition, in MSetIntervals]
+Ops.partitionZ_single_aux [definition, in MSetIntervals]
+Ops.partitionZ_fold_fun [definition, in MSetIntervals]
+Ops.partitionZ_fold_skip [definition, in MSetIntervals]
+Ops.partitionZ_fold_insert [definition, in MSetIntervals]
Ops.partition_aux [definition, in MSetListWithDups]
Ops.RDFS [module, in MSetListWithDups]
Ops.remove [definition, in MSetIntervals]
Ops.remove [definition, in MSetListWithDups]
Ops.removeZ [definition, in MSetIntervals]
Ops.removeZ_aux [definition, in MSetIntervals]
-Ops.removeZ_aux_insert_guarded [definition, in MSetIntervals]
Ops.remove_list [definition, in MSetIntervals]
Ops.rev_filter [definition, in MSetListWithDups]
Ops.rev_filter_aux [definition, in MSetListWithDups]
@@ -770,46 +823,48 @@

Global Index

Ops.t [definition, in MSetListWithDups]
Ops.union [definition, in MSetIntervals]
Ops.union [definition, in MSetListWithDups]
+Ops.union_aux [definition, in MSetIntervals]


R

Raw [module, in MSetIntervals]
-Raw.addZ_isok [lemma, in MSetIntervals]
-Raw.addZ_spec [lemma, in MSetIntervals]
+Raw.addZ_invariant [lemma, in MSetIntervals]
+Raw.addZ_InZ [lemma, in MSetIntervals]
Raw.addZ_alt_def [lemma, in MSetIntervals]
Raw.addZ_aux_alt_def [lemma, in MSetIntervals]
-Raw.addZ_ind [lemma, in MSetIntervals]
Raw.add_list_spec [lemma, in MSetIntervals]
Raw.add_list_ok [lemma, in MSetIntervals]
Raw.add_spec [lemma, in MSetIntervals]
Raw.add_ok [instance, in MSetIntervals]
Raw.cardinalN_spec [lemma, in MSetIntervals]
Raw.cardinal_spec [lemma, in MSetIntervals]
-Raw.chooseZ_spec3 [lemma, in MSetIntervals]
-Raw.chooseZ_InZ [lemma, in MSetIntervals]
-Raw.chooseZ_min [lemma, in MSetIntervals]
+Raw.choose_spec3 [lemma, in MSetIntervals]
Raw.choose_spec2 [definition, in MSetIntervals]
Raw.choose_spec1 [definition, in MSetIntervals]
-Raw.choose_alt_def [lemma, in MSetIntervals]
+Raw.compare_spec [lemma, in MSetIntervals]
+Raw.compare_antisym [lemma, in MSetIntervals]
+Raw.compare_eq_Lt_cons [lemma, in MSetIntervals]
+Raw.compare_eq_Lt_nil_r [lemma, in MSetIntervals]
+Raw.compare_eq_Lt_nil_l [lemma, in MSetIntervals]
+Raw.compare_eq_Eq [lemma, in MSetIntervals]
Raw.diff_spec [lemma, in MSetIntervals]
Raw.diff_ok [instance, in MSetIntervals]
-Raw.elementsZ_single_equiv [lemma, in MSetIntervals]
-Raw.elementsZ_cons_equiv_hd [lemma, in MSetIntervals]
-Raw.elementsZ_cons_le_end [lemma, in MSetIntervals]
-Raw.elementsZ_cons_le_start [lemma, in MSetIntervals]
+Raw.diff_invariant [lemma, in MSetIntervals]
+Raw.diff_InZ [lemma, in MSetIntervals]
+Raw.diff_aux2_props [lemma, in MSetIntervals]
+Raw.diff_aux_props [lemma, in MSetIntervals]
+Raw.diff_aux_alt_def [lemma, in MSetIntervals]
+Raw.elementsZ_sorted [lemma, in MSetIntervals]
+Raw.elementsZ_single_sorted [lemma, in MSetIntervals]
Raw.elementsZ_spec2w [lemma, in MSetIntervals]
-Raw.elementsZ_removeZ_aux_insert_guarded [lemma, in MSetIntervals]
+Raw.elementsZ_insert_intervalZ_guarded [lemma, in MSetIntervals]
+Raw.elementsZ_app [lemma, in MSetIntervals]
Raw.elementsZ_cons [lemma, in MSetIntervals]
-Raw.elementsZ_aux_alt_def [lemma, in MSetIntervals]
-Raw.elementsZ_single_intro [lemma, in MSetIntervals]
-Raw.elementsZ_aux'_succ [lemma, in MSetIntervals]
-Raw.elementsZ_aux'_zero [lemma, in MSetIntervals]
-Raw.elementsZ_aux'_pos [lemma, in MSetIntervals]
-Raw.elementsZ_aux''_irrel [lemma, in MSetIntervals]
Raw.elementsZ_single_succ_front [lemma, in MSetIntervals]
Raw.elementsZ_single_add [lemma, in MSetIntervals]
Raw.elementsZ_single_succ [lemma, in MSetIntervals]
Raw.elementsZ_single_base [lemma, in MSetIntervals]
Raw.elementsZ_single [definition, in MSetIntervals]
Raw.elementsZ_nil [lemma, in MSetIntervals]
+Raw.elements_sorted [lemma, in MSetIntervals]
Raw.elements_spec2w [lemma, in MSetIntervals]
Raw.elements_spec1 [lemma, in MSetIntervals]
Raw.elements_cons [lemma, in MSetIntervals]
@@ -819,43 +874,115 @@

Global Index

Raw.empty_spec [lemma, in MSetIntervals]
Raw.empty_spec' [lemma, in MSetIntervals]
Raw.empty_ok [instance, in MSetIntervals]
+Raw.encode_decode_eq [lemma, in MSetIntervals]
Raw.Equal [definition, in MSetIntervals]
Raw.equal_spec [lemma, in MSetIntervals]
Raw.equal_elementsZ [lemma, in MSetIntervals]
Raw.equal_alt_def [lemma, in MSetIntervals]
Raw.Exists [definition, in MSetIntervals]
Raw.exists_spec [lemma, in MSetIntervals]
+Raw.filterZ_props [lemma, in MSetIntervals]
+Raw.filterZ_aux_props [lemma, in MSetIntervals]
+Raw.filterZ_single_props [lemma, in MSetIntervals]
+Raw.filterZ_single_aux_props [lemma, in MSetIntervals]
Raw.filter_spec [lemma, in MSetIntervals]
Raw.filter_ok [instance, in MSetIntervals]
Raw.fold_spec [lemma, in MSetIntervals]
+Raw.fold_elementsZ_alt_def [lemma, in MSetIntervals]
+Raw.fold_elementsZ_alt_def_aux [lemma, in MSetIntervals]
+Raw.fold_elementsZ_cons [lemma, in MSetIntervals]
+Raw.fold_elementsZ_nil [lemma, in MSetIntervals]
+Raw.fold_elementsZ_single_alt_def [lemma, in MSetIntervals]
+Raw.fold_opt_app [lemma, in MSetIntervals]
+Raw.fold_opt_list_cons [lemma, in MSetIntervals]
+Raw.fold_opt [definition, in MSetIntervals]
+Raw.fold_elementsZ_single_succ [lemma, in MSetIntervals]
+Raw.fold_elementsZ_single_zero [lemma, in MSetIntervals]
+Raw.fold_elementsZ_single_pos [lemma, in MSetIntervals]
+Raw.fold_elementsZ_aux_irrel [lemma, in MSetIntervals]
Raw.ForNotations [section, in MSetIntervals]
Raw.for_all_spec [lemma, in MSetIntervals]
Raw.For_all [definition, in MSetIntervals]
Raw.In [definition, in MSetIntervals]
-Raw.inf [definition, in MSetIntervals]
-Raw.inf_removeZ_aux_insert_guarded [lemma, in MSetIntervals]
-Raw.inf_impl [lemma, in MSetIntervals]
+Raw.insert_interval_begin_spec [lemma, in MSetIntervals]
+Raw.insert_intervalZ_guarded_rev_nil_app [lemma, in MSetIntervals]
+Raw.insert_intervalZ_guarded_app [lemma, in MSetIntervals]
+Raw.interval_list_elements_greater_insert_intervalZ_guarded [lemma, in MSetIntervals]
+Raw.interval_list_invariant_insert_intervalZ_guarded [lemma, in MSetIntervals]
+Raw.interval_list_invariant_snoc_intro [lemma, in MSetIntervals]
+Raw.interval_list_invariant_app_iff [lemma, in MSetIntervals]
+Raw.interval_list_invariant_app_elim [lemma, in MSetIntervals]
+Raw.interval_list_invariant_app_intro [lemma, in MSetIntervals]
+Raw.interval_list_elements_greater_app_elim_1 [lemma, in MSetIntervals]
+Raw.interval_list_elements_greater_intro [lemma, in MSetIntervals]
+Raw.interval_list_elements_greater_alt2_def [lemma, in MSetIntervals]
+Raw.interval_list_elements_greater_alt_def [lemma, in MSetIntervals]
+Raw.interval_list_invariant_sing [lemma, in MSetIntervals]
+Raw.interval_list_invariant_cons [lemma, in MSetIntervals]
+Raw.interval_list_invariant_nil [lemma, in MSetIntervals]
+Raw.interval_list_elements_greater_impl [lemma, in MSetIntervals]
+Raw.interval_list_elements_greater_cons [lemma, in MSetIntervals]
+Raw.interval_1_compare_elim [lemma, in MSetIntervals]
+Raw.interval_1_compare_alt_def [lemma, in MSetIntervals]
+Raw.interval_compare_swap [lemma, in MSetIntervals]
+Raw.interval_compare_elim [lemma, in MSetIntervals]
+Raw.interval_list_invariant [definition, in MSetIntervals]
+Raw.interval_list_elements_greater [definition, in MSetIntervals]
Raw.inter_spec [lemma, in MSetIntervals]
Raw.inter_ok [instance, in MSetIntervals]
+Raw.inter_invariant [lemma, in MSetIntervals]
+Raw.inter_InZ [lemma, in MSetIntervals]
+Raw.inter_aux2_props [lemma, in MSetIntervals]
+Raw.inter_aux_props [lemma, in MSetIntervals]
+Raw.inter_aux_alt_def [lemma, in MSetIntervals]
Raw.InZ [definition, in MSetIntervals]
+Raw.InZ_partitionZ_fold_skip [lemma, in MSetIntervals]
+Raw.InZ_partitionZ_fold_insert [lemma, in MSetIntervals]
+Raw.InZ_partitionZ_fold_current_Some [lemma, in MSetIntervals]
+Raw.InZ_insert_intervalZ_guarded [lemma, in MSetIntervals]
+Raw.InZ_In [lemma, in MSetIntervals]
+Raw.InZ_dec [lemma, in MSetIntervals]
+Raw.InZ_rev [lemma, in MSetIntervals]
+Raw.InZ_app [lemma, in MSetIntervals]
+Raw.InZ_cons [lemma, in MSetIntervals]
+Raw.InZ_nil [lemma, in MSetIntervals]
+Raw.In_merge_interval [lemma, in MSetIntervals]
Raw.In_InZ [lemma, in MSetIntervals]
Raw.In_alt_def [lemma, in MSetIntervals]
Raw.In_elementsZ_single_hd [lemma, in MSetIntervals]
+Raw.In_elementsZ_single_dec [lemma, in MSetIntervals]
Raw.In_elementsZ_single1 [lemma, in MSetIntervals]
Raw.In_elementsZ_single [lemma, in MSetIntervals]
Raw.IsOk [definition, in MSetIntervals]
-Raw.isok [definition, in MSetIntervals]
-Raw.isok_removeZ_aux_insert_guarded [lemma, in MSetIntervals]
-Raw.isok_inf_nin [lemma, in MSetIntervals]
-Raw.isok_cons [lemma, in MSetIntervals]
Raw.IsOk_Ok [instance, in MSetIntervals]
Raw.is_empty_spec [lemma, in MSetIntervals]
Raw.is_encoded_elems_list_rev [lemma, in MSetIntervals]
Raw.is_encoded_elems_list_app [lemma, in MSetIntervals]
Raw.is_encoded_elems_list [definition, in MSetIntervals]
Raw.length_elementsZ_single [lemma, in MSetIntervals]
+Raw.lt [definition, in MSetIntervals]
+Raw.lt_Transitive [lemma, in MSetIntervals]
+Raw.lt_Irreflexive [lemma, in MSetIntervals]
+Raw.max_elt_spec3 [definition, in MSetIntervals]
+Raw.max_elt_spec2 [definition, in MSetIntervals]
+Raw.max_elt_spec1 [definition, in MSetIntervals]
+Raw.max_eltZ_spec3 [definition, in MSetIntervals]
+Raw.max_eltZ_eq_None [lemma, in MSetIntervals]
+Raw.max_eltZ_spec2 [lemma, in MSetIntervals]
+Raw.max_eltZ_spec1 [definition, in MSetIntervals]
Raw.memZ_spec [lemma, in MSetIntervals]
Raw.mem_spec [lemma, in MSetIntervals]
+Raw.merge_interval_size_invariant [lemma, in MSetIntervals]
+Raw.merge_interval_size_eq_max [lemma, in MSetIntervals]
+Raw.merge_interval_size_add [lemma, in MSetIntervals]
+Raw.merge_interval_size_neq_0 [lemma, in MSetIntervals]
+Raw.min_elt_spec3 [definition, in MSetIntervals]
+Raw.min_elt_spec2 [definition, in MSetIntervals]
+Raw.min_elt_spec1 [definition, in MSetIntervals]
+Raw.min_eltZ_spec3 [definition, in MSetIntervals]
+Raw.min_eltZ_spec2 [lemma, in MSetIntervals]
+Raw.min_eltZ_spec1 [definition, in MSetIntervals]
+Raw.Nin_elements_greater_equal [lemma, in MSetIntervals]
Raw.Nin_elements_greater [lemma, in MSetIntervals]
Raw.NoDupA_elementsZ_single [lemma, in MSetIntervals]
Raw.ok [projection, in MSetIntervals]
@@ -864,15 +991,21 @@

Global Index

Raw.Ok [inductive, in MSetIntervals]
Raw.Ok_cons [lemma, in MSetIntervals]
Raw.Ok_nil [lemma, in MSetIntervals]
+Raw.partitionZ_alt_def [lemma, in MSetIntervals]
+Raw.partitionZ_aux_alt_def [lemma, in MSetIntervals]
+Raw.partitionZ_single_aux_alt_def [lemma, in MSetIntervals]
+Raw.partitionZ_fold_current [definition, in MSetIntervals]
+Raw.partitionZ_aux_invariant_skip [lemma, in MSetIntervals]
+Raw.partitionZ_aux_invariant_insert [lemma, in MSetIntervals]
+Raw.partitionZ_aux_invariant [definition, in MSetIntervals]
Raw.partition_spec2 [lemma, in MSetIntervals]
Raw.partition_spec1 [lemma, in MSetIntervals]
Raw.partition_ok2 [instance, in MSetIntervals]
Raw.partition_ok1 [instance, in MSetIntervals]
+Raw.partition_alt_def [lemma, in MSetIntervals]
Raw.removeZ_spec [lemma, in MSetIntervals]
-Raw.removeZ_isok [lemma, in MSetIntervals]
-Raw.removeZ_counter_pos_aux [lemma, in MSetIntervals]
+Raw.removeZ_interval_list_invariant [lemma, in MSetIntervals]
Raw.removeZ_alt_def [lemma, in MSetIntervals]
-Raw.removeZ_aux_alt_def [lemma, in MSetIntervals]
Raw.remove_list_spec [lemma, in MSetIntervals]
Raw.remove_list_ok [lemma, in MSetIntervals]
Raw.remove_spec [lemma, in MSetIntervals]
@@ -881,8 +1014,17 @@

Global Index

Raw.singleton_ok [instance, in MSetIntervals]
Raw.Subset [definition, in MSetIntervals]
Raw.subset_spec [lemma, in MSetIntervals]
+Raw.subset_props [lemma, in MSetIntervals]
+Raw.subset_props_aux_before [lemma, in MSetIntervals]
+Raw.subset_props_aux [lemma, in MSetIntervals]
+Raw.subset_flatten_alt_def [lemma, in MSetIntervals]
Raw.union_spec [lemma, in MSetIntervals]
Raw.union_ok [instance, in MSetIntervals]
+Raw.union_invariant [lemma, in MSetIntervals]
+Raw.union_InZ [lemma, in MSetIntervals]
+Raw.union_alt_def [lemma, in MSetIntervals]
+Raw.union_aux_alt_def [lemma, in MSetIntervals]
+Raw.union_aux_flatten_alt_def [lemma, in MSetIntervals]
RemoveDupsFromSorted [module, in MSetListWithDups]
RemoveDupsFromSorted.eqb_eq_alt [lemma, in MSetListWithDups]
RemoveDupsFromSorted.MX [module, in MSetListWithDups]
@@ -961,6 +1103,9 @@

Global Index

WSetsWithFoldA [module, in MSetFoldWithAbort]
WSetsWithFoldA.E [module, in MSetFoldWithAbort]


Z

+Z_to_N_minus_neq_0 [lemma, in MSetIntervals]
+Z_lt_le_add_r [lemma, in MSetIntervals]
+Z_lt_add_r [lemma, in MSetIntervals]
Z_le_add_r [lemma, in MSetIntervals]


Notation Index

@@ -1052,10 +1197,13 @@

Lemma Index

add_add_sub_eq [in MSetIntervals]


E

ElementEncodeNat.decode_encode_ok [in MSetIntervals]
+ElementEncodeNat.encode_lt [in MSetIntervals]
ElementEncodeNat.encode_eq [in MSetIntervals]
ElementEncodeN.decode_encode_ok [in MSetIntervals]
+ElementEncodeN.encode_lt [in MSetIntervals]
ElementEncodeN.encode_eq [in MSetIntervals]
ElementEncodeZ.decode_encode_ok [in MSetIntervals]
+ElementEncodeZ.encode_lt [in MSetIntervals]
ElementEncodeZ.encode_eq [in MSetIntervals]


H

HasFoldWithAbortOps.choose_with_abort_spec [in MSetFoldWithAbort]
@@ -1105,9 +1253,12 @@

Lemma Index

Make.union_spec [in MSetListWithDups]
MSetIntervals.add_spec [in MSetIntervals]
MSetIntervals.cardinal_spec [in MSetIntervals]
+MSetIntervals.choose_spec3 [in MSetIntervals]
MSetIntervals.choose_spec2 [in MSetIntervals]
MSetIntervals.choose_spec1 [in MSetIntervals]
+MSetIntervals.compare_spec [in MSetIntervals]
MSetIntervals.diff_spec [in MSetIntervals]
+MSetIntervals.elements_spec2 [in MSetIntervals]
MSetIntervals.elements_spec2w [in MSetIntervals]
MSetIntervals.elements_spec1 [in MSetIntervals]
MSetIntervals.empty_spec [in MSetIntervals]
@@ -1118,7 +1269,13 @@

Lemma Index

MSetIntervals.for_all_spec [in MSetIntervals]
MSetIntervals.inter_spec [in MSetIntervals]
MSetIntervals.is_empty_spec [in MSetIntervals]
+MSetIntervals.max_elt_spec3 [in MSetIntervals]
+MSetIntervals.max_elt_spec2 [in MSetIntervals]
+MSetIntervals.max_elt_spec1 [in MSetIntervals]
MSetIntervals.mem_spec [in MSetIntervals]
+MSetIntervals.min_elt_spec3 [in MSetIntervals]
+MSetIntervals.min_elt_spec2 [in MSetIntervals]
+MSetIntervals.min_elt_spec1 [in MSetIntervals]
MSetIntervals.partition_spec2 [in MSetIntervals]
MSetIntervals.partition_spec1 [in MSetIntervals]
MSetIntervals.remove_spec [in MSetIntervals]
@@ -1131,86 +1288,160 @@

Lemma Index

Ops.acc_pred [in MSetIntervals]
Ops.singleton_alt_def [in MSetIntervals]


R

-Raw.addZ_isok [in MSetIntervals]
-Raw.addZ_spec [in MSetIntervals]
+Raw.addZ_invariant [in MSetIntervals]
+Raw.addZ_InZ [in MSetIntervals]
Raw.addZ_alt_def [in MSetIntervals]
Raw.addZ_aux_alt_def [in MSetIntervals]
-Raw.addZ_ind [in MSetIntervals]
Raw.add_list_spec [in MSetIntervals]
Raw.add_list_ok [in MSetIntervals]
Raw.add_spec [in MSetIntervals]
Raw.cardinalN_spec [in MSetIntervals]
Raw.cardinal_spec [in MSetIntervals]
-Raw.chooseZ_spec3 [in MSetIntervals]
-Raw.chooseZ_InZ [in MSetIntervals]
-Raw.chooseZ_min [in MSetIntervals]
-Raw.choose_alt_def [in MSetIntervals]
+Raw.choose_spec3 [in MSetIntervals]
+Raw.compare_spec [in MSetIntervals]
+Raw.compare_antisym [in MSetIntervals]
+Raw.compare_eq_Lt_cons [in MSetIntervals]
+Raw.compare_eq_Lt_nil_r [in MSetIntervals]
+Raw.compare_eq_Lt_nil_l [in MSetIntervals]
+Raw.compare_eq_Eq [in MSetIntervals]
Raw.diff_spec [in MSetIntervals]
-Raw.elementsZ_single_equiv [in MSetIntervals]
-Raw.elementsZ_cons_equiv_hd [in MSetIntervals]
-Raw.elementsZ_cons_le_end [in MSetIntervals]
-Raw.elementsZ_cons_le_start [in MSetIntervals]
+Raw.diff_invariant [in MSetIntervals]
+Raw.diff_InZ [in MSetIntervals]
+Raw.diff_aux2_props [in MSetIntervals]
+Raw.diff_aux_props [in MSetIntervals]
+Raw.diff_aux_alt_def [in MSetIntervals]
+Raw.elementsZ_sorted [in MSetIntervals]
+Raw.elementsZ_single_sorted [in MSetIntervals]
Raw.elementsZ_spec2w [in MSetIntervals]
-Raw.elementsZ_removeZ_aux_insert_guarded [in MSetIntervals]
+Raw.elementsZ_insert_intervalZ_guarded [in MSetIntervals]
+Raw.elementsZ_app [in MSetIntervals]
Raw.elementsZ_cons [in MSetIntervals]
-Raw.elementsZ_aux_alt_def [in MSetIntervals]
-Raw.elementsZ_single_intro [in MSetIntervals]
-Raw.elementsZ_aux'_succ [in MSetIntervals]
-Raw.elementsZ_aux'_zero [in MSetIntervals]
-Raw.elementsZ_aux'_pos [in MSetIntervals]
-Raw.elementsZ_aux''_irrel [in MSetIntervals]
Raw.elementsZ_single_succ_front [in MSetIntervals]
Raw.elementsZ_single_add [in MSetIntervals]
Raw.elementsZ_single_succ [in MSetIntervals]
Raw.elementsZ_single_base [in MSetIntervals]
Raw.elementsZ_nil [in MSetIntervals]
+Raw.elements_sorted [in MSetIntervals]
Raw.elements_spec2w [in MSetIntervals]
Raw.elements_spec1 [in MSetIntervals]
Raw.elements_cons [in MSetIntervals]
Raw.elements_nil [in MSetIntervals]
Raw.empty_spec [in MSetIntervals]
Raw.empty_spec' [in MSetIntervals]
+Raw.encode_decode_eq [in MSetIntervals]
Raw.equal_spec [in MSetIntervals]
Raw.equal_elementsZ [in MSetIntervals]
Raw.equal_alt_def [in MSetIntervals]
Raw.exists_spec [in MSetIntervals]
+Raw.filterZ_props [in MSetIntervals]
+Raw.filterZ_aux_props [in MSetIntervals]
+Raw.filterZ_single_props [in MSetIntervals]
+Raw.filterZ_single_aux_props [in MSetIntervals]
Raw.filter_spec [in MSetIntervals]
Raw.fold_spec [in MSetIntervals]
+Raw.fold_elementsZ_alt_def [in MSetIntervals]
+Raw.fold_elementsZ_alt_def_aux [in MSetIntervals]
+Raw.fold_elementsZ_cons [in MSetIntervals]
+Raw.fold_elementsZ_nil [in MSetIntervals]
+Raw.fold_elementsZ_single_alt_def [in MSetIntervals]
+Raw.fold_opt_app [in MSetIntervals]
+Raw.fold_opt_list_cons [in MSetIntervals]
+Raw.fold_elementsZ_single_succ [in MSetIntervals]
+Raw.fold_elementsZ_single_zero [in MSetIntervals]
+Raw.fold_elementsZ_single_pos [in MSetIntervals]
+Raw.fold_elementsZ_aux_irrel [in MSetIntervals]
Raw.for_all_spec [in MSetIntervals]
-Raw.inf_removeZ_aux_insert_guarded [in MSetIntervals]
-Raw.inf_impl [in MSetIntervals]
+Raw.insert_interval_begin_spec [in MSetIntervals]
+Raw.insert_intervalZ_guarded_rev_nil_app [in MSetIntervals]
+Raw.insert_intervalZ_guarded_app [in MSetIntervals]
+Raw.interval_list_elements_greater_insert_intervalZ_guarded [in MSetIntervals]
+Raw.interval_list_invariant_insert_intervalZ_guarded [in MSetIntervals]
+Raw.interval_list_invariant_snoc_intro [in MSetIntervals]
+Raw.interval_list_invariant_app_iff [in MSetIntervals]
+Raw.interval_list_invariant_app_elim [in MSetIntervals]
+Raw.interval_list_invariant_app_intro [in MSetIntervals]
+Raw.interval_list_elements_greater_app_elim_1 [in MSetIntervals]
+Raw.interval_list_elements_greater_intro [in MSetIntervals]
+Raw.interval_list_elements_greater_alt2_def [in MSetIntervals]
+Raw.interval_list_elements_greater_alt_def [in MSetIntervals]
+Raw.interval_list_invariant_sing [in MSetIntervals]
+Raw.interval_list_invariant_cons [in MSetIntervals]
+Raw.interval_list_invariant_nil [in MSetIntervals]
+Raw.interval_list_elements_greater_impl [in MSetIntervals]
+Raw.interval_list_elements_greater_cons [in MSetIntervals]
+Raw.interval_1_compare_elim [in MSetIntervals]
+Raw.interval_1_compare_alt_def [in MSetIntervals]
+Raw.interval_compare_swap [in MSetIntervals]
+Raw.interval_compare_elim [in MSetIntervals]
Raw.inter_spec [in MSetIntervals]
+Raw.inter_invariant [in MSetIntervals]
+Raw.inter_InZ [in MSetIntervals]
+Raw.inter_aux2_props [in MSetIntervals]
+Raw.inter_aux_props [in MSetIntervals]
+Raw.inter_aux_alt_def [in MSetIntervals]
+Raw.InZ_partitionZ_fold_skip [in MSetIntervals]
+Raw.InZ_partitionZ_fold_insert [in MSetIntervals]
+Raw.InZ_partitionZ_fold_current_Some [in MSetIntervals]
+Raw.InZ_insert_intervalZ_guarded [in MSetIntervals]
+Raw.InZ_In [in MSetIntervals]
+Raw.InZ_dec [in MSetIntervals]
+Raw.InZ_rev [in MSetIntervals]
+Raw.InZ_app [in MSetIntervals]
+Raw.InZ_cons [in MSetIntervals]
+Raw.InZ_nil [in MSetIntervals]
+Raw.In_merge_interval [in MSetIntervals]
Raw.In_InZ [in MSetIntervals]
Raw.In_alt_def [in MSetIntervals]
Raw.In_elementsZ_single_hd [in MSetIntervals]
+Raw.In_elementsZ_single_dec [in MSetIntervals]
Raw.In_elementsZ_single1 [in MSetIntervals]
Raw.In_elementsZ_single [in MSetIntervals]
-Raw.isok_removeZ_aux_insert_guarded [in MSetIntervals]
-Raw.isok_inf_nin [in MSetIntervals]
-Raw.isok_cons [in MSetIntervals]
Raw.is_empty_spec [in MSetIntervals]
Raw.is_encoded_elems_list_rev [in MSetIntervals]
Raw.is_encoded_elems_list_app [in MSetIntervals]
Raw.length_elementsZ_single [in MSetIntervals]
+Raw.lt_Transitive [in MSetIntervals]
+Raw.lt_Irreflexive [in MSetIntervals]
+Raw.max_eltZ_eq_None [in MSetIntervals]
+Raw.max_eltZ_spec2 [in MSetIntervals]
Raw.memZ_spec [in MSetIntervals]
Raw.mem_spec [in MSetIntervals]
+Raw.merge_interval_size_invariant [in MSetIntervals]
+Raw.merge_interval_size_eq_max [in MSetIntervals]
+Raw.merge_interval_size_add [in MSetIntervals]
+Raw.merge_interval_size_neq_0 [in MSetIntervals]
+Raw.min_eltZ_spec2 [in MSetIntervals]
+Raw.Nin_elements_greater_equal [in MSetIntervals]
Raw.Nin_elements_greater [in MSetIntervals]
Raw.NoDupA_elementsZ_single [in MSetIntervals]
Raw.Ok_cons [in MSetIntervals]
Raw.Ok_nil [in MSetIntervals]
+Raw.partitionZ_alt_def [in MSetIntervals]
+Raw.partitionZ_aux_alt_def [in MSetIntervals]
+Raw.partitionZ_single_aux_alt_def [in MSetIntervals]
+Raw.partitionZ_aux_invariant_skip [in MSetIntervals]
+Raw.partitionZ_aux_invariant_insert [in MSetIntervals]
Raw.partition_spec2 [in MSetIntervals]
Raw.partition_spec1 [in MSetIntervals]
+Raw.partition_alt_def [in MSetIntervals]
Raw.removeZ_spec [in MSetIntervals]
-Raw.removeZ_isok [in MSetIntervals]
-Raw.removeZ_counter_pos_aux [in MSetIntervals]
+Raw.removeZ_interval_list_invariant [in MSetIntervals]
Raw.removeZ_alt_def [in MSetIntervals]
-Raw.removeZ_aux_alt_def [in MSetIntervals]
Raw.remove_list_spec [in MSetIntervals]
Raw.remove_list_ok [in MSetIntervals]
Raw.remove_spec [in MSetIntervals]
Raw.singleton_spec [in MSetIntervals]
Raw.subset_spec [in MSetIntervals]
+Raw.subset_props [in MSetIntervals]
+Raw.subset_props_aux_before [in MSetIntervals]
+Raw.subset_props_aux [in MSetIntervals]
+Raw.subset_flatten_alt_def [in MSetIntervals]
Raw.union_spec [in MSetIntervals]
+Raw.union_invariant [in MSetIntervals]
+Raw.union_InZ [in MSetIntervals]
+Raw.union_alt_def [in MSetIntervals]
+Raw.union_aux_alt_def [in MSetIntervals]
+Raw.union_aux_flatten_alt_def [in MSetIntervals]
RemoveDupsFromSorted.eqb_eq_alt [in MSetListWithDups]
RemoveDupsFromSorted.remove_dups_by_sortingA_spec [in MSetListWithDups]
RemoveDupsFromSorted.remove_dups_from_sortedA_spec [in MSetListWithDups]
@@ -1224,6 +1455,9 @@

Lemma Index

WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w [in MSetWithDups]
WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1 [in MSetWithDups]


Z

+Z_to_N_minus_neq_0 [in MSetIntervals]
+Z_lt_le_add_r [in MSetIntervals]
+Z_lt_add_r [in MSetIntervals]
Z_le_add_r [in MSetIntervals]


Axiom Index

@@ -1231,6 +1465,7 @@

Axiom Index

ElementEncode.decode [in MSetIntervals]
ElementEncode.decode_encode_ok [in MSetIntervals]
ElementEncode.encode [in MSetIntervals]
+ElementEncode.encode_lt [in MSetIntervals]
ElementEncode.encode_eq [in MSetIntervals]


H

HasFoldWithAbort.foldWithAbortPrecompute [in MSetFoldWithAbort]
@@ -1265,6 +1500,16 @@

Axiom Index

Constructor Index

M

MSetIntervals.Mkt [in MSetIntervals]
+

O

+Ops.ICR_after_touch [in MSetIntervals]
+Ops.ICR_after [in MSetIntervals]
+Ops.ICR_subsume_2 [in MSetIntervals]
+Ops.ICR_subsume_1 [in MSetIntervals]
+Ops.ICR_equal [in MSetIntervals]
+Ops.ICR_overlap_after [in MSetIntervals]
+Ops.ICR_overlap_before [in MSetIntervals]
+Ops.ICR_before_touch [in MSetIntervals]
+Ops.ICR_before [in MSetIntervals]


R

Raw.ok [in MSetIntervals]


@@ -1281,7 +1526,9 @@

Projection Index

Raw.ok [in MSetIntervals]


Inductive Index

-

R

+

O

+Ops.interval_compare_result [in MSetIntervals]
+

R

Raw.Ok [in MSetIntervals]


Section Index

@@ -1297,6 +1544,8 @@

Instance Index

Make.In_compat [in MSetListWithDups]
MSetIntervals.eq_equiv [in MSetIntervals]
MSetIntervals.In_compat [in MSetIntervals]
+MSetIntervals.lt_compat [in MSetIntervals]
+MSetIntervals.lt_strorder [in MSetIntervals]


R

Raw.add_ok [in MSetIntervals]
Raw.diff_ok [in MSetIntervals]
@@ -1363,6 +1612,7 @@

Definition Index

MSetIntervals.add [in MSetIntervals]
MSetIntervals.cardinal [in MSetIntervals]
MSetIntervals.choose [in MSetIntervals]
+MSetIntervals.compare [in MSetIntervals]
MSetIntervals.diff [in MSetIntervals]
MSetIntervals.elements [in MSetIntervals]
MSetIntervals.elt [in MSetIntervals]
@@ -1381,7 +1631,10 @@

Definition Index

MSetIntervals.In [in MSetIntervals]
MSetIntervals.inter [in MSetIntervals]
MSetIntervals.is_empty [in MSetIntervals]
+MSetIntervals.lt [in MSetIntervals]
+MSetIntervals.max_elt [in MSetIntervals]
MSetIntervals.mem [in MSetIntervals]
+MSetIntervals.min_elt [in MSetIntervals]
MSetIntervals.partition [in MSetIntervals]
MSetIntervals.remove [in MSetIntervals]
MSetIntervals.singleton [in MSetIntervals]
@@ -1400,15 +1653,14 @@

Definition Index

Ops.cardinalN [in MSetIntervals]
Ops.choose [in MSetIntervals]
Ops.choose [in MSetListWithDups]
-Ops.chooseZ [in MSetIntervals]
+Ops.compare [in MSetIntervals]
Ops.diff [in MSetIntervals]
Ops.diff [in MSetListWithDups]
+Ops.diff_aux2 [in MSetIntervals]
+Ops.diff_aux [in MSetIntervals]
Ops.elements [in MSetIntervals]
Ops.elements [in MSetListWithDups]
Ops.elementsZ [in MSetIntervals]
-Ops.elementsZ_aux [in MSetIntervals]
-Ops.elementsZ_aux' [in MSetIntervals]
-Ops.elementsZ_aux'' [in MSetIntervals]
Ops.elements_dist [in MSetListWithDups]
Ops.elt [in MSetIntervals]
Ops.elt [in MSetListWithDups]
@@ -1420,26 +1672,51 @@

Definition Index

Ops.exists_ [in MSetListWithDups]
Ops.filter [in MSetIntervals]
Ops.filter [in MSetListWithDups]
+Ops.filterZ [in MSetIntervals]
+Ops.filterZ_aux [in MSetIntervals]
+Ops.filterZ_single [in MSetIntervals]
+Ops.filterZ_single_aux [in MSetIntervals]
+Ops.filterZ_fold_fun [in MSetIntervals]
Ops.fold [in MSetIntervals]
Ops.fold [in MSetListWithDups]
+Ops.fold_elementsZ [in MSetIntervals]
+Ops.fold_elementsZ_single [in MSetIntervals]
+Ops.fold_elementsZ_aux [in MSetIntervals]
Ops.for_all [in MSetIntervals]
Ops.for_all [in MSetListWithDups]
Ops.from_elements [in MSetIntervals]
+Ops.insert_intervalZ_guarded [in MSetIntervals]
+Ops.insert_interval_begin [in MSetIntervals]
Ops.inter [in MSetIntervals]
Ops.inter [in MSetListWithDups]
+Ops.interval_1_compare [in MSetIntervals]
+Ops.interval_compare [in MSetIntervals]
+Ops.inter_aux2 [in MSetIntervals]
+Ops.inter_aux [in MSetIntervals]
Ops.is_empty [in MSetIntervals]
Ops.is_empty [in MSetListWithDups]
+Ops.max_elt [in MSetIntervals]
+Ops.max_eltZ [in MSetIntervals]
Ops.mem [in MSetIntervals]
Ops.mem [in MSetListWithDups]
Ops.memZ [in MSetIntervals]
+Ops.merge_interval_size [in MSetIntervals]
+Ops.min_elt [in MSetIntervals]
+Ops.min_eltZ [in MSetIntervals]
Ops.partition [in MSetIntervals]
Ops.partition [in MSetListWithDups]
+Ops.partitionZ [in MSetIntervals]
+Ops.partitionZ_aux [in MSetIntervals]
+Ops.partitionZ_single [in MSetIntervals]
+Ops.partitionZ_single_aux [in MSetIntervals]
+Ops.partitionZ_fold_fun [in MSetIntervals]
+Ops.partitionZ_fold_skip [in MSetIntervals]
+Ops.partitionZ_fold_insert [in MSetIntervals]
Ops.partition_aux [in MSetListWithDups]
Ops.remove [in MSetIntervals]
Ops.remove [in MSetListWithDups]
Ops.removeZ [in MSetIntervals]
Ops.removeZ_aux [in MSetIntervals]
-Ops.removeZ_aux_insert_guarded [in MSetIntervals]
Ops.remove_list [in MSetIntervals]
Ops.rev_filter [in MSetListWithDups]
Ops.rev_filter_aux [in MSetListWithDups]
@@ -1451,6 +1728,7 @@

Definition Index

Ops.t [in MSetListWithDups]
Ops.union [in MSetIntervals]
Ops.union [in MSetListWithDups]
+Ops.union_aux [in MSetIntervals]


R

Raw.choose_spec2 [in MSetIntervals]
Raw.choose_spec1 [in MSetIntervals]
@@ -1459,13 +1737,27 @@

Definition Index

Raw.Empty [in MSetIntervals]
Raw.Equal [in MSetIntervals]
Raw.Exists [in MSetIntervals]
+Raw.fold_opt [in MSetIntervals]
Raw.For_all [in MSetIntervals]
Raw.In [in MSetIntervals]
-Raw.inf [in MSetIntervals]
+Raw.interval_list_invariant [in MSetIntervals]
+Raw.interval_list_elements_greater [in MSetIntervals]
Raw.InZ [in MSetIntervals]
Raw.IsOk [in MSetIntervals]
-Raw.isok [in MSetIntervals]
Raw.is_encoded_elems_list [in MSetIntervals]
+Raw.lt [in MSetIntervals]
+Raw.max_elt_spec3 [in MSetIntervals]
+Raw.max_elt_spec2 [in MSetIntervals]
+Raw.max_elt_spec1 [in MSetIntervals]
+Raw.max_eltZ_spec3 [in MSetIntervals]
+Raw.max_eltZ_spec1 [in MSetIntervals]
+Raw.min_elt_spec3 [in MSetIntervals]
+Raw.min_elt_spec2 [in MSetIntervals]
+Raw.min_elt_spec1 [in MSetIntervals]
+Raw.min_eltZ_spec3 [in MSetIntervals]
+Raw.min_eltZ_spec1 [in MSetIntervals]
+Raw.partitionZ_fold_current [in MSetIntervals]
+Raw.partitionZ_aux_invariant [in MSetIntervals]
Raw.Subset [in MSetIntervals]
RemoveDupsFromSorted.remove_dups_by_sortingA [in MSetListWithDups]
RemoveDupsFromSorted.remove_dups_from_sortedA [in MSetListWithDups]
@@ -1522,7 +1814,7 @@

Record Index

Z _ other -(451 entries) +(596 entries) Notation Index @@ -1682,7 +1974,7 @@

Record Index

Z _ other -(168 entries) +(257 entries) Axiom Index @@ -1714,7 +2006,7 @@

Record Index

Z _ other -(31 entries) +(32 entries) Constructor Index @@ -1732,7 +2024,7 @@

Record Index

L M N -O +O P Q R @@ -1746,7 +2038,7 @@

Record Index

Z _ other -(2 entries) +(11 entries) Projection Index @@ -1796,7 +2088,7 @@

Record Index

L M N -O +O P Q R @@ -1810,7 +2102,7 @@

Record Index

Z _ other -(1 entry) +(2 entries) Section Index @@ -1874,7 +2166,7 @@

Record Index

Z _ other -(15 entries) +(17 entries) Abbreviation Index @@ -1938,7 +2230,7 @@

Record Index

Z _ other -(156 entries) +(199 entries) Record Index diff --git a/coqdoc/mset_doc.pdf b/coqdoc/mset_doc.pdf index 86d6833445fd1083b5c254fbd97763689b2a94aa..049c0419d74c4aba7f3674592cafa2b06b587a91 100644 GIT binary patch delta 617892 zcmZs?V|So!(5@NVX2-T|+qP}?Gt7>F)^i;pQ(%5%K&k#m1fiul3Rz1{l6?qXK z1`zneU!gJiO!($LGuLn)ZHQ!43i`Z%wd8L>oDLnDw!U&jnIoyTl@Zl(9W9L#%0};- z+IygQP&14kRm_pVptd7jkue#N_imD8gv#WcWK-dH^*;xmmv_`3)n4h}JfOtd)Axr4 z=J@J#VQ<=|Cb^27K_&&V7)}iuAC!Kb1F$-6gh-Ii7f7KuwM;EUq9F}s?~Q_x8aDB8F#pqQa``ZaoJK#?tZoP}=ZiR1C2qIa(O4!yJ$Ykn2&{9H49lex*z zvL2$#J4{_xap`!mw)p7`g<`8v2h8pT=9JHPCrvn<%y9IYT$##SW_t0G#1VFKLREA= ztjG+GsG^;TDjt%Y{?^psFBU`0tv;oHDDQ1>p;MDV^EH<{OPr#`9YJWa!?f-}Fwj^& zlE`CM;p{F>q#BI`Ke#e_oA^=sO)L9(FbCgu5f$c4ohBhg-j?cHWSf*)4~WVBQ3XAy+H0OZzBcvr%I3w&NsS3>U{l-0RV6y3|u8C0U{slX2GQ$8lPrv{Jn(@OiIW zwIxViY0B|zHD;M@Ex@El0EmzA8TUN*KaUDXjG}!Ffh;@+hWj`dpf>q&;Q}Yw3(Q_N z0@>6CB44;P<`o=bhO6_|dA_>OAxn0d=wB}54oTvYPtV5g$jE7#Yo5>x^$s{@eyhx# 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z5q5>>FZl2CyD^xb-AQ6^cSBG&_14N-7h(0VZB92&VP>Chv4Q{s=Zx1kZRFYr8+r$Z z-tG;e5oYP_6|udZ-P07$Nen}3^T*!nm<-R;>oQorFNIKTed4zZv;Ih2qVslmcrW@5 ztKq&GlWotOIT33p7ioJa7a!Fh+k^c!SCMVrZmeRT;BUCUY@~I4QGQHoCj*W^D7Q21 za9&z&YsfDyj=mw(;f|oI|KB9@gdg7*%`<E$^UYXr*8eY_uF5)%l{m9 zRQCQ~us`PK#{>MWv#E6_|gCKkv!ja-;ez*?5-c>%iHRd zY?QjMAJQ%Oqkg#8eUDsT?qmC>e@vIL)ZNMR7Bm3XQjpqNwepDW< zyL0nz>xTup|GKMwE-h#l`u}uxWBkvhFEW1gz3P3x;@+E%qxOP%<&XD^y^DC{>;87~ z!+Ey%jc`{re{5v)%iV_TcQih(G$fE@oCIPj;`@1|AUi|L_Ne|Gn%#^B2t*t(@@x z*mtS?MgR9VzPG>k!|#dj7yCsYI)3|B9I>^(bJox9XSVV`k;58li9Cfq$3%RNn)n>& zIdeqk%rTiWM=hQyHvWugXpLw&8qu^gqU~u! zc<<=Z-qDr4qfdHAXY`I<>m69fwQH<+>hJH4IOi{Im-}PO!^>^MlX3X*jg#*`O(_1? z@=%?}KJf?VVSkCbLm!0O<{SK>mXLJvWP5F!X|w*Y?soqtzr7^M7;~dVaYelements list properties -

  • Alternative definition of addZ +
  • comparing intervals
    • -
  • Alternative definition of removeZ +
  • Alternative definition of addZ
  • Auxiliary Lemmata about Invariant @@ -142,81 +142,93 @@
  • Membership specification
    • -
  • add specification +
  • insert if length not 0
    • -
  • remove specification +
  • Merging intervals
    • -
  • empty specification +
  • add specification
    • -
  • is_empty specification +
  • empty specification
    • -
  • singleton specification +
  • is_empty specification
    • -
  • add_list specification +
  • singleton specification
    • -
  • remove_list specification +
  • add_list specification
  • union specification
    • -
  • filter specification +
  • inter specification
    • -
  • inter specification +
  • diff specification
    • -
  • diff specification +
  • remove specification
    • -
  • subset specification +
  • remove_list specification
    • -
  • elements and elementsZ specification +
  • subset specification
    • -
  • equal specification +
  • elements and elementsZ specification
    • -
  • choose specification +
  • equal specification
    • -
  • fold specification +
  • compare
    • -
  • cardinal specification +
  • elements is sorted
    • -
  • for_all specification +
  • choose specification
    • -
  • exists specification +
  • fold specification
    • -
  • partition specification +
  • cardinal specification + +
    • +
  • for_all specification + +
    • +
  • exists specification + +
    • +
  • filter specification + +
    • +
  • partition specification
    • -
  • Main Module +
  • Main Module
    • -
  • Instantiations +
  • Instantiations @@ -228,18 +240,18 @@

    Library MSetListWithDups
    • -

    Fold with abort for sets

    +

    Fold with abort for sets

    @@ -38,18 +38,18 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Require Export MSetInterface.
    -Require Import ssreflect.
    +Require Export MSetInterface.
    +Require Import mathcomp.ssreflect.ssreflect.
    Require Import MSetWithDups.
    -Require Import Int.
    -Require Import MSetGenTree MSetAVL MSetRBT.
    -Require Import MSetList MSetWeakList.
    +Require Import Int.
    +Require Import MSetGenTree MSetAVL MSetRBT.
    +Require Import MSetList MSetWeakList.

    -

    Fold With Abort Operations

    +

    Fold With Abort Operations

    @@ -82,25 +82,25 @@

    Library MSetsExtra.MSetFoldWithAbort

    :=
    -    (elt → A → A) →
    +    (elt → A → A) →
    f
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_abort
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value @@ -110,8 +110,8 @@

    Library MSetsExtra.MSetFoldWithAbort


    Definition foldWithAbortSpecPred {elt t : Type}
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
        (foldWithAbort : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=

        âˆ€
    @@ -123,7 +123,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (i i´ : A)
    +      (i i' : A)
          
    @@ -131,7 +131,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → A → A) (f´ : elt → A → A)
    +      (f : elt → A → A) (f' : elt → A → A)
          
    @@ -139,7 +139,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_abort : elt → A → bool)
    +      (f_abort : elt → A → bool)
          
    @@ -154,7 +154,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    equivalence relation on results @@ -165,39 +165,39 @@

    Library MSetsExtra.MSetFoldWithAbort

        
    -P is an equivalence relation +P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →
          
        
    -f is for the elements of s compatible with the equivalence relation P +f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st´ e, In e s → P st st´ → P (f e st) (f e st´)) →
    +    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →

        
    -f and f´ agree for the elements of s +f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e st) (f´ e st))) →
    +    (∀ e st, In e s → (P (f e st) (f' e st))) →

        
    -f_abort is OK, i.e. all other elements can be skipped without +f_abort is OK, i.e. all other elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_abort e1 st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → e2 ≠ e1 →
    -                        P st (f e2 st´))) →
    +    (∀ e1 st,
    +        In e1 s → f_abort e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → e2 ≠ e1 →
    +                        P st (f e2 st'))) →

        
    @@ -205,7 +205,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i´ →
    +    P i i' →

        
    @@ -213,7 +213,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The results are in equivalence relation
    -    P (foldWithAbort f f_abort s i) (fold f´ s i´).
    +    P (foldWithAbort f f_abort s i) (fold f' s i').

    @@ -248,25 +248,25 @@

    Library MSetsExtra.MSetFoldWithAbort

    :=
    -    (elt → A → A) →
    +    (elt → A → A) →
    f
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_gt
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value @@ -276,9 +276,9 @@

    Library MSetsExtra.MSetFoldWithAbort


    Definition foldWithAbortGtSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (lt : elt → elt → Prop)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
        (foldWithAbortGt : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=

        âˆ€
    @@ -290,7 +290,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (i i´ : A)
    +      (i i' : A)
          
    @@ -298,7 +298,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → A → A) (f´ : elt → A → A)
    +      (f : elt → A → A) (f' : elt → A → A)
          
    @@ -306,7 +306,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_gt : elt → A → bool)
    +      (f_gt : elt → A → bool)
          
    @@ -321,7 +321,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    equivalence relation on results @@ -333,39 +333,39 @@

    Library MSetsExtra.MSetFoldWithAbort

        
    -P is an equivalence relation +P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →

        
    -f is for the elements of s compatible with the equivalence relation P +f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st´ e, In e s → P st st´ → P (f e st) (f e st´)) →
    +    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →

        
    -f and f´ agree for the elements of s +f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e st) (f´ e st))) →
    +    (∀ e st, In e s → (P (f e st) (f' e st))) →

        
    -f_abort is OK, i.e. all other elements can be skipped without +f_abort is OK, i.e. all other elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 st´))) →
    +    (∀ e1 st,
    +        In e1 s → f_gt e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e1 e2 →
    +                        P st (f e2 st'))) →

        
    @@ -373,7 +373,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i´ →
    +    P i i' →

        
    @@ -381,7 +381,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The results are in equivalence relation
    -    P (foldWithAbortGt f f_gt s i) (fold f´ s i´).
    +    P (foldWithAbortGt f f_gt s i) (fold f' s i').

    @@ -424,31 +424,31 @@

    Library MSetsExtra.MSetFoldWithAbort

    :=
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_lt
    -    (elt → A → A) →
    +    (elt → A → A) →
    f
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_gt
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value @@ -458,9 +458,9 @@

    Library MSetsExtra.MSetFoldWithAbort


    Definition foldWithAbortGtLtSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (lt : elt → elt → Prop)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
        (foldWithAbortGtLt : ∀ {A : Type}, foldWithAbortGtLtType elt t A) : Prop :=

        âˆ€
    @@ -472,7 +472,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (i i´ : A)
    +      (i i' : A)
          
    @@ -480,7 +480,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → A → A) (f´ : elt → A → A)
    +      (f : elt → A → A) (f' : elt → A → A)
          
    @@ -488,7 +488,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_lt f_gt : elt → A → bool)
    +      (f_lt f_gt : elt → A → bool)
          
    @@ -503,7 +503,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    equivalence relation on results @@ -515,52 +515,52 @@

    Library MSetsExtra.MSetFoldWithAbort

        
    -P is an equivalence relation +P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →

        
    -f is for the elements of s compatible with the equivalence relation P +f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st´ e, In e s → P st st´ → P (f e st) (f e st´)) →
    +    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →

        
    -f and f´ agree for the elements of s +f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e st) (f´ e st))) →
    +    (∀ e st, In e s → (P (f e st) (f' e st))) →

        
    -f_lt is OK, i.e. smaller elements can be skipped without +f_lt is OK, i.e. smaller elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_lt e1 st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e2 e1 →
    -                        P st (f e2 st´))) →
    +    (∀ e1 st,
    +        In e1 s → f_lt e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e2 e1 →
    +                        P st (f e2 st'))) →

        
    -f_gt is OK, i.e. greater elements can be skipped without +f_gt is OK, i.e. greater elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 st´))) →
    +    (∀ e1 st,
    +        In e1 s → f_gt e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e1 e2 →
    +                        P st (f e2 st'))) →

        
    @@ -568,7 +568,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i´ →
    +    P i i' →

        
    @@ -576,7 +576,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The results are in equivalence relation
    -    P (foldWithAbortGtLt f_lt f f_gt s i) (fold f´ s i´).
    +    P (foldWithAbortGtLt f_lt f f_gt s i) (fold f' s i').

    @@ -584,14 +584,14 @@

    Library MSetsExtra.MSetFoldWithAbort

    We are interested in folding with abort mainly for runtime performance reasons of extracted code. The argument functions - f_lt, f_gt and f of foldWithAbortGtLt often share a large, + f_lt, f_gt and f of foldWithAbortGtLt often share a large, comparably expensive part of their computation.
    In order to further improve runtime performance, therefore another - version foldWithAbortPrecompute f_precompute f_lt f f_gt that - uses an extra function f_precompute to allows to compute the + version foldWithAbortPrecompute f_precompute f_lt f f_gt that + uses an extra function f_precompute to allows to compute the commonly used parts of these functions only once. This leads to the following definitions.
    @@ -625,37 +625,37 @@

    Library MSetsExtra.MSetFoldWithAbort

    :=

    -    (elt → B) →
    +    (elt → B) →
    f_precompute
    -    (elt → B → A → bool) →
    +    (elt → B → A → bool) →
    f_lt
    -    (elt → B → A → A) →
    +    (elt → B → A → A) →
    f
    -    (elt → B → A → bool) →
    +    (elt → B → A → bool) →
    f_gt
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value @@ -668,13 +668,13 @@

    Library MSetsExtra.MSetFoldWithAbort

    The specification is similar to the one without precompute, - but uses f_precompute so avoid doing computations multiple times + but uses f_precompute so avoid doing computations multiple times
    Definition foldWithAbortPrecomputeSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (lt : elt → elt → Prop)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
        (foldWithAbortPrecompute : ∀ {A B : Type}, foldWithAbortPrecomputeType elt t A B) : Prop :=

        âˆ€
    @@ -686,7 +686,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (i i´ : A)
    +      (i i' : A)
          
    @@ -694,7 +694,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_precompute : elt → B)
    +      (f_precompute : elt → B)
          
    @@ -702,7 +702,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → B → A → A) (f´ : elt → A → A)
    +      (f : elt → B → A → A) (f' : elt → A → A)
          
    @@ -710,7 +710,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_lt f_gt : elt → B → A → bool)
    +      (f_lt f_gt : elt → B → A → bool)
          
    @@ -725,7 +725,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    equivalence relation on results @@ -737,52 +737,52 @@

    Library MSetsExtra.MSetFoldWithAbort

        
    -P is an equivalence relation +P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →

        
    -f is for the elements of s compatible with the equivalence relation P +f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st´ e, In e s → P st st´ → P (f e (f_precompute e) st) (f e (f_precompute e) st´)) →
    +    (∀ st st' e, In e s → P st st' → P (f e (f_precompute e) st) (f e (f_precompute e) st')) →

        
    -f and f´ agree for the elements of s +f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e (f_precompute e) st) (f´ e st))) →
    +    (∀ e st, In e s → (P (f e (f_precompute e) st) (f' e st))) →

        
    -f_lt is OK, i.e. smaller elements can be skipped without +f_lt is OK, i.e. smaller elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_lt e1 (f_precompute e1) st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e2 e1 →
    -                        P st (f e2 (f_precompute e2) st´))) →
    +    (∀ e1 st,
    +        In e1 s → f_lt e1 (f_precompute e1) st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e2 e1 →
    +                        P st (f e2 (f_precompute e2) st'))) →

        
    -f_gt is OK, i.e. greater elements can be skipped without +f_gt is OK, i.e. greater elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 (f_precompute e1) st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 (f_precompute e2) st´))) →
    +    (∀ e1 st,
    +        In e1 s → f_gt e1 (f_precompute e1) st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e1 e2 →
    +                        P st (f e2 (f_precompute e2) st'))) →

        
    @@ -790,7 +790,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i´ →
    +    P i i' →

        
    @@ -798,13 +798,13 @@

    Library MSetsExtra.MSetFoldWithAbort

    The results are in equivalence relation
    -    P (foldWithAbortPrecompute f_precompute f_lt f f_gt s i) (fold f´ s i´).
    +    P (foldWithAbortPrecompute f_precompute f_lt f f_gt s i) (fold f' s i').

    -

    Module Types

    +

    Module Types

    @@ -815,7 +815,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Module Type HasFoldWithAbort (E : OrderedType) (Import C : WSetsOnWithDups E).
    +Module Type HasFoldWithAbort (E : OrderedType) (Import C : WSetsOnWithDups E).

      Parameter foldWithAbortPrecompute : ∀ {A B : Type},
    @@ -832,7 +832,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Derived operations

    +

    Derived operations

    @@ -844,7 +844,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Module HasFoldWithAbortOps (E : OrderedType) (C : WSetsOnWithDups E)
    +Module HasFoldWithAbortOps (E : OrderedType) (C : WSetsOnWithDups E)
                               (FT : HasFoldWithAbort E C).

    @@ -855,72 +855,43 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    First lets define the other folding with abort variants

    +

    First lets define the other folding with abort variants


    -  Definition foldWithAbortGtLt {A} f_lt (f : (elt → A → A)) f_gt :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_lt e st)
    +  Definition foldWithAbortGtLt {A} f_lt (f : (elt → A → A)) f_gt :=
    +    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_lt e st)
          (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).

      Lemma foldWithAbortGtLtSpec :
         foldWithAbortGtLtSpecPred E.lt In (@fold) (@foldWithAbortGtLt).
    -  Proof.
    -    rewrite /foldWithAbortGtLt /foldWithAbortGtLtSpecPred.
    -    intros A i i´ f f´ f_lt f_gt s P.
    -    move ⇒ H_f_compat H_ff´ H_lt H_gt H_ii´.
    -    apply foldWithAbortPrecomputeSpec ⇒ //.
    -  Qed.

    -  Definition foldWithAbortGt {A} (f : (elt → A → A)) f_gt :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun _ _ _ ⇒ false)
    +  Definition foldWithAbortGt {A} (f : (elt → A → A)) f_gt :=
    +    foldWithAbortPrecompute (fun _ ⇒ tt) (fun _ _ _ ⇒ false)
          (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).

      Lemma foldWithAbortGtSpec :
         foldWithAbortGtSpecPred E.lt In (@fold) (@foldWithAbortGt).
    -  Proof.
    -    rewrite /foldWithAbortGt /foldWithAbortGtSpecPred.
    -    intros A i i´ f f´ f_gt s P.
    -    move ⇒ H_f_compat H_ff´ H_gt H_ii´.
    -    apply foldWithAbortPrecomputeSpec ⇒ //.
    -  Qed.

    -  Definition foldWithAbort {A} (f : (elt → A → A)) f_abort :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_abort e st)
    +  Definition foldWithAbort {A} (f : (elt → A → A)) f_abort :=
    +    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_abort e st)
          (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_abort e st).

      Lemma foldWithAbortSpec :
         foldWithAbortSpecPred In (@fold) (@foldWithAbort).
    -  Proof.
    -    rewrite /foldWithAbort /foldWithAbortGtSpecPred.
    -    intros A i i´ f f´ f_abort s P.
    -    move ⇒ H_equiv_P H_f_compat H_ff´ H_abort H_ii´.
    -    have H_lt_neq: (∀ e1 e2, E.lt e1 e2 → e1 ≠ e2). {
    -      move ⇒ e1 e2 H_lt H_e12_eq.
    -      rewrite H_e12_eq in H_lt.
    -      have : ( Irreflexive E.lt) by apply StrictOrder_Irreflexive.
    -      rewrite /Irreflexive /Reflexive /complement ⇒ H.
    -      eapply H, H_lt.
    -    }
    -    apply foldWithAbortPrecomputeSpec ⇒ //; (
    -      move ⇒ e1 st H_in_e1 H_abort_e1 st´ e2 H_P H_in_e2 /H_lt_neq H_lt;
    -      apply (H_abort e1 st H_in_e1 H_abort_e1 st´ e2 H_P H_in_e2);
    -      by auto
    -    ).
    -  Qed.

    -

    Specialisations for equality

    +

    Specialisations for equality

    @@ -928,179 +899,115 @@

    Library MSetsExtra.MSetFoldWithAbort

    of an arbitrary equivalence relation on results.
    -  Lemma foldWithAbortPrecomputeSpec_Equal : ∀ (A B : Type) (i : A) (f_pre : elt → B)
    -      (f : elt → B → A → A) (f´ : elt → A → A) (f_lt f_gt : elt → B → A → bool) (s : t),
    +  Lemma foldWithAbortPrecomputeSpec_Equal : ∀ (A B : Type) (i : A) (f_pre : elt → B)
    +      (f : elt → B → A → A) (f' : elt → A → A) (f_lt f_gt : elt → B → A → bool) (s : t),

    -      (∀ e st, In e s → (f e (f_pre e) st = f´ e st)) →
    +      (∀ e st, In e s → (f e (f_pre e) st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_lt e1 (f_pre e1) st = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 →
    -                      (f e2 (f_pre e2) st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_lt e1 (f_pre e1) st = true →
    +          (∀ e2, In e2 s → E.lt e2 e1 →
    +                      (f e2 (f_pre e2) st = st))) →

          
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 (f_pre e1) st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 (f_pre e2) st = st))) →
    -
    -      (foldWithAbortPrecompute f_pre f_lt f f_gt s i) = (fold f´ s i).
    -  Proof.
    -    intros A B i f_pre f f´ f_lt f_gt s H_f´ H_lt H_gt.
    - -
    -     eapply (foldWithAbortPrecomputeSpec A B i i f_pre f f´); eauto. {
    -       apply eq_equivalence.
    -     } {
    -       move ⇒ st1 st2 e H_in → //.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_smaller st´ e2 <-.
    -       move : (H_lt e1 st H_e1_in H_do_smaller e2).
    -       intuition.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_greater st´ e2 <-.
    -       move : (H_gt e1 st H_e1_in H_do_greater e2).
    -       intuition.
    -     }
    -  Qed.
    - +      (∀ e1 st,
    +          In e1 s → f_gt e1 (f_pre e1) st = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 →
    +                      (f e2 (f_pre e2) st = st))) →
    +
    +      (foldWithAbortPrecompute f_pre f_lt f f_gt s i) = (fold f' s i).
    +
      Lemma foldWithAbortGtLtSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f´ : elt → A → A) (f_lt f_gt : elt → A → bool) (s : t),
    +      (f : elt → A → A) (f' : elt → A → A) (f_lt f_gt : elt → A → bool) (s : t),

    -      (∀ e st, In e s → (f e st = f´ e st)) →
    +      (∀ e st, In e s → (f e st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_lt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_lt e1 st = true →
    +          (∀ e2, In e2 s → E.lt e2 e1 →
    +                      (f e2 st = st))) →

          
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 st = st))) →
    -
    -      (foldWithAbortGtLt f_lt f f_gt s i) = (fold f´ s i).
    -  Proof.
    -    intros A i f f´ f_lt f_gt s H_f´ H_lt H_gt.
    - -
    -     eapply (foldWithAbortGtLtSpec A i i f f´); eauto. {
    -       apply eq_equivalence.
    -     } {
    -       move ⇒ st1 st2 e H_in → //.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_smaller st´ e2 <-.
    -       move : (H_lt e1 st H_e1_in H_do_smaller e2).
    -       intuition.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_greater st´ e2 <-.
    -       move : (H_gt e1 st H_e1_in H_do_greater e2).
    -       intuition.
    -     }
    -  Qed.
    - +      (∀ e1 st,
    +          In e1 s → f_gt e1 st = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 →
    +                      (f e2 st = st))) →
    +
    +      (foldWithAbortGtLt f_lt f f_gt s i) = (fold f' s i).
    +
      Lemma foldWithAbortGtSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f´ : elt → A → A) (f_gt : elt → A → bool) (s : t),
    +      (f : elt → A → A) (f' : elt → A → A) (f_gt : elt → A → bool) (s : t),

    -      (∀ e st, In e s → (f e st = f´ e st)) →
    +      (∀ e st, In e s → (f e st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_gt e1 st = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 →
    +                      (f e2 st = st))) →

    -      (foldWithAbortGt f f_gt s i) = (fold f´ s i).
    -  Proof.
    -    intros A i f f´ f_gt s H_f´ H_gt.
    - -
    -     eapply (foldWithAbortGtSpec A i i f f´); eauto. {
    -       apply eq_equivalence.
    -     } {
    -       move ⇒ st1 st2 e H_in → //.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_greater st´ e2 <-.
    -       move : (H_gt e1 st H_e1_in H_do_greater e2).
    -       intuition.
    -     }
    -  Qed.
    - +      (foldWithAbortGt f f_gt s i) = (fold f' s i).
    +
      Lemma foldWithAbortSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f´ : elt → A → A) (f_abort : elt → A → bool) (s : t),
    +      (f : elt → A → A) (f' : elt → A → A) (f_abort : elt → A → bool) (s : t),

    -      (∀ e st, In e s → (f e st = f´ e st)) →
    +      (∀ e st, In e s → (f e st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_abort e1 st = true →
    -          (∀ e2, In e2 s → e1 ≠ e2 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_abort e1 st = true →
    +          (∀ e2, In e2 s → e1 ≠ e2 →
    +                      (f e2 st = st))) →

    -      (foldWithAbort f f_abort s i) = (fold f´ s i).
    -  Proof.
    -     intros A i f f´ f_abort s H_f´ H_abort.
    - -
    -     eapply (foldWithAbortSpec A i i f f´); eauto. {
    -       apply eq_equivalence.
    -     } {
    -       move ⇒ st1 st2 e H_in → //.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_abort st´ e2 <-.
    -       move : (H_abort e1 st H_e1_in H_do_abort e2).
    -       intuition.
    -     }
    -  Qed.
    - +      (foldWithAbort f f_abort s i) = (fold f' s i).
    +
    -

    FoldWithAbortSpecArgs

    +

    FoldWithAbortSpecArgs

    While folding, we are often interested in skipping elements that do not - satisfy a certain property P. This needs expressing in terms of + satisfy a certain property P. This needs expressing in terms of skips of smaller of larger elements in order to be done efficiently by our folding functions. Formally, this leads to the definition of foldWithAbortSpecForPred.
    - Given a FoldWithAbortSpecArg for a predicate P and a - set s, many operations can be implemented efficiently. Below we will provide - efficient versions of filter, choose, ∃, ∀ and more. + Given a FoldWithAbortSpecArg for a predicate P and a + set s, many operations can be implemented efficiently. Below we will provide + efficient versions of filter, choose, ∃, ∀ and more.
      Record FoldWithAbortSpecArg {B} := {
    -     fwasa_f_pre : (elt → B);
    +     fwasa_f_pre : (elt → B);
    The precompute function
    -     fwasa_f_lt : (elt → B → bool);
    +     fwasa_f_lt : (elt → B → bool);
    f_lt without state argument
    -     fwasa_f_gt : (elt → B → bool);
    +     fwasa_f_gt : (elt → B → bool);
    f_gt without state argument
    -     fwasa_P´ : (elt → B → bool)
    +     fwasa_P' : (elt → B → bool)
    the predicate P @@ -1112,20 +1019,20 @@

    Library MSetsExtra.MSetFoldWithAbort

    -foldWithAbortSpecForPred s P fwasa holds, if - the argument fwasa fits the predicate P for set s. +foldWithAbortSpecForPred s P fwasa holds, if + the argument fwasa fits the predicate P for set s.
      Definition foldWithAbortSpecArgsForPred {A : Type}
    -     (s : t) (P : elt → bool) (fwasa : @FoldWithAbortSpecArg A) :=
    +     (s : t) (P : elt → bool) (fwasa : @FoldWithAbortSpecArg A) :=

         
    -the predicate P´ coincides for s and the given precomputation with P +the predicate P' coincides for s and the given precomputation with P
    -     (∀ e, In e s → (fwasa_P´ fwasa e (fwasa_f_pre fwasa e) = P e)) ∧
    +     (∀ e, In e s → (fwasa_P' fwasa e (fwasa_f_pre fwasa e) = P e)) ∧

         
    @@ -1134,9 +1041,9 @@

    Library MSetsExtra.MSetFoldWithAbort

    don't satisfy predicate P.
    -     (∀ e1,
    -          In e1 s → fwasa_f_lt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 → (P e2 = false))) ∧
    +     (∀ e1,
    +          In e1 s → fwasa_f_lt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    +          (∀ e2, In e2 s → E.lt e2 e1 → (P e2 = false))) ∧

         
    @@ -1145,9 +1052,9 @@

    Library MSetsExtra.MSetFoldWithAbort

    don't satisfy predicate P.
    -     (∀ e1,
    -          In e1 s → fwasa_f_gt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 → (P e2 = false))).
    +     (∀ e1,
    +          In e1 s → fwasa_f_gt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 → (P e2 = false))).

    @@ -1155,237 +1062,69 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Filter with abort

    +

    Filter with abort

      Definition filter_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
         @foldWithAbortPrecompute t B (fwasa_f_pre fwasa) (fun e p _ ⇒ fwasa_f_lt fwasa e p)
    -       (fun e e_pre s ⇒ if fwasa_P´ fwasa e e_pre then add e s else s)
    +       (fun e e_pre s ⇒ if fwasa_P' fwasa e e_pre then add e s else s)
           (fun e p _ ⇒ fwasa_f_gt fwasa e p) s empty.

      Lemma filter_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
        Equal (filter_with_abort fwasa s)
              (filter P s).
    -  Proof.
    -    unfold foldWithAbortSpecArgsForPred.
    -    move ⇒ [] f_pre f_lt f_gt P´ P s /=.
    -    move ⇒ [H_f´] [H_lt] H_gt H_proper.
    -    rewrite /filter_with_abort /=.
    - -
    -    have → : (foldWithAbortPrecompute f_pre (fun e p _ ⇒ f_lt e p)
    -     (fun (e : elt) (e_pre : B) (s0 : t) ⇒
    -      if P´ e e_pre then add e s0 else s0) (fun e p _ ⇒ f_gt e p) s empty =
    -     (fold (fun e s0 ⇒ if P e then add e s0 else s0) s empty)). {
    -      apply foldWithAbortPrecomputeSpec_Equal. {
    -        intros e st H_e_in.
    -        rewrite H_f´ //.
    -      } {
    -        intros e1 st H_e1_in H_f_lt_eq e2 H_e2_in H_lt_e2_e1.
    -        rewrite (H_f´ _ H_e2_in).
    -        suff → : (P e2 = false) by done.
    -        apply (H_lt e1); eauto.
    -      } {
    -        intros e1 st H_e1_in H_f_gt_eq e2 H_e2_in H_gt_e2_e1.
    -        rewrite (H_f´ _ H_e2_in).
    -        suff → : (P e2 = false) by done.
    -        apply (H_gt e1); eauto.
    -      }
    -    }
    -
    -    rewrite /Equal ⇒ e.
    -    rewrite fold_spec.
    -    setoid_rewrite filter_spec ⇒ //.
    - -
    -    suff → : ∀ acc, In e
    -      (fold_left
    -         (flip (fun (e0 : elt) (s0 : t) ⇒ if P e0 then add e0 s0 else s0))
    -         (elements s) acc) ↔ (InA E.eq e (elements s) ∧ P e = true) ∨ (In e acc). {
    -      rewrite elements_spec1.
    -      suff : (¬In e empty) by tauto.
    -      apply empty_spec.
    -    }
    -    induction (elements s) as [| x xs IH] ⇒ acc. {
    -      rewrite /= InA_nil. tauto.
    -    } {
    -      rewrite /= /flip IH InA_cons.
    -      case_eq (P x). {
    -        rewrite add_spec.
    -        intuition.
    -        left.
    -        rewrite H0.
    -        split ⇒ //.
    -        left.
    -        apply Equivalence_Reflexive.
    -      } {
    -        intuition.
    -        contradict H2.
    -        setoid_rewrite H1.
    -        by rewrite H.
    -      }
    -    }
    -  Qed.
    - -
    -
    - -
    -

    Choose with abort

    + +
    +
    + +
    +

    Choose with abort

      Definition choose_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
         foldWithAbortPrecompute (fwasa_f_pre fwasa)
    -       (fun e p st ⇒ match st with None ⇒ (fwasa_f_lt fwasa e p) | _ ⇒ true end)
    -       (fun e e_pre st ⇒ match st with None ⇒
    -          if (fwasa_P´ fwasa e e_pre) then Some e else None | _ ⇒ st end)
    +       (fun e p st ⇒ match st with None ⇒ (fwasa_f_lt fwasa e p) | _ ⇒ true end)
    +       (fun e e_pre st ⇒ match st with None ⇒
    +          if (fwasa_P' fwasa e e_pre) then Some e else None | _ ⇒ st end)

    -       (fun e p st ⇒ match st with None ⇒ (fwasa_f_gt fwasa e p) | _ ⇒ true end)
    -       s None.
    +       (fun e p st ⇒ match st with None ⇒ (fwasa_f_gt fwasa e p) | _ ⇒ true end)
    +       s None.

      Lemma choose_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (match (choose_with_abort fwasa s) with
    -       | None ⇒ (∀ e, In e s → P e = false)
    -       | Some e ⇒ In e s ∧ (P e = true)
    -     end).
    -  Proof.
    -    rewrite /foldWithAbortSpecArgsForPred.
    -    move ⇒ [] f_pre f_lt f_gt P´ P s /=.
    -    move ⇒ [H_f´] [H_lt] H_gt H_proper.
    - -
    -    set fwasa := {|
    -       fwasa_f_pre := f_pre;
    -       fwasa_f_lt := f_lt;
    -       fwasa_f_gt := f_gt;
    -       fwasa_P´ := P´ |}.
    - -
    -    suff : (match (choose_with_abort fwasa s) with
    -       | None ⇒ (∀ e, InA E.eq e (elements s) → P e = false)
    -       | Some e ⇒ InA E.eq e (elements s) ∧ (P e = true)
    -     end). {
    -       case (choose_with_abort fwasa s). {
    -         move ⇒ e.
    -         rewrite elements_spec1 //.
    -       } {
    -         move ⇒ H e H_in.
    -         apply H.
    -         rewrite elements_spec1 //.
    -       }
    -    }
    -
    -    have → : (choose_with_abort fwasa s =
    -      (fold (fun e st ⇒
    -        match st with
    -          | None ⇒ if P e then Some e else None
    -          | _ ⇒ st end) s None)). {
    -      apply foldWithAbortPrecomputeSpec_Equal. {
    -        intros e st H_e_in.
    -        case st ⇒ //=.
    -        rewrite H_f´ //.
    -      } {
    -        move ⇒ e1 [] //= H_e1_in H_f_lt_eq e2 H_e2_in H_lt_e2_e1.
    -        rewrite (H_f´ _ H_e2_in).
    -        case_eq (P e2) ⇒ // H_P_e2.
    -        contradict H_P_e2.
    -        apply not_true_iff_false, (H_lt e1); auto.
    -      } {
    -        move ⇒ e1 [] //= H_e1_in H_f_gt_eq e2 H_e2_in H_gt_e2_e1.
    -        rewrite (H_f´ _ H_e2_in).
    -        case_eq (P e2) ⇒ // H_P_e2.
    -        contradict H_P_e2.
    -        apply not_true_iff_false, (H_gt e1); auto.
    -      }
    -    }
    -
    -    rewrite fold_spec /flip.
    -    induction (elements s) as [| x xs IH]. {
    -      rewrite /=.
    -      move ⇒ e /InA_nil //.
    -    } {
    -      case_eq (P x) ⇒ H_Px; rewrite /= H_Px. {
    -        have → : ∀ xs, fold_left (fun (x0 : option elt) (y : elt) ⇒
    -                    match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    -                    end) xs (Some x) = Some x. {
    -          move ⇒ ys.
    -          induction ys ⇒ //.
    -        }
    -        split; last assumption.
    -        apply InA_cons_hd.
    -        apply E.eq_equiv.
    -      } {
    -        move : IH.
    -        case (fold_left
    -          (fun (x0 : option elt) (y : elt) ⇒
    -             match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    -             end) xs None). {
    -
    -             move ⇒ e [H_e_in] H_Pe.
    -             split; last assumption.
    -             apply InA_cons_tl ⇒ //.
    -        } {
    -          move ⇒ H_e_nin e H_e_in.
    -          have : (InA E.eq e xs ∨ (E.eq e x)). {
    -            inversion H_e_in; tauto.
    -          }
    -          move ⇒ []. {
    -            apply H_e_nin.
    -          } {
    -            move ⇒ → //.
    -          }
    -        }
    -      }
    -    }
    -  Qed.
    - -
    -
    - -
    -

    Exists and Forall with abort

    +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    (match (choose_with_abort fwasa s) with
    +       | None ⇒ (∀ e, In e s → P e = false)
    +       | Some e ⇒ In e s ∧ (P e = true)
    +     end).
    + +
    +
    + +
    +

    Exists and Forall with abort

      Definition exists_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
        match choose_with_abort fwasa s with
    -      | None ⇒ false
    -      | Some _ ⇒ true
    +      | None ⇒ false
    +      | Some _ ⇒ true
        end.

      Lemma exists_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (exists_with_abort fwasa s =
    -     exists_ P s).
    -  Proof.
    -    intros fwasa P s H_fwasa H_proper.
    -    apply Logic.eq_sym.
    -    rewrite /exists_with_abort.
    -    move : (choose_with_abort_spec _ _ _ H_fwasa H_proper).
    -    case (choose_with_abort fwasa s). {
    -      move ⇒ e [H_e_in] H_Pe.
    -      rewrite exists_spec /Exists.
    -      by ∃ e.
    -    } {
    -      move ⇒ H_not_ex.
    -      apply not_true_iff_false.
    -      rewrite exists_spec /Exists.
    -      move ⇒ [e] [H_in] H_pe.
    -      move : (H_not_ex e H_in).
    -      rewrite H_pe //.
    -    }
    -  Qed.
    - +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    (exists_with_abort fwasa s =
    +     exists_ P s).
    +
    @@ -1394,38 +1133,15 @@

    Library MSetsExtra.MSetFoldWithAbort

      Definition forall_with_abort {B} fwasa s :=
    -     negb (@exists_with_abort B fwasa s).
    +     negb (@exists_with_abort B fwasa s).

      Lemma forall_with_abort_spec {B} : ∀ fwasa s P,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (forall_with_abort fwasa s =
    -     for_all (fun e ⇒ negb (P e)) s).
    -  Proof.
    -    intros fwasa s P H_ok H_proper.
    -    rewrite /forall_with_abort exists_with_abort_spec; auto.
    - -
    -    rewrite eq_iff_eq_true negb_true_iff -not_true_iff_false.
    -    rewrite exists_spec.
    -    setoid_rewrite for_all_spec; last solve_proper.
    - -
    -    rewrite /Exists /For_all.
    -    split. {
    -      move ⇒ H_pre x H_x_in.
    -      rewrite negb_true_iff -not_true_iff_false ⇒ H_Px.
    -      apply H_pre.
    -      by ∃ x.
    -    } {
    -      move ⇒ H_pre [x] [H_x_in] H_P_x.
    -      move : (H_pre x H_x_in).
    -      rewrite H_P_x.
    -      done.
    -    }
    -  Qed.
    - +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    (forall_with_abort fwasa s =
    +     for_all (fun e ⇒ negb (P e)) s).
    +
    End HasFoldWithAbortOps.
    @@ -1433,29 +1149,29 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Modules Types For Sets with Fold with Abort

    +

    Modules Types For Sets with Fold with Abort

    Module Type WSetsWithDupsFoldA.
    -  Declare Module E : OrderedType.
    +  Declare Module E : OrderedType.
      Include WSetsOnWithDups E.
      Include HasFoldWithAbort E.
      Include HasFoldWithAbortOps E.
    End WSetsWithDupsFoldA.

    -Module Type WSetsWithFoldA <: WSets.
    -  Declare Module E : OrderedType.
    -  Include WSetsOn E.
    +Module Type WSetsWithFoldA <: WSets.
    +  Declare Module E : OrderedType.
    +  Include WSetsOn E.
      Include HasFoldWithAbort E.
      Include HasFoldWithAbortOps E.
    End WSetsWithFoldA.

    -Module Type SetsWithFoldA <: Sets.
    -  Declare Module E : OrderedType.
    -  Include SetsOn E.
    +Module Type SetsWithFoldA <: Sets.
    +  Declare Module E : OrderedType.
    +  Include SetsOn E.
      Include HasFoldWithAbort E.
      Include HasFoldWithAbortOps E.
    End SetsWithFoldA.
    @@ -1464,21 +1180,21 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Implementations

    +

    Implementations

    -

    GenTree implementation

    +

    GenTree implementation

    Finally, provide such a fold with abort operation for generic trees.
    -Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
    -  (Import Raw:Ops E I)
    -  (M : MSetGenTree.Props E I Raw).
    +Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
    +  (Import Raw:Ops E I)
    +  (M : MSetGenTree.Props E I Raw).

    -  Fixpoint foldWithAbort_Raw {A B: Type} (f_pre : E.t → B) f_lt (f: E.t → B → A → A) f_gt t (base: A) : A :=
    +  Fixpoint foldWithAbort_Raw {A B: Type} (f_pre : E.t → B) f_lt (f: E.t → B → A → A) f_gt t (base: A) : A :=
        match t with
        | Raw.Leaf ⇒ base
        | Raw.Node _ l x r ⇒
    @@ -1490,225 +1206,54 @@

    Library MSetsExtra.MSetFoldWithAbort

        end.

    -  Lemma foldWithAbort_RawSpec : ∀ (A B : Type) (i i´ : A) (f_pre : E.t → B)
    -      (f : E.t → B → A → A) (f´ : E.t → A → A) (f_lt f_gt : E.t → B → A → bool) (s : Raw.tree)
    -      (P : A → A → Prop),
    +  Lemma foldWithAbort_RawSpec : ∀ (A B : Type) (i i' : A) (f_pre : E.t → B)
    +      (f : E.t → B → A → A) (f' : E.t → A → A) (f_lt f_gt : E.t → B → A → bool) (s : Raw.tree)
    +      (P : A → A → Prop),

    -      (M.bst s) →
    -      Equivalence P →
    -      (∀ st st´ e, M.In e s → P st st´ → P (f e (f_pre e) st) (f e (f_pre e) st´)) →
    -      (∀ e st, M.In e s → P (f e (f_pre e) st) (f´ e st)) →
    +      (M.bst s) →
    +      Equivalence P →
    +      (∀ st st' e, M.In e s → P st st' → P (f e (f_pre e) st) (f e (f_pre e) st')) →
    +      (∀ e st, M.In e s → P (f e (f_pre e) st) (f' e st)) →

          
    -      (∀ e1 st,
    -          M.In e1 s → f_lt e1 (f_pre e1) st = true →
    -          (∀ st´ e2, P st st´ →
    -                          M.In e2 s → E.lt e2 e1 →
    -                          P st (f e2 (f_pre e2) st´))) →
    +      (∀ e1 st,
    +          M.In e1 s → f_lt e1 (f_pre e1) st = true →
    +          (∀ st' e2, P st st' →
    +                          M.In e2 s → E.lt e2 e1 →
    +                          P st (f e2 (f_pre e2) st'))) →

          
    -      (∀ e1 st,
    -          M.In e1 s → f_gt e1 (f_pre e1) st = true →
    -          (∀ st´ e2, P st st´ →
    -                          M.In e2 s → E.lt e1 e2 →
    -                          P st (f e2 (f_pre e2) st´))) →
    -
    -      P i i´ →
    -      P (foldWithAbort_Raw f_pre f_lt f f_gt s i) (fold f´ s i´).
    -   Proof.
    -     intros A B i i´ f_pre f f´ f_lt f_gt s P.
    -     move ⇒ H_bst H_equiv_P H_P_f H_f´ H_RL H_RG.
    - -
    -     set base := s.
    - -
    -     move : i i´.
    -     have : (∀ e, M.In e base → M.In e s). {
    -       rewrite /In /base //.
    -     }
    -     have : M.bst base. {
    -       apply H_bst.
    -     }
    -     move : base.
    -     clear H_bst.
    - -
    -     induction base as [|c l IHl e r IHr] using M.tree_ind. {
    -       rewrite /foldWithAbort_Raw /Raw.fold.
    -       move ⇒ _ _ i i´ //.
    -     } {
    -       move ⇒ H_bst H_sub i i´ H_P_ii´.
    - -
    -       have [H_bst_l [H_bst_r [H_lt_tree_l H_gt_tree_r]]]:
    -         M.bst l ∧ M.bst r ∧ M.lt_tree e l ∧ M.gt_tree e r. {
    -         inversion H_bst. done.
    -       }
    -       
    -       have H_sub_l : (∀ e0 : E.t, M.In e0 l → M.In e0 s ∧ E.lt e0 e). {
    -         intros e0 H_in_l.
    -         split; last by apply H_lt_tree_l.
    -         eapply H_sub.
    -         rewrite /M.In M.In_node_iff.
    -         tauto.
    -       }
    -       move : (IHl H_bst_l) ⇒ {IHl} IHl {H_bst_l} {H_lt_tree_l}.
    -       have H_sub_r : (∀ e0 : E.t, M.In e0 r → M.In e0 s ∧ E.lt e e0). {
    -         intros e0 H_in_r.
    -         split; last by apply H_gt_tree_r.
    -         eapply H_sub.
    -         rewrite /M.In M.In_node_iff.
    -         tauto.
    -       }
    -       move : (IHr H_bst_r) ⇒ {IHr} IHr {H_bst_r} {H_gt_tree_r}.
    -       have H_in_e : M.In e s. {
    -         eapply H_sub.
    -         rewrite /M.In M.In_node_iff.
    -         right; left.
    -         apply Equivalence_Reflexive.
    -       }
    -       move ⇒ {H_sub}.
    - -
    -       rewrite /=.
    -       set st0 := if f_lt e (f_pre e) i then i else foldWithAbort_Raw f_pre f_lt f f_gt l i.
    -       set st0´ := Raw.fold f´ l i´.
    -       set st1 := f e (f_pre e) st0.
    -       set st1´ := f´ e st0´.
    -       set st2 := if f_gt e (f_pre e) st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1.
    -       set st2´ := Raw.fold f´ r st1´.
    - -
    -       have H_P_st0 : P st0 st0´. {
    -         rewrite /st0 /st0´.
    -         case_eq (f_lt e (f_pre e) i). {
    -           move ⇒ H_fl_eq.
    -           move : H_P_ii´ H_sub_l.
    -           move : H_equiv_P H_f´ (H_RL _ _ H_in_e H_fl_eq).
    -           rewrite /M.lt_tree. clear.
    -           move ⇒ H_equiv_P H_f´ H_RL.
    -           move : i´.
    -           induction l as [|c l IHl e´ r IHr] using M.tree_ind. {
    -             done.
    -           } {
    -             intros i´ H_P_ii´ H_sub_l.
    -             rewrite /=.
    -             apply IHr; last first. {
    -               move ⇒ y H_y_in.
    -               apply H_sub_l.
    -               rewrite /M.In M.In_node_iff. tauto.
    -             }
    -             have [] : (M.In e´ s ∧ E.lt e´ e). {
    -               apply H_sub_l.
    -               rewrite /M.In M.In_node_iff.
    -               right; left.
    -               apply Equivalence_Reflexive.
    -             }
    -             move ⇒ H_e´_in H_lt_in.
    -             suff H_P_i : (P i (f e´ (f_pre e´) (fold f´ l i´))). {
    -               eapply Equivalence_Transitive; first apply H_P_i.
    -               by apply H_f´.
    -             }
    -             eapply H_RL ⇒ //.
    -             apply IHl; last first. {
    -               move ⇒ y H_y_in.
    -               apply H_sub_l.
    -               rewrite /M.In M.In_node_iff. tauto.
    -             }
    -             assumption.
    -           }
    -         } {
    -           move ⇒ _.
    -           apply IHl ⇒ //.
    -           eapply H_sub_l.
    -         }
    -       }
    -       have H_P_st1 : P st1 st1´. {
    -         rewrite /st1 /st1´.
    -         rewrite -H_f´ //.
    -         apply H_P_f ⇒ //.
    -       }
    -       have H_P_st2 : P st2 st2´. {
    -         rewrite /st2 /st2´.
    -         clearbody st1 st1´.
    -         case_eq (f_gt e (f_pre e) st1). {
    -           move ⇒ H_gt_eq.
    -           move : H_P_st1 H_sub_r.
    -           move : H_equiv_P (H_RG _ _ H_in_e H_gt_eq) H_f´.
    -           unfold M.gt_tree. clear.
    -           move ⇒ H_equiv_P H_RG H_f´.
    -           move : st1´.
    -           induction r as [|c l IHl e´ r IHr] using M.tree_ind. {
    -             done.
    -           } {
    -             intros st1´ H_P_st1 H_sub_r.
    -             rewrite /=.
    -             apply IHr; last first. {
    -               move ⇒ y H_y_in.
    -               apply H_sub_r.
    -               rewrite /M.In M.In_node_iff. tauto.
    -             }
    -             have [] : (M.In e´ s ∧ E.lt e e´). {
    -               apply H_sub_r.
    -               rewrite /M.In M.In_node_iff.
    -               right; left.
    -               apply Equivalence_Reflexive.
    -             }
    -             move ⇒ H_e´_in H_lt_ee´.
    -             suff H_P_st1_aux : (P st1 (f e´ (f_pre e´) (fold f´ l st1´))). {
    -               eapply Equivalence_Transitive; first apply H_P_st1_aux.
    -               by apply H_f´.
    -             }
    -             eapply H_RG ⇒ //.
    -             apply IHl; last first. {
    -               move ⇒ y H_y_in.
    -               apply H_sub_r.
    -               rewrite /M.In M.In_node_iff. tauto.
    -             }
    -             assumption.
    -           }
    -         } {
    -           move ⇒ _.
    -           apply IHr ⇒ //.
    -           eapply H_sub_r.
    -         }
    -       }
    -       done.
    -     }
    -  Qed.
    -End MakeGenTreeFoldA.
    - -
    -
    - -
    -

    AVL implementation

    +      (∀ e1 st,
    +          M.In e1 s → f_gt e1 (f_pre e1) st = true →
    +          (∀ st' e2, P st st' →
    +                          M.In e2 s → E.lt e1 e2 →
    +                          P st (f e2 (f_pre e2) st'))) →
    +
    +      P i i' →
    +      P (foldWithAbort_Raw f_pre f_lt f f_gt s i) (fold f' s i').
    + End MakeGenTreeFoldA.
    + +
    +
    + +
    +

    AVL implementation

    The generic tree implementation naturally leads to an AVL one.

    -Module MakeAVLSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetAVL.Make X.
    -  Include MakeGenTreeFoldA X Z_as_Int Raw Raw.
    +Module MakeAVLSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    +  Include MSetAVL.Make X.
    +  Include MakeGenTreeFoldA X Z_as_Int Raw Raw.

    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    +  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
        foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.

      Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    -  Proof.
    -    intros A B i i´ f_pre f f´ f_lt f_gt s P.
    -    move ⇒ H_P_f H_f´ H_RL H_RG H_P_ii´.
    - -
    -    rewrite /foldWithAbortPrecompute /fold.
    -    apply foldWithAbort_RawSpec ⇒ //.
    -    case s. rewrite /this /Raw.Ok //.
    -  Qed.

      Include HasFoldWithAbortOps X.
    @@ -1720,30 +1265,21 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    RBT implementation

    +

    RBT implementation

    The generic tree implementation naturally leads to an RBT one.
    -Module MakeRBTSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetRBT.Make X.
    -  Include MakeGenTreeFoldA X Color Raw Raw.
    +Module MakeRBTSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    +  Include MSetRBT.Make X.
    +  Include MakeGenTreeFoldA X Color Raw Raw.

    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    +  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
        foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.

      Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    -  Proof.
    -    intros A B i i´ f_pre f f´ f_lt f_gt s P.
    -    move ⇒ H_P_f H_f´ H_RL H_RG H_P_ii´.
    - -
    -    rewrite /foldWithAbortPrecompute /fold.
    -    apply foldWithAbort_RawSpec ⇒ //.
    -    case s. rewrite /this /Raw.Ok //.
    -  Qed.

      Include HasFoldWithAbortOps X.
    @@ -1755,19 +1291,19 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Sorted Lists Implementation

    +

    Sorted Lists Implementation

    -Module MakeListSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetList.Make X.
    +Module MakeListSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    +  Include MSetList.Make X.

    -  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    -    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    +  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    +    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
      match t with
    -    | nil ⇒ acc
    -    | x :: xs ⇒ (
    +    | nil ⇒ acc
    +    | x :: xs ⇒ (
            let pre_x := f_pre x in
            let acc := f x (pre_x) acc in
            if (f_gt x pre_x acc) then
    @@ -1783,92 +1319,7 @@

    Library MSetsExtra.MSetFoldWithAbort


      Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    -  Proof.
    -    intros A B i i´ f_pre f f´ f_lt f_gt.
    -    move ⇒ [] l H_is_ok_l P H_equiv_P.
    -    rewrite /fold /foldWithAbortPrecompute /In /this /Raw.In /Raw.fold.
    -    move ⇒ H_P_f H_f´ H_RL H_RG.
    - -
    -    set base := l.
    -    move : i i´.
    -    have : (∀ e, InA X.eq e base → InA X.eq e l). {
    -      rewrite /base //.
    -    }
    -    have : sort X.lt base. {
    -      rewrite Raw.isok_iff /base //.
    -    }
    -    clear H_is_ok_l.
    - -
    -    induction base as [| x xs IH]. {
    -      by simpl.
    -    }
    -    move ⇒ H_sort H_in_xxs i i´ Pii´ /=.
    - -
    -    have [H_sort_xs H_hd_rel {H_sort}] : Sorted X.lt xs ∧ HdRel X.lt x xs. {
    -      by inversion H_sort.
    -    }
    -
    -    move : H_hd_rel.
    -    rewrite (Raw.ML.Inf_alt x H_sort_xs) ⇒ H_lt_xs.
    - -
    -    have H_x_in_l : InA X.eq x l. {
    -      apply H_in_xxs.
    -      apply InA_cons_hd.
    -      apply X.eq_equiv.
    -    }
    -    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
    -      intros e H_in.
    -      apply H_in_xxs, InA_cons_tl ⇒ //.
    -    }
    -   
    -    have H_P_next : P (f x (f_pre x) i) (flip f´ i´ x). {
    -      rewrite /flip -H_f´ //.
    -      apply H_P_f ⇒ //.
    -    }
    -
    -    case_eq (f_gt x (f_pre x) (f x (f_pre x) i)); last first. {
    -      move ⇒ _.
    -      apply IH ⇒ //.
    -    } {
    -      move ⇒ H_gt.
    -      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    -         P (f x (f_pre x) i) (fold_left (flip f´) xs st)). {
    -         apply H_suff ⇒ //.
    -      }
    -      move : H_in_xs H_lt_xs.
    - -
    -      clear IH H_in_xxs H_sort_xs.
    -      move : (H_RG x _ H_x_in_l H_gt) ⇒ H_RG_x.
    -      induction xs as [| x´ xs´ IH´]. {
    -        done.
    -      } {
    -        intros H_in_xs H_lt_xs st H_P_st.
    -        rewrite /=.
    -        have H_x´_in_l : InA X.eq x´ l. {
    -          apply H_in_xs.
    -          apply InA_cons_hd, X.eq_equiv.
    -        }
    -        apply IH´. {
    -          intros e H.
    -          apply H_in_xs, InA_cons_tl ⇒ //.
    -        } {
    -          intros e H.
    -          apply H_lt_xs, InA_cons_tl ⇒ //.
    -        } {
    -          rewrite /flip -H_f´ //.
    -          apply H_RG_x ⇒ //.
    -          apply H_lt_xs.
    -          apply InA_cons_hd, X.eq_equiv.
    -        }
    -      }
    -    }
    -  Qed.
    - +
      Include HasFoldWithAbortOps X.
    @@ -1879,24 +1330,24 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Unsorted Lists without Dups Implementation

    +

    Unsorted Lists without Dups Implementation

    -Module MakeWeakListSetsWithFoldA (X : OrderedType) <: WSetsWithFoldA with Module E := X.
    -  Module Raw := MSetWeakList.MakeRaw X.
    +Module MakeWeakListSetsWithFoldA (X : OrderedType) <: WSetsWithFoldA with Module E := X.
    +  Module Raw := MSetWeakList.MakeRaw X.
      Module E := X.
    -  Include WRaw2SetsOn E Raw.
    +  Include WRaw2SetsOn E Raw.

    -  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    -    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    +  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    +    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
      match t with
    -    | nil ⇒ acc
    -    | x :: xs ⇒ (
    +    | nil ⇒ acc
    +    | x :: xs ⇒ (
            let pre_x := f_pre x in
            let acc := f x (pre_x) acc in
    -        if (f_gt x pre_x acc) && (f_lt x pre_x acc) then
    +        if (f_gt x pre_x acc) && (f_lt x pre_x acc) then
              acc
            else
              foldWithAbortRaw f_pre f_lt f f_gt xs acc
    @@ -1909,114 +1360,7 @@

    Library MSetsExtra.MSetFoldWithAbort


      Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    -  Proof.
    -    intros A B i i´ f_pre f f´ f_lt f_gt.
    -    move ⇒ [] l H_is_ok_l P H_P_equiv.
    -    rewrite /fold /foldWithAbortPrecompute /In /this /Raw.In /Raw.fold.
    -    move ⇒ H_P_f H_f´ H_RL H_RG.
    - -
    -    set base := l.
    -    move : i i´.
    -    have : (∀ e, InA X.eq e base → InA X.eq e l). {
    -      rewrite /base //.
    -    }
    -    have : NoDupA X.eq base. {
    -      apply H_is_ok_l.
    -    }
    -    clear H_is_ok_l.
    - -
    -    induction base as [| x xs IH]. {
    -      by simpl.
    -    }
    -    move ⇒ H_nodup_xxs H_in_xxs i i´ Pii´ /=.
    - -
    -    have [H_nin_x_xs H_nodup_xs {H_nodup_xxs}] : ¬ InA X.eq x xs ∧ NoDupA X.eq xs. {
    -      by inversion H_nodup_xxs.
    -    }
    -
    -    have H_x_in_l : InA X.eq x l. {
    -      apply H_in_xxs.
    -      apply InA_cons_hd.
    -      apply X.eq_equiv.
    -    }
    -    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
    -      intros e H_in.
    -      apply H_in_xxs, InA_cons_tl ⇒ //.
    -    }
    -   
    -    have H_P_next : P (f x (f_pre x) i) (flip f´ i´ x). {
    -      rewrite /flip -H_f´ //.
    -      apply H_P_f ⇒ //.
    -    }
    -
    -    case_eq (f_gt x (f_pre x) (f x (f_pre x) i) &&
    -             f_lt x (f_pre x) (f x (f_pre x) i)); last first. {
    -      move ⇒ _.
    -      apply IH ⇒ //.
    -    } {
    -      move ⇒ /andb_true_iff [H_gt H_lt].
    -      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    -         P (f x (f_pre x) i) (fold_left (flip f´) xs st)). {
    -         apply H_suff ⇒ //.
    -      }
    -
    -      have H_neq_xs : ∀ e, List.In e xs → X.lt x e ∨ X.lt e x. {
    -        intros e H_in.
    - -
    -        move : (X.compare_spec x e).
    -        case (X.compare x e) ⇒ H_cmp; inversion H_cmp. {
    -          contradict H_nin_x_xs.
    -          rewrite InA_alt.
    -          by ∃ e.
    -        } {
    -          by left.
    -        } {
    -          by right.
    -        }
    -      }
    -      move : H_in_xs H_neq_xs.
    - -
    -      clear IH H_in_xxs H_nodup_xs.
    -      move : (H_RG x _ H_x_in_l H_gt) ⇒ H_RG_x.
    -      move : (H_RL x _ H_x_in_l H_lt) ⇒ H_RL_x.
    -      induction xs as [| x´ xs´ IH´]. {
    -        done.
    -      } {
    -        intros H_in_xs H_neq_xs st H_P_st.
    -        rewrite /=.
    -        have H_x´_in_xxs´ : List.In x´ (x´ :: xs´). {
    -          simpl; by left.
    -        }
    -        have H_x´_in_l : InA X.eq x´ l. {
    -          apply H_in_xs.
    -          apply InA_cons_hd, X.eq_equiv.
    -        }
    -        apply IH´. {
    -          intros H.
    -          apply H_nin_x_xs, InA_cons_tl ⇒ //.
    -        } {
    -          intros e H.
    -          apply H_in_xs, InA_cons_tl ⇒ //.
    -        } {
    -          intros e H.
    -          apply H_neq_xs, in_cons ⇒ //.
    -        } {
    -          rewrite /flip -H_f´ //.
    -          move : (H_neq_xs x´ H_x´_in_xxs´) ⇒ [] H_cmp. {
    -            apply H_RG_x ⇒ //.
    -          } {
    -            apply H_RL_x ⇒ //.
    -          }
    -        }
    -      }
    -    }
    -  Qed.
    - +
      Include HasFoldWithAbortOps X.
    diff --git a/html/MSetsExtra.MSetIntervals.html b/html/MSetsExtra.MSetIntervals.html index e4b6c9b..696aaf2 100644 --- a/html/MSetsExtra.MSetIntervals.html +++ b/html/MSetsExtra.MSetIntervals.html @@ -26,7 +26,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Weak sets implemented by interval lists

    +

    Weak sets implemented by interval lists

    @@ -47,129 +47,60 @@

    Library MSetsExtra.MSetIntervals


    -Require Import MSetInterface OrdersFacts OrdersLists.
    -Require Import BinNat.
    -Require Import ssreflect.
    -Require Import NArith.
    -Require Import ZArith.
    -Require Import NOrder.
    -Require Import DecidableTypeEx.
    -Module Import NOP := NOrderProp N.
    +Require Import MSetInterface OrdersFacts OrdersLists.
    +Require Import BinNat.
    +Require Import mathcomp.ssreflect.ssreflect.
    +Require Import NArith.
    +Require Import ZArith.
    +Require Import NOrder.
    +Require Import DecidableTypeEx.
    +Module Import NOP := NOrderProp N.
    Open Scope Z_scope.

    -

    Auxiliary stuff

    +

    Auxiliary stuff

    Simple auxiliary lemmata
    -Lemma Z_le_add_r : ∀ (z : Z) (n : N),
    -  z ≤ z + Z.of_N n.
    -Proof.
    -  intros z n.
    -  suff : (z + 0 ≤ z + Z.of_N n). {
    -    rewrite Z.add_0_r //.
    -  }
    -  apply Zplus_le_compat_l.
    -  apply N2Z.is_nonneg.
    -Qed.
    - -
    -Lemma Z_lt_add_r : ∀ (z : Z) (n : N),
    -  (n ≠ 0)%N →
    -  z < z + Z.of_N n.
    -Proof.
    -  move ⇒ z n H_neq_0.
    -  suff : (z + Z.of_N 0 < z + Z.of_N n). {
    -    rewrite Z.add_0_r //.
    -  }
    -  apply Z.add_lt_mono_l, N2Z.inj_lt.
    -  by apply N.neq_0_lt_0.
    -Qed.
    +Lemma Z_le_add_r : ∀ (z : Z) (n : N),
    +  z ≤ z + Z.of_N n.
    + +
    +Lemma Z_lt_add_r : ∀ (z : Z) (n : N),
    +  (n ≠ 0)%N →
    +  z < z + Z.of_N n.

    Lemma Z_lt_le_add_r : ∀ y1 y2 c,
    -  y1 < y2 →
    -  y1 ≤ y2 + Z.of_N c.
    -Proof.
    -  intros y1 y2 c H.
    -  apply Z.le_trans with (m := y2). {
    -    by apply Z.lt_le_incl.
    -  } {
    -    apply Z_le_add_r.
    -  }
    -Qed.
    - -
    -Lemma Z_to_N_minus_neq_0 : ∀ (x y : Z),
    -    y < x →
    -    Z.to_N (x - y) ≠ 0%N.
    -Proof.
    -  intros x y H_y_lt.
    -  apply N.neq_0_lt_0.
    -  apply N2Z.inj_lt.
    -  suff H : 0 < x - y. {
    -    rewrite Z2N.id ⇒ //.
    -    by apply Z.lt_le_incl.
    -  }
    -  by apply Z.lt_0_sub.
    -Qed.
    - -
    -Lemma add_add_sub_eq : ∀ (x y : Z), (x + (y - x) = y).
    -Proof.
    -  intros x y.
    -  rewrite Z.add_sub_assoc ⇒ //.
    -  rewrite Z.add_sub_swap Z.sub_diag Z.add_0_l //.
    -Qed.
    - -
    -Lemma NoDupA_map {A B} : ∀ (eqA : A → A → Prop) (eqB : B → B → Prop) (f : A → B) l,
    -  NoDupA eqA l →
    -  (∀ x1 x2, List.In x1 l → List.In x2 l →
    -                  eqB (f x1) (f x2) → eqA x1 x2) →
    -  NoDupA eqB (map f l).
    -Proof.
    -  intros eqA eqB f.
    -  induction l as [| x xs IH]. {
    -    move ⇒ _ _; rewrite /=.
    -    apply NoDupA_nil.
    -  } {
    -    move ⇒ H_pre H_eqA_impl.
    -    have [H_nin_x H_no_dup_xs] : ¬ InA eqA x xs ∧ NoDupA eqA xs. {
    -      by inversion_clear H_pre.
    -    }
    -    simpl.
    -    apply NoDupA_cons; last first. {
    -      apply IH ⇒ //.
    -      intros x1 x2 H_in_x1 H_in_x2 H_eqB.
    -      apply H_eqA_impl ⇒ //=; by right.
    -    }
    -    move ⇒ H_in_map; apply H_nin_x.
    -    move : H_in_map.
    -    rewrite !InA_alt ⇒ [[y] [H_eqB_y]].
    -    rewrite in_map_iff ⇒ [[y´] [H_y_eq] H_y´_in].
    -    subst.
    -    âˆƒ y´.
    -    split ⇒ //.
    -    apply H_eqA_impl ⇒ //. {
    -      by simpl; left.
    -    } {
    -      by simpl; right.
    -    }
    -  }
    -Qed.
    +  y1 < y2 →
    +  y1 ≤ y2 + Z.of_N c.
    + +
    +Lemma Z_to_N_minus_neq_0 : ∀ (x y : Z),
    +    y < x →
    +    Z.to_N (x - y) ≠ 0%N.
    +
    +Lemma add_add_sub_eq : ∀ (x y : Z), (x + (y - x) = y).
    + +
    +Lemma NoDupA_map {A B} : ∀ (eqA : A → A → Prop) (eqB : B → B → Prop) (f : A → B) l,
    +  NoDupA eqA l →
    +  (∀ x1 x2, List.In x1 l → List.In x2 l →
    +                  eqB (f x1) (f x2) → eqA x1 x2) →
    +  NoDupA eqB (map f l).
    +
    -

    rev_map

    +

    rev_map

    @@ -177,14 +108,14 @@

    Library MSetsExtra.MSetIntervals

    rev_map is used for efficiency.
    -Fixpoint rev_map_aux {A B} (f : A → B) (acc : list B) (l : list A) :=
    +Fixpoint rev_map_aux {A B} (f : A → B) (acc : list B) (l : list A) :=
      match l with
    -   | nil ⇒ acc
    -   | x :: xs ⇒ rev_map_aux f ((f x)::acc) xs
    +   | nil ⇒ acc
    +   | x :: xs ⇒ rev_map_aux f ((f x)::acc) xs
      end.

    -Definition rev_map {A B} (f : A → B) (l : list A) : list B := rev_map_aux f nil l.
    +Definition rev_map {A B} (f : A → B) (l : list A) : list B := rev_map_aux f nil l.

    @@ -193,32 +124,18 @@

    Library MSetsExtra.MSetIntervals

    Lemmata about rev_map
    -Lemma rev_map_aux_alt_def {A B} : ∀ (f : A → B) l acc,
    -  rev_map_aux f acc l = List.rev_append (List.map f l) acc.
    -Proof.
    -  intro f.
    -  induction l as [| x xs IH]. {
    -    intros acc.
    -    by simpl.
    -  } {
    -    intros acc.
    -    rewrite /= IH //.
    -  }
    -Qed.
    - +Lemma rev_map_aux_alt_def {A B} : ∀ (f : A → B) l acc,
    +  rev_map_aux f acc l = List.rev_append (List.map f l) acc.
    +
    -Lemma rev_map_alt_def {A B} : ∀ (f : A → B) l,
    -  rev_map f l = List.rev (List.map f l).
    -Proof.
    -  intros f l.
    -  rewrite /rev_map rev_map_aux_alt_def -rev_alt //.
    -Qed.
    +Lemma rev_map_alt_def {A B} : ∀ (f : A → B) l,
    +  rev_map f l = List.rev (List.map f l).

    -

    Encoding Elements

    +

    Encoding Elements

    @@ -231,11 +148,11 @@

    Library MSetsExtra.MSetIntervals


    Module Type ElementEncode.
    -  Declare Module E : OrderedType.
    +  Declare Module E : OrderedType.

    -  Parameter encode : E.t → Z.
    -  Parameter decode : Z → E.t.
    +  Parameter encode : E.t → Z.
    +  Parameter decode : Z → E.t.

    @@ -244,11 +161,11 @@

    Library MSetsExtra.MSetIntervals

    Decoding is the inverse of encoding. Notice that the reverse is not demanded. This means that we do need to provide for all integers z an - element e with encode v = z. + element e with encode v = z.
      Axiom decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    +    decode (encode e) = e.

    @@ -258,7 +175,7 @@

    Library MSetsExtra.MSetIntervals

      Axiom encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.

    @@ -268,7 +185,7 @@

    Library MSetsExtra.MSetIntervals

      Axiom encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.

    End ElementEncode.
    @@ -277,7 +194,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Set Operations

    +

    Set Operations

    @@ -301,7 +218,7 @@

    Library MSetsExtra.MSetIntervals

    - Intervals we represent by tuples (Z, N). The tuple (z, c) represents the + Intervals we represent by tuples (Z, N). The tuple (z, c) represents the interval z-(z+c).
    @@ -312,9 +229,9 @@

    Library MSetsExtra.MSetIntervals

    -Module Ops (Enc : ElementEncode) <: Ops Enc.E.
    +Module Ops (Enc : ElementEncode) <: Ops Enc.E.
      Definition elt := Enc.E.t.
    -  Definition t := list (Z × N).
    +  Definition t := list (Z × N).

    @@ -323,8 +240,8 @@

    Library MSetsExtra.MSetIntervals

    The empty list is trivial to define and check for.
    -  Definition empty : t := nil.
    -  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.
    +  Definition empty : t := nil.
    +  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.

    @@ -336,45 +253,38 @@

    Library MSetsExtra.MSetIntervals

    if it needs to be executable.
    -  Lemma acc_pred : ∀ n p, n = Npos p → Acc N.lt n → Acc N.lt (N.pred n).
    -  Proof.
    -    intros n p H0 H1.
    -    apply H1.
    -    rewrite H0.
    -    apply N.lt_pred_l.
    -    discriminate.
    -  Defined.
    - -
    -  Fixpoint fold_elementsZ_aux {A} (f : A → Z → option A) (acc : A) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : (bool × A) :=
    -    match c as c0 return c = c0 → (bool × A) with
    -    | N0 ⇒ fun _ ⇒ (false, acc)
    +  Lemma acc_pred : ∀ n p, n = Npos p → Acc N.lt n → Acc N.lt (N.pred n).
    + +
    +  Fixpoint fold_elementsZ_aux {A} (f : A → Z → option A) (acc : A) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : (bool × A) :=
    +    match c as c0 return c = c0 → (bool × A) with
    +    | N0 ⇒ fun _ ⇒ (false, acc)
        | c ⇒ fun Heq ⇒ match (f acc x) with
    -        | None ⇒ (true, acc)
    -        | Some acc´ ⇒
    -          fold_elementsZ_aux f acc´ (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end
    -    end (refl_equal _).
    +        | None ⇒ (true, acc)
    +        | Some acc' ⇒
    +          fold_elementsZ_aux f acc' (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end
    +    end (refl_equal _).

      Definition fold_elementsZ_single {A} f (acc : A) x c := fold_elementsZ_aux f acc x c (lt_wf_0 _).

    -  Fixpoint fold_elementsZ {A} f (acc : A) (s : t) : (bool × A) :=
    +  Fixpoint fold_elementsZ {A} f (acc : A) (s : t) : (bool × A) :=
        match s with
    -    | nil ⇒ (false, acc)
    -    | (x, c) :: s´ ⇒
    +    | nil ⇒ (false, acc)
    +    | (x, c) :: s' ⇒
          match fold_elementsZ_single f acc x c with
    -          (false, acc´) ⇒ fold_elementsZ f acc´ s´
    -        | (true, acc´) ⇒ (true, acc´)
    +          (false, acc') ⇒ fold_elementsZ f acc' s'
    +        | (true, acc') ⇒ (true, acc')
          end
        end.

    -  Definition elementsZ (s : t) : list Z :=
    -    snd (fold_elementsZ (fun l x ⇒ Some (x :: l)) nil s).
    +  Definition elementsZ (s : t) : list Z :=
    +    snd (fold_elementsZ (fun l x ⇒ Some (x :: l)) nil s).

    -  Definition elements (s : t) : list elt :=
    +  Definition elements (s : t) : list elt :=
        rev_map Enc.decode (elementsZ s).

    @@ -384,12 +294,12 @@

    Library MSetsExtra.MSetIntervals

    membership is easily defined
    -  Fixpoint memZ (x : Z) (s : t) :=
    +  Fixpoint memZ (x : Z) (s : t) :=
        match s with
    -    | nil ⇒ false
    -    | (y, c) :: l ⇒
    -        if (Z.ltb x y) then false else
    -        if (Z.ltb x (y+Z.of_N c)) then true else
    +    | nil ⇒ false
    +    | (y, c) :: l ⇒
    +        if (Z.ltb x y) then false else
    +        if (Z.ltb x (y+Z.of_N c)) then true else
            memZ x l
        end.
    @@ -415,59 +325,59 @@

    Library MSetsExtra.MSetIntervals

        | ICR_after_touch.

    -  Definition interval_compare (i1 i2 : (Z × N)) : interval_compare_result :=
    -    match (i1, i2) with ((y1, c1), (y2, c2)) ⇒
    -      let yc2 := (y2+Z.of_N c2) in
    -      match (Z.compare yc2 y1) with
    -        | Lt ⇒ ICR_after
    -        | Eq ⇒ ICR_after_touch
    -        | Gt ⇒ let yc1 := (y1+Z.of_N c1) in
    -                match (Z.compare yc1 y2) with
    -                | Lt ⇒ ICR_before
    -                | Eq ⇒ ICR_before_touch
    -                | Gt ⇒
    -                        match (Z.compare y1 y2, Z.compare yc1 yc2) with
    -                        | (Lt, Lt) ⇒ ICR_overlap_before
    -                        | (Lt, _) ⇒ ICR_subsume_2
    -                        | (Eq, Lt) ⇒ ICR_subsume_1
    -                        | (Eq, Gt) ⇒ ICR_subsume_2
    -                        | (Eq, Eq) ⇒ ICR_equal
    -                        | (Gt, Gt) ⇒ ICR_overlap_after
    -                        | (Gt, _) ⇒ ICR_subsume_1
    +  Definition interval_compare (i1 i2 : (Z × N)) : interval_compare_result :=
    +    match (i1, i2) with ((y1, c1), (y2, c2)) ⇒
    +      let yc2 := (y2+Z.of_N c2) in
    +      match (Z.compare yc2 y1) with
    +        | Lt ⇒ ICR_after
    +        | Eq ⇒ ICR_after_touch
    +        | Gt ⇒ let yc1 := (y1+Z.of_N c1) in
    +                match (Z.compare yc1 y2) with
    +                | Lt ⇒ ICR_before
    +                | Eq ⇒ ICR_before_touch
    +                | Gt ⇒
    +                        match (Z.compare y1 y2, Z.compare yc1 yc2) with
    +                        | (Lt, Lt) ⇒ ICR_overlap_before
    +                        | (Lt, _) ⇒ ICR_subsume_2
    +                        | (Eq, Lt) ⇒ ICR_subsume_1
    +                        | (Eq, Gt) ⇒ ICR_subsume_2
    +                        | (Eq, Eq) ⇒ ICR_equal
    +                        | (Gt, Gt) ⇒ ICR_overlap_after
    +                        | (Gt, _) ⇒ ICR_subsume_1
                            end
                    end
          end
        end.

    -  Definition interval_1_compare (y1 : Z) (i : (Z × N)) : interval_compare_result :=
    -    match i with (y2, c2) ⇒
    -      let yc2 := (y2+Z.of_N c2) in
    -      match (Z.compare yc2 y1) with
    -        | Lt ⇒ ICR_after
    -        | Eq ⇒ ICR_after_touch
    -        | Gt ⇒ match (Z.compare (Z.succ y1) y2) with
    -                | Lt ⇒ ICR_before
    -                | Eq ⇒ ICR_before_touch
    -                | Gt ⇒ ICR_subsume_1
    +  Definition interval_1_compare (y1 : Z) (i : (Z × N)) : interval_compare_result :=
    +    match i with (y2, c2) ⇒
    +      let yc2 := (y2+Z.of_N c2) in
    +      match (Z.compare yc2 y1) with
    +        | Lt ⇒ ICR_after
    +        | Eq ⇒ ICR_after_touch
    +        | Gt ⇒ match (Z.compare (Z.succ y1) y2) with
    +                | Lt ⇒ ICR_before
    +                | Eq ⇒ ICR_before_touch
    +                | Gt ⇒ ICR_subsume_1
                    end
          end
        end.

      Fixpoint compare (s1 s2 : t) :=
    -    match (s1, s2) with
    -      | (nil, nil) ⇒ Eq
    -      | (nil, _ :: _) ⇒ Lt
    -      | (_ :: _, nil) ⇒ Gt
    -      | ((y1, c1)::s1´, (y2, c2)::s2´) ⇒
    -        match (Z.compare y1 y2) with
    -          | Lt ⇒ Lt
    -          | Gt ⇒ Gt
    -          | Eq ⇒ match N.compare c1 c2 with
    -                    | Lt ⇒ Lt
    -                    | Gt ⇒ Gt
    -                    | Eq ⇒ compare s1´ s2´
    +    match (s1, s2) with
    +      | (nil, nil) ⇒ Eq
    +      | (nil, _ :: _) ⇒ Lt
    +      | (_ :: _, nil) ⇒ Gt
    +      | ((y1, c1)::s1', (y2, c2)::s2') ⇒
    +        match (Z.compare y1 y2) with
    +          | Lt ⇒ Lt
    +          | Gt ⇒ Gt
    +          | Eq ⇒ match N.compare c1 c2 with
    +                    | Lt ⇒ Lt
    +                    | Gt ⇒ Gt
    +                    | Eq ⇒ compare s1' s2'
                      end
            end
        end.
    @@ -479,18 +389,18 @@

    Library MSetsExtra.MSetIntervals

    Auxiliary functions for inserting at front and merging intervals
    -  Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N :=
    -    (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))).
    +  Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N :=
    +    (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))).

    -  Fixpoint insert_interval_begin (x : Z) (c : N) (l : t) :=
    +  Fixpoint insert_interval_begin (x : Z) (c : N) (l : t) :=
        match l with
    -    | nil ⇒ (x,c)::nil
    -    | (y, c´)::l´ ⇒
    -         match (Z.compare (x + Z.of_N c) y) with
    -         | Lt ⇒ (x, c) :: l
    -         | Eq ⇒ (x, (c+c´)%N) :: l´
    -         | Gt ⇒ insert_interval_begin x (merge_interval_size x c y c´) l´
    +    | nil ⇒ (x,c)::nil
    +    | (y, c')::l' ⇒
    +         match (Z.compare (x + Z.of_N c) y) with
    +         | Lt ⇒ (x, c) :: l
    +         | Eq ⇒ (x, (c+c')%N) :: l'
    +         | Gt ⇒ insert_interval_begin x (merge_interval_size x c y c') l'
             end
        end.
    @@ -502,21 +412,21 @@

    Library MSetsExtra.MSetIntervals

    generate efficient code
    -  Fixpoint addZ_aux (acc : list (Z × N)) (x : Z) (s : t) :=
    +  Fixpoint addZ_aux (acc : list (Z × N)) (x : Z) (s : t) :=
        match s with
    -    | nil ⇒ List.rev´ ((x, (1%N))::acc)
    -    | (y, c) :: l ⇒
    -        match (interval_1_compare x (y,c)) with
    -          | ICR_before ⇒ List.rev_append ((x, (1%N))::acc) s
    -          | ICR_before_touch ⇒ List.rev_append ((x, N.succ c)::acc) l
    -          | ICR_after ⇒ addZ_aux ((y,c) :: acc) x l
    -          | ICR_after_touch ⇒ List.rev_append acc (insert_interval_begin y (N.succ c) l)
    -          | _ ⇒ List.rev_append ((y, c)::acc) l
    +    | nil ⇒ List.rev' ((x, (1%N))::acc)
    +    | (y, c) :: l ⇒
    +        match (interval_1_compare x (y,c)) with
    +          | ICR_before ⇒ List.rev_append ((x, (1%N))::acc) s
    +          | ICR_before_touch ⇒ List.rev_append ((x, N.succ c)::acc) l
    +          | ICR_after ⇒ addZ_aux ((y,c) :: acc) x l
    +          | ICR_after_touch ⇒ List.rev_append acc (insert_interval_begin y (N.succ c) l)
    +          | _ ⇒ List.rev_append ((y, c)::acc) l
            end
        end.

    -  Definition addZ x s := addZ_aux nil x s.
    +  Definition addZ x s := addZ_aux nil x s.
      Definition add x s := addZ (Enc.encode x) s.

    @@ -527,25 +437,24 @@

    Library MSetsExtra.MSetIntervals

    This is used to define the function from_elements.
    -  Definition add_list (l : list elt) (s : t) : t :=
    -     List.fold_left (fun s x ⇒ add x s) l s.
    +  Definition add_list (l : list elt) (s : t) : t :=
    +     List.fold_left (fun s x ⇒ add x s) l s.

    -  Definition from_elements (l : list elt) : t := add_list l empty.
    +  Definition from_elements (l : list elt) : t := add_list l empty.

    -singleton is trivial to define +singleton is trivial to define
    -  Definition singleton (x : elt) : t := (Enc.encode x, 1%N) :: nil.
    +  Definition singleton (x : elt) : t := (Enc.encode x, 1%N) :: nil.

    -  Lemma singleton_alt_def : ∀ x, singleton x = add x empty.
    -  Proof. by []. Qed.
    - +  Lemma singleton_alt_def : ∀ x, singleton x = add x empty.
    +
    @@ -553,29 +462,29 @@

    Library MSetsExtra.MSetIntervals

    removing needs to be done with code extraction in mind again.
    -  Definition insert_intervalZ_guarded (x : Z) (c : N) s :=
    -     if (N.eqb c 0) then s else (x, c) :: s.
    +  Definition insert_intervalZ_guarded (x : Z) (c : N) s :=
    +     if (N.eqb c 0) then s else (x, c) :: s.

    -  Fixpoint removeZ_aux (acc : list (Z × N)) (x : Z) (s : t) : t :=
    +  Fixpoint removeZ_aux (acc : list (Z × N)) (x : Z) (s : t) : t :=
        match s with
    -    | nil ⇒ List.rev´ acc
    -    | (y, c) :: l ⇒
    -        if (Z.ltb x y) then List.rev_append acc s else
    -        if (Z.ltb x (y+Z.of_N c)) then (
    -           List.rev_append (insert_intervalZ_guarded (Z.succ x)
    -              (Z.to_N ((y+Z.of_N c)- (Z.succ x)))
    -             (insert_intervalZ_guarded y (Z.to_N (x-y)) acc)) l
    -        ) else removeZ_aux ((y,c)::acc) x l
    +    | nil ⇒ List.rev' acc
    +    | (y, c) :: l ⇒
    +        if (Z.ltb x y) then List.rev_append acc s else
    +        if (Z.ltb x (y+Z.of_N c)) then (
    +           List.rev_append (insert_intervalZ_guarded (Z.succ x)
    +              (Z.to_N ((y+Z.of_N c)- (Z.succ x)))
    +             (insert_intervalZ_guarded y (Z.to_N (x-y)) acc)) l
    +        ) else removeZ_aux ((y,c)::acc) x l
        end.

    -  Definition removeZ (x : Z) (s : t) : t := removeZ_aux nil x s.
    +  Definition removeZ (x : Z) (s : t) : t := removeZ_aux nil x s.
      Definition remove (x : elt) (s : t) : t := removeZ (Enc.encode x) s.

    -  Definition remove_list (l : list elt) (s : t) : t :=
    -     List.fold_left (fun s x ⇒ remove x s) l s.
    +  Definition remove_list (l : list elt) (s : t) : t :=
    +     List.fold_left (fun s x ⇒ remove x s) l s.

    @@ -585,19 +494,19 @@

    Library MSetsExtra.MSetIntervals

      Fixpoint union_aux (s1 : t) :=
    -    fix aux (s2 : t) (acc : list (Z × N)) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ List.rev_append acc s2
    -    | (_, nil) ⇒ List.rev_append acc s1
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
    +    fix aux (s2 : t) (acc : list (Z × N)) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ List.rev_append acc s2
    +    | (_, nil) ⇒ List.rev_append acc s1
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
              | ICR_before_touch ⇒
                  union_aux l1 (
    -               insert_interval_begin y1 ((c1+c2)%N) l2) acc
    -          | ICR_after ⇒ aux l2 ((y2, c2)::acc)
    +               insert_interval_begin y1 ((c1+c2)%N) l2) acc
    +          | ICR_after ⇒ aux l2 ((y2, c2)::acc)
              | ICR_after_touch ⇒ union_aux l1 (
    -              insert_interval_begin y2 ((c1+c2)%N) l2) acc
    +              insert_interval_begin y2 ((c1+c2)%N) l2) acc
              | ICR_overlap_before ⇒
                  union_aux l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2) acc
              | ICR_overlap_after ⇒
    @@ -609,7 +518,7 @@

    Library MSetsExtra.MSetIntervals

        end.

    -  Definition union s1 s2 := union_aux s1 s2 nil.
    +  Definition union s1 s2 := union_aux s1 s2 nil.

    @@ -620,38 +529,38 @@

    Library MSetsExtra.MSetIntervals


    -  Fixpoint diff_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
    +  Fixpoint diff_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
        match s with
    -    | nil ⇒ (acc, nil)
    -    | ((y1, c1) :: l1) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    -          | ICR_before_touch ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    -          | ICR_after ⇒ (acc, s)
    -          | ICR_after_touch ⇒ (acc, s)
    -          | ICR_overlap_before ⇒ diff_aux y2 c2 ((y1, Z.to_N (y2 - y1))::acc) l1
    -          | ICR_overlap_after ⇒ (acc, (y2+Z.of_N c2, Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) :: l1)
    -          | ICR_equal ⇒ (acc, l1)
    +    | nil ⇒ (acc, nil)
    +    | ((y1, c1) :: l1) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    +          | ICR_before_touch ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    +          | ICR_after ⇒ (acc, s)
    +          | ICR_after_touch ⇒ (acc, s)
    +          | ICR_overlap_before ⇒ diff_aux y2 c2 ((y1, Z.to_N (y2 - y1))::acc) l1
    +          | ICR_overlap_after ⇒ (acc, (y2+Z.of_N c2, Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) :: l1)
    +          | ICR_equal ⇒ (acc, l1)
              | ICR_subsume_1 ⇒ diff_aux y2 c2 acc l1
    -          | ICR_subsume_2 ⇒ ((insert_intervalZ_guarded y1
    -                (Z.to_N (y2 - y1)) acc),
    -              insert_intervalZ_guarded (y2+Z.of_N c2) (Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) l1)
    +          | ICR_subsume_2 ⇒ ((insert_intervalZ_guarded y1
    +                (Z.to_N (y2 - y1)) acc),
    +              insert_intervalZ_guarded (y2+Z.of_N c2) (Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) l1)
            end
        end.

    -  Fixpoint diff_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ rev_append acc s1
    -    | (_, nil) ⇒ rev_append acc s1
    -    | (_, (y2, c2) :: l2) ⇒
    +  Fixpoint diff_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ rev_append acc s1
    +    | (_, nil) ⇒ rev_append acc s1
    +    | (_, (y2, c2) :: l2) ⇒
          match diff_aux y2 c2 acc s1 with
    -        (acc´, s1´) ⇒ diff_aux2 acc´ s1´ l2
    +        (acc', s1') ⇒ diff_aux2 acc' s1' l2
          end
        end.

    -  Definition diff s1 s2 := diff_aux2 nil s1 s2.
    +  Definition diff s1 s2 := diff_aux2 nil s1 s2.

    @@ -662,20 +571,20 @@

    Library MSetsExtra.MSetIntervals

      Fixpoint subset (s1 : t) :=
        fix aux (s2 : t) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ true
    -    | (_ :: _, nil) ⇒ false
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ false
    -          | ICR_before_touch ⇒ false
    +    match (s1, s2) with
    +    | (nil, _) ⇒ true
    +    | (_ :: _, nil) ⇒ false
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ false
    +          | ICR_before_touch ⇒ false
              | ICR_after ⇒ aux l2
    -          | ICR_after_touch ⇒ false
    -          | ICR_overlap_before ⇒ false
    -          | ICR_overlap_after ⇒ false
    +          | ICR_after_touch ⇒ false
    +          | ICR_overlap_before ⇒ false
    +          | ICR_overlap_after ⇒ false
              | ICR_equal ⇒ subset l1 l2
              | ICR_subsume_1 ⇒ subset l1 s2
    -          | ICR_subsume_2 ⇒ false
    +          | ICR_subsume_2 ⇒ false
            end
        end.
    @@ -686,10 +595,10 @@

    Library MSetsExtra.MSetIntervals

    equal
    -  Fixpoint equal (s s´ : t) : bool := match s, s´ with
    -    | nil, nil ⇒ true
    -    | ((x,cx)::xs), ((y,cy)::ys) ⇒ andb (Z.eqb x y) (andb (N.eqb cx cy) (equal xs ys))
    -    | _, _ ⇒ false
    +  Fixpoint equal (s s' : t) : bool := match s, s' with
    +    | nil, nil ⇒ true
    +    | ((x,cx)::xs), ((y,cy)::ys) ⇒ andb (Z.eqb x y) (andb (N.eqb cx cy) (equal xs ys))
    +    | _, _ ⇒ false
      end.

    @@ -699,36 +608,36 @@

    Library MSetsExtra.MSetIntervals

    inter
    -  Fixpoint inter_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
    +  Fixpoint inter_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
        match s with
    -    | nil ⇒ (acc, nil)
    -    | ((y1, c1) :: l1) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    +    | nil ⇒ (acc, nil)
    +    | ((y1, c1) :: l1) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
              | ICR_before ⇒ inter_aux y2 c2 acc l1
              | ICR_before_touch ⇒ inter_aux y2 c2 acc l1
    -          | ICR_after ⇒ (acc, s)
    -          | ICR_after_touch ⇒ (acc, s)
    -          | ICR_overlap_before ⇒ inter_aux y2 c2 ((y2, Z.to_N (y1 + Z.of_N c1 - y2))::acc) l1
    -          | ICR_overlap_after ⇒ ((y1, Z.to_N (y2 + Z.of_N c2 - y1))::acc, s)
    -          | ICR_equal ⇒ ((y1,c1)::acc, l1)
    -          | ICR_subsume_1 ⇒ inter_aux y2 c2 ((y1, c1)::acc) l1
    -          | ICR_subsume_2 ⇒ ((y2, c2)::acc, s)
    +          | ICR_after ⇒ (acc, s)
    +          | ICR_after_touch ⇒ (acc, s)
    +          | ICR_overlap_before ⇒ inter_aux y2 c2 ((y2, Z.to_N (y1 + Z.of_N c1 - y2))::acc) l1
    +          | ICR_overlap_after ⇒ ((y1, Z.to_N (y2 + Z.of_N c2 - y1))::acc, s)
    +          | ICR_equal ⇒ ((y1,c1)::acc, l1)
    +          | ICR_subsume_1 ⇒ inter_aux y2 c2 ((y1, c1)::acc) l1
    +          | ICR_subsume_2 ⇒ ((y2, c2)::acc, s)
            end
        end.

    -  Fixpoint inter_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ List.rev´ acc
    -    | (_, nil) ⇒ List.rev´ acc
    -    | (_, (y2, c2) :: l2) ⇒
    +  Fixpoint inter_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ List.rev' acc
    +    | (_, nil) ⇒ List.rev' acc
    +    | (_, (y2, c2) :: l2) ⇒
          match inter_aux y2 c2 acc s1 with
    -        (acc´, s1´) ⇒ inter_aux2 acc´ s1´ l2
    +        (acc', s1') ⇒ inter_aux2 acc' s1' l2
          end
        end.

    -  Definition inter s1 s2 := inter_aux2 nil s1 s2.
    +  Definition inter s1 s2 := inter_aux2 nil s1 s2.

    @@ -740,93 +649,93 @@

    Library MSetsExtra.MSetIntervals


      Definition partitionZ_fold_insert
    -             (cur : option (Z × N)) (x : Z) :=
    +             (cur : option (Z × N)) (x : Z) :=
        match cur with
    -       None ⇒ (x, 1%N)
    -     | Some (y, c) ⇒ (y, N.succ c)
    +       None ⇒ (x, 1%N)
    +     | Some (y, c) ⇒ (y, N.succ c)
        end.

    -  Definition partitionZ_fold_skip (acc : list (Z × N))
    -             (cur : option (Z × N)) : (list (Z × N)) :=
    +  Definition partitionZ_fold_skip (acc : list (Z × N))
    +             (cur : option (Z × N)) : (list (Z × N)) :=
        match cur with
    -       None ⇒ acc
    -     | Some yc ⇒ yc::acc
    +       None ⇒ acc
    +     | Some yc ⇒ yc::acc
        end.

    -  Definition partitionZ_fold_fun f st (x : Z) :=
    -    match st with ((acc_t, c_t), (acc_f, c_f)) ⇒
    +  Definition partitionZ_fold_fun f st (x : Z) :=
    +    match st with ((acc_t, c_t), (acc_f, c_f)) ⇒
          if (f x) then
    -        ((acc_t, Some (partitionZ_fold_insert c_t x)),
    -         (partitionZ_fold_skip acc_f c_f, None))
    +        ((acc_t, Some (partitionZ_fold_insert c_t x)),
    +         (partitionZ_fold_skip acc_f c_f, None))
          else
    -        ((partitionZ_fold_skip acc_t c_t, None),
    -         (acc_f, Some (partitionZ_fold_insert c_f x)))
    +        ((partitionZ_fold_skip acc_t c_t, None),
    +         (acc_f, Some (partitionZ_fold_insert c_f x)))
        end.

    -  Definition partitionZ_single_aux f st (x : Z) (c : N) :=
    -    snd (fold_elementsZ_single (fun st x ⇒ Some (partitionZ_fold_fun f st x)) st x c).
    +  Definition partitionZ_single_aux f st (x : Z) (c : N) :=
    +    snd (fold_elementsZ_single (fun st x ⇒ Some (partitionZ_fold_fun f st x)) st x c).

      Definition partitionZ_single f acc_t acc_f x c :=
    -    match partitionZ_single_aux f ((acc_t, None), (acc_f, None)) x c with
    -    | ((acc_t, c_t), (acc_f, c_f)) ⇒
    -        (partitionZ_fold_skip acc_t c_t,
    -         partitionZ_fold_skip acc_f c_f)
    +    match partitionZ_single_aux f ((acc_t, None), (acc_f, None)) x c with
    +    | ((acc_t, c_t), (acc_f, c_f)) ⇒
    +        (partitionZ_fold_skip acc_t c_t,
    +         partitionZ_fold_skip acc_f c_f)
        end.

      Fixpoint partitionZ_aux acc_t acc_f f s :=
        match s with
    -    | nil ⇒ (List.rev acc_t, List.rev acc_f)
    -    | (y, c) :: s´ ⇒
    +    | nil ⇒ (List.rev acc_t, List.rev acc_f)
    +    | (y, c) :: s' ⇒
          match partitionZ_single f acc_t acc_f y c with
    -      | (acc_t´, acc_f´) ⇒ partitionZ_aux acc_t´ acc_f´ f s´
    +      | (acc_t', acc_f') ⇒ partitionZ_aux acc_t' acc_f' f s'
          end
        end.

    -  Definition partitionZ := partitionZ_aux nil nil.
    +  Definition partitionZ := partitionZ_aux nil nil.

    -  Definition partition (f : elt → bool) : t → (t × t) :=
    +  Definition partition (f : elt → bool) : t → (t × t) :=
        partitionZ (fun z ⇒ f (Enc.decode z)).

    -  Definition filterZ_fold_fun f st (x : Z) :=
    -    match st with (acc_t, c_t) ⇒
    +  Definition filterZ_fold_fun f st (x : Z) :=
    +    match st with (acc_t, c_t) ⇒
          if (f x) then
    -        (acc_t, Some (partitionZ_fold_insert c_t x))
    +        (acc_t, Some (partitionZ_fold_insert c_t x))
          else
    -        (partitionZ_fold_skip acc_t c_t, None)
    +        (partitionZ_fold_skip acc_t c_t, None)
        end.

    -  Definition filterZ_single_aux f st (x : Z) (c : N) :=
    -    snd (fold_elementsZ_single (fun st x ⇒ Some (filterZ_fold_fun f st x)) st x c).
    +  Definition filterZ_single_aux f st (x : Z) (c : N) :=
    +    snd (fold_elementsZ_single (fun st x ⇒ Some (filterZ_fold_fun f st x)) st x c).

      Definition filterZ_single f acc x c :=
    -    match filterZ_single_aux f (acc, None) x c with
    -    | (acc, c) ⇒
    +    match filterZ_single_aux f (acc, None) x c with
    +    | (acc, c) ⇒
            (partitionZ_fold_skip acc c)
        end.

      Fixpoint filterZ_aux acc f s :=
        match s with
    -    | nil ⇒ (List.rev acc)
    -    | (y, c) :: s´ ⇒
    -      filterZ_aux (filterZ_single f acc y c) f s´
    +    | nil ⇒ (List.rev acc)
    +    | (y, c) :: s' ⇒
    +      filterZ_aux (filterZ_single f acc y c) f s'
        end.

    -  Definition filterZ := filterZ_aux nil.
    +  Definition filterZ := filterZ_aux nil.

    -  Definition filter (f : elt → bool) : t → t :=
    +  Definition filter (f : elt → bool) : t → t :=
        filterZ (fun z ⇒ f (Enc.decode z)).

    @@ -840,62 +749,62 @@

    Library MSetsExtra.MSetIntervals


    -  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    -    snd (fold_elementsZ (fun b z ⇒ Some (f (Enc.decode z) b)) i s).
    +  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    +    snd (fold_elementsZ (fun b z ⇒ Some (f (Enc.decode z) b)) i s).

    -  Definition for_all (f : elt → bool) (s : t) : bool :=
    -    snd (fold_elementsZ (fun b z ⇒
    +  Definition for_all (f : elt → bool) (s : t) : bool :=
    +    snd (fold_elementsZ (fun b z ⇒
          if b then
    -        Some (f (Enc.decode z))
    -      else None) true s).
    +        Some (f (Enc.decode z))
    +      else None) true s).

    -  Definition exists_ (f : elt → bool) (s : t) : bool :=
    -    snd (fold_elementsZ (fun b z ⇒
    +  Definition exists_ (f : elt → bool) (s : t) : bool :=
    +    snd (fold_elementsZ (fun b z ⇒
          if b then
    -        None
    -      else Some (f (Enc.decode z))) false s).
    +        None
    +      else Some (f (Enc.decode z))) false s).

    -  Fixpoint cardinalN c (s : t) : N := match s with
    -    | nil ⇒ c
    -    | (_,cx)::xs ⇒ cardinalN (c + cx)%N xs
    +  Fixpoint cardinalN c (s : t) : N := match s with
    +    | nil ⇒ c
    +    | (_,cx)::xs ⇒ cardinalN (c + cx)%N xs
      end.

    -  Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s).
    +  Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s).

    -  Definition min_eltZ (s : t) : option Z :=
    +  Definition min_eltZ (s : t) : option Z :=
        match s with
    -    | nil ⇒ None
    -    | (x, _) :: _ ⇒ Some x
    +    | nil ⇒ None
    +    | (x, _) :: _ ⇒ Some x
        end.

    -  Definition min_elt (s : t) : option elt :=
    +  Definition min_elt (s : t) : option elt :=
        match (min_eltZ s) with
    -    | None ⇒ None
    -    | Some x ⇒ Some (Enc.decode x)
    +    | None ⇒ None
    +    | Some x ⇒ Some (Enc.decode x)
        end.

      Definition choose := min_elt.

    -  Fixpoint max_eltZ (s : t) : option Z :=
    +  Fixpoint max_eltZ (s : t) : option Z :=
        match s with
    -    | nil ⇒ None
    -    | (x, c) :: nil ⇒ Some (Z.pred (x + Z.of_N c))
    -    | (x, _) :: s´ ⇒ max_eltZ s´
    +    | nil ⇒ None
    +    | (x, c) :: nil ⇒ Some (Z.pred (x + Z.of_N c))
    +    | (x, _) :: s' ⇒ max_eltZ s'
        end.

    -  Definition max_elt (s : t) : option elt :=
    +  Definition max_elt (s : t) : option elt :=
        match (max_eltZ s) with
    -    | None ⇒ None
    -    | Some x ⇒ Some (Enc.decode x)
    +    | None ⇒ None
    +    | Some x ⇒ Some (Enc.decode x)
        end.

    @@ -905,7 +814,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Raw Module

    +

    Raw Module

    @@ -924,36 +833,36 @@

    Library MSetsExtra.MSetIntervals

    -

    Defining invariant IsOk

    +

    Defining invariant IsOk

    -  Definition is_encoded_elems_list (l : list Z) : Prop :=
    -    (∀ x, List.In x l → ∃ e, Enc.encode e = x).
    +  Definition is_encoded_elems_list (l : list Z) : Prop :=
    +    (∀ x, List.In x l → ∃ e, Enc.encode e = x).

    -  Definition interval_list_elements_greater (x : Z) (l : t) : bool :=
    +  Definition interval_list_elements_greater (x : Z) (l : t) : bool :=
        match l with
    -      | nil ⇒ true
    -      | (y, _)::_ ⇒ Z.ltb x y
    +      | nil ⇒ true
    +      | (y, _)::_ ⇒ Z.ltb x y
        end.

      Fixpoint interval_list_invariant (l : t) :=
       match l with
    -   | nil ⇒ true
    -   | (x, c) :: l´ ⇒
    -       interval_list_elements_greater (x + (Z.of_N c)) l´ && negb (N.eqb c 0) && interval_list_invariant l´
    +   | nil ⇒ true
    +   | (x, c) :: l' ⇒
    +       interval_list_elements_greater (x + (Z.of_N c)) l' && negb (N.eqb c 0) && interval_list_invariant l'
       end.

    -  Definition IsOk s := (interval_list_invariant s = true ∧ is_encoded_elems_list (elementsZ s)).
    +  Definition IsOk s := (interval_list_invariant s = true ∧ is_encoded_elems_list (elementsZ s)).

    -

    Defining notations

    +

    Defining notations

    @@ -966,13 +875,13 @@

    Library MSetsExtra.MSetIntervals

        Instance IsOk_Ok s `(Hs : IsOk s) : Ok s := { ok := Hs }.

    -    Definition In x s := (SetoidList.InA Enc.E.eq x (elements s)).
    -    Definition InZ x s := (List.In x (elementsZ s)).
    -    Definition Equal s s´ := ∀ a : elt, In a s ↔ In a s´.
    -    Definition Subset s s´ := ∀ a : elt, In a s → In a s´.
    -    Definition Empty s := ∀ a : elt, ¬ In a s.
    -    Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    -    Definition Exists (P : elt → Prop) (s : t) := ∃ x, In x s ∧ P x.
    +    Definition In x s := (SetoidList.InA Enc.E.eq x (elements s)).
    +    Definition InZ x s := (List.In x (elementsZ s)).
    +    Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    +    Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    +    Definition Empty s := ∀ a : elt, ¬ In a s.
    +    Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    +    Definition Exists (P : elt → Prop) (s : t) := ∃ x, In x s ∧ P x.

      End ForNotations.
    @@ -981,487 +890,225 @@

    Library MSetsExtra.MSetIntervals

    -

    elements list properties

    +

    elements list properties

    The functions elementsZ, elementsZ_single, - elements and elements_single are crucial and used + elements and elements_single are crucial and used everywhere. Therefore, we first establish a few properties of these important functions.

    -  Lemma elementsZ_nil : (elementsZ (nil : t) = nil).
    -  Proof. done. Qed.
    - +  Lemma elementsZ_nil : (elementsZ (nil : t) = nil).
    +
    -  Lemma elements_nil : (elements (nil : t) = nil).
    -  Proof. done. Qed.
    - +  Lemma elements_nil : (elements (nil : t) = nil).
    +
    -  Definition elementsZ_single (x:Z) (c:N) :=
    -      List.rev´ (N.peano_rec (fun _ ⇒ list Z)
    -                  nil (fun n ls ⇒ (x+Z.of_N n)%Z :: ls) c).
    +  Definition elementsZ_single (x:Z) (c:N) :=
    +      List.rev' (N.peano_rec (fun _ ⇒ list Z)
    +                  nil (fun n ls ⇒ (x+Z.of_N n)%Z :: ls) c).

      Definition elements_single x c :=
    -    List.map Enc.decode (elementsZ_single x c).
    +    List.map Enc.decode (elementsZ_single x c).

      Lemma elementsZ_single_base : ∀ x,
    -    elementsZ_single x (0%N) = nil.
    -  Proof. done. Qed.
    - +    elementsZ_single x (0%N) = nil.
    +
      Lemma elementsZ_single_succ : ∀ x c,
    -    elementsZ_single x (N.succ c) =
    -    elementsZ_single x c ++ (x+Z.of_N c)::nil.
    -  Proof.
    -    intros x c.
    -    rewrite /elementsZ_single N.peano_rec_succ /rev´ -!rev_alt //.
    -  Qed.
    - +    elementsZ_single x (N.succ c) =
    +    elementsZ_single x c ++ (x+Z.of_N c)::nil.
    +
      Lemma elementsZ_single_add : ∀ x c2 c1,
    -    elementsZ_single x (c1 + c2)%N =
    -    elementsZ_single x c1 ++ elementsZ_single (x+Z.of_N c1) c2.
    -  Proof.
    -    intros x.
    -    induction c2 as [| c2´ IH] using N.peano_ind. {
    -      move ⇒ c1.
    -      rewrite elementsZ_single_base /= app_nil_r N.add_0_r //.
    -    } {
    -      move ⇒ c1.
    -      rewrite N.add_succ_r !elementsZ_single_succ IH app_assoc N2Z.inj_add Z.add_assoc //.
    -    }
    -  Qed.
    - +    elementsZ_single x (c1 + c2)%N =
    +    elementsZ_single x c1 ++ elementsZ_single (x+Z.of_N c1) c2.
    +
      Lemma elementsZ_single_succ_front : ∀ x c,
    -    elementsZ_single x (N.succ c) =
    -    x :: elementsZ_single (Z.succ x) c.
    -  Proof.
    -    intros x c.
    -    rewrite -N.add_1_l elementsZ_single_add.
    -    rewrite N.one_succ elementsZ_single_succ elementsZ_single_base /= Z.add_0_r.
    -    by rewrite Z.add_1_r.
    -  Qed.
    - +    elementsZ_single x (N.succ c) =
    +    x :: elementsZ_single (Z.succ x) c.
    +
      Lemma In_elementsZ_single : ∀ c y x,
    -    List.In y (elementsZ_single x c) ↔
    -    (x ≤ y) ∧ (y < (x+Z.of_N c)).
    -  Proof.
    -    induction c as [| c´ IH] using N.peano_ind. {
    -      intros y x.
    -      rewrite elementsZ_single_base Z.add_0_r /=.
    -      omega.
    -    } {
    -      intros y x.
    -      rewrite elementsZ_single_succ in_app_iff IH /= N2Z.inj_succ Z.add_succ_r Z.lt_succ_r.
    -      split. {
    -        move ⇒ [ | []] //. {
    -          move ⇒ [H_x_le H_y_le].
    -          omega.
    -        } {
    -          move ⇒ <-.
    -          split.
    -            - by apply Z_le_add_r.
    -            - by apply Z.le_refl.
    -        }
    -      } {
    -        move ⇒ [H_x_le] H_y_lt.
    -        omega.
    -      }
    -    }
    -  Qed.
    - +    List.In y (elementsZ_single x c) ↔
    +    (x ≤ y) ∧ (y < (x+Z.of_N c)).
    +
      Lemma In_elementsZ_single1 : ∀ y x,
    -    List.In y (elementsZ_single x (1%N)) ↔
    -    (x = y).
    -  Proof.
    -    intros y x.
    -    rewrite In_elementsZ_single /= Z.add_1_r Z.lt_succ_r.
    -    omega.
    -  Qed.
    +    List.In y (elementsZ_single x (1%N)) ↔
    +    (x = y).

      Lemma length_elementsZ_single : ∀ cx x,
    -    length (elementsZ_single x cx) = N.to_nat cx.
    -  Proof.
    -    induction cx as [| cx´ IH] using N.peano_ind. {
    -      by simpl.
    -    } {
    -      intros x.
    -      rewrite elementsZ_single_succ_front /=.
    -      rewrite IH N2Nat.inj_succ //.
    -    }
    -  Qed.
    - +    length (elementsZ_single x cx) = N.to_nat cx.
    +
      Lemma fold_elementsZ_aux_irrel {A} :
        âˆ€ f c (acc : A) x H1 H2,
    -      fold_elementsZ_aux f acc x c H1 =
    +      fold_elementsZ_aux f acc x c H1 =
          fold_elementsZ_aux f acc x c H2.
    -  Proof.
    -    intros f c.
    -    induction c as [c IH] using (well_founded_ind lt_wf_0).
    -    case_eq c. {
    -      intros H_c acc x; case; intro H_H1; case; intro H_H2.
    -      reflexivity.
    -    } {
    -      intros p H_c acc x; case; intro H_H1; case; intro H_H2.
    -      unfold fold_elementsZ_aux; fold (@fold_elementsZ_aux A).
    -      case (f acc x) ⇒ // acc´.
    -      apply IH.
    -      rewrite H_c.
    -      apply N.lt_pred_l.
    -      discriminate.
    -    }
    -  Qed.
    - +
      Lemma fold_elementsZ_single_pos {A} : ∀ f (acc : A) x p,
    -    fold_elementsZ_single f acc x (N.pos p) =
    +    fold_elementsZ_single f acc x (N.pos p) =
        match f acc x with
    -    | Some acc´ ⇒
    -        fold_elementsZ_single f acc´ (Z.succ x)
    -         (N.pred (N.pos p))
    -    | None ⇒ (true, acc)
    +    | Some acc' ⇒
    +        fold_elementsZ_single f acc' (Z.succ x)
    +         (N.pred (N.pos p))
    +    | None ⇒ (true, acc)
        end.
    -  Proof.
    -    intros f acc x p.
    -    unfold fold_elementsZ_single.
    -    unfold fold_elementsZ_aux.
    -    case: (lt_wf_0 _).
    -    fold (@fold_elementsZ_aux A).
    -    intro.
    -    case (f acc x) ⇒ // acc´.
    -    apply fold_elementsZ_aux_irrel.
    -  Qed.

      Lemma fold_elementsZ_single_zero {A} : ∀ f (acc : A) x,
    -      fold_elementsZ_single f acc x (0%N) = (false, acc).
    -  Proof.
    -    intros f acc x.
    -    unfold fold_elementsZ_single.
    -    case (lt_wf_0 (0%N)); intro.
    -    unfold fold_elementsZ_aux.
    -    reflexivity.
    -  Qed.
    +      fold_elementsZ_single f acc x (0%N) = (false, acc).

      Lemma fold_elementsZ_single_succ {A} : ∀ f (acc : A) x c,
    -    fold_elementsZ_single f acc x (N.succ c) =
    +    fold_elementsZ_single f acc x (N.succ c) =
        match f acc x with
    -      | Some acc´ ⇒
    -          fold_elementsZ_single f acc´ (Z.succ x) c
    -      | None ⇒ (true, acc)
    +      | Some acc' ⇒
    +          fold_elementsZ_single f acc' (Z.succ x) c
    +      | None ⇒ (true, acc)
        end.
    -  Proof.
    -    intros f acc x c.
    -    case c. {
    -      by rewrite fold_elementsZ_single_pos.
    -    } {
    -      intro p; simpl.
    -      rewrite fold_elementsZ_single_pos.
    -      case (f acc x) ⇒ // acc´ /=.
    -      by rewrite Pos.pred_N_succ.
    -    }
    -  Qed.
    - -
    -  Fixpoint fold_opt {A B} f (acc : A) (bs : list B) : (bool × A) :=
    + +
    +  Fixpoint fold_opt {A B} f (acc : A) (bs : list B) : (bool × A) :=
        match bs with
    -      | nil ⇒ (false, acc)
    -      | (b :: bs´) ⇒
    +      | nil ⇒ (false, acc)
    +      | (b :: bs') ⇒
            match f acc b with
    -        | Some acc´ ⇒ fold_opt f acc´ bs´
    -        | None ⇒ (true, acc)
    +        | Some acc' ⇒ fold_opt f acc' bs'
    +        | None ⇒ (true, acc)
            end
        end.

    -  Lemma fold_opt_list_cons : ∀ {A} (bs : list A) (acc : list A),
    -    fold_opt (fun l x ⇒ Some (x :: l)) acc bs =
    -    (false, List.rev bs ++ acc).
    -  Proof.
    -    induction bs as [| b bs´ IH] ⇒ acc. {
    -      by simpl.
    -    } {
    -      rewrite /= IH -app_assoc //.
    -    }
    -  Qed.
    - +  Lemma fold_opt_list_cons : ∀ {A} (bs : list A) (acc : list A),
    +    fold_opt (fun l x ⇒ Some (x :: l)) acc bs =
    +    (false, List.rev bs ++ acc).
    +
    -  Lemma fold_opt_app {A B} : ∀ f (acc : A) (l1 l2 : list B),
    -    fold_opt f acc (l1 ++ l2) =
    -    (let (ab, acc´) := fold_opt f acc l1 in
    -     if ab then (true, acc´) else fold_opt f acc´ l2).
    -  Proof.
    -    intros f acc l1 l2.
    -    move : acc.
    -    induction l1 as [| b l1´ IH] ⇒ acc. {
    -      rewrite app_nil_l //.
    -    } {
    -      rewrite /=.
    -      case (f acc b); last done.
    -      intro acc´.
    -      rewrite IH //.
    -    }
    -  Qed.
    - +  Lemma fold_opt_app {A B} : ∀ f (acc : A) (l1 l2 : list B),
    +    fold_opt f acc (l1 ++ l2) =
    +    (let (ab, acc') := fold_opt f acc l1 in
    +     if ab then (true, acc') else fold_opt f acc' l2).
    +

      Lemma fold_elementsZ_single_alt_def {A} : ∀ f c (acc : A) x,
    -     fold_elementsZ_single f acc x c =
    +     fold_elementsZ_single f acc x c =
         fold_opt f acc (elementsZ_single x c).
    -  Proof.
    -    intro f.
    -    induction c as [| c´ IH] using N.peano_ind. {
    -      intros acc x.
    -      rewrite fold_elementsZ_single_zero
    -              elementsZ_single_base /fold_opt //.
    -    } {
    -      intros acc x.
    -      rewrite fold_elementsZ_single_succ
    -              elementsZ_single_succ_front /=.
    -      case (f acc x); last reflexivity.
    -      intro acc´.
    -      apply IH.
    -    }
    -  Qed.
    - +
      Lemma fold_elementsZ_nil {A} : ∀ f (acc : A),
    -     fold_elementsZ f acc nil = (false, acc).
    -  Proof. done. Qed.
    - +     fold_elementsZ f acc nil = (false, acc).
    +
      Lemma fold_elementsZ_cons {A} : ∀ f (acc : A) y c s,
    -    fold_elementsZ f acc ((y, c)::s) =
    -    (let (ab, acc´) := fold_elementsZ_single f acc y c in
    -     if ab then (true, acc´) else fold_elementsZ f acc´ s).
    -  Proof.
    -    intros f acc y c s.
    -    done.
    -  Qed.
    +    fold_elementsZ f acc ((y, c)::s) =
    +    (let (ab, acc') := fold_elementsZ_single f acc y c in
    +     if ab then (true, acc') else fold_elementsZ f acc' s).

      Lemma fold_elementsZ_alt_def_aux : ∀ (s : t) base,
    -    (snd (fold_elementsZ
    -      (fun (l : list Z) (x : Z) ⇒ Some (x :: l)) base s)) =
    -    elementsZ s ++ base.
    -  Proof.
    -    induction s as [| [y1 c1] s´ IH] ⇒ base. {
    -      rewrite elementsZ_nil /fold_elementsZ /fold_opt /snd
    -        app_nil_l //.
    -    } {
    -      rewrite /elementsZ !fold_elementsZ_cons.
    -      rewrite !fold_elementsZ_single_alt_def !fold_opt_list_cons.
    -      rewrite !IH app_nil_r app_assoc //.
    -    }
    -  Qed.
    - +    (snd (fold_elementsZ
    +      (fun (l : list Z) (x : Z) ⇒ Some (x :: l)) base s)) =
    +    elementsZ s ++ base.
    +

      Lemma fold_elementsZ_alt_def {A} : ∀ f s (acc : A),
    -     fold_elementsZ f acc s =
    -     fold_opt f acc (rev (elementsZ s)).
    -  Proof.
    -    intro f.
    -    induction s as [| [y1 c1] s´ IH] ⇒ acc. {
    -      by simpl.
    -    } {
    -      rewrite /elementsZ !fold_elementsZ_cons.
    -      rewrite !fold_elementsZ_single_alt_def
    -              fold_opt_list_cons app_nil_r
    -              fold_elementsZ_alt_def_aux rev_app_distr
    -              rev_involutive fold_opt_app.
    -      case (fold_opt f acc (elementsZ_single y1 c1)).
    -      move ⇒ [] //.
    -    }
    -  Qed.
    - -
    -  Lemma elementsZ_cons : ∀ x c s, elementsZ (((x, c) :: s) : t) =
    -     ((elementsZ s) ++ (List.rev (elementsZ_single x c))).
    -  Proof.
    -    intros x c s.
    -    rewrite /elementsZ fold_elementsZ_cons
    -            !fold_elementsZ_alt_def
    -            fold_elementsZ_single_alt_def.
    -    rewrite !fold_opt_list_cons.
    -    rewrite !app_nil_r !rev_involutive /=.
    -    rewrite fold_elementsZ_alt_def_aux //.
    -  Qed.
    - -
    -  Lemma elements_cons : ∀ x c s, elements (((x, c) :: s) : t) =
    -     ((elements_single x c) ++ elements s).
    -  Proof.
    -    intros x c s.
    -    rewrite /elements /elements_single elementsZ_cons.
    -    rewrite !rev_map_alt_def map_app rev_app_distr map_rev rev_involutive //.
    -  Qed.
    - -
    -  Lemma elementsZ_app : ∀ (s1 s2 : t), elementsZ (s1 ++ s2) =
    -     ((elementsZ s2) ++ (elementsZ s1)).
    -  Proof.
    -    induction s1 as [| [x1 c1] s1 IH1]. {
    -      move ⇒ s2.
    -      rewrite elementsZ_nil app_nil_r //.
    -    }
    -    move ⇒ s2.
    -    rewrite -app_comm_cons !elementsZ_cons IH1 -app_assoc //.
    -  Qed.
    - -
    -  Lemma InZ_nil : ∀ y, InZ y nil ↔ False.
    -  Proof.
    -    intro y.
    -    done.
    -  Qed.
    - -
    -  Lemma InZ_cons : ∀ y x c s, InZ y (((x, c) :: s) : t) ↔
    -     List.In y (elementsZ_single x c) ∨ InZ y s.
    -  Proof.
    -    intros y x c s.
    -    rewrite /InZ elementsZ_cons in_app_iff -in_rev.
    -    firstorder.
    -  Qed.
    - +     fold_elementsZ f acc s =
    +     fold_opt f acc (rev (elementsZ s)).
    + +
    +  Lemma elementsZ_cons : ∀ x c s, elementsZ (((x, c) :: s) : t) =
    +     ((elementsZ s) ++ (List.rev (elementsZ_single x c))).
    + +
    +  Lemma elements_cons : ∀ x c s, elements (((x, c) :: s) : t) =
    +     ((elements_single x c) ++ elements s).
    + +
    +  Lemma elementsZ_app : ∀ (s1 s2 : t), elementsZ (s1 ++ s2) =
    +     ((elementsZ s2) ++ (elementsZ s1)).
    + +
    +  Lemma InZ_nil : ∀ y, InZ y nil ↔ False.
    + +
    +  Lemma InZ_cons : ∀ y x c s, InZ y (((x, c) :: s) : t) ↔
    +     List.In y (elementsZ_single x c) ∨ InZ y s.
    +
      Lemma InZ_app : ∀ s1 s2 y,
    -     InZ y (s1 ++ s2) ↔ InZ y s1 ∨ InZ y s2.
    -  Proof.
    -    intros s1 s2 y.
    -    rewrite /InZ elementsZ_app in_app_iff.
    -    tauto.
    -  Qed.
    - +     InZ y (s1 ++ s2) ↔ InZ y s1 ∨ InZ y s2.
    +
      Lemma InZ_rev : ∀ s y,
    -     InZ y (List.rev s) ↔ InZ y s.
    -  Proof.
    -    intros s x.
    -    rewrite /InZ.
    -    induction s as [| [y c] s´ IH]; first done.
    -    simpl.
    -    rewrite elementsZ_app in_app_iff IH.
    -    rewrite !elementsZ_cons !in_app_iff elementsZ_nil
    -            -!in_rev /=.
    -    tauto.
    -  Qed.
    - +     InZ y (List.rev s) ↔ InZ y s.
    +
      Lemma In_elementsZ_single_dec : ∀ y x c,
    -    {List.In y (elementsZ_single x c)} +
    -    {¬ List.In y (elementsZ_single x c)}.
    -  Proof.
    -    intros y x c.
    -    case (Z_le_dec x y); last first. {
    -      right; rewrite In_elementsZ_single; tauto.
    -    }
    -    case (Z_lt_dec y (x + Z.of_N c)); last first. {
    -      right; rewrite In_elementsZ_single; tauto.
    -    }
    -    left; rewrite In_elementsZ_single; tauto.
    -  Qed.
    - +    {List.In y (elementsZ_single x c)} +
    +    {¬ List.In y (elementsZ_single x c)}.
    +
      Lemma InZ_dec : ∀ y s,
    -     {InZ y s} + {¬InZ y s}.
    -  Proof.
    -    intros y.
    -    induction s as [| [x c] s IH]. {
    -      by right.
    -    }
    -    move : IH ⇒ [] IH. {
    -      by left; rewrite InZ_cons; right.
    -    }
    -    case (In_elementsZ_single_dec y x c). {
    -      by left; rewrite InZ_cons; left.
    -    } {
    -      by right; rewrite InZ_cons; tauto.
    -    }
    -  Qed.
    - -
    -  Lemma In_elementsZ_single_hd : ∀ (c : N) x, (c ≠ 0)%N → List.In x (elementsZ_single x c).
    -  Proof.
    -    intros c x H_c_neq.
    -    rewrite In_elementsZ_single.
    -    split. {
    -      apply Z.le_refl.
    -    } {
    -      apply Z.lt_add_pos_r.
    -      have → : 0 = Z.of_N (0%N) by [].
    -      apply N2Z.inj_lt.
    -      by apply N.neq_0_lt_0.
    -    }
    -  Qed.
    - +     {InZ y s} + {¬InZ y s}.
    + +
    +  Lemma In_elementsZ_single_hd : ∀ (c : N) x, (c ≠ 0)%N → List.In x (elementsZ_single x c).
    +

    -

    comparing intervals

    +

    comparing intervals


      Ltac Z_named_compare_cases H := match goal with
    -    | [ |- context [Z.compare ?z1 ?z2] ] ⇒
    -      case_eq (Z.compare z1 z2); [move ⇒ /Z.compare_eq_iff | move ⇒ /Z.compare_lt_iff | move ⇒ /Z.compare_gt_iff]; move ⇒ H //
    +    | [ |- context [Z.compare ?z1 ?z2] ] ⇒
    +      case_eq (Z.compare z1 z2); [move ⇒ /Z.compare_eq_iff | move ⇒ /Z.compare_lt_iff | move ⇒ /Z.compare_gt_iff]; move ⇒ H //
      end.

      Ltac Z_compare_cases := let H := fresh "H" in Z_named_compare_cases H.

    -  Lemma interval_compare_elim : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    -    match (interval_compare (y1, c1) (y2, c2)) with
    -      | ICR_before ⇒ (y1 + Z.of_N c1) < y2
    -      | ICR_before_touch ⇒ (y1 + Z.of_N c1) = y2
    -      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    -      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    -      | ICR_equal ⇒ (y1 = y2) ∧ (c1 = c2)
    -      | ICR_overlap_before ⇒ (y1 < y2) ∧ (y2 < y1 + Z.of_N c1) ∧ (y1 + Z.of_N c1 < y2 + Z.of_N c2)
    -      | ICR_overlap_after ⇒ (y2 < y1) ∧ (y1 < y2 + Z.of_N c2) ∧ (y2 + Z.of_N c2 < y1 + Z.of_N c1)
    -      | ICR_subsume_1 ⇒ (y2 ≤ y1) ∧ (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) ∧ (y2 < y1 ∨ y1 + Z.of_N c1 < y2 + Z.of_N c2)
    -      | ICR_subsume_2 ⇒ (y1 ≤ y2) ∧ (y2 + Z.of_N c2 ≤ y1 + Z.of_N c1) ∧ (y1 < y2 ∨ y2 + Z.of_N c2 < y1 + Z.of_N c1)
    +  Lemma interval_compare_elim : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    +    match (interval_compare (y1, c1) (y2, c2)) with
    +      | ICR_before ⇒ (y1 + Z.of_N c1) < y2
    +      | ICR_before_touch ⇒ (y1 + Z.of_N c1) = y2
    +      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    +      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    +      | ICR_equal ⇒ (y1 = y2) ∧ (c1 = c2)
    +      | ICR_overlap_before ⇒ (y1 < y2) ∧ (y2 < y1 + Z.of_N c1) ∧ (y1 + Z.of_N c1 < y2 + Z.of_N c2)
    +      | ICR_overlap_after ⇒ (y2 < y1) ∧ (y1 < y2 + Z.of_N c2) ∧ (y2 + Z.of_N c2 < y1 + Z.of_N c1)
    +      | ICR_subsume_1 ⇒ (y2 ≤ y1) ∧ (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) ∧ (y2 < y1 ∨ y1 + Z.of_N c1 < y2 + Z.of_N c2)
    +      | ICR_subsume_2 ⇒ (y1 ≤ y2) ∧ (y2 + Z.of_N c2 ≤ y1 + Z.of_N c1) ∧ (y1 < y2 ∨ y2 + Z.of_N c2 < y1 + Z.of_N c1)
        end.
    -  Proof.
    -    intros y1 c1 y2 c2.
    -    rewrite /interval_compare.
    -    (repeat Z_compare_cases); subst; repeat split;
    -       try (by apply Z.eq_le_incl);
    -       try (by apply Z.lt_le_incl);
    -       try (by left); try (by right).
    - -
    -    apply Z.add_reg_l in H2.
    -    by apply N2Z.inj.
    -  Qed.

    -  Lemma interval_compare_swap : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    -    (c1 ≠ 0%N) ∨ (c2 ≠ 0%N) →
    -    interval_compare (y2, c2) (y1, c1) =
    -    match (interval_compare (y1, c1) (y2, c2)) with
    +  Lemma interval_compare_swap : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    +    (c1 ≠ 0%N) ∨ (c2 ≠ 0%N) →
    +    interval_compare (y2, c2) (y1, c1) =
    +    match (interval_compare (y1, c1) (y2, c2)) with
          | ICR_before ⇒ ICR_after
          | ICR_before_touch ⇒ ICR_after_touch
          | ICR_after ⇒ ICR_before
    @@ -1472,176 +1119,55 @@

    Library MSetsExtra.MSetIntervals

          | ICR_subsume_1 ⇒ ICR_subsume_2
          | ICR_subsume_2 ⇒ ICR_subsume_1
        end.
    -  Proof.
    -    intros y1 c1 y2 c2 H_c12_neq_0.
    -    rewrite /interval_compare.
    -    move : (Z.compare_antisym y1 y2) ⇒ →.
    -    move : (Z.compare_antisym (y1 + Z.of_N c1) (y2 + Z.of_N c2)) ⇒ →.
    -    have H_suff : y1 + Z.of_N c1 ≤ y2 → y2 + Z.of_N c2 ≤ y1 → False. {
    -      move ⇒ H1 H2.
    -      case H_c12_neq_0 ⇒ H_c_neq_0. {
    -        suff : (y1 + Z.of_N c1 ≤ y1). {
    -          apply Z.nle_gt.
    -          by apply Z_lt_add_r.
    -        }
    -        eapply Z.le_trans; eauto.
    -        eapply Z.le_trans; eauto.
    -        apply Z_le_add_r.
    -      } {
    -        suff : (y2 + Z.of_N c2 ≤ y2). {
    -          apply Z.nle_gt.
    -          by apply Z_lt_add_r.
    -        }
    -        eapply Z.le_trans; eauto.
    -        eapply Z.le_trans; eauto.
    -        apply Z_le_add_r.
    -      }
    -    }
    -    repeat Z_compare_cases. {
    -      exfalso; apply H_suff.
    -        - by rewrite H; apply Z.le_refl.
    -        - by rewrite H0; apply Z.le_refl.
    -    } {
    -      exfalso; apply H_suff.
    -        - by rewrite H; apply Z.le_refl.
    -        - by apply Z.lt_le_incl.
    -    } {
    -      exfalso; apply H_suff.
    -        - by apply Z.lt_le_incl.
    -        - by rewrite H0; apply Z.le_refl.
    -    } {
    -      exfalso; apply H_suff.
    -        - by apply Z.lt_le_incl.
    -        - by apply Z.lt_le_incl.
    -    }
    -  Qed.
    - -
    -  Lemma interval_1_compare_alt_def : ∀ (y : Z) (i : (Z × N)),
    -    interval_1_compare y i = match (interval_compare (y, (1%N)) i) with
    + +
    +  Lemma interval_1_compare_alt_def : ∀ (y : Z) (i : (Z × N)),
    +    interval_1_compare y i = match (interval_compare (y, (1%N)) i) with
          | ICR_equal ⇒ ICR_subsume_1
          | ICR_subsume_1 ⇒ ICR_subsume_1
          | ICR_subsume_2 ⇒ ICR_subsume_1
          | r ⇒ r
        end.
    -  Proof.
    -    move ⇒ y1 [y2 c2].
    -    rewrite /interval_1_compare /interval_compare.
    -    replace (y1 + Z.of_N 1) with (Z.succ y1); last done.
    -    repeat Z_compare_cases. {
    -      contradict H1.
    -      by apply Zle_not_lt, Zlt_succ_le.
    -    } {
    -      contradict H.
    -      by apply Zle_not_lt, Zlt_succ_le.
    -    }
    -  Qed.
    - -
    -  Lemma interval_1_compare_elim : ∀ (y1 : Z) (y2 : Z) (c2 : N),
    -    match (interval_1_compare y1 (y2, c2)) with
    -      | ICR_before ⇒ Z.succ y1 < y2
    -      | ICR_before_touch ⇒ y2 = Z.succ y1
    -      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    -      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    -      | ICR_equal ⇒ False
    -      | ICR_overlap_before ⇒ False
    -      | ICR_overlap_after ⇒ False
    -      | ICR_subsume_1 ⇒ (c2 = 0%N) ∨ ((y2 ≤ y1) ∧ (y1 < y2 + Z.of_N c2))
    -      | ICR_subsume_2 ⇒ False
    + +
    +  Lemma interval_1_compare_elim : ∀ (y1 : Z) (y2 : Z) (c2 : N),
    +    match (interval_1_compare y1 (y2, c2)) with
    +      | ICR_before ⇒ Z.succ y1 < y2
    +      | ICR_before_touch ⇒ y2 = Z.succ y1
    +      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    +      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    +      | ICR_equal ⇒ False
    +      | ICR_overlap_before ⇒ False
    +      | ICR_overlap_after ⇒ False
    +      | ICR_subsume_1 ⇒ (c2 = 0%N) ∨ ((y2 ≤ y1) ∧ (y1 < y2 + Z.of_N c2))
    +      | ICR_subsume_2 ⇒ False
        end.
    -  Proof.
    -    intros y1 y2 c2.
    -    move : (interval_compare_elim y1 (1%N) y2 c2).
    -    rewrite interval_1_compare_alt_def.
    -    have H_succ: ∀ z, z + Z.of_N 1 = Z.succ z by done.
    - -
    -    case_eq (interval_compare (y1, 1%N) (y2, c2)) ⇒ H_comp;
    -      rewrite ?H_succ ?Z.lt_succ_r ?Z.le_succ_l //. {
    -      move ⇒ [H_lt] [H_le] _.
    -      contradict H_lt.
    -      by apply Zle_not_lt.
    -    } {
    -      move ⇒ [_] [H_lt] H_le.
    -      contradict H_lt.
    -      by apply Zle_not_lt.
    -    } {
    -      move ⇒ [->] <-.
    -      rewrite ?Z.lt_succ_r.
    -      right.
    -      split; apply Z.le_refl.
    -    } {
    -      tauto.
    -    } {
    -      case (N.zero_or_succ c2). {
    -        move ⇒ → _; by left.
    -      } {
    -        move ⇒ [c2´] →.
    -        rewrite !N2Z.inj_succ Z.add_succ_r -Z.succ_le_mono Z.le_succ_l.
    -        move ⇒ [H_y1_le] [H_le_y1].
    -        suff → : y1 = y2. {
    -          move ⇒ [] H_pre; contradict H_pre. {
    -            apply Z.lt_irrefl.
    -          } {
    -            apply Zle_not_lt, Z_le_add_r.
    -          }
    -        }
    -        apply Z.le_antisymm ⇒ //.
    -        eapply Z.le_trans; last apply H_le_y1.
    -        apply Z_le_add_r.
    -      }
    -    }
    -  Qed.
    - +
    -

    Alternative definition of addZ

    +

    Alternative definition of addZ

      Lemma addZ_aux_alt_def : ∀ x s acc,
    -    addZ_aux acc x s = (List.rev acc) ++ addZ x s.
    -  Proof.
    -    intros y1 s.
    -    unfold addZ.
    -    induction s as [| [y2 c2] s´ IH] ⇒ acc. {
    -      rewrite /addZ_aux /addZ /= /rev´ !rev_append_rev /= app_nil_r //.
    -    } {
    -      unfold addZ_aux.
    -      case (interval_1_compare y1 (y2, c2)); fold addZ_aux;
    -        rewrite ?rev_append_rev /= ?app_assoc_reverse //.
    -      rewrite (IH ((y2,c2)::acc)) (IH ((y2,c2)::nil)).
    -      rewrite /= app_assoc_reverse //.
    -    }
    -  Qed.
    - +    addZ_aux acc x s = (List.rev acc) ++ addZ x s.
    +
      Lemma addZ_alt_def : ∀ x s,
    -    addZ x s =
    +    addZ x s =
        match s with
    -    | nil ⇒ (x, (1%N))::nil
    -    | (y, c) :: l ⇒
    -        match (interval_1_compare x (y,c)) with
    -          | ICR_before ⇒ (x, (1%N))::s
    -          | ICR_before_touch ⇒ (x, N.succ c)::l
    -          | ICR_after ⇒ (y,c)::(addZ x l)
    -          | ICR_after_touch ⇒ insert_interval_begin y (N.succ c) l
    -          | _ ⇒ (y, c)::l
    +    | nil ⇒ (x, (1%N))::nil
    +    | (y, c) :: l ⇒
    +        match (interval_1_compare x (y,c)) with
    +          | ICR_before ⇒ (x, (1%N))::s
    +          | ICR_before_touch ⇒ (x, N.succ c)::l
    +          | ICR_after ⇒ (y,c)::(addZ x l)
    +          | ICR_after_touch ⇒ insert_interval_begin y (N.succ c) l
    +          | _ ⇒ (y, c)::l
            end
        end.
    -  Proof.
    -    intros x s.
    -    rewrite /addZ.
    -    case s ⇒ //.
    -    move ⇒ [y c] s´.
    -    unfold addZ_aux.
    -    case (interval_1_compare x (y, c)); fold addZ_aux;
    -      rewrite ?rev_append_rev /= ?app_assoc_reverse //.
    -    rewrite addZ_aux_alt_def //.
    -  Qed.

    @@ -1649,413 +1175,145 @@

    Library MSetsExtra.MSetIntervals

    -

    Auxiliary Lemmata about Invariant

    +

    Auxiliary Lemmata about Invariant


      Lemma interval_list_elements_greater_cons : ∀ z x c s,
    -    interval_list_elements_greater z ((x, c) :: s) = true ↔
    -    (z < x).
    -  Proof.
    -    intros z x c s.
    -    rewrite /=.
    -    apply Z.ltb_lt.
    -  Qed.
    +    interval_list_elements_greater z ((x, c) :: s) = true ↔
    +    (z < x).

      Lemma interval_list_elements_greater_impl : ∀ x y s,
    -    (y ≤ x) →
    -    interval_list_elements_greater x s = true →
    -    interval_list_elements_greater y s = true.
    -  Proof.
    -    intros x y s.
    -    case s ⇒ //.
    -    move ⇒ [z c] s´.
    -    rewrite /interval_list_elements_greater.
    -    move ⇒ H_y_leq /Z.ltb_lt H_x_lt.
    -    apply Z.ltb_lt.
    -    eapply Z.le_lt_trans; eauto.
    -  Qed.
    +    (y ≤ x) →
    +    interval_list_elements_greater x s = true →
    +    interval_list_elements_greater y s = true.

    -  Lemma interval_list_invariant_nil : interval_list_invariant nil = true.
    -  Proof.
    -    by [].
    -  Qed.
    +  Lemma interval_list_invariant_nil : interval_list_invariant nil = true.

    -  Lemma Ok_nil : Ok nil ↔ True.
    -  Proof.
    -    rewrite /Ok /IsOk /interval_list_invariant /is_encoded_elems_list //.
    -  Qed.
    +  Lemma Ok_nil : Ok nil ↔ True.

      Lemma is_encoded_elems_list_app : ∀ l1 l2,
    -    is_encoded_elems_list (l1 ++ l2) ↔
    -    (is_encoded_elems_list l1 ∧ is_encoded_elems_list l2).
    -  Proof.
    -    intros l1 l2.
    -    rewrite /is_encoded_elems_list.
    -    setoid_rewrite in_app_iff.
    -    split; firstorder.
    -  Qed.
    +    is_encoded_elems_list (l1 ++ l2) ↔
    +    (is_encoded_elems_list l1 ∧ is_encoded_elems_list l2).

      Lemma is_encoded_elems_list_rev : ∀ l,
    -    is_encoded_elems_list (List.rev l) ↔
    +    is_encoded_elems_list (List.rev l) ↔
        is_encoded_elems_list l.
    -  Proof.
    -    intros l.
    -    rewrite /is_encoded_elems_list.
    -    split; (
    -      move ⇒ H x H_in;
    -      apply H;
    -      move : H_in;
    -      rewrite -in_rev ⇒ //
    -    ).
    -  Qed.
    - -
    -  Lemma interval_list_invariant_cons : ∀ y c s´,
    -    interval_list_invariant ((y, c) :: s´) = true ↔
    -    (interval_list_elements_greater (y+Z.of_N c) s´ = true ∧
    -      ((c ≠ 0)%N) ∧ interval_list_invariant s´ = true).
    -  Proof.
    -    rewrite /interval_list_invariant -/interval_list_invariant.
    -    intros y c s´.
    -    rewrite !Bool.andb_true_iff negb_true_iff.
    -    split. {
    -      move ⇒ [] [H_inf] /N.eqb_neq H_c H_s´. tauto.
    -    } {
    -      move ⇒ [H_inf] [/N.eqb_neq H_c] H_s´. tauto.
    -    }
    -  Qed.
    +
    +  Lemma interval_list_invariant_cons : ∀ y c s',
    +    interval_list_invariant ((y, c) :: s') = true ↔
    +    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧
    +      ((c ≠ 0)%N) ∧ interval_list_invariant s' = true).
    +
      Lemma interval_list_invariant_sing : ∀ x c,
    -    interval_list_invariant ((x, c)::nil) = true ↔ (c ≠ 0)%N.
    -  Proof.
    -    intros x c.
    -    rewrite interval_list_invariant_cons.
    -    split; tauto.
    -  Qed.
    +    interval_list_invariant ((x, c)::nil) = true ↔ (c ≠ 0)%N.

    -  Lemma Ok_cons : ∀ y c s´, Ok ((y, c) :: s´) ↔
    -    (interval_list_elements_greater (y+Z.of_N c) s´ = true ∧ ((c ≠ 0)%N) ∧
    -     is_encoded_elems_list (elementsZ_single y c) ∧ Ok s´).
    -  Proof.
    -    intros y c s´.
    -    rewrite /Ok /IsOk interval_list_invariant_cons elementsZ_cons is_encoded_elems_list_app
    -       is_encoded_elems_list_rev.
    -    tauto.
    -  Qed.
    +  Lemma Ok_cons : ∀ y c s', Ok ((y, c) :: s') ↔
    +    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧ ((c ≠ 0)%N) ∧
    +     is_encoded_elems_list (elementsZ_single y c) ∧ Ok s').

      Lemma Nin_elements_greater : ∀ s y,
    -     interval_list_elements_greater y s = true →
    -     interval_list_invariant s = true →
    -     âˆ€ x, x ≤ y → ~(InZ x s).
    -  Proof.
    -    induction s as [| [z c] s´ IH]. {
    -      intros y _ _ x _.
    -      by simpl.
    -    } {
    -      move ⇒ y /interval_list_elements_greater_cons H_y_lt
    -        /interval_list_invariant_cons [H_gr] [H_c] H_s´
    -        x H_x_le.
    -      rewrite InZ_cons In_elementsZ_single.
    -      have H_x_lt : x < z by eapply Z.le_lt_trans; eauto.
    - -
    -      move ⇒ []. {
    -        move ⇒ [H_z_leq] _; contradict H_z_leq.
    -        by apply Z.nle_gt.
    -      } {
    -        eapply IH; eauto.
    -        by apply Z_lt_le_add_r.
    -      }
    -    }
    -  Qed.
    - +     interval_list_elements_greater y s = true →
    +     interval_list_invariant s = true →
    +     âˆ€ x, x ≤ y → ~(InZ x s).
    +
      Lemma Nin_elements_greater_equal :
         âˆ€ x s,
    -       interval_list_elements_greater x s = true →
    -       interval_list_invariant s = true →
    -       Â¬ (InZ x s).
    -  Proof.
    -    move ⇒ x s H_inv H_gr.
    -    apply (Nin_elements_greater s x) ⇒ //.
    -    apply Z.le_refl.
    -  Qed.
    +       interval_list_elements_greater x s = true →
    +       interval_list_invariant s = true →
    +       Â¬ (InZ x s).

      Lemma interval_list_elements_greater_alt_def : ∀ s y,
    -     interval_list_invariant s = true →
    -
    -     (interval_list_elements_greater y s = true ↔
    -      (∀ x, x ≤ y → ~(InZ x s))).
    -  Proof.
    -    intros s y H_inv.
    -    split. {
    -      move ⇒ H_gr.
    -      apply Nin_elements_greater ⇒ //.
    -    } {
    -      move : H_inv.
    -      case s as [| [x2 c] s´] ⇒ //.
    -      rewrite interval_list_invariant_cons interval_list_elements_greater_cons.
    -      move ⇒ [_] [H_c_neq] _ H.
    -      apply Z.nle_gt ⇒ H_ge.
    -      apply (H x2) ⇒ //.
    -      rewrite InZ_cons; left.
    -      apply In_elementsZ_single_hd ⇒ //.
    -    }
    -  Qed.
    - +     interval_list_invariant s = true →
    +
    +     (interval_list_elements_greater y s = true ↔
    +      (∀ x, x ≤ y → ~(InZ x s))).
    +
      Lemma interval_list_elements_greater_alt2_def : ∀ s y,
    -     interval_list_invariant s = true →
    +     interval_list_invariant s = true →

    -     (interval_list_elements_greater y s = true ↔
    -      (∀ x, InZ x s → y < x)).
    -  Proof.
    -    intros s y H.
    -    rewrite interval_list_elements_greater_alt_def //.
    -    firstorder.
    -    apply Z.nle_gt.
    -    move ⇒ H_lt.
    -    eapply H0; eauto.
    -  Qed.
    +     (interval_list_elements_greater y s = true ↔
    +      (∀ x, InZ x s → y < x)).

      Lemma interval_list_elements_greater_intro : ∀ s y,
    -     interval_list_invariant s = true →
    -     (∀ x, InZ x s → y < x) →
    -     interval_list_elements_greater y s = true.
    -  Proof.
    -    intros s y H1 H2.
    -    rewrite interval_list_elements_greater_alt2_def //.
    -  Qed.
    +     interval_list_invariant s = true →
    +     (∀ x, InZ x s → y < x) →
    +     interval_list_elements_greater y s = true.

      Lemma interval_list_elements_greater_app_elim_1 : ∀ s1 s2 y,
    -    interval_list_elements_greater y (s1 ++ s2) = true →
    -    interval_list_elements_greater y s1 = true.
    -  Proof.
    -    intros s1 s2 y.
    -    case s1 ⇒ //.
    -  Qed.
    +    interval_list_elements_greater y (s1 ++ s2) = true →
    +    interval_list_elements_greater y s1 = true.

      Lemma interval_list_invariant_app_intro : ∀ s1 s2,
    -      interval_list_invariant s1 = true →
    -      interval_list_invariant s2 = true →
    -      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    -      interval_list_invariant (s1 ++ s2) = true.
    -  Proof.
    -    induction s1 as [| [y1 c1] s1´ IH]. {
    -      move ⇒ s2 _ //.
    -    } {
    -      move ⇒ s2.
    -      rewrite -app_comm_cons !interval_list_invariant_cons.
    -      move ⇒ [H_gr] [H_c1_neq] H_inv_s1´ H_inv_s2 H_inz_s2.
    -      split; last split. {
    -        move : H_gr H_inz_s2.
    -        case s1´ as [| [y1´ c1´] s1´´]; last done.
    -        move ⇒ _ H_inz_s2.
    -        rewrite app_nil_l.
    -        apply interval_list_elements_greater_intro ⇒ //.
    -        move ⇒ x H_x_in_s2.
    -        suff H_inz : InZ (Z.pred (y1 + Z.of_N c1)) ((y1, c1) :: nil). {
    -          move : (H_inz_s2 _ _ H_inz H_x_in_s2).
    -          by rewrite Z.succ_pred.
    -        }
    -        rewrite InZ_cons In_elementsZ_single -Z.lt_le_pred; left.
    -        split. {
    -          by apply Z_lt_add_r.
    -        } {
    -          apply Z.lt_pred_l.
    -        }
    -      } {
    -        assumption.
    -      } {
    -        apply IH ⇒ //.
    -        intros x1 x2 H_in_x1 H_in_x2.
    -        apply H_inz_s2 ⇒ //.
    -        rewrite InZ_cons; by right.
    -      }
    -    }
    -  Qed.
    - +      interval_list_invariant s1 = true →
    +      interval_list_invariant s2 = true →
    +      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    +      interval_list_invariant (s1 ++ s2) = true.
    +
      Lemma interval_list_invariant_app_elim : ∀ s1 s2,
    -      interval_list_invariant (s1 ++ s2) = true →
    -      interval_list_invariant s1 = true ∧
    -      interval_list_invariant s2 = true ∧
    -      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2).
    -  Proof.
    -    move ⇒ s1 s2.
    -    induction s1 as [| [y1 c1] s1´ IH]; first done.
    -    rewrite -app_comm_cons !interval_list_invariant_cons.
    - -
    -    move ⇒ [H_gr] [H_c1_neq_0] /IH [H_inv_s1´] [H_inv_s2] H_in_s1´_s2.
    -    repeat split; try assumption. {
    -      move : H_gr.
    -      case s1´; first done.
    -      move ⇒ [y2 c2] s1´´.
    -      rewrite interval_list_elements_greater_cons //.
    -    } {
    -      move ⇒ x1 x2.
    -      rewrite InZ_cons In_elementsZ_single.
    -      move ⇒ []; last by apply H_in_s1´_s2.
    -      move ⇒ [] H_y1_le H_x1_lt H_x2_in.
    -      move : H_gr.
    -      rewrite interval_list_elements_greater_alt2_def; last first. {
    -          by apply interval_list_invariant_app_intro.
    -      }
    -      move ⇒ H_in_s12´.
    -      have : (y1 + Z.of_N c1 < x2). {
    -        apply H_in_s12´.
    -        rewrite InZ_app.
    -        by right.
    -      }
    -      move ⇒ H_lt_x2.
    -      apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -      by apply Zlt_le_succ.
    -    }
    -  Qed.
    - +      interval_list_invariant (s1 ++ s2) = true →
    +      interval_list_invariant s1 = true ∧
    +      interval_list_invariant s2 = true ∧
    +      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2).
    +
      Lemma interval_list_invariant_app_iff : ∀ s1 s2,
    -      interval_list_invariant (s1 ++ s2) = true ↔
    -      (interval_list_invariant s1 = true ∧
    -      interval_list_invariant s2 = true ∧
    -      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2)).
    -  Proof.
    -    intros s1 s2.
    -    split. {
    -      by apply interval_list_invariant_app_elim.
    -    } {
    -      move ⇒ [] H_inv_s1 [].
    -      by apply interval_list_invariant_app_intro.
    -    }
    -  Qed.
    - +      interval_list_invariant (s1 ++ s2) = true ↔
    +      (interval_list_invariant s1 = true ∧
    +      interval_list_invariant s2 = true ∧
    +      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2)).
    +
      Lemma interval_list_invariant_snoc_intro : ∀ s1 y2 c2,
    -      interval_list_invariant s1 = true →
    -      (c2 ≠ 0)%N →
    -      (∀ x, InZ x s1 → Z.succ x < y2) →
    -      interval_list_invariant (s1 ++ ((y2, c2)::nil)) = true.
    -  Proof.
    -    intros s1 y2 c2 H_inv_s1 H_c2_neq H_in_s1.
    -    apply interval_list_invariant_app_intro ⇒ //. {
    -      rewrite interval_list_invariant_cons; done.
    -    } {
    -      intros x1 x2 H_in_x1.
    -      rewrite InZ_cons.
    -      move ⇒ [] //.
    -      rewrite In_elementsZ_single.
    -      move ⇒ [H_y2_le] _.
    -      eapply Z.lt_le_trans; eauto.
    -    }
    -  Qed.
    - +      interval_list_invariant s1 = true →
    +      (c2 ≠ 0)%N →
    +      (∀ x, InZ x s1 → Z.succ x < y2) →
    +      interval_list_invariant (s1 ++ ((y2, c2)::nil)) = true.
    +

    -

    Properties of In and InZ

    +

    Properties of In and InZ


    -  Lemma encode_decode_eq : ∀ x s, Ok s → InZ x s →
    -    (Enc.encode (Enc.decode x) = x).
    -  Proof.
    -    intros x s.
    -    rewrite /Ok /IsOk /InZ.
    -    move ⇒ [_] H_enc H_in_x.
    -    move : (H_enc _ H_in_x) ⇒ [x´] <-.
    -    rewrite Enc.decode_encode_ok //.
    -  Qed.
    - -
    -  Lemma In_alt_def : ∀ x s, Ok s →
    -    (In x s ↔ List.In x (elements s)).
    -  Proof.
    -    intros x s H_ok.
    -    rewrite /In InA_alt /elements rev_map_alt_def.
    -    split. {
    -      move ⇒ [y] [H_y_eq].
    -      rewrite -!in_rev !in_map_iff.
    -      move ⇒ [x´] [H_y_eq´] H_x´_in.
    -      suff H_x´_eq : (Enc.encode x = x´). {
    -        âˆƒ x´.
    -        split ⇒ //.
    -        rewrite -H_x´_eq Enc.decode_encode_ok //.
    -      }
    -      have H_enc_list : is_encoded_elems_list (elementsZ s). {
    -        move : H_ok.
    -        rewrite /Ok /IsOk ⇒ [] [] //.
    -      }
    -      move : (H_enc_list _ H_x´_in) ⇒ [x´´] H_x´_eq.
    -      move : H_y_eq´.
    -      rewrite -!H_x´_eq Enc.decode_encode_ok ⇒ H_y_eq´.
    -      subst.
    -      suff → : Z.eq (Enc.encode x) (Enc.encode y) by done.
    -      by rewrite Enc.encode_eq.
    -    } {
    -      move ⇒ H_enc_in.
    -      âˆƒ x.
    -      split ⇒ //.
    -      apply Enc.E.eq_equiv.
    -    }
    -  Qed.
    - -
    -  Lemma In_InZ : ∀ x s, Ok s →
    -    (In x s ↔ InZ (Enc.encode x) s).
    -  Proof.
    -    intros x s H_ok.
    -    rewrite /InZ In_alt_def /elements rev_map_alt_def -in_rev in_map_iff.
    -    split; last first. {
    -      âˆƒ (Enc.encode x).
    -      by rewrite Enc.decode_encode_ok.
    -    }
    -    move ⇒ [y] [<-] H_y_in.
    -    suff : ∃ z, (Enc.encode z = y). {
    -      move ⇒ [z] H_y_eq.
    -      move : H_y_in.
    -      by rewrite -!H_y_eq Enc.decode_encode_ok.
    -    }
    -    suff H_enc_list : is_encoded_elems_list (elementsZ s). {
    -      by apply H_enc_list.
    -    }
    -    apply H_ok.
    -  Qed.
    - -
    -  Lemma InZ_In : ∀ x s, Ok s →
    -    (InZ x s → In (Enc.decode x) s).
    -  Proof.
    -    intros x s H_ok.
    -    rewrite In_InZ /InZ.
    -    move : H_ok.
    -    rewrite /Ok /IsOk /is_encoded_elems_list.
    -    move ⇒ [_] H_enc.
    -    move ⇒ H_in.
    -    move : (H_enc _ H_in) ⇒ [e] H_x.
    -    subst.
    -    by rewrite Enc.decode_encode_ok.
    -  Qed.
    +  Lemma encode_decode_eq : ∀ x s, Ok s → InZ x s →
    +    (Enc.encode (Enc.decode x) = x).
    + +
    +  Lemma In_alt_def : ∀ x s, Ok s →
    +    (In x s ↔ List.In x (elements s)).
    + +
    +  Lemma In_InZ : ∀ x s, Ok s →
    +    (In x s ↔ InZ (Enc.encode x) s).
    + +
    +  Lemma InZ_In : ∀ x s, Ok s →
    +    (InZ x s → In (Enc.decode x) s).

    @@ -2063,625 +1321,219 @@

    Library MSetsExtra.MSetIntervals

    -

    Membership specification

    +

    Membership specification


      Lemma memZ_spec :
    -    âˆ€ (s : t) (x : Z) (Hs : Ok s), memZ x s = true ↔ InZ x s.
    -  Proof.
    -    induction s as [| [y c] s´ IH]. {
    -      intros x _.
    -      rewrite /InZ elementsZ_nil //.
    -    } {
    -      move ⇒ x /Ok_cons [H_inf] [H_c] [H_is_enc] H_s´.
    -      rewrite /InZ /memZ elementsZ_cons -/memZ.
    -      rewrite in_app_iff -!in_rev In_elementsZ_single.
    - -
    -      case_eq (x <? y). {
    -        move ⇒ /Z.ltb_lt H_x_lt.
    -        split; first done.
    -        move ⇒ []. {
    -          move ⇒ H_x_in; contradict H_x_in.
    -          apply Nin_elements_greater with (y := (y + Z.of_N c)) ⇒ //. {
    -            apply H_s´.
    -          } {
    -            apply Z_lt_le_add_r ⇒ //.
    -          }
    -        } {
    -          move ⇒ [H_y_le]; contradict H_y_le.
    -          by apply Z.nle_gt.
    -        }
    -      } {
    -        move ⇒ /Z.ltb_ge H_y_le.
    -        case_eq (x <? y + Z.of_N c). {
    -          move ⇒ /Z.ltb_lt H_x_lt.
    -          split; last done.
    -          move ⇒ _.
    -          by right.
    -        } {
    -          move ⇒ /Z.ltb_ge H_yc_le.
    -          rewrite IH.
    -          split; first tauto.
    -          move ⇒ [] //.
    -          move ⇒ [_] H_x_lt; contradict H_x_lt.
    -          by apply Z.nlt_ge.
    -        }
    -      }
    -    }
    -  Qed.
    - +    âˆ€ (s : t) (x : Z) (Hs : Ok s), memZ x s = true ↔ InZ x s.
    +
      Lemma mem_spec :
    -   âˆ€ (s : t) (x : elt) (Hs : Ok s), mem x s = true ↔ In x s.
    -  Proof.
    -    intros s x Hs.
    -    rewrite /mem memZ_spec In_InZ //.
    -  Qed.
    +   âˆ€ (s : t) (x : elt) (Hs : Ok s), mem x s = true ↔ In x s.

      Lemma merge_interval_size_neq_0 : ∀ x1 c1 x2 c2,
    -     (c1 ≠ 0%N) →
    -     (merge_interval_size x1 c1 x2 c2 ≠ 0)%N.
    -  Proof.
    -    intros x1 c1 x2 c2.
    -    rewrite /merge_interval_size !N.neq_0_lt_0 N.max_lt_iff.
    -    by left.
    -  Qed.
    +     (c1 ≠ 0%N) →
    +     (merge_interval_size x1 c1 x2 c2 ≠ 0)%N.

    -

    insert if length not 0

    +

    insert if length not 0


      Lemma interval_list_invariant_insert_intervalZ_guarded : ∀ x c s,
    -    interval_list_invariant s = true →
    -    interval_list_elements_greater (x + Z.of_N c) s = true →
    -    interval_list_invariant (insert_intervalZ_guarded x c s) = true.
    -  Proof.
    -    intros x c s.
    -    rewrite /insert_intervalZ_guarded.
    -    case_eq (c =? 0)%N ⇒ //.
    -    move ⇒ /N.eqb_neq.
    -    rewrite interval_list_invariant_cons.
    -    tauto.
    -  Qed.
    +    interval_list_invariant s = true →
    +    interval_list_elements_greater (x + Z.of_N c) s = true →
    +    interval_list_invariant (insert_intervalZ_guarded x c s) = true.

      Lemma interval_list_elements_greater_insert_intervalZ_guarded : ∀ x c y s,
    -    interval_list_elements_greater y (insert_intervalZ_guarded x c s) = true ↔
    -    (if (c =? 0)%N then (interval_list_elements_greater y s = true) else (y < x)).
    -  Proof.
    -    intros x c y s.
    -    rewrite /insert_intervalZ_guarded.
    -    case (c =? 0)%N ⇒ //.
    -    rewrite /interval_list_elements_greater Z.ltb_lt //.
    -  Qed.
    +    interval_list_elements_greater y (insert_intervalZ_guarded x c s) = true ↔
    +    (if (c =? 0)%N then (interval_list_elements_greater y s = true) else (y < x)).

      Lemma insert_intervalZ_guarded_app : ∀ x c s1 s2,
    -    (insert_intervalZ_guarded x c s1) ++ s2 =
    -    insert_intervalZ_guarded x c (s1 ++ s2).
    -  Proof.
    -    intros x c s1 s2.
    -    rewrite /insert_intervalZ_guarded.
    -    case (N.eqb c 0) ⇒ //.
    -  Qed.
    +    (insert_intervalZ_guarded x c s1) ++ s2 =
    +    insert_intervalZ_guarded x c (s1 ++ s2).

      Lemma insert_intervalZ_guarded_rev_nil_app : ∀ x c s,
    -    rev (insert_intervalZ_guarded x c nil) ++ s =
    +    rev (insert_intervalZ_guarded x c nil) ++ s =
        insert_intervalZ_guarded x c s.
    -  Proof.
    -    intros x c s.
    -    rewrite /insert_intervalZ_guarded.
    -    case (N.eqb c 0) ⇒ //.
    -  Qed.

    Lemma elementsZ_insert_intervalZ_guarded : ∀ x c s,
    -    elementsZ (insert_intervalZ_guarded x c s) = elementsZ ((x, c) :: s).
    -  Proof.
    -    intros x c s.
    -    rewrite /insert_intervalZ_guarded.
    -    case_eq (c =? 0)%N ⇒ //.
    -    move ⇒ /N.eqb_eq →.
    -    rewrite elementsZ_cons elementsZ_single_base /= app_nil_r //.
    -  Qed.
    +    elementsZ (insert_intervalZ_guarded x c s) = elementsZ ((x, c) :: s).

      Lemma InZ_insert_intervalZ_guarded : ∀ y x c s,
    -    InZ y (insert_intervalZ_guarded x c s) = InZ y ((x, c) :: s).
    -  Proof.
    -    intros y x c s.
    -    rewrite /InZ elementsZ_insert_intervalZ_guarded //.
    -  Qed.
    +    InZ y (insert_intervalZ_guarded x c s) = InZ y ((x, c) :: s).

    -

    Merging intervals

    +

    Merging intervals


      Lemma merge_interval_size_add : ∀ x c1 c2,
    -     (merge_interval_size x c1 (x + Z.of_N c1) c2 = (c1 + c2))%N.
    -  Proof.
    -    intros x c1 c2.
    -    rewrite /merge_interval_size.
    -    replace (x + Z.of_N c1 + Z.of_N c2 - x) with
    -            (Z.of_N c1 + Z.of_N c2) by omega.
    -    rewrite -N2Z.inj_add N2Z.id.
    -    apply N.max_r, N.le_add_r.
    -  Qed.
    +     (merge_interval_size x c1 (x + Z.of_N c1) c2 = (c1 + c2))%N.

      Lemma merge_interval_size_eq_max : ∀ y1 c1 y2 c2,
    -     y1 ≤ y2 + Z.of_N c2 →
    -     y1 + Z.of_N (merge_interval_size y1 c1 y2 c2) =
    -     Z.max (y1 + Z.of_N c1) (y2 + Z.of_N c2).
    -  Proof.
    -    intros y1 c1 y2 c2 H_y1_le.
    -    rewrite /merge_interval_size N2Z.inj_max Z2N.id; last first. {
    -      by apply Zle_minus_le_0.
    -    }
    -    rewrite -Z.add_max_distr_l.
    -    replace (y1 + (y2 + Z.of_N c2 - y1)) with (y2 + Z.of_N c2) by omega.
    -    done.
    -  Qed.
    +     y1 ≤ y2 + Z.of_N c2 →
    +     y1 + Z.of_N (merge_interval_size y1 c1 y2 c2) =
    +     Z.max (y1 + Z.of_N c1) (y2 + Z.of_N c2).

      Lemma merge_interval_size_invariant : ∀ y1 c1 y2 c2 z s,
    -    interval_list_invariant s = true →
    -    y1 + Z.of_N c1 ≤ y2 + Z.of_N c2 →
    -    y2 + Z.of_N c2 ≤ z →
    -    interval_list_elements_greater z s = true →
    -    (c1 ≠ 0)%N →
    -    interval_list_invariant ((y1, merge_interval_size y1 c1 y2 c2) :: s) =
    -   true.
    -  Proof.
    -    intros y1 c1 y2 c2 z s H_inv H_le H_le_z H_gr H_c1_neq_0.
    -    rewrite interval_list_invariant_cons.
    -    split; last split. {
    -      rewrite merge_interval_size_eq_max; last first. {
    -        eapply Z.le_trans; last apply H_le.
    -        apply Z_le_add_r.
    -      } {
    -        rewrite Z.max_r ⇒ //.
    -        eapply interval_list_elements_greater_impl; first apply H_le_z.
    -        done.
    -      }
    -    } {
    -      apply merge_interval_size_neq_0.
    -      assumption.
    -    } {
    -      assumption.
    -    }
    -  Qed.
    - +    interval_list_invariant s = true →
    +    y1 + Z.of_N c1 ≤ y2 + Z.of_N c2 →
    +    y2 + Z.of_N c2 ≤ z →
    +    interval_list_elements_greater z s = true →
    +    (c1 ≠ 0)%N →
    +    interval_list_invariant ((y1, merge_interval_size y1 c1 y2 c2) :: s) =
    +   true.
    +
      Lemma In_merge_interval : ∀ x1 c1 x2 c2 y,
    -    x1 ≤ x2 →
    -    x2 ≤ x1 + Z.of_N c1 → (
    -    List.In y (elementsZ_single x1 (merge_interval_size x1 c1 x2 c2)) ↔
    -    List.In y (elementsZ_single x1 c1) ∨ List.In y (elementsZ_single x2 c2)).
    -  Proof.
    -    intros x1 c1 x2 c2 y H_x1_le H_x2_le.
    -    rewrite !In_elementsZ_single merge_interval_size_eq_max;
    -      last first. {
    -      eapply Z.le_trans; eauto.
    -      by apply Z_le_add_r.
    -    }
    -    rewrite Z.max_lt_iff.
    -    split. {
    -      move ⇒ [H_x_le] [] H_y_lt. {
    -        by left.
    -      } {
    -        case_eq (Z.leb x2 y). {
    -          move ⇒ /Z.leb_le H_y´_le.
    -          by right.
    -        } {
    -          move ⇒ /Z.leb_gt H_y_lt_x2.
    -          left.
    -          split ⇒ //.
    -          eapply Z.lt_le_trans; eauto.
    -        }
    -      }
    -    } {
    -      move ⇒ []. {
    -        tauto.
    -      } {
    -        move ⇒ [H_x2_le´] H_y_lt.
    -        split. {
    -          eapply Z.le_trans; eauto.
    -        } {
    -          by right.
    -        }
    -      }
    -    }
    -  Qed.
    - +    x1 ≤ x2 →
    +    x2 ≤ x1 + Z.of_N c1 → (
    +    List.In y (elementsZ_single x1 (merge_interval_size x1 c1 x2 c2)) ↔
    +    List.In y (elementsZ_single x1 c1) ∨ List.In y (elementsZ_single x2 c2)).
    +
      Lemma insert_interval_begin_spec : ∀ y s x c,
    -     interval_list_invariant s = true →
    -     interval_list_elements_greater x s = true →
    -     (c ≠ 0)%N →
    -      (
    -     interval_list_invariant (insert_interval_begin x c s) = true ∧
    -     (InZ y (insert_interval_begin x c s) ↔
    -     (List.In y (elementsZ_single x c) ∨ InZ y s))).
    -  Proof.
    -    intros y.
    -    induction s as [| [y´ c´] s´ IH]. {
    -      intros x c _ H_c_neq H_z_lt.
    -      rewrite /insert_interval_begin InZ_cons interval_list_invariant_cons //.
    -    } {
    -      intros x c.
    -      rewrite interval_list_invariant_cons
    -       interval_list_elements_greater_cons.
    -      move ⇒ [H_gr] [H_c´_neq_0] H_inv_s´ H_x_lt H_c_neq_0.
    -      unfold insert_interval_begin.
    -      Z_named_compare_cases H_y´; fold insert_interval_begin. {
    -        subst.
    -        rewrite !InZ_cons elementsZ_single_add in_app_iff.
    -        split; last tauto.
    -        rewrite interval_list_invariant_cons N2Z.inj_add
    -          Z.add_assoc N.eq_add_0.
    -        tauto.
    -      } {
    -        rewrite !InZ_cons !interval_list_invariant_cons
    -          interval_list_elements_greater_cons.
    -        repeat split ⇒ //.
    -      } {
    -        set c´´ := merge_interval_size x c y´ c´.
    -        have H_x_lt´ : x < y´ + Z.of_N c´. {
    -          eapply Z.lt_le_trans with (m := y´) ⇒ //.
    -          by apply Z_le_add_r.
    -        }
    -
    -        have H_pre : interval_list_elements_greater x s´ = true. {
    -          eapply interval_list_elements_greater_impl; eauto.
    -          by apply Z.lt_le_incl.
    -        }
    -        have H_pre2 : c´´ ≠ 0%N. {
    -          by apply merge_interval_size_neq_0.
    -        }
    -        move : (IH x c´´ H_inv_s´ H_pre H_pre2) ⇒ {IH} {H_pre} {H_pre2} [->] →.
    - -
    -        split; first reflexivity.
    -        unfold c´´; clear c´´.
    -        rewrite In_merge_interval. {
    -          rewrite InZ_cons.
    -          tauto.
    -        } {
    -          by apply Z.lt_le_incl.
    -        } {
    -          by apply Z.lt_le_incl.
    -        }
    -      }
    -    }
    -  Qed.
    - +     interval_list_invariant s = true →
    +     interval_list_elements_greater x s = true →
    +     (c ≠ 0)%N →
    +      (
    +     interval_list_invariant (insert_interval_begin x c s) = true ∧
    +     (InZ y (insert_interval_begin x c s) ↔
    +     (List.In y (elementsZ_single x c) ∨ InZ y s))).
    +
    -

    add specification

    +

    add specification

      Lemma addZ_InZ :
    -   âˆ€ (s : t) (x y : Z),
    -    interval_list_invariant s = true →
    -    (InZ y (addZ x s) ↔ x = y ∨ InZ y s).
    -  Proof.
    -    move ⇒ s x y.
    -    induction s as [| [z c] s´ IH]. {
    -      move ⇒ _.
    -      rewrite /InZ addZ_alt_def
    -              elementsZ_cons elementsZ_nil app_nil_l -in_rev
    -              In_elementsZ_single1 /=.
    -      firstorder.
    -    } {
    -      move ⇒ /interval_list_invariant_cons [H_greater] [H_c_neq_0] H_inv_c´.
    -      move : (IH H_inv_c´) ⇒ {IH} IH.
    -      rewrite addZ_alt_def.
    -      have H_succ : ∀ z, z + Z.of_N 1 = Z.succ z by done.
    -      move : (interval_1_compare_elim x z c).
    -      case (interval_1_compare x (z, c));
    -        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
    -        move ⇒ →.
    -        rewrite elementsZ_single_succ_front /=.
    -        tauto.
    -      } {
    -        move ⇒ [] // H_x_in.
    -        split; first tauto.
    -        move ⇒ [] // <-.
    -        left.
    -        by rewrite In_elementsZ_single.
    -      } {
    -        rewrite IH.
    -        tauto.
    -      } {
    -        move ⇒ H_x_eq.
    -        have → : (InZ y (insert_interval_begin z (N.succ c) s´) ↔
    -                   List.In y (elementsZ_single z (N.succ c)) ∨ InZ y s´). {
    -          eapply insert_interval_begin_spec. {
    -            by apply H_inv_c´.
    -          } {
    -            eapply interval_list_elements_greater_impl; eauto.
    -            apply Z_le_add_r.
    -          } {
    -            by apply N.neq_succ_0.
    -          }
    -        }
    -        rewrite -H_x_eq elementsZ_single_succ in_app_iff /=.
    -        tauto.
    -      }
    -    }
    -  Qed.
    - +   âˆ€ (s : t) (x y : Z),
    +    interval_list_invariant s = true →
    +    (InZ y (addZ x s) ↔ x = y ∨ InZ y s).
    +
      Lemma addZ_invariant : ∀ s x,
    -    interval_list_invariant s = true →
    -    interval_list_invariant (addZ x s) = true.
    -  Proof.
    -    move ⇒ s x.
    -    induction s as [| [z c] s´ IH]. {
    -      move ⇒ _.
    -      by simpl.
    -    } {
    -      move ⇒ /interval_list_invariant_cons [H_greater] [H_c_neq_0]
    -              H_inv_c´.
    -      move : (IH H_inv_c´) ⇒ {IH} IH.
    -      rewrite addZ_alt_def.
    -      have H_succ : ∀ z, z + Z.of_N 1 = Z.succ z by done.
    -      move : (interval_1_compare_elim x z c).
    -      case_eq (interval_1_compare x (z, c)) ⇒ H_comp;
    -        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
    -        move ⇒ H_z_gt.
    -        rewrite interval_list_invariant_cons /= !andb_true_iff !H_succ.
    -        repeat split ⇒ //. {
    -          by apply Z.ltb_lt.
    -        } {
    -          apply negb_true_iff, N.eqb_neq ⇒ //.
    -        }
    -      } {
    -        move ⇒ ?; subst.
    -        rewrite /= !andb_true_iff.
    -        repeat split ⇒ //. {
    -          move : H_greater.
    -          rewrite Z.add_succ_l -Z.add_succ_r N2Z.inj_succ //.
    -        } {
    -          apply negb_true_iff, N.eqb_neq ⇒ //.
    -          apply N.neq_succ_0.
    -        }
    -      } {
    -        move ⇒ [] // _.
    -        rewrite interval_list_invariant_cons /=.
    -        tauto.
    -      } {
    -        rewrite interval_list_invariant_cons.
    -        move ⇒ H_lt_x.
    -        repeat split ⇒ //.
    -        apply interval_list_elements_greater_intro ⇒ //.
    -        move ⇒ xx.
    -        rewrite addZ_InZ ⇒ //.
    -        move ⇒ [<- //|].
    -        apply interval_list_elements_greater_alt2_def ⇒ //.
    -      } {
    -        move ⇒ H_x_eq.
    -        apply insert_interval_begin_spec ⇒ //. {
    -          eapply interval_list_elements_greater_impl; eauto.
    -          apply Z_le_add_r.
    -        } {
    -          by apply N.neq_succ_0.
    -        }
    -      }
    -    }
    -  Qed.
    - +    interval_list_invariant s = true →
    +    interval_list_invariant (addZ x s) = true.
    +
      Global Instance add_ok s x : ∀ `(Ok s), Ok (add x s).
    -  Proof.
    -    move ⇒ H_ok_s.
    -    move : (H_ok_s).
    -    rewrite /Ok /IsOk /is_encoded_elems_list /add.
    -    move ⇒ [H_isok_s] H_pre.
    -    split. {
    -      apply addZ_invariant ⇒ //.
    -    } {
    -      intros y.
    -      move : (addZ_InZ s (Enc.encode x) y H_isok_s).
    -      rewrite /InZ ⇒ →.
    -      move ⇒ []. {
    -        move ⇒ <-.
    -        by ∃ x.
    -      } {
    -        move ⇒ /H_pre //.
    -      }
    -    }
    -  Qed.
    - +
      Lemma add_spec :
       âˆ€ (s : t) (x y : elt) (Hs : Ok s),
    -     In y (add x s) ↔ Enc.E.eq y x ∨ In y s.
    -  Proof.
    -    intros s x y Hs.
    -    have Hs´ := (add_ok s x Hs).
    -    rewrite !In_InZ.
    -    rewrite /add addZ_InZ. {
    -      rewrite -Enc.encode_eq; firstorder.
    -    } {
    -      apply Hs.
    -    }
    -  Qed.
    - +     In y (add x s) ↔ Enc.E.eq y x ∨ In y s.
    +
    -

    empty specification

    +

    empty specification


      Global Instance empty_ok : Ok empty.
    -  Proof.
    -    rewrite /empty Ok_nil //.
    -  Qed.

    -  Lemma empty_spec´ : ∀ x, (In x empty ↔ False).
    -    rewrite /Empty /empty /In elements_nil.
    -    intros a.
    -    rewrite InA_nil //.
    -  Qed.
    +  Lemma empty_spec' : ∀ x, (In x empty ↔ False).

      Lemma empty_spec : Empty empty.
    -  Proof.
    -    rewrite /Empty ⇒ a.
    -    rewrite empty_spec´ //.
    -  Qed.

    -

    is_empty specification

    +

    is_empty specification


    -  Lemma is_empty_spec : ∀ (s : t) (Hs : Ok s), is_empty s = true ↔ Empty s.
    -  Proof.
    -    intros [ | [x c] s]. {
    -      split ⇒ // _.
    -      apply empty_spec.
    -    } {
    -      rewrite /= /Empty Ok_cons.
    -      move ⇒ [_] [H_c_neq] [H_enc] _.
    -      split ⇒ //.
    -      move ⇒ H.
    -      contradiction (H (Enc.decode x)) ⇒ {H}.
    -      rewrite /In InA_alt elements_cons.
    -      âˆƒ (Enc.decode x).
    -      split; first by apply Enc.E.eq_equiv.
    -      rewrite in_app_iff; left.
    -      rewrite /elements_single in_map_iff.
    -      âˆƒ x.
    -      split ⇒ //.
    -      apply In_elementsZ_single_hd ⇒ //.
    -    }
    -  Qed.
    - +  Lemma is_empty_spec : ∀ (s : t) (Hs : Ok s), is_empty s = true ↔ Empty s.
    +
    -

    singleton specification

    +

    singleton specification


      Global Instance singleton_ok x : Ok (singleton x).
    -  Proof.
    -    rewrite singleton_alt_def.
    -    apply add_ok.
    -    apply empty_ok.
    -  Qed.

    -  Lemma singleton_spec : ∀ x y : elt, In y (singleton x) ↔ Enc.E.eq y x.
    -  Proof.
    -    intros x y.
    -    rewrite singleton_alt_def.
    -    rewrite (add_spec empty x y) /empty /In elements_nil InA_nil.
    -    split. {
    -      move ⇒ [] //.
    -    } {
    -      by left.
    -    }
    -  Qed.
    - +  Lemma singleton_spec : ∀ x y : elt, In y (singleton x) ↔ Enc.E.eq y x.
    +
    -

    add_list specification

    +

    add_list specification


    -  Lemma add_list_ok : ∀ l s, Ok s → Ok (add_list l s).
    -  Proof.
    -    induction l as [| x l´ IH]. {
    -      done.
    -    } {
    -      move ⇒ s H_s_ok /=.
    -      apply IH.
    -      by apply add_ok.
    -    }
    -  Qed.
    - +  Lemma add_list_ok : ∀ l s, Ok s → Ok (add_list l s).
    +
    -  Lemma add_list_spec : ∀ x l s, Ok s →
    -     (In x (add_list l s) ↔ (SetoidList.InA Enc.E.eq x l) ∨ In x s).
    -  Proof.
    -    move ⇒ x.
    -    induction l as [| y l´ IH]. {
    -      intros s H.
    -      rewrite /= InA_nil.
    -      tauto.
    -    } {
    -      move ⇒ s H_ok /=.
    -      rewrite IH add_spec InA_cons.
    -      tauto.
    -    }
    -  Qed.
    - +  Lemma add_list_spec : ∀ x l s, Ok s →
    +     (In x (add_list l s) ↔ (SetoidList.InA Enc.E.eq x l) ∨ In x s).
    +
    -

    union specification

    +

    union specification


      Lemma union_aux_flatten_alt_def : ∀ (s1 s2 : t) acc,
    -    union_aux s1 s2 acc =
    -    match (s1, s2) with
    -    | (nil, _) ⇒ List.rev_append acc s2
    -    | (_, nil) ⇒ List.rev_append acc s1
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
    +    union_aux s1 s2 acc =
    +    match (s1, s2) with
    +    | (nil, _) ⇒ List.rev_append acc s2
    +    | (_, nil) ⇒ List.rev_append acc s1
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
              | ICR_before_touch ⇒
                  union_aux l1 (
    -                insert_interval_begin y1 ((c1+c2)%N) l2) acc
    -          | ICR_after ⇒ union_aux s1 l2 ((y2, c2)::acc)
    +                insert_interval_begin y1 ((c1+c2)%N) l2) acc
    +          | ICR_after ⇒ union_aux s1 l2 ((y2, c2)::acc)
              | ICR_after_touch ⇒ union_aux l1 (
    -              insert_interval_begin y2 ((c1+c2)%N) l2) acc
    +              insert_interval_begin y2 ((c1+c2)%N) l2) acc
              | ICR_overlap_before ⇒
                  union_aux l1 (
                    insert_interval_begin y1
    @@ -2695,48 +1547,26 @@

    Library MSetsExtra.MSetIntervals

              | ICR_subsume_2 ⇒ union_aux s1 l2 acc
            end
        end.
    -  Proof.
    -    intros s1 s2 acc.
    -    case s1, s2 ⇒ //.
    -  Qed.

      Lemma union_aux_alt_def : ∀ (s1 s2 : t) acc,
    -    union_aux s1 s2 acc =
    -    List.rev_append acc (union s1 s2).
    -  Proof.
    -    rewrite /union.
    -    intros s1 s2 acc.
    -    move : acc s2.
    -    induction s1 as [| [y1 c1] l1 IH1]. {
    -      intros acc s2.
    -      rewrite !union_aux_flatten_alt_def.
    -      rewrite !rev_append_rev //.
    -    }
    -    intros acc s2; move : acc.
    -    induction s2 as [| [y2 c2] l2 IH2]; first by simpl.
    -    move ⇒ acc.
    -    rewrite !(union_aux_flatten_alt_def ((y1, c1)::l1) ((y2, c2)::l2)).
    -    case (interval_compare (y1, c1) (y2, c2));
    -      rewrite ?(IH1 ((y1, c1) :: acc)) ?(IH1 ((y1, c1) :: nil))
    -              ?(IH2 ((y2, c2) :: acc)) ?(IH2 ((y2, c2) :: nil))
    -              ?(IH1 acc) //.
    -  Qed.
    +    union_aux s1 s2 acc =
    +    List.rev_append acc (union s1 s2).

      Lemma union_alt_def : ∀ (s1 s2 : t),
    -    union s1 s2 =
    -    match (s1, s2) with
    -    | (nil, _) ⇒ s2
    -    | (_, nil) ⇒ s1
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ (y1, c1) :: (union l1 s2)
    +    union s1 s2 =
    +    match (s1, s2) with
    +    | (nil, _) ⇒ s2
    +    | (_, nil) ⇒ s1
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ (y1, c1) :: (union l1 s2)
              | ICR_before_touch ⇒
    -              union l1 (insert_interval_begin y1 ((c1+c2)%N) l2)
    -          | ICR_after ⇒ (y2, c2) :: union s1 l2
    +              union l1 (insert_interval_begin y1 ((c1+c2)%N) l2)
    +          | ICR_after ⇒ (y2, c2) :: union s1 l2
              | ICR_after_touch ⇒ union l1
    -              (insert_interval_begin y2 ((c1+c2)%N) l2)
    +              (insert_interval_begin y2 ((c1+c2)%N) l2)
              | ICR_overlap_before ⇒
                  union l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2)
              | ICR_overlap_after ⇒
    @@ -2746,3587 +1576,704 @@

    Library MSetsExtra.MSetIntervals

              | ICR_subsume_2 ⇒ union s1 l2
            end
         end.
    -  Proof.
    -    intros s1 s2.
    -    rewrite /union union_aux_flatten_alt_def.
    -    case s1 as [| [y1 c1] l1] ⇒ //.
    -    case s2 as [| [y2 c2] l2] ⇒ //.
    -    case (interval_compare (y1, c1) (y2, c2));
    -      rewrite union_aux_alt_def //.
    -  Qed.

      Lemma union_InZ :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    âˆ€ y, (InZ y (union s1 s2) ↔ InZ y s1 ∨ InZ y s2).
    -  Proof.
    -    intro s1.
    -    induction s1 as [| [y1 c1] l1 IH1]. {
    -      intros s2 _ _ y.
    -      rewrite union_alt_def /InZ /=.
    -      tauto.
    -    } {
    -      induction s2 as [| [y2 c2] l2 IH2]. {
    -        intros _ _ y.
    -        rewrite union_alt_def /InZ /=.
    -        tauto.
    -      } {
    -        move ⇒ H_inv_s1 H_inv_s2.
    -        move : (H_inv_s1) (H_inv_s2).
    -        rewrite !interval_list_invariant_cons.
    -        move ⇒ [H_gr_l1] [H_c1_neq_0] H_inv_l1.
    -        move ⇒ [H_gr_l2] [H_c2_neq_0] H_inv_l2.
    -        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {IH2} IH2.
    -        have : ∀ s2 : t,
    -          interval_list_invariant s2 = true →
    -          âˆ€ y : Z, InZ y (union l1 s2) ↔ InZ y l1 ∨ InZ y s2. {
    -          intros. by apply IH1.
    -        }
    -        move ⇒ {IH1} IH1 y.
    -        rewrite union_alt_def.
    -        move : (interval_compare_elim y1 c1 y2 c2).
    -        case (interval_compare (y1, c1) (y2, c2)). {
    -          rewrite !InZ_cons IH1 // InZ_cons.
    -          tauto.
    -        } {
    -          move ⇒ H_y2_eq.
    -          replace (c1 + c2)%N with (merge_interval_size y1 c1 y2 c2);
    -            last first. {
    -            rewrite -H_y2_eq merge_interval_size_add //.
    -          }
    -          set c´´ := merge_interval_size y1 c1 y2 c2.
    -          have [H_inv_insert H_InZ_insert] :
    -               interval_list_invariant (insert_interval_begin y1 c´´ l2) = true ∧
    -                (InZ y (insert_interval_begin y1 c´´ l2) ↔
    -                List.In y (elementsZ_single y1 c´´) ∨ InZ y l2). {
    -            apply insert_interval_begin_spec ⇒ //. {
    -              eapply interval_list_elements_greater_impl; eauto.
    -              rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
    -              apply Z_le_add_r.
    -            } {
    -              by apply merge_interval_size_neq_0.
    -            }
    -          }
    -  
    -          rewrite IH1 ⇒ //.
    -          rewrite H_InZ_insert !InZ_cons /c´´.
    -          rewrite -H_y2_eq In_merge_interval. {
    -            tauto.
    -          } {
    -            apply Z_le_add_r.
    -          } {
    -            by apply Z.le_refl.
    -          }
    -        } {
    -          move ⇒ [H_y1_lt] [H_y2_lt] H_y1_c1_lt.
    -          set c´´ := merge_interval_size y1 c1 y2 c2.
    -          have [H_inv_insert H_InZ_insert] :
    -               interval_list_invariant (insert_interval_begin y1 c´´ l2) = true ∧
    -                (InZ y (insert_interval_begin y1 c´´ l2) ↔
    -                List.In y (elementsZ_single y1 c´´) ∨ InZ y l2). {
    -            apply insert_interval_begin_spec ⇒ //. {
    -              eapply interval_list_elements_greater_impl; eauto.
    -              apply Z_lt_le_add_r ⇒ //.
    -            } {
    -              by apply merge_interval_size_neq_0.
    -            }
    -          }
    -          rewrite IH1 ⇒ //.
    -          rewrite H_InZ_insert !InZ_cons /c´´.
    -          rewrite In_merge_interval. {
    -            tauto.
    -          } {
    -            by apply Z.lt_le_incl.
    -          } {
    -            by apply Z.lt_le_incl.
    -          }
    -        } {
    -          move ⇒ [H_y2_lt] [H_y1_lt] H_y2_c_lt.
    -          set c´´ := merge_interval_size y2 c2 y1 c1.
    -          have [H_inv_insert H_InZ_insert] :
    -               interval_list_invariant (insert_interval_begin y2 c´´ l2) = true ∧
    -                (InZ y (insert_interval_begin y2 c´´ l2) ↔
    -                List.In y (elementsZ_single y2 c´´) ∨ InZ y l2). {
    -            apply insert_interval_begin_spec ⇒ //. {
    -              eapply interval_list_elements_greater_impl; eauto.
    -              apply Z_le_add_r.
    -            } {
    -              by apply merge_interval_size_neq_0.
    -            }
    -          }
    -  
    -          rewrite IH1 ⇒ //.
    -          rewrite H_InZ_insert !InZ_cons /c´´.
    -          rewrite In_merge_interval. {
    -            tauto.
    -          } {
    -            by apply Z.lt_le_incl.
    -          } {
    -            by apply Z.lt_le_incl.
    -          }
    -        } {
    -          move ⇒ [? ?]; subst.
    -          rewrite IH1 ⇒ //.
    -          rewrite !InZ_cons.
    -          tauto.
    -        } {
    -          move ⇒ [H_y2_le] [H_y1_c1_le] _.
    -          rewrite IH1 ⇒ //.
    -          rewrite !InZ_cons.
    -          suff : (List.In y (elementsZ_single y1 c1) →
    -                  List.In y (elementsZ_single y2 c2)). {
    -            tauto.
    -          }
    -          rewrite !In_elementsZ_single.
    -          move ⇒ [H_y1_le H_y_lt].
    -          split; omega.
    -        } {
    -          move ⇒ [H_y1_le] [H_y2_c2_le] _.
    -          rewrite IH2.
    -          rewrite !InZ_cons.
    -          suff : (List.In y (elementsZ_single y2 c2) →
    -                  List.In y (elementsZ_single y1 c1)). {
    -            tauto.
    -          }
    -          rewrite !In_elementsZ_single.
    -          move ⇒ [H_y2_le H_y_lt].
    -          split; omega.
    -        } {
    -          rewrite !InZ_cons IH2 !InZ_cons.
    -          tauto.
    -        } {
    -          move ⇒ H_y1_eq.
    -          replace (c1 + c2)%N with (merge_interval_size y2 c2 y1 c1);
    -            last first. {
    -            rewrite -H_y1_eq merge_interval_size_add N.add_comm //.
    -          }
    -          set c´´ := merge_interval_size y2 c2 y1 c1.
    -          have [H_inv_insert H_InZ_insert] :
    -               interval_list_invariant (insert_interval_begin y2 c´´ l2) = true ∧
    -                (InZ y (insert_interval_begin y2 c´´ l2) ↔
    -                List.In y (elementsZ_single y2 c´´) ∨ InZ y l2). {
    -            apply insert_interval_begin_spec ⇒ //. {
    -              eapply interval_list_elements_greater_impl; eauto.
    -              apply Z_le_add_r.
    -            } {
    -              by apply merge_interval_size_neq_0.
    -            }
    -          }
    -  
    -          rewrite IH1 ⇒ //.
    -          rewrite H_InZ_insert !InZ_cons /c´´.
    -          rewrite -H_y1_eq In_merge_interval. {
    -            tauto.
    -          } {
    -            apply Z_le_add_r.
    -          } {
    -            by apply Z.le_refl.
    -          }
    -        }
    -      }
    -    }
    -  Qed.
    - +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    âˆ€ y, (InZ y (union s1 s2) ↔ InZ y s1 ∨ InZ y s2).
    +
      Lemma union_invariant :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    interval_list_invariant (union s1 s2) = true.
    -  Proof.
    -    intro s1.
    -    induction s1 as [| [y1 c1] l1 IH1]. {
    -      intros s2 _ H_inv_s2.
    -      rewrite union_alt_def /InZ //.
    -    } {
    -      induction s2 as [| [y2 c2] l2 IH2]. {
    -        intros H_inv_s1 _.
    -        rewrite union_alt_def /InZ //.
    -      } {
    -        move ⇒ H_inv_s1 H_inv_s2.
    -        move : (H_inv_s1) (H_inv_s2).
    -        rewrite !interval_list_invariant_cons.
    -        move ⇒ [H_gr_l1] [H_c1_neq_0] H_inv_l1.
    -        move ⇒ [H_gr_l2] [H_c2_neq_0] H_inv_l2.
    -        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {IH2} IH2.
    -        have : ∀ s2 : t,
    -          interval_list_invariant s2 = true →
    -          interval_list_invariant (union l1 s2) = true. {
    -          intros. by apply IH1.
    -        }
    -        move ⇒ {IH1} IH1.
    - -
    -        rewrite union_alt_def.
    -        move : (interval_compare_elim y1 c1 y2 c2).
    -        case (interval_compare (y1, c1) (y2, c2)). {
    -          move ⇒ H_lt_y2.
    -          have H_inv´ : interval_list_invariant (union l1 ((y2, c2) :: l2)) = true. {
    -            by apply IH1.
    -          }
    -
    -          rewrite interval_list_invariant_cons.
    -          repeat split ⇒ //.
    -          apply interval_list_elements_greater_intro ⇒ // x.
    -          rewrite union_InZ ⇒ //.
    -          move ⇒ []. {
    -            apply interval_list_elements_greater_alt2_def ⇒ //.
    -          } {
    -            apply interval_list_elements_greater_alt2_def ⇒ //.
    -            rewrite interval_list_elements_greater_cons //.
    -          }
    -        } {
    -          move ⇒ H_y2_eq.
    -          apply IH1.
    -          apply insert_interval_begin_spec ⇒ //. {
    -            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
    -            rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
    -            apply Z_le_add_r.
    -          } {
    -            rewrite N.eq_add_0.
    -            tauto.
    -          }
    -        } {
    -          move ⇒ [H_y1_lt] _.
    -          apply IH1.
    -          apply insert_interval_begin_spec ⇒ //. {
    -            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
    -            apply Z_lt_le_add_r ⇒ //.
    -          } {
    -            apply merge_interval_size_neq_0 ⇒ //.
    -          }
    -        } {
    -          move ⇒ [H_y2_lt] _.
    -          apply IH1.
    -          apply insert_interval_begin_spec ⇒ //. {
    -            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
    -            apply Z_le_add_r ⇒ //.
    -          } {
    -            apply merge_interval_size_neq_0 ⇒ //.
    -          }
    -        } {
    -          move ⇒ [? ?]; subst.
    -          apply IH1 ⇒ //.
    -        } {
    -          move ⇒ _.
    -          apply IH1 ⇒ //.
    -        } {
    -          move ⇒ _.
    -          apply IH2 ⇒ //.
    -        } {
    -          move ⇒ H_lt_y1.
    -          rewrite interval_list_invariant_cons ⇒ //.
    -          repeat split ⇒ //.
    -          apply interval_list_elements_greater_intro ⇒ // x.
    -          rewrite union_InZ ⇒ //.
    -          move ⇒ []. {
    -            apply interval_list_elements_greater_alt2_def ⇒ //.
    -            rewrite interval_list_elements_greater_cons //.
    -          } {
    -            apply interval_list_elements_greater_alt2_def ⇒ //.
    -          }
    -        } {
    -          move ⇒ H_y1_eq.
    -          apply IH1 ⇒ //.
    -          apply insert_interval_begin_spec ⇒ //. {
    -            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
    -            apply Z_le_add_r.
    -          } {
    -            rewrite N.eq_add_0.
    -            tauto.
    -          }
    -        }
    -      }
    -    }
    -  Qed.
    - +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    interval_list_invariant (union s1 s2) = true.
    +

      Global Instance union_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (union s1 s2).
    -  Proof.
    -    move ⇒ H_ok_s1 H_ok_s2.
    -    move : (H_ok_s1) (H_ok_s2).
    -    rewrite /Ok /IsOk /is_encoded_elems_list /add.
    -    move ⇒ [H_inv_s1] H_pre1.
    -    move ⇒ [H_inv_s2] H_pre2.
    -    split. {
    -      apply union_invariant ⇒ //.
    -    } {
    -      intros y.
    -      move : (union_InZ s1 s2 H_inv_s1 H_inv_s2).
    -      rewrite /InZ ⇒ →.
    -      move ⇒ []. {
    -        apply H_pre1.
    -      } {
    -        apply H_pre2.
    -      }
    -    }
    -  Qed.
    - +
      Lemma union_spec :
    -   âˆ€ (s s´ : t) (x : elt) (Hs : Ok s) (Hs´ : Ok s´),
    -   In x (union s s´) ↔ In x s ∨ In x s´.
    -  Proof.
    -    intros s s´ x H_ok H_ok´.
    -    rewrite !In_InZ.
    -    rewrite union_InZ ⇒ //. {
    -      apply H_ok.
    -    } {
    -      apply H_ok´.
    -    }
    -  Qed.
    - +   âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    +   In x (union s s') ↔ In x s ∨ In x s'.
    +
    -

    inter specification

    +

    inter specification


      Lemma inter_aux_alt_def :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      inter_aux y2 c2 acc s = match inter_aux y2 c2 nil s with
    -                               (acc´, s´) ⇒ (acc´ ++ acc, s´)
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      inter_aux y2 c2 acc s = match inter_aux y2 c2 nil s with
    +                               (acc', s') ⇒ (acc' ++ acc, s')
                                 end.
    -  Proof.
    -    intros y2 c2.
    -    induction s as [| [y1 c1] s´ IH] ⇒ acc. {
    -      rewrite /inter_aux app_nil_l //.
    -    } {
    -      simpl.
    -      case_eq (inter_aux y2 c2 nil s´) ⇒ acc´´ s´´ H_eq.
    -      case (interval_compare (y1, c1) (y2, c2));
    -        rewrite ?(IH acc)
    -                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: acc))
    -                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: nil))
    -                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: acc))
    -                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: nil))
    -                ?(IH ((y1, c1) :: acc))
    -                ?(IH ((y1, c1) :: nil))
    -                
    -                ?H_eq -?app_assoc -?app_comm_cons //.
    -    }
    -  Qed.
    - +
      Lemma inter_aux_props :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      interval_list_invariant (rev acc) = true →
    -      interval_list_invariant s = true →
    -      (∀ x1 x2, InZ x1 acc → InZ x2 s →
    -                     List.In x2 (elementsZ_single y2 c2) →
    -                     Z.succ x1 < x2) →
    -      (c2 ≠ 0%N) →
    -      match (inter_aux y2 c2 acc s) with (acc´, s´) ⇒
    -        (∀ y, (InZ y acc´ ↔
    -                   (InZ y acc ∨ (InZ y s ∧ (List.In y (elementsZ_single y2 c2)))))) ∧
    -        (∀ y, InZ y s´ → InZ y s) ∧
    -        (∀ y, InZ y s → y2 + Z.of_N c2 < y → InZ y s´) ∧
    -        interval_list_invariant (rev acc´) = true ∧
    -        interval_list_invariant s´ = true
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      interval_list_invariant (rev acc) = true →
    +      interval_list_invariant s = true →
    +      (∀ x1 x2, InZ x1 acc → InZ x2 s →
    +                     List.In x2 (elementsZ_single y2 c2) →
    +                     Z.succ x1 < x2) →
    +      (c2 ≠ 0%N) →
    +      match (inter_aux y2 c2 acc s) with (acc', s') ⇒
    +        (∀ y, (InZ y acc' ↔
    +                   (InZ y acc ∨ (InZ y s ∧ (List.In y (elementsZ_single y2 c2)))))) ∧
    +        (∀ y, InZ y s' → InZ y s) ∧
    +        (∀ y, InZ y s → y2 + Z.of_N c2 < y → InZ y s') ∧
    +        interval_list_invariant (rev acc') = true ∧
    +        interval_list_invariant s' = true
          end.
    -  Proof.
    -    intros y2 c2.
    -    induction s as [| [y1 c1] s1´ IH] ⇒ acc. {
    -      rewrite /inter_aux.
    -      move ⇒ H_inv_acc _ _ _.
    -      split; last split; try done.
    -      move ⇒ y. rewrite InZ_nil.
    -      tauto.
    -    } {
    -      rewrite interval_list_invariant_cons.
    -      move ⇒ H_inv_acc [H_gr_s1´] [H_c1_neq_0] H_inv_s1´.
    -      move ⇒ H_in_acc_lt H_c2_neq_0.
    - -
    -      rewrite inter_aux_alt_def.
    -      case_eq (inter_aux y2 c2 nil ((y1, c1) :: s1´)).
    -      move ⇒ acc´ s´ H_inter_aux_eq.
    - -
    -      set P1 := ∀ y : Z,
    -        (InZ y acc´ ↔
    -        ((InZ y ((y1, c1) :: s1´) ∧ List.In y (elementsZ_single y2 c2)))).
    -      set P2 := (∀ y,
    -                  (InZ y s´ → InZ y ((y1, c1) :: s1´)) ∧
    -                  (InZ y ((y1, c1) :: s1´) →
    -                     y2 + Z.of_N c2 < y → InZ y s´)).
    -      set P3 := interval_list_invariant (rev acc´) = true.
    -      set P4 := interval_list_invariant s´ = true.
    - -
    -      suff : (P1 ∧ P2 ∧ P3 ∧ P4). {
    -        move ⇒ [H_P1] [H_P2] [H_P3] H_P4.
    -        split; last split; last split; last split. {
    -          move ⇒ y.
    -          move : (H_P1 y).
    -          rewrite !InZ_app InZ_cons !In_elementsZ_single.
    -          move ⇒ <-.
    -          tauto.
    -        } {
    -          move ⇒ y H_y_in.
    -          by apply H_P2.
    -        } {
    -          move ⇒ y H_y_in.
    -          by apply H_P2.
    -        } {
    -          rewrite rev_app_distr.
    -          apply interval_list_invariant_app_intro ⇒ //.
    -          move ⇒ x1 x2.
    -          rewrite !InZ_rev.
    -          move ⇒ H_x1_in /H_P1 [H_x2_in1] H_x2_in2.
    -          apply H_in_acc_lt ⇒ //.
    -        } {
    -          apply H_P4.
    -        }
    -      }
    -
    -      move : (H_gr_s1´).
    -      rewrite interval_list_elements_greater_alt2_def ⇒ // ⇒ H_gr_s1´_alt.
    - -
    -      have : ∀ (acc : list (Z × N)),
    -        interval_list_invariant (rev acc) = true →
    -        (∀ y, InZ y acc ↔ (
    -           y1 ≤ y < y1 + Z.of_N c1 ∧
    -           y2 ≤ y < y2 + Z.of_N c2)) →
    -        (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) →
    -        (inter_aux y2 c2 acc s1´ = (acc´, s´)) →
    -        P1 ∧ P2 ∧ P3 ∧ P4. {
    -        
    -        intros acc0 H_inv_acc0 H_in_acc0 H_y2c_lt H_inter_aux_eq0.
    -        have H_in_acc0_lt : (∀ x1 x2 : Z,
    -          InZ x1 acc0 →
    -          InZ x2 s1´ →
    -          List.In x2 (elementsZ_single y2 c2) →
    -          Z.succ x1 < x2). {
    -
    -          intros x1 x2 H_x1_in_acc0 H_x2_in_s1´ H_x2_in_yc2.
    - -
    -          suff H_yc1_lt_x2 : Z.succ x1 ≤ y1 + Z.of_N c1. {
    -            apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -            by apply H_gr_s1´_alt.
    -          }
    -          move : (H_x1_in_acc0).
    -          rewrite H_in_acc0 Z.le_succ_l.
    -          tauto.
    -        }
    -        
    -        move : (IH acc0 H_inv_acc0 H_inv_s1´ H_in_acc0_lt H_c2_neq_0).
    -        rewrite H_inter_aux_eq0.
    -        move ⇒ [H1] [H2] [H3] [H4] H5.
    -        split; last split ⇒ //. {
    -          move ⇒ y.
    -          rewrite (H1 y).
    -          rewrite InZ_cons !In_elementsZ_single
    -                  H_in_acc0.
    -          tauto.
    -        } {
    -          move ⇒ y.
    -          split. {
    -            move ⇒ /H2.
    -            rewrite InZ_cons.
    -            by right.
    -          } {
    -            rewrite InZ_cons In_elementsZ_single.
    -            move ⇒ []. {
    -              move ⇒ [_] H_y_lt H_lt_y.
    -              exfalso.
    -              suff : (y < y) by apply Z.lt_irrefl.
    -              apply Z.lt_le_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -              apply Z.le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    - -
    -              by apply Z.lt_le_incl.
    -            } {
    -              apply H3.
    -            }
    -          }
    -        }
    -      }
    -      move ⇒ {IH} IH.
    - -
    -      clear H_inv_acc H_in_acc_lt acc.
    -      move : (interval_compare_elim y1 c1 y2 c2) H_inter_aux_eq.
    -      unfold inter_aux.
    -      case_eq (interval_compare (y1, c1) (y2, c2)) ⇒ H_comp;
    -         fold inter_aux. {
    -        move ⇒ H_lt_y2.
    -        apply IH. {
    -          done.
    -        } {
    -          move ⇒ x.
    -          rewrite InZ_nil.
    -          split ⇒ //.
    -          omega.
    -        } {
    -          apply Z.le_trans with (m := y2). {
    -            by apply Z.lt_le_incl.
    -          } {
    -            apply Z_le_add_r.
    -          }
    -        }
    -      } {
    -        move ⇒ H_eq_y2.
    -        apply IH. {
    -          done.
    -        } {
    -          move ⇒ x.
    -          rewrite InZ_nil.
    -          split ⇒ //.
    -          omega.
    -        } {
    -          rewrite H_eq_y2.
    -          apply Z_le_add_r.
    -        }
    -      } {
    -        move ⇒ [H_y1_lt_y2] [H_y2_lt_yc1] H_yc1_lt_yc2.
    -        apply IH. {
    -          rewrite interval_list_invariant_sing.
    -          by apply Z_to_N_minus_neq_0.
    -        } {
    -          move ⇒ x.
    -          rewrite InZ_cons InZ_nil In_elementsZ_single Z2N.id; last omega.
    -          replace (y1 + (y2 - y1)) with y2 by omega.
    -          split; omega.
    -        } {
    -          by apply Z.lt_le_incl.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3 /P4.
    -        move ⇒ [H_y2_lt] [H_y1_lt] H_yc1_lt.
    -        move ⇒ [] H_acc´ H_s´.
    -        clear IH P1 P2 P3 P4 H_comp.
    -        subst s´ acc´.
    -        split; last split; last split. {
    -          move ⇒ y.
    -          have H_yc2_intro : y1 + Z.of_N (Z.to_N (y2 + Z.of_N c2 - y1)) =
    -                             y2 + Z.of_N c2. {
    -            rewrite Z2N.id; omega.
    -          }
    -          
    -          rewrite !InZ_cons !In_elementsZ_single InZ_nil H_yc2_intro.
    -          split. {
    -            move ⇒ [] //.
    -            move ⇒ [H_y1_le] H_y_lt.
    -            split; last by omega.
    -            left. omega.
    -          } {
    -            move ⇒ [H_in_s] [H_y2_le] H_y_lt.
    -            left.
    -            split; last assumption.
    -            move : H_in_s ⇒ []. {
    -              tauto.
    -            } {
    -              move ⇒ /H_gr_s1´_alt H_lt_y.
    -              apply Z.le_trans with (m := y1 + Z.of_N c1). {
    -                by apply Z_le_add_r.
    -              } {
    -                by apply Z.lt_le_incl.
    -              }
    -            }
    -          }
    -        } {
    -          move ⇒ y.
    -          split; done.
    -        } {
    -          rewrite interval_list_invariant_sing.
    -          by apply Z_to_N_minus_neq_0.
    -        } {
    -          by rewrite interval_list_invariant_cons.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3 /P4.
    -        move ⇒ [H_y12_eq] H_c12_eq [] H_acc´ H_s´.
    -        clear IH P1 P2 P3 P4 H_comp.
    -        subst.
    -        split; last split; last split. {
    -          move ⇒ y.
    -          rewrite !InZ_cons InZ_nil In_elementsZ_single.
    -          split; last by tauto. {
    -            move ⇒ [] //.
    -            tauto.
    -          }
    -        } {
    -          move ⇒ y.
    -          rewrite InZ_cons In_elementsZ_single.
    -          split; first by right.
    -          move ⇒ [] //.
    -          move ⇒ [_] H_y_lt H_lt_y.
    -          exfalso.
    -          suff : (y2 + Z.of_N c2 < y2 + Z.of_N c2) by
    -              apply Z.lt_irrefl.
    -          apply Z.lt_trans with (m := y) ⇒ //.
    -        } {
    -          rewrite interval_list_invariant_sing //.
    -        } {
    -          assumption.
    -        }
    -      } {
    -        move ⇒ [H_y2_le_y1] [H_yc1_le_yc2] _.
    -        apply IH. {
    -          by rewrite interval_list_invariant_sing.
    -        } {
    -          move ⇒ y.
    -          rewrite InZ_cons InZ_nil In_elementsZ_single.
    -          split. {
    -            move ⇒ [] //.
    -            move ⇒ [H_y1_le] H_y_lt.
    -            split; first done.
    -            split; omega.
    -          } {
    -            move ⇒ [?] _.
    -            by left.
    -          }
    -        } {
    -          assumption.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3 /P4.
    -        move ⇒ [H_y1_le] [H_yc2_le] _.
    -        move ⇒ [] H_acc´ H_s´.
    -        clear IH P1 P2 P3 P4 H_comp.
    -        subst.
    -        split; last split; last split. {
    -          move ⇒ y.
    -          rewrite !InZ_cons !In_elementsZ_single InZ_nil.
    -          split. {
    -            move ⇒ [] //.
    -            move ⇒ [H_y2_le] H_y_lt.
    -            split; last by omega.
    -            left. omega.
    -          } {
    -            move ⇒ [H_in_s] [H_y2_le] H_y_lt.
    -            by left.
    -          }
    -        } {
    -          tauto.
    -        } {
    -          by rewrite interval_list_invariant_sing.
    -        } {
    -          by rewrite interval_list_invariant_cons.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3 /P4.
    -        move ⇒ H_yc2_lt [] H_acc´ H_s´.
    -        clear IH P1 P2 P3 P4 H_comp.
    -        subst.
    -        split; last split; last split. {
    -          move ⇒ y.
    -          rewrite InZ_cons !In_elementsZ_single InZ_nil.
    -          split; first done.
    -          move ⇒ [] []. {
    -            move ⇒ [H_y1_le_y] H_y_lt_yc1.
    -            move ⇒ [H_y2_le_y] H_y_lt_yc2.
    -            omega.
    -          } {
    -            move ⇒ /H_gr_s1´_alt H_lt_y [_] H_y_lt.
    -            suff : (y1 + Z.of_N c1 < y1). {
    -              apply Z.nlt_ge.
    -              apply Z_le_add_r.
    -            }
    -            omega.
    -          }
    -        } {
    -          tauto.
    -        } {
    -          done.
    -        } {
    -          by rewrite interval_list_invariant_cons.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3 /P4.
    -        move ⇒ H_y1_eq [] H_acc´ H_s´.
    -        clear IH P1 P2 P3 P4 H_comp.
    -        subst acc´ s´.
    -        split; last split; last split. {
    -          move ⇒ y.
    -          rewrite InZ_cons !In_elementsZ_single InZ_nil.
    -          split; first done.
    -          move ⇒ [] []. {
    -            move ⇒ [H_y1_le_y] H_y_lt_yc1.
    -            move ⇒ [H_y2_le_y] H_y_lt_yc2.
    -            omega.
    -          } {
    -            move ⇒ /H_gr_s1´_alt H_lt_y [_] H_y_lt.
    -            suff : (y1 + Z.of_N c1 < y1). {
    -              apply Z.nlt_ge.
    -              apply Z_le_add_r.
    -            }
    -            omega.
    -          }
    -        } {
    -          tauto.
    -        } {
    -          done.
    -        } {
    -          by rewrite interval_list_invariant_cons.
    -        }
    -      }
    -    }
    -  Qed.
    - +
      Lemma inter_aux2_props :
       âˆ€ (s2 s1 acc : t),
    -    interval_list_invariant (rev acc) = true →
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    (∀ x1 x2, InZ x1 acc → InZ x2 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    -    ((∀ y, (InZ y (inter_aux2 acc s1 s2) ↔
    -               (InZ y acc) ∨ ((InZ y s1) ∧ InZ y s2))) ∧
    -    (interval_list_invariant (inter_aux2 acc s1 s2) = true)).
    -  Proof.
    -    induction s2 as [| [y2 c2] s2´ IH]. {
    -      move ⇒ s1 acc.
    -      move ⇒ H_inv_acc _ _ _.
    -      unfold inter_aux2.
    -      replace (match s1 with
    -        | nil ⇒ rev´ acc
    -        | _ :: _ ⇒ rev´ acc
    -               end) with (rev´ acc); last by case s1.
    -      rewrite /rev´ rev_append_rev app_nil_r.
    -      split; last done.
    -      move ⇒ y.
    -      rewrite InZ_nil InZ_rev.
    -      tauto.
    -    } {
    -      intros s1 acc H_inv_acc H_inv_s1.
    -      rewrite interval_list_invariant_cons.
    -      move ⇒ [H_gr_s2´] [H_c2_neq_0] H_inv_s2´.
    -      move ⇒ H_acc_s12.
    - -
    -      move : H_gr_s2´.
    -      rewrite interval_list_elements_greater_alt2_def //.
    -      move ⇒ H_gr_s2´.
    - -
    -      rewrite /inter_aux2; fold inter_aux2.
    -      case_eq s1. {
    -        move ⇒ H_s1_eq.
    -        split. {
    -          move ⇒ y.
    -          rewrite /rev´ rev_append_rev app_nil_r InZ_nil
    -                  InZ_rev.
    -          tauto.
    -        } {
    -          rewrite /rev´ rev_append_rev app_nil_r //.
    -        }
    -      } {
    -        move ⇒ [_ _] _ <-.
    -        case_eq (inter_aux y2 c2 acc s1).
    -        move ⇒ acc´ s1´ H_inter_aux_eq.
    - -
    -        have H_acc_s1_yc2 : ∀ x1 x2 : Z,
    -          InZ x1 acc →
    -          InZ x2 s1 →
    -          List.In x2 (elementsZ_single y2 c2) →
    -          Z.succ x1 < x2. {
    -          intros x1 x2 H_x1_in H_x2_in1 H_x2_in2.
    -          apply H_acc_s12 ⇒ //.
    -          rewrite InZ_cons; by left.
    -        }
    -
    -        move : (inter_aux_props y2 c2 s1 acc H_inv_acc H_inv_s1 H_acc_s1_yc2 H_c2_neq_0).
    -        rewrite H_inter_aux_eq.
    -        move ⇒ [H_in_acc´] [H_in_s1´_elim] [H_in_s1´_intro]
    -                [H_inv_acc´] H_inv_s1´.
    - -
    -        have H_in_acc´_s2´ : ∀ x1 x2 : Z,
    -            InZ x1 acc´ → InZ x2 s1´ → InZ x2 s2´ → Z.succ x1 < x2. {
    -          move ⇒ x1 x2 /H_in_acc´.
    -          move ⇒ []. {
    -            move ⇒ H_in_acc H_in_s1´ H_in_s2´.
    -            apply H_acc_s12 ⇒ //. {
    -              by apply H_in_s1´_elim.
    -            } {
    -              rewrite InZ_cons; by right.
    -            }
    -          } {
    -            rewrite In_elementsZ_single.
    -            move ⇒ [H_in_s1] [_] H_x1_lt _.
    -            move ⇒ /H_gr_s2´ H_lt_x2.
    -            apply Z.le_lt_trans with (m := y2 + Z.of_N c2) ⇒ //.
    -            by apply Z.le_succ_l.
    -          }
    -        }
    -       
    -        move : (IH s1´ acc´ H_inv_acc´ H_inv_s1´ H_inv_s2´ H_in_acc´_s2´).
    -        move ⇒ [H_inZ_res] H_inv_res.
    -        split; last assumption.
    -        move ⇒ y.
    -        rewrite H_inZ_res H_in_acc´ InZ_cons
    -                In_elementsZ_single.
    -        split. {
    -          move ⇒ []; first by tauto.
    -          move ⇒ [H_y_in_s1´ H_y_in_s2´].
    -          right.
    -          split; last by right.
    -          by apply H_in_s1´_elim.
    -        } {
    -          move ⇒ []. {
    -            move ⇒ H_y_in_acc.
    -            by left; left.
    -          } {
    -            move ⇒ [H_y_in_s1].
    -            move ⇒ []. {
    -              move ⇒ H_in_yc2.
    -              by left; right.
    -            } {
    -              right.
    -              split; last assumption.
    -              apply H_in_s1´_intro ⇒ //.
    -              by apply H_gr_s2´.
    -            }
    -          }
    -        }
    -      }
    -    }
    -  Qed.
    - +    interval_list_invariant (rev acc) = true →
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    (∀ x1 x2, InZ x1 acc → InZ x2 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    +    ((∀ y, (InZ y (inter_aux2 acc s1 s2) ↔
    +               (InZ y acc) ∨ ((InZ y s1) ∧ InZ y s2))) ∧
    +    (interval_list_invariant (inter_aux2 acc s1 s2) = true)).
    +
      Lemma inter_InZ :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    âˆ€ y, (InZ y (inter s1 s2) ↔ InZ y s1 ∧ InZ y s2).
    -  Proof.
    -    intros s1 s2 H_inv_s1 H_inv_s2 y.
    -    rewrite /inter.
    - -
    -    move : (inter_aux2_props s2 s1 nil).
    -    move ⇒ [] //.
    -    move ⇒ H_in_inter _.
    -    rewrite H_in_inter InZ_nil.
    -    tauto.
    -  Qed.
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    âˆ€ y, (InZ y (inter s1 s2) ↔ InZ y s1 ∧ InZ y s2).

      Lemma inter_invariant :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    interval_list_invariant (inter s1 s2) = true.
    -  Proof.
    -    intros s1 s2 H_inv_s1 H_inv_s2.
    -    rewrite /inter.
    - -
    -    move : (inter_aux2_props s2 s1 nil).
    -    move ⇒ [] //.
    -  Qed.
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    interval_list_invariant (inter s1 s2) = true.


      Global Instance inter_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (inter s1 s2).
    -  Proof.
    -    move ⇒ H_ok_s1 H_ok_s2.
    -    move : (H_ok_s1) (H_ok_s2).
    -    rewrite /Ok /IsOk /is_encoded_elems_list /add.
    -    move ⇒ [H_inv_s1] H_pre1.
    -    move ⇒ [H_inv_s2] H_pre2.
    -    split. {
    -      apply inter_invariant ⇒ //.
    -    } {
    -      intros y.
    -      move : (inter_InZ s1 s2 H_inv_s1 H_inv_s2).
    -      rewrite /InZ ⇒ →.
    -      move ⇒ [].
    -      move ⇒ /H_pre1 //.
    -    }
    -  Qed.
    - +
      Lemma inter_spec :
    -   âˆ€ (s s´ : t) (x : elt) (Hs : Ok s) (Hs´ : Ok s´),
    -   In x (inter s s´) ↔ In x s ∧ In x s´.
    -  Proof.
    -    intros s s´ x H_ok H_ok´.
    -    rewrite !In_InZ.
    -    rewrite inter_InZ ⇒ //. {
    -      apply H_ok.
    -    } {
    -      apply H_ok´.
    -    }
    -  Qed.
    - +   âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    +   In x (inter s s') ↔ In x s ∧ In x s'.
    +

    -

    diff specification

    +

    diff specification


      Lemma diff_aux_alt_def :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      diff_aux y2 c2 acc s = match diff_aux y2 c2 nil s with
    -                               (acc´, s´) ⇒ (acc´ ++ acc, s´)
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      diff_aux y2 c2 acc s = match diff_aux y2 c2 nil s with
    +                               (acc', s') ⇒ (acc' ++ acc, s')
                                 end.
    -  Proof.
    -    intros y2 c2.
    -    induction s as [| [y1 c1] acc´ IH] ⇒ acc. {
    -      rewrite /diff_aux app_nil_l //.
    -    } {
    -      simpl.
    -      case_eq (diff_aux y2 c2 nil acc´) ⇒ acc´´ s´´ H_eq.
    -      case (interval_compare (y1, c1) (y2, c2));
    -        rewrite ?(IH ((y1, c1)::acc)) ?(IH ((y1, c1)::nil))
    -                ?(IH acc) ?(IH ((y1, Z.to_N (y2 - y1)) :: acc))
    -                ?(IH ((y1, Z.to_N (y2 - y1)) :: nil)) ?H_eq;
    -        rewrite ?insert_intervalZ_guarded_app -?app_assoc -?app_comm_cons //.
    -    }
    -  Qed.
    - +

      Lemma diff_aux_props :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      interval_list_invariant (List.rev acc) = true →
    -      interval_list_invariant s = true →
    -      (∀ x1 x2, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    -      (∀ x, InZ x acc → x < y2) →
    -      (c2 ≠ 0%N) →
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      interval_list_invariant (List.rev acc) = true →
    +      interval_list_invariant s = true →
    +      (∀ x1 x2, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    +      (∀ x, InZ x acc → x < y2) →
    +      (c2 ≠ 0%N) →
          match (diff_aux y2 c2 acc s) with
    -        (acc´, s´) ⇒ (∀ y, InZ y (List.rev_append acc´ s´) ↔
    -                                 InZ y (List.rev_append acc s) ∧ ~(List.In y (elementsZ_single y2 c2))) ∧
    -                      (interval_list_invariant (List.rev_append acc´ s´) = true) ∧
    -                      (∀ x, InZ x acc´ → x < y2 + Z.of_N c2)
    +        (acc', s') ⇒ (∀ y, InZ y (List.rev_append acc' s') ↔
    +                                 InZ y (List.rev_append acc s) ∧ ~(List.In y (elementsZ_single y2 c2))) ∧
    +                      (interval_list_invariant (List.rev_append acc' s') = true) ∧
    +                      (∀ x, InZ x acc' → x < y2 + Z.of_N c2)
          end.
    -  Proof.
    -    intros y2 c2.
    -    induction s as [| [y1 c1] s1´ IH] ⇒ acc. {
    -      rewrite /diff_aux -rev_alt.
    -      move ⇒ H_inv_acc _ _ H_in_acc H_c2_neq.
    -      split; last split. {
    -        move ⇒ y; split; last by move ⇒ [] //.
    -        rewrite InZ_rev.
    -        move ⇒ H_in. split ⇒ //.
    -        move ⇒ /In_elementsZ_single ⇒ [] [] /Z.nlt_ge H_neq.
    -        contradict H_neq.
    -        by apply H_in_acc.
    -      } {
    -        assumption.
    -      } {
    -        intros x H_in_acc´.
    -        apply Z.lt_le_trans with (m := y2). {
    -          by apply H_in_acc.
    -        } {
    -          by apply Z_le_add_r.
    -        }
    -      }
    -    } {
    -      rewrite interval_list_invariant_cons.
    -      move ⇒ H_inv_acc [H_gr_s1´] [H_c1_neq_0] H_inv_s1´.
    -      move ⇒ H_in_s1 H_in_acc H_c2_neq_0.
    - -
    -      rewrite diff_aux_alt_def.
    -      case_eq (diff_aux y2 c2 nil ((y1, c1) :: s1´)).
    -      move ⇒ acc´ s´ H_diff_aux_eq.
    - -
    -      set P1 := ∀ y : Z,
    -        (InZ y acc´ ∨ InZ y s´) ↔
    -        InZ y ((y1, c1) :: s1´) ∧ ¬ List.In y (elementsZ_single y2 c2).
    -      set P2 := interval_list_invariant (rev acc´ ++ s´) = true.
    -      set P3 := ∀ x : Z, InZ x acc´ → (x < y2 + Z.of_N c2).
    - -
    -      suff : (P1 ∧ P2 ∧ P3). {
    -        move ⇒ [H_P1] [H_P2] H_P3.
    -        split; last split. {
    -          move ⇒ y.
    -          move : (H_P1 y).
    -          rewrite !rev_append_rev rev_app_distr !InZ_app
    -                  !InZ_rev In_elementsZ_single.
    -          suff : (InZ y acc → ¬ y2 ≤ y < y2 + Z.of_N c2). {
    -            tauto.
    -          }
    -          move ⇒ /H_in_acc H_y_lt [H_y_ge] _.
    -          contradict H_y_ge.
    -          by apply Zlt_not_le.
    -        } {
    -          rewrite rev_append_rev rev_app_distr -app_assoc.
    -          apply interval_list_invariant_app_intro ⇒ //.
    -          move ⇒ x1 x2.
    -          rewrite InZ_app !InZ_rev.
    -          move ⇒ H_in_acc´ H_x2_in_s´.
    -          suff : (InZ x2 ((y1, c1)::s1´)). {
    -            by apply H_in_s1.
    -          }
    -          move : (H_P1 x2).
    -          tauto.
    -        } {
    -          move ⇒ x.
    -          rewrite InZ_app.
    -          move ⇒ []. {
    -            apply H_P3.
    -          } {
    -            move ⇒ /H_in_acc H_x_lt.
    -            eapply Z.lt_trans; eauto.
    -            by apply Z_lt_add_r.
    -          }
    -        }
    -      }
    -
    -      move : (H_gr_s1´).
    -      rewrite interval_list_elements_greater_alt2_def ⇒ // ⇒ H_gr_s1´_alt.
    - -
    -      have : ∀ (acc : list (Z × N)),
    -        interval_list_invariant (rev acc) = true →
    -        (∀ x : Z,
    -            InZ x acc ↔
    -            ((y1 ≤ x < y1 + Z.of_N c1) ∧ (x < y2))) →
    -        (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) →
    -        (diff_aux y2 c2 acc s1´ = (acc´, s´)) →
    -        P1 ∧ P2 ∧ P3. {
    -        
    -        intros acc0 H_inv_acc0 H_in_acc0 H_c1_before H_diff_aux_eq0.
    -        have H_in_s1´ : (∀ x1 x2 : Z,
    -                            InZ x1 acc0 → InZ x2 s1´ → Z.succ x1 < x2). {
    -          intros x1 x2 H_x1_in_acc0.
    -          move ⇒ /H_gr_s1´_alt.
    - -
    -          eapply Z.le_lt_trans.
    -          move : H_x1_in_acc0.
    -          rewrite Z.le_succ_l H_in_acc0.
    -          tauto.
    -        }
    -        have H_in_acc0´ : (∀ x : Z, InZ x acc0 → x < y2). {
    -          move ⇒ x.
    -          rewrite H_in_acc0.
    -          move ⇒ [_] //.
    -        }
    -        move : (IH acc0 H_inv_acc0 H_inv_s1´ H_in_s1´ H_in_acc0´ H_c2_neq_0).
    -        rewrite H_diff_aux_eq0 !rev_append_rev.
    -        move ⇒ [H1] [H2] H3.
    -        split; last split ⇒ //. {
    -          move ⇒ y.
    -          move : (H1 y).
    -          rewrite !InZ_app !InZ_rev In_elementsZ_single.
    -          move ⇒ →.
    -          rewrite InZ_cons In_elementsZ_single.
    -          split. {
    -            rewrite H_in_acc0 -(Z.nle_gt y2 y).
    -            tauto.
    -          } {
    -            rewrite H_in_acc0 -(Z.nle_gt y2 y).
    -            move ⇒ [] H_in H_nin_i2.
    -            split; last by assumption.
    -            move : H_in ⇒ [] H_in; last by right.
    -            left.
    -            omega.
    -          }
    -        }
    -      }
    -      move ⇒ {IH} IH.
    - -
    -      clear H_inv_acc H_in_s1 H_in_acc acc.
    -      move : (interval_compare_elim y1 c1 y2 c2) H_diff_aux_eq.
    -      unfold diff_aux.
    -      case_eq (interval_compare (y1, c1) (y2, c2)) ⇒ H_comp;
    -                                                        fold diff_aux. {
    -        move ⇒ H_lt_y2.
    -        apply IH. {
    -          by rewrite interval_list_invariant_sing.
    -        } {
    -          move ⇒ x.
    -          rewrite InZ_cons In_elementsZ_single.
    -          split; last by tauto.
    -          move ⇒ []; last done.
    -          move ⇒ [H_y1_le H_x_lt].
    -          split; first done.
    -          eapply Z.lt_trans; eauto.
    -        } {
    -          apply Z.le_trans with (m := y2).
    -            - by apply Z.lt_le_incl.
    -            - by apply Z_le_add_r.
    -        }
    -      } {
    -        move ⇒ H_eq_y2.
    -        apply IH. {
    -          by rewrite interval_list_invariant_sing.
    -        } {
    -          move ⇒ x.
    -          rewrite InZ_cons In_elementsZ_single -H_eq_y2.
    -          split; last by tauto.
    -          move ⇒ []; last done.
    -          move ⇒ []. done.
    -        } {
    -          rewrite H_eq_y2.
    -          by apply Z_le_add_r.
    -        }
    -      } {
    -        move ⇒ [H_y1_lt_y2] [H_y2_lt_yc1] H_yc1_lt_yc2.
    -        apply IH. {
    -          rewrite interval_list_invariant_sing.
    -          by apply Z_to_N_minus_neq_0.
    -        } {
    -          move ⇒ x.
    -          rewrite InZ_cons In_elementsZ_single Z2N.id; last omega.
    -          replace (y1 + (y2 - y1)) with y2 by omega.
    -          split; last tauto.
    -          move ⇒ [] //.
    -          move ⇒ [H_y1_le] H_x_lt.
    -          repeat split ⇒ //.
    -          apply Z.lt_trans with (m := y2) ⇒ //.
    -        } {
    -          by apply Z.lt_le_incl.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3.
    -        move ⇒ [H_y2_lt] [H_y1_lt] H_yc1_lt [H_acc´] H_s´.
    -        clear IH P1 P2 P3 H_comp.
    -        subst.
    -        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
    -          rewrite Z2N.id; omega.
    -        }
    -        have H_nin_yc2 : ∀ y,
    -            InZ y s1´ → ¬ y2 ≤ y < y2 + Z.of_N c2. {
    -          move ⇒ y /H_gr_s1´_alt H_lt_y.
    -          move ⇒ [H_y2_le_y].
    -          apply Z.le_ngt, Z.lt_le_incl.
    -          by apply Z.lt_trans with (m := y1 + Z.of_N c1).
    -        }
    -        split; last split. {
    -          move ⇒ y.
    -          rewrite !InZ_cons !In_elementsZ_single H_yc1_intro.
    -          split. {
    -            move ⇒ [] //.
    -            move ⇒ []. {
    -              move ⇒ [H_le_y] H_y_lt.
    -              split. {
    -                left; omega.
    -              } {
    -                move ⇒ [_].
    -                by apply Z.nlt_ge.
    -              }
    -            } {
    -              move : (H_nin_yc2 y). tauto.
    -            }
    -          } {
    -            move ⇒ [] []; last by right; right.
    -            move ⇒ [H_y_ge] H_y_lt_yc1 H_nin_yc2´.
    -            right; left. omega.
    -          }
    -        } {
    -          rewrite interval_list_invariant_cons H_yc1_intro.
    -          split ⇒ //.
    -          split ⇒ //.
    -          by apply Z_to_N_minus_neq_0.
    -        } {
    -          move ⇒ [] //.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3.
    -        move ⇒ [H_y12_eq] H_c12_eq [] H_acc´ H_s´.
    -        clear IH P1 P2 P3 H_comp.
    -        subst.
    -        split; last split. {
    -          move ⇒ y.
    -          rewrite InZ_cons In_elementsZ_single.
    -          split; last by tauto. {
    -            move ⇒ [] //.
    -            move ⇒ H_y_in.
    -            split; first by right.
    -            move ⇒ [] _.
    -            by apply Z.nlt_ge, Z.lt_le_incl, H_gr_s1´_alt.
    -          }
    -        } {
    -          apply H_inv_s1´.
    -        } {
    -          move ⇒ x [] //.
    -        }
    -      } {
    -        move ⇒ [H_y2_le_y1] [H_yc1_le_yc2] _.
    -        apply IH. {
    -          done.
    -        } {
    -          move ⇒ x.
    -          split; first done.
    -          omega.
    -        } {
    -          assumption.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3.
    -        move ⇒ [H_y1_le] [H_yc2_le_yc1] _ [] H_acc´ H_s´.
    -        clear IH P1 P2 P3 H_comp.
    -        subst.
    -        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
    -          rewrite Z2N.id; omega.
    -        }
    -        have H_y1_intro : y1 + Z.of_N (Z.to_N (y2 - y1)) = y2. {
    -          rewrite Z2N.id; omega.
    -        }
    -        split; last split. {
    -          move ⇒ y.
    -          rewrite !InZ_insert_intervalZ_guarded
    -                  !InZ_cons !In_elementsZ_single
    -                  H_yc1_intro H_y1_intro InZ_nil.
    -          split. {
    -            rewrite -!or_assoc.
    -            move ⇒ [[[]|]|] //. {
    -              move ⇒ [H_y1_le_y] H_y_lt.
    -              split. {
    -                left.
    -                split ⇒ //.
    -                apply Z.lt_le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    -                apply Z.lt_trans with (m := y2) ⇒ //.
    -                by apply Z_lt_add_r.
    -              } {
    -                move ⇒ [] /Z.le_ngt //.
    -              }
    -            } {
    -              move ⇒ [H_y2c_le_y] H_y_lt_yc1.
    -              split. {
    -                left.
    -                split ⇒ //.
    -                apply Z.le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    -                apply Z.le_trans with (m := y2) ⇒ //.
    -                apply Z_le_add_r.
    -              } {
    -                move ⇒ [] _ /Z.lt_nge //.
    -              }
    -            } {
    -              move ⇒ H_y_in_s1´.
    -              split; first by right.
    -              suff H_suff : y2 + Z.of_N c2 ≤ y. {
    -                move ⇒ [] _ /Z.lt_nge //.
    -              }
    -              apply Z.le_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -              apply Z.lt_le_incl.
    -              by apply H_gr_s1´_alt.
    -            }
    -          } {
    -            move ⇒ [] []; last by tauto.
    -            move ⇒ [H_y1_le_y] H_y_lt H_neq_y2.
    -            apply not_and in H_neq_y2; last by apply Z.le_decidable.
    -            case H_neq_y2. {
    -              move ⇒ /Z.nle_gt H_y_lt´.
    -              left; left; done.
    -            } {
    -              move ⇒ /Z.nlt_ge H_le_y.
    -              right; left; done.
    -            }
    -          }
    -        } {
    -          rewrite insert_intervalZ_guarded_rev_nil_app.
    -          rewrite !interval_list_invariant_insert_intervalZ_guarded ⇒ //. {
    -            rewrite H_yc1_intro ⇒ //.
    -          } {
    -            rewrite /insert_intervalZ_guarded.
    -            case_eq ((Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2)) =? 0)%N). {
    -              rewrite H_y1_intro.
    -              move ⇒ /N.eqb_eq /N2Z.inj_iff.
    -              rewrite Z2N.id; last first. {
    -                by apply Z.le_0_sub.
    -              }
    -              move ⇒ /Zminus_eq H_yc1_eq.
    -              eapply interval_list_elements_greater_impl;
    -                last apply H_gr_s1´.
    -              rewrite H_yc1_eq.
    -              apply Z_le_add_r.
    -            } {
    -              move ⇒ _.
    -              rewrite interval_list_elements_greater_cons
    -                      H_y1_intro.
    -              by apply Z_lt_add_r.
    -            }
    -          }
    -        } {
    -          move ⇒ x.
    -          rewrite InZ_insert_intervalZ_guarded InZ_cons In_elementsZ_single H_y1_intro InZ_nil.
    -          move ⇒ [] //.
    -          move ⇒ [_] H_x_lt.
    -          apply Z.lt_le_trans with (m := y2) ⇒ //.
    -          apply Z_le_add_r.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3.
    -        move ⇒ H_yc2_lt [] H_acc´ H_s´.
    -        clear IH P1 P2 P3 H_comp.
    -        subst.
    -        split; last split. {
    -          move ⇒ y.
    -          rewrite InZ_cons In_elementsZ_single.
    -          split; last by tauto. {
    -            move ⇒ [] //.
    -            move ⇒ H_y_in.
    -            split; first assumption.
    -            rewrite In_elementsZ_single.
    -            move ⇒ [] H_y2_le H_y_lt.
    -            case H_y_in; first by omega.
    -            move ⇒ /H_gr_s1´_alt H_lt_y.
    -            suff : y2 + Z.of_N c2 < y. {
    -              move ⇒ ?. omega.
    -            }
    -            apply Z.lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -            apply Z.lt_le_trans with (m := y1) ⇒ //.
    -            apply Z_le_add_r.
    -          }
    -        } {
    -          by rewrite interval_list_invariant_cons.
    -        } {
    -          done.
    -        }
    -      } {
    -        rewrite /P1 /P2 /P3.
    -        move ⇒ H_yc2_eq [] H_acc´ H_s´.
    -        clear IH P1 P2 P3 H_comp.
    -        subst.
    -        split; last split. {
    -          move ⇒ y.
    -          rewrite InZ_cons In_elementsZ_single.
    -          split; last by tauto. {
    -            move ⇒ [] //.
    -            move ⇒ H_y_in.
    -            split; first assumption.
    -            rewrite In_elementsZ_single.
    -            move ⇒ [] H_y2_le H_y_lt.
    -            case H_y_in; first by omega.
    -            move ⇒ /H_gr_s1´_alt H_lt_y.
    -            suff : y2 + Z.of_N c2 < y. {
    -              move ⇒ ?. omega.
    -            }
    -            apply Z.lt_trans with (m := (y2 + Z.of_N c2) + Z.of_N c1) ⇒ //.
    -            by apply Z_lt_add_r.
    -          }
    -        } {
    -          by rewrite interval_list_invariant_cons.
    -        } {
    -          done.
    -        }
    -      }
    -    }
    -  Qed.
    - +
      Lemma diff_aux2_props :
       âˆ€ (s2 s1 acc : t),
    -    interval_list_invariant (rev_append acc s1) = true →
    -    interval_list_invariant s2 = true →
    -    (∀ x1 x2, InZ x1 acc → InZ x2 s2 → Z.succ x1 < x2) →
    -    ((∀ y, (InZ y (diff_aux2 acc s1 s2) ↔
    -               ((InZ y acc) ∨ (InZ y s1)) ∧ ¬InZ y s2)) ∧
    -    (interval_list_invariant (diff_aux2 acc s1 s2) = true)).
    -  Proof.
    -    induction s2 as [| [y2 c2] s2´ IH]. {
    -      move ⇒ s1 acc H_inv_acc_s1 _ _.
    -      rewrite /diff_aux2.
    -      replace (match s1 with
    -        | nil ⇒ rev_append acc s1
    -        | _ :: _ ⇒ rev_append acc s1
    -       end) with (rev_append acc s1); last by case s1.
    -      split. {
    -        move ⇒ y.
    -        rewrite rev_append_rev InZ_app InZ_rev InZ_nil.
    -        tauto.
    -      } {
    -        assumption.
    -      }
    -    } {
    -      intros s1 acc H_inv_acc_s1.
    -      rewrite interval_list_invariant_cons.
    -      move ⇒ [H_gr_s2´] [H_c2_neq_0] H_inv_s2´.
    -      move ⇒ H_acc_s2.
    -      rewrite /diff_aux2; fold diff_aux2.
    -      case_eq s1. {
    -        move ⇒ H_s1_eq.
    -        split. {
    -          move ⇒ y.
    -          rewrite rev_append_rev InZ_app InZ_nil InZ_rev.
    -          split; last tauto.
    -          move ⇒ [] // H_y_in.
    -          split; first by left.
    -          move ⇒ H_y_in´.
    -          move : (H_acc_s2 _ _ H_y_in H_y_in´).
    -          apply Z.nlt_succ_diag_l.
    -        } {
    -          move : H_inv_acc_s1.
    -           by rewrite H_s1_eq.
    -        }
    -      } {
    -        move ⇒ [_ _] _ <-.
    -        case_eq (diff_aux y2 c2 acc s1).
    -        move ⇒ acc´ s1´ H_diff_aux_eq.
    - -
    -        have H_acc_lt_y2 : (∀ x : Z, InZ x acc → x < y2). {
    -          move ⇒ x H_x_in.
    -          have H_y2_in: (InZ y2 ((y2,c2) :: s2´)). {
    -            rewrite InZ_cons.
    -            left.
    -            by apply In_elementsZ_single_hd.
    -          }
    -          move : (H_acc_s2 _ _ H_x_in H_y2_in).
    -          apply Z.lt_trans, Z.lt_succ_diag_r.
    -        }
    -
    -        have [H_inv_acc [H_inv_s1 H_acc_s1]] :
    -          interval_list_invariant (rev acc) = true ∧
    -          interval_list_invariant s1 = true ∧
    -          (∀ x1 x2 : Z,
    -             InZ x1 acc → InZ x2 s1 → Z.succ x1 < x2). {
    -
    -          move : H_inv_acc_s1.
    -          rewrite rev_append_rev.
    -          move ⇒ /interval_list_invariant_app_elim.
    -          move ⇒ [?] [?] H_x.
    -          split; first assumption.
    -          split; first assumption.
    -          move ⇒ x1 x2 H_in_x1.
    -          apply H_x.
    -          by rewrite InZ_rev.
    -        }
    -
    -        move : (diff_aux_props y2 c2 s1 acc H_inv_acc H_inv_s1 H_acc_s1 H_acc_lt_y2 H_c2_neq_0).
    -        rewrite !H_diff_aux_eq.
    -        move ⇒ [H_inZ_res] [H_inv_res] H_inZ_acc´.
    - -
    -        have H_acc´_s2´ : (∀ x1 x2 : Z,
    -                              InZ x1 acc´ → InZ x2 s2´ → Z.succ x1 < x2). {
    -          move ⇒ x1 x2 H_x1_in H_x2_in.
    -          apply Z.le_lt_trans with (m := y2 + Z.of_N c2). {
    -            apply Z.le_succ_l.
    -            by apply H_inZ_acc´.
    -          } {
    -            move : H_gr_s2´.
    -            rewrite interval_list_elements_greater_alt2_def //.
    -            move ⇒ H_gr_s2´.
    -            by apply H_gr_s2´.
    -          }
    -        }
    -
    -        move : (IH s1´ acc´ H_inv_res H_inv_s2´ H_acc´_s2´).
    -        move ⇒ [] H_inZ_diff_res →.
    -        split; last done.
    -        move ⇒ y.
    -        rewrite H_inZ_diff_res.
    -        move : (H_inZ_res y).
    -        rewrite !rev_append_rev !InZ_app !InZ_rev InZ_cons.
    -        move ⇒ →.
    -        tauto.
    -      }
    -    }
    -  Qed.
    - +    interval_list_invariant (rev_append acc s1) = true →
    +    interval_list_invariant s2 = true →
    +    (∀ x1 x2, InZ x1 acc → InZ x2 s2 → Z.succ x1 < x2) →
    +    ((∀ y, (InZ y (diff_aux2 acc s1 s2) ↔
    +               ((InZ y acc) ∨ (InZ y s1)) ∧ ¬InZ y s2)) ∧
    +    (interval_list_invariant (diff_aux2 acc s1 s2) = true)).
    +
      Lemma diff_InZ :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    âˆ€ y, (InZ y (diff s1 s2) ↔ InZ y s1 ∧ ¬InZ y s2).
    -  Proof.
    -    intros s1 s2 H_inv_s1 H_inv_s2 y.
    -    rewrite /diff.
    - -
    -    move : (diff_aux2_props s2 s1 nil).
    -    move ⇒ [] //.
    -    move ⇒ H_in_diff _.
    -    rewrite H_in_diff InZ_nil.
    -    tauto.
    -  Qed.
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    âˆ€ y, (InZ y (diff s1 s2) ↔ InZ y s1 ∧ ¬InZ y s2).

      Lemma diff_invariant :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    interval_list_invariant (diff s1 s2) = true.
    -  Proof.
    -    intros s1 s2 H_inv_s1 H_inv_s2.
    -    rewrite /diff.
    - -
    -    move : (diff_aux2_props s2 s1 nil).
    -    move ⇒ [] //.
    -  Qed.
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    interval_list_invariant (diff s1 s2) = true.


      Global Instance diff_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (diff s1 s2).
    -  Proof.
    -    move ⇒ H_ok_s1 H_ok_s2.
    -    move : (H_ok_s1) (H_ok_s2).
    -    rewrite /Ok /IsOk /is_encoded_elems_list /add.
    -    move ⇒ [H_inv_s1] H_pre1.
    -    move ⇒ [H_inv_s2] H_pre2.
    -    split. {
    -      apply diff_invariant ⇒ //.
    -    } {
    -      intros y.
    -      move : (diff_InZ s1 s2 H_inv_s1 H_inv_s2).
    -      rewrite /InZ ⇒ →.
    -      move ⇒ [].
    -      move ⇒ /H_pre1 //.
    -    }
    -  Qed.
    - +
      Lemma diff_spec :
    -   âˆ€ (s s´ : t) (x : elt) (Hs : Ok s) (Hs´ : Ok s´),
    -   In x (diff s s´) ↔ In x s ∧ ¬In x s´.
    -  Proof.
    -    intros s s´ x H_ok H_ok´.
    -    rewrite !In_InZ.
    -    rewrite diff_InZ ⇒ //. {
    -      apply H_ok.
    -    } {
    -      apply H_ok´.
    -    }
    -  Qed.
    - +   âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    +   In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +
    -

    remove specification

    +

    remove specification


      Lemma removeZ_alt_def : ∀ x s acc,
    -    interval_list_invariant s = true →
    -    removeZ_aux acc x s = match diff_aux x (1%N) acc s with
    -                    (acc´, s´) ⇒ rev_append acc´ s´
    +    interval_list_invariant s = true →
    +    removeZ_aux acc x s = match diff_aux x (1%N) acc s with
    +                    (acc', s') ⇒ rev_append acc' s'
                      end.
    -  Proof.
    -    intros y2.
    -    induction s as [| [y1 c1] s´ IH]; first done.
    - -
    -    move ⇒ acc.
    -    rewrite interval_list_invariant_cons /=.
    -    move ⇒ [H_gr] [H_c1_neq_0] H_inv_s´.
    -    move : (interval_1_compare_elim y2 y1 c1).
    -    rewrite interval_1_compare_alt_def.
    -    rewrite !(interval_compare_swap y1 c1 y2); last first. {
    -      right. done.
    -    }
    -    move : (interval_compare_elim y1 c1 y2 (1%N)).
    -    case_eq (interval_compare (y1, c1) (y2, (1%N))) ⇒ H_eq. {
    -      move ⇒ H_lt_y2 _.
    -      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
    -        by apply Z.nlt_ge, Z.lt_le_incl.
    -      }
    -      have H_y2_nlt : ~(y2 < y1). {
    -        move ⇒ H_y2_y1.
    -        apply H_yc2_nlt.
    -        apply Z.lt_le_trans with (m := y1) ⇒ //.
    -        apply Z_le_add_r.
    -      }
    -      move : (H_y2_nlt) (H_yc2_nlt) ⇒ /Z.ltb_nlt → /Z.ltb_nlt →.
    -      rewrite IH //.
    -    } {
    -      move ⇒ H_lt_y2 _.
    -      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
    -        apply Z.nlt_ge.
    -        rewrite H_lt_y2.
    -        apply Z.le_refl.
    -      }
    -      have H_y2_nlt : ~(y2 < y1). {
    -        move ⇒ H_y2_y1.
    -        apply H_yc2_nlt.
    -        apply Z.lt_le_trans with (m := y1) ⇒ //.
    -        apply Z_le_add_r.
    -      }
    -      move : (H_y2_nlt) (H_yc2_nlt) ⇒ /Z.ltb_nlt → /Z.ltb_nlt →.
    -      rewrite IH //.
    -    } {
    -      done.
    -    } {
    -      done.
    -    } {
    -      move ⇒ [H_y1_eq] H_c1_eq.
    -      move ⇒ [] // .
    -      move ⇒ [H_lt_y2] H_y2_lt.
    -      have H_y2_nlt : ~(y2 < y1). {
    -        apply Z.nlt_ge ⇒ //.
    -      }
    -      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    -      rewrite /insert_intervalZ_guarded.
    - -
    -      have → : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
    -        rewrite H_y1_eq H_c1_eq Z.add_1_r Z.sub_diag //.
    -      }
    -
    -      have → : (Z.to_N (y2 - y1) =? 0 = true)%N. {
    -        rewrite H_y1_eq Z.sub_diag //.
    -      }
    -      done.
    -    } {
    -      move ⇒ [H_y2_le] [H_yc1_le] _.
    -      move ⇒ [] //.
    -      move ⇒ [H_y1_le] H_y2_lt.
    -      have H_y2_nlt : ~(y2 < y1). {
    -        apply Z.nlt_ge ⇒ //.
    -      }
    -      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    - -
    -      have H_y1_eq : (y1 = y2) by omega.
    -      have H_yc1_eq : (y1 + Z.of_N c1 = Z.succ y2). {
    -        apply Z.le_antisymm. {
    -          move : H_yc1_le.
    -          rewrite Z.add_1_r //.
    -        } {
    -          by apply Z.le_succ_l.
    -        }
    -      }
    -
    -      rewrite /insert_intervalZ_guarded.
    -      have → : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
    -        rewrite H_yc1_eq Z.sub_diag //.
    -      }
    -
    -      have → : (Z.to_N (y2 - y1) =? 0 = true)%N. {
    -        rewrite H_y1_eq Z.sub_diag //.
    -      }
    -
    -      suff → : diff_aux y2 (1%N) acc s´ = (acc, s´) by done.
    - -
    -      move : H_gr.
    -      rewrite H_yc1_eq.
    -      case s´ as [| [y´ c´] s´´]. {
    -        done.
    -      } {
    -        rewrite interval_list_elements_greater_cons /=
    -                /interval_compare.
    -        move ⇒ H_lt_y´.
    -        have → : y2 + Z.of_N 1 ?= y´ = Lt. {
    -          apply Z.compare_lt_iff.
    -          by rewrite Z.add_1_r.
    -        }
    -        done.
    -      }
    -    } {
    -      move ⇒ [H_y1_le] [H_yc2_le] _.
    -      move ⇒ [] //.
    -      move ⇒ [_] H_y2_lt.
    -      have H_y2_nlt : ~(y2 < y1). {
    -        apply Z.nlt_ge ⇒ //.
    -      }
    -      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    -      rewrite !rev_append_rev /insert_intervalZ_guarded Z.add_1_r.
    -      case ((Z.to_N (y2 - y1) =? 0)%N), (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0)%N. {
    -        reflexivity.
    -      } {
    -        rewrite /= -!app_assoc //.
    -      } {
    -        reflexivity.
    -      } {
    -        rewrite /= -!app_assoc //.
    -      }
    -    } {
    -      move ⇒ _ H_y2_lt´.
    -      have H_y2_lt : (y2 < y1). {
    -        apply Z.lt_trans with (m := Z.succ y2) ⇒ //.
    -        apply Z.lt_succ_diag_r.
    -      }
    -      move : (H_y2_lt) ⇒ /Z.ltb_lt → //.
    -    } {
    -      move ⇒ _ H_y1_eq.
    -      have H_y2_lt : (y2 < y1). {
    -        rewrite H_y1_eq.
    -        apply Z.lt_succ_diag_r.
    -      }
    -      move : (H_y2_lt) ⇒ /Z.ltb_lt → //.
    -    }
    -  Qed.
    - -
    -  Lemma removeZ_interval_list_invariant : ∀ s x, interval_list_invariant s = true → interval_list_invariant (removeZ x s) = true.
    -  Proof.
    -    intros s x H_inv.
    -    rewrite /removeZ removeZ_alt_def //.
    -    move : (diff_aux_props x (1%N) s nil).
    -    case_eq (diff_aux x 1%N nil s).
    -    move ⇒ acc´ s´ H_eq [] //.
    -    move ⇒ _ [] //.
    -  Qed.
    + +
    +  Lemma removeZ_interval_list_invariant : ∀ s x, interval_list_invariant s = true → interval_list_invariant (removeZ x s) = true.

      Lemma removeZ_spec :
    -   âˆ€ (s : t) (x y : Z) (Hs : interval_list_invariant s = true),
    -    InZ y (removeZ x s) ↔ InZ y s ∧ ¬Z.eq y x.
    -  Proof.
    -    intros s x y H_inv.
    -    rewrite /removeZ removeZ_alt_def //.
    -    move : (diff_aux_props x (1%N) s nil).
    -    case_eq (diff_aux x 1%N nil s).
    -    move ⇒ acc´ s´ H_eq [] //.
    -    move ⇒ → _.
    -    rewrite rev_append_rev InZ_app InZ_rev InZ_nil
    -            In_elementsZ_single1.
    -    split; move ⇒ [H1 H2]; split ⇒ //;
    -      move ⇒ H3; apply H2; by rewrite H3.
    -  Qed.
    +   âˆ€ (s : t) (x y : Z) (Hs : interval_list_invariant s = true),
    +    InZ y (removeZ x s) ↔ InZ y s ∧ ¬Z.eq y x.

      Global Instance remove_ok s x : ∀ `(Ok s), Ok (remove x s).
    -  Proof.
    -    rewrite /Ok /interval_list_invariant /remove.
    -    move ⇒ [H_is_ok_s H_enc_s].
    -    split. {
    -      by apply removeZ_interval_list_invariant.
    -    } {
    -      rewrite /is_encoded_elems_list ⇒ y.
    -      move : (removeZ_spec s (Enc.encode x) y H_is_ok_s).
    -      rewrite /InZ ⇒ → [] H_y_in _.
    -      apply H_enc_s ⇒ //.
    -    }
    -  Qed.
    - +
      Lemma remove_spec :
       âˆ€ (s : t) (x y : elt) (Hs : Ok s),
    -    In y (remove x s) ↔ In y s ∧ ¬Enc.E.eq y x.
    -  Proof.
    -    intros s x y Hs.
    -    have H_rs := (remove_ok s x Hs).
    -    rewrite /remove !In_InZ.
    -    rewrite removeZ_spec. {
    -      rewrite Enc.encode_eq //.
    -    } {
    -      apply Hs.
    -    }
    -  Qed.
    - +    In y (remove x s) ↔ In y s ∧ ¬Enc.E.eq y x.
    +
    -

    remove_list specification

    +

    remove_list specification


    -  Lemma remove_list_ok : ∀ l s, Ok s → Ok (remove_list l s).
    -  Proof.
    -    induction l as [| x l´ IH]. {
    -      done.
    -    } {
    -      move ⇒ s H_s_ok /=.
    -      apply IH.
    -      by apply remove_ok.
    -    }
    -  Qed.
    - +  Lemma remove_list_ok : ∀ l s, Ok s → Ok (remove_list l s).
    +
    -  Lemma remove_list_spec : ∀ x l s, Ok s →
    -     (In x (remove_list l s) ↔ ~(InA Enc.E.eq x l) ∧ In x s).
    -  Proof.
    -    move ⇒ x.
    -    induction l as [| y l´ IH]. {
    -      intros s H.
    -      rewrite /= InA_nil.
    -      tauto.
    -    } {
    -      move ⇒ s H_ok /=.
    -      rewrite IH remove_spec InA_cons.
    -      tauto.
    -    }
    -  Qed.
    - +  Lemma remove_list_spec : ∀ x l s, Ok s →
    +     (In x (remove_list l s) ↔ ~(InA Enc.E.eq x l) ∧ In x s).
    +

    -

    subset specification

    +

    subset specification


      Lemma subset_flatten_alt_def : ∀ (s1 s2 : t),
    -    subset s1 s2 =
    -    match (s1, s2) with
    -    | (nil, _) ⇒ true
    -    | (_ :: _, nil) ⇒ false
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ false
    -          | ICR_before_touch ⇒ false
    +    subset s1 s2 =
    +    match (s1, s2) with
    +    | (nil, _) ⇒ true
    +    | (_ :: _, nil) ⇒ false
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ false
    +          | ICR_before_touch ⇒ false
              | ICR_after ⇒ subset s1 l2
    -          | ICR_after_touch ⇒ false
    -          | ICR_overlap_before ⇒ false
    -          | ICR_overlap_after ⇒ false
    +          | ICR_after_touch ⇒ false
    +          | ICR_overlap_before ⇒ false
    +          | ICR_overlap_after ⇒ false
              | ICR_equal ⇒ subset l1 l2
              | ICR_subsume_1 ⇒ subset l1 s2
    -          | ICR_subsume_2 ⇒ false
    +          | ICR_subsume_2 ⇒ false
            end
        end.
    -  Proof.
    -    intros s1 s2.
    -    case s1, s2 ⇒ //.
    -  Qed.

      Lemma subset_props_aux : ∀ y1 c1 l1 y2 c2 l2,
    -    (∃ y, InZ y ((y1, c1) :: l1) ∧ ¬InZ y ((y2, c2) :: l2)) →
    -    (false = true ↔
    -    (∀ y : Z,
    -        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
    -  Proof.
    -    intros y1 c1 l1 y2 c2 l2.
    -    move ⇒ [y] [H_y_in H_y_nin].
    -    split; first done.
    -    move ⇒ H.
    -    contradict H_y_nin.
    -    by apply H.
    -  Qed.
    +    (∃ y, InZ y ((y1, c1) :: l1) ∧ ¬InZ y ((y2, c2) :: l2)) →
    +    (false = true ↔
    +    (∀ y : Z,
    +        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).

      Lemma subset_props_aux_before : ∀ y1 c1 l1 y2 c2 l2,
    -    (c1 ≠ 0%N) →
    -    interval_list_invariant ((y2, c2) :: l2) = true →
    -    (y1 < y2) →
    -    (false = true ↔
    -    (∀ y : Z,
    -        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
    -  Proof.
    -    intros y1 c1 l1 y2 c2 l2.
    -    rewrite interval_list_invariant_cons.
    -    move ⇒ H_c1_neq_0 [H_gr] [H_inv_l2] H_c2_neq_0 H_y1_lt.
    -    apply subset_props_aux.
    -    âˆƒ y1.
    -    split. {
    -      rewrite InZ_cons.
    -      left.
    -      by apply In_elementsZ_single_hd.
    -    } {
    -      rewrite InZ_cons.
    -      suff : ¬ (List.In y1 (elementsZ_single y2 c2)) ∧ ¬ InZ y1 l2 by tauto.
    -      split. {
    -        rewrite In_elementsZ_single.
    -        move ⇒ [] /Z.le_ngt //.
    -      } {
    -        eapply Nin_elements_greater; eauto.
    -        apply Z.le_trans with (m := y2). {
    -          by apply Z.lt_le_incl.
    -        } {
    -          apply Z_le_add_r.
    -        }
    -      }
    -    }
    -  Qed.
    - +    (c1 ≠ 0%N) →
    +    interval_list_invariant ((y2, c2) :: l2) = true →
    +    (y1 < y2) →
    +    (false = true ↔
    +    (∀ y : Z,
    +        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
    +

      Lemma subset_props : ∀ s1 s2 : t,
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    (subset s1 s2 = true ↔
    -     (∀ y, InZ y s1 → InZ y s2)).
    -  Proof.
    -    induction s1 as [| [y1 c1] l1 IH1]. {
    -      move ⇒ s2 _ _.
    -      rewrite subset_flatten_alt_def.
    -      split; done.
    -    } {
    -      induction s2 as [| [y2 c2] l2 IH2]. {
    -        rewrite interval_list_invariant_cons
    -                subset_flatten_alt_def.
    -        move ⇒ [_] [H_c1_neq_0] _ _.
    -        split ⇒ //.
    -        move ⇒ H; move : (H y1).
    -        rewrite InZ_nil ⇒ {H} H.
    -        contradict H.
    -        rewrite InZ_cons; left.
    -        by apply In_elementsZ_single_hd.
    -      } {
    -        move ⇒ H_inv_s1 H_inv_s2.
    -        move : (H_inv_s1) (H_inv_s2).
    -        rewrite !interval_list_invariant_cons.
    -        move ⇒ [H_gr_l1] [H_c1_neq_0] H_inv_l1.
    -        move ⇒ [H_gr_l2] [H_c2_neq_0] H_inv_l2.
    -        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {IH2} IH2.
    -        have : ∀ s2 : t,
    -          interval_list_invariant s2 = true →
    -          (subset l1 s2 = true ↔
    -          (∀ y : Z, InZ y l1 → InZ y s2)). {
    -          intros. by apply IH1.
    -        }
    -        move ⇒ {IH1} IH1.
    - -
    -        have H_yc2_nin : ¬ InZ (y2 + Z.of_N c2) ((y2, c2) :: l2). {
    -          rewrite !InZ_cons !In_elementsZ_single.
    -          move ⇒ []. {
    -            move ⇒ [_] /Z.lt_irrefl //.
    -          } {
    -            eapply Nin_elements_greater; eauto.
    -            apply Z.le_refl.
    -          }
    -        }
    -          
    -        rewrite subset_flatten_alt_def.
    -        move : (interval_compare_elim y1 c1 y2 c2).
    -        case (interval_compare (y1, c1) (y2, c2)). {
    -          move ⇒ H_lt_y2.
    -          apply subset_props_aux_before ⇒ //.
    -          apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -          apply Z_le_add_r.
    -        } {
    -          move ⇒ H_y2_eq.
    -          apply subset_props_aux_before ⇒ //.
    -          rewrite -H_y2_eq.
    -          by apply Z_lt_add_r.
    -        } {
    -          move ⇒ [H_y1_lt] _.
    -          apply subset_props_aux_before ⇒ //.
    -        } {
    -          move ⇒ [H_y2_lt] [H_y1_lt] H_yc2_lt.
    -          apply subset_props_aux.
    -          âˆƒ (y2 + Z.of_N c2).
    -          split ⇒ //.
    -          rewrite !InZ_cons !In_elementsZ_single.
    -          left.
    -          split ⇒ //.
    -          by apply Z.lt_le_incl.
    -        } {
    -          move ⇒ [H_y1_eq] H_c1_eq; subst.
    -          rewrite IH1 ⇒ //.
    -          split; move ⇒ H_pre y; move : (H_pre y) ⇒ {H_pre};
    -            rewrite !InZ_cons. {
    -            tauto.
    -          } {
    -            move ⇒ H_pre H_y_in_l1.
    -            suff : ~(List.In y (elementsZ_single y2 c2)). {
    -              tauto.
    -            }
    -            move : H_gr_l1.
    -            rewrite interval_list_elements_greater_alt2_def
    -              // In_elementsZ_single.
    -            move ⇒ H; move : (H y H_y_in_l1) ⇒ {H}.
    -            move ⇒ /Z.lt_ngt H_neq [_] //.
    -          }
    -        } {
    -          move ⇒ [H_y2_lt_y1] [H_yc1_le] _.
    -          rewrite IH1.
    -          split; move ⇒ H_pre y; move : (H_pre y) ⇒ {H_pre};
    -            rewrite !InZ_cons. {
    -            move ⇒ H []; last apply H. move ⇒ {H}.
    -            rewrite !In_elementsZ_single.
    -            move ⇒ [H_y1_le] H_y_lt.
    -            left.
    -            omega.
    -          } {
    -            move ⇒ H_pre H_y_in_l1.
    -            apply H_pre.
    -            by right.
    -          } {
    -            assumption.
    -          }
    -        } {
    -          move ⇒ [H_y1_le] [_] []. {
    -            apply subset_props_aux_before ⇒ //.
    -          } {
    -            move ⇒ H_yc2_lt.
    -            apply subset_props_aux.
    -            âˆƒ (y2 + Z.of_N c2).
    -            split ⇒ //.
    -            rewrite !InZ_cons !In_elementsZ_single.
    -            left.
    -            split ⇒ //.
    -            apply Z.le_trans with (m := y2) ⇒ //.
    -            apply Z_le_add_r.
    -          }
    -        } {
    -          move ⇒ H_yc2_lt_y1.
    -          rewrite IH2.
    -          split; move ⇒ H_pre y; move : (H_pre y) ⇒ {H_pre};
    -                                                        rewrite !InZ_cons. {
    -            tauto.
    -          } {
    -            rewrite !In_elementsZ_single.
    -            move ⇒ H_pre H_y_in.
    -            suff : ~(y2 ≤ y < y2 + Z.of_N c2). {
    -              tauto.
    -            }
    -            move ⇒ [H_y2_le H_y_lt].
    -            move : H_y_in ⇒ []. {
    -              move ⇒ [H_y1_le] H_y_lt´.
    -              omega.
    -            } {
    -              eapply Nin_elements_greater; eauto.
    -              apply Z.le_trans with (m := y1); last first. {
    -                apply Z_le_add_r.
    -              }
    -              apply Z.lt_le_incl.
    -              apply Z.lt_trans with (m := y2 + Z.of_N c2) ⇒ //.
    -            }
    -          }
    -        } {
    -          move ⇒ H_y1_eq.
    -          apply subset_props_aux.
    -          âˆƒ y1.
    -          rewrite !InZ_cons.
    -          split. {
    -            left.
    -            by apply In_elementsZ_single_hd.
    -          } {
    -            rewrite !In_elementsZ_single H_y1_eq.
    -            move ⇒ []. {
    -              move ⇒ [_] /Z.lt_irrefl //.
    -            } {
    -              eapply Nin_elements_greater; eauto.
    -              rewrite H_y1_eq.
    -              apply Z.le_refl.
    -            }
    -          }
    -        }
    -      }
    -    }
    -  Qed.
    - +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    (subset s1 s2 = true ↔
    +     (∀ y, InZ y s1 → InZ y s2)).
    +
      Lemma subset_spec :
    -   âˆ€ (s s´ : t) (Hs : Ok s) (Hs´ : Ok s´),
    -   subset s s´ = true ↔ Subset s s´.
    -  Proof.
    -    intros s s´ Hs Hs´.
    -    move : (Hs) (Hs´).
    -    rewrite /Ok /IsOk.
    -    move ⇒ [H_inv_s H_enc_s] [H_inv_s´ H_enc_s´].
    -    rewrite (subset_props s s´ H_inv_s H_inv_s´).
    -    rewrite /Subset.
    -    split. {
    -      move ⇒ H_pre enc_y.
    -      rewrite !In_InZ.
    -      apply H_pre.
    -    } {
    -      move ⇒ H_pre y H_y_in.
    -      move : (H_enc_s _ H_y_in) ⇒ [e] H_e. subst.
    -      move : (H_pre e) H_y_in.
    -      rewrite !In_InZ //.
    -    }
    -  Qed.
    - +   âˆ€ (s s' : t) (Hs : Ok s) (Hs' : Ok s'),
    +   subset s s' = true ↔ Subset s s'.
    +
    -

    elements and elementsZ specification

    +

    elements and elementsZ specification


    -  Lemma elements_spec1 : ∀ (s : t) (x : elt) (Hs : Ok s), List.In x (elements s) ↔ In x s.
    -  Proof.
    -    intros s x Hs.
    -    by rewrite In_alt_def.
    -  Qed.
    +  Lemma elements_spec1 : ∀ (s : t) (x : elt) (Hs : Ok s), List.In x (elements s) ↔ In x s.

      Lemma NoDupA_elementsZ_single: ∀ c x,
    -    NoDupA Z.eq (elementsZ_single x c).
    -  Proof.
    -    induction c as [| c´ IH] using N.peano_ind. {
    -      intros x.
    -      rewrite elementsZ_single_base //.
    -    } {
    -      intros x.
    -      rewrite elementsZ_single_succ.
    -      apply NoDupA_app. {
    -        apply Z.eq_equiv.
    -      } {
    -        apply IH.
    -      } {
    -        apply NoDupA_singleton.
    -      } {
    -        move ⇒ y.
    -        rewrite !InA_alt.
    -        move ⇒ [_] [<-] H_y_in.
    -        move ⇒ [_] [<-] H_y_in´.
    -        move : H_y_in H_y_in´.
    -        rewrite In_elementsZ_single /=.
    -        move ⇒ [H_x_le] H_y_lt [] // H_y_eq.
    -        contradict H_y_lt.
    -        rewrite H_y_eq.
    -        apply Z.lt_irrefl.
    -      }
    -    }
    -  Qed.
    - -
    -  Lemma elementsZ_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Z.eq (elementsZ s).
    -  Proof.
    -    induction s as [| [x c] s´ IH]. {
    -      move ⇒ _.
    -      rewrite elementsZ_nil.
    -      apply NoDupA_nil.
    -    } {
    -      move ⇒ H_ok_s.
    -      move : (H_ok_s) ⇒ /Ok_cons [H_interval_list_elements_greater] [H_c] [H_enc] H_s´.
    -      rewrite elementsZ_cons.
    -      apply NoDupA_app. {
    -        apply Z.eq_equiv.
    -      } {
    -        by apply IH.
    -      } {
    -        apply NoDupA_rev. {
    -          apply Z.eq_equiv.
    -        } {
    -          apply NoDupA_elementsZ_single.
    -        }
    -      } {
    -        move ⇒ y.
    -        rewrite !InA_alt.
    -        move ⇒ [_] [<-] H_y_in.
    -        move ⇒ [_] [<-] H_y_in´.
    -        move : H_y_in´.
    -        rewrite -in_rev In_elementsZ_single /=.
    -        move ⇒ [H_x_le] H_y_lt.
    -        eapply (Nin_elements_greater s´ (x + Z.of_N c)) ⇒ //. {
    -          apply H_s´.
    -        } {
    -          apply Z.lt_le_incl, H_y_lt.
    -        } {
    -          apply H_y_in.
    -        }
    -      }
    -    }
    -  Qed.
    - -
    -  Lemma elements_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Enc.E.eq (elements s).
    -  Proof.
    -    intros s Hs.
    -    rewrite /elements rev_map_alt_def.
    -    apply NoDupA_rev. {
    -      apply Enc.E.eq_equiv.
    -    } {
    -      eapply NoDupA_map; first by apply elementsZ_spec2w.
    - -
    -      intros x1 x2 H_x1_in H_x2_in H_dec_eq.
    -      have H_is_enc: is_encoded_elems_list (elementsZ s). {
    -        apply Hs.
    -      }
    -      move : (H_is_enc _ H_x1_in) ⇒ [y1 H_x1_eq].
    -      move : (H_is_enc _ H_x2_in) ⇒ [y2 H_x2_eq].
    -      move : H_dec_eq.
    -      rewrite -H_x1_eq -H_x2_eq !Enc.decode_encode_ok Enc.encode_eq //.
    -    }
    -  Qed.
    - +    NoDupA Z.eq (elementsZ_single x c).
    + +
    +  Lemma elementsZ_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Z.eq (elementsZ s).
    + +
    +  Lemma elements_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Enc.E.eq (elements s).
    +
    -

    equal specification

    +

    equal specification


      Lemma equal_alt_def : ∀ s1 s2,
    -    equal s1 s2 = true ↔ (s1 = s2).
    -  Proof.
    -    induction s1 as [| [x cx] xs IH]. {
    -      move ⇒ [] //.
    -    } {
    -      move ⇒ [] //=.
    -      move ⇒ [y cy] ys.
    -      rewrite !andb_true_iff IH N.eqb_eq Z.eqb_eq.
    -      split. {
    -        move ⇒ [->] [->] → //.
    -      } {
    -        move ⇒ [->] → → //.
    -      }
    -    }
    -  Qed.
    - +    equal s1 s2 = true ↔ (s1 = s2).
    +
      Lemma equal_elementsZ :
    -    âˆ€ (s s´ : t) {Hs : Ok s} {Hs´ : Ok s´},
    -    (∀ x, (InZ x s ↔ InZ x s´)) → (s = s´).
    -  Proof.
    -    intros s s´.
    -    move ⇒ H_ok_s H_ok_s´ H_InZ_eq.
    -    have [] : ((subset s s´ = true) ∧ (subset s´ s = true)). {
    -      rewrite !subset_spec /Subset.
    -      split ⇒ x; rewrite !In_InZ H_InZ_eq //.
    -    }
    -    have : interval_list_invariant s´ = true by apply H_ok_s´.
    -    have : interval_list_invariant s = true by apply H_ok_s.
    -    clear H_ok_s H_ok_s´ H_InZ_eq.
    -    move : s s´.
    -    induction s as [| [x1 c1] s1 IH];
    -      case s´ as [| [x2 c2] s2] ⇒ //.
    -    rewrite !interval_list_invariant_cons.
    -    move ⇒ [H_gr_s1] [H_c1_neq_0] H_inv_s1.
    -    move ⇒ [H_gr_s2] [H_c2_neq_0] H_inv_s2.
    -    rewrite subset_flatten_alt_def
    -            (subset_flatten_alt_def ((x2, c2)::s2)).
    -    rewrite (interval_compare_swap x1 c1); last by left.
    -    move : (interval_compare_elim x1 c1 x2 c2).
    -    case (interval_compare (x1, c1) (x2, c2)) ⇒ //.
    -    move ⇒ [->] → H_sub_s12 H_sub_s21.
    - -
    -    suff → : s1 = s2 by done.
    -    by apply IH.
    -  Qed.
    +    âˆ€ (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
    +    (∀ x, (InZ x s ↔ InZ x s')) → (s = s').

      Lemma equal_spec :
    -    âˆ€ (s s´ : t) {Hs : Ok s} {Hs´ : Ok s´},
    -    equal s s´ = true ↔ Equal s s´.
    -  Proof.
    -    intros s s´ Hs Hs´.
    -    rewrite equal_alt_def /Equal.
    -    split. {
    -      move ⇒ → //.
    -    } {
    -      move ⇒ H.
    -      apply equal_elementsZ ⇒ // x.
    -      move : (H (Enc.decode x)).
    -      rewrite !In_InZ.
    - -
    -      suff H_ex : (∀ s´´, Ok s´´ → InZ x s´´ → (∃ z, Enc.encode z = x)). {
    -        move ⇒ HH.
    -        split. {
    -          move ⇒ H3.
    -          move : HH (H3).
    -          move : (H_ex s Hs H3) ⇒ [z] <-.
    -          rewrite Enc.decode_encode_ok ⇒ <- //.
    -        } {
    -          move ⇒ H3.
    -          move : HH (H3).
    -          move : (H_ex s´ Hs´ H3) ⇒ [z] <-.
    -          rewrite Enc.decode_encode_ok ⇒ <- //.
    -        }
    -      }
    -      clear.
    -      intros s´´ H_ok H_in_x.
    -      have H_enc : is_encoded_elems_list (elementsZ s´´). {
    -        apply H_ok.
    -      }
    -      apply H_enc.
    -      apply H_in_x.
    -    }
    -  Qed.
    - +    âˆ€ (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
    +    equal s s' = true ↔ Equal s s'.
    +
    -

    compare

    +

    compare

    -  Definition lt (s1 s2 : t) : Prop := (compare s1 s2 = Lt).
    +  Definition lt (s1 s2 : t) : Prop := (compare s1 s2 = Lt).

      Lemma compare_eq_Eq : ∀ s1 s2,
    -    (compare s1 s2 = Eq ↔ equal s1 s2 = true).
    -  Proof.
    -    induction s1 as [| [y1 c1] s1´ IH];
    -      case s2 as [| [y2 c2] s2´] ⇒ //.
    -    rewrite /= !andb_true_iff -IH Z.eqb_eq N.eqb_eq.
    -    move : (Z.compare_eq_iff y1 y2).
    -    case (Z.compare y1 y2). {
    -      move ⇒ H.
    -      have → : y1 = y2. by apply H.
    -      clear H.
    - -
    -      move : (N.compare_eq_iff c1 c2).
    -      case (N.compare c1 c2). {
    -        move ⇒ H.
    -        have → : c1 = c2. by apply H.
    -        tauto.
    -      } {
    -        move ⇒ H.
    -        have H_neq : ~(c1 = c2). by rewrite -H ⇒ {H}.
    -        tauto.
    -      } {
    -        move ⇒ H.
    -        have H_neq : ~(c1 = c2). by rewrite -H ⇒ {H}.
    -        tauto.
    -      }
    -    } {
    -      move ⇒ H.
    -      have H_neq : ~(y1 = y2). by rewrite -H ⇒ {H}.
    -      tauto.
    -    } {
    -      move ⇒ H.
    -      have H_neq : ~(y1 = y2). by rewrite -H ⇒ {H}.
    -      tauto.
    -    }
    -  Qed.
    - +    (compare s1 s2 = Eq ↔ equal s1 s2 = true).
    +
      Lemma compare_eq_Lt_nil_l : ∀ s,
    -    compare nil s = Lt ↔ s ≠ nil.
    -  Proof.
    -    intros s.
    -    case s ⇒ //=.
    -    split ⇒ //.
    -  Qed.
    +    compare nil s = Lt ↔ s ≠ nil.

      Lemma compare_eq_Lt_nil_r : ∀ s,
    -    ~(compare s nil = Lt).
    -  Proof.
    -    intros s.
    -    case s as [| [y1 c1] s´] ⇒ //=.
    -  Qed.
    +    ~(compare s nil = Lt).

      Lemma compare_eq_Lt_cons : ∀ y1 y2 c1 c2 s1 s2,
    -    compare ((y1, c1)::s1) ((y2, c2)::s2) = Lt ↔
    -    (y1 < y2) ∨ ((y1 = y2) ∧ (c1 < c2)%N) ∨
    -    ((y1 = y2) ∧ (c1 = c2) ∧ compare s1 s2 = Lt).
    -  Proof.
    -    intros y1 y2 c1 c2 s1 s2.
    -    rewrite /=.
    -    case_eq (Z.compare y1 y2). {
    -      move ⇒ /Z.compare_eq_iff →.
    -      case_eq (N.compare c1 c2). {
    -        move ⇒ /N.compare_eq_iff →.
    -        split. {
    -          move ⇒ H.
    -          right; right.
    -          done.
    -        } {
    -          move ⇒ [| []]. {
    -            move ⇒ /Z.lt_irrefl //.
    -          } {
    -            move ⇒ [_] /N.lt_irrefl //.
    -          } {
    -            move ⇒ [_] [_] → //.
    -          }
    -        }
    -      } {
    -        move ⇒ /N.compare_lt_iff H_c1_lt.
    -        split ⇒ //.
    -        move ⇒ _.
    -        right; left. done.
    -      } {
    -        move ⇒ /N.compare_gt_iff H_c2_lt.
    -        split ⇒ //.
    -        move ⇒ [| []]. {
    -          move ⇒ /Z.lt_irrefl //.
    -        } {
    -          move ⇒ [_] /N.lt_asymm //.
    -        } {
    -          move ⇒ [_] [] H_c1_eq.
    -          contradict H_c2_lt.
    -          subst c1.
    -          by apply N.lt_irrefl.
    -        }
    -      }
    -    } {
    -      move ⇒ /Z.compare_lt_iff.
    -      tauto.
    -    } {
    -      move ⇒ /Z.compare_gt_iff H_y2_lt.
    -      split ⇒ //.
    -      move ⇒ [| []]. {
    -        move ⇒ /Z.lt_asymm //.
    -      } {
    -        move ⇒ [] H_y1_eq.
    -        exfalso. omega.
    -      } {
    -        move ⇒ [] H_y1_eq.
    -        exfalso. omega.
    -      }
    -    }
    -  Qed.
    - +    compare ((y1, c1)::s1) ((y2, c2)::s2) = Lt ↔
    +    (y1 < y2) ∨ ((y1 = y2) ∧ (c1 < c2)%N) ∨
    +    ((y1 = y2) ∧ (c1 = c2) ∧ compare s1 s2 = Lt).
    +

      Lemma compare_antisym: ∀ (s1 s2 : t),
    -    (compare s1 s2) = CompOpp (compare s2 s1).
    -  Proof.
    -    induction s1 as [| [y1 c1] s1´ IH];
    -      case s2 as [| [y2 c2] s2´] ⇒ //.
    -    rewrite /= (Z.compare_antisym y1 y2) (N.compare_antisym c1 c2).
    -    case (Z.compare y1 y2) ⇒ //=.
    -    case (N.compare c1 c2) ⇒ //=.
    -  Qed.
    +    (compare s1 s2) = CompOpp (compare s2 s1).

      Lemma compare_spec : ∀ s1 s2,
    -    CompSpec eq lt s1 s2 (compare s1 s2).
    -  Proof.
    -    intros s1 s2.
    -    rewrite /CompSpec /lt (compare_antisym s2 s1).
    -    case_eq (compare s1 s2). {
    -      rewrite compare_eq_Eq equal_alt_def ⇒ →.
    -      by apply CompEq.
    -    } {
    -      move ⇒ _.
    -      by apply CompLt.
    -    } {
    -      move ⇒ _.
    -      by apply CompGt.
    -    }
    -  Qed.
    - -
    -  Lemma lt_Irreflexive : Irreflexive lt.
    -  Proof.
    -    rewrite /Irreflexive /Reflexive /complement /lt.
    -    intros x.
    -    suff → : compare x x = Eq by done.
    -    rewrite compare_eq_Eq equal_alt_def //.
    -  Qed.
    - -
    -  Lemma lt_Transitive : Transitive lt.
    -  Proof.
    -    rewrite /Transitive /lt.
    -    induction x as [| [y1 c1] s1´ IH];
    -      case y as [| [y2 c2] s2´];
    -      case z as [| [y3 c3] s3´] ⇒ //.
    - -
    -    rewrite !compare_eq_Lt_cons.
    -    move ⇒ [H_y1_lt | [[->] H_c1_lt | [->] [->] H_comp]]
    - [H_y2_lt | [[<-] H_c2_lt | [<-] [<-] H_comp´]]. {
    -      left.
    -      by apply Z.lt_trans with (m := y2).
    -    } {
    -      by left.
    -    } {
    -      by left.
    -    } {
    -      by left.
    -    } {
    -      right; left.
    -      split ⇒ //.
    -      by apply N.lt_trans with (m := c2).
    -    } {
    -      by right; left.
    -    } {
    -      by left.
    -    } {
    -      by right; left.
    -    } {
    -      right; right.
    -      split ⇒ //.
    -      split ⇒ //.
    -      by apply (IH s2´).
    -    }
    -  Qed.
    +    CompSpec eq lt s1 s2 (compare s1 s2).
    + +
    +  Lemma lt_Irreflexive : Irreflexive lt.
    +
    +  Lemma lt_Transitive : Transitive lt.
    +
    -

    elements is sorted

    +

    elements is sorted


      Lemma elementsZ_single_sorted : ∀ c x,
    -    sort Z.lt (elementsZ_single x c).
    -  Proof.
    -    induction c as [| c´ IH] using N.peano_ind. {
    -      intro x.
    -      rewrite elementsZ_single_base.
    -      apply Sorted_nil.
    -    } {
    -      intro x.
    -      rewrite elementsZ_single_succ_front.
    -      apply Sorted_cons. {
    -        apply IH.
    -      } {
    -        case (N.zero_or_succ c´). {
    -          move ⇒ →.
    -          rewrite elementsZ_single_base //.
    -        } {
    -          move ⇒ [c´´] →.
    -          rewrite elementsZ_single_succ_front.
    -          constructor.
    -          apply Z.lt_succ_diag_r.
    -        }
    -      }
    -    }
    -  Qed.
    - +    sort Z.lt (elementsZ_single x c).
    +
      Lemma elementsZ_sorted : ∀ s,
    -    interval_list_invariant s = true →
    -    sort Z.lt (rev (elementsZ s)).
    -  Proof.
    -    induction s as [| [y c] s´ IH]. {
    -      move ⇒ _.
    -      rewrite elementsZ_nil.
    -      apply Sorted_nil.
    -    } {
    -      rewrite interval_list_invariant_cons elementsZ_cons
    -              rev_app_distr rev_involutive.
    -      move ⇒ [H_gr] [H_c_neq_0] H_inv_s´.
    -      apply SortA_app with (eqA := Logic.eq). {
    -        apply eq_equivalence.
    -      } {
    -        apply Z.lt_strorder.
    -      } {
    -        apply elementsZ_single_sorted.
    -      } {
    -        by apply IH.
    -      } {
    -        intros x1 x2.
    -        move ⇒ /InA_alt [_] [<-] /In_elementsZ_single [_ H_x1_lt].
    -        move ⇒ /InA_alt [_] [<-].
    -        rewrite -In_rev ⇒ H_x2_in.
    - -
    -        apply Z.lt_trans with (m := (y + Z.of_N c)) ⇒ //.
    -        eapply interval_list_elements_greater_alt2_def;
    -          eauto.
    -      }
    -    }
    -  Qed.
    - +    interval_list_invariant s = true →
    +    sort Z.lt (rev (elementsZ s)).
    +

      Lemma elements_sorted : ∀ s,
    -    Ok s →
    -    sort Enc.E.lt (elements s).
    -  Proof.
    -    move ⇒ s [H_inv] H_enc.
    -    rewrite /elements rev_map_alt_def -map_rev.
    -    have : (∀ x : Z, List.In x (rev (elementsZ s)) →
    -           âˆƒ e : Enc.E.t, Enc.encode e = x). {
    -      move ⇒ x.
    -      move : (H_enc x).
    -      rewrite In_rev //.
    -    }
    -    move : (elementsZ_sorted s H_inv) ⇒ {H_enc}.
    -    generalize (rev (elementsZ s)).
    -    induction l as [| x xs IH]. {
    -      rewrite /= ⇒ _ _.
    -      apply Sorted_nil.
    -    } {
    -      move ⇒ H_sort H_enc.
    -      apply Sorted_inv in H_sort as [H_sort H_hd_rel].
    -      simpl.
    -      apply Sorted_cons. {
    -        apply IH ⇒ //.
    -        move ⇒ xx H_xx_in.
    -        apply H_enc.
    -        by apply in_cons.
    -      } {
    -        move : H_hd_rel H_enc.
    -        case xs ⇒ //=.
    -        move ⇒ x´ xs´ H_hd_rel H_enc.
    -        apply HdRel_inv in H_hd_rel.
    -        apply HdRel_cons.
    -        rewrite -Enc.encode_lt.
    -        have [y H_y] : (∃ y, Enc.encode y = x). {
    -          apply H_enc. by left.
    -        }
    -        have [y´ H_y´] : (∃ y´, Enc.encode y´ = x´). {
    -          apply H_enc. by right; left.
    -        }
    -        move : H_hd_rel.
    -        rewrite -!H_y -!H_y´ !Enc.decode_encode_ok //.
    -      }
    -    }
    -  Qed.
    - +    Ok s →
    +    sort Enc.E.lt (elements s).
    +

    -

    choose specification

    +

    choose specification


      Definition min_eltZ_spec1 :
    -    âˆ€ (s : t) (x : Z),
    -      interval_list_invariant s = true →
    -      min_eltZ s = Some x → InZ x s.
    -  Proof.
    -    intros s x.
    -    case s as [| [x´ c] s´]. {
    -      rewrite /min_eltZ //.
    -    } {
    -      rewrite /min_eltZ InZ_cons interval_list_invariant_cons.
    -      move ⇒ [_] [H_c_neq] _ [->].
    -      left.
    -      by apply In_elementsZ_single_hd.
    -    }
    -  Qed.
    - +    âˆ€ (s : t) (x : Z),
    +      interval_list_invariant s = true →
    +      min_eltZ s = Some x → InZ x s.
    +
      Lemma min_eltZ_spec2 :
    -    âˆ€ (s : t) (x y : Z) (Hs : Ok s),
    -    min_eltZ s = Some x → InZ y s → ¬ Z.lt y x.
    -  Proof.
    -    intros s x y H_ok H_min H_in H_y_lt_x.
    -    eapply (Nin_elements_greater s (Z.pred x)) ⇒ //; last apply H_in. {
    -      move : H_ok H_min.
    -      case s ⇒ //.
    -      move ⇒ [z c] s´ _ [<-].
    -      rewrite interval_list_elements_greater_cons.
    -      apply Z.lt_pred_l.
    -    } {
    -      apply H_ok.
    -    } {
    -      by apply Z.lt_le_pred.
    -    }
    -  Qed.
    - +    âˆ€ (s : t) (x y : Z) (Hs : Ok s),
    +    min_eltZ s = Some x → InZ y s → ¬ Z.lt y x.
    +
      Definition min_eltZ_spec3 :
        âˆ€ (s : t),
    -      min_eltZ s = None → ∀ x, ¬InZ x s.
    -  Proof.
    -    intros s.
    -    case s as [| [x´ c] s´];
    -      rewrite /min_eltZ //.
    -    move ⇒ _ x //.
    -  Qed.
    - +      min_eltZ s = None → ∀ x, ¬InZ x s.
    +
      Definition min_elt_spec1 :
    -    âˆ€ (s : t) (x : elt) (Hs : Ok s), min_elt s = Some x → In x s.
    -  Proof.
    -    rewrite /min_elt.
    -    move ⇒ s x H_ok.
    -    case_eq (min_eltZ s) ⇒ //.
    -    move ⇒ z H_min_elt [<-].
    -    apply InZ_In ⇒ //.
    -    apply min_eltZ_spec1 ⇒ //.
    -    apply H_ok.
    -  Qed.
    - +    âˆ€ (s : t) (x : elt) (Hs : Ok s), min_elt s = Some x → In x s.
    +
      Definition min_elt_spec2 :
    -    âˆ€ (s : t) (x y : elt) (Hs : Ok s), min_elt s = Some x → In y s → ~(Enc.E.lt y x).
    -  Proof.
    -    rewrite /min_elt.
    -    move ⇒ s x y H_ok.
    -    case_eq (min_eltZ s) ⇒ //.
    -    move ⇒ z H_min_elt [<-].
    -    rewrite In_InZ ⇒ H_inZ.
    -    have H_y_eq : y = Enc.decode (Enc.encode y). {
    -      by rewrite Enc.decode_encode_ok.
    -    }
    -    rewrite H_y_eq -Enc.encode_lt.
    -    apply (min_eltZ_spec2 _ _ _ H_ok); last first. {
    -      by rewrite Enc.decode_encode_ok.
    -    }
    -    suff → : Enc.encode (Enc.decode z) = z by assumption.
    -    apply encode_decode_eq with (s := s) ⇒ //.
    -    apply min_eltZ_spec1 ⇒ //.
    -    apply H_ok.
    -  Qed.
    - +    âˆ€ (s : t) (x y : elt) (Hs : Ok s), min_elt s = Some x → In y s → ~(Enc.E.lt y x).
    +
      Definition min_elt_spec3 :
    -    âˆ€ s : t, min_elt s = None → Empty s.
    -  Proof.
    -    rewrite /min_elt /min_eltZ /Empty /In.
    -    case s as [| [x´ c] s´] ⇒ //.
    -    move ⇒ _ e.
    -    rewrite elements_nil InA_nil //.
    -  Qed.
    - +    âˆ€ s : t, min_elt s = None → Empty s.
    +
      Definition choose_spec1 :
    -    âˆ€ (s : t) (x : elt) (Hs : Ok s), choose s = Some x → In x s.
    -  Proof.
    -    rewrite /choose.
    -    apply min_elt_spec1.
    -  Qed.
    - +    âˆ€ (s : t) (x : elt) (Hs : Ok s), choose s = Some x → In x s.
    +
      Definition choose_spec2 :
    -    âˆ€ s : t, choose s = None → Empty s.
    -  Proof.
    -    rewrite /choose.
    -    apply min_elt_spec3.
    -  Qed.
    - +    âˆ€ s : t, choose s = None → Empty s.
    +
    -  Lemma choose_spec3: ∀ s s´ x x´, Ok s → Ok s´ →
    -    choose s = Some x → choose s´ = Some x´ → Equal s s´ → x = x´.
    -  Proof.
    -    intros s s´ x x´ Hs Hs´ Hx Hx´.
    -    rewrite -equal_spec equal_alt_def ⇒ H_s_eq.
    -    move : Hx Hx´.
    -    rewrite H_s_eq ⇒ → [] //.
    -  Qed.
    +  Lemma choose_spec3: ∀ s s' x x', Ok s → Ok s' →
    +    choose s = Some x → choose s' = Some x' → Equal s s' → x = x'.


      Definition max_eltZ_spec1 :
    -    âˆ€ (s : t) (x : Z),
    -      interval_list_invariant s = true →
    -      max_eltZ s = Some x → InZ x s.
    -  Proof.
    -    intros s x.
    -    induction s as [| [x´ c] s´ IH]. {
    -      rewrite /max_eltZ //.
    -    } {
    -      rewrite InZ_cons interval_list_invariant_cons /=.
    -      move ⇒ [_] [H_c_neq].
    -      case s´ as [| [y´ c´] s´´]. {
    -        move ⇒ _ [<-].
    -        left. {
    -          rewrite In_elementsZ_single.
    -          split. {
    -            rewrite -Z.lt_le_pred.
    -            by apply Z_lt_add_r.
    -          } {
    -            apply Z.lt_pred_l.
    -          }
    -        }
    -      } {
    -        move ⇒ H_inv H_max_eq.
    -        right.
    -        by apply IH.
    -      }
    -    }
    -  Qed.
    - +    âˆ€ (s : t) (x : Z),
    +      interval_list_invariant s = true →
    +      max_eltZ s = Some x → InZ x s.
    +
      Lemma max_eltZ_spec2 :
    -    âˆ€ (s : t) (x y : Z),
    -    interval_list_invariant s = true →
    -    max_eltZ s = Some x → InZ y s → ¬ Z.lt x y.
    -  Proof.
    -    induction s as [| [y c] s´ IH]. {
    -      done.
    -    } {
    -      move ⇒ x x´.
    -      rewrite interval_list_invariant_cons.
    -      move ⇒ [H_gr] [H_c_neq_0] H_inv_s´.
    -      have H_gr´ : (∀ xx : Z, InZ xx (s´) → y + Z.of_N c < xx). {
    -        apply interval_list_elements_greater_alt2_def ⇒ //.
    -      }
    -
    -      case s´ as [| [y´ c´] s´´]. {
    -        move ⇒ [<-].
    -        rewrite InZ_cons InZ_nil In_elementsZ_single.
    -        omega.
    -      } {
    -        move ⇒ H_max_eq.
    -        rewrite InZ_cons.
    -        move ⇒ []; last by apply IH.
    -        rewrite In_elementsZ_single.
    -        move ⇒ [_] H_x´_lt H_lt_x´.
    - -
    -        have H_x_in : InZ x ((y´, c´)::s´´). {
    -          by apply max_eltZ_spec1.
    -        }
    -        
    -        move : (H_gr´ _ H_x_in).
    -        apply Z.nlt_ge, Z.lt_le_incl.
    -        by apply Z.lt_trans with (m := x´).
    -      }
    -    }
    -  Qed.
    - +    âˆ€ (s : t) (x y : Z),
    +    interval_list_invariant s = true →
    +    max_eltZ s = Some x → InZ y s → ¬ Z.lt x y.
    +
      Lemma max_eltZ_eq_None :
        âˆ€ (s : t),
    -      max_eltZ s = None → s = nil.
    -  Proof.
    -    induction s as [| [x´ c] s´ IH] ⇒ //.
    -    move : IH.
    -    case s´ as [| [y´ c´] s´´] ⇒ //=.
    -    move ⇒ H H_pre.
    -    move : (H H_pre) ⇒ //.
    -  Qed.
    +      max_eltZ s = None → s = nil.

      Definition max_eltZ_spec3 :
        âˆ€ (s : t),
    -      max_eltZ s = None → ∀ x, ¬InZ x s.
    -  Proof.
    -    move ⇒ s /max_eltZ_eq_None → x /InZ_nil //.
    -  Qed.
    +      max_eltZ s = None → ∀ x, ¬InZ x s.

      Definition max_elt_spec1 :
    -    âˆ€ (s : t) (x : elt) (Hs : Ok s), max_elt s = Some x → In x s.
    -  Proof.
    -    rewrite /max_elt.
    -    move ⇒ s x H_ok.
    -    case_eq (max_eltZ s) ⇒ //.
    -    move ⇒ z H_max_elt [<-].
    -    apply InZ_In ⇒ //.
    -    apply max_eltZ_spec1 ⇒ //.
    -    apply H_ok.
    -  Qed.
    - +    âˆ€ (s : t) (x : elt) (Hs : Ok s), max_elt s = Some x → In x s.
    +
      Definition max_elt_spec2 :
    -    âˆ€ (s : t) (x y : elt) (Hs : Ok s), max_elt s = Some x → In y s → ~(Enc.E.lt x y).
    -  Proof.
    -    rewrite /max_elt.
    -    move ⇒ s x y H_ok.
    -    move : (H_ok) ⇒ [H_inv] _.
    -    case_eq (max_eltZ s) ⇒ //.
    -    move ⇒ z H_max_elt [<-].
    -    rewrite In_InZ ⇒ H_inZ.
    -    rewrite -Enc.encode_lt.
    -    apply (max_eltZ_spec2 _ _ _ H_inv) ⇒ //.
    -    suff → : Enc.encode (Enc.decode z) = z ⇒ //.
    -    apply encode_decode_eq with (s := s) ⇒ //.
    -    apply max_eltZ_spec1 ⇒ //.
    -  Qed.
    - +    âˆ€ (s : t) (x y : elt) (Hs : Ok s), max_elt s = Some x → In y s → ~(Enc.E.lt x y).
    +
      Definition max_elt_spec3 :
    -    âˆ€ s : t, max_elt s = None → Empty s.
    -  Proof.
    -    intro s.
    -    rewrite /max_elt /Empty.
    -    case_eq (max_eltZ s) ⇒ //.
    -    move ⇒ /max_eltZ_eq_None → _ x.
    -    rewrite /In elements_nil InA_nil //.
    -  Qed.
    +    âˆ€ s : t, max_elt s = None → Empty s.

    -

    fold specification

    +

    fold specification


      Lemma fold_spec :
    -   âˆ€ (s : t) (A : Type) (i : A) (f : elt → A → A),
    -   fold f s i = fold_left (flip f) (elements s) i.
    -  Proof.
    -    intros s A i f.
    -    rewrite /fold fold_elementsZ_alt_def /elements
    -            rev_map_alt_def -map_rev.
    -    move : i.
    -    generalize (rev (elementsZ s)).
    -    induction l as [| x xs IH]. {
    -      done.
    -    } {
    -      move ⇒ i.
    -      rewrite /= IH //.
    -    }
    -  Qed.
    - +   âˆ€ (s : t) (A : Type) (i : A) (f : elt → A → A),
    +   fold f s i = fold_left (flip f) (elements s) i.
    +

    -

    cardinal specification

    +

    cardinal specification


    -  Lemma cardinalN_spec : ∀ (s : t) (c : N),
    -    cardinalN c s = (c + N.of_nat (length (elements s)))%N.
    -  Proof.
    -    induction s as [| [x cx] xs IH]. {
    -      intros c.
    -      rewrite elements_nil /= N.add_0_r //.
    -    } {
    -      intros c.
    -      rewrite /= IH.
    -      rewrite /elements !rev_map_alt_def !rev_length !map_length.
    -      rewrite elementsZ_cons app_length Nat2N.inj_add rev_length.
    -      rewrite length_elementsZ_single N2Nat.id.
    -      ring.
    -    }
    -  Qed.
    - +  Lemma cardinalN_spec : ∀ (s : t) (c : N),
    +    cardinalN c s = (c + N.of_nat (length (elements s)))%N.
    +
      Lemma cardinal_spec :
       âˆ€ (s : t),
    -   cardinal s = length (elements s).
    -  Proof.
    -    intros s.
    -    rewrite /cardinal cardinalN_spec N.add_0_l Nat2N.id //.
    -  Qed.
    +   cardinal s = length (elements s).

    -

    for_all specification

    +

    for_all specification


      Lemma for_all_spec :
    -   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    -   Proper (Enc.E.eq==>eq) f →
    -   (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    -  Proof.
    -    intros s f Hs H.
    -    rewrite /for_all /For_all /In fold_elementsZ_alt_def
    -            /elements rev_map_alt_def -map_rev.
    -    generalize (rev (elementsZ s)).
    -    induction l as [| x xs IH]. {
    -      split ⇒ // _ x /= /InA_nil //.
    -    } {
    -      rewrite /=.
    -      case_eq (f (Enc.decode x)) ⇒ H_f_eq. {
    -        rewrite IH.
    -        split. {
    -          move ⇒ HH x´ /InA_cons []. {
    -            by move ⇒ →.
    -          } {
    -            apply HH.
    -          }
    -        } {
    -          move ⇒ HH x´ H_in.
    -          apply HH.
    -          apply InA_cons.
    -          by right.
    -        }
    -      } {
    -        split; move ⇒ HH. {
    -          contradict HH.
    -          case xs ⇒ //.
    -        } {
    -          exfalso.
    -          have H_in: (InA Enc.E.eq (Enc.decode x) (Enc.decode x :: map Enc.decode xs)). {
    -            apply InA_cons.
    -            left.
    -            apply Enc.E.eq_equiv.
    -          }
    -          move : (HH _ H_in).
    -          rewrite H_f_eq ⇒ //.
    -        }
    -      }
    -    }
    -  Qed.
    - +   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    +   Proper (Enc.E.eq==>eq) f →
    +   (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +
    -

    exists specification

    +

    exists specification


      Lemma exists_spec :
    -   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    -   Proper (Enc.E.eq==>eq) f →
    -   (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    -  Proof.
    -    intros s f Hs H.
    -    rewrite /exists_ /Exists /In fold_elementsZ_alt_def
    -            /elements rev_map_alt_def -map_rev.
    -    generalize (rev (elementsZ s)).
    -    induction l as [| x xs IH]. {
    -      split ⇒ //.
    -      move ⇒ [x] /= [] /InA_nil //.
    -    } {
    -      rewrite /=.
    -      case_eq (f (Enc.decode x)) ⇒ H_f_eq. {
    -        split ⇒ _. {
    -          âˆƒ (Enc.decode x).
    -          split ⇒ //.
    -          apply InA_cons.
    -          left.
    -          apply Enc.E.eq_equiv.
    -        } {
    -          case xs ⇒ //.
    -        }
    -      } {
    -        rewrite IH.
    -        split. {
    -          move ⇒ [x0] [H_in] H_f_x0.
    -          âˆƒ x0.
    -          split ⇒ //.
    -          apply InA_cons.
    -          by right.
    -        } {
    -          move ⇒ [x0] [] /InA_cons H_in H_f_x0.
    -          âˆƒ x0.
    -          split ⇒ //.
    -          move : H_in ⇒ [] // H_in.
    -          contradict H_f_x0.
    -          rewrite H_in H_f_eq //.
    -        }
    -      }
    -    }
    -  Qed.
    - +   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    +   Proper (Enc.E.eq==>eq) f →
    +   (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +
    -

    filter specification

    +

    filter specification


    -  Definition partitionZ_aux_invariant (x : Z) acc c :=
    -    interval_list_invariant (List.rev (partitionZ_fold_skip acc c)) = true ∧
    +  Definition partitionZ_aux_invariant (x : Z) acc c :=
    +    interval_list_invariant (List.rev (partitionZ_fold_skip acc c)) = true ∧
        match c with
    -      None ⇒ (∀ y´, InZ y´ acc → Z.succ y´ < x)
    -    | Some (y, c´) ⇒ (x = y + Z.of_N c´)
    +      None ⇒ (∀ y', InZ y' acc → Z.succ y' < x)
    +    | Some (y, c') ⇒ (x = y + Z.of_N c')
        end.

      Lemma partitionZ_aux_invariant_insert : ∀ x acc c,
    -    partitionZ_aux_invariant x acc c →
    -    partitionZ_aux_invariant (Z.succ x) acc
    -      (Some (partitionZ_fold_insert c x)).
    -  Proof.
    -    intros x acc c.
    -    rewrite /partitionZ_fold_insert /partitionZ_aux_invariant
    -            /partitionZ_fold_skip.
    -    case c; last first. {
    -      move ⇒ [H_inv] H_in.
    -      rewrite /= interval_list_invariant_app_iff Z.add_1_r.
    -      split; last done.
    -      split; first done.
    -      split; first done.
    -      move ⇒ x1 x2.
    -      rewrite InZ_rev InZ_cons InZ_nil In_elementsZ_single1.
    -      move ⇒ H_x1_in [] // <-.
    -      by apply H_in.
    -    } {
    -      move ⇒ [y c´].
    -      rewrite /= !interval_list_invariant_app_iff
    -        N2Z.inj_succ Z.add_succ_r .
    -      rewrite !interval_list_invariant_cons !interval_list_invariant_nil.
    -      move ⇒ [] [H_inv_acc] [] [] _ [H_c_neq_0] _
    -        H_in_c →.
    -      split; last done.
    -      split; first done.
    -      split. {
    -        split; first done.
    -        split; last done.
    -        apply N.neq_succ_0.
    -      } {
    -        move ⇒ x1 x2.
    -        rewrite InZ_cons InZ_nil In_elementsZ_single.
    -        move ⇒ H_x1_in [] // [H_y_le] H_x2_lt.
    -        apply Z.lt_le_trans with (m := y) ⇒ //.
    -        apply H_in_c ⇒ //.
    -        rewrite InZ_cons In_elementsZ_single.
    -        left.
    -        split. {
    -          apply Z.le_refl.
    -        } {
    -          by apply Z_lt_add_r.
    -        }
    -      }
    -    }
    -  Qed.
    - +    partitionZ_aux_invariant x acc c →
    +    partitionZ_aux_invariant (Z.succ x) acc
    +      (Some (partitionZ_fold_insert c x)).
    +
      Lemma partitionZ_aux_invariant_skip : ∀ x acc c,
    -    partitionZ_aux_invariant x acc c →
    -    partitionZ_aux_invariant (Z.succ x) (partitionZ_fold_skip acc c) None.
    -  Proof.
    -    intros x acc c.
    -    rewrite /partitionZ_fold_skip /partitionZ_aux_invariant
    -            /partitionZ_fold_skip.
    -    case c; last first. {
    -      move ⇒ [H_inv] H_in.
    -      split; first done.
    -      move ⇒ y´ H_y´_in.
    -      apply Z.lt_trans with (m := x). {
    -        by apply H_in.
    -      } {
    -        apply Z.lt_succ_diag_r.
    -      }
    -    } {
    -      move ⇒ [y c´] [H_inv] →.
    -      split ⇒ //.
    -      move ⇒ y´.
    -      rewrite InZ_cons In_elementsZ_single.
    -      move ⇒ []. {
    -        move ⇒ [_].
    -        rewrite -Z.succ_lt_mono //.
    -      } {
    -        move : H_inv.
    -        rewrite /= !interval_list_invariant_app_iff interval_list_invariant_cons.
    -        move ⇒ [_] [] [_] [H_c´_neq] _ H_pre H_y´_in.
    -        apply Z.lt_trans with (m := y). {
    -          apply H_pre. {
    -            by rewrite InZ_rev.
    -          } {
    -            rewrite InZ_cons.
    -            left.
    -            by apply In_elementsZ_single_hd.
    -          }
    -        }
    -        apply Z.lt_succ_r, Z_le_add_r.
    -      }
    -    }
    -  Qed.
    - -
    -  Definition partitionZ_fold_current (c : option (Z × N)) :=
    +    partitionZ_aux_invariant x acc c →
    +    partitionZ_aux_invariant (Z.succ x) (partitionZ_fold_skip acc c) None.
    + +
    +  Definition partitionZ_fold_current (c : option (Z × N)) :=
        match c with
    -       None ⇒ nil
    -     | Some yc ⇒ yc::nil
    +       None ⇒ nil
    +     | Some yc ⇒ yc::nil
        end.

      Lemma InZ_partitionZ_fold_current_Some : ∀ yc y,
    -     InZ y (partitionZ_fold_current (Some yc)) ↔
    -     InZ y (yc :: nil).
    -  Proof. done. Qed.
    - +     InZ y (partitionZ_fold_current (Some yc)) ↔
    +     InZ y (yc :: nil).
    +
      Lemma InZ_partitionZ_fold_insert : ∀ c x y l,
       match c with
    -   | Some (y, c´) ⇒ x = y + Z.of_N c´
    -   | None ⇒ True
    -   end → (
    -   InZ y (partitionZ_fold_insert c x :: l) ↔
    -    ((x = y) ∨ InZ y (partitionZ_fold_current c) ∨
    -       InZ y l)).
    -  Proof.
    -    intros c x y l.
    -    rewrite /partitionZ_fold_insert /partitionZ_fold_current
    -            /partitionZ_fold_skip.
    -    case c. {
    -      move ⇒ [y´ c´] →.
    -      rewrite !InZ_cons elementsZ_single_succ in_app_iff
    -              InZ_nil /=.
    -      tauto.
    -    } {
    -      rewrite InZ_cons InZ_nil In_elementsZ_single1.
    -      tauto.
    -    }
    -  Qed.
    - +   | Some (y, c') ⇒ x = y + Z.of_N c'
    +   | None ⇒ True
    +   end → (
    +   InZ y (partitionZ_fold_insert c x :: l) ↔
    +    ((x = y) ∨ InZ y (partitionZ_fold_current c) ∨
    +       InZ y l)).
    +
      Lemma InZ_partitionZ_fold_skip : ∀ c acc y,
    -    InZ y (partitionZ_fold_skip acc c) ↔
    -    (InZ y (partitionZ_fold_current c) ∨ InZ y acc).
    -  Proof.
    -    intros c acc y.
    -    rewrite /partitionZ_fold_skip /partitionZ_fold_current
    -            /partitionZ_fold_skip.
    -    case c. {
    -      move ⇒ [y´ c´].
    -      rewrite !InZ_cons InZ_nil /=.
    -      tauto.
    -    } {
    -      rewrite InZ_nil.
    -      tauto.
    -    }
    -  Qed.
    - +    InZ y (partitionZ_fold_skip acc c) ↔
    +    (InZ y (partitionZ_fold_current c) ∨ InZ y acc).
    +
      Lemma filterZ_single_aux_props :
        âˆ€ f c x acc cur,
    -      partitionZ_aux_invariant x acc cur →
    -      match (filterZ_single_aux f (acc, cur) x c) with
    -        (acc´, c´) ⇒
    -        let r := partitionZ_fold_skip acc´ c´ in
    -        interval_list_invariant (List.rev r) = true ∧
    -        (∀ y´, InZ y´ r ↔ (InZ y´ (partitionZ_fold_skip acc cur) ∨
    -                                    (f y´ = true ∧ List.In y´ (elementsZ_single x c))))
    +      partitionZ_aux_invariant x acc cur →
    +      match (filterZ_single_aux f (acc, cur) x c) with
    +        (acc', c') ⇒
    +        let r := partitionZ_fold_skip acc' c' in
    +        interval_list_invariant (List.rev r) = true ∧
    +        (∀ y', InZ y' r ↔ (InZ y' (partitionZ_fold_skip acc cur) ∨
    +                                    (f y' = true ∧ List.In y' (elementsZ_single x c))))

          end.
    -  Proof.
    -    intro f.
    -    induction c as [| c´ IH] using N.peano_ind. {
    -      intros x acc cur.
    -      rewrite /partitionZ_aux_invariant.
    -      move ⇒ [H_inv] _.
    -      rewrite /filterZ_single_aux fold_elementsZ_single_zero /=.
    -      tauto.
    -    }
    -    intros x acc cur H_inv.
    -    have → : filterZ_single_aux f (acc, cur) x (N.succ c´) =
    -               filterZ_single_aux f (filterZ_fold_fun f (acc, cur) x) (Z.succ x) c´. {
    -        by rewrite /filterZ_single_aux fold_elementsZ_single_succ.
    -    }
    -    case_eq (filterZ_fold_fun f (acc, cur) x).
    -    move ⇒ acc´ cur´ H_fold_eq.
    - -
    -    case_eq (filterZ_single_aux f (acc´, cur´) (Z.succ x) c´).
    -    move ⇒ acc´´ cur´´ H_succ_eq.
    - -
    -    have H_inv´ : partitionZ_aux_invariant (Z.succ x) acc´ cur´. {
    -      move : H_fold_eq H_inv.
    -      rewrite /filterZ_fold_fun.
    -      case (f x); move ⇒ [<-] <-. {
    -        apply partitionZ_aux_invariant_insert.
    -      } {
    -        apply partitionZ_aux_invariant_skip.
    -      }
    -    }
    -   
    -    move : (IH (Z.succ x) acc´ cur´ H_inv´) ⇒ {IH}.
    -    rewrite H_succ_eq /=.
    -    set r := partitionZ_fold_skip acc´´ cur´´.
    -    move ⇒ [H_inv_r] H_in_r.
    -    split; first assumption.
    - -
    -    move ⇒ y´.
    -    move : H_fold_eq.
    -    rewrite H_in_r /filterZ_fold_fun.
    -    case_eq (f x) ⇒ H_fx [<-] <-. {
    -      rewrite InZ_partitionZ_fold_skip InZ_partitionZ_fold_current_Some InZ_partitionZ_fold_skip elementsZ_single_succ_front.
    -      rewrite InZ_partitionZ_fold_insert; last first. {
    -        move : H_inv.
    -        rewrite /partitionZ_aux_invariant ⇒ [[_]].
    -        case cur ⇒ //.
    -      }
    -      rewrite InZ_nil /=.
    -      split; last by tauto.
    -      move ⇒ []; last by tauto.
    -      move ⇒ []; last by tauto.
    -      move ⇒ []. {
    -        move ⇒ <-.
    -        tauto.
    -      } {
    -        tauto.
    -      }
    -    } {
    -      rewrite InZ_partitionZ_fold_skip /partitionZ_fold_current InZ_partitionZ_fold_skip elementsZ_single_succ_front !InZ_nil /=.
    -      split; first by tauto.
    -      move ⇒ []; first by tauto.
    -      move ⇒ [] H_fy´ []. {
    -        move ⇒ H_x_eq; subst y´.
    -        contradict H_fy´.
    -        by rewrite H_fx.
    -      } {
    -        tauto.
    -      }
    -    }
    -  Qed.
    - +
      Lemma filterZ_single_props :
        âˆ€ f c x acc,
    -      interval_list_invariant (rev acc) = true →
    -      (∀ y´ : Z, InZ y´ acc → Z.succ y´ < x) →
    +      interval_list_invariant (rev acc) = true →
    +      (∀ y' : Z, InZ y' acc → Z.succ y' < x) →
          match (filterZ_single f acc x c) with
            r ⇒
    -        interval_list_invariant (List.rev r) = true ∧
    -        (∀ y´, InZ y´ r ↔ (InZ y´ acc ∨
    -                                 (f y´ = true ∧ List.In y´ (elementsZ_single x c))))
    +        interval_list_invariant (List.rev r) = true ∧
    +        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    +                                 (f y' = true ∧ List.In y' (elementsZ_single x c))))

          end.
    -  Proof.
    -    intros f c x acc.
    -    move ⇒ H_inv H_acc.
    -    rewrite /filterZ_single.
    - -
    -    have H_inv´ : partitionZ_aux_invariant x acc None. {
    -      by rewrite /partitionZ_aux_invariant /=.
    -    }
    -    move : (filterZ_single_aux_props f c x acc None H_inv´).
    -    case_eq (filterZ_single_aux f (acc, None) x c).
    -    move ⇒ acc´ cur´ /= H_res.
    -    tauto.
    -  Qed.


      Lemma filterZ_aux_props :
        âˆ€ f s acc,
    -      interval_list_invariant s = true →
    -      interval_list_invariant (rev acc) = true →
    -      (∀ x1 x2 : Z, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    +      interval_list_invariant s = true →
    +      interval_list_invariant (rev acc) = true →
    +      (∀ x1 x2 : Z, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
          match (filterZ_aux acc f s) with
            r ⇒
    -        interval_list_invariant r = true ∧
    -        (∀ y´, InZ y´ r ↔ (InZ y´ acc ∨
    -                                 (f y´ = true ∧ InZ y´ s)))
    +        interval_list_invariant r = true ∧
    +        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    +                                 (f y' = true ∧ InZ y' s)))
          end.
    -  Proof.
    -    intro f.
    -    induction s as [| [y c] s´ IH]. {
    -      intros acc.
    -      move ⇒ _ H_inv _.
    -      rewrite /filterZ_aux.
    -      split; first assumption.
    -      move ⇒ y´; rewrite InZ_rev InZ_nil; tauto.
    -    } {
    -      intros acc.
    -      rewrite interval_list_invariant_cons.
    -      move ⇒ [H_gr] [H_c_neq_0] H_inv_s´ H_inv H_in_acc /=.
    -      move : H_gr.
    -      rewrite interval_list_elements_greater_alt2_def ⇒ // H_gr.
    - -
    -      have H_pre : (∀ y´ : Z, InZ y´ acc → Z.succ y´ < y). {
    -        move ⇒ x1 H_x1_in.
    -        apply H_in_acc ⇒ //.
    -        rewrite InZ_cons.
    -        by left; apply In_elementsZ_single_hd.
    -      }
    -      
    -      move : (filterZ_single_props f c y acc H_inv H_pre) ⇒ {H_pre}.
    -      set acc´ := filterZ_single f acc y c.
    -      move ⇒ [H_inv´] H_in_acc´.
    - -
    -      have H_pre : (∀ x1 x2 : Z,
    -                      InZ x1 acc´ → InZ x2 s´ → Z.succ x1 < x2). {
    -        move ⇒ x1 x2.
    -        rewrite H_in_acc´ In_elementsZ_single.
    -        move ⇒ []. {
    -          move ⇒ H_x1_in H_x2_in.
    -          apply H_in_acc ⇒ //.
    -          rewrite InZ_cons.
    -          by right.
    -        } {
    -          move ⇒ [_] [_] H_x1_lt H_x2_in.
    -          apply Z.le_lt_trans with (m := y + Z.of_N c).
    -            - by apply Z.le_succ_l.
    -            - by apply H_gr.
    -        }
    -      }
    -      move : (IH acc´ H_inv_s´ H_inv´ H_pre) ⇒ {H_pre}.
    - -
    -      move ⇒ [H_inv_r] H_in_r.
    -      split; first assumption.
    -      move ⇒ y´.
    -      rewrite H_in_r H_in_acc´ InZ_cons.
    -      tauto.
    -    }
    -  Qed.
    - +
      Lemma filterZ_props :
        âˆ€ f s,
    -      interval_list_invariant s = true →
    +      interval_list_invariant s = true →
          match (filterZ f s) with r ⇒
    -        interval_list_invariant r = true ∧
    -        (∀ y´, InZ y´ r ↔ (f y´ = true ∧ InZ y´ s))
    +        interval_list_invariant r = true ∧
    +        (∀ y', InZ y' r ↔ (f y' = true ∧ InZ y' s))
          end.
    -  Proof.
    -    intros f s H_inv_s.
    -    rewrite /filterZ.
    - -
    -    have H_pre_1 : interval_list_invariant (rev nil) = true by done.
    -    have H_pre_2 : (∀ x1 x2 : Z, InZ x1 nil → InZ x2 s → Z.succ x1 < x2) by done.
    - -
    -    move : (filterZ_aux_props f s nil H_inv_s H_pre_1 H_pre_2) ⇒ {H_pre_1} {H_pre_2}.
    -    move ⇒ [H_inv´] H_in_r.
    -    split; first assumption.
    -    move ⇒ y´.
    -    rewrite H_in_r InZ_nil.
    -    tauto.
    -  Qed.

      Global Instance filter_ok s f : ∀ `(Ok s), Ok (filter f s).
    -  Proof.
    -    move ⇒ [H_inv H_enc].
    -    rewrite /filter.
    -    set f´ := (fun z : Z ⇒ f (Enc.decode z)).
    -    move : (filterZ_props f´ s H_inv).
    -    move ⇒ [H_inv´] H_in_r.
    -    rewrite /Ok /IsOk /is_encoded_elems_list.
    -    split; first assumption.
    -    move ⇒ x /H_in_r [_] H_x_in.
    -    by apply H_enc.
    -  Qed.

      Lemma filter_spec :
    -   âˆ€ (s : t) (x : elt) (f : elt → bool),
    -   Ok s →
    -   (In x (filter f s) ↔ In x s ∧ f x = true).
    -  Proof.
    -    move ⇒ s x f H_ok.
    -    suff H_suff :
    -      (∀ x, (InZ x (filter f s)) ↔
    -                 InZ x s ∧ f (Enc.decode x) = true). {
    -      rewrite !In_alt_def /elements !rev_map_alt_def
    -              -!in_rev !in_map_iff.
    -      setoid_rewrite H_suff.
    -      rewrite /InZ.
    -      split. {
    -        move ⇒ [y] [<-] [?] ?.
    -        split ⇒ //.
    -        by ∃ y.
    -      } {
    -        move ⇒ [] [y] [<-] ? ?.
    -        by ∃ y.
    -      }
    -    }
    -    rewrite /filter.
    -    set f´ := (fun z : Z ⇒ f (Enc.decode z)).
    -    move : (H_ok) ⇒ [H_inv _].
    -    move : (filterZ_props f´ s H_inv).
    -    move ⇒ [H_inv´] H_in_r.
    -    move ⇒ y; rewrite H_in_r; tauto.
    -  Qed.
    - +   âˆ€ (s : t) (x : elt) (f : elt → bool),
    +   Ok s →
    +   (In x (filter f s) ↔ In x s ∧ f x = true).
    +
    -

    partition specification

    +

    partition specification


      Lemma partitionZ_single_aux_alt_def : ∀ f c y acc_t c_t acc_f c_f,
    -    partitionZ_single_aux f ((acc_t, c_t), (acc_f, c_f)) y c =
    -    (filterZ_single_aux f (acc_t, c_t) y c,
    -     filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y c).
    -  Proof.
    -    intros f.
    -    rewrite /partitionZ_single_aux /filterZ_single_aux.
    -    induction c as [| c´ IH] using N.peano_ind. {
    -      intros y acc_t c_t acc_f c_f.
    -      rewrite !fold_elementsZ_single_zero //. } {
    -      intros y acc_t c_t acc_f c_f.
    -      rewrite !fold_elementsZ_single_succ.
    -      case_eq (partitionZ_fold_fun f (acc_t, c_t, (acc_f, c_f)) y) ⇒ [] [acc_t´ c_t´] [acc_f´ c_f´] H_fold_eq.
    -      rewrite IH ⇒ {IH}.
    -      suff : (filterZ_fold_fun f (acc_t, c_t) y = (acc_t´, c_t´)) ∧
    -             (filterZ_fold_fun (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y = (acc_f´, c_f´)). {
    -        move ⇒ [->] → //.
    -      }
    -      move : H_fold_eq.
    -      rewrite /partitionZ_fold_fun /filterZ_fold_fun.
    -      case (f y); move ⇒ [<-] <- <- <- //.
    -    }
    -  Qed.
    +    partitionZ_single_aux f ((acc_t, c_t), (acc_f, c_f)) y c =
    +    (filterZ_single_aux f (acc_t, c_t) y c,
    +     filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y c).

      Lemma partitionZ_aux_alt_def : ∀ f s acc_t acc_f,
    -   partitionZ_aux acc_t acc_f f s =
    -   (filterZ_aux acc_t f s,
    -    filterZ_aux acc_f (fun x : Z ⇒ negb (f x)) s).
    -  Proof.
    -    intros f.
    -    induction s as [| [y c] s´ IH]. {
    -      done.
    -    } {
    -      intros acc_t acc_f.
    -      rewrite /= /partitionZ_single /filterZ_single
    -              partitionZ_single_aux_alt_def.
    -      case (filterZ_single_aux f (acc_t, None) y c) ⇒ acc_t´ c_t´.
    -      case (filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, None) y c) ⇒ acc_f´ c_f´.
    -      rewrite IH //.
    -    }
    -  Qed.
    - +   partitionZ_aux acc_t acc_f f s =
    +   (filterZ_aux acc_t f s,
    +    filterZ_aux acc_f (fun x : Z ⇒ negb (f x)) s).
    +
      Lemma partitionZ_alt_def : ∀ f s,
    -    partitionZ f s = (filterZ f s,
    -                      filterZ (fun x ⇒ negb (f x)) s).
    -  Proof.
    -    intros f s.
    -    rewrite /partitionZ /filterZ
    -            partitionZ_aux_alt_def //.
    -  Qed.
    +    partitionZ f s = (filterZ f s,
    +                      filterZ (fun x ⇒ negb (f x)) s).

      Lemma partition_alt_def : ∀ f s,
    -    partition f s = (filter f s,
    -                     filter (fun x ⇒ negb (f x)) s).
    -  Proof.
    -    intros f s.
    -    rewrite /partition /filter partitionZ_alt_def.
    -    done.
    -  Qed.
    +    partition f s = (filter f s,
    +                     filter (fun x ⇒ negb (f x)) s).

    -  Global Instance partition_ok1 s f : ∀ `(Ok s), Ok (fst (partition f s)).
    -  Proof.
    -    move ⇒ H_ok.
    -    rewrite partition_alt_def /fst.
    -    by apply filter_ok.
    -  Qed.
    +  Global Instance partition_ok1 s f : ∀ `(Ok s), Ok (fst (partition f s)).

    -  Global Instance partition_ok2 s f : ∀ `(Ok s), Ok (snd (partition f s)).
    -  Proof.
    -    move ⇒ H_ok.
    -    rewrite partition_alt_def /snd.
    -    by apply filter_ok.
    -  Qed.
    +  Global Instance partition_ok2 s f : ∀ `(Ok s), Ok (snd (partition f s)).

      Lemma partition_spec1 :
    -   âˆ€ (s : t) (f : elt → bool),
    -   Equal (fst (partition f s)) (filter f s).
    -  Proof.
    -    intros s f.
    -    rewrite partition_alt_def /fst /Equal //.
    -  Qed.
    +   âˆ€ (s : t) (f : elt → bool),
    +   Equal (fst (partition f s)) (filter f s).

      Lemma partition_spec2 :
    -   âˆ€ (s : t) (f : elt → bool),
    -   Ok s →
    -   Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
    -  Proof.
    -    intros s f.
    -    rewrite partition_alt_def /snd /Equal //.
    -  Qed.
    +   âˆ€ (s : t) (f : elt → bool),
    +   Ok s →
    +   Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).

    End Raw.
    @@ -6335,7 +2282,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Main Module

    +

    Main Module

    @@ -6346,217 +2293,123 @@

    Library MSetsExtra.MSetIntervals


    -Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E.
    +Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E.
      Module E := Enc.E.
      Module Raw := Raw Enc.

    - Local Unset Elimination Schemes.
    - Local Unset Case Analysis Schemes.

     Definition elt := Raw.elt.
     Record t_ := Mkt {this :> Raw.t; is_ok : Raw.Ok this}.
     Definition t := t_.
    - Arguments Mkt this {is_ok}.
    - Hint Resolve is_ok : typeclass_instances.
    + Hint Resolve is_ok : typeclass_instances.

     Definition In (x : elt)(s : t) := Raw.In x s.(this).
    - Definition Equal (s s´ : t) := ∀ a : elt, In a s ↔ In a s´.
    - Definition Subset (s s´ : t) := ∀ a : elt, In a s → In a s´.
    - Definition Empty (s : t) := ∀ a : elt, ¬ In a s.
    - Definition For_all (P : elt → Prop)(s : t) := ∀ x, In x s → P x.
    - Definition Exists (P : elt → Prop)(s : t) := ∃ x, In x s ∧ P x.
    + Definition Equal (s s' : t) := ∀ a : elt, In a s ↔ In a s'.
    + Definition Subset (s s' : t) := ∀ a : elt, In a s → In a s'.
    + Definition Empty (s : t) := ∀ a : elt, ¬ In a s.
    + Definition For_all (P : elt → Prop)(s : t) := ∀ x, In x s → P x.
    + Definition Exists (P : elt → Prop)(s : t) := ∃ x, In x s ∧ P x.

     Definition mem (x : elt)(s : t) := Raw.mem x s.(this).
     Definition add (x : elt)(s : t) : t := Mkt (Raw.add x s.(this)).
     Definition remove (x : elt)(s : t) : t := Mkt (Raw.remove x s.(this)).
     Definition singleton (x : elt) : t := Mkt (Raw.singleton x).
    - Definition union (s s´ : t) : t := Mkt (Raw.union s s´).
    - Definition inter (s s´ : t) : t := Mkt (Raw.inter s s´).
    - Definition diff (s s´ : t) : t := Mkt (Raw.diff s s´).
    - Definition equal (s s´ : t) := Raw.equal s s´.
    - Definition subset (s s´ : t) := Raw.subset s s´.(this).
    + Definition union (s s' : t) : t := Mkt (Raw.union s s').
    + Definition inter (s s' : t) : t := Mkt (Raw.inter s s').
    + Definition diff (s s' : t) : t := Mkt (Raw.diff s s').
    + Definition equal (s s' : t) := Raw.equal s s'.
    + Definition subset (s s' : t) := Raw.subset s s'.(this).
     Definition empty : t := Mkt Raw.empty.
     Definition is_empty (s : t) := Raw.is_empty s.
    - Definition elements (s : t) : list elt := Raw.elements s.
    - Definition min_elt (s : t) : option elt := Raw.min_elt s.
    - Definition max_elt (s : t) : option elt := Raw.max_elt s.
    - Definition choose (s : t) : option elt := Raw.choose s.
    - Definition compare (s1 s2 : t) : comparison := Raw.compare s1 s2.
    - Definition fold {A : Type}(f : elt → A → A)(s : t) : A → A := Raw.fold f s.
    + Definition elements (s : t) : list elt := Raw.elements s.
    + Definition min_elt (s : t) : option elt := Raw.min_elt s.
    + Definition max_elt (s : t) : option elt := Raw.max_elt s.
    + Definition choose (s : t) : option elt := Raw.choose s.
    + Definition compare (s1 s2 : t) : comparison := Raw.compare s1 s2.
    + Definition fold {A : Type}(f : elt → A → A)(s : t) : A → A := Raw.fold f s.
     Definition cardinal (s : t) := Raw.cardinal s.
    - Definition filter (f : elt → bool)(s : t) : t := Mkt (Raw.filter f s).
    - Definition for_all (f : elt → bool)(s : t) := Raw.for_all f s.
    - Definition exists_ (f : elt → bool)(s : t) := Raw.exists_ f s.
    - Definition partition (f : elt → bool)(s : t) : t × t :=
    -   let p := Raw.partition f s in (Mkt (fst p), Mkt (snd p)).
    - -
    - Instance In_compat : Proper (E.eq==>eq==>iff) In.
    - Proof.
    -   repeat red.
    -   move ⇒ x y H_eq_xy x´ y´ →.
    -   rewrite /In /Raw.In.
    -   setoid_rewrite H_eq_xy.
    -   done.
    - Qed.
    - -
    - Definition eq : t → t → Prop := Equal.
    - -
    - Instance eq_equiv : Equivalence eq.
    - Proof. firstorder. Qed.
    - -
    - Definition eq_dec : ∀ (s s´:t), { eq s s´ }+{ ¬eq s s´ }.
    - Proof.
    -  intros (s,Hs) (s´,Hs´).
    -  change ({Raw.Equal s s´}+{¬Raw.Equal s s´}).
    -  destruct (Raw.equal s s´) eqn:H; [left|right];
    -   rewrite <- Raw.equal_spec; congruence.
    - Defined.
    - -
    - Definition lt : t → t → Prop := Raw.lt.
    - -
    - Instance lt_strorder : StrictOrder lt.
    - Proof.
    -   unfold lt.
    -   constructor. {
    -     move : Raw.lt_Irreflexive.
    -     rewrite /Irreflexive /complement /Reflexive.
    -     move ⇒ H x.
    -     apply H.
    -   } {
    -     move : Raw.lt_Transitive.
    -     rewrite /Transitive.
    -     move ⇒ H x y z.
    -     apply H.
    -   }
    - Qed.
    - -
    - Instance lt_compat : Proper (eq==>eq==>iff) lt.
    - Proof.
    -   repeat red.
    -   move ⇒ [x1 H_x1_ok] [y1 H_y1_ok] H_eq.
    -   move ⇒ [x2 H_x2_ok] [y2 H_y2_ok].
    -   move : H_eq.
    -   rewrite /eq /lt /Equal /In /=.
    -   replace (∀ a : elt, Raw.In a x1 ↔ Raw.In a y1) with
    -     (Raw.Equal x1 y1) by done.
    -   replace (∀ a : elt, Raw.In a x2 ↔ Raw.In a y2) with
    -     (Raw.Equal x2 y2) by done.
    -   rewrite -!Raw.equal_spec !Raw.equal_alt_def.
    -   move ⇒ → → //.
    - Qed.
    + Definition filter (f : elt → bool)(s : t) : t := Mkt (Raw.filter f s).
    + Definition for_all (f : elt → bool)(s : t) := Raw.for_all f s.
    + Definition exists_ (f : elt → bool)(s : t) := Raw.exists_ f s.
    + Definition partition (f : elt → bool)(s : t) : t × t :=
    +   let p := Raw.partition f s in (Mkt (fst p), Mkt (snd p)).
    + +
    + Instance In_compat : Proper (E.eq==>eq==>iff) In.
    + +
    + Definition eq : t → t → Prop := Equal.
    + +
    + Instance eq_equiv : Equivalence eq.
    + +
    + Definition eq_dec : ∀ (s s':t), { eq s s' }+{ ¬eq s s' }.
    + +
    + Definition lt : t → t → Prop := Raw.lt.
    + +
    + Instance lt_strorder : StrictOrder lt.
    + +
    + Instance lt_compat : Proper (eq==>eq==>iff) lt.

     Section Spec.
    -  Variable s s´ : t.
    +  Variable s s' : t.
      Variable x y : elt.
    -  Variable f : elt → bool.
    -  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    - -
    -  Lemma mem_spec : mem x s = true ↔ In x s.
    -  Proof. exact (Raw.mem_spec _ _ _). Qed.
    -  Lemma equal_spec : equal s s´ = true ↔ Equal s s´.
    -  Proof. rewrite Raw.equal_spec //. Qed.
    -  Lemma subset_spec : subset s s´ = true ↔ Subset s s´.
    -  Proof. exact (Raw.subset_spec _ _ _ _). Qed.
    -  Lemma empty_spec : Empty empty.
    -  Proof. exact Raw.empty_spec. Qed.
    -  Lemma is_empty_spec : is_empty s = true ↔ Empty s.
    -  Proof. rewrite Raw.is_empty_spec //. Qed.
    -  Lemma add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    -  Proof. exact (Raw.add_spec _ _ _ _). Qed.
    -  Lemma remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    -  Proof. exact (Raw.remove_spec _ _ _ _). Qed.
    -  Lemma singleton_spec : In y (singleton x) ↔ E.eq y x.
    -  Proof. exact (Raw.singleton_spec _ _). Qed.
    -  Lemma union_spec : In x (union s s´) ↔ In x s ∨ In x s´.
    -  Proof. exact (Raw.union_spec _ _ _ _ _). Qed.
    -  Lemma inter_spec : In x (inter s s´) ↔ In x s ∧ In x s´.
    -  Proof. exact (Raw.inter_spec _ _ _ _ _). Qed.
    -  Lemma diff_spec : In x (diff s s´) ↔ In x s ∧ ¬In x s´.
    -  Proof. exact (Raw.diff_spec _ _ _ _ _). Qed.
    -  Lemma fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    -      fold f s i = fold_left (fun a e ⇒ f e a) (elements s) i.
    -  Proof. exact (@Raw.fold_spec _). Qed.
    -  Lemma cardinal_spec : cardinal s = length (elements s).
    -  Proof. exact (@Raw.cardinal_spec s). Qed.
    -  Lemma filter_spec : compatb f →
    -    (In x (filter f s) ↔ In x s ∧ f x = true).
    -  Proof. move ⇒ _; exact (@Raw.filter_spec _ _ _ _). Qed.
    -  Lemma for_all_spec : compatb f →
    -    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    -  Proof. exact (@Raw.for_all_spec _ _ _). Qed.
    -  Lemma exists_spec : compatb f →
    -    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    -  Proof. exact (@Raw.exists_spec _ _ _). Qed.
    -  Lemma partition_spec1 : compatb f → Equal (fst (partition f s)) (filter f s).
    -  Proof. move ⇒ _; exact (@Raw.partition_spec1 _ _). Qed.
    -  Lemma partition_spec2 : compatb f →
    -      Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
    -  Proof. move ⇒ _; exact (@Raw.partition_spec2 _ _ _). Qed.
    -  Lemma elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    -  Proof. rewrite /In /Raw.In /elements //. Qed.
    -  Lemma elements_spec2w : NoDupA E.eq (elements s).
    -  Proof. exact (Raw.elements_spec2w _ _). Qed.
    -  Lemma elements_spec2 : sort E.lt (elements s).
    -  Proof. exact (Raw.elements_sorted _ _). Qed.
    -  Lemma choose_spec1 : choose s = Some x → In x s.
    -  Proof. exact (Raw.choose_spec1 _ _ _). Qed.
    -  Lemma choose_spec2 : choose s = None → Empty s.
    -  Proof. exact (Raw.choose_spec2 _). Qed.
    -  Lemma choose_spec3 : choose s = Some x → choose s´ = Some y →
    -    Equal s s´ → E.eq x y.
    -  Proof.
    -    intros H1 H2 H3.
    -    suff → : x = y. {
    -      apply E.eq_equiv.
    -    }
    -    move : H1 H2 H3.
    -    exact (Raw.choose_spec3 _ _ _ _ _ _).
    -  Qed.
    - -
    -  Lemma min_elt_spec1 : choose s = Some x → In x s.
    -  Proof. exact (Raw.min_elt_spec1 _ _ _). Qed.
    -  Lemma min_elt_spec2 : min_elt s = Some x → In y s → ¬ E.lt y x.
    -  Proof. exact (Raw.min_elt_spec2 _ _ _ _). Qed.
    -  Lemma min_elt_spec3 : choose s = None → Empty s.
    -  Proof. exact (Raw.min_elt_spec3 _). Qed.
    - -
    -  Lemma max_elt_spec1 : max_elt s = Some x → In x s.
    -  Proof. exact (Raw.max_elt_spec1 _ _ _). Qed.
    -  Lemma max_elt_spec2 : max_elt s = Some x → In y s → ¬ E.lt x y.
    -  Proof. exact (Raw.max_elt_spec2 _ _ _ _). Qed.
    -  Lemma max_elt_spec3 : max_elt s = None → Empty s.
    -  Proof. exact (Raw.max_elt_spec3 _). Qed.
    - -
    -  Lemma compare_spec : CompSpec eq lt s s´ (compare s s´).
    -  Proof.
    -    generalize s s´.
    -    move ⇒ [s1 H_ok_s1] [s2 H_ok_s2].
    -    move : (Raw.compare_spec s1 s2).
    -    rewrite /CompSpec /eq /Equal /In /lt /compare /=.
    -    replace (∀ a : elt, Raw.In a s1 ↔ Raw.In a s2) with
    -    (Raw.Equal s1 s2) by done.
    -    suff H_eq : (Raw.Equal s1 s2) ↔ (s1 = s2). {
    -      move ⇒ [] H; constructor ⇒ //.
    -      by rewrite H_eq.
    -    }
    -    rewrite -Raw.equal_spec Raw.equal_alt_def //.
    -  Qed.
    +  Variable f : elt → bool.
    +  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    + +
    +  Lemma mem_spec : mem x s = true ↔ In x s.
    +   Lemma equal_spec : equal s s' = true ↔ Equal s s'.
    +   Lemma subset_spec : subset s s' = true ↔ Subset s s'.
    +   Lemma empty_spec : Empty empty.
    +   Lemma is_empty_spec : is_empty s = true ↔ Empty s.
    +   Lemma add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    +   Lemma remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    +   Lemma singleton_spec : In y (singleton x) ↔ E.eq y x.
    +   Lemma union_spec : In x (union s s') ↔ In x s ∨ In x s'.
    +   Lemma inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
    +   Lemma diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +   Lemma fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    +      fold f s i = fold_left (fun a e ⇒ f e a) (elements s) i.
    +   Lemma cardinal_spec : cardinal s = length (elements s).
    +   Lemma filter_spec : compatb f →
    +    (In x (filter f s) ↔ In x s ∧ f x = true).
    +   Lemma for_all_spec : compatb f →
    +    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +   Lemma exists_spec : compatb f →
    +    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +   Lemma partition_spec1 : compatb f → Equal (fst (partition f s)) (filter f s).
    +   Lemma partition_spec2 : compatb f →
    +      Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
    +   Lemma elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    +   Lemma elements_spec2w : NoDupA E.eq (elements s).
    +   Lemma elements_spec2 : sort E.lt (elements s).
    +   Lemma choose_spec1 : choose s = Some x → In x s.
    +   Lemma choose_spec2 : choose s = None → Empty s.
    +   Lemma choose_spec3 : choose s = Some x → choose s' = Some y →
    +    Equal s s' → E.eq x y.
    + +
    +  Lemma min_elt_spec1 : choose s = Some x → In x s.
    +   Lemma min_elt_spec2 : min_elt s = Some x → In y s → ¬ E.lt y x.
    +   Lemma min_elt_spec3 : choose s = None → Empty s.
    + +
    +  Lemma max_elt_spec1 : max_elt s = Some x → In x s.
    +   Lemma max_elt_spec2 : max_elt s = Some x → In y s → ¬ E.lt x y.
    +   Lemma max_elt_spec3 : max_elt s = None → Empty s.
    + +
    +  Lemma compare_spec : CompSpec eq lt s s' (compare s s').

     End Spec.
    @@ -6568,7 +2421,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Instantiations

    +

    Instantiations

    @@ -6576,131 +2429,104 @@

    Library MSetsExtra.MSetIntervals

    It remains to provide instantiations for commonly used datatypes.
    -

    Z

    +

    Z


    Module ElementEncodeZ <: ElementEncode.
    -  Module E := Z.
    +  Module E := Z.

    -  Definition encode (z : Z) := z.
    -  Definition decode (z : Z) := z.
    +  Definition encode (z : Z) := z.
    +  Definition decode (z : Z) := z.

      Lemma decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    -  Proof. by []. Qed.
    - +    decode (encode e) = e.
    +
      Lemma encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    -  Proof. by []. Qed.
    - +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +
      Lemma encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    -  Proof. by []. Qed.
    - +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +
    End ElementEncodeZ.

    -Module MSetIntervalsZ <: SetsOn Z := MSetIntervals ElementEncodeZ.
    +Module MSetIntervalsZ <: SetsOn Z := MSetIntervals ElementEncodeZ.

    -

    N

    +

    N


    Module ElementEncodeN <: ElementEncode.
    -  Module E := N.
    +  Module E := N.

    -  Definition encode (n : N) := Z.of_N n.
    -  Definition decode (z : Z) := Z.to_N z.
    +  Definition encode (n : N) := Z.of_N n.
    +  Definition decode (z : Z) := Z.to_N z.

      Lemma decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    -  Proof.
    -    intros e.
    -    rewrite /encode /decode N2Z.id //.
    -  Qed.
    - +    decode (encode e) = e.
    +
      Lemma encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    -  Proof.
    -    intros e1 e2.
    -    rewrite /encode /Z.eq N2Z.inj_iff /E.eq //.
    -  Qed.
    - +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +
      Lemma encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    -  Proof.
    -    intros e1 e2.
    -    rewrite /encode -N2Z.inj_lt //.
    -  Qed.
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.

    End ElementEncodeN.

    -Module MSetIntervalsN <: SetsOn N := MSetIntervals ElementEncodeN.
    +Module MSetIntervalsN <: SetsOn N := MSetIntervals ElementEncodeN.

    -

    nat

    +

    nat

    Module ElementEncodeNat <: ElementEncode.
    -  Module E := NPeano.Nat.
    +  Module E := Nat.

    -  Definition encode (n : nat) := Z.of_nat n.
    -  Definition decode (z : Z) := Z.to_nat z.
    +  Definition encode (n : nat) := Z.of_nat n.
    +  Definition decode (z : Z) := Z.to_nat z.

      Lemma decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    -  Proof.
    -    intros e.
    -    rewrite /encode /decode Nat2Z.id //.
    -  Qed.
    - +    decode (encode e) = e.
    +
      Lemma encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    -  Proof.
    -    intros e1 e2.
    -    rewrite /encode /Z.eq Nat2Z.inj_iff /E.eq //.
    -  Qed.
    - +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +
      Lemma encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    -  Proof.
    -    intros e1 e2.
    -    rewrite /encode -Nat2Z.inj_lt //.
    -  Qed.
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.

    End ElementEncodeNat.

    -Module MSetIntervalsNat <: SetsOn NPeano.Nat := MSetIntervals ElementEncodeNat.
    +Module MSetIntervalsNat <: SetsOn Nat := MSetIntervals ElementEncodeNat.

    diff --git a/html/MSetsExtra.MSetIterator.html b/html/MSetsExtra.MSetIterator.html deleted file mode 100644 index b03af4d..0000000 --- a/html/MSetsExtra.MSetIterator.html +++ /dev/null @@ -1,2037 +0,0 @@ - - - - - -MSetsExtra.MSetIterator - - - - -
    - - - -
    - -

    Library MSetsExtra.MSetIterator

    - -
    - -
    -
    - -
    -

    Fold with abort for sets

    - - -
    - - This file provided an efficient fold operation for set interfaces. - The standard fold iterates over all elements of the set. The - efficient one - called foldWithAbort - is allowed to skip - certain elements and thereby abort early. - -
    -
    - -
    -Require Export MSetInterface.
    -Require Import ssreflect.
    -Require Import MSetWithDups.
    -Require Import Int.
    -Require Import MSetGenTree MSetAVL MSetRBT.
    -Require Import MSetList MSetWeakList.
    - -
    -
    - -
    -

    Fold With Abort Operations

    - - -
    - - We want to provide an efficient folding operation. Efficieny is - gained by aborting the folding early, if we know that continuing - would not have an effect any more. Formalising this leads to the following - specification of foldWithAbort. -
    -
    - -
    -Definition foldWithAbortType
    -    elt
    - -
    -element type of set -
    -
    -    t
    - -
    -type of set -
    -
    -    A
    - -
    -return type -
    -
    - :=
    -    (elt → A → A) →
    - -
    -f -
    -
    -    (elt → A → bool) →
    - -
    -f_abort -
    -
    -    t →
    - -
    -input set -
    -
    -    A →
    - -
    -base value -
    -
    -    A.
    - -
    -Definition foldWithAbortSpecPred {elt t : Type}
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    -    (foldWithAbort : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=
    -
    -    âˆ€
    -      (A : Type)
    -      
    - -
    -result type -
    -
    -
    -      (i i´ : A)
    -      
    - -
    -base values for foldWithAbort and fold -
    -
    -
    -      (f : elt → A → A) (f´ : elt → A → A)
    -      
    - -
    -fold functions for foldWithAbort and fold -
    -
    -
    -      (f_abort : elt → A → bool)
    -      
    - -
    -abort function -
    -
    -
    -      (s : t)
    - -
    -sets to fold over -
    -
    -
    -      (P : A → A → Prop)
    - -
    -equivalence relation on results -
    -
    -,
    -
    -    
    - -
    -P is an equivalence relation -
    -
    -    Equivalence P →
    -      
    -    
    - -
    -f is for the elements of s compatible with the equivalence relation P -
    -
    -    (∀ st st´ e, In e s → P st st´ → P (f e st) (f e st´)) →
    -
    -    
    - -
    -f and f´ agree for the elements of s -
    -
    -    (∀ e st, In e s → (P (f e st) (f´ e st))) →
    -
    -    
    - -
    -f_abort is OK, i.e. all other elements can be skipped without - leaving the equivalence relation. -
    -
    -    (∀ e1 st,
    -        In e1 s → f_abort e1 st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → e2 ≠ e1 →
    -                        P st (f e2 st´))) →
    -
    -    
    - -
    -The base values are in equivalence relation -
    -
    -    P i i´ →
    -
    -    
    - -
    -The results are in equivalence relation -
    -
    -    P (foldWithAbort f f_abort s i) (fold f´ s i´).
    - -
    -
    - -
    -The specification of folding for ordered sets (as represented by - interface Sets) demands that elements are visited in increasing - order. For ordered sets we can therefore abort folding based on - the weaker knowledge that greater elements have no effect on the - result. The following definition captures this. -
    -
    - -
    -Definition foldWithAbortGtType
    -    elt
    - -
    -element type of set -
    -
    -    t
    - -
    -type of set -
    -
    -    A
    - -
    -return type -
    -
    - :=
    -    (elt → A → A) →
    - -
    -f -
    -
    -    (elt → A → bool) →
    - -
    -f_gt -
    -
    -    t →
    - -
    -input set -
    -
    -    A →
    - -
    -base value -
    -
    -    A.
    - -
    -Definition foldWithAbortGtSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    -    (foldWithAbortGt : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=
    -
    -    âˆ€
    -      (A : Type)
    -      
    - -
    -result type -
    -
    -
    -      (i i´ : A)
    -      
    - -
    -base values for foldWithAbort and fold -
    -
    -
    -      (f : elt → A → A) (f´ : elt → A → A)
    -      
    - -
    -fold functions for foldWithAbort and fold -
    -
    -
    -      (f_gt : elt → A → bool)
    -      
    - -
    -abort function -
    -
    -
    -      (s : t)
    - -
    -sets to fold over -
    -
    -
    -      (P : A → A → Prop)
    - -
    -equivalence relation on results -
    -
    -,
    -
    -    
    -    
    - -
    -P is an equivalence relation -
    -
    -    Equivalence P →
    -
    -    
    - -
    -f is for the elements of s compatible with the equivalence relation P -
    -
    -    (∀ st st´ e, In e s → P st st´ → P (f e st) (f e st´)) →
    -
    -    
    - -
    -f and f´ agree for the elements of s -
    -
    -    (∀ e st, In e s → (P (f e st) (f´ e st))) →
    -
    -    
    - -
    -f_abort is OK, i.e. all other elements can be skipped without - leaving the equivalence relation. -
    -
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 st´))) →
    -
    -    
    - -
    -The base values are in equivalence relation -
    -
    -    P i i´ →
    -
    -    
    - -
    -The results are in equivalence relation -
    -
    -    P (foldWithAbortGt f f_gt s i) (fold f´ s i´).
    - -
    -
    - -
    -For ordered sets, we can safely skip elements at the end - based on the knowledge that they are all greater than the current element. - This leads to serious performance improvements for operations like - filtering. It is tempting to try the symmetric operation and skip elements at - the beginning based on the knowledge that they are too small to be interesting. - So, we would like to start late as well as abort early. - -
    - - This is indeed a very natural and efficient operation for set implementations - based on binary search trees (i.e. the AVL and RBT sets). We can completely symmetrically - to skipping greater elements also skip smaller elements. This leads to the following - specification. -
    -
    - -
    -Definition foldWithAbortGtLtType
    -    elt
    - -
    -element type of set -
    -
    -    t
    - -
    -type of set -
    -
    -    A
    - -
    -return type -
    -
    - :=
    -    (elt → A → bool) →
    - -
    -f_lt -
    -
    -    (elt → A → A) →
    - -
    -f -
    -
    -    (elt → A → bool) →
    - -
    -f_gt -
    -
    -    t →
    - -
    -input set -
    -
    -    A →
    - -
    -base value -
    -
    -    A.
    - -
    -Definition foldWithAbortGtLtSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    -    (foldWithAbortGtLt : ∀ {A : Type}, foldWithAbortGtLtType elt t A) : Prop :=
    -
    -    âˆ€
    -      (A : Type)
    -      
    - -
    -result type -
    -
    -
    -      (i i´ : A)
    -      
    - -
    -base values for foldWithAbort and fold -
    -
    -
    -      (f : elt → A → A) (f´ : elt → A → A)
    -      
    - -
    -fold functions for foldWithAbort and fold -
    -
    -
    -      (f_lt f_gt : elt → A → bool)
    -      
    - -
    -abort functions -
    -
    -
    -      (s : t)
    - -
    -sets to fold over -
    -
    -
    -      (P : A → A → Prop)
    - -
    -equivalence relation on results -
    -
    -,
    -
    -    
    -    
    - -
    -P is an equivalence relation -
    -
    -    Equivalence P →
    -
    -    
    - -
    -f is for the elements of s compatible with the equivalence relation P -
    -
    -    (∀ st st´ e, In e s → P st st´ → P (f e st) (f e st´)) →
    -
    -    
    - -
    -f and f´ agree for the elements of s -
    -
    -    (∀ e st, In e s → (P (f e st) (f´ e st))) →
    -
    -    
    - -
    -f_lt is OK, i.e. smaller elements can be skipped without - leaving the equivalence relation. -
    -
    -    (∀ e1 st,
    -        In e1 s → f_lt e1 st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e2 e1 →
    -                        P st (f e2 st´))) →
    -
    -    
    - -
    -f_gt is OK, i.e. greater elements can be skipped without - leaving the equivalence relation. -
    -
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 st´))) →
    -
    -    
    - -
    -The base values are in equivalence relation -
    -
    -    P i i´ →
    -
    -    
    - -
    -The results are in equivalence relation -
    -
    -    P (foldWithAbortGtLt f_lt f f_gt s i) (fold f´ s i´).
    - -
    -
    - -
    -We are interested in folding with abort mainly for runtime - performance reasons of extracted code. The argument functions - f_lt, f_gt and f of foldWithAbortGtLt often share a large, - comparably expensive part of their computation. - -
    - - In order to further improve runtime performance, therefore another - version foldWithAbortPrecompute f_precompute f_lt f f_gt that - uses an extra function f_precompute to allows to compute the - commonly used parts of these functions only once. This leads to - the following definitions. -
    -
    - -
    -Definition foldWithAbortPrecomputeType
    -    elt
    - -
    -element type of set -
    -
    -    t
    - -
    -type of set -
    -
    -    A
    - -
    -return type -
    -
    -    B
    - -
    -type of precomputed results -
    -
    - :=
    -
    -    (elt → B) →
    - -
    -f_precompute -
    -
    -    (elt → B → A → bool) →
    - -
    -f_lt -
    -
    -    (elt → B → A → A) →
    - -
    -f -
    -
    -    (elt → B → A → bool) →
    - -
    -f_gt -
    -
    -    t →
    - -
    -input set -
    -
    -    A →
    - -
    -base value -
    -
    -    A.
    - -
    -
    - -
    -The specification is similar to the one without precompute, - but uses f_precompute so avoid doing computations multiple times -
    -
    -Definition foldWithAbortPrecomputeSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    -    (foldWithAbortPrecompute : ∀ {A B : Type}, foldWithAbortPrecomputeType elt t A B) : Prop :=
    -
    -    âˆ€
    -      (A B : Type)
    -      
    - -
    -result type -
    -
    -
    -      (i i´ : A)
    -      
    - -
    -base values for foldWithAbortPrecompute and fold -
    -
    -
    -      (f_precompute : elt → B)
    -      
    - -
    -precompute function -
    -
    -
    -      (f : elt → B → A → A) (f´ : elt → A → A)
    -      
    - -
    -fold functions for foldWithAbortPrecompute and fold -
    -
    -
    -      (f_lt f_gt : elt → B → A → bool)
    -      
    - -
    -abort functions -
    -
    -
    -      (s : t)
    - -
    -sets to fold over -
    -
    -
    -      (P : A → A → Prop)
    - -
    -equivalence relation on results -
    -
    -,
    -
    -    
    -    
    - -
    -P is an equivalence relation -
    -
    -    Equivalence P →
    -
    -    
    - -
    -f is for the elements of s compatible with the equivalence relation P -
    -
    -    (∀ st st´ e, In e s → P st st´ → P (f e (f_precompute e) st) (f e (f_precompute e) st´)) →
    -
    -    
    - -
    -f and f´ agree for the elements of s -
    -
    -    (∀ e st, In e s → (P (f e (f_precompute e) st) (f´ e st))) →
    -
    -    
    - -
    -f_lt is OK, i.e. smaller elements can be skipped without - leaving the equivalence relation. -
    -
    -    (∀ e1 st,
    -        In e1 s → f_lt e1 (f_precompute e1) st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e2 e1 →
    -                        P st (f e2 (f_precompute e2) st´))) →
    -
    -    
    - -
    -f_gt is OK, i.e. greater elements can be skipped without - leaving the equivalence relation. -
    -
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 (f_precompute e1) st = true →
    -        (∀ st´ e2, P st st´ →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 (f_precompute e2) st´))) →
    -
    -    
    - -
    -The base values are in equivalence relation -
    -
    -    P i i´ →
    -
    -    
    - -
    -The results are in equivalence relation -
    -
    -    P (foldWithAbortPrecompute f_precompute f_lt f f_gt s i) (fold f´ s i´).
    - -
    -
    - -
    -

    Module Types

    - -
    - - We now define a module type for foldWithAbort. This module - type demands only the existence of the precompute version, since - the other ones can be easily defined via this most efficient one. -
    -
    - -
    -Module Type HasFoldWithAbort (E : OrderedType) (Import C : WSetsOnWithDups E).
    - -
    -  Parameter foldWithAbortPrecompute : ∀ {A B : Type},
    -    foldWithAbortPrecomputeType elt t A B.
    - -
    -  Parameter foldWithAbortPrecomputeSpec :
    -     foldWithAbortPrecomputeSpecPred E.lt In (@fold) (@foldWithAbortPrecompute).
    - -
    -End HasFoldWithAbort.
    - -
    -
    - -
    -

    Derived operations

    - - -
    - - Using these efficient fold operations, many operations can - be implemented efficiently. We provide lemmata and efficient implementations - of useful algorithms via module HasFoldWithAbortOps. -
    -
    - -
    -Module HasFoldWithAbortOps (E : OrderedType) (C : WSetsOnWithDups E)
    -                           (FT : HasFoldWithAbort E C).
    - -
    -  Import FT.
    -  Import C.
    - -
    -
    - -
    -

    First lets define the other folding with abort variants

    - -
    -
    - -
    -  Definition foldWithAbortGtLt {A} f_lt (f : (elt → A → A)) f_gt :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_lt e st)
    -      (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).
    - -
    -  Lemma foldWithAbortGtLtSpec :
    -     foldWithAbortGtLtSpecPred E.lt In (@fold) (@foldWithAbortGtLt).
    -  Proof.
    -    rewrite /foldWithAbortGtLt /foldWithAbortGtLtSpecPred.
    -    intros A i i´ f f´ f_lt f_gt s P.
    -    move ⇒ H_f_compat H_ff´ H_lt H_gt H_ii´.
    -    apply foldWithAbortPrecomputeSpec ⇒ //.
    -  Qed.
    - -
    -  Definition foldWithAbortGt {A} (f : (elt → A → A)) f_gt :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun _ _ _ ⇒ false)
    -      (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).
    - -
    -  Lemma foldWithAbortGtSpec :
    -     foldWithAbortGtSpecPred E.lt In (@fold) (@foldWithAbortGt).
    -  Proof.
    -    rewrite /foldWithAbortGt /foldWithAbortGtSpecPred.
    -    intros A i i´ f f´ f_gt s P.
    -    move ⇒ H_f_compat H_ff´ H_gt H_ii´.
    -    apply foldWithAbortPrecomputeSpec ⇒ //.
    -  Qed.
    - -
    -  Definition foldWithAbort {A} (f : (elt → A → A)) f_abort :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_abort e st)
    -      (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_abort e st).
    - -
    -  Lemma foldWithAbortSpec :
    -     foldWithAbortSpecPred In (@fold) (@foldWithAbort).
    -  Proof.
    -    rewrite /foldWithAbort /foldWithAbortGtSpecPred.
    -    intros A i i´ f f´ f_abort s P.
    -    move ⇒ H_equiv_P H_f_compat H_ff´ H_abort H_ii´.
    -    have H_lt_neq: (∀ e1 e2, E.lt e1 e2 → e1 ≠ e2). {
    -      move ⇒ e1 e2 H_lt H_e12_eq.
    -      rewrite H_e12_eq in H_lt.
    -      have : ( Irreflexive E.lt) by apply StrictOrder_Irreflexive.
    -      rewrite /Irreflexive /Reflexive /complement ⇒ H.
    -      eapply H, H_lt.
    -    }
    -    apply foldWithAbortPrecomputeSpec ⇒ //; (
    -      move ⇒ e1 st H_in_e1 H_abort_e1 st´ e2 H_P H_in_e2 /H_lt_neq H_lt;
    -      apply (H_abort e1 st H_in_e1 H_abort_e1 st´ e2 H_P H_in_e2);
    -      by auto
    -    ).
    -  Qed.
    - -
    -
    - -
    -

    Specialisations for equality

    - -
    - - Let's provide simplified specifications, which use equality instead - of an arbitrary equivalence relation on results. -
    -
    -  Lemma foldWithAbortPrecomputeSpec_Equal : ∀ (A B : Type) (i : A) (f_pre : elt → B)
    -      (f : elt → B → A → A) (f´ : elt → A → A) (f_lt f_gt : elt → B → A → bool) (s : t),
    -
    -      (∀ e st, In e s → (f e (f_pre e) st = f´ e st)) →
    -
    -      
    -      (∀ e1 st,
    -          In e1 s → f_lt e1 (f_pre e1) st = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 →
    -                      (f e2 (f_pre e2) st = st))) →
    -
    -      
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 (f_pre e1) st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 (f_pre e2) st = st))) →
    -
    -      (foldWithAbortPrecompute f_pre f_lt f f_gt s i) = (fold f´ s i).
    -  Proof.
    -    intros A B i f_pre f f´ f_lt f_gt s H_f´ H_lt H_gt.
    - -
    -     eapply (foldWithAbortPrecomputeSpec A B i i f_pre f f´); eauto. {
    -       apply eq_equivalence.
    -     } {
    -       move ⇒ st1 st2 e H_in → //.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_smaller st´ e2 <-.
    -       move : (H_lt e1 st H_e1_in H_do_smaller e2).
    -       intuition.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_greater st´ e2 <-.
    -       move : (H_gt e1 st H_e1_in H_do_greater e2).
    -       intuition.
    -     }
    -  Qed.
    - -
    -  Lemma foldWithAbortGtLtSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f´ : elt → A → A) (f_lt f_gt : elt → A → bool) (s : t),
    -
    -      (∀ e st, In e s → (f e st = f´ e st)) →
    -
    -      
    -      (∀ e1 st,
    -          In e1 s → f_lt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 →
    -                      (f e2 st = st))) →
    -
    -      
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 st = st))) →
    -
    -      (foldWithAbortGtLt f_lt f f_gt s i) = (fold f´ s i).
    -  Proof.
    -    intros A i f f´ f_lt f_gt s H_f´ H_lt H_gt.
    - -
    -     eapply (foldWithAbortGtLtSpec A i i f f´); eauto. {
    -       apply eq_equivalence.
    -     } {
    -       move ⇒ st1 st2 e H_in → //.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_smaller st´ e2 <-.
    -       move : (H_lt e1 st H_e1_in H_do_smaller e2).
    -       intuition.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_greater st´ e2 <-.
    -       move : (H_gt e1 st H_e1_in H_do_greater e2).
    -       intuition.
    -     }
    -  Qed.
    - -
    -  Lemma foldWithAbortGtSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f´ : elt → A → A) (f_gt : elt → A → bool) (s : t),
    -
    -      (∀ e st, In e s → (f e st = f´ e st)) →
    -
    -      
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 st = st))) →
    -
    -      (foldWithAbortGt f f_gt s i) = (fold f´ s i).
    -  Proof.
    -    intros A i f f´ f_gt s H_f´ H_gt.
    - -
    -     eapply (foldWithAbortGtSpec A i i f f´); eauto. {
    -       apply eq_equivalence.
    -     } {
    -       move ⇒ st1 st2 e H_in → //.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_greater st´ e2 <-.
    -       move : (H_gt e1 st H_e1_in H_do_greater e2).
    -       intuition.
    -     }
    -  Qed.
    - -
    -  Lemma foldWithAbortSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f´ : elt → A → A) (f_abort : elt → A → bool) (s : t),
    -
    -      (∀ e st, In e s → (f e st = f´ e st)) →
    -
    -      
    -      (∀ e1 st,
    -          In e1 s → f_abort e1 st = true →
    -          (∀ e2, In e2 s → e1 ≠ e2 →
    -                      (f e2 st = st))) →
    -
    -      (foldWithAbort f f_abort s i) = (fold f´ s i).
    -  Proof.
    -     intros A i f f´ f_abort s H_f´ H_abort.
    - -
    -     eapply (foldWithAbortSpec A i i f f´); eauto. {
    -       apply eq_equivalence.
    -     } {
    -       move ⇒ st1 st2 e H_in → //.
    -     } {
    -       move ⇒ e1 st H_e1_in H_do_abort st´ e2 <-.
    -       move : (H_abort e1 st H_e1_in H_do_abort e2).
    -       intuition.
    -     }
    -  Qed.
    - -
    -
    - -
    -

    FoldWithAbortSpecArgs

    - -
    - - While folding, we are often interested in skipping elements that do not - satisfy a certain property P. This needs expressing in terms of - skips of smaller of larger elements in order to be done efficiently by - our folding functions. Formally, this leads to the definition - of foldWithAbortSpecForPred. - -
    - - Given a FoldWithAbortSpecArg for a predicate P and a - set s, many operations can be implemented efficiently. Below we will provide - efficient versions of filter, choose, ∃, ∀ and more. - -
    -
    -  Record FoldWithAbortSpecArg {B} := {
    -     fwasa_f_pre : (elt → B);
    - -
    -The precompute function -
    -
    -     fwasa_f_lt : (elt → B → bool);
    - -
    -f_lt without state argument -
    -
    -     fwasa_f_gt : (elt → B → bool);
    - -
    -f_gt without state argument -
    -
    -     fwasa_P´ : (elt → B → bool)
    - -
    -the predicate P -
    -
    -  }.
    - -
    -
    - -
    -foldWithAbortSpecForPred s P fwasa holds, if - the argument fwasa fits the predicate P for set s. -
    -
    -  Definition foldWithAbortSpecArgsForPred {A : Type}
    -     (s : t) (P : elt → bool) (fwasa : @FoldWithAbortSpecArg A) :=
    -
    -     
    - -
    -the predicate P´ coincides for s and the given precomputation with P -
    -
    -     (∀ e, In e s → (fwasa_P´ fwasa e (fwasa_f_pre fwasa e) = P e)) ∧
    -
    -     
    - -
    -If fwasa_f_lt holds, all elements smaller than the current one - don't satisfy predicate P. -
    -
    -     (∀ e1,
    -          In e1 s → fwasa_f_lt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 → (P e2 = false))) ∧
    -
    -     
    - -
    -If fwasa_f_gt holds, all elements greater than the current one - don't satisfy predicate P. -
    -
    -     (∀ e1,
    -          In e1 s → fwasa_f_gt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 → (P e2 = false))).
    - -
    - -
    -
    - -
    -

    Filter with abort

    - -
    -
    -  Definition filter_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
    -     @foldWithAbortPrecompute t B (fwasa_f_pre fwasa) (fun e p _ ⇒ fwasa_f_lt fwasa e p)
    -       (fun e e_pre s ⇒ if fwasa_P´ fwasa e e_pre then add e s else s)
    -       (fun e p _ ⇒ fwasa_f_gt fwasa e p) s empty.
    - -
    -  Lemma filter_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    Equal (filter_with_abort fwasa s)
    -          (filter P s).
    -  Proof.
    -    unfold foldWithAbortSpecArgsForPred.
    -    move ⇒ [] f_pre f_lt f_gt P´ P s /=.
    -    move ⇒ [H_f´] [H_lt] H_gt H_proper.
    -    rewrite /filter_with_abort /=.
    - -
    -    have → : (foldWithAbortPrecompute f_pre (fun e p _ ⇒ f_lt e p)
    -     (fun (e : elt) (e_pre : B) (s0 : t) ⇒
    -      if P´ e e_pre then add e s0 else s0) (fun e p _ ⇒ f_gt e p) s empty =
    -     (fold (fun e s0 ⇒ if P e then add e s0 else s0) s empty)). {
    -      apply foldWithAbortPrecomputeSpec_Equal. {
    -        intros e st H_e_in.
    -        rewrite H_f´ //.
    -      } {
    -        intros e1 st H_e1_in H_f_lt_eq e2 H_e2_in H_lt_e2_e1.
    -        rewrite (H_f´ _ H_e2_in).
    -        suff → : (P e2 = false) by done.
    -        apply (H_lt e1); eauto.
    -      } {
    -        intros e1 st H_e1_in H_f_gt_eq e2 H_e2_in H_gt_e2_e1.
    -        rewrite (H_f´ _ H_e2_in).
    -        suff → : (P e2 = false) by done.
    -        apply (H_gt e1); eauto.
    -      }
    -    }
    -
    -    rewrite /Equal ⇒ e.
    -    rewrite fold_spec.
    -    setoid_rewrite filter_spec ⇒ //.
    - -
    -    suff → : ∀ acc, In e
    -      (fold_left
    -         (flip (fun (e0 : elt) (s0 : t) ⇒ if P e0 then add e0 s0 else s0))
    -         (elements s) acc) ↔ (InA E.eq e (elements s) ∧ P e = true) ∨ (In e acc). {
    -      rewrite elements_spec1.
    -      suff : (¬In e empty) by tauto.
    -      apply empty_spec.
    -    }
    -    induction (elements s) as [| x xs IH] ⇒ acc. {
    -      rewrite /= InA_nil. tauto.
    -    } {
    -      rewrite /= /flip IH InA_cons.
    -      case_eq (P x). {
    -        rewrite add_spec.
    -        intuition.
    -        left.
    -        rewrite H0.
    -        split ⇒ //.
    -        left.
    -        apply Equivalence_Reflexive.
    -      } {
    -        intuition.
    -        contradict H2.
    -        setoid_rewrite H1.
    -        by rewrite H.
    -      }
    -    }
    -  Qed.
    - -
    -
    - -
    -

    Choose with abort

    - -
    -
    -  Definition choose_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
    -     foldWithAbortPrecompute (fwasa_f_pre fwasa)
    -       (fun e p st ⇒ match st with None ⇒ (fwasa_f_lt fwasa e p) | _ ⇒ true end)
    -       (fun e e_pre st ⇒ match st with None ⇒
    -          if (fwasa_P´ fwasa e e_pre) then Some e else None | _ ⇒ st end)
    -
    -       (fun e p st ⇒ match st with None ⇒ (fwasa_f_gt fwasa e p) | _ ⇒ true end)
    -       s None.
    - -
    -  Lemma choose_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (match (choose_with_abort fwasa s) with
    -       | None ⇒ (∀ e, In e s → P e = false)
    -       | Some e ⇒ In e s ∧ (P e = true)
    -     end).
    -  Proof.
    -    rewrite /foldWithAbortSpecArgsForPred.
    -    move ⇒ [] f_pre f_lt f_gt P´ P s /=.
    -    move ⇒ [H_f´] [H_lt] H_gt H_proper.
    - -
    -    set fwasa := {|
    -       fwasa_f_pre := f_pre;
    -       fwasa_f_lt := f_lt;
    -       fwasa_f_gt := f_gt;
    -       fwasa_P´ := P´ |}.
    - -
    -    suff : (match (choose_with_abort fwasa s) with
    -       | None ⇒ (∀ e, InA E.eq e (elements s) → P e = false)
    -       | Some e ⇒ InA E.eq e (elements s) ∧ (P e = true)
    -     end). {
    -       case (choose_with_abort fwasa s). {
    -         move ⇒ e.
    -         rewrite elements_spec1 //.
    -       } {
    -         move ⇒ H e H_in.
    -         apply H.
    -         rewrite elements_spec1 //.
    -       }
    -    }
    -
    -    have → : (choose_with_abort fwasa s =
    -      (fold (fun e st ⇒
    -        match st with
    -          | None ⇒ if P e then Some e else None
    -          | _ ⇒ st end) s None)). {
    -      apply foldWithAbortPrecomputeSpec_Equal. {
    -        intros e st H_e_in.
    -        case st ⇒ //=.
    -        rewrite H_f´ //.
    -      } {
    -        move ⇒ e1 [] //= H_e1_in H_f_lt_eq e2 H_e2_in H_lt_e2_e1.
    -        rewrite (H_f´ _ H_e2_in).
    -        case_eq (P e2) ⇒ // H_P_e2.
    -        contradict H_P_e2.
    -        apply not_true_iff_false, (H_lt e1); auto.
    -      } {
    -        move ⇒ e1 [] //= H_e1_in H_f_gt_eq e2 H_e2_in H_gt_e2_e1.
    -        rewrite (H_f´ _ H_e2_in).
    -        case_eq (P e2) ⇒ // H_P_e2.
    -        contradict H_P_e2.
    -        apply not_true_iff_false, (H_gt e1); auto.
    -      }
    -    }
    -
    -    rewrite fold_spec /flip.
    -    induction (elements s) as [| x xs IH]. {
    -      rewrite /=.
    -      move ⇒ e /InA_nil //.
    -    } {
    -      case_eq (P x) ⇒ H_Px; rewrite /= H_Px. {
    -        have → : ∀ xs, fold_left (fun (x0 : option elt) (y : elt) ⇒
    -                    match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    -                    end) xs (Some x) = Some x. {
    -          move ⇒ ys.
    -          induction ys ⇒ //.
    -        }
    -        split; last assumption.
    -        apply InA_cons_hd.
    -        apply E.eq_equiv.
    -      } {
    -        move : IH.
    -        case (fold_left
    -          (fun (x0 : option elt) (y : elt) ⇒
    -             match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    -             end) xs None). {
    -
    -             move ⇒ e [H_e_in] H_Pe.
    -             split; last assumption.
    -             apply InA_cons_tl ⇒ //.
    -        } {
    -          move ⇒ H_e_nin e H_e_in.
    -          have : (InA E.eq e xs ∨ (E.eq e x)). {
    -            inversion H_e_in; tauto.
    -          }
    -          move ⇒ []. {
    -            apply H_e_nin.
    -          } {
    -            move ⇒ → //.
    -          }
    -        }
    -      }
    -    }
    -  Qed.
    - -
    -
    - -
    -

    Exists and Forall with abort

    - -
    -
    -  Definition exists_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
    -    match choose_with_abort fwasa s with
    -      | None ⇒ false
    -      | Some _ ⇒ true
    -    end.
    - -
    -  Lemma exists_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (exists_with_abort fwasa s =
    -     exists_ P s).
    -  Proof.
    -    intros fwasa P s H_fwasa H_proper.
    -    apply Logic.eq_sym.
    -    rewrite /exists_with_abort.
    -    move : (choose_with_abort_spec _ _ _ H_fwasa H_proper).
    -    case (choose_with_abort fwasa s). {
    -      move ⇒ e [H_e_in] H_Pe.
    -      rewrite exists_spec /Exists.
    -      by ∃ e.
    -    } {
    -      move ⇒ H_not_ex.
    -      apply not_true_iff_false.
    -      rewrite exists_spec /Exists.
    -      move ⇒ [e] [H_in] H_pe.
    -      move : (H_not_ex e H_in).
    -      rewrite H_pe //.
    -    }
    -  Qed.
    - -
    -
    - -
    -Negation leads to forall. -
    -
    -  Definition forall_with_abort {B} fwasa s :=
    -     negb (@exists_with_abort B fwasa s).
    - -
    -  Lemma forall_with_abort_spec {B} : ∀ fwasa s P,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (forall_with_abort fwasa s =
    -     for_all (fun e ⇒ negb (P e)) s).
    -  Proof.
    -    intros fwasa s P H_ok H_proper.
    -    rewrite /forall_with_abort exists_with_abort_spec; auto.
    - -
    -    rewrite eq_iff_eq_true negb_true_iff -not_true_iff_false.
    -    rewrite exists_spec.
    -    setoid_rewrite for_all_spec; last solve_proper.
    - -
    -    rewrite /Exists /For_all.
    -    split. {
    -      move ⇒ H_pre x H_x_in.
    -      rewrite negb_true_iff -not_true_iff_false ⇒ H_Px.
    -      apply H_pre.
    -      by ∃ x.
    -    } {
    -      move ⇒ H_pre [x] [H_x_in] H_P_x.
    -      move : (H_pre x H_x_in).
    -      rewrite H_P_x.
    -      done.
    -    }
    -  Qed.
    - -
    -End HasFoldWithAbortOps.
    - -
    -
    - -
    -

    Modules Types For Sets with Fold with Abort

    - -
    -
    -Module Type WSetsWithDupsFoldA.
    -  Declare Module E : OrderedType.
    -  Include WSetsOnWithDups E.
    -  Include HasFoldWithAbort E.
    -  Include HasFoldWithAbortOps E.
    -End WSetsWithDupsFoldA.
    - -
    -Module Type WSetsWithFoldA <: WSets.
    -  Declare Module E : OrderedType.
    -  Include WSetsOn E.
    -  Include HasFoldWithAbort E.
    -  Include HasFoldWithAbortOps E.
    -End WSetsWithFoldA.
    - -
    -Module Type SetsWithFoldA <: Sets.
    -  Declare Module E : OrderedType.
    -  Include SetsOn E.
    -  Include HasFoldWithAbort E.
    -  Include HasFoldWithAbortOps E.
    -End SetsWithFoldA.
    - -
    -
    - -
    -

    Implementations

    - -
    - -

    GenTree implementation

    - - Finally, provide such a fold with abort operation for generic trees. -
    -
    -Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
    -  (Import Raw:Ops E I)
    -  (M : MSetGenTree.Props E I Raw).
    - -
    -  Fixpoint foldWithAbort_Raw {A B: Type} (f_pre : E.t → B) f_lt (f: E.t → B → A → A) f_gt t (base: A) : A :=
    -    match t with
    -    | Raw.Leaf ⇒ base
    -    | Raw.Node _ l x r ⇒
    -        let x_pre := f_pre x in
    -        let st0 := if f_lt x x_pre base then base else foldWithAbort_Raw f_pre f_lt f f_gt l base in
    -        let st1 := f x x_pre st0 in
    -        let st2 := if f_gt x x_pre st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1 in
    -        st2
    -    end.
    - -
    -  Lemma foldWithAbort_RawSpec : ∀ (A B : Type) (i i´ : A) (f_pre : E.t → B)
    -      (f : E.t → B → A → A) (f´ : E.t → A → A) (f_lt f_gt : E.t → B → A → bool) (s : Raw.tree)
    -      (P : A → A → Prop),
    -
    -      (M.bst s) →
    -      Equivalence P →
    -      (∀ st st´ e, M.In e s → P st st´ → P (f e (f_pre e) st) (f e (f_pre e) st´)) →
    -      (∀ e st, M.In e s → P (f e (f_pre e) st) (f´ e st)) →
    -
    -      
    -      (∀ e1 st,
    -          M.In e1 s → f_lt e1 (f_pre e1) st = true →
    -          (∀ st´ e2, P st st´ →
    -                          M.In e2 s → E.lt e2 e1 →
    -                          P st (f e2 (f_pre e2) st´))) →
    -
    -      
    -      (∀ e1 st,
    -          M.In e1 s → f_gt e1 (f_pre e1) st = true →
    -          (∀ st´ e2, P st st´ →
    -                          M.In e2 s → E.lt e1 e2 →
    -                          P st (f e2 (f_pre e2) st´))) →
    -
    -      P i i´ →
    -      P (foldWithAbort_Raw f_pre f_lt f f_gt s i) (fold f´ s i´).
    -   Proof.
    -     intros A B i i´ f_pre f f´ f_lt f_gt s P.
    -     move ⇒ H_bst H_equiv_P H_P_f H_f´ H_RL H_RG.
    - -
    -     set base := s.
    - -
    -     move : i i´.
    -     have : (∀ e, M.In e base → M.In e s). {
    -       rewrite /In /base //.
    -     }
    -     have : M.bst base. {
    -       apply H_bst.
    -     }
    -     move : base.
    -     clear H_bst.
    - -
    -     induction base as [|c l IHl e r IHr] using M.tree_ind. {
    -       rewrite /foldWithAbort_Raw /Raw.fold.
    -       move ⇒ _ _ i i´ //.
    -     } {
    -       move ⇒ H_bst H_sub i i´ H_P_ii´.
    - -
    -       have [H_bst_l [H_bst_r [H_lt_tree_l H_gt_tree_r]]]:
    -         M.bst l ∧ M.bst r ∧ M.lt_tree e l ∧ M.gt_tree e r. {
    -         inversion H_bst. done.
    -       }
    -       
    -       have H_sub_l : (∀ e0 : E.t, M.In e0 l → M.In e0 s ∧ E.lt e0 e). {
    -         intros e0 H_in_l.
    -         split; last by apply H_lt_tree_l.
    -         eapply H_sub.
    -         rewrite /M.In M.In_node_iff.
    -         tauto.
    -       }
    -       move : (IHl H_bst_l) ⇒ {IHl} IHl {H_bst_l} {H_lt_tree_l}.
    -       have H_sub_r : (∀ e0 : E.t, M.In e0 r → M.In e0 s ∧ E.lt e e0). {
    -         intros e0 H_in_r.
    -         split; last by apply H_gt_tree_r.
    -         eapply H_sub.
    -         rewrite /M.In M.In_node_iff.
    -         tauto.
    -       }
    -       move : (IHr H_bst_r) ⇒ {IHr} IHr {H_bst_r} {H_gt_tree_r}.
    -       have H_in_e : M.In e s. {
    -         eapply H_sub.
    -         rewrite /M.In M.In_node_iff.
    -         right; left.
    -         apply Equivalence_Reflexive.
    -       }
    -       move ⇒ {H_sub}.
    - -
    -       rewrite /=.
    -       set st0 := if f_lt e (f_pre e) i then i else foldWithAbort_Raw f_pre f_lt f f_gt l i.
    -       set st0´ := Raw.fold f´ l i´.
    -       set st1 := f e (f_pre e) st0.
    -       set st1´ := f´ e st0´.
    -       set st2 := if f_gt e (f_pre e) st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1.
    -       set st2´ := Raw.fold f´ r st1´.
    - -
    -       have H_P_st0 : P st0 st0´. {
    -         rewrite /st0 /st0´.
    -         case_eq (f_lt e (f_pre e) i). {
    -           move ⇒ H_fl_eq.
    -           move : H_P_ii´ H_sub_l.
    -           move : H_equiv_P H_f´ (H_RL _ _ H_in_e H_fl_eq).
    -           rewrite /M.lt_tree. clear.
    -           move ⇒ H_equiv_P H_f´ H_RL.
    -           move : i´.
    -           induction l as [|c l IHl e´ r IHr] using M.tree_ind. {
    -             done.
    -           } {
    -             intros i´ H_P_ii´ H_sub_l.
    -             rewrite /=.
    -             apply IHr; last first. {
    -               move ⇒ y H_y_in.
    -               apply H_sub_l.
    -               rewrite /M.In M.In_node_iff. tauto.
    -             }
    -             have [] : (M.In e´ s ∧ E.lt e´ e). {
    -               apply H_sub_l.
    -               rewrite /M.In M.In_node_iff.
    -               right; left.
    -               apply Equivalence_Reflexive.
    -             }
    -             move ⇒ H_e´_in H_lt_in.
    -             suff H_P_i : (P i (f e´ (f_pre e´) (fold f´ l i´))). {
    -               eapply Equivalence_Transitive; first apply H_P_i.
    -               by apply H_f´.
    -             }
    -             eapply H_RL ⇒ //.
    -             apply IHl; last first. {
    -               move ⇒ y H_y_in.
    -               apply H_sub_l.
    -               rewrite /M.In M.In_node_iff. tauto.
    -             }
    -             assumption.
    -           }
    -         } {
    -           move ⇒ _.
    -           apply IHl ⇒ //.
    -           eapply H_sub_l.
    -         }
    -       }
    -       have H_P_st1 : P st1 st1´. {
    -         rewrite /st1 /st1´.
    -         rewrite -H_f´ //.
    -         apply H_P_f ⇒ //.
    -       }
    -       have H_P_st2 : P st2 st2´. {
    -         rewrite /st2 /st2´.
    -         clearbody st1 st1´.
    -         case_eq (f_gt e (f_pre e) st1). {
    -           move ⇒ H_gt_eq.
    -           move : H_P_st1 H_sub_r.
    -           move : H_equiv_P (H_RG _ _ H_in_e H_gt_eq) H_f´.
    -           unfold M.gt_tree. clear.
    -           move ⇒ H_equiv_P H_RG H_f´.
    -           move : st1´.
    -           induction r as [|c l IHl e´ r IHr] using M.tree_ind. {
    -             done.
    -           } {
    -             intros st1´ H_P_st1 H_sub_r.
    -             rewrite /=.
    -             apply IHr; last first. {
    -               move ⇒ y H_y_in.
    -               apply H_sub_r.
    -               rewrite /M.In M.In_node_iff. tauto.
    -             }
    -             have [] : (M.In e´ s ∧ E.lt e e´). {
    -               apply H_sub_r.
    -               rewrite /M.In M.In_node_iff.
    -               right; left.
    -               apply Equivalence_Reflexive.
    -             }
    -             move ⇒ H_e´_in H_lt_ee´.
    -             suff H_P_st1_aux : (P st1 (f e´ (f_pre e´) (fold f´ l st1´))). {
    -               eapply Equivalence_Transitive; first apply H_P_st1_aux.
    -               by apply H_f´.
    -             }
    -             eapply H_RG ⇒ //.
    -             apply IHl; last first. {
    -               move ⇒ y H_y_in.
    -               apply H_sub_r.
    -               rewrite /M.In M.In_node_iff. tauto.
    -             }
    -             assumption.
    -           }
    -         } {
    -           move ⇒ _.
    -           apply IHr ⇒ //.
    -           eapply H_sub_r.
    -         }
    -       }
    -       done.
    -     }
    -  Qed.
    -End MakeGenTreeFoldA.
    - -
    -
    - -
    -

    AVL implementation

    - - The generic tree implementation naturally leads to an AVL one. -
    -
    - -
    -Module MakeAVLSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetAVL.Make X.
    -  Include MakeGenTreeFoldA X Z_as_Int Raw Raw.
    - -
    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    -    foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.
    - -
    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    -  Proof.
    -    intros A B i i´ f_pre f f´ f_lt f_gt s P.
    -    move ⇒ H_P_f H_f´ H_RL H_RG H_P_ii´.
    - -
    -    rewrite /foldWithAbortPrecompute /fold.
    -    apply foldWithAbort_RawSpec ⇒ //.
    -    case s. rewrite /this /Raw.Ok //.
    -  Qed.
    - -
    -  Include HasFoldWithAbortOps X.
    - -
    -End MakeAVLSetsWithFoldA.
    - -
    -
    - -
    -

    RBT implementation

    - - The generic tree implementation naturally leads to an RBT one. -
    -
    -Module MakeRBTSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetRBT.Make X.
    -  Include MakeGenTreeFoldA X Color Raw Raw.
    - -
    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    -    foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.
    - -
    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    -  Proof.
    -    intros A B i i´ f_pre f f´ f_lt f_gt s P.
    -    move ⇒ H_P_f H_f´ H_RL H_RG H_P_ii´.
    - -
    -    rewrite /foldWithAbortPrecompute /fold.
    -    apply foldWithAbort_RawSpec ⇒ //.
    -    case s. rewrite /this /Raw.Ok //.
    -  Qed.
    - -
    -  Include HasFoldWithAbortOps X.
    - -
    -End MakeRBTSetsWithFoldA.
    - -
    -
    - -
    -

    Sorted Lists Implementation

    - -
    -
    -Module MakeListSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetList.Make X.
    - -
    -  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    -    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    -  match t with
    -    | nil ⇒ acc
    -    | x :: xs ⇒ (
    -        let pre_x := f_pre x in
    -        let acc := f x (pre_x) acc in
    -        if (f_gt x pre_x acc) then
    -          acc
    -        else
    -          foldWithAbortRaw f_pre f_lt f f_gt xs acc
    -      )
    -  end.
    - -
    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt f f_gt t acc :=
    -    @foldWithAbortRaw A B f_pre f_lt f f_gt t.(this) acc.
    - -
    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    -  Proof.
    -    intros A B i i´ f_pre f f´ f_lt f_gt.
    -    move ⇒ [] l H_is_ok_l P H_equiv_P.
    -    rewrite /fold /foldWithAbortPrecompute /In /this /Raw.In /Raw.fold.
    -    move ⇒ H_P_f H_f´ H_RL H_RG.
    - -
    -    set base := l.
    -    move : i i´.
    -    have : (∀ e, InA X.eq e base → InA X.eq e l). {
    -      rewrite /base //.
    -    }
    -    have : sort X.lt base. {
    -      rewrite Raw.isok_iff /base //.
    -    }
    -    clear H_is_ok_l.
    - -
    -    induction base as [| x xs IH]. {
    -      by simpl.
    -    }
    -    move ⇒ H_sort H_in_xxs i i´ Pii´ /=.
    - -
    -    have [H_sort_xs H_hd_rel {H_sort}] : Sorted X.lt xs ∧ HdRel X.lt x xs. {
    -      by inversion H_sort.
    -    }
    -
    -    move : H_hd_rel.
    -    rewrite (Raw.ML.Inf_alt x H_sort_xs) ⇒ H_lt_xs.
    - -
    -    have H_x_in_l : InA X.eq x l. {
    -      apply H_in_xxs.
    -      apply InA_cons_hd.
    -      apply X.eq_equiv.
    -    }
    -    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
    -      intros e H_in.
    -      apply H_in_xxs, InA_cons_tl ⇒ //.
    -    }
    -   
    -    have H_P_next : P (f x (f_pre x) i) (flip f´ i´ x). {
    -      rewrite /flip -H_f´ //.
    -      apply H_P_f ⇒ //.
    -    }
    -
    -    case_eq (f_gt x (f_pre x) (f x (f_pre x) i)); last first. {
    -      move ⇒ _.
    -      apply IH ⇒ //.
    -    } {
    -      move ⇒ H_gt.
    -      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    -         P (f x (f_pre x) i) (fold_left (flip f´) xs st)). {
    -         apply H_suff ⇒ //.
    -      }
    -      move : H_in_xs H_lt_xs.
    - -
    -      clear IH H_in_xxs H_sort_xs.
    -      move : (H_RG x _ H_x_in_l H_gt) ⇒ H_RG_x.
    -      induction xs as [| x´ xs´ IH´]. {
    -        done.
    -      } {
    -        intros H_in_xs H_lt_xs st H_P_st.
    -        rewrite /=.
    -        have H_x´_in_l : InA X.eq x´ l. {
    -          apply H_in_xs.
    -          apply InA_cons_hd, X.eq_equiv.
    -        }
    -        apply IH´. {
    -          intros e H.
    -          apply H_in_xs, InA_cons_tl ⇒ //.
    -        } {
    -          intros e H.
    -          apply H_lt_xs, InA_cons_tl ⇒ //.
    -        } {
    -          rewrite /flip -H_f´ //.
    -          apply H_RG_x ⇒ //.
    -          apply H_lt_xs.
    -          apply InA_cons_hd, X.eq_equiv.
    -        }
    -      }
    -    }
    -  Qed.
    - -
    -  Include HasFoldWithAbortOps X.
    - -
    -End MakeListSetsWithFoldA.
    - -
    -
    - -
    -

    Unsorted Lists without Dups Implementation

    - -
    -
    -Module MakeWeakListSetsWithFoldA (X : OrderedType) <: WSetsWithFoldA with Module E := X.
    -  Module Raw := MSetWeakList.MakeRaw X.
    -  Module E := X.
    -  Include WRaw2SetsOn E Raw.
    - -
    -  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    -    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    -  match t with
    -    | nil ⇒ acc
    -    | x :: xs ⇒ (
    -        let pre_x := f_pre x in
    -        let acc := f x (pre_x) acc in
    -        if (f_gt x pre_x acc) && (f_lt x pre_x acc) then
    -          acc
    -        else
    -          foldWithAbortRaw f_pre f_lt f f_gt xs acc
    -      )
    -  end.
    - -
    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt f f_gt t acc :=
    -    @foldWithAbortRaw A B f_pre f_lt f f_gt t.(this) acc.
    - -
    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    -  Proof.
    -    intros A B i i´ f_pre f f´ f_lt f_gt.
    -    move ⇒ [] l H_is_ok_l P H_P_equiv.
    -    rewrite /fold /foldWithAbortPrecompute /In /this /Raw.In /Raw.fold.
    -    move ⇒ H_P_f H_f´ H_RL H_RG.
    - -
    -    set base := l.
    -    move : i i´.
    -    have : (∀ e, InA X.eq e base → InA X.eq e l). {
    -      rewrite /base //.
    -    }
    -    have : NoDupA X.eq base. {
    -      apply H_is_ok_l.
    -    }
    -    clear H_is_ok_l.
    - -
    -    induction base as [| x xs IH]. {
    -      by simpl.
    -    }
    -    move ⇒ H_nodup_xxs H_in_xxs i i´ Pii´ /=.
    - -
    -    have [H_nin_x_xs H_nodup_xs {H_nodup_xxs}] : ¬ InA X.eq x xs ∧ NoDupA X.eq xs. {
    -      by inversion H_nodup_xxs.
    -    }
    -
    -    have H_x_in_l : InA X.eq x l. {
    -      apply H_in_xxs.
    -      apply InA_cons_hd.
    -      apply X.eq_equiv.
    -    }
    -    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
    -      intros e H_in.
    -      apply H_in_xxs, InA_cons_tl ⇒ //.
    -    }
    -   
    -    have H_P_next : P (f x (f_pre x) i) (flip f´ i´ x). {
    -      rewrite /flip -H_f´ //.
    -      apply H_P_f ⇒ //.
    -    }
    -
    -    case_eq (f_gt x (f_pre x) (f x (f_pre x) i) &&
    -             f_lt x (f_pre x) (f x (f_pre x) i)); last first. {
    -      move ⇒ _.
    -      apply IH ⇒ //.
    -    } {
    -      move ⇒ /andb_true_iff [H_gt H_lt].
    -      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    -         P (f x (f_pre x) i) (fold_left (flip f´) xs st)). {
    -         apply H_suff ⇒ //.
    -      }
    -
    -      have H_neq_xs : ∀ e, List.In e xs → X.lt x e ∨ X.lt e x. {
    -        intros e H_in.
    - -
    -        move : (X.compare_spec x e).
    -        case (X.compare x e) ⇒ H_cmp; inversion H_cmp. {
    -          contradict H_nin_x_xs.
    -          rewrite InA_alt.
    -          by ∃ e.
    -        } {
    -          by left.
    -        } {
    -          by right.
    -        }
    -      }
    -      move : H_in_xs H_neq_xs.
    - -
    -      clear IH H_in_xxs H_nodup_xs.
    -      move : (H_RG x _ H_x_in_l H_gt) ⇒ H_RG_x.
    -      move : (H_RL x _ H_x_in_l H_lt) ⇒ H_RL_x.
    -      induction xs as [| x´ xs´ IH´]. {
    -        done.
    -      } {
    -        intros H_in_xs H_neq_xs st H_P_st.
    -        rewrite /=.
    -        have H_x´_in_xxs´ : List.In x´ (x´ :: xs´). {
    -          simpl; by left.
    -        }
    -        have H_x´_in_l : InA X.eq x´ l. {
    -          apply H_in_xs.
    -          apply InA_cons_hd, X.eq_equiv.
    -        }
    -        apply IH´. {
    -          intros H.
    -          apply H_nin_x_xs, InA_cons_tl ⇒ //.
    -        } {
    -          intros e H.
    -          apply H_in_xs, InA_cons_tl ⇒ //.
    -        } {
    -          intros e H.
    -          apply H_neq_xs, in_cons ⇒ //.
    -        } {
    -          rewrite /flip -H_f´ //.
    -          move : (H_neq_xs x´ H_x´_in_xxs´) ⇒ [] H_cmp. {
    -            apply H_RG_x ⇒ //.
    -          } {
    -            apply H_RL_x ⇒ //.
    -          }
    -        }
    -      }
    -    }
    -  Qed.
    - -
    -  Include HasFoldWithAbortOps X.
    - -
    -End MakeWeakListSetsWithFoldA.
    - -
    -
    -
    - - - -
    - - - \ No newline at end of file diff --git a/html/MSetsExtra.MSetListWithDups.html b/html/MSetsExtra.MSetListWithDups.html index 2186d08..7ec5cc9 100644 --- a/html/MSetsExtra.MSetListWithDups.html +++ b/html/MSetsExtra.MSetListWithDups.html @@ -24,7 +24,7 @@

    Library MSetsExtra.MSetListWithDups

    -

    Weak sets implemented as lists with duplicates

    +

    Weak sets implemented as lists with duplicates

    @@ -37,7 +37,7 @@

    Library MSetsExtra.MSetListWithDups

    This implementation is useful, if one needs very efficient insert and union operation, and can guarantee that one does not - add too many duplicates. The operation elements_dist is implemented + add too many duplicates. The operation elements_dist is implemented by sorting the list first. Therefore this instantiation can only be used if the element type is ordered. @@ -45,17 +45,17 @@

    Library MSetsExtra.MSetListWithDups


    -Require Export MSetInterface.
    -Require Import ssreflect.
    -Require Import List OrdersFacts OrdersLists.
    -Require Import Sorting Permutation.
    +Require Export MSetInterface.
    +Require Import mathcomp.ssreflect.ssreflect.
    +Require Import List OrdersFacts OrdersLists.
    +Require Import Sorting Permutation.
    Require Import MSetWithDups.

    -

    Removing duplicates from sorted lists

    +

    Removing duplicates from sorted lists

    @@ -72,7 +72,7 @@

    Library MSetsExtra.MSetListWithDups

    duplicates.
    -Module RemoveDupsFromSorted (Import X:OrderedType).
    +Module RemoveDupsFromSorted (Import X:OrderedType).

    @@ -81,41 +81,30 @@

    Library MSetsExtra.MSetListWithDups

    First, we need some infrastructure for our ordered type
    -  Module Import MX := OrderedTypeFacts X.
    +  Module Import MX := OrderedTypeFacts X.

    -  Module Import XTotalLeBool <: TotalLeBool.
    +  Module Import XTotalLeBool <: TotalLeBool.
        Definition t := X.t.
        Definition leb x y :=
          match X.compare x y with
    -        | Lt ⇒ true
    -        | Eq ⇒ true
    -        | Gt ⇒ false
    +        | Lt ⇒ true
    +        | Eq ⇒ true
    +        | Gt ⇒ false
          end.

        Infix "<=?" := leb (at level 35).

    -    Theorem leb_total : ∀ (a1 a2 : t), (a1 <=? a2 = true) ∨ (a2 <=? a1 = true).
    -    Proof.
    -      intros a1 a2.
    -      unfold leb.
    -      rewrite (compare_antisym a1 a2).
    -      case (X.compare a1 a2); rewrite /=; tauto.
    -    Qed.
    +    Theorem leb_total : ∀ (a1 a2 : t), (a1 <=? a2 = true) ∨ (a2 <=? a1 = true).

    -    Definition le x y := (leb x y = true).
    +    Definition le x y := (leb x y = true).
      End XTotalLeBool.

    -  Lemma eqb_eq_alt : ∀ x y, eqb x y = true ↔ eq x y.
    -  Proof.
    -    intros x y.
    -    rewrite eqb_alt -compare_eq_iff.
    -    case (compare x y) ⇒ //.
    -  Qed.
    +  Lemma eqb_eq_alt : ∀ x y, eqb x y = true ↔ eq x y.

    @@ -124,22 +113,22 @@

    Library MSetsExtra.MSetListWithDups

    Now we can define our main function
    -  Fixpoint remove_dups_from_sortedA_aux (acc : list t) (l : list t) : list t :=
    +  Fixpoint remove_dups_from_sortedA_aux (acc : list t) (l : list t) : list t :=
        match l with
    -    | nil ⇒ List.rev´ acc
    -    | x :: xs ⇒
    +    | nil ⇒ List.rev' acc
    +    | x :: xs ⇒
           match xs with
    -       | nil ⇒ List.rev´ (x :: acc)
    -       | y :: ys ⇒
    +       | nil ⇒ List.rev' (x :: acc)
    +       | y :: ys ⇒
               if eqb x y then
                 remove_dups_from_sortedA_aux acc xs
               else
    -             remove_dups_from_sortedA_aux (x::acc) xs
    +             remove_dups_from_sortedA_aux (x::acc) xs
           end
        end.

    -  Definition remove_dups_from_sortedA := remove_dups_from_sortedA_aux (nil : list t).
    +  Definition remove_dups_from_sortedA := remove_dups_from_sortedA_aux (nil : list t).

    @@ -148,87 +137,34 @@

    Library MSetsExtra.MSetListWithDups

    We can prove some technical lemmata
    -  Lemma remove_dups_from_sortedA_aux_alt : ∀ (l : list X.t) acc,
    -    remove_dups_from_sortedA_aux acc l =
    -    List.rev acc ++ (remove_dups_from_sortedA l).
    -  Proof.
    -    unfold remove_dups_from_sortedA.
    -    induction l as [| x xs IH] ⇒ acc. {
    -      rewrite /remove_dups_from_sortedA_aux /rev´ -!rev_alt /= app_nil_r //.
    -    } {
    -      rewrite /=.
    -      case_eq xs. {
    -        rewrite /rev´ -!rev_alt //.
    -      } {
    -        move ⇒ y ys H_xs_eq.
    -        rewrite -!H_xs_eq !(IH acc) !(IH (x :: acc)) (IH (x::nil)).
    -        case (eqb x y) ⇒ //.
    -        rewrite /= -app_assoc //.
    -      }
    -    }
    -  Qed.
    - +  Lemma remove_dups_from_sortedA_aux_alt : ∀ (l : list X.t) acc,
    +    remove_dups_from_sortedA_aux acc l =
    +    List.rev acc ++ (remove_dups_from_sortedA l).
    +
      Lemma remove_dups_from_sortedA_alt :
    -    âˆ€ (l : list t),
    -    remove_dups_from_sortedA l =
    +    âˆ€ (l : list t),
    +    remove_dups_from_sortedA l =
        match l with
    -    | nil ⇒ nil
    -    | x :: xs ⇒
    +    | nil ⇒ nil
    +    | x :: xs ⇒
           match xs with
    -       | nil ⇒ l
    -       | y :: ys ⇒
    +       | nil ⇒ l
    +       | y :: ys ⇒
               if eqb x y then
                 remove_dups_from_sortedA xs
               else
    -             x :: remove_dups_from_sortedA xs
    +             x :: remove_dups_from_sortedA xs
           end
        end.
    -  Proof.
    -    case. {
    -      done.
    -    } {
    -      intros x xs.
    -      rewrite /remove_dups_from_sortedA /= /rev´ /=.
    -      case xs ⇒ //.
    -      move ⇒ y ys.
    -      rewrite !remove_dups_from_sortedA_aux_alt /= //.
    -    }
    -  Qed.
    - +
      Lemma remove_dups_from_sortedA_hd :
          âˆ€ x xs,
    -      âˆƒ (x´:t) xs´,
    -        remove_dups_from_sortedA (x :: xs) =
    -        (x´ :: xs´) ∧ (eqb x x´ = true).
    -  Proof.
    -    intros x xs.
    -    move : x;
    -    induction xs as [| y ys IH] ⇒ x. {
    -      rewrite remove_dups_from_sortedA_alt.
    -      âˆƒ x, nil.
    -      split; first reflexivity.
    -      rewrite eqb_alt compare_refl //.
    -    } {
    -      rewrite remove_dups_from_sortedA_alt.
    -      case_eq (eqb x y); last first. {
    -        move ⇒ _.
    -        âˆƒ x, (remove_dups_from_sortedA (y :: ys)).
    -        split; first reflexivity.
    -        rewrite eqb_alt compare_refl //.
    -      } {
    -        move ⇒ H_eqb_xy.
    -        move : (IH y) ⇒ {IH} [x´] [xs´] [->] H_eqb_yx´.
    -        âˆƒ x´, xs´.
    -        split; first done.
    -        move : H_eqb_xy H_eqb_yx´.
    -        rewrite !eqb_eq_alt.
    -        apply MX.eq_trans.
    -      }
    -    }
    -  Qed.
    - +      âˆƒ (x':t) xs',
    +        remove_dups_from_sortedA (x :: xs) =
    +        (x' :: xs') ∧ (eqb x x' = true).
    +
    @@ -237,102 +173,14 @@

    Library MSetsExtra.MSetListWithDups

      Lemma remove_dups_from_sortedA_spec :
    -    âˆ€ (l : list t),
    -      Sorted le l →
    -      let l´ := remove_dups_from_sortedA l in (
    +    âˆ€ (l : list t),
    +      Sorted le l →
    +      let l' := remove_dups_from_sortedA l in (
        
    -      Sorted lt l´ ∧
    -      NoDupA eq l´ ∧
    -      (∀ x, InA eq x l ↔ InA eq x l´)).
    -  Proof.
    -    simpl.
    -    induction l as [| x xs IH]. {
    -      rewrite remove_dups_from_sortedA_alt.
    -      done.
    -    } {
    -      rewrite remove_dups_from_sortedA_alt.
    -      move : IH.
    -      case xs ⇒ {xs}. {
    -        move ⇒ _.
    -        split; last split. {
    -          apply Sorted_cons ⇒ //.
    -        } {
    -          apply NoDupA_singleton.
    -        } {
    -          done.
    -        }
    -      } {
    -        move ⇒ y ys IH H_sorted_x_y_ys.
    -        apply Sorted_inv in H_sorted_x_y_ys as [H_sorted_y_ys H_hd_rel].
    -        apply HdRel_inv in H_hd_rel.
    - -
    -        have : ∃ y´ ys´,
    -          remove_dups_from_sortedA (y :: ys) = y´ :: ys´ ∧
    -          eqb y y´ = true. {
    -          apply remove_dups_from_sortedA_hd ⇒ //.
    -        }
    -        move ⇒ [y´] [ys´] [H_yys´_intro] /eqb_eq_alt H_eq_y_y´.
    - -
    -        move : (IH H_sorted_y_ys).
    -        rewrite !H_yys´_intro.
    -        move ⇒ {IH} [IH1] [IH2] IH3.
    - -
    -        case_eq (eqb x y). {
    -          rewrite eqb_eq_alt ⇒ H_eq_x_y.
    -          split ⇒ //.
    -          split ⇒ //.
    -          move ⇒ x´.
    -          rewrite InA_cons IH3.
    -          split; last tauto.
    -          move ⇒ [] //.
    -          move ⇒ H_eq_x´_x.
    -          apply InA_cons_hd.
    -          apply eq_trans with (y := x) ⇒ //.
    -          apply eq_trans with (y := y) ⇒ //.
    -        }
    -        move ⇒ H_neqb_x_y.
    - -
    -        have H_sorted : Sorted lt (x :: y´ :: ys´). {
    -          apply Sorted_cons ⇒ //.
    -          apply HdRel_cons.
    -          rewrite -compare_lt_iff.
    -          suff : (compare x y = Lt). {
    -            setoid_rewrite compare_compat; eauto;
    -              apply eq_refl.
    -          }
    -          move : H_hd_rel H_neqb_x_y.
    -          rewrite eqb_alt /le /leb.
    -          case (compare x y) ⇒ //.
    -        }
    -        split; last split. {
    -          assumption.
    -        } {
    -          apply NoDupA_cons ⇒ //.
    - -
    -          move ⇒ /InA_alt [x´] [H_eq_xx´] H_in_x´.
    -          have : Forall (lt x) (y´ :: ys´). {
    -            apply Sorted_extends ⇒ //.
    -            rewrite /Relations_1.Transitive.
    -            by apply lt_trans.
    -          }
    -          rewrite Forall_forall ⇒ H_forall.
    -          move : (H_forall _ H_in_x´) ⇒ {H_forall}.
    -          move : H_eq_xx´.
    -          rewrite -compare_lt_iff -compare_eq_iff.
    -          move ⇒ → //.
    -        } {
    -          move ⇒ x0.
    -          rewrite !(InA_cons eq x0 x) IH3 //.
    -        }
    -      }
    -    }
    -  Qed.
    - +      Sorted lt l' ∧
    +      NoDupA eq l' ∧
    +      (∀ x, InA eq x l ↔ InA eq x l')).
    +
    @@ -343,54 +191,18 @@

    Library MSetsExtra.MSetListWithDups

      Module Import XSort := Sort XTotalLeBool.

    -  Definition remove_dups_by_sortingA (l : list t) : list t :=
    +  Definition remove_dups_by_sortingA (l : list t) : list t :=
        remove_dups_from_sortedA (XSort.sort l).

      Lemma remove_dups_by_sortingA_spec :
    -    âˆ€ (l : list t),
    -      let l´ := remove_dups_by_sortingA l in (
    +    âˆ€ (l : list t),
    +      let l' := remove_dups_by_sortingA l in (
        
    -      Sorted lt l´ ∧
    -      NoDupA eq l´ ∧
    -      (∀ x, InA eq x l ↔ InA eq x l´)).
    -  Proof.
    -    intro l.
    - -
    -    suff : (∀ x : X.t, InA eq x (sort l) ↔ InA eq x l) ∧
    -           Sorted le (sort l). {
    -      
    -      unfold remove_dups_by_sortingA.
    -      move : (remove_dups_from_sortedA_spec (sort l)).
    -      simpl.
    -      move ⇒ H_spec [H_in_sort H_sorted_sort].
    -      move : (H_spec H_sorted_sort).
    -      move ⇒ [H1] [H2] H3.
    -      split ⇒ //.
    -      split ⇒ //.
    -      move ⇒ x.
    -      rewrite -H_in_sort H3 //.
    -    }
    -
    -    split. {
    -      have H_in_sort : ∀ x, List.In x (XSort.sort l) ↔ List.In x l. {
    -        intros x.
    -        have H_perm := (XSort.Permuted_sort l).
    -        split; apply Permutation_in ⇒ //.
    -        by apply Permutation_sym.
    -      }
    -
    -      intros x.
    -      rewrite !InA_alt.
    -      setoid_rewrite H_in_sort ⇒ //.
    -    } {
    -
    -      move : (LocallySorted_sort l).
    -      rewrite /is_true /le /leb //.
    -    }
    -  Qed.
    - +      Sorted lt l' ∧
    +      NoDupA eq l' ∧
    +      (∀ x, InA eq x l ↔ InA eq x l')).
    +
    End RemoveDupsFromSorted.
    @@ -398,7 +210,7 @@

    Library MSetsExtra.MSetListWithDups

    -

    Operations Module

    +

    Operations Module

    @@ -410,113 +222,113 @@

    Library MSetsExtra.MSetListWithDups


    -Module Ops (X:OrderedType) <: WOps X.
    +Module Ops (X:OrderedType) <: WOps X.

      Module RDFS := RemoveDupsFromSorted X.
    -  Module Import MX := OrderedTypeFacts X.
    +  Module Import MX := OrderedTypeFacts X.

      Definition elt := X.t.
    -  Definition t := list elt.
    +  Definition t := list elt.

    -  Definition empty : t := nil.
    +  Definition empty : t := nil.

    -  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.
    -  Fixpoint mem (x : elt) (s : t) : bool :=
    +  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.
    +  Fixpoint mem (x : elt) (s : t) : bool :=
        match s with
    -    | nil ⇒ false
    -    | y :: l ⇒
    +    | nil ⇒ false
    +    | y :: l ⇒
               match X.compare x y with
    -               Eq ⇒ true
    +               Eq ⇒ true
                 | _ ⇒ mem x l
               end
        end.

    -  Definition add x (s : t) := x :: s.
    -  Definition singleton (x : elt) := x :: nil.
    +  Definition add x (s : t) := x :: s.
    +  Definition singleton (x : elt) := x :: nil.

    -  Fixpoint rev_filter_aux acc (f : elt → bool) s :=
    +  Fixpoint rev_filter_aux acc (f : elt → bool) s :=
        match s with
    -       nil ⇒ acc
    -     | (x :: xs) ⇒ rev_filter_aux (if (f x) then (x :: acc) else acc) f xs
    +       nil ⇒ acc
    +     | (x :: xs) ⇒ rev_filter_aux (if (f x) then (x :: acc) else acc) f xs
        end.
    -  Definition rev_filter := rev_filter_aux nil.
    +  Definition rev_filter := rev_filter_aux nil.

    -  Definition filter (f : elt → bool) (s : t) : t := rev_filter f s.
    +  Definition filter (f : elt → bool) (s : t) : t := rev_filter f s.

      Definition remove x s :=
    -    rev_filter (fun y ⇒ match X.compare x y with Eq ⇒ false | _ ⇒ true end) s.
    +    rev_filter (fun y ⇒ match X.compare x y with Eq ⇒ false | _ ⇒ true end) s.

      Definition union (s1 s2 : t) : t :=
    -    List.rev_append s2 s1.
    +    List.rev_append s2 s1.

      Definition inter (s1 s2 : t) : t :=
        rev_filter (fun y ⇒ mem y s2) s1.

    -  Definition elements (x : t) : list elt := x.
    +  Definition elements (x : t) : list elt := x.

    -  Definition elements_dist (x : t) : list elt :=
    +  Definition elements_dist (x : t) : list elt :=
        RDFS.remove_dups_by_sortingA x.

    -  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    -    fold_left (flip f) (elements s) i.
    +  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    +    fold_left (flip f) (elements s) i.

    -  Definition diff (s s´ : t) : t := fold remove s´ s.
    +  Definition diff (s s' : t) : t := fold remove s' s.

    -  Definition subset (s s´ : t) : bool :=
    -    List.forallb (fun x ⇒ mem x s´) s.
    +  Definition subset (s s' : t) : bool :=
    +    List.forallb (fun x ⇒ mem x s') s.

    -  Definition equal (s s´ : t) : bool := andb (subset s s´) (subset s´ s).
    +  Definition equal (s s' : t) : bool := andb (subset s s') (subset s' s).

    -  Fixpoint for_all (f : elt → bool) (s : t) : bool :=
    +  Fixpoint for_all (f : elt → bool) (s : t) : bool :=
        match s with
    -    | nil ⇒ true
    -    | x :: l ⇒ if f x then for_all f l else false
    +    | nil ⇒ true
    +    | x :: l ⇒ if f x then for_all f l else false
        end.

    -  Fixpoint exists_ (f : elt → bool) (s : t) : bool :=
    +  Fixpoint exists_ (f : elt → bool) (s : t) : bool :=
        match s with
    -    | nil ⇒ false
    -    | x :: l ⇒ if f x then true else exists_ f l
    +    | nil ⇒ false
    +    | x :: l ⇒ if f x then true else exists_ f l
        end.

    -  Fixpoint partition_aux (a1 a2 : t) (f : elt → bool) (s : t) : t × t :=
    +  Fixpoint partition_aux (a1 a2 : t) (f : elt → bool) (s : t) : t × t :=
        match s with
    -    | nil ⇒ (a1, a2)
    -    | x :: l ⇒
    -        if f x then partition_aux (x :: a1) a2 f l else
    -                    partition_aux a1 (x :: a2) f l
    +    | nil ⇒ (a1, a2)
    +    | x :: l ⇒
    +        if f x then partition_aux (x :: a1) a2 f l else
    +                    partition_aux a1 (x :: a2) f l
        end.

    -  Definition partition := partition_aux nil nil.
    +  Definition partition := partition_aux nil nil.

    -  Definition cardinal (s : t) : nat := length (elements_dist s).
    +  Definition cardinal (s : t) : nat := length (elements_dist s).

    -  Definition choose (s : t) : option elt :=
    +  Definition choose (s : t) : option elt :=
         match s with
    -      | nil ⇒ None
    -      | x::_ ⇒ Some x
    +      | nil ⇒ None
    +      | x::_ ⇒ Some x
         end.

    @@ -526,7 +338,7 @@

    Library MSetsExtra.MSetListWithDups

    -

    Main Module

    +

    Main Module

    @@ -540,7 +352,7 @@

    Library MSetsExtra.MSetListWithDups

    -Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
    +Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
      Include Ops E.
      Import MX.
    @@ -548,40 +360,30 @@

    Library MSetsExtra.MSetListWithDups

    -

    Proofs of set operation specifications.

    +

    Proofs of set operation specifications.

    Logical predicates
    -  Definition In x (s : t) := SetoidList.InA E.eq x s.
    +  Definition In x (s : t) := SetoidList.InA E.eq x s.

    -  Instance In_compat : Proper (E.eq==>eq==>iff) In.
    -  Proof. repeat red. intros. rewrite H H0. auto. Qed.
    - +  Instance In_compat : Proper (E.eq==>eq==>iff) In.
    +
    -  Definition Equal s s´ := ∀ a : elt, In a s ↔ In a s´.
    -  Definition Subset s s´ := ∀ a : elt, In a s → In a s´.
    -  Definition Empty s := ∀ a : elt, ¬ In a s.
    -  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    -  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.
    +  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    +  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    +  Definition Empty s := ∀ a : elt, ¬ In a s.
    +  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    +  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.

      Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
      Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).

    -  Definition eq : t → t → Prop := Equal.
    -  Lemma eq_equiv : Equivalence eq.
    -  Proof.
    -    constructor. {
    -      done.
    -    } {
    -      by constructor; rewrite H.
    -    } {
    -      by constructor; rewrite H H0.
    -    }
    -  Qed.
    - +  Definition eq : t → t → Prop := Equal.
    +  Lemma eq_equiv : Equivalence eq.
    +

    @@ -593,376 +395,107 @@

    Library MSetsExtra.MSetListWithDups


    -  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    - -
    -  Lemma mem_spec : ∀ s x, mem x s = true ↔ In x s.
    -  Proof.
    -    induction s as [| y s´ IH]. {
    -      move ⇒ x.
    -      rewrite /= /In InA_nil.
    -      split ⇒ //.
    -    } {
    -      move ⇒ x.
    -      rewrite /= /In InA_cons.
    -      move : (MX.compare_eq_iff x y).
    -      case (E.compare x y). {
    -        tauto.
    -      } {
    -        rewrite IH; intuition; inversion H.
    -      } {
    -        rewrite IH; intuition; inversion H.
    -      }
    -    }
    - Qed.
    - -
    -  Lemma subset_spec : ∀ s s´, subset s s´ = true ↔ s[<=]s´.
    -  Proof.
    -    intros s s´.
    -    rewrite /subset forallb_forall /Subset /In.
    -    split. {
    -      move ⇒ H z /InA_alt [] x [H_z_eq] H_in.
    -      move : (H _ H_in).
    -      rewrite mem_spec.
    -      setoid_replace z with x ⇒ //.
    -    } {
    -      move ⇒ H z H_in.
    -      rewrite mem_spec.
    -      apply H, In_InA ⇒ //.
    -      apply E.eq_equiv.
    -    }
    -  Qed.
    - -
    -  Lemma equal_spec : ∀ s s´, equal s s´ = true ↔ s[=]s´.
    -  Proof.
    -    intros s s´.
    -    rewrite /Equal /equal Bool.andb_true_iff !subset_spec /Subset.
    -    split. {
    -      move ⇒ [H1 H2] a.
    -      split.
    -        - by apply H1.
    -        - by apply H2.
    -    } {
    -      move ⇒ H.
    -      split; move ⇒ a; rewrite H //.
    -    }
    -  Qed.
    - -
    -  Lemma eq_dec : ∀ x y : t, {eq x y} + {¬ eq x y}.
    -  Proof.
    -    intros x y.
    -    change ({Equal x y}+{¬Equal x y}).
    -    destruct (equal x y) eqn:H; [left|right];
    -     rewrite <- equal_spec; congruence.
    -  Qed.
    +  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).

    -  Lemma empty_spec : Empty empty.
    -  Proof. rewrite /Empty /empty /In. move ⇒ a /InA_nil //. Qed.
    - -
    -  Lemma is_empty_spec : ∀ s, is_empty s = true ↔ Empty s.
    -  Proof.
    -    rewrite /is_empty /Empty /In.
    -    case; split ⇒ //. {
    -      move ⇒ _ a.
    -      rewrite InA_nil //.
    -    } {
    -      move ⇒ H; contradiction (H a).
    -      apply InA_cons_hd.
    -      apply Equivalence_Reflexive.
    -    }
    -  Qed.
    - -
    -  Lemma add_spec : ∀ s x y, In y (add x s) ↔ E.eq y x ∨ In y s.
    -  Proof.
    -    intros s x y.
    -    rewrite /add /In InA_cons //.
    -  Qed.
    - -
    -  Lemma singleton_spec : ∀ x y, In y (singleton x) ↔ E.eq y x.
    -  Proof.
    -    intros x y.
    -    rewrite /singleton /In InA_cons.
    -    split. {
    -      move ⇒ [] // /InA_nil //.
    -    } {
    -      by left.
    -    }
    -  Qed.
    - -
    -  Hint Resolve (@Equivalence_Reflexive _ _ E.eq_equiv).
    -  Hint Immediate (@Equivalence_Symmetric _ _ E.eq_equiv).
    -  Hint Resolve (@Equivalence_Transitive _ _ E.eq_equiv).
    - -
    -  Lemma rev_filter_aux_spec : ∀ s acc x f, compatb f →
    -    (In x (rev_filter_aux acc f s) ↔ (In x s ∧ f x = true) ∨ (In x acc)).
    -  Proof.
    -    intros s acc x f H_compat.
    -    move : x acc.
    -    induction s as [| y s´ IH]. {
    -      intros x acc.
    -      rewrite /rev_filter_aux /In InA_nil.
    -      tauto.
    -    } {
    -      intros x acc.
    -      rewrite /= IH /In.
    -      case_eq (f y) ⇒ H_fy; rewrite !InA_cons; intuition. {
    -        left.
    -        split; first by left.
    -        setoid_replace x with y ⇒ //.
    -      } {
    -        contradict H1.
    -        setoid_replace x with y ⇒ //.
    -        by rewrite H_fy.
    -      }
    -    }
    -  Qed.
    - -
    -  Lemma filter_spec : ∀ s x f, compatb f →
    -    (In x (filter f s) ↔ In x s ∧ f x = true).
    -  Proof.
    -    intros s x f H_compat.
    -    rewrite /filter /rev_filter rev_filter_aux_spec /In InA_nil.
    -    tauto.
    -  Qed.
    - -
    -  Lemma remove_spec : ∀ s x y, In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    -  Proof.
    -    intros s x y.
    -    rewrite /remove /rev_filter.
    -    have H_compat : compatb ((fun y0 : elt ⇒
    -         match E.compare x y0 with
    -         | Eq ⇒ false
    -         | _ ⇒ true
    -         end)). {
    -      repeat red; intros.
    -      setoid_replace x0 with y0 ⇒ //.
    -    }
    -    rewrite rev_filter_aux_spec /In InA_nil.
    -    have → : (E.eq y x ↔ E.eq x y). {
    -      split; move ⇒ ?; by apply Equivalence_Symmetric.
    -    }
    -    rewrite -compare_eq_iff.
    -    case (E.compare x y). {
    -      intuition.
    -    } {
    -      intuition.
    -      inversion H0.
    -    } {
    -      intuition.
    -      inversion H0.
    -    }
    -  Qed.
    - -
    -  Lemma union_spec : ∀ s s´ x, In x (union s s´) ↔ In x s ∨ In x s´.
    -  Proof.
    -    intros s s´ x.
    -    rewrite /union /In rev_append_rev InA_app_iff InA_rev; tauto.
    -  Qed.
    - -
    -  Lemma inter_spec : ∀ s s´ x, In x (inter s s´) ↔ In x s ∧ In x s´.
    -  Proof.
    -    intros s s´ x.
    -    have H_compat : compatb (fun y : elt ⇒ mem y s´). {
    -      repeat red; intros.
    -      suff : ( mem x0 s´ = true ↔ mem y s´ = true). {
    -        case (mem y s´), (mem x0 s´); intuition.
    -      }
    -      rewrite !mem_spec /In.
    -      setoid_replace x0 with y ⇒ //.
    -    }
    -    rewrite /inter rev_filter_aux_spec mem_spec /In InA_nil.
    -    tauto.
    -  Qed.
    - -
    -  Lemma fold_spec : ∀ s (A : Type) (i : A) (f : elt → A → A),
    -    fold f s i = fold_left (flip f) (elements s) i.
    -  Proof. done. Qed.
    - -
    -  Lemma elements_spec1 : ∀ s x, InA E.eq x (elements s) ↔ In x s.
    -  Proof.
    -    intros s x.
    -    rewrite /elements /In //.
    -  Qed.
    - -
    -  Lemma diff_spec : ∀ s s´ x, In x (diff s s´) ↔ In x s ∧ ¬In x s´.
    -  Proof.
    -    intros s s´ x.
    -    rewrite /diff fold_spec -(elements_spec1 s´).
    -    move : s.
    -    induction (elements s´) as [| y ys IH] ⇒ s. {
    -      rewrite InA_nil /=; tauto.
    -    } {
    -      rewrite /= IH InA_cons /flip remove_spec.
    -      tauto.
    -    }
    -  Qed.
    - -
    -  Lemma cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    -  Proof. rewrite /cardinal //. Qed.
    - -
    -  Lemma for_all_spec : ∀ s f, compatb f →
    -    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    -  Proof.
    -    intros s f H_compat.
    -    rewrite /For_all.
    -    induction s as [| x xs IH]. {
    -      rewrite /= /In.
    -      split ⇒ //.
    -      move ⇒ _ x /InA_nil //.
    -    } {
    -      rewrite /=.
    -      case_eq (f x) ⇒ H_fx. {
    -        rewrite IH.
    -        split. {
    -          move ⇒ H x´ /InA_cons []. {
    -            move ⇒ → //.
    -          } {
    -            apply H.
    -          }
    -        } {
    -          move ⇒ H x´ H_in.
    -          apply H.
    -          apply InA_cons.
    -          by right.
    -        }
    -      } {
    -        split ⇒ //.
    -        move ⇒ H.
    -        suff : f x = true. {
    -          rewrite H_fx //.
    -        }
    -        apply H.
    -        apply InA_cons_hd.
    -        done.
    -      }
    -    }
    -  Qed.
    - -
    -  Lemma exists_spec : ∀ s f, compatb f →
    -    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    -  Proof.
    -    intros s f H_compat.
    -    rewrite /Exists.
    -    induction s as [| x xs IH]. {
    -      rewrite /= /In.
    -      split ⇒ //.
    -      move ⇒ [x] [] /InA_nil //.
    -    } {
    -      rewrite /=.
    -      case_eq (f x) ⇒ H_fx. {
    -        split ⇒ // _.
    -        âˆƒ x.
    -        split ⇒ //.
    -        apply InA_cons_hd.
    -        done.
    -      } {
    -        rewrite IH.
    -        split. {
    -          move ⇒ [x´] [H_in] H_fx´.
    -          âˆƒ x´.
    -          split ⇒ //.
    -          apply InA_cons.
    -          by right.
    -        } {
    -          move ⇒ [x´] [] /InA_cons []. {
    -            move ⇒ →.
    -            rewrite H_fx //.
    -          } {
    -            by ∃ x´.
    -          }
    -        }
    -      }
    -    }
    -  Qed.
    - +  Lemma mem_spec : ∀ s x, mem x s = true ↔ In x s.
    +
    -  Lemma partition_aux_spec : ∀ a1 a2 s f,
    -    (partition_aux a1 a2 f s = (rev_filter_aux a1 f s, rev_filter_aux a2 (fun x ⇒ negb (f x)) s)).
    -  Proof.
    -    move ⇒ a1 a2 s f.
    -    move : a1 a2.
    -    induction s as [| x xs IH]. {
    -      rewrite /partition_aux /rev_filter_aux //.
    -    } {
    -      intros a1 a2.
    -      rewrite /= IH.
    -      case (f x) ⇒ //.
    - +  Lemma subset_spec : ∀ s s', subset s s' = true ↔ s[<=]s'.
    + +
    +  Lemma equal_spec : ∀ s s', equal s s' = true ↔ s[=]s'.
    +
    -    }
    -  Qed.
    +  Lemma eq_dec : ∀ x y : t, {eq x y} + {¬ eq x y}.

    -  Lemma partition_spec1 : ∀ s f, compatb f →
    -    fst (partition f s) [=] filter f s.
    -  Proof.
    -    move ⇒ s f _.
    -    rewrite /partition partition_aux_spec /fst /filter /rev_filter //.
    -  Qed.
    +  Lemma empty_spec : Empty empty.
    + +
    +  Lemma is_empty_spec : ∀ s, is_empty s = true ↔ Empty s.
    + +
    +  Lemma add_spec : ∀ s x y, In y (add x s) ↔ E.eq y x ∨ In y s.
    + +
    +  Lemma singleton_spec : ∀ x y, In y (singleton x) ↔ E.eq y x.
    + +
    +  Hint Resolve (@Equivalence_Reflexive _ _ E.eq_equiv).
    +  Hint Immediate (@Equivalence_Symmetric _ _ E.eq_equiv).
    +  Hint Resolve (@Equivalence_Transitive _ _ E.eq_equiv).

    -  Lemma partition_spec2 : ∀ s f, compatb f →
    -    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    -  Proof.
    -    move ⇒ s f _.
    -    rewrite /partition partition_aux_spec /snd /filter /rev_filter //.
    -  Qed.
    +  Lemma rev_filter_aux_spec : ∀ s acc x f, compatb f →
    +    (In x (rev_filter_aux acc f s) ↔ (In x s ∧ f x = true) ∨ (In x acc)).
    + +
    +  Lemma filter_spec : ∀ s x f, compatb f →
    +    (In x (filter f s) ↔ In x s ∧ f x = true).
    + +
    +  Lemma remove_spec : ∀ s x y, In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    + +
    +  Lemma union_spec : ∀ s s' x, In x (union s s') ↔ In x s ∨ In x s'.

    -  Lemma choose_spec1 : ∀ s x, choose s = Some x → In x s.
    -  Proof.
    -   move ⇒ [] // y s´ x [->].
    -   rewrite /In.
    -   apply InA_cons_hd.
    -   apply Equivalence_Reflexive.
    -  Qed.
    +  Lemma inter_spec : ∀ s s' x, In x (inter s s') ↔ In x s ∧ In x s'.

    -  Lemma choose_spec2 : ∀ s, choose s = None → Empty s.
    -  Proof. move ⇒ [] // _ a. rewrite /In InA_nil //. Qed.
    +  Lemma fold_spec : ∀ s (A : Type) (i : A) (f : elt → A → A),
    +    fold f s i = fold_left (flip f) (elements s) i.
    + +
    +  Lemma elements_spec1 : ∀ s x, InA E.eq x (elements s) ↔ In x s.
    +
    +  Lemma diff_spec : ∀ s s' x, In x (diff s s') ↔ In x s ∧ ¬In x s'.
    + +
    +  Lemma cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    + +
    +  Lemma for_all_spec : ∀ s f, compatb f →
    +    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    + +
    +  Lemma exists_spec : ∀ s f, compatb f →
    +    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    + +
    +  Lemma partition_aux_spec : ∀ a1 a2 s f,
    +    (partition_aux a1 a2 f s = (rev_filter_aux a1 f s, rev_filter_aux a2 (fun x ⇒ negb (f x)) s)).
    + +
    +  Lemma partition_spec1 : ∀ s f, compatb f →
    +    fst (partition f s) [=] filter f s.
    + +
    +  Lemma partition_spec2 : ∀ s f, compatb f →
    +    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    + +
    +  Lemma choose_spec1 : ∀ s x, choose s = Some x → In x s.
    + +
    +  Lemma choose_spec2 : ∀ s, choose s = None → Empty s.
    +
      Lemma elements_dist_spec_full :
        âˆ€ s,
    -      Sorted E.lt (elements_dist s) ∧
    -      NoDupA E.eq (elements_dist s) ∧
    -      (∀ x, InA E.eq x (elements_dist s) ↔ InA E.eq x (elements s)).
    -  Proof.
    -    move ⇒ s.
    -    rewrite /elements_dist /elements.
    -    move : (RDFS.remove_dups_by_sortingA_spec s).
    -    simpl.
    -    firstorder.
    -  Qed.
    +      Sorted E.lt (elements_dist s) ∧
    +      NoDupA E.eq (elements_dist s) ∧
    +      (∀ x, InA E.eq x (elements_dist s) ↔ InA E.eq x (elements s)).

    -  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    -                                          InA E.eq x (elements s).
    -  Proof. intros; apply elements_dist_spec_full. Qed.
    - +  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    +                                          InA E.eq x (elements s).
    +
    -  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    -  Proof. intros; apply elements_dist_spec_full. Qed.
    - +  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    +
    End Make.
    diff --git a/html/MSetsExtra.MSetWithDups.html b/html/MSetsExtra.MSetWithDups.html index 95c6aeb..f4d3ac0 100644 --- a/html/MSetsExtra.MSetWithDups.html +++ b/html/MSetsExtra.MSetWithDups.html @@ -24,14 +24,14 @@

    Library MSetsExtra.MSetWithDups

    -

    Signature for weak sets which may contain duplicates

    +

    Signature for weak sets which may contain duplicates

    - The interface WSetsOn demands that elements returns a list + The interface WSetsOn demands that elements returns a list without duplicates and that the fold function iterates over this - result. Another potential problem is that the function cardinal + result. Another potential problem is that the function cardinal is supposed to return the length of the elements list.
    @@ -50,36 +50,36 @@

    Library MSetsExtra.MSetWithDups

    In order to allow efficient weak set implementations that use duplicates internally, we provide new module types in this file. There is WSetsOnWithDups, which is a proper subset of - WSetsOn. It just removes the problematic properties of elements - and cardinal. + WSetsOn. It just removes the problematic properties of elements + and cardinal.
    Since one is of course interested in specifying the cardinality and in computing a list of elements without duplicates, there is also an extension WSetsOnWithDupsExtra of WSetsOnWithDups. This - extension introduces a new operation elements_dist, which is a - version of elements without duplicates. This allows to - specify cardinality with respect to elements_dist. + extension introduces a new operation elements_dist, which is a + version of elements without duplicates. This allows to + specify cardinality with respect to elements_dist.

    -Require Import Coq.MSets.MSetInterface.
    -Require Import ssreflect.
    +Require Import Coq.MSets.MSetInterface.
    +Require Import mathcomp.ssreflect.ssreflect.

    -

    WSetsOnWithDups

    +

    WSetsOnWithDups

    The module type WSetOnWithDups is a proper subset of WSetsOn; - the problematic parameters cardinal_spec and elements_spec2w + the problematic parameters cardinal_spec and elements_spec2w are missing.
    @@ -91,68 +91,68 @@

    Library MSetsExtra.MSetWithDups

    -Module Type WSetsOnWithDups (E : DecidableType).
    -  Include WOps E.
    +Module Type WSetsOnWithDups (E : DecidableType).
    +  Include WOps E.

    -  Parameter In : elt → t → Prop.
    -  Declare Instance In_compat : Proper (E.eq==>eq==>iff) In.
    +  Parameter In : elt → t → Prop.
    +  Declare Instance In_compat : Proper (E.eq==>eq==>iff) In.

    -  Definition Equal s s´ := ∀ a : elt, In a s ↔ In a s´.
    -  Definition Subset s s´ := ∀ a : elt, In a s → In a s´.
    -  Definition Empty s := ∀ a : elt, ¬ In a s.
    -  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    -  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.
    +  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    +  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    +  Definition Empty s := ∀ a : elt, ¬ In a s.
    +  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    +  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.

      Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
      Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).

    -  Definition eq : t → t → Prop := Equal.
    -  Include IsEq.
    +  Definition eq : t → t → Prop := Equal.
    +  Include IsEq.
    -eq is obviously an equivalence, for subtyping only +eq is obviously an equivalence, for subtyping only
    -  Include HasEqDec.
    +  Include HasEqDec.

      Section Spec.
    -  Variable s s´: t.
    +  Variable s s': t.
      Variable x y : elt.
    -  Variable f : elt → bool.
    -  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    +  Variable f : elt → bool.
    +  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).

    -  Parameter mem_spec : mem x s = true ↔ In x s.
    -  Parameter equal_spec : equal s s´ = true ↔ s[=]s´.
    -  Parameter subset_spec : subset s s´ = true ↔ s[<=]s´.
    +  Parameter mem_spec : mem x s = true ↔ In x s.
    +  Parameter equal_spec : equal s s' = true ↔ s[=]s'.
    +  Parameter subset_spec : subset s s' = true ↔ s[<=]s'.
      Parameter empty_spec : Empty empty.
    -  Parameter is_empty_spec : is_empty s = true ↔ Empty s.
    -  Parameter add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    -  Parameter remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    -  Parameter singleton_spec : In y (singleton x) ↔ E.eq y x.
    -  Parameter union_spec : In x (union s s´) ↔ In x s ∨ In x s´.
    -  Parameter inter_spec : In x (inter s s´) ↔ In x s ∧ In x s´.
    -  Parameter diff_spec : In x (diff s s´) ↔ In x s ∧ ¬In x s´.
    -  Parameter fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    -    fold f s i = fold_left (flip f) (elements s) i.
    -  Parameter filter_spec : compatb f →
    -    (In x (filter f s) ↔ In x s ∧ f x = true).
    -  Parameter for_all_spec : compatb f →
    -    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    -  Parameter exists_spec : compatb f →
    -    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    -  Parameter partition_spec1 : compatb f →
    -    fst (partition f s) [=] filter f s.
    -  Parameter partition_spec2 : compatb f →
    -    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    -  Parameter elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    -  Parameter choose_spec1 : choose s = Some x → In x s.
    -  Parameter choose_spec2 : choose s = None → Empty s.
    +  Parameter is_empty_spec : is_empty s = true ↔ Empty s.
    +  Parameter add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    +  Parameter remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    +  Parameter singleton_spec : In y (singleton x) ↔ E.eq y x.
    +  Parameter union_spec : In x (union s s') ↔ In x s ∨ In x s'.
    +  Parameter inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
    +  Parameter diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +  Parameter fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    +    fold f s i = fold_left (flip f) (elements s) i.
    +  Parameter filter_spec : compatb f →
    +    (In x (filter f s) ↔ In x s ∧ f x = true).
    +  Parameter for_all_spec : compatb f →
    +    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +  Parameter exists_spec : compatb f →
    +    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +  Parameter partition_spec1 : compatb f →
    +    fst (partition f s) [=] filter f s.
    +  Parameter partition_spec2 : compatb f →
    +    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    +  Parameter elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    +  Parameter choose_spec1 : choose s = Some x → In x s.
    +  Parameter choose_spec2 : choose s = None → Empty s.

      End Spec.
    @@ -164,17 +164,17 @@

    Library MSetsExtra.MSetWithDups

    -

    WSetsOnWithDupsExtra

    +

    WSetsOnWithDupsExtra

    - WSetsOnWithDupsExtra introduces elements_dist in order to + WSetsOnWithDupsExtra introduces elements_dist in order to specify cardinality and in order to get an operation similar to - the original behavior of elements. + the original behavior of elements.
    -Module Type WSetsOnWithDupsExtra (E : DecidableType).
    +Module Type WSetsOnWithDupsExtra (E : DecidableType).
      Include WSetsOnWithDups E.

    @@ -184,30 +184,30 @@

    Library MSetsExtra.MSetWithDups

    An operation for getting an elements list without duplicates
    -  Parameter elements_dist : t → list elt.
    +  Parameter elements_dist : t → list elt.

    -  Parameter elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    -                                              InA E.eq x (elements s).
    +  Parameter elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    +                                              InA E.eq x (elements s).

    -  Parameter elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    +  Parameter elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).

    -Cardinality can then be specified with respect to elements_dist. +Cardinality can then be specified with respect to elements_dist.
    -  Parameter cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    +  Parameter cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    End WSetsOnWithDupsExtra.

    -

    WSetOn to WSetsOnWithDupsExtra

    +

    WSetOn to WSetsOnWithDupsExtra

    @@ -219,7 +219,7 @@

    Library MSetsExtra.MSetWithDups


    -Module WSetsOn_TO_WSetsOnWithDupsExtra (E : DecidableType) (W : WSetsOn E) <:
    +Module WSetsOn_TO_WSetsOnWithDupsExtra (E : DecidableType) (W : WSetsOn E) <:
      WSetsOnWithDupsExtra E.

    @@ -229,14 +229,12 @@

    Library MSetsExtra.MSetWithDups

      Definition elements_dist := W.elements.

    -  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    -                                          InA E.eq x (elements s).
    -  Proof. done. Qed.
    - +  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    +                                          InA E.eq x (elements s).
    +
    -  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    -  Proof. apply elements_spec2w. Qed.
    - +  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    +
    End WSetsOn_TO_WSetsOnWithDupsExtra.
    diff --git a/html/index.html b/html/index.html index 8080f7b..51c254e 100644 --- a/html/index.html +++ b/html/index.html @@ -47,7 +47,7 @@ Z _ other -(658 entries) +(596 entries) Notation Index @@ -111,7 +111,7 @@ Z _ other -(58 entries) +(43 entries) Variable Index @@ -175,7 +175,7 @@ Z _ other -(5 entries) +(4 entries) Lemma Index @@ -207,7 +207,7 @@ Z _ other -(273 entries) +(257 entries) Axiom Index @@ -239,7 +239,7 @@ Z _ other -(34 entries) +(32 entries) Constructor Index @@ -303,7 +303,7 @@ Z _ other -(11 entries) +(7 entries) Inductive Index @@ -463,7 +463,7 @@ Z _ other -(222 entries) +(199 entries) Record Index @@ -495,7 +495,7 @@ Z _ other -(4 entries) +(3 entries)
    @@ -532,118 +532,63 @@

    Global Index

    ElementEncode.encode_lt [axiom, in MSetsExtra.MSetIntervals]
    ElementEncode.encode_eq [axiom, in MSetsExtra.MSetIntervals]


    F

    -foldWithAbortGtLtSpecPred [definition, in MSetsExtra.MSetIterator]
    foldWithAbortGtLtSpecPred [definition, in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortGtLtType [definition, in MSetsExtra.MSetIterator]
    foldWithAbortGtLtType [definition, in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortGtSpecPred [definition, in MSetsExtra.MSetIterator]
    foldWithAbortGtSpecPred [definition, in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortGtType [definition, in MSetsExtra.MSetIterator]
    foldWithAbortGtType [definition, in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortPrecomputeSpecPred [definition, in MSetsExtra.MSetIterator]
    foldWithAbortPrecomputeSpecPred [definition, in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortPrecomputeType [definition, in MSetsExtra.MSetIterator]
    foldWithAbortPrecomputeType [definition, in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortSpecPred [definition, in MSetsExtra.MSetIterator]
    foldWithAbortSpecPred [definition, in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortType [definition, in MSetsExtra.MSetIterator]
    foldWithAbortType [definition, in MSetsExtra.MSetFoldWithAbort]


    H

    -HasFoldWithAbort [module, in MSetsExtra.MSetIterator]
    HasFoldWithAbort [module, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps [module, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps [module, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.choose_with_abort_spec [lemma, in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.choose_with_abort [definition, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.choose_with_abort_spec [lemma, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.choose_with_abort [definition, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.exists_with_abort_spec [lemma, in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.exists_with_abort [definition, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.exists_with_abort_spec [lemma, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.exists_with_abort [definition, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.filter_with_abort_spec [lemma, in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.filter_with_abort [definition, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.filter_with_abort_spec [lemma, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.filter_with_abort [definition, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbort [definition, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbort [definition, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGt [definition, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGt [definition, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtLt [definition, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtLt [definition, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtLtSpec [lemma, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtLtSpec [lemma, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal [lemma, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal [lemma, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtSpec [lemma, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtSpec [lemma, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtSpec_Equal [lemma, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtSpec_Equal [lemma, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal [lemma, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal [lemma, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortSpec [lemma, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortSpec [lemma, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.FoldWithAbortSpecArg [record, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.FoldWithAbortSpecArg [record, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortSpecArgsForPred [definition, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortSpecArgsForPred [definition, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortSpec_Equal [lemma, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortSpec_Equal [lemma, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.forall_with_abort_spec [lemma, in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.forall_with_abort [definition, in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.forall_with_abort_spec [lemma, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.forall_with_abort [definition, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.fwasa_P' [projection, in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.fwasa_f_gt [projection, in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.fwasa_f_lt [projection, in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.fwasa_f_pre [projection, in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.fwasa_P' [projection, in MSetsExtra.MSetFoldWithAbort]
    +HasFoldWithAbortOps.fwasa_P' [projection, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.fwasa_f_gt [projection, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.fwasa_f_lt [projection, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.fwasa_f_pre [projection, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbort.foldWithAbortPrecompute [axiom, in MSetsExtra.MSetIterator]
    HasFoldWithAbort.foldWithAbortPrecompute [axiom, in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbort.foldWithAbortPrecomputeSpec [axiom, in MSetsExtra.MSetIterator]
    HasFoldWithAbort.foldWithAbortPrecomputeSpec [axiom, in MSetsExtra.MSetFoldWithAbort]


    M

    Make [module, in MSetsExtra.MSetListWithDups]
    -MakeAVLSetsWithFoldA [module, in MSetsExtra.MSetIterator]
    MakeAVLSetsWithFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    -MakeAVLSetsWithFoldA.foldWithAbortPrecompute [definition, in MSetsExtra.MSetIterator]
    MakeAVLSetsWithFoldA.foldWithAbortPrecompute [definition, in MSetsExtra.MSetFoldWithAbort]
    -MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec [lemma, in MSetsExtra.MSetIterator]
    MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec [lemma, in MSetsExtra.MSetFoldWithAbort]
    -MakeGenTreeFoldA [module, in MSetsExtra.MSetIterator]
    MakeGenTreeFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    -MakeGenTreeFoldA.foldWithAbort_RawSpec [lemma, in MSetsExtra.MSetIterator]
    -MakeGenTreeFoldA.foldWithAbort_Raw [definition, in MSetsExtra.MSetIterator]
    MakeGenTreeFoldA.foldWithAbort_RawSpec [lemma, in MSetsExtra.MSetFoldWithAbort]
    MakeGenTreeFoldA.foldWithAbort_Raw [definition, in MSetsExtra.MSetFoldWithAbort]
    -MakeListSetsWithFoldA [module, in MSetsExtra.MSetIterator]
    MakeListSetsWithFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    -MakeListSetsWithFoldA.foldWithAbortPrecompute [definition, in MSetsExtra.MSetIterator]
    MakeListSetsWithFoldA.foldWithAbortPrecompute [definition, in MSetsExtra.MSetFoldWithAbort]
    -MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec [lemma, in MSetsExtra.MSetIterator]
    MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec [lemma, in MSetsExtra.MSetFoldWithAbort]
    -MakeListSetsWithFoldA.foldWithAbortRaw [definition, in MSetsExtra.MSetIterator]
    MakeListSetsWithFoldA.foldWithAbortRaw [definition, in MSetsExtra.MSetFoldWithAbort]
    -MakeRBTSetsWithFoldA [module, in MSetsExtra.MSetIterator]
    MakeRBTSetsWithFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    -MakeRBTSetsWithFoldA.foldWithAbortPrecompute [definition, in MSetsExtra.MSetIterator]
    MakeRBTSetsWithFoldA.foldWithAbortPrecompute [definition, in MSetsExtra.MSetFoldWithAbort]
    -MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec [lemma, in MSetsExtra.MSetIterator]
    MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec [lemma, in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA [module, in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.E [module, in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.E [module, in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.foldWithAbortPrecompute [definition, in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.foldWithAbortPrecompute [definition, in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec [lemma, in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec [lemma, in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.foldWithAbortRaw [definition, in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.foldWithAbortRaw [definition, in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.Raw [module, in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.Raw [module, in MSetsExtra.MSetFoldWithAbort]
    Make.add_spec [lemma, in MSetsExtra.MSetListWithDups]
    Make.cardinal_spec [lemma, in MSetsExtra.MSetListWithDups]
    @@ -760,7 +705,7 @@

    Global Index

    MSetIntervals.Spec [section, in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.f [variable, in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.s [variable, in MSetsExtra.MSetIntervals]
    -MSetIntervals.Spec.s' [variable, in MSetsExtra.MSetIntervals]
    +MSetIntervals.Spec.s' [variable, in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.x [variable, in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.y [variable, in MSetsExtra.MSetIntervals]
    MSetIntervals.subset [definition, in MSetsExtra.MSetIntervals]
    @@ -771,7 +716,6 @@

    Global Index

    MSetIntervals.t_ [record, in MSetsExtra.MSetIntervals]
    MSetIntervals.union [definition, in MSetsExtra.MSetIntervals]
    MSetIntervals.union_spec [lemma, in MSetsExtra.MSetIntervals]
    -MSetIterator [library]
    MSetListWithDups [library]
    MSetWithDups [library]


    N

    @@ -928,7 +872,7 @@

    Global Index

    Raw.elements_nil [lemma, in MSetsExtra.MSetIntervals]
    Raw.Empty [definition, in MSetsExtra.MSetIntervals]
    Raw.empty_spec [lemma, in MSetsExtra.MSetIntervals]
    -Raw.empty_spec' [lemma, in MSetsExtra.MSetIntervals]
    +Raw.empty_spec' [lemma, in MSetsExtra.MSetIntervals]
    Raw.empty_ok [instance, in MSetsExtra.MSetIntervals]
    Raw.encode_decode_eq [lemma, in MSetsExtra.MSetIntervals]
    Raw.Equal [definition, in MSetsExtra.MSetIntervals]
    @@ -1104,9 +1048,7 @@

    Global Index

    rev_map [definition, in MSetsExtra.MSetIntervals]
    rev_map_aux [definition, in MSetsExtra.MSetIntervals]


    S

    -SetsWithFoldA [module, in MSetsExtra.MSetIterator]
    SetsWithFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    -SetsWithFoldA.E [module, in MSetsExtra.MSetIterator]
    SetsWithFoldA.E [module, in MSetsExtra.MSetFoldWithAbort]


    W

    WSetsOnWithDups [module, in MSetsExtra.MSetWithDups]
    @@ -1144,7 +1086,7 @@

    Global Index

    WSetsOnWithDups.Spec [section, in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Spec.f [variable, in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Spec.s [variable, in MSetsExtra.MSetWithDups]
    -WSetsOnWithDups.Spec.s' [variable, in MSetsExtra.MSetWithDups]
    +WSetsOnWithDups.Spec.s' [variable, in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Spec.x [variable, in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Spec.y [variable, in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Subset [definition, in MSetsExtra.MSetWithDups]
    @@ -1156,13 +1098,9 @@

    Global Index

    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1 [lemma, in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist [definition, in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra [module, in MSetsExtra.MSetWithDups]
    -WSetsWithDupsFoldA [module, in MSetsExtra.MSetIterator]
    WSetsWithDupsFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    -WSetsWithDupsFoldA.E [module, in MSetsExtra.MSetIterator]
    WSetsWithDupsFoldA.E [module, in MSetsExtra.MSetFoldWithAbort]
    -WSetsWithFoldA [module, in MSetsExtra.MSetIterator]
    WSetsWithFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    -WSetsWithFoldA.E [module, in MSetsExtra.MSetIterator]
    WSetsWithFoldA.E [module, in MSetsExtra.MSetFoldWithAbort]


    Z

    Z_to_N_minus_neq_0 [lemma, in MSetsExtra.MSetIntervals]
    @@ -1191,25 +1129,16 @@

    Module Index

    ElementEncodeZ.E [in MSetsExtra.MSetIntervals]
    ElementEncode.E [in MSetsExtra.MSetIntervals]


    H

    -HasFoldWithAbort [in MSetsExtra.MSetIterator]
    HasFoldWithAbort [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps [in MSetsExtra.MSetFoldWithAbort]


    M

    Make [in MSetsExtra.MSetListWithDups]
    -MakeAVLSetsWithFoldA [in MSetsExtra.MSetIterator]
    MakeAVLSetsWithFoldA [in MSetsExtra.MSetFoldWithAbort]
    -MakeGenTreeFoldA [in MSetsExtra.MSetIterator]
    MakeGenTreeFoldA [in MSetsExtra.MSetFoldWithAbort]
    -MakeListSetsWithFoldA [in MSetsExtra.MSetIterator]
    MakeListSetsWithFoldA [in MSetsExtra.MSetFoldWithAbort]
    -MakeRBTSetsWithFoldA [in MSetsExtra.MSetIterator]
    MakeRBTSetsWithFoldA [in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA [in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA [in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.E [in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.E [in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.Raw [in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.Raw [in MSetsExtra.MSetFoldWithAbort]
    MSetIntervals [in MSetsExtra.MSetIntervals]
    MSetIntervalsN [in MSetsExtra.MSetIntervals]
    @@ -1231,34 +1160,28 @@

    Module Index

    RemoveDupsFromSorted.XSort [in MSetsExtra.MSetListWithDups]
    RemoveDupsFromSorted.XTotalLeBool [in MSetsExtra.MSetListWithDups]


    S

    -SetsWithFoldA [in MSetsExtra.MSetIterator]
    SetsWithFoldA [in MSetsExtra.MSetFoldWithAbort]
    -SetsWithFoldA.E [in MSetsExtra.MSetIterator]
    SetsWithFoldA.E [in MSetsExtra.MSetFoldWithAbort]


    W

    WSetsOnWithDups [in MSetsExtra.MSetWithDups]
    WSetsOnWithDupsExtra [in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra [in MSetsExtra.MSetWithDups]
    -WSetsWithDupsFoldA [in MSetsExtra.MSetIterator]
    WSetsWithDupsFoldA [in MSetsExtra.MSetFoldWithAbort]
    -WSetsWithDupsFoldA.E [in MSetsExtra.MSetIterator]
    WSetsWithDupsFoldA.E [in MSetsExtra.MSetFoldWithAbort]
    -WSetsWithFoldA [in MSetsExtra.MSetIterator]
    WSetsWithFoldA [in MSetsExtra.MSetFoldWithAbort]
    -WSetsWithFoldA.E [in MSetsExtra.MSetIterator]
    WSetsWithFoldA.E [in MSetsExtra.MSetFoldWithAbort]


    Variable Index

    M

    MSetIntervals.Spec.f [in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.s [in MSetsExtra.MSetIntervals]
    -MSetIntervals.Spec.s' [in MSetsExtra.MSetIntervals]
    +MSetIntervals.Spec.s' [in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.x [in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.y [in MSetsExtra.MSetIntervals]


    W

    WSetsOnWithDups.Spec.f [in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Spec.s [in MSetsExtra.MSetWithDups]
    -WSetsOnWithDups.Spec.s' [in MSetsExtra.MSetWithDups]
    +WSetsOnWithDups.Spec.s' [in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Spec.x [in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Spec.y [in MSetsExtra.MSetWithDups]


    @@ -1266,7 +1189,6 @@

    Library Index

    M

    MSetFoldWithAbort
    MSetIntervals
    -MSetIterator
    MSetListWithDups
    MSetWithDups


    @@ -1284,38 +1206,22 @@

    Lemma Index

    ElementEncodeZ.encode_lt [in MSetsExtra.MSetIntervals]
    ElementEncodeZ.encode_eq [in MSetsExtra.MSetIntervals]


    H

    -HasFoldWithAbortOps.choose_with_abort_spec [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.choose_with_abort_spec [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.exists_with_abort_spec [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.exists_with_abort_spec [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.filter_with_abort_spec [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.filter_with_abort_spec [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtLtSpec [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtLtSpec [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtSpec [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtSpec [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtSpec_Equal [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtSpec_Equal [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortSpec [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortSpec [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortSpec_Equal [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortSpec_Equal [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.forall_with_abort_spec [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.forall_with_abort_spec [in MSetsExtra.MSetFoldWithAbort]


    M

    -MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetIterator]
    MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetFoldWithAbort]
    -MakeGenTreeFoldA.foldWithAbort_RawSpec [in MSetsExtra.MSetIterator]
    MakeGenTreeFoldA.foldWithAbort_RawSpec [in MSetsExtra.MSetFoldWithAbort]
    -MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetIterator]
    MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetFoldWithAbort]
    -MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetIterator]
    MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetFoldWithAbort]
    Make.add_spec [in MSetsExtra.MSetListWithDups]
    Make.cardinal_spec [in MSetsExtra.MSetListWithDups]
    @@ -1421,7 +1327,7 @@

    Lemma Index

    Raw.elements_cons [in MSetsExtra.MSetIntervals]
    Raw.elements_nil [in MSetsExtra.MSetIntervals]
    Raw.empty_spec [in MSetsExtra.MSetIntervals]
    -Raw.empty_spec' [in MSetsExtra.MSetIntervals]
    +Raw.empty_spec' [in MSetsExtra.MSetIntervals]
    Raw.encode_decode_eq [in MSetsExtra.MSetIntervals]
    Raw.equal_spec [in MSetsExtra.MSetIntervals]
    Raw.equal_elementsZ [in MSetsExtra.MSetIntervals]
    @@ -1562,9 +1468,7 @@

    Axiom Index

    ElementEncode.encode_lt [in MSetsExtra.MSetIntervals]
    ElementEncode.encode_eq [in MSetsExtra.MSetIntervals]


    H

    -HasFoldWithAbort.foldWithAbortPrecompute [in MSetsExtra.MSetIterator]
    HasFoldWithAbort.foldWithAbortPrecompute [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbort.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetIterator]
    HasFoldWithAbort.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetFoldWithAbort]


    W

    WSetsOnWithDupsExtra.cardinal_spec [in MSetsExtra.MSetWithDups]
    @@ -1611,11 +1515,7 @@

    Constructor Index



    Projection Index

    H

    -HasFoldWithAbortOps.fwasa_P' [in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.fwasa_f_gt [in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.fwasa_f_lt [in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.fwasa_f_pre [in MSetsExtra.MSetIterator]
    -HasFoldWithAbortOps.fwasa_P' [in MSetsExtra.MSetFoldWithAbort]
    +HasFoldWithAbortOps.fwasa_P' [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.fwasa_f_gt [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.fwasa_f_lt [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.fwasa_f_pre [in MSetsExtra.MSetFoldWithAbort]
    @@ -1677,53 +1577,30 @@

    Definition Index

    ElementEncodeZ.decode [in MSetsExtra.MSetIntervals]
    ElementEncodeZ.encode [in MSetsExtra.MSetIntervals]


    F

    -foldWithAbortGtLtSpecPred [in MSetsExtra.MSetIterator]
    foldWithAbortGtLtSpecPred [in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortGtLtType [in MSetsExtra.MSetIterator]
    foldWithAbortGtLtType [in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortGtSpecPred [in MSetsExtra.MSetIterator]
    foldWithAbortGtSpecPred [in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortGtType [in MSetsExtra.MSetIterator]
    foldWithAbortGtType [in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortPrecomputeSpecPred [in MSetsExtra.MSetIterator]
    foldWithAbortPrecomputeSpecPred [in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortPrecomputeType [in MSetsExtra.MSetIterator]
    foldWithAbortPrecomputeType [in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortSpecPred [in MSetsExtra.MSetIterator]
    foldWithAbortSpecPred [in MSetsExtra.MSetFoldWithAbort]
    -foldWithAbortType [in MSetsExtra.MSetIterator]
    foldWithAbortType [in MSetsExtra.MSetFoldWithAbort]


    H

    -HasFoldWithAbortOps.choose_with_abort [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.choose_with_abort [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.exists_with_abort [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.exists_with_abort [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.filter_with_abort [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.filter_with_abort [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbort [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbort [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGt [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGt [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortGtLt [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortGtLt [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.foldWithAbortSpecArgsForPred [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.foldWithAbortSpecArgsForPred [in MSetsExtra.MSetFoldWithAbort]
    -HasFoldWithAbortOps.forall_with_abort [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.forall_with_abort [in MSetsExtra.MSetFoldWithAbort]


    M

    -MakeAVLSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetIterator]
    MakeAVLSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetFoldWithAbort]
    -MakeGenTreeFoldA.foldWithAbort_Raw [in MSetsExtra.MSetIterator]
    MakeGenTreeFoldA.foldWithAbort_Raw [in MSetsExtra.MSetFoldWithAbort]
    -MakeListSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetIterator]
    MakeListSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetFoldWithAbort]
    -MakeListSetsWithFoldA.foldWithAbortRaw [in MSetsExtra.MSetIterator]
    MakeListSetsWithFoldA.foldWithAbortRaw [in MSetsExtra.MSetFoldWithAbort]
    -MakeRBTSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetIterator]
    MakeRBTSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetFoldWithAbort]
    -MakeWeakListSetsWithFoldA.foldWithAbortRaw [in MSetsExtra.MSetIterator]
    MakeWeakListSetsWithFoldA.foldWithAbortRaw [in MSetsExtra.MSetFoldWithAbort]
    Make.Empty [in MSetsExtra.MSetListWithDups]
    Make.eq [in MSetsExtra.MSetListWithDups]
    @@ -1901,7 +1778,6 @@

    Definition Index



    Record Index

    H

    -HasFoldWithAbortOps.FoldWithAbortSpecArg [in MSetsExtra.MSetIterator]
    HasFoldWithAbortOps.FoldWithAbortSpecArg [in MSetsExtra.MSetFoldWithAbort]


    M

    MSetIntervals.t_ [in MSetsExtra.MSetIntervals]
    @@ -1938,7 +1814,7 @@

    Record Index

    Z _ other -(658 entries) +(596 entries) Notation Index @@ -2002,7 +1878,7 @@

    Record Index

    Z _ other -(58 entries) +(43 entries) Variable Index @@ -2066,7 +1942,7 @@

    Record Index

    Z _ other -(5 entries) +(4 entries) Lemma Index @@ -2098,7 +1974,7 @@

    Record Index

    Z _ other -(273 entries) +(257 entries) Axiom Index @@ -2130,7 +2006,7 @@

    Record Index

    Z _ other -(34 entries) +(32 entries) Constructor Index @@ -2194,7 +2070,7 @@

    Record Index

    Z _ other -(11 entries) +(7 entries) Inductive Index @@ -2354,7 +2230,7 @@

    Record Index

    Z _ other -(222 entries) +(199 entries) Record Index @@ -2386,7 +2262,7 @@

    Record Index

    Z _ other -(4 entries) +(3 entries)
    This page has been generated by coqdoc diff --git a/html/toc.html b/html/toc.html index d253a00..cb185ba 100644 --- a/html/toc.html +++ b/html/toc.html @@ -17,98 +17,61 @@
    -

    Library MSetsExtra.MSetWithDups

     - -

    Library MSetsExtra.MSetListWithDups

     - -

    Library MSetsExtra.MSetIterator

     +

    Library MSetsExtra.MSetFoldWithAbort
    :=
    -    (elt → A → A) →
    +    (elt → A → A) →
    f
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_abort
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value @@ -110,8 +110,8 @@

    Library MSetsExtra.MSetFoldWithAbort


    Definition foldWithAbortSpecPred {elt t : Type}
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
        (foldWithAbort : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=

        âˆ€
    @@ -131,7 +131,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → A → A) (f' : elt → A → A)
    +      (f : elt → A → A) (f' : elt → A → A)
          
    @@ -139,7 +139,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_abort : elt → A → bool)
    +      (f_abort : elt → A → bool)
          
    @@ -154,7 +154,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    equivalence relation on results @@ -168,7 +168,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →
          
        
    @@ -176,7 +176,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →
    +    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →

        
    @@ -184,7 +184,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e st) (f' e st))) →
    +    (∀ e st, In e s → (P (f e st) (f' e st))) →

        
    @@ -193,11 +193,11 @@

    Library MSetsExtra.MSetFoldWithAbort

    leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_abort e1 st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → e2 ≠ e1 →
    -                        P st (f e2 st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_abort e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → e2 ≠ e1 →
    +                        P st (f e2 st'))) →

        
    @@ -205,7 +205,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i' →
    +    P i i' →

        
    @@ -248,25 +248,25 @@

    Library MSetsExtra.MSetFoldWithAbort

    :=
    -    (elt → A → A) →
    +    (elt → A → A) →
    f
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_gt
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value @@ -276,9 +276,9 @@

    Library MSetsExtra.MSetFoldWithAbort


    Definition foldWithAbortGtSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (lt : elt → elt → Prop)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
        (foldWithAbortGt : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=

        âˆ€
    @@ -298,7 +298,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → A → A) (f' : elt → A → A)
    +      (f : elt → A → A) (f' : elt → A → A)
          
    @@ -306,7 +306,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_gt : elt → A → bool)
    +      (f_gt : elt → A → bool)
          
    @@ -321,7 +321,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    equivalence relation on results @@ -336,7 +336,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →

        
    @@ -344,7 +344,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →
    +    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →

        
    @@ -352,7 +352,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e st) (f' e st))) →
    +    (∀ e st, In e s → (P (f e st) (f' e st))) →

        
    @@ -361,11 +361,11 @@

    Library MSetsExtra.MSetFoldWithAbort

    leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_gt e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e1 e2 →
    +                        P st (f e2 st'))) →

        
    @@ -373,7 +373,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i' →
    +    P i i' →

        
    @@ -424,31 +424,31 @@

    Library MSetsExtra.MSetFoldWithAbort

    :=
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_lt
    -    (elt → A → A) →
    +    (elt → A → A) →
    f
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_gt
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value @@ -458,9 +458,9 @@

    Library MSetsExtra.MSetFoldWithAbort


    Definition foldWithAbortGtLtSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (lt : elt → elt → Prop)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
        (foldWithAbortGtLt : ∀ {A : Type}, foldWithAbortGtLtType elt t A) : Prop :=

        âˆ€
    @@ -480,7 +480,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → A → A) (f' : elt → A → A)
    +      (f : elt → A → A) (f' : elt → A → A)
          
    @@ -488,7 +488,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_lt f_gt : elt → A → bool)
    +      (f_lt f_gt : elt → A → bool)
          
    @@ -503,7 +503,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    equivalence relation on results @@ -518,7 +518,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →

        
    @@ -526,7 +526,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →
    +    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →

        
    @@ -534,7 +534,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e st) (f' e st))) →
    +    (∀ e st, In e s → (P (f e st) (f' e st))) →

        
    @@ -543,11 +543,11 @@

    Library MSetsExtra.MSetFoldWithAbort

    leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_lt e1 st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e2 e1 →
    -                        P st (f e2 st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_lt e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e2 e1 →
    +                        P st (f e2 st'))) →

        
    @@ -556,11 +556,11 @@

    Library MSetsExtra.MSetFoldWithAbort

    leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_gt e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e1 e2 →
    +                        P st (f e2 st'))) →

        
    @@ -568,7 +568,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i' →
    +    P i i' →

        
    @@ -625,37 +625,37 @@

    Library MSetsExtra.MSetFoldWithAbort

    :=

    -    (elt → B) →
    +    (elt → B) →
    f_precompute
    -    (elt → B → A → bool) →
    +    (elt → B → A → bool) →
    f_lt
    -    (elt → B → A → A) →
    +    (elt → B → A → A) →
    f
    -    (elt → B → A → bool) →
    +    (elt → B → A → bool) →
    f_gt
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value @@ -672,9 +672,9 @@

    Library MSetsExtra.MSetFoldWithAbort

    Definition foldWithAbortPrecomputeSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (lt : elt → elt → Prop)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
        (foldWithAbortPrecompute : ∀ {A B : Type}, foldWithAbortPrecomputeType elt t A B) : Prop :=

        âˆ€
    @@ -694,7 +694,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_precompute : elt → B)
    +      (f_precompute : elt → B)
          
    @@ -702,7 +702,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → B → A → A) (f' : elt → A → A)
    +      (f : elt → B → A → A) (f' : elt → A → A)
          
    @@ -710,7 +710,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_lt f_gt : elt → B → A → bool)
    +      (f_lt f_gt : elt → B → A → bool)
          
    @@ -725,7 +725,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    equivalence relation on results @@ -740,7 +740,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →

        
    @@ -748,7 +748,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st' e, In e s → P st st' → P (f e (f_precompute e) st) (f e (f_precompute e) st')) →
    +    (∀ st st' e, In e s → P st st' → P (f e (f_precompute e) st) (f e (f_precompute e) st')) →

        
    @@ -756,7 +756,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e (f_precompute e) st) (f' e st))) →
    +    (∀ e st, In e s → (P (f e (f_precompute e) st) (f' e st))) →

        
    @@ -765,11 +765,11 @@

    Library MSetsExtra.MSetFoldWithAbort

    leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_lt e1 (f_precompute e1) st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e2 e1 →
    -                        P st (f e2 (f_precompute e2) st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_lt e1 (f_precompute e1) st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e2 e1 →
    +                        P st (f e2 (f_precompute e2) st'))) →

        
    @@ -778,11 +778,11 @@

    Library MSetsExtra.MSetFoldWithAbort

    leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 (f_precompute e1) st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 (f_precompute e2) st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_gt e1 (f_precompute e1) st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e1 e2 →
    +                        P st (f e2 (f_precompute e2) st'))) →

        
    @@ -790,7 +790,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i' →
    +    P i i' →

        
    @@ -815,15 +815,15 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Module Type HasFoldWithAbort (E : OrderedType) (Import C : WSetsOnWithDups E).
    +Module Type HasFoldWithAbort (E : OrderedType) (Import C : WSetsOnWithDups E).

      Parameter foldWithAbortPrecompute : ∀ {A B : Type},
    -    foldWithAbortPrecomputeType elt t A B.
    +    foldWithAbortPrecomputeType elt t A B.

      Parameter foldWithAbortPrecomputeSpec :
    -     foldWithAbortPrecomputeSpecPred E.lt In (@fold) (@foldWithAbortPrecompute).
    +     foldWithAbortPrecomputeSpecPred E.lt In (@fold) (@foldWithAbortPrecompute).

    End HasFoldWithAbort.
    @@ -844,7 +844,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Module HasFoldWithAbortOps (E : OrderedType) (C : WSetsOnWithDups E)
    +Module HasFoldWithAbortOps (E : OrderedType) (C : WSetsOnWithDups E)
                               (FT : HasFoldWithAbort E C).

    @@ -861,31 +861,60 @@

    Library MSetsExtra.MSetFoldWithAbort


    -  Definition foldWithAbortGtLt {A} f_lt (f : (elt → A → A)) f_gt :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_lt e st)
    +  Definition foldWithAbortGtLt {A} f_lt (f : (elt → A → A)) f_gt :=
    +    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_lt e st)
          (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).

      Lemma foldWithAbortGtLtSpec :
    -     foldWithAbortGtLtSpecPred E.lt In (@fold) (@foldWithAbortGtLt).
    +     foldWithAbortGtLtSpecPred E.lt In (@fold) (@foldWithAbortGtLt).
    +  Proof.
    +    rewrite /foldWithAbortGtLt /foldWithAbortGtLtSpecPred.
    +    intros A i i' f f' f_lt f_gt s P.
    +    move ⇒ H_f_compat H_ff' H_lt H_gt H_ii'.
    +    apply foldWithAbortPrecomputeSpec ⇒ //.
    +  Qed.

    -  Definition foldWithAbortGt {A} (f : (elt → A → A)) f_gt :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun _ _ _ ⇒ false)
    +  Definition foldWithAbortGt {A} (f : (elt → A → A)) f_gt :=
    +    foldWithAbortPrecompute (fun _ ⇒ tt) (fun _ _ _ ⇒ false)
          (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).

      Lemma foldWithAbortGtSpec :
    -     foldWithAbortGtSpecPred E.lt In (@fold) (@foldWithAbortGt).
    +     foldWithAbortGtSpecPred E.lt In (@fold) (@foldWithAbortGt).
    +  Proof.
    +    rewrite /foldWithAbortGt /foldWithAbortGtSpecPred.
    +    intros A i i' f f' f_gt s P.
    +    move ⇒ H_f_compat H_ff' H_gt H_ii'.
    +    apply foldWithAbortPrecomputeSpec ⇒ //.
    +  Qed.

    -  Definition foldWithAbort {A} (f : (elt → A → A)) f_abort :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_abort e st)
    +  Definition foldWithAbort {A} (f : (elt → A → A)) f_abort :=
    +    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_abort e st)
          (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_abort e st).

      Lemma foldWithAbortSpec :
    -     foldWithAbortSpecPred In (@fold) (@foldWithAbort).
    +     foldWithAbortSpecPred In (@fold) (@foldWithAbort).
    +  Proof.
    +    rewrite /foldWithAbort /foldWithAbortGtSpecPred.
    +    intros A i i' f f' f_abort s P.
    +    move ⇒ H_equiv_P H_f_compat H_ff' H_abort H_ii'.
    +    have H_lt_neq: (∀ e1 e2, E.lt e1 e2 → e1 ≠ e2). {
    +      move ⇒ e1 e2 H_lt H_e12_eq.
    +      rewrite H_e12_eq in H_lt.
    +      have : ( Irreflexive E.lt) by apply StrictOrder_Irreflexive.
    +      rewrite /Irreflexive /Reflexive /complement ⇒ H.
    +      eapply H, H_lt.
    +    }
    +    apply foldWithAbortPrecomputeSpec ⇒ //; (
    +      move ⇒ e1 st H_in_e1 H_abort_e1 st' e2 H_P H_in_e2 /H_lt_neq H_lt;
    +      apply (H_abort e1 st H_in_e1 H_abort_e1 st' e2 H_P H_in_e2);
    +      by auto
    +    ).
    +  Qed.

    @@ -899,73 +928,137 @@

    Library MSetsExtra.MSetFoldWithAbort

    of an arbitrary equivalence relation on results.
    -  Lemma foldWithAbortPrecomputeSpec_Equal : ∀ (A B : Type) (i : A) (f_pre : elt → B)
    -      (f : elt → B → A → A) (f' : elt → A → A) (f_lt f_gt : elt → B → A → bool) (s : t),
    +  Lemma foldWithAbortPrecomputeSpec_Equal : ∀ (A B : Type) (i : A) (f_pre : elt → B)
    +      (f : elt → B → A → A) (f' : elt → A → A) (f_lt f_gt : elt → B → A → bool) (s : t),

    -      (∀ e st, In e s → (f e (f_pre e) st = f' e st)) →
    +      (∀ e st, In e s → (f e (f_pre e) st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_lt e1 (f_pre e1) st = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 →
    -                      (f e2 (f_pre e2) st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_lt e1 (f_pre e1) st = true →
    +          (∀ e2, In e2 s → E.lt e2 e1 →
    +                      (f e2 (f_pre e2) st = st))) →

          
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 (f_pre e1) st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 (f_pre e2) st = st))) →
    -
    -      (foldWithAbortPrecompute f_pre f_lt f f_gt s i) = (fold f' s i).
    - +      (∀ e1 st,
    +          In e1 s → f_gt e1 (f_pre e1) st = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 →
    +                      (f e2 (f_pre e2) st = st))) →
    +
    +      (foldWithAbortPrecompute f_pre f_lt f f_gt s i) = (fold f' s i).
    +  Proof.
    +    intros A B i f_pre f f' f_lt f_gt s H_f' H_lt H_gt.
    + +
    +     eapply (foldWithAbortPrecomputeSpec A B i i f_pre f f'); eauto. {
    +       apply eq_equivalence.
    +     } {
    +       move ⇒ st1 st2 e H_in → //.
    +     } {
    +       move ⇒ e1 st H_e1_in H_do_smaller st' e2 <-.
    +       move : (H_lt e1 st H_e1_in H_do_smaller e2).
    +       intuition.
    +     } {
    +       move ⇒ e1 st H_e1_in H_do_greater st' e2 <-.
    +       move : (H_gt e1 st H_e1_in H_do_greater e2).
    +       intuition.
    +     }
    +  Qed.
    +
      Lemma foldWithAbortGtLtSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f' : elt → A → A) (f_lt f_gt : elt → A → bool) (s : t),
    +      (f : elt → A → A) (f' : elt → A → A) (f_lt f_gt : elt → A → bool) (s : t),

    -      (∀ e st, In e s → (f e st = f' e st)) →
    +      (∀ e st, In e s → (f e st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_lt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_lt e1 st = true →
    +          (∀ e2, In e2 s → E.lt e2 e1 →
    +                      (f e2 st = st))) →

          
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 st = st))) →
    -
    -      (foldWithAbortGtLt f_lt f f_gt s i) = (fold f' s i).
    - +      (∀ e1 st,
    +          In e1 s → f_gt e1 st = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 →
    +                      (f e2 st = st))) →
    +
    +      (foldWithAbortGtLt f_lt f f_gt s i) = (fold f' s i).
    +  Proof.
    +    intros A i f f' f_lt f_gt s H_f' H_lt H_gt.
    + +
    +     eapply (foldWithAbortGtLtSpec A i i f f'); eauto. {
    +       apply eq_equivalence.
    +     } {
    +       move ⇒ st1 st2 e H_in → //.
    +     } {
    +       move ⇒ e1 st H_e1_in H_do_smaller st' e2 <-.
    +       move : (H_lt e1 st H_e1_in H_do_smaller e2).
    +       intuition.
    +     } {
    +       move ⇒ e1 st H_e1_in H_do_greater st' e2 <-.
    +       move : (H_gt e1 st H_e1_in H_do_greater e2).
    +       intuition.
    +     }
    +  Qed.
    +
      Lemma foldWithAbortGtSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f' : elt → A → A) (f_gt : elt → A → bool) (s : t),
    +      (f : elt → A → A) (f' : elt → A → A) (f_gt : elt → A → bool) (s : t),

    -      (∀ e st, In e s → (f e st = f' e st)) →
    +      (∀ e st, In e s → (f e st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_gt e1 st = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 →
    +                      (f e2 st = st))) →

    -      (foldWithAbortGt f f_gt s i) = (fold f' s i).
    - +      (foldWithAbortGt f f_gt s i) = (fold f' s i).
    +  Proof.
    +    intros A i f f' f_gt s H_f' H_gt.
    + +
    +     eapply (foldWithAbortGtSpec A i i f f'); eauto. {
    +       apply eq_equivalence.
    +     } {
    +       move ⇒ st1 st2 e H_in → //.
    +     } {
    +       move ⇒ e1 st H_e1_in H_do_greater st' e2 <-.
    +       move : (H_gt e1 st H_e1_in H_do_greater e2).
    +       intuition.
    +     }
    +  Qed.
    +
      Lemma foldWithAbortSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f' : elt → A → A) (f_abort : elt → A → bool) (s : t),
    +      (f : elt → A → A) (f' : elt → A → A) (f_abort : elt → A → bool) (s : t),

    -      (∀ e st, In e s → (f e st = f' e st)) →
    +      (∀ e st, In e s → (f e st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_abort e1 st = true →
    -          (∀ e2, In e2 s → e1 ≠ e2 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_abort e1 st = true →
    +          (∀ e2, In e2 s → e1 ≠ e2 →
    +                      (f e2 st = st))) →

    -      (foldWithAbort f f_abort s i) = (fold f' s i).
    - +      (foldWithAbort f f_abort s i) = (fold f' s i).
    +  Proof.
    +     intros A i f f' f_abort s H_f' H_abort.
    + +
    +     eapply (foldWithAbortSpec A i i f f'); eauto. {
    +       apply eq_equivalence.
    +     } {
    +       move ⇒ st1 st2 e H_in → //.
    +     } {
    +       move ⇒ e1 st H_e1_in H_do_abort st' e2 <-.
    +       move : (H_abort e1 st H_e1_in H_do_abort e2).
    +       intuition.
    +     }
    +  Qed.
    +
    @@ -989,25 +1082,25 @@

    Library MSetsExtra.MSetFoldWithAbort

      Record FoldWithAbortSpecArg {B} := {
    -     fwasa_f_pre : (elt → B);
    +     fwasa_f_pre : (elt → B);
    The precompute function
    -     fwasa_f_lt : (elt → B → bool);
    +     fwasa_f_lt : (elt → B → bool);
    f_lt without state argument
    -     fwasa_f_gt : (elt → B → bool);
    +     fwasa_f_gt : (elt → B → bool);
    f_gt without state argument
    -     fwasa_P' : (elt → B → bool)
    +     fwasa_P' : (elt → B → bool)
    the predicate P @@ -1024,7 +1117,7 @@

    Library MSetsExtra.MSetFoldWithAbort

      Definition foldWithAbortSpecArgsForPred {A : Type}
    -     (s : t) (P : elt → bool) (fwasa : @FoldWithAbortSpecArg A) :=
    +     (s : t) (P : elt → bool) (fwasa : @FoldWithAbortSpecArg A) :=

         
    @@ -1032,7 +1125,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    the predicate P' coincides for s and the given precomputation with P
    -     (∀ e, In e s → (fwasa_P' fwasa e (fwasa_f_pre fwasa e) = P e)) ∧
    +     (∀ e, In e s → (fwasa_P' fwasa e (fwasa_f_pre fwasa e) = P e)) ∧

         
    @@ -1041,9 +1134,9 @@

    Library MSetsExtra.MSetFoldWithAbort

    don't satisfy predicate P.
    -     (∀ e1,
    -          In e1 s → fwasa_f_lt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 → (P e2 = false))) ∧
    +     (∀ e1,
    +          In e1 s → fwasa_f_lt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    +          (∀ e2, In e2 s → E.lt e2 e1 → (P e2 = false))) ∧

         
    @@ -1052,9 +1145,9 @@

    Library MSetsExtra.MSetFoldWithAbort

    don't satisfy predicate P.
    -     (∀ e1,
    -          In e1 s → fwasa_f_gt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 → (P e2 = false))).
    +     (∀ e1,
    +          In e1 s → fwasa_f_gt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 → (P e2 = false))).

    @@ -1068,16 +1161,76 @@

    Library MSetsExtra.MSetFoldWithAbort

      Definition filter_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
         @foldWithAbortPrecompute t B (fwasa_f_pre fwasa) (fun e p _ ⇒ fwasa_f_lt fwasa e p)
    -       (fun e e_pre s ⇒ if fwasa_P' fwasa e e_pre then add e s else s)
    +       (fun e e_pre s ⇒ if fwasa_P' fwasa e e_pre then add e s else s)
           (fun e p _ ⇒ fwasa_f_gt fwasa e p) s empty.

      Lemma filter_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
        Equal (filter_with_abort fwasa s)
              (filter P s).
    - +  Proof.
    +    unfold foldWithAbortSpecArgsForPred.
    +    move ⇒ [] f_pre f_lt f_gt P' P s /=.
    +    move ⇒ [H_f'] [H_lt] H_gt H_proper.
    +    rewrite /filter_with_abort /=.
    + +
    +    have → : (foldWithAbortPrecompute f_pre (fun e p _ ⇒ f_lt e p)
    +     (fun (e : elt) (e_pre : B) (s0 : t) ⇒
    +      if P' e e_pre then add e s0 else s0) (fun e p _ ⇒ f_gt e p) s empty =
    +     (fold (fun e s0 ⇒ if P e then add e s0 else s0) s empty)). {
    +      apply foldWithAbortPrecomputeSpec_Equal. {
    +        intros e st H_e_in.
    +        rewrite H_f' //.
    +      } {
    +        intros e1 st H_e1_in H_f_lt_eq e2 H_e2_in H_lt_e2_e1.
    +        rewrite (H_f' _ H_e2_in).
    +        suff → : (P e2 = false) by done.
    +        apply (H_lt e1); eauto.
    +      } {
    +        intros e1 st H_e1_in H_f_gt_eq e2 H_e2_in H_gt_e2_e1.
    +        rewrite (H_f' _ H_e2_in).
    +        suff → : (P e2 = false) by done.
    +        apply (H_gt e1); eauto.
    +      }
    +    }
    +
    +    rewrite /Equal ⇒ e.
    +    rewrite fold_spec.
    +    setoid_rewrite filter_spec ⇒ //.
    + +
    +    suff → : ∀ acc, In e
    +      (fold_left
    +         (flip (fun (e0 : elt) (s0 : t) ⇒ if P e0 then add e0 s0 else s0))
    +         (elements s) acc) ↔ (InA E.eq e (elements s) ∧ P e = true) ∨ (In e acc). {
    +      rewrite elements_spec1.
    +      suff : (¬In e empty) by tauto.
    +      apply empty_spec.
    +    }
    +    induction (elements s) as [| x xs IH] ⇒ acc. {
    +      rewrite /= InA_nil. tauto.
    +    } {
    +      rewrite /= /flip IH InA_cons.
    +      case_eq (P x). {
    +        rewrite add_spec.
    +        intuition.
    +        left.
    +        rewrite H0.
    +        split ⇒ //.
    +        left.
    +        apply Equivalence_Reflexive.
    +      } {
    +        intuition.
    +        contradict H2.
    +        setoid_rewrite H1.
    +        by rewrite H.
    +      }
    +    }
    +  Qed.
    +
    @@ -1088,22 +1241,112 @@

    Library MSetsExtra.MSetFoldWithAbort

      Definition choose_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
         foldWithAbortPrecompute (fwasa_f_pre fwasa)
    -       (fun e p st ⇒ match st with None ⇒ (fwasa_f_lt fwasa e p) | _ ⇒ true end)
    -       (fun e e_pre st ⇒ match st with None ⇒
    -          if (fwasa_P' fwasa e e_pre) then Some e else None | _ ⇒ st end)
    +       (fun e p st ⇒ match st with None ⇒ (fwasa_f_lt fwasa e p) | _ ⇒ true end)
    +       (fun e e_pre st ⇒ match st with None ⇒
    +          if (fwasa_P' fwasa e e_pre) then Some e else None | _ ⇒ st end)

    -       (fun e p st ⇒ match st with None ⇒ (fwasa_f_gt fwasa e p) | _ ⇒ true end)
    -       s None.
    +       (fun e p st ⇒ match st with None ⇒ (fwasa_f_gt fwasa e p) | _ ⇒ true end)
    +       s None.

      Lemma choose_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (match (choose_with_abort fwasa s) with
    -       | None ⇒ (∀ e, In e s → P e = false)
    -       | Some e ⇒ In e s ∧ (P e = true)
    -     end).
    - +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    (match (choose_with_abort fwasa s) with
    +       | None ⇒ (∀ e, In e s → P e = false)
    +       | Some e ⇒ In e s ∧ (P e = true)
    +     end).
    +  Proof.
    +    rewrite /foldWithAbortSpecArgsForPred.
    +    move ⇒ [] f_pre f_lt f_gt P' P s /=.
    +    move ⇒ [H_f'] [H_lt] H_gt H_proper.
    + +
    +    set fwasa := {|
    +       fwasa_f_pre := f_pre;
    +       fwasa_f_lt := f_lt;
    +       fwasa_f_gt := f_gt;
    +       fwasa_P' := P' |}.
    + +
    +    suff : (match (choose_with_abort fwasa s) with
    +       | None ⇒ (∀ e, InA E.eq e (elements s) → P e = false)
    +       | Some e ⇒ InA E.eq e (elements s) ∧ (P e = true)
    +     end). {
    +       case (choose_with_abort fwasa s). {
    +         move ⇒ e.
    +         rewrite elements_spec1 //.
    +       } {
    +         move ⇒ H e H_in.
    +         apply H.
    +         rewrite elements_spec1 //.
    +       }
    +    }
    +
    +    have → : (choose_with_abort fwasa s =
    +      (fold (fun e st ⇒
    +        match st with
    +          | None ⇒ if P e then Some e else None
    +          | _ ⇒ st end) s None)). {
    +      apply foldWithAbortPrecomputeSpec_Equal. {
    +        intros e st H_e_in.
    +        case st ⇒ //=.
    +        rewrite H_f' //.
    +      } {
    +        move ⇒ e1 [] //= H_e1_in H_f_lt_eq e2 H_e2_in H_lt_e2_e1.
    +        rewrite (H_f' _ H_e2_in).
    +        case_eq (P e2) ⇒ // H_P_e2.
    +        contradict H_P_e2.
    +        apply not_true_iff_false, (H_lt e1); auto.
    +      } {
    +        move ⇒ e1 [] //= H_e1_in H_f_gt_eq e2 H_e2_in H_gt_e2_e1.
    +        rewrite (H_f' _ H_e2_in).
    +        case_eq (P e2) ⇒ // H_P_e2.
    +        contradict H_P_e2.
    +        apply not_true_iff_false, (H_gt e1); auto.
    +      }
    +    }
    +
    +    rewrite fold_spec /flip.
    +    induction (elements s) as [| x xs IH]. {
    +      rewrite /=.
    +      move ⇒ e /InA_nil //.
    +    } {
    +      case_eq (P x) ⇒ H_Px; rewrite /= H_Px. {
    +        have → : ∀ xs, fold_left (fun (x0 : option elt) (y : elt) ⇒
    +                    match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    +                    end) xs (Some x) = Some x. {
    +          move ⇒ ys.
    +          induction ys ⇒ //.
    +        }
    +        split; last assumption.
    +        apply InA_cons_hd.
    +        apply E.eq_equiv.
    +      } {
    +        move : IH.
    +        case (fold_left
    +          (fun (x0 : option elt) (y : elt) ⇒
    +             match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    +             end) xs None). {
    +
    +             move ⇒ e [H_e_in] H_Pe.
    +             split; last assumption.
    +             apply InA_cons_tl ⇒ //.
    +        } {
    +          move ⇒ H_e_nin e H_e_in.
    +          have : (InA E.eq e xs ∨ (E.eq e x)). {
    +            inversion H_e_in; tauto.
    +          }
    +          move ⇒ []. {
    +            apply H_e_nin.
    +          } {
    +            move ⇒ → //.
    +          }
    +        }
    +      }
    +    }
    +  Qed.
    +
    @@ -1114,17 +1357,35 @@

    Library MSetsExtra.MSetFoldWithAbort

      Definition exists_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
        match choose_with_abort fwasa s with
    -      | None ⇒ false
    -      | Some _ ⇒ true
    +      | None ⇒ false
    +      | Some _ ⇒ true
        end.

      Lemma exists_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (exists_with_abort fwasa s =
    -     exists_ P s).
    - +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    (exists_with_abort fwasa s =
    +     exists_ P s).
    +  Proof.
    +    intros fwasa P s H_fwasa H_proper.
    +    apply Logic.eq_sym.
    +    rewrite /exists_with_abort.
    +    move : (choose_with_abort_spec _ _ _ H_fwasa H_proper).
    +    case (choose_with_abort fwasa s). {
    +      move ⇒ e [H_e_in] H_Pe.
    +      rewrite exists_spec /Exists.
    +      by ∃ e.
    +    } {
    +      move ⇒ H_not_ex.
    +      apply not_true_iff_false.
    +      rewrite exists_spec /Exists.
    +      move ⇒ [e] [H_in] H_pe.
    +      move : (H_not_ex e H_in).
    +      rewrite H_pe //.
    +    }
    +  Qed.
    +
    @@ -1133,15 +1394,38 @@

    Library MSetsExtra.MSetFoldWithAbort

      Definition forall_with_abort {B} fwasa s :=
    -     negb (@exists_with_abort B fwasa s).
    +     negb (@exists_with_abort B fwasa s).

      Lemma forall_with_abort_spec {B} : ∀ fwasa s P,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (forall_with_abort fwasa s =
    -     for_all (fun e ⇒ negb (P e)) s).
    - +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    (forall_with_abort fwasa s =
    +     for_all (fun e ⇒ negb (P e)) s).
    +  Proof.
    +    intros fwasa s P H_ok H_proper.
    +    rewrite /forall_with_abort exists_with_abort_spec; auto.
    + +
    +    rewrite eq_iff_eq_true negb_true_iff -not_true_iff_false.
    +    rewrite exists_spec.
    +    setoid_rewrite for_all_spec; last solve_proper.
    + +
    +    rewrite /Exists /For_all.
    +    split. {
    +      move ⇒ H_pre x H_x_in.
    +      rewrite negb_true_iff -not_true_iff_false ⇒ H_Px.
    +      apply H_pre.
    +      by ∃ x.
    +    } {
    +      move ⇒ H_pre [x] [H_x_in] H_P_x.
    +      move : (H_pre x H_x_in).
    +      rewrite H_P_x.
    +      done.
    +    }
    +  Qed.
    +
    End HasFoldWithAbortOps.
    @@ -1154,24 +1438,24 @@

    Library MSetsExtra.MSetFoldWithAbort

    Module Type WSetsWithDupsFoldA.
    -  Declare Module E : OrderedType.
    +  Declare Module E : OrderedType.
      Include WSetsOnWithDups E.
      Include HasFoldWithAbort E.
      Include HasFoldWithAbortOps E.
    End WSetsWithDupsFoldA.

    -Module Type WSetsWithFoldA <: WSets.
    -  Declare Module E : OrderedType.
    -  Include WSetsOn E.
    +Module Type WSetsWithFoldA <: WSets.
    +  Declare Module E : OrderedType.
    +  Include WSetsOn E.
      Include HasFoldWithAbort E.
      Include HasFoldWithAbortOps E.
    End WSetsWithFoldA.

    -Module Type SetsWithFoldA <: Sets.
    -  Declare Module E : OrderedType.
    -  Include SetsOn E.
    +Module Type SetsWithFoldA <: Sets.
    +  Declare Module E : OrderedType.
    +  Include SetsOn E.
      Include HasFoldWithAbort E.
      Include HasFoldWithAbortOps E.
    End SetsWithFoldA.
    @@ -1189,49 +1473,211 @@

    Library MSetsExtra.MSetFoldWithAbort

    Finally, provide such a fold with abort operation for generic trees.
    -Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
    -  (Import Raw:Ops E I)
    -  (M : MSetGenTree.Props E I Raw).
    +Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
    +  (Import Raw:Ops E I)
    +  (M : MSetGenTree.Props E I Raw).

    -  Fixpoint foldWithAbort_Raw {A B: Type} (f_pre : E.t → B) f_lt (f: E.t → B → A → A) f_gt t (base: A) : A :=
    +  Fixpoint foldWithAbort_Raw {A B: Type} (f_pre : E.t → B) f_lt (f: E.t → B → A → A) f_gt t (base: A) : A :=
        match t with
        | Raw.Leaf ⇒ base
        | Raw.Node _ l x r ⇒
    -        let x_pre := f_pre x in
    -        let st0 := if f_lt x x_pre base then base else foldWithAbort_Raw f_pre f_lt f f_gt l base in
    -        let st1 := f x x_pre st0 in
    -        let st2 := if f_gt x x_pre st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1 in
    +        let x_pre := f_pre x in
    +        let st0 := if f_lt x x_pre base then base else foldWithAbort_Raw f_pre f_lt f f_gt l base in
    +        let st1 := f x x_pre st0 in
    +        let st2 := if f_gt x x_pre st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1 in
            st2
        end.

    -  Lemma foldWithAbort_RawSpec : ∀ (A B : Type) (i i' : A) (f_pre : E.t → B)
    -      (f : E.t → B → A → A) (f' : E.t → A → A) (f_lt f_gt : E.t → B → A → bool) (s : Raw.tree)
    -      (P : A → A → Prop),
    +  Lemma foldWithAbort_RawSpec : ∀ (A B : Type) (i i' : A) (f_pre : E.t → B)
    +      (f : E.t → B → A → A) (f' : E.t → A → A) (f_lt f_gt : E.t → B → A → bool) (s : Raw.tree)
    +      (P : A → A → Prop),

    -      (M.bst s) →
    -      Equivalence P →
    -      (∀ st st' e, M.In e s → P st st' → P (f e (f_pre e) st) (f e (f_pre e) st')) →
    -      (∀ e st, M.In e s → P (f e (f_pre e) st) (f' e st)) →
    +      (M.bst s) →
    +      Equivalence P →
    +      (∀ st st' e, M.In e s → P st st' → P (f e (f_pre e) st) (f e (f_pre e) st')) →
    +      (∀ e st, M.In e s → P (f e (f_pre e) st) (f' e st)) →

          
    -      (∀ e1 st,
    -          M.In e1 s → f_lt e1 (f_pre e1) st = true →
    -          (∀ st' e2, P st st' →
    -                          M.In e2 s → E.lt e2 e1 →
    -                          P st (f e2 (f_pre e2) st'))) →
    +      (∀ e1 st,
    +          M.In e1 s → f_lt e1 (f_pre e1) st = true →
    +          (∀ st' e2, P st st' →
    +                          M.In e2 s → E.lt e2 e1 →
    +                          P st (f e2 (f_pre e2) st'))) →

          
    -      (∀ e1 st,
    -          M.In e1 s → f_gt e1 (f_pre e1) st = true →
    -          (∀ st' e2, P st st' →
    -                          M.In e2 s → E.lt e1 e2 →
    -                          P st (f e2 (f_pre e2) st'))) →
    +      (∀ e1 st,
    +          M.In e1 s → f_gt e1 (f_pre e1) st = true →
    +          (∀ st' e2, P st st' →
    +                          M.In e2 s → E.lt e1 e2 →
    +                          P st (f e2 (f_pre e2) st'))) →

    -      P i i' →
    +      P i i' →
          P (foldWithAbort_Raw f_pre f_lt f f_gt s i) (fold f' s i').
    - End MakeGenTreeFoldA.
    +   Proof.
    +     intros A B i i' f_pre f f' f_lt f_gt s P.
    +     move ⇒ H_bst H_equiv_P H_P_f H_f' H_RL H_RG.
    + +
    +     set base := s.
    + +
    +     move : i i'.
    +     have : (∀ e, M.In e base → M.In e s). {
    +       rewrite /In /base //.
    +     }
    +     have : M.bst base. {
    +       apply H_bst.
    +     }
    +     move : base.
    +     clear H_bst.
    + +
    +     induction base as [|c l IHl e r IHr] using M.tree_ind. {
    +       rewrite /foldWithAbort_Raw /Raw.fold.
    +       move ⇒ _ _ i i' //.
    +     } {
    +       move ⇒ H_bst H_sub i i' H_P_ii'.
    + +
    +       have [H_bst_l [H_bst_r [H_lt_tree_l H_gt_tree_r]]]:
    +         M.bst l ∧ M.bst r ∧ M.lt_tree e l ∧ M.gt_tree e r. {
    +         inversion H_bst. done.
    +       }
    +       
    +       have H_sub_l : (∀ e0 : E.t, M.In e0 l → M.In e0 s ∧ E.lt e0 e). {
    +         intros e0 H_in_l.
    +         split; last by apply H_lt_tree_l.
    +         eapply H_sub.
    +         rewrite /M.In M.In_node_iff.
    +         tauto.
    +       }
    +       move : (IHl H_bst_l) ⇒ {IHl} IHl {H_bst_l} {H_lt_tree_l}.
    +       have H_sub_r : (∀ e0 : E.t, M.In e0 r → M.In e0 s ∧ E.lt e e0). {
    +         intros e0 H_in_r.
    +         split; last by apply H_gt_tree_r.
    +         eapply H_sub.
    +         rewrite /M.In M.In_node_iff.
    +         tauto.
    +       }
    +       move : (IHr H_bst_r) ⇒ {IHr} IHr {H_bst_r} {H_gt_tree_r}.
    +       have H_in_e : M.In e s. {
    +         eapply H_sub.
    +         rewrite /M.In M.In_node_iff.
    +         right; left.
    +         apply Equivalence_Reflexive.
    +       }
    +       move ⇒ {H_sub}.
    + +
    +       rewrite /=.
    +       set st0 := if f_lt e (f_pre e) i then i else foldWithAbort_Raw f_pre f_lt f f_gt l i.
    +       set st0' := Raw.fold f' l i'.
    +       set st1 := f e (f_pre e) st0.
    +       set st1' := f' e st0'.
    +       set st2 := if f_gt e (f_pre e) st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1.
    +       set st2' := Raw.fold f' r st1'.
    + +
    +       have H_P_st0 : P st0 st0'. {
    +         rewrite /st0 /st0'.
    +         case_eq (f_lt e (f_pre e) i). {
    +           move ⇒ H_fl_eq.
    +           move : H_P_ii' H_sub_l.
    +           move : H_equiv_P H_f' (H_RL _ _ H_in_e H_fl_eq).
    +           rewrite /M.lt_tree. clear.
    +           move ⇒ H_equiv_P H_f' H_RL.
    +           move : i'.
    +           induction l as [|c l IHl e' r IHr] using M.tree_ind. {
    +             done.
    +           } {
    +             intros i' H_P_ii' H_sub_l.
    +             rewrite /=.
    +             apply IHr; last first. {
    +               move ⇒ y H_y_in.
    +               apply H_sub_l.
    +               rewrite /M.In M.In_node_iff. tauto.
    +             }
    +             have [] : (M.In e' s ∧ E.lt e' e). {
    +               apply H_sub_l.
    +               rewrite /M.In M.In_node_iff.
    +               right; left.
    +               apply Equivalence_Reflexive.
    +             }
    +             move ⇒ H_e'_in H_lt_in.
    +             suff H_P_i : (P i (f e' (f_pre e') (fold f' l i'))). {
    +               eapply Equivalence_Transitive; first apply H_P_i.
    +               by apply H_f'.
    +             }
    +             eapply H_RL ⇒ //.
    +             apply IHl; last first. {
    +               move ⇒ y H_y_in.
    +               apply H_sub_l.
    +               rewrite /M.In M.In_node_iff. tauto.
    +             }
    +             assumption.
    +           }
    +         } {
    +           move ⇒ _.
    +           apply IHl ⇒ //.
    +           eapply H_sub_l.
    +         }
    +       }
    +       have H_P_st1 : P st1 st1'. {
    +         rewrite /st1 /st1'.
    +         rewrite -H_f' //.
    +         apply H_P_f ⇒ //.
    +       }
    +       have H_P_st2 : P st2 st2'. {
    +         rewrite /st2 /st2'.
    +         clearbody st1 st1'.
    +         case_eq (f_gt e (f_pre e) st1). {
    +           move ⇒ H_gt_eq.
    +           move : H_P_st1 H_sub_r.
    +           move : H_equiv_P (H_RG _ _ H_in_e H_gt_eq) H_f'.
    +           unfold M.gt_tree. clear.
    +           move ⇒ H_equiv_P H_RG H_f'.
    +           move : st1'.
    +           induction r as [|c l IHl e' r IHr] using M.tree_ind. {
    +             done.
    +           } {
    +             intros st1' H_P_st1 H_sub_r.
    +             rewrite /=.
    +             apply IHr; last first. {
    +               move ⇒ y H_y_in.
    +               apply H_sub_r.
    +               rewrite /M.In M.In_node_iff. tauto.
    +             }
    +             have [] : (M.In e' s ∧ E.lt e e'). {
    +               apply H_sub_r.
    +               rewrite /M.In M.In_node_iff.
    +               right; left.
    +               apply Equivalence_Reflexive.
    +             }
    +             move ⇒ H_e'_in H_lt_ee'.
    +             suff H_P_st1_aux : (P st1 (f e' (f_pre e') (fold f' l st1'))). {
    +               eapply Equivalence_Transitive; first apply H_P_st1_aux.
    +               by apply H_f'.
    +             }
    +             eapply H_RG ⇒ //.
    +             apply IHl; last first. {
    +               move ⇒ y H_y_in.
    +               apply H_sub_r.
    +               rewrite /M.In M.In_node_iff. tauto.
    +             }
    +             assumption.
    +           }
    +         } {
    +           move ⇒ _.
    +           apply IHr ⇒ //.
    +           eapply H_sub_r.
    +         }
    +       }
    +       done.
    +     }
    +  Qed.
    +End MakeGenTreeFoldA.

    @@ -1244,16 +1690,25 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Module MakeAVLSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetAVL.Make X.
    -  Include MakeGenTreeFoldA X Z_as_Int Raw Raw.
    +Module MakeAVLSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    +  Include MSetAVL.Make X.
    +  Include MakeGenTreeFoldA X Z_as_Int Raw Raw.

    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    +  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
        foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.

    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Proof.
    +    intros A B i i' f_pre f f' f_lt f_gt s P.
    +    move ⇒ H_P_f H_f' H_RL H_RG H_P_ii'.
    + +
    +    rewrite /foldWithAbortPrecompute /fold.
    +    apply foldWithAbort_RawSpec ⇒ //.
    +    case s. rewrite /this /Raw.Ok //.
    +  Qed.

      Include HasFoldWithAbortOps X.
    @@ -1270,16 +1725,25 @@

    Library MSetsExtra.MSetFoldWithAbort

    The generic tree implementation naturally leads to an RBT one.
    -Module MakeRBTSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetRBT.Make X.
    -  Include MakeGenTreeFoldA X Color Raw Raw.
    +Module MakeRBTSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    +  Include MSetRBT.Make X.
    +  Include MakeGenTreeFoldA X Color Raw Raw.

    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    +  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
        foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.

    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Proof.
    +    intros A B i i' f_pre f f' f_lt f_gt s P.
    +    move ⇒ H_P_f H_f' H_RL H_RG H_P_ii'.
    + +
    +    rewrite /foldWithAbortPrecompute /fold.
    +    apply foldWithAbort_RawSpec ⇒ //.
    +    case s. rewrite /this /Raw.Ok //.
    +  Qed.

      Include HasFoldWithAbortOps X.
    @@ -1295,20 +1759,20 @@

    Library MSetsExtra.MSetFoldWithAbort

    -Module MakeListSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetList.Make X.
    +Module MakeListSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    +  Include MSetList.Make X.

    -  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    -    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    +  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    +    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
      match t with
    -    | nil ⇒ acc
    -    | x :: xs ⇒ (
    -        let pre_x := f_pre x in
    -        let acc := f x (pre_x) acc in
    -        if (f_gt x pre_x acc) then
    +    | nil ⇒ acc
    +    | x :: xs ⇒ (
    +        let pre_x := f_pre x in
    +        let acc := f x (pre_x) acc in
    +        if (f_gt x pre_x acc) then
              acc
    -        else
    +        else
              foldWithAbortRaw f_pre f_lt f f_gt xs acc
          )
      end.
    @@ -1318,8 +1782,93 @@

    Library MSetsExtra.MSetFoldWithAbort

        @foldWithAbortRaw A B f_pre f_lt f f_gt t.(this) acc.

    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    - +  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Proof.
    +    intros A B i i' f_pre f f' f_lt f_gt.
    +    move ⇒ [] l H_is_ok_l P H_equiv_P.
    +    rewrite /fold /foldWithAbortPrecompute /In /this /Raw.In /Raw.fold.
    +    move ⇒ H_P_f H_f' H_RL H_RG.
    + +
    +    set base := l.
    +    move : i i'.
    +    have : (∀ e, InA X.eq e base → InA X.eq e l). {
    +      rewrite /base //.
    +    }
    +    have : sort X.lt base. {
    +      rewrite Raw.isok_iff /base //.
    +    }
    +    clear H_is_ok_l.
    + +
    +    induction base as [| x xs IH]. {
    +      by simpl.
    +    }
    +    move ⇒ H_sort H_in_xxs i i' Pii' /=.
    + +
    +    have [H_sort_xs H_hd_rel {H_sort}] : Sorted X.lt xs ∧ HdRel X.lt x xs. {
    +      by inversion H_sort.
    +    }
    +
    +    move : H_hd_rel.
    +    rewrite (Raw.ML.Inf_alt x H_sort_xs) ⇒ H_lt_xs.
    + +
    +    have H_x_in_l : InA X.eq x l. {
    +      apply H_in_xxs.
    +      apply InA_cons_hd.
    +      apply X.eq_equiv.
    +    }
    +    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
    +      intros e H_in.
    +      apply H_in_xxs, InA_cons_tl ⇒ //.
    +    }
    +   
    +    have H_P_next : P (f x (f_pre x) i) (flip f' i' x). {
    +      rewrite /flip -H_f' //.
    +      apply H_P_f ⇒ //.
    +    }
    +
    +    case_eq (f_gt x (f_pre x) (f x (f_pre x) i)); last first. {
    +      move ⇒ _.
    +      apply IH ⇒ //.
    +    } {
    +      move ⇒ H_gt.
    +      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    +         P (f x (f_pre x) i) (fold_left (flip f') xs st)). {
    +         apply H_suff ⇒ //.
    +      }
    +      move : H_in_xs H_lt_xs.
    + +
    +      clear IH H_in_xxs H_sort_xs.
    +      move : (H_RG x _ H_x_in_l H_gt) ⇒ H_RG_x.
    +      induction xs as [| x' xs' IH']. {
    +        done.
    +      } {
    +        intros H_in_xs H_lt_xs st H_P_st.
    +        rewrite /=.
    +        have H_x'_in_l : InA X.eq x' l. {
    +          apply H_in_xs.
    +          apply InA_cons_hd, X.eq_equiv.
    +        }
    +        apply IH'. {
    +          intros e H.
    +          apply H_in_xs, InA_cons_tl ⇒ //.
    +        } {
    +          intros e H.
    +          apply H_lt_xs, InA_cons_tl ⇒ //.
    +        } {
    +          rewrite /flip -H_f' //.
    +          apply H_RG_x ⇒ //.
    +          apply H_lt_xs.
    +          apply InA_cons_hd, X.eq_equiv.
    +        }
    +      }
    +    }
    +  Qed.
    +
      Include HasFoldWithAbortOps X.
    @@ -1334,22 +1883,22 @@

    Library MSetsExtra.MSetFoldWithAbort

    -Module MakeWeakListSetsWithFoldA (X : OrderedType) <: WSetsWithFoldA with Module E := X.
    -  Module Raw := MSetWeakList.MakeRaw X.
    +Module MakeWeakListSetsWithFoldA (X : OrderedType) <: WSetsWithFoldA with Module E := X.
    +  Module Raw := MSetWeakList.MakeRaw X.
      Module E := X.
    -  Include WRaw2SetsOn E Raw.
    +  Include WRaw2SetsOn E Raw.

    -  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    -    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    +  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    +    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
      match t with
    -    | nil ⇒ acc
    -    | x :: xs ⇒ (
    -        let pre_x := f_pre x in
    -        let acc := f x (pre_x) acc in
    -        if (f_gt x pre_x acc) && (f_lt x pre_x acc) then
    +    | nil ⇒ acc
    +    | x :: xs ⇒ (
    +        let pre_x := f_pre x in
    +        let acc := f x (pre_x) acc in
    +        if (f_gt x pre_x acc) && (f_lt x pre_x acc) then
              acc
    -        else
    +        else
              foldWithAbortRaw f_pre f_lt f f_gt xs acc
          )
      end.
    @@ -1359,8 +1908,115 @@

    Library MSetsExtra.MSetFoldWithAbort

        @foldWithAbortRaw A B f_pre f_lt f f_gt t.(this) acc.

    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    - +  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Proof.
    +    intros A B i i' f_pre f f' f_lt f_gt.
    +    move ⇒ [] l H_is_ok_l P H_P_equiv.
    +    rewrite /fold /foldWithAbortPrecompute /In /this /Raw.In /Raw.fold.
    +    move ⇒ H_P_f H_f' H_RL H_RG.
    + +
    +    set base := l.
    +    move : i i'.
    +    have : (∀ e, InA X.eq e base → InA X.eq e l). {
    +      rewrite /base //.
    +    }
    +    have : NoDupA X.eq base. {
    +      apply H_is_ok_l.
    +    }
    +    clear H_is_ok_l.
    + +
    +    induction base as [| x xs IH]. {
    +      by simpl.
    +    }
    +    move ⇒ H_nodup_xxs H_in_xxs i i' Pii' /=.
    + +
    +    have [H_nin_x_xs H_nodup_xs {H_nodup_xxs}] : ¬ InA X.eq x xs ∧ NoDupA X.eq xs. {
    +      by inversion H_nodup_xxs.
    +    }
    +
    +    have H_x_in_l : InA X.eq x l. {
    +      apply H_in_xxs.
    +      apply InA_cons_hd.
    +      apply X.eq_equiv.
    +    }
    +    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
    +      intros e H_in.
    +      apply H_in_xxs, InA_cons_tl ⇒ //.
    +    }
    +   
    +    have H_P_next : P (f x (f_pre x) i) (flip f' i' x). {
    +      rewrite /flip -H_f' //.
    +      apply H_P_f ⇒ //.
    +    }
    +
    +    case_eq (f_gt x (f_pre x) (f x (f_pre x) i) &&
    +             f_lt x (f_pre x) (f x (f_pre x) i)); last first. {
    +      move ⇒ _.
    +      apply IH ⇒ //.
    +    } {
    +      move ⇒ /andb_true_iff [H_gt H_lt].
    +      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    +         P (f x (f_pre x) i) (fold_left (flip f') xs st)). {
    +         apply H_suff ⇒ //.
    +      }
    +
    +      have H_neq_xs : ∀ e, List.In e xs → X.lt x e ∨ X.lt e x. {
    +        intros e H_in.
    + +
    +        move : (X.compare_spec x e).
    +        case (X.compare x e) ⇒ H_cmp; inversion H_cmp. {
    +          contradict H_nin_x_xs.
    +          rewrite InA_alt.
    +          by ∃ e.
    +        } {
    +          by left.
    +        } {
    +          by right.
    +        }
    +      }
    +      move : H_in_xs H_neq_xs.
    + +
    +      clear IH H_in_xxs H_nodup_xs.
    +      move : (H_RG x _ H_x_in_l H_gt) ⇒ H_RG_x.
    +      move : (H_RL x _ H_x_in_l H_lt) ⇒ H_RL_x.
    +      induction xs as [| x' xs' IH']. {
    +        done.
    +      } {
    +        intros H_in_xs H_neq_xs st H_P_st.
    +        rewrite /=.
    +        have H_x'_in_xxs' : List.In x' (x' :: xs'). {
    +          simpl; by left.
    +        }
    +        have H_x'_in_l : InA X.eq x' l. {
    +          apply H_in_xs.
    +          apply InA_cons_hd, X.eq_equiv.
    +        }
    +        apply IH'. {
    +          intros H.
    +          apply H_nin_x_xs, InA_cons_tl ⇒ //.
    +        } {
    +          intros e H.
    +          apply H_in_xs, InA_cons_tl ⇒ //.
    +        } {
    +          intros e H.
    +          apply H_neq_xs, in_cons ⇒ //.
    +        } {
    +          rewrite /flip -H_f' //.
    +          move : (H_neq_xs x' H_x'_in_xxs') ⇒ [] H_cmp. {
    +            apply H_RG_x ⇒ //.
    +          } {
    +            apply H_RL_x ⇒ //.
    +          }
    +        }
    +      }
    +    }
    +  Qed.
    +
      Include HasFoldWithAbortOps X.
    diff --git a/html/MSetsExtra.MSetIntervals.html b/html/MSetsExtra.MSetIntervals.html index 696aaf2..649c296 100644 --- a/html/MSetsExtra.MSetIntervals.html +++ b/html/MSetsExtra.MSetIntervals.html @@ -47,14 +47,14 @@

    Library MSetsExtra.MSetIntervals


    -Require Import MSetInterface OrdersFacts OrdersLists.
    -Require Import BinNat.
    +Require Import MSetInterface OrdersFacts OrdersLists.
    +Require Import BinNat.
    Require Import mathcomp.ssreflect.ssreflect.
    -Require Import NArith.
    -Require Import ZArith.
    -Require Import NOrder.
    -Require Import DecidableTypeEx.
    -Module Import NOP := NOrderProp N.
    +Require Import NArith.
    +Require Import ZArith.
    +Require Import NOrder.
    +Require Import DecidableTypeEx.
    +Module Import NOP := NOrderProp N.
    Open Scope Z_scope.

    @@ -68,34 +68,103 @@

    Library MSetsExtra.MSetIntervals

    Simple auxiliary lemmata
    -Lemma Z_le_add_r : ∀ (z : Z) (n : N),
    -  z ≤ z + Z.of_N n.
    - -
    -Lemma Z_lt_add_r : ∀ (z : Z) (n : N),
    -  (n ≠ 0)%N →
    -  z < z + Z.of_N n.
    +Lemma Z_le_add_r : ∀ (z : Z) (n : N),
    +  z ≤ z + Z.of_N n.
    +Proof.
    +  intros z n.
    +  suff : (z + 0 ≤ z + Z.of_N n). {
    +    rewrite Z.add_0_r //.
    +  }
    +  apply Zplus_le_compat_l.
    +  apply N2Z.is_nonneg.
    +Qed.
    + +
    +Lemma Z_lt_add_r : ∀ (z : Z) (n : N),
    +  (n ≠ 0)%N →
    +  z < z + Z.of_N n.
    +Proof.
    +  move ⇒ z n H_neq_0.
    +  suff : (z + Z.of_N 0 < z + Z.of_N n). {
    +    rewrite Z.add_0_r //.
    +  }
    +  apply Z.add_lt_mono_l, N2Z.inj_lt.
    +  by apply N.neq_0_lt_0.
    +Qed.

    Lemma Z_lt_le_add_r : ∀ y1 y2 c,
    -  y1 < y2 →
    -  y1 ≤ y2 + Z.of_N c.
    - -
    -Lemma Z_to_N_minus_neq_0 : ∀ (x y : Z),
    -    y < x →
    -    Z.to_N (x - y) ≠ 0%N.
    +  y1 < y2 →
    +  y1 ≤ y2 + Z.of_N c.
    +Proof.
    +  intros y1 y2 c H.
    +  apply Z.le_trans with (m := y2). {
    +    by apply Z.lt_le_incl.
    +  } {
    +    apply Z_le_add_r.
    +  }
    +Qed.
    + +
    +Lemma Z_to_N_minus_neq_0 : ∀ (x y : Z),
    +    y < x →
    +    Z.to_N (x - y) ≠ 0%N.
    +Proof.
    +  intros x y H_y_lt.
    +  apply N.neq_0_lt_0.
    +  apply N2Z.inj_lt.
    +  suff H : 0 < x - y. {
    +    rewrite Z2N.id ⇒ //.
    +    by apply Z.lt_le_incl.
    +  }
    +  by apply Z.lt_0_sub.
    +Qed.
    + +
    +Lemma add_add_sub_eq : ∀ (x y : Z), (x + (y - x) = y).
    +Proof.
    +  intros x y.
    +  rewrite Z.add_sub_assoc ⇒ //.
    +  rewrite Z.add_sub_swap Z.sub_diag Z.add_0_l //.
    +Qed.
    + +
    +Lemma NoDupA_map {A B} : ∀ (eqA : A → A → Prop) (eqB : B → B → Prop) (f : A → B) l,
    +  NoDupA eqA l →
    +  (∀ x1 x2, List.In x1 l → List.In x2 l →
    +                  eqB (f x1) (f x2) → eqA x1 x2) →
    +  NoDupA eqB (map f l).
    +Proof.
    +  intros eqA eqB f.
    +  induction l as [| x xs IH]. {
    +    move ⇒ _ _; rewrite /=.
    +    apply NoDupA_nil.
    +  } {
    +    move ⇒ H_pre H_eqA_impl.
    +    have [H_nin_x H_no_dup_xs] : ¬ InA eqA x xs ∧ NoDupA eqA xs. {
    +      by inversion_clear H_pre.
    +    }
    +    simpl.
    +    apply NoDupA_cons; last first. {
    +      apply IH ⇒ //.
    +      intros x1 x2 H_in_x1 H_in_x2 H_eqB.
    +      apply H_eqA_impl ⇒ //=; by right.
    +    }
    +    move ⇒ H_in_map; apply H_nin_x.
    +    move : H_in_map.
    +    rewrite !InA_alt ⇒ [[y] [H_eqB_y]].
    +    rewrite in_map_iff ⇒ [[y'] [H_y_eq] H_y'_in].
    +    subst.
    +    âˆƒ y'.
    +    split ⇒ //.
    +    apply H_eqA_impl ⇒ //. {
    +      by simpl; left.
    +    } {
    +      by simpl; right.
    +    }
    +  }
    +Qed.
    -
    -Lemma add_add_sub_eq : ∀ (x y : Z), (x + (y - x) = y).
    - -
    -Lemma NoDupA_map {A B} : ∀ (eqA : A → A → Prop) (eqB : B → B → Prop) (f : A → B) l,
    -  NoDupA eqA l →
    -  (∀ x1 x2, List.In x1 l → List.In x2 l →
    -                  eqB (f x1) (f x2) → eqA x1 x2) →
    -  NoDupA eqB (map f l).
    -
    @@ -108,14 +177,14 @@

    Library MSetsExtra.MSetIntervals

    rev_map is used for efficiency.
    -Fixpoint rev_map_aux {A B} (f : A → B) (acc : list B) (l : list A) :=
    +Fixpoint rev_map_aux {A B} (f : A → B) (acc : list B) (l : list A) :=
      match l with
    -   | nil ⇒ acc
    -   | x :: xs ⇒ rev_map_aux f ((f x)::acc) xs
    +   | nil ⇒ acc
    +   | x :: xs ⇒ rev_map_aux f ((f x)::acc) xs
      end.

    -Definition rev_map {A B} (f : A → B) (l : list A) : list B := rev_map_aux f nil l.
    +Definition rev_map {A B} (f : A → B) (l : list A) : list B := rev_map_aux f nil l.

    @@ -124,12 +193,26 @@

    Library MSetsExtra.MSetIntervals

    Lemmata about rev_map
    -Lemma rev_map_aux_alt_def {A B} : ∀ (f : A → B) l acc,
    -  rev_map_aux f acc l = List.rev_append (List.map f l) acc.
    - +Lemma rev_map_aux_alt_def {A B} : ∀ (f : A → B) l acc,
    +  rev_map_aux f acc l = List.rev_append (List.map f l) acc.
    +Proof.
    +  intro f.
    +  induction l as [| x xs IH]. {
    +    intros acc.
    +    by simpl.
    +  } {
    +    intros acc.
    +    rewrite /= IH //.
    +  }
    +Qed.
    +
    -Lemma rev_map_alt_def {A B} : ∀ (f : A → B) l,
    -  rev_map f l = List.rev (List.map f l).
    +Lemma rev_map_alt_def {A B} : ∀ (f : A → B) l,
    +  rev_map f l = List.rev (List.map f l).
    +Proof.
    +  intros f l.
    +  rewrite /rev_map rev_map_aux_alt_def -rev_alt //.
    +Qed.

    @@ -148,11 +231,11 @@

    Library MSetsExtra.MSetIntervals


    Module Type ElementEncode.
    -  Declare Module E : OrderedType.
    +  Declare Module E : OrderedType.

    -  Parameter encode : E.t → Z.
    -  Parameter decode : Z → E.t.
    +  Parameter encode : E.t → Z.
    +  Parameter decode : Z → E.t.

    @@ -164,8 +247,8 @@

    Library MSetsExtra.MSetIntervals

    element e with encode v = z.
    -  Axiom decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    +  Axiom decode_encode_ok: ∀ (e : E.t),
    +    decode (encode e) = e.

    @@ -174,8 +257,8 @@

    Library MSetsExtra.MSetIntervals

    Encoding is compatible with the equality of elements.
    -  Axiom encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +  Axiom encode_eq : ∀ (e1 e2 : E.t),
    +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.

    @@ -184,8 +267,8 @@

    Library MSetsExtra.MSetIntervals

    Encoding is compatible with the order of elements.
    -  Axiom encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Axiom encode_lt : ∀ (e1 e2 : E.t),
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.

    End ElementEncode.
    @@ -218,7 +301,7 @@

    Library MSetsExtra.MSetIntervals

    - Intervals we represent by tuples (Z, N). The tuple (z, c) represents the + Intervals we represent by tuples (Z, N). The tuple (z, c) represents the interval z-(z+c).
    @@ -229,9 +312,9 @@

    Library MSetsExtra.MSetIntervals

    -Module Ops (Enc : ElementEncode) <: Ops Enc.E.
    +Module Ops (Enc : ElementEncode) <: Ops Enc.E.
      Definition elt := Enc.E.t.
    -  Definition t := list (Z × N).
    +  Definition t := list (Z × N).

    @@ -240,8 +323,8 @@

    Library MSetsExtra.MSetIntervals

    The empty list is trivial to define and check for.
    -  Definition empty : t := nil.
    -  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.
    +  Definition empty : t := nil.
    +  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.

    @@ -253,39 +336,46 @@

    Library MSetsExtra.MSetIntervals

    if it needs to be executable.
    -  Lemma acc_pred : ∀ n p, n = Npos p → Acc N.lt n → Acc N.lt (N.pred n).
    - -
    -  Fixpoint fold_elementsZ_aux {A} (f : A → Z → option A) (acc : A) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : (bool × A) :=
    -    match c as c0 return c = c0 → (bool × A) with
    -    | N0 ⇒ fun _ ⇒ (false, acc)
    +  Lemma acc_pred : ∀ n p, n = Npos p → Acc N.lt n → Acc N.lt (N.pred n).
    +  Proof.
    +    intros n p H0 H1.
    +    apply H1.
    +    rewrite H0.
    +    apply N.lt_pred_l.
    +    discriminate.
    +  Defined.
    + +
    +  Fixpoint fold_elementsZ_aux {A} (f : A → Z → option A) (acc : A) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : (bool × A) :=
    +    match c as c0 return c = c0 → (bool × A) with
    +    | N0 ⇒ fun _ ⇒ (false, acc)
        | c ⇒ fun Heq ⇒ match (f acc x) with
    -        | None ⇒ (true, acc)
    -        | Some acc' ⇒
    -          fold_elementsZ_aux f acc' (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end
    -    end (refl_equal _).
    +        | None ⇒ (true, acc)
    +        | Some acc' ⇒
    +          fold_elementsZ_aux f acc' (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end
    +    end (refl_equal _).

      Definition fold_elementsZ_single {A} f (acc : A) x c := fold_elementsZ_aux f acc x c (lt_wf_0 _).

    -  Fixpoint fold_elementsZ {A} f (acc : A) (s : t) : (bool × A) :=
    +  Fixpoint fold_elementsZ {A} f (acc : A) (s : t) : (bool × A) :=
        match s with
    -    | nil ⇒ (false, acc)
    -    | (x, c) :: s' ⇒
    +    | nil ⇒ (false, acc)
    +    | (x, c) :: s' ⇒
          match fold_elementsZ_single f acc x c with
    -          (false, acc') ⇒ fold_elementsZ f acc' s'
    -        | (true, acc') ⇒ (true, acc')
    +          (false, acc') ⇒ fold_elementsZ f acc' s'
    +        | (true, acc') ⇒ (true, acc')
          end
        end.

    -  Definition elementsZ (s : t) : list Z :=
    -    snd (fold_elementsZ (fun l x ⇒ Some (x :: l)) nil s).
    +  Definition elementsZ (s : t) : list Z :=
    +    snd (fold_elementsZ (fun l x ⇒ Some (x :: l)) nil s).

    -  Definition elements (s : t) : list elt :=
    -    rev_map Enc.decode (elementsZ s).
    +  Definition elements (s : t) : list elt :=
    +    rev_map Enc.decode (elementsZ s).

    @@ -294,12 +384,12 @@

    Library MSetsExtra.MSetIntervals

    membership is easily defined
    -  Fixpoint memZ (x : Z) (s : t) :=
    +  Fixpoint memZ (x : Z) (s : t) :=
        match s with
    -    | nil ⇒ false
    -    | (y, c) :: l ⇒
    -        if (Z.ltb x y) then false else
    -        if (Z.ltb x (y+Z.of_N c)) then true else
    +    | nil ⇒ false
    +    | (y, c) :: l ⇒
    +        if (Z.ltb x y) then false else
    +        if (Z.ltb x (y+Z.of_N c)) then true else
            memZ x l
        end.
    @@ -325,59 +415,59 @@

    Library MSetsExtra.MSetIntervals

        | ICR_after_touch.

    -  Definition interval_compare (i1 i2 : (Z × N)) : interval_compare_result :=
    -    match (i1, i2) with ((y1, c1), (y2, c2)) ⇒
    -      let yc2 := (y2+Z.of_N c2) in
    -      match (Z.compare yc2 y1) with
    -        | Lt ⇒ ICR_after
    -        | Eq ⇒ ICR_after_touch
    -        | Gt ⇒ let yc1 := (y1+Z.of_N c1) in
    -                match (Z.compare yc1 y2) with
    -                | Lt ⇒ ICR_before
    -                | Eq ⇒ ICR_before_touch
    -                | Gt ⇒
    -                        match (Z.compare y1 y2, Z.compare yc1 yc2) with
    -                        | (Lt, Lt) ⇒ ICR_overlap_before
    -                        | (Lt, _) ⇒ ICR_subsume_2
    -                        | (Eq, Lt) ⇒ ICR_subsume_1
    -                        | (Eq, Gt) ⇒ ICR_subsume_2
    -                        | (Eq, Eq) ⇒ ICR_equal
    -                        | (Gt, Gt) ⇒ ICR_overlap_after
    -                        | (Gt, _) ⇒ ICR_subsume_1
    +  Definition interval_compare (i1 i2 : (Z × N)) : interval_compare_result :=
    +    match (i1, i2) with ((y1, c1), (y2, c2)) ⇒
    +      let yc2 := (y2+Z.of_N c2) in
    +      match (Z.compare yc2 y1) with
    +        | Lt ⇒ ICR_after
    +        | Eq ⇒ ICR_after_touch
    +        | Gt ⇒ let yc1 := (y1+Z.of_N c1) in
    +                match (Z.compare yc1 y2) with
    +                | Lt ⇒ ICR_before
    +                | Eq ⇒ ICR_before_touch
    +                | Gt ⇒
    +                        match (Z.compare y1 y2, Z.compare yc1 yc2) with
    +                        | (Lt, Lt) ⇒ ICR_overlap_before
    +                        | (Lt, _) ⇒ ICR_subsume_2
    +                        | (Eq, Lt) ⇒ ICR_subsume_1
    +                        | (Eq, Gt) ⇒ ICR_subsume_2
    +                        | (Eq, Eq) ⇒ ICR_equal
    +                        | (Gt, Gt) ⇒ ICR_overlap_after
    +                        | (Gt, _) ⇒ ICR_subsume_1
                            end
                    end
          end
        end.

    -  Definition interval_1_compare (y1 : Z) (i : (Z × N)) : interval_compare_result :=
    -    match i with (y2, c2) ⇒
    -      let yc2 := (y2+Z.of_N c2) in
    -      match (Z.compare yc2 y1) with
    -        | Lt ⇒ ICR_after
    -        | Eq ⇒ ICR_after_touch
    -        | Gt ⇒ match (Z.compare (Z.succ y1) y2) with
    -                | Lt ⇒ ICR_before
    -                | Eq ⇒ ICR_before_touch
    -                | Gt ⇒ ICR_subsume_1
    +  Definition interval_1_compare (y1 : Z) (i : (Z × N)) : interval_compare_result :=
    +    match i with (y2, c2) ⇒
    +      let yc2 := (y2+Z.of_N c2) in
    +      match (Z.compare yc2 y1) with
    +        | Lt ⇒ ICR_after
    +        | Eq ⇒ ICR_after_touch
    +        | Gt ⇒ match (Z.compare (Z.succ y1) y2) with
    +                | Lt ⇒ ICR_before
    +                | Eq ⇒ ICR_before_touch
    +                | Gt ⇒ ICR_subsume_1
                    end
          end
        end.

      Fixpoint compare (s1 s2 : t) :=
    -    match (s1, s2) with
    -      | (nil, nil) ⇒ Eq
    -      | (nil, _ :: _) ⇒ Lt
    -      | (_ :: _, nil) ⇒ Gt
    -      | ((y1, c1)::s1', (y2, c2)::s2') ⇒
    -        match (Z.compare y1 y2) with
    -          | Lt ⇒ Lt
    -          | Gt ⇒ Gt
    -          | Eq ⇒ match N.compare c1 c2 with
    -                    | Lt ⇒ Lt
    -                    | Gt ⇒ Gt
    -                    | Eq ⇒ compare s1' s2'
    +    match (s1, s2) with
    +      | (nil, nil) ⇒ Eq
    +      | (nil, _ :: _) ⇒ Lt
    +      | (_ :: _, nil) ⇒ Gt
    +      | ((y1, c1)::s1', (y2, c2)::s2') ⇒
    +        match (Z.compare y1 y2) with
    +          | Lt ⇒ Lt
    +          | Gt ⇒ Gt
    +          | Eq ⇒ match N.compare c1 c2 with
    +                    | Lt ⇒ Lt
    +                    | Gt ⇒ Gt
    +                    | Eq ⇒ compare s1' s2'
                      end
            end
        end.
    @@ -389,18 +479,18 @@

    Library MSetsExtra.MSetIntervals

    Auxiliary functions for inserting at front and merging intervals
    -  Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N :=
    -    (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))).
    +  Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N :=
    +    (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))).

    -  Fixpoint insert_interval_begin (x : Z) (c : N) (l : t) :=
    +  Fixpoint insert_interval_begin (x : Z) (c : N) (l : t) :=
        match l with
    -    | nil ⇒ (x,c)::nil
    -    | (y, c')::l' ⇒
    -         match (Z.compare (x + Z.of_N c) y) with
    -         | Lt ⇒ (x, c) :: l
    -         | Eq ⇒ (x, (c+c')%N) :: l'
    -         | Gt ⇒ insert_interval_begin x (merge_interval_size x c y c') l'
    +    | nil ⇒ (x,c)::nil
    +    | (y, c')::l' ⇒
    +         match (Z.compare (x + Z.of_N c) y) with
    +         | Lt ⇒ (x, c) :: l
    +         | Eq ⇒ (x, (c+c')%N) :: l'
    +         | Gt ⇒ insert_interval_begin x (merge_interval_size x c y c') l'
             end
        end.
    @@ -412,21 +502,21 @@

    Library MSetsExtra.MSetIntervals

    generate efficient code
    -  Fixpoint addZ_aux (acc : list (Z × N)) (x : Z) (s : t) :=
    +  Fixpoint addZ_aux (acc : list (Z × N)) (x : Z) (s : t) :=
        match s with
    -    | nil ⇒ List.rev' ((x, (1%N))::acc)
    -    | (y, c) :: l ⇒
    -        match (interval_1_compare x (y,c)) with
    -          | ICR_before ⇒ List.rev_append ((x, (1%N))::acc) s
    -          | ICR_before_touch ⇒ List.rev_append ((x, N.succ c)::acc) l
    -          | ICR_after ⇒ addZ_aux ((y,c) :: acc) x l
    -          | ICR_after_touch ⇒ List.rev_append acc (insert_interval_begin y (N.succ c) l)
    -          | _ ⇒ List.rev_append ((y, c)::acc) l
    +    | nil ⇒ List.rev' ((x, (1%N))::acc)
    +    | (y, c) :: l ⇒
    +        match (interval_1_compare x (y,c)) with
    +          | ICR_before ⇒ List.rev_append ((x, (1%N))::acc) s
    +          | ICR_before_touch ⇒ List.rev_append ((x, N.succ c)::acc) l
    +          | ICR_after ⇒ addZ_aux ((y,c) :: acc) x l
    +          | ICR_after_touch ⇒ List.rev_append acc (insert_interval_begin y (N.succ c) l)
    +          | _ ⇒ List.rev_append ((y, c)::acc) l
            end
        end.

    -  Definition addZ x s := addZ_aux nil x s.
    +  Definition addZ x s := addZ_aux nil x s.
      Definition add x s := addZ (Enc.encode x) s.

    @@ -437,11 +527,11 @@

    Library MSetsExtra.MSetIntervals

    This is used to define the function from_elements.
    -  Definition add_list (l : list elt) (s : t) : t :=
    -     List.fold_left (fun s x ⇒ add x s) l s.
    +  Definition add_list (l : list elt) (s : t) : t :=
    +     List.fold_left (fun s x ⇒ add x s) l s.

    -  Definition from_elements (l : list elt) : t := add_list l empty.
    +  Definition from_elements (l : list elt) : t := add_list l empty.

    @@ -450,11 +540,12 @@

    Library MSetsExtra.MSetIntervals

    singleton is trivial to define
    -  Definition singleton (x : elt) : t := (Enc.encode x, 1%N) :: nil.
    +  Definition singleton (x : elt) : t := (Enc.encode x, 1%N) :: nil.

    -  Lemma singleton_alt_def : ∀ x, singleton x = add x empty.
    - +  Lemma singleton_alt_def : ∀ x, singleton x = add x empty.
    +  Proof. by []. Qed.
    +
    @@ -462,29 +553,29 @@

    Library MSetsExtra.MSetIntervals

    removing needs to be done with code extraction in mind again.
    -  Definition insert_intervalZ_guarded (x : Z) (c : N) s :=
    -     if (N.eqb c 0) then s else (x, c) :: s.
    +  Definition insert_intervalZ_guarded (x : Z) (c : N) s :=
    +     if (N.eqb c 0) then s else (x, c) :: s.

    -  Fixpoint removeZ_aux (acc : list (Z × N)) (x : Z) (s : t) : t :=
    +  Fixpoint removeZ_aux (acc : list (Z × N)) (x : Z) (s : t) : t :=
        match s with
    -    | nil ⇒ List.rev' acc
    -    | (y, c) :: l ⇒
    -        if (Z.ltb x y) then List.rev_append acc s else
    -        if (Z.ltb x (y+Z.of_N c)) then (
    -           List.rev_append (insert_intervalZ_guarded (Z.succ x)
    -              (Z.to_N ((y+Z.of_N c)- (Z.succ x)))
    -             (insert_intervalZ_guarded y (Z.to_N (x-y)) acc)) l
    -        ) else removeZ_aux ((y,c)::acc) x l
    +    | nil ⇒ List.rev' acc
    +    | (y, c) :: l ⇒
    +        if (Z.ltb x y) then List.rev_append acc s else
    +        if (Z.ltb x (y+Z.of_N c)) then (
    +           List.rev_append (insert_intervalZ_guarded (Z.succ x)
    +              (Z.to_N ((y+Z.of_N c)- (Z.succ x)))
    +             (insert_intervalZ_guarded y (Z.to_N (x-y)) acc)) l
    +        ) else removeZ_aux ((y,c)::acc) x l
        end.

    -  Definition removeZ (x : Z) (s : t) : t := removeZ_aux nil x s.
    +  Definition removeZ (x : Z) (s : t) : t := removeZ_aux nil x s.
      Definition remove (x : elt) (s : t) : t := removeZ (Enc.encode x) s.

    -  Definition remove_list (l : list elt) (s : t) : t :=
    -     List.fold_left (fun s x ⇒ remove x s) l s.
    +  Definition remove_list (l : list elt) (s : t) : t :=
    +     List.fold_left (fun s x ⇒ remove x s) l s.

    @@ -494,19 +585,19 @@

    Library MSetsExtra.MSetIntervals

      Fixpoint union_aux (s1 : t) :=
    -    fix aux (s2 : t) (acc : list (Z × N)) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ List.rev_append acc s2
    -    | (_, nil) ⇒ List.rev_append acc s1
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
    +    fix aux (s2 : t) (acc : list (Z × N)) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ List.rev_append acc s2
    +    | (_, nil) ⇒ List.rev_append acc s1
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
              | ICR_before_touch ⇒
                  union_aux l1 (
    -               insert_interval_begin y1 ((c1+c2)%N) l2) acc
    -          | ICR_after ⇒ aux l2 ((y2, c2)::acc)
    +               insert_interval_begin y1 ((c1+c2)%N) l2) acc
    +          | ICR_after ⇒ aux l2 ((y2, c2)::acc)
              | ICR_after_touch ⇒ union_aux l1 (
    -              insert_interval_begin y2 ((c1+c2)%N) l2) acc
    +              insert_interval_begin y2 ((c1+c2)%N) l2) acc
              | ICR_overlap_before ⇒
                  union_aux l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2) acc
              | ICR_overlap_after ⇒
    @@ -518,7 +609,7 @@

    Library MSetsExtra.MSetIntervals

        end.

    -  Definition union s1 s2 := union_aux s1 s2 nil.
    +  Definition union s1 s2 := union_aux s1 s2 nil.

    @@ -529,38 +620,38 @@

    Library MSetsExtra.MSetIntervals


    -  Fixpoint diff_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
    +  Fixpoint diff_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
        match s with
    -    | nil ⇒ (acc, nil)
    -    | ((y1, c1) :: l1) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    -          | ICR_before_touch ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    -          | ICR_after ⇒ (acc, s)
    -          | ICR_after_touch ⇒ (acc, s)
    -          | ICR_overlap_before ⇒ diff_aux y2 c2 ((y1, Z.to_N (y2 - y1))::acc) l1
    -          | ICR_overlap_after ⇒ (acc, (y2+Z.of_N c2, Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) :: l1)
    -          | ICR_equal ⇒ (acc, l1)
    +    | nil ⇒ (acc, nil)
    +    | ((y1, c1) :: l1) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    +          | ICR_before_touch ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    +          | ICR_after ⇒ (acc, s)
    +          | ICR_after_touch ⇒ (acc, s)
    +          | ICR_overlap_before ⇒ diff_aux y2 c2 ((y1, Z.to_N (y2 - y1))::acc) l1
    +          | ICR_overlap_after ⇒ (acc, (y2+Z.of_N c2, Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) :: l1)
    +          | ICR_equal ⇒ (acc, l1)
              | ICR_subsume_1 ⇒ diff_aux y2 c2 acc l1
    -          | ICR_subsume_2 ⇒ ((insert_intervalZ_guarded y1
    -                (Z.to_N (y2 - y1)) acc),
    -              insert_intervalZ_guarded (y2+Z.of_N c2) (Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) l1)
    +          | ICR_subsume_2 ⇒ ((insert_intervalZ_guarded y1
    +                (Z.to_N (y2 - y1)) acc),
    +              insert_intervalZ_guarded (y2+Z.of_N c2) (Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) l1)
            end
        end.

    -  Fixpoint diff_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ rev_append acc s1
    -    | (_, nil) ⇒ rev_append acc s1
    -    | (_, (y2, c2) :: l2) ⇒
    +  Fixpoint diff_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ rev_append acc s1
    +    | (_, nil) ⇒ rev_append acc s1
    +    | (_, (y2, c2) :: l2) ⇒
          match diff_aux y2 c2 acc s1 with
    -        (acc', s1') ⇒ diff_aux2 acc' s1' l2
    +        (acc', s1') ⇒ diff_aux2 acc' s1' l2
          end
        end.

    -  Definition diff s1 s2 := diff_aux2 nil s1 s2.
    +  Definition diff s1 s2 := diff_aux2 nil s1 s2.

    @@ -571,20 +662,20 @@

    Library MSetsExtra.MSetIntervals

      Fixpoint subset (s1 : t) :=
        fix aux (s2 : t) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ true
    -    | (_ :: _, nil) ⇒ false
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ false
    -          | ICR_before_touch ⇒ false
    +    match (s1, s2) with
    +    | (nil, _) ⇒ true
    +    | (_ :: _, nil) ⇒ false
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ false
    +          | ICR_before_touch ⇒ false
              | ICR_after ⇒ aux l2
    -          | ICR_after_touch ⇒ false
    -          | ICR_overlap_before ⇒ false
    -          | ICR_overlap_after ⇒ false
    +          | ICR_after_touch ⇒ false
    +          | ICR_overlap_before ⇒ false
    +          | ICR_overlap_after ⇒ false
              | ICR_equal ⇒ subset l1 l2
              | ICR_subsume_1 ⇒ subset l1 s2
    -          | ICR_subsume_2 ⇒ false
    +          | ICR_subsume_2 ⇒ false
            end
        end.
    @@ -595,10 +686,10 @@

    Library MSetsExtra.MSetIntervals

    equal
    -  Fixpoint equal (s s' : t) : bool := match s, s' with
    -    | nil, nil ⇒ true
    -    | ((x,cx)::xs), ((y,cy)::ys) ⇒ andb (Z.eqb x y) (andb (N.eqb cx cy) (equal xs ys))
    -    | _, _ ⇒ false
    +  Fixpoint equal (s s' : t) : bool := match s, s' with
    +    | nil, nil ⇒ true
    +    | ((x,cx)::xs), ((y,cy)::ys) ⇒ andb (Z.eqb x y) (andb (N.eqb cx cy) (equal xs ys))
    +    | _, _ ⇒ false
      end.

    @@ -608,36 +699,36 @@

    Library MSetsExtra.MSetIntervals

    inter
    -  Fixpoint inter_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
    +  Fixpoint inter_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
        match s with
    -    | nil ⇒ (acc, nil)
    -    | ((y1, c1) :: l1) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    +    | nil ⇒ (acc, nil)
    +    | ((y1, c1) :: l1) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
              | ICR_before ⇒ inter_aux y2 c2 acc l1
              | ICR_before_touch ⇒ inter_aux y2 c2 acc l1
    -          | ICR_after ⇒ (acc, s)
    -          | ICR_after_touch ⇒ (acc, s)
    -          | ICR_overlap_before ⇒ inter_aux y2 c2 ((y2, Z.to_N (y1 + Z.of_N c1 - y2))::acc) l1
    -          | ICR_overlap_after ⇒ ((y1, Z.to_N (y2 + Z.of_N c2 - y1))::acc, s)
    -          | ICR_equal ⇒ ((y1,c1)::acc, l1)
    -          | ICR_subsume_1 ⇒ inter_aux y2 c2 ((y1, c1)::acc) l1
    -          | ICR_subsume_2 ⇒ ((y2, c2)::acc, s)
    +          | ICR_after ⇒ (acc, s)
    +          | ICR_after_touch ⇒ (acc, s)
    +          | ICR_overlap_before ⇒ inter_aux y2 c2 ((y2, Z.to_N (y1 + Z.of_N c1 - y2))::acc) l1
    +          | ICR_overlap_after ⇒ ((y1, Z.to_N (y2 + Z.of_N c2 - y1))::acc, s)
    +          | ICR_equal ⇒ ((y1,c1)::acc, l1)
    +          | ICR_subsume_1 ⇒ inter_aux y2 c2 ((y1, c1)::acc) l1
    +          | ICR_subsume_2 ⇒ ((y2, c2)::acc, s)
            end
        end.

    -  Fixpoint inter_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ List.rev' acc
    -    | (_, nil) ⇒ List.rev' acc
    -    | (_, (y2, c2) :: l2) ⇒
    +  Fixpoint inter_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ List.rev' acc
    +    | (_, nil) ⇒ List.rev' acc
    +    | (_, (y2, c2) :: l2) ⇒
          match inter_aux y2 c2 acc s1 with
    -        (acc', s1') ⇒ inter_aux2 acc' s1' l2
    +        (acc', s1') ⇒ inter_aux2 acc' s1' l2
          end
        end.

    -  Definition inter s1 s2 := inter_aux2 nil s1 s2.
    +  Definition inter s1 s2 := inter_aux2 nil s1 s2.

    @@ -649,93 +740,93 @@

    Library MSetsExtra.MSetIntervals


      Definition partitionZ_fold_insert
    -             (cur : option (Z × N)) (x : Z) :=
    +             (cur : option (Z × N)) (x : Z) :=
        match cur with
    -       None ⇒ (x, 1%N)
    -     | Some (y, c) ⇒ (y, N.succ c)
    +       None ⇒ (x, 1%N)
    +     | Some (y, c) ⇒ (y, N.succ c)
        end.

    -  Definition partitionZ_fold_skip (acc : list (Z × N))
    -             (cur : option (Z × N)) : (list (Z × N)) :=
    +  Definition partitionZ_fold_skip (acc : list (Z × N))
    +             (cur : option (Z × N)) : (list (Z × N)) :=
        match cur with
    -       None ⇒ acc
    -     | Some yc ⇒ yc::acc
    +       None ⇒ acc
    +     | Some yc ⇒ yc::acc
        end.

    -  Definition partitionZ_fold_fun f st (x : Z) :=
    -    match st with ((acc_t, c_t), (acc_f, c_f)) ⇒
    -      if (f x) then
    -        ((acc_t, Some (partitionZ_fold_insert c_t x)),
    -         (partitionZ_fold_skip acc_f c_f, None))
    -      else
    -        ((partitionZ_fold_skip acc_t c_t, None),
    -         (acc_f, Some (partitionZ_fold_insert c_f x)))
    +  Definition partitionZ_fold_fun f st (x : Z) :=
    +    match st with ((acc_t, c_t), (acc_f, c_f)) ⇒
    +      if (f x) then
    +        ((acc_t, Some (partitionZ_fold_insert c_t x)),
    +         (partitionZ_fold_skip acc_f c_f, None))
    +      else
    +        ((partitionZ_fold_skip acc_t c_t, None),
    +         (acc_f, Some (partitionZ_fold_insert c_f x)))
        end.

    -  Definition partitionZ_single_aux f st (x : Z) (c : N) :=
    -    snd (fold_elementsZ_single (fun st x ⇒ Some (partitionZ_fold_fun f st x)) st x c).
    +  Definition partitionZ_single_aux f st (x : Z) (c : N) :=
    +    snd (fold_elementsZ_single (fun st x ⇒ Some (partitionZ_fold_fun f st x)) st x c).

      Definition partitionZ_single f acc_t acc_f x c :=
    -    match partitionZ_single_aux f ((acc_t, None), (acc_f, None)) x c with
    -    | ((acc_t, c_t), (acc_f, c_f)) ⇒
    -        (partitionZ_fold_skip acc_t c_t,
    -         partitionZ_fold_skip acc_f c_f)
    +    match partitionZ_single_aux f ((acc_t, None), (acc_f, None)) x c with
    +    | ((acc_t, c_t), (acc_f, c_f)) ⇒
    +        (partitionZ_fold_skip acc_t c_t,
    +         partitionZ_fold_skip acc_f c_f)
        end.

      Fixpoint partitionZ_aux acc_t acc_f f s :=
        match s with
    -    | nil ⇒ (List.rev acc_t, List.rev acc_f)
    -    | (y, c) :: s' ⇒
    +    | nil ⇒ (List.rev acc_t, List.rev acc_f)
    +    | (y, c) :: s' ⇒
          match partitionZ_single f acc_t acc_f y c with
    -      | (acc_t', acc_f') ⇒ partitionZ_aux acc_t' acc_f' f s'
    +      | (acc_t', acc_f') ⇒ partitionZ_aux acc_t' acc_f' f s'
          end
        end.

    -  Definition partitionZ := partitionZ_aux nil nil.
    +  Definition partitionZ := partitionZ_aux nil nil.

    -  Definition partition (f : elt → bool) : t → (t × t) :=
    +  Definition partition (f : elt → bool) : t → (t × t) :=
        partitionZ (fun z ⇒ f (Enc.decode z)).

    -  Definition filterZ_fold_fun f st (x : Z) :=
    -    match st with (acc_t, c_t) ⇒
    -      if (f x) then
    -        (acc_t, Some (partitionZ_fold_insert c_t x))
    -      else
    -        (partitionZ_fold_skip acc_t c_t, None)
    +  Definition filterZ_fold_fun f st (x : Z) :=
    +    match st with (acc_t, c_t) ⇒
    +      if (f x) then
    +        (acc_t, Some (partitionZ_fold_insert c_t x))
    +      else
    +        (partitionZ_fold_skip acc_t c_t, None)
        end.

    -  Definition filterZ_single_aux f st (x : Z) (c : N) :=
    -    snd (fold_elementsZ_single (fun st x ⇒ Some (filterZ_fold_fun f st x)) st x c).
    +  Definition filterZ_single_aux f st (x : Z) (c : N) :=
    +    snd (fold_elementsZ_single (fun st x ⇒ Some (filterZ_fold_fun f st x)) st x c).

      Definition filterZ_single f acc x c :=
    -    match filterZ_single_aux f (acc, None) x c with
    -    | (acc, c) ⇒
    +    match filterZ_single_aux f (acc, None) x c with
    +    | (acc, c) ⇒
            (partitionZ_fold_skip acc c)
        end.

      Fixpoint filterZ_aux acc f s :=
        match s with
    -    | nil ⇒ (List.rev acc)
    -    | (y, c) :: s' ⇒
    +    | nil ⇒ (List.rev acc)
    +    | (y, c) :: s' ⇒
          filterZ_aux (filterZ_single f acc y c) f s'
        end.

    -  Definition filterZ := filterZ_aux nil.
    +  Definition filterZ := filterZ_aux nil.

    -  Definition filter (f : elt → bool) : t → t :=
    +  Definition filter (f : elt → bool) : t → t :=
        filterZ (fun z ⇒ f (Enc.decode z)).

    @@ -749,62 +840,62 @@

    Library MSetsExtra.MSetIntervals


    -  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    -    snd (fold_elementsZ (fun b z ⇒ Some (f (Enc.decode z) b)) i s).
    +  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    +    snd (fold_elementsZ (fun b z ⇒ Some (f (Enc.decode z) b)) i s).

    -  Definition for_all (f : elt → bool) (s : t) : bool :=
    -    snd (fold_elementsZ (fun b z ⇒
    -      if b then
    -        Some (f (Enc.decode z))
    -      else None) true s).
    +  Definition for_all (f : elt → bool) (s : t) : bool :=
    +    snd (fold_elementsZ (fun b z ⇒
    +      if b then
    +        Some (f (Enc.decode z))
    +      else None) true s).

    -  Definition exists_ (f : elt → bool) (s : t) : bool :=
    -    snd (fold_elementsZ (fun b z ⇒
    -      if b then
    -        None
    -      else Some (f (Enc.decode z))) false s).
    +  Definition exists_ (f : elt → bool) (s : t) : bool :=
    +    snd (fold_elementsZ (fun b z ⇒
    +      if b then
    +        None
    +      else Some (f (Enc.decode z))) false s).

    -  Fixpoint cardinalN c (s : t) : N := match s with
    -    | nil ⇒ c
    -    | (_,cx)::xs ⇒ cardinalN (c + cx)%N xs
    +  Fixpoint cardinalN c (s : t) : N := match s with
    +    | nil ⇒ c
    +    | (_,cx)::xs ⇒ cardinalN (c + cx)%N xs
      end.

    -  Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s).
    +  Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s).

    -  Definition min_eltZ (s : t) : option Z :=
    +  Definition min_eltZ (s : t) : option Z :=
        match s with
    -    | nil ⇒ None
    -    | (x, _) :: _ ⇒ Some x
    +    | nil ⇒ None
    +    | (x, _) :: _ ⇒ Some x
        end.

    -  Definition min_elt (s : t) : option elt :=
    +  Definition min_elt (s : t) : option elt :=
        match (min_eltZ s) with
    -    | None ⇒ None
    -    | Some x ⇒ Some (Enc.decode x)
    +    | None ⇒ None
    +    | Some x ⇒ Some (Enc.decode x)
        end.

      Definition choose := min_elt.

    -  Fixpoint max_eltZ (s : t) : option Z :=
    +  Fixpoint max_eltZ (s : t) : option Z :=
        match s with
    -    | nil ⇒ None
    -    | (x, c) :: nil ⇒ Some (Z.pred (x + Z.of_N c))
    -    | (x, _) :: s' ⇒ max_eltZ s'
    +    | nil ⇒ None
    +    | (x, c) :: nil ⇒ Some (Z.pred (x + Z.of_N c))
    +    | (x, _) :: s' ⇒ max_eltZ s'
        end.

    -  Definition max_elt (s : t) : option elt :=
    +  Definition max_elt (s : t) : option elt :=
        match (max_eltZ s) with
    -    | None ⇒ None
    -    | Some x ⇒ Some (Enc.decode x)
    +    | None ⇒ None
    +    | Some x ⇒ Some (Enc.decode x)
        end.

    @@ -837,26 +928,26 @@

    Library MSetsExtra.MSetIntervals

    -  Definition is_encoded_elems_list (l : list Z) : Prop :=
    -    (∀ x, List.In x l → ∃ e, Enc.encode e = x).
    +  Definition is_encoded_elems_list (l : list Z) : Prop :=
    +    (∀ x, List.In x l → ∃ e, Enc.encode e = x).

    -  Definition interval_list_elements_greater (x : Z) (l : t) : bool :=
    +  Definition interval_list_elements_greater (x : Z) (l : t) : bool :=
        match l with
    -      | nil ⇒ true
    -      | (y, _)::_ ⇒ Z.ltb x y
    +      | nil ⇒ true
    +      | (y, _)::_ ⇒ Z.ltb x y
        end.

      Fixpoint interval_list_invariant (l : t) :=
       match l with
    -   | nil ⇒ true
    -   | (x, c) :: l' ⇒
    -       interval_list_elements_greater (x + (Z.of_N c)) l' && negb (N.eqb c 0) && interval_list_invariant l'
    +   | nil ⇒ true
    +   | (x, c) :: l' ⇒
    +       interval_list_elements_greater (x + (Z.of_N c)) l' && negb (N.eqb c 0) && interval_list_invariant l'
       end.

    -  Definition IsOk s := (interval_list_invariant s = true ∧ is_encoded_elems_list (elementsZ s)).
    +  Definition IsOk s := (interval_list_invariant s = true ∧ is_encoded_elems_list (elementsZ s)).

    @@ -875,13 +966,13 @@

    Library MSetsExtra.MSetIntervals

        Instance IsOk_Ok s `(Hs : IsOk s) : Ok s := { ok := Hs }.

    -    Definition In x s := (SetoidList.InA Enc.E.eq x (elements s)).
    -    Definition InZ x s := (List.In x (elementsZ s)).
    -    Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    -    Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    -    Definition Empty s := ∀ a : elt, ¬ In a s.
    -    Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    -    Definition Exists (P : elt → Prop) (s : t) := ∃ x, In x s ∧ P x.
    +    Definition In x s := (SetoidList.InA Enc.E.eq x (elements s)).
    +    Definition InZ x s := (List.In x (elementsZ s)).
    +    Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    +    Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    +    Definition Empty s := ∀ a : elt, ¬ In a s.
    +    Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    +    Definition Exists (P : elt → Prop) (s : t) := ∃ x, In x s ∧ P x.

      End ForNotations.
    @@ -903,173 +994,423 @@

    Library MSetsExtra.MSetIntervals


    -  Lemma elementsZ_nil : (elementsZ (nil : t) = nil).
    - +  Lemma elementsZ_nil : (elementsZ (nil : t) = nil).
    +  Proof. done. Qed.
    +
    -  Lemma elements_nil : (elements (nil : t) = nil).
    - +  Lemma elements_nil : (elements (nil : t) = nil).
    +  Proof. done. Qed.
    +
    -  Definition elementsZ_single (x:Z) (c:N) :=
    -      List.rev' (N.peano_rec (fun _ ⇒ list Z)
    -                  nil (fun n ls ⇒ (x+Z.of_N n)%Z :: ls) c).
    +  Definition elementsZ_single (x:Z) (c:N) :=
    +      List.rev' (N.peano_rec (fun _ ⇒ list Z)
    +                  nil (fun n ls ⇒ (x+Z.of_N n)%Z :: ls) c).

      Definition elements_single x c :=
    -    List.map Enc.decode (elementsZ_single x c).
    +    List.map Enc.decode (elementsZ_single x c).

      Lemma elementsZ_single_base : ∀ x,
    -    elementsZ_single x (0%N) = nil.
    - +    elementsZ_single x (0%N) = nil.
    +  Proof. done. Qed.
    +
      Lemma elementsZ_single_succ : ∀ x c,
    -    elementsZ_single x (N.succ c) =
    -    elementsZ_single x c ++ (x+Z.of_N c)::nil.
    - +    elementsZ_single x (N.succ c) =
    +    elementsZ_single x c ++ (x+Z.of_N c)::nil.
    +  Proof.
    +    intros x c.
    +    rewrite /elementsZ_single N.peano_rec_succ /rev' -!rev_alt //.
    +  Qed.
    +
      Lemma elementsZ_single_add : ∀ x c2 c1,
    -    elementsZ_single x (c1 + c2)%N =
    -    elementsZ_single x c1 ++ elementsZ_single (x+Z.of_N c1) c2.
    - +    elementsZ_single x (c1 + c2)%N =
    +    elementsZ_single x c1 ++ elementsZ_single (x+Z.of_N c1) c2.
    +  Proof.
    +    intros x.
    +    induction c2 as [| c2' IH] using N.peano_ind. {
    +      move ⇒ c1.
    +      rewrite elementsZ_single_base /= app_nil_r N.add_0_r //.
    +    } {
    +      move ⇒ c1.
    +      rewrite N.add_succ_r !elementsZ_single_succ IH app_assoc N2Z.inj_add Z.add_assoc //.
    +    }
    +  Qed.
    +
      Lemma elementsZ_single_succ_front : ∀ x c,
    -    elementsZ_single x (N.succ c) =
    -    x :: elementsZ_single (Z.succ x) c.
    - +    elementsZ_single x (N.succ c) =
    +    x :: elementsZ_single (Z.succ x) c.
    +  Proof.
    +    intros x c.
    +    rewrite -N.add_1_l elementsZ_single_add.
    +    rewrite N.one_succ elementsZ_single_succ elementsZ_single_base /= Z.add_0_r.
    +    by rewrite Z.add_1_r.
    +  Qed.
    +
      Lemma In_elementsZ_single : ∀ c y x,
    -    List.In y (elementsZ_single x c) ↔
    -    (x ≤ y) ∧ (y < (x+Z.of_N c)).
    - +    List.In y (elementsZ_single x c) ↔
    +    (x ≤ y) ∧ (y < (x+Z.of_N c)).
    +  Proof.
    +    induction c as [| c' IH] using N.peano_ind. {
    +      intros y x.
    +      rewrite elementsZ_single_base Z.add_0_r /=.
    +      omega.
    +    } {
    +      intros y x.
    +      rewrite elementsZ_single_succ in_app_iff IH /= N2Z.inj_succ Z.add_succ_r Z.lt_succ_r.
    +      split. {
    +        move ⇒ [ | []] //. {
    +          move ⇒ [H_x_le H_y_le].
    +          omega.
    +        } {
    +          move ⇒ <-.
    +          split.
    +            - by apply Z_le_add_r.
    +            - by apply Z.le_refl.
    +        }
    +      } {
    +        move ⇒ [H_x_le] H_y_lt.
    +        omega.
    +      }
    +    }
    +  Qed.
    +
      Lemma In_elementsZ_single1 : ∀ y x,
    -    List.In y (elementsZ_single x (1%N)) ↔
    -    (x = y).
    +    List.In y (elementsZ_single x (1%N)) ↔
    +    (x = y).
    +  Proof.
    +    intros y x.
    +    rewrite In_elementsZ_single /= Z.add_1_r Z.lt_succ_r.
    +    omega.
    +  Qed.

      Lemma length_elementsZ_single : ∀ cx x,
    -    length (elementsZ_single x cx) = N.to_nat cx.
    - +    length (elementsZ_single x cx) = N.to_nat cx.
    +  Proof.
    +    induction cx as [| cx' IH] using N.peano_ind. {
    +      by simpl.
    +    } {
    +      intros x.
    +      rewrite elementsZ_single_succ_front /=.
    +      rewrite IH N2Nat.inj_succ //.
    +    }
    +  Qed.
    +
      Lemma fold_elementsZ_aux_irrel {A} :
        âˆ€ f c (acc : A) x H1 H2,
    -      fold_elementsZ_aux f acc x c H1 =
    +      fold_elementsZ_aux f acc x c H1 =
          fold_elementsZ_aux f acc x c H2.
    - +  Proof.
    +    intros f c.
    +    induction c as [c IH] using (well_founded_ind lt_wf_0).
    +    case_eq c. {
    +      intros H_c acc x; case; intro H_H1; case; intro H_H2.
    +      reflexivity.
    +    } {
    +      intros p H_c acc x; case; intro H_H1; case; intro H_H2.
    +      unfold fold_elementsZ_aux; fold (@fold_elementsZ_aux A).
    +      case (f acc x) ⇒ // acc'.
    +      apply IH.
    +      rewrite H_c.
    +      apply N.lt_pred_l.
    +      discriminate.
    +    }
    +  Qed.
    +
      Lemma fold_elementsZ_single_pos {A} : ∀ f (acc : A) x p,
    -    fold_elementsZ_single f acc x (N.pos p) =
    +    fold_elementsZ_single f acc x (N.pos p) =
        match f acc x with
    -    | Some acc' ⇒
    -        fold_elementsZ_single f acc' (Z.succ x)
    -         (N.pred (N.pos p))
    -    | None ⇒ (true, acc)
    +    | Some acc' ⇒
    +        fold_elementsZ_single f acc' (Z.succ x)
    +         (N.pred (N.pos p))
    +    | None ⇒ (true, acc)
        end.
    +  Proof.
    +    intros f acc x p.
    +    unfold fold_elementsZ_single.
    +    unfold fold_elementsZ_aux.
    +    case: (lt_wf_0 _).
    +    fold (@fold_elementsZ_aux A).
    +    intro.
    +    case (f acc x) ⇒ // acc'.
    +    apply fold_elementsZ_aux_irrel.
    +  Qed.

      Lemma fold_elementsZ_single_zero {A} : ∀ f (acc : A) x,
    -      fold_elementsZ_single f acc x (0%N) = (false, acc).
    +      fold_elementsZ_single f acc x (0%N) = (false, acc).
    +  Proof.
    +    intros f acc x.
    +    unfold fold_elementsZ_single.
    +    case (lt_wf_0 (0%N)); intro.
    +    unfold fold_elementsZ_aux.
    +    reflexivity.
    +  Qed.

      Lemma fold_elementsZ_single_succ {A} : ∀ f (acc : A) x c,
    -    fold_elementsZ_single f acc x (N.succ c) =
    +    fold_elementsZ_single f acc x (N.succ c) =
        match f acc x with
    -      | Some acc' ⇒
    -          fold_elementsZ_single f acc' (Z.succ x) c
    -      | None ⇒ (true, acc)
    +      | Some acc' ⇒
    +          fold_elementsZ_single f acc' (Z.succ x) c
    +      | None ⇒ (true, acc)
        end.
    - -
    -  Fixpoint fold_opt {A B} f (acc : A) (bs : list B) : (bool × A) :=
    +  Proof.
    +    intros f acc x c.
    +    case c. {
    +      by rewrite fold_elementsZ_single_pos.
    +    } {
    +      intro p; simpl.
    +      rewrite fold_elementsZ_single_pos.
    +      case (f acc x) ⇒ // acc' /=.
    +      by rewrite Pos.pred_N_succ.
    +    }
    +  Qed.
    + +
    +  Fixpoint fold_opt {A B} f (acc : A) (bs : list B) : (bool × A) :=
        match bs with
    -      | nil ⇒ (false, acc)
    -      | (b :: bs') ⇒
    +      | nil ⇒ (false, acc)
    +      | (b :: bs') ⇒
            match f acc b with
    -        | Some acc' ⇒ fold_opt f acc' bs'
    -        | None ⇒ (true, acc)
    +        | Some acc' ⇒ fold_opt f acc' bs'
    +        | None ⇒ (true, acc)
            end
        end.

    -  Lemma fold_opt_list_cons : ∀ {A} (bs : list A) (acc : list A),
    -    fold_opt (fun l x ⇒ Some (x :: l)) acc bs =
    -    (false, List.rev bs ++ acc).
    - +  Lemma fold_opt_list_cons : ∀ {A} (bs : list A) (acc : list A),
    +    fold_opt (fun l x ⇒ Some (x :: l)) acc bs =
    +    (false, List.rev bs ++ acc).
    +  Proof.
    +    induction bs as [| b bs' IH] ⇒ acc. {
    +      by simpl.
    +    } {
    +      rewrite /= IH -app_assoc //.
    +    }
    +  Qed.
    +
    -  Lemma fold_opt_app {A B} : ∀ f (acc : A) (l1 l2 : list B),
    -    fold_opt f acc (l1 ++ l2) =
    -    (let (ab, acc') := fold_opt f acc l1 in
    -     if ab then (true, acc') else fold_opt f acc' l2).
    - +  Lemma fold_opt_app {A B} : ∀ f (acc : A) (l1 l2 : list B),
    +    fold_opt f acc (l1 ++ l2) =
    +    (let (ab, acc') := fold_opt f acc l1 in
    +     if ab then (true, acc') else fold_opt f acc' l2).
    +  Proof.
    +    intros f acc l1 l2.
    +    move : acc.
    +    induction l1 as [| b l1' IH] ⇒ acc. {
    +      rewrite app_nil_l //.
    +    } {
    +      rewrite /=.
    +      case (f acc b); last done.
    +      intro acc'.
    +      rewrite IH //.
    +    }
    +  Qed.
    +

      Lemma fold_elementsZ_single_alt_def {A} : ∀ f c (acc : A) x,
    -     fold_elementsZ_single f acc x c =
    +     fold_elementsZ_single f acc x c =
         fold_opt f acc (elementsZ_single x c).
    - +  Proof.
    +    intro f.
    +    induction c as [| c' IH] using N.peano_ind. {
    +      intros acc x.
    +      rewrite fold_elementsZ_single_zero
    +              elementsZ_single_base /fold_opt //.
    +    } {
    +      intros acc x.
    +      rewrite fold_elementsZ_single_succ
    +              elementsZ_single_succ_front /=.
    +      case (f acc x); last reflexivity.
    +      intro acc'.
    +      apply IH.
    +    }
    +  Qed.
    +
      Lemma fold_elementsZ_nil {A} : ∀ f (acc : A),
    -     fold_elementsZ f acc nil = (false, acc).
    - +     fold_elementsZ f acc nil = (false, acc).
    +  Proof. done. Qed.
    +
      Lemma fold_elementsZ_cons {A} : ∀ f (acc : A) y c s,
    -    fold_elementsZ f acc ((y, c)::s) =
    -    (let (ab, acc') := fold_elementsZ_single f acc y c in
    -     if ab then (true, acc') else fold_elementsZ f acc' s).
    +    fold_elementsZ f acc ((y, c)::s) =
    +    (let (ab, acc') := fold_elementsZ_single f acc y c in
    +     if ab then (true, acc') else fold_elementsZ f acc' s).
    +  Proof.
    +    intros f acc y c s.
    +    done.
    +  Qed.

      Lemma fold_elementsZ_alt_def_aux : ∀ (s : t) base,
    -    (snd (fold_elementsZ
    -      (fun (l : list Z) (x : Z) ⇒ Some (x :: l)) base s)) =
    -    elementsZ s ++ base.
    - +    (snd (fold_elementsZ
    +      (fun (l : list Z) (x : Z) ⇒ Some (x :: l)) base s)) =
    +    elementsZ s ++ base.
    +  Proof.
    +    induction s as [| [y1 c1] s' IH] ⇒ base. {
    +      rewrite elementsZ_nil /fold_elementsZ /fold_opt /snd
    +        app_nil_l //.
    +    } {
    +      rewrite /elementsZ !fold_elementsZ_cons.
    +      rewrite !fold_elementsZ_single_alt_def !fold_opt_list_cons.
    +      rewrite !IH app_nil_r app_assoc //.
    +    }
    +  Qed.
    +

      Lemma fold_elementsZ_alt_def {A} : ∀ f s (acc : A),
    -     fold_elementsZ f acc s =
    -     fold_opt f acc (rev (elementsZ s)).
    - -
    -  Lemma elementsZ_cons : ∀ x c s, elementsZ (((x, c) :: s) : t) =
    -     ((elementsZ s) ++ (List.rev (elementsZ_single x c))).
    - -
    -  Lemma elements_cons : ∀ x c s, elements (((x, c) :: s) : t) =
    -     ((elements_single x c) ++ elements s).
    - -
    -  Lemma elementsZ_app : ∀ (s1 s2 : t), elementsZ (s1 ++ s2) =
    -     ((elementsZ s2) ++ (elementsZ s1)).
    - -
    -  Lemma InZ_nil : ∀ y, InZ y nil ↔ False.
    - -
    -  Lemma InZ_cons : ∀ y x c s, InZ y (((x, c) :: s) : t) ↔
    -     List.In y (elementsZ_single x c) ∨ InZ y s.
    - +     fold_elementsZ f acc s =
    +     fold_opt f acc (rev (elementsZ s)).
    +  Proof.
    +    intro f.
    +    induction s as [| [y1 c1] s' IH] ⇒ acc. {
    +      by simpl.
    +    } {
    +      rewrite /elementsZ !fold_elementsZ_cons.
    +      rewrite !fold_elementsZ_single_alt_def
    +              fold_opt_list_cons app_nil_r
    +              fold_elementsZ_alt_def_aux rev_app_distr
    +              rev_involutive fold_opt_app.
    +      case (fold_opt f acc (elementsZ_single y1 c1)).
    +      move ⇒ [] //.
    +    }
    +  Qed.
    + +
    +  Lemma elementsZ_cons : ∀ x c s, elementsZ (((x, c) :: s) : t) =
    +     ((elementsZ s) ++ (List.rev (elementsZ_single x c))).
    +  Proof.
    +    intros x c s.
    +    rewrite /elementsZ fold_elementsZ_cons
    +            !fold_elementsZ_alt_def
    +            fold_elementsZ_single_alt_def.
    +    rewrite !fold_opt_list_cons.
    +    rewrite !app_nil_r !rev_involutive /=.
    +    rewrite fold_elementsZ_alt_def_aux //.
    +  Qed.
    + +
    +  Lemma elements_cons : ∀ x c s, elements (((x, c) :: s) : t) =
    +     ((elements_single x c) ++ elements s).
    +  Proof.
    +    intros x c s.
    +    rewrite /elements /elements_single elementsZ_cons.
    +    rewrite !rev_map_alt_def map_app rev_app_distr map_rev rev_involutive //.
    +  Qed.
    + +
    +  Lemma elementsZ_app : ∀ (s1 s2 : t), elementsZ (s1 ++ s2) =
    +     ((elementsZ s2) ++ (elementsZ s1)).
    +  Proof.
    +    induction s1 as [| [x1 c1] s1 IH1]. {
    +      move ⇒ s2.
    +      rewrite elementsZ_nil app_nil_r //.
    +    }
    +    move ⇒ s2.
    +    rewrite -app_comm_cons !elementsZ_cons IH1 -app_assoc //.
    +  Qed.
    + +
    +  Lemma InZ_nil : ∀ y, InZ y nil ↔ False.
    +  Proof.
    +    intro y.
    +    done.
    +  Qed.
    + +
    +  Lemma InZ_cons : ∀ y x c s, InZ y (((x, c) :: s) : t) ↔
    +     List.In y (elementsZ_single x c) ∨ InZ y s.
    +  Proof.
    +    intros y x c s.
    +    rewrite /InZ elementsZ_cons in_app_iff -in_rev.
    +    firstorder.
    +  Qed.
    +
      Lemma InZ_app : ∀ s1 s2 y,
    -     InZ y (s1 ++ s2) ↔ InZ y s1 ∨ InZ y s2.
    - +     InZ y (s1 ++ s2) ↔ InZ y s1 ∨ InZ y s2.
    +  Proof.
    +    intros s1 s2 y.
    +    rewrite /InZ elementsZ_app in_app_iff.
    +    tauto.
    +  Qed.
    +
      Lemma InZ_rev : ∀ s y,
    -     InZ y (List.rev s) ↔ InZ y s.
    - +     InZ y (List.rev s) ↔ InZ y s.
    +  Proof.
    +    intros s x.
    +    rewrite /InZ.
    +    induction s as [| [y c] s' IH]; first done.
    +    simpl.
    +    rewrite elementsZ_app in_app_iff IH.
    +    rewrite !elementsZ_cons !in_app_iff elementsZ_nil
    +            -!in_rev /=.
    +    tauto.
    +  Qed.
    +
      Lemma In_elementsZ_single_dec : ∀ y x c,
    -    {List.In y (elementsZ_single x c)} +
    -    {¬ List.In y (elementsZ_single x c)}.
    - +    {List.In y (elementsZ_single x c)} +
    +    {¬ List.In y (elementsZ_single x c)}.
    +  Proof.
    +    intros y x c.
    +    case (Z_le_dec x y); last first. {
    +      right; rewrite In_elementsZ_single; tauto.
    +    }
    +    case (Z_lt_dec y (x + Z.of_N c)); last first. {
    +      right; rewrite In_elementsZ_single; tauto.
    +    }
    +    left; rewrite In_elementsZ_single; tauto.
    +  Qed.
    +
      Lemma InZ_dec : ∀ y s,
    -     {InZ y s} + {¬InZ y s}.
    - -
    -  Lemma In_elementsZ_single_hd : ∀ (c : N) x, (c ≠ 0)%N → List.In x (elementsZ_single x c).
    - +     {InZ y s} + {¬InZ y s}.
    +  Proof.
    +    intros y.
    +    induction s as [| [x c] s IH]. {
    +      by right.
    +    }
    +    move : IH ⇒ [] IH. {
    +      by left; rewrite InZ_cons; right.
    +    }
    +    case (In_elementsZ_single_dec y x c). {
    +      by left; rewrite InZ_cons; left.
    +    } {
    +      by right; rewrite InZ_cons; tauto.
    +    }
    +  Qed.
    + +
    +  Lemma In_elementsZ_single_hd : ∀ (c : N) x, (c ≠ 0)%N → List.In x (elementsZ_single x c).
    +  Proof.
    +    intros c x H_c_neq.
    +    rewrite In_elementsZ_single.
    +    split. {
    +      apply Z.le_refl.
    +    } {
    +      apply Z.lt_add_pos_r.
    +      have → : 0 = Z.of_N (0%N) by [].
    +      apply N2Z.inj_lt.
    +      by apply N.neq_0_lt_0.
    +    }
    +  Qed.
    +

    @@ -1083,32 +1424,44 @@

    Library MSetsExtra.MSetIntervals


      Ltac Z_named_compare_cases H := match goal with
    -    | [ |- context [Z.compare ?z1 ?z2] ] ⇒
    -      case_eq (Z.compare z1 z2); [move ⇒ /Z.compare_eq_iff | move ⇒ /Z.compare_lt_iff | move ⇒ /Z.compare_gt_iff]; move ⇒ H //
    +    | [ |- context [Z.compare ?z1 ?z2] ] ⇒
    +      case_eq (Z.compare z1 z2); [move ⇒ /Z.compare_eq_iff | move ⇒ /Z.compare_lt_iff | move ⇒ /Z.compare_gt_iff]; move ⇒ H //
      end.

    -  Ltac Z_compare_cases := let H := fresh "H" in Z_named_compare_cases H.
    +  Ltac Z_compare_cases := let H := fresh "H" in Z_named_compare_cases H.

    -  Lemma interval_compare_elim : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    -    match (interval_compare (y1, c1) (y2, c2)) with
    -      | ICR_before ⇒ (y1 + Z.of_N c1) < y2
    -      | ICR_before_touch ⇒ (y1 + Z.of_N c1) = y2
    -      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    -      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    -      | ICR_equal ⇒ (y1 = y2) ∧ (c1 = c2)
    -      | ICR_overlap_before ⇒ (y1 < y2) ∧ (y2 < y1 + Z.of_N c1) ∧ (y1 + Z.of_N c1 < y2 + Z.of_N c2)
    -      | ICR_overlap_after ⇒ (y2 < y1) ∧ (y1 < y2 + Z.of_N c2) ∧ (y2 + Z.of_N c2 < y1 + Z.of_N c1)
    -      | ICR_subsume_1 ⇒ (y2 ≤ y1) ∧ (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) ∧ (y2 < y1 ∨ y1 + Z.of_N c1 < y2 + Z.of_N c2)
    -      | ICR_subsume_2 ⇒ (y1 ≤ y2) ∧ (y2 + Z.of_N c2 ≤ y1 + Z.of_N c1) ∧ (y1 < y2 ∨ y2 + Z.of_N c2 < y1 + Z.of_N c1)
    +  Lemma interval_compare_elim : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    +    match (interval_compare (y1, c1) (y2, c2)) with
    +      | ICR_before ⇒ (y1 + Z.of_N c1) < y2
    +      | ICR_before_touch ⇒ (y1 + Z.of_N c1) = y2
    +      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    +      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    +      | ICR_equal ⇒ (y1 = y2) ∧ (c1 = c2)
    +      | ICR_overlap_before ⇒ (y1 < y2) ∧ (y2 < y1 + Z.of_N c1) ∧ (y1 + Z.of_N c1 < y2 + Z.of_N c2)
    +      | ICR_overlap_after ⇒ (y2 < y1) ∧ (y1 < y2 + Z.of_N c2) ∧ (y2 + Z.of_N c2 < y1 + Z.of_N c1)
    +      | ICR_subsume_1 ⇒ (y2 ≤ y1) ∧ (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) ∧ (y2 < y1 ∨ y1 + Z.of_N c1 < y2 + Z.of_N c2)
    +      | ICR_subsume_2 ⇒ (y1 ≤ y2) ∧ (y2 + Z.of_N c2 ≤ y1 + Z.of_N c1) ∧ (y1 < y2 ∨ y2 + Z.of_N c2 < y1 + Z.of_N c1)
        end.
    +  Proof.
    +    intros y1 c1 y2 c2.
    +    rewrite /interval_compare.
    +    (repeat Z_compare_cases); subst; repeat split;
    +       try (by apply Z.eq_le_incl);
    +       try (by apply Z.lt_le_incl);
    +       try (by left); try (by right).
    + +
    +    apply Z.add_reg_l in H2.
    +    by apply N2Z.inj.
    +  Qed.

    -  Lemma interval_compare_swap : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    -    (c1 ≠ 0%N) ∨ (c2 ≠ 0%N) →
    -    interval_compare (y2, c2) (y1, c1) =
    -    match (interval_compare (y1, c1) (y2, c2)) with
    +  Lemma interval_compare_swap : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    +    (c1 ≠ 0%N) ∨ (c2 ≠ 0%N) →
    +    interval_compare (y2, c2) (y1, c1) =
    +    match (interval_compare (y1, c1) (y2, c2)) with
          | ICR_before ⇒ ICR_after
          | ICR_before_touch ⇒ ICR_after_touch
          | ICR_after ⇒ ICR_before
    @@ -1119,30 +1472,128 @@

    Library MSetsExtra.MSetIntervals

          | ICR_subsume_1 ⇒ ICR_subsume_2
          | ICR_subsume_2 ⇒ ICR_subsume_1
        end.
    - -
    -  Lemma interval_1_compare_alt_def : ∀ (y : Z) (i : (Z × N)),
    -    interval_1_compare y i = match (interval_compare (y, (1%N)) i) with
    +  Proof.
    +    intros y1 c1 y2 c2 H_c12_neq_0.
    +    rewrite /interval_compare.
    +    move : (Z.compare_antisym y1 y2) ⇒ →.
    +    move : (Z.compare_antisym (y1 + Z.of_N c1) (y2 + Z.of_N c2)) ⇒ →.
    +    have H_suff : y1 + Z.of_N c1 ≤ y2 → y2 + Z.of_N c2 ≤ y1 → False. {
    +      move ⇒ H1 H2.
    +      case H_c12_neq_0 ⇒ H_c_neq_0. {
    +        suff : (y1 + Z.of_N c1 ≤ y1). {
    +          apply Z.nle_gt.
    +          by apply Z_lt_add_r.
    +        }
    +        eapply Z.le_trans; eauto.
    +        eapply Z.le_trans; eauto.
    +        apply Z_le_add_r.
    +      } {
    +        suff : (y2 + Z.of_N c2 ≤ y2). {
    +          apply Z.nle_gt.
    +          by apply Z_lt_add_r.
    +        }
    +        eapply Z.le_trans; eauto.
    +        eapply Z.le_trans; eauto.
    +        apply Z_le_add_r.
    +      }
    +    }
    +    repeat Z_compare_cases. {
    +      exfalso; apply H_suff.
    +        - by rewrite H; apply Z.le_refl.
    +        - by rewrite H0; apply Z.le_refl.
    +    } {
    +      exfalso; apply H_suff.
    +        - by rewrite H; apply Z.le_refl.
    +        - by apply Z.lt_le_incl.
    +    } {
    +      exfalso; apply H_suff.
    +        - by apply Z.lt_le_incl.
    +        - by rewrite H0; apply Z.le_refl.
    +    } {
    +      exfalso; apply H_suff.
    +        - by apply Z.lt_le_incl.
    +        - by apply Z.lt_le_incl.
    +    }
    +  Qed.
    + +
    +  Lemma interval_1_compare_alt_def : ∀ (y : Z) (i : (Z × N)),
    +    interval_1_compare y i = match (interval_compare (y, (1%N)) i) with
          | ICR_equal ⇒ ICR_subsume_1
          | ICR_subsume_1 ⇒ ICR_subsume_1
          | ICR_subsume_2 ⇒ ICR_subsume_1
          | r ⇒ r
        end.
    - -
    -  Lemma interval_1_compare_elim : ∀ (y1 : Z) (y2 : Z) (c2 : N),
    -    match (interval_1_compare y1 (y2, c2)) with
    -      | ICR_before ⇒ Z.succ y1 < y2
    -      | ICR_before_touch ⇒ y2 = Z.succ y1
    -      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    -      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    -      | ICR_equal ⇒ False
    -      | ICR_overlap_before ⇒ False
    -      | ICR_overlap_after ⇒ False
    -      | ICR_subsume_1 ⇒ (c2 = 0%N) ∨ ((y2 ≤ y1) ∧ (y1 < y2 + Z.of_N c2))
    -      | ICR_subsume_2 ⇒ False
    +  Proof.
    +    move ⇒ y1 [y2 c2].
    +    rewrite /interval_1_compare /interval_compare.
    +    replace (y1 + Z.of_N 1) with (Z.succ y1); last done.
    +    repeat Z_compare_cases. {
    +      contradict H1.
    +      by apply Zle_not_lt, Zlt_succ_le.
    +    } {
    +      contradict H.
    +      by apply Zle_not_lt, Zlt_succ_le.
    +    }
    +  Qed.
    + +
    +  Lemma interval_1_compare_elim : ∀ (y1 : Z) (y2 : Z) (c2 : N),
    +    match (interval_1_compare y1 (y2, c2)) with
    +      | ICR_before ⇒ Z.succ y1 < y2
    +      | ICR_before_touch ⇒ y2 = Z.succ y1
    +      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    +      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    +      | ICR_equal ⇒ False
    +      | ICR_overlap_before ⇒ False
    +      | ICR_overlap_after ⇒ False
    +      | ICR_subsume_1 ⇒ (c2 = 0%N) ∨ ((y2 ≤ y1) ∧ (y1 < y2 + Z.of_N c2))
    +      | ICR_subsume_2 ⇒ False
        end.
    - +  Proof.
    +    intros y1 y2 c2.
    +    move : (interval_compare_elim y1 (1%N) y2 c2).
    +    rewrite interval_1_compare_alt_def.
    +    have H_succ: ∀ z, z + Z.of_N 1 = Z.succ z by done.
    + +
    +    case_eq (interval_compare (y1, 1%N) (y2, c2)) ⇒ H_comp;
    +      rewrite ?H_succ ?Z.lt_succ_r ?Z.le_succ_l //. {
    +      move ⇒ [H_lt] [H_le] _.
    +      contradict H_lt.
    +      by apply Zle_not_lt.
    +    } {
    +      move ⇒ [_] [H_lt] H_le.
    +      contradict H_lt.
    +      by apply Zle_not_lt.
    +    } {
    +      move ⇒ [->] <-.
    +      rewrite ?Z.lt_succ_r.
    +      right.
    +      split; apply Z.le_refl.
    +    } {
    +      tauto.
    +    } {
    +      case (N.zero_or_succ c2). {
    +        move ⇒ → _; by left.
    +      } {
    +        move ⇒ [c2'] →.
    +        rewrite !N2Z.inj_succ Z.add_succ_r -Z.succ_le_mono Z.le_succ_l.
    +        move ⇒ [H_y1_le] [H_le_y1].
    +        suff → : y1 = y2. {
    +          move ⇒ [] H_pre; contradict H_pre. {
    +            apply Z.lt_irrefl.
    +          } {
    +            apply Zle_not_lt, Z_le_add_r.
    +          }
    +        }
    +        apply Z.le_antisymm ⇒ //.
    +        eapply Z.le_trans; last apply H_le_y1.
    +        apply Z_le_add_r.
    +      }
    +    }
    +  Qed.
    +
    @@ -1152,22 +1603,45 @@

    Library MSetsExtra.MSetIntervals

      Lemma addZ_aux_alt_def : ∀ x s acc,
    -    addZ_aux acc x s = (List.rev acc) ++ addZ x s.
    - +    addZ_aux acc x s = (List.rev acc) ++ addZ x s.
    +  Proof.
    +    intros y1 s.
    +    unfold addZ.
    +    induction s as [| [y2 c2] s' IH] ⇒ acc. {
    +      rewrite /addZ_aux /addZ /= /rev' !rev_append_rev /= app_nil_r //.
    +    } {
    +      unfold addZ_aux.
    +      case (interval_1_compare y1 (y2, c2)); fold addZ_aux;
    +        rewrite ?rev_append_rev /= ?app_assoc_reverse //.
    +      rewrite (IH ((y2,c2)::acc)) (IH ((y2,c2)::nil)).
    +      rewrite /= app_assoc_reverse //.
    +    }
    +  Qed.
    +
      Lemma addZ_alt_def : ∀ x s,
    -    addZ x s =
    +    addZ x s =
        match s with
    -    | nil ⇒ (x, (1%N))::nil
    -    | (y, c) :: l ⇒
    -        match (interval_1_compare x (y,c)) with
    -          | ICR_before ⇒ (x, (1%N))::s
    -          | ICR_before_touch ⇒ (x, N.succ c)::l
    -          | ICR_after ⇒ (y,c)::(addZ x l)
    -          | ICR_after_touch ⇒ insert_interval_begin y (N.succ c) l
    -          | _ ⇒ (y, c)::l
    +    | nil ⇒ (x, (1%N))::nil
    +    | (y, c) :: l ⇒
    +        match (interval_1_compare x (y,c)) with
    +          | ICR_before ⇒ (x, (1%N))::s
    +          | ICR_before_touch ⇒ (x, N.succ c)::l
    +          | ICR_after ⇒ (y,c)::(addZ x l)
    +          | ICR_after_touch ⇒ insert_interval_begin y (N.succ c) l
    +          | _ ⇒ (y, c)::l
            end
        end.
    +  Proof.
    +    intros x s.
    +    rewrite /addZ.
    +    case s ⇒ //.
    +    move ⇒ [y c] s'.
    +    unfold addZ_aux.
    +    case (interval_1_compare x (y, c)); fold addZ_aux;
    +      rewrite ?rev_append_rev /= ?app_assoc_reverse //.
    +    rewrite addZ_aux_alt_def //.
    +  Qed.

    @@ -1182,112 +1656,315 @@

    Library MSetsExtra.MSetIntervals


      Lemma interval_list_elements_greater_cons : ∀ z x c s,
    -    interval_list_elements_greater z ((x, c) :: s) = true ↔
    -    (z < x).
    +    interval_list_elements_greater z ((x, c) :: s) = true ↔
    +    (z < x).
    +  Proof.
    +    intros z x c s.
    +    rewrite /=.
    +    apply Z.ltb_lt.
    +  Qed.

      Lemma interval_list_elements_greater_impl : ∀ x y s,
    -    (y ≤ x) →
    -    interval_list_elements_greater x s = true →
    -    interval_list_elements_greater y s = true.
    +    (y ≤ x) →
    +    interval_list_elements_greater x s = true →
    +    interval_list_elements_greater y s = true.
    +  Proof.
    +    intros x y s.
    +    case s ⇒ //.
    +    move ⇒ [z c] s'.
    +    rewrite /interval_list_elements_greater.
    +    move ⇒ H_y_leq /Z.ltb_lt H_x_lt.
    +    apply Z.ltb_lt.
    +    eapply Z.le_lt_trans; eauto.
    +  Qed.

    -  Lemma interval_list_invariant_nil : interval_list_invariant nil = true.
    +  Lemma interval_list_invariant_nil : interval_list_invariant nil = true.
    +  Proof.
    +    by [].
    +  Qed.

    -  Lemma Ok_nil : Ok nil ↔ True.
    +  Lemma Ok_nil : Ok nil ↔ True.
    +  Proof.
    +    rewrite /Ok /IsOk /interval_list_invariant /is_encoded_elems_list //.
    +  Qed.

      Lemma is_encoded_elems_list_app : ∀ l1 l2,
    -    is_encoded_elems_list (l1 ++ l2) ↔
    -    (is_encoded_elems_list l1 ∧ is_encoded_elems_list l2).
    +    is_encoded_elems_list (l1 ++ l2) ↔
    +    (is_encoded_elems_list l1 ∧ is_encoded_elems_list l2).
    +  Proof.
    +    intros l1 l2.
    +    rewrite /is_encoded_elems_list.
    +    setoid_rewrite in_app_iff.
    +    split; firstorder.
    +  Qed.

      Lemma is_encoded_elems_list_rev : ∀ l,
    -    is_encoded_elems_list (List.rev l) ↔
    +    is_encoded_elems_list (List.rev l) ↔
        is_encoded_elems_list l.
    +  Proof.
    +    intros l.
    +    rewrite /is_encoded_elems_list.
    +    split; (
    +      move ⇒ H x H_in;
    +      apply H;
    +      move : H_in;
    +      rewrite -in_rev ⇒ //
    +    ).
    +  Qed.

      Lemma interval_list_invariant_cons : ∀ y c s',
    -    interval_list_invariant ((y, c) :: s') = true ↔
    -    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧
    -      ((c ≠ 0)%N) ∧ interval_list_invariant s' = true).
    - +    interval_list_invariant ((y, c) :: s') = true ↔
    +    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧
    +      ((c ≠ 0)%N) ∧ interval_list_invariant s' = true).
    +  Proof.
    +    rewrite /interval_list_invariant -/interval_list_invariant.
    +    intros y c s'.
    +    rewrite !Bool.andb_true_iff negb_true_iff.
    +    split. {
    +      move ⇒ [] [H_inf] /N.eqb_neq H_c H_s'. tauto.
    +    } {
    +      move ⇒ [H_inf] [/N.eqb_neq H_c] H_s'. tauto.
    +    }
    +  Qed.
    +
      Lemma interval_list_invariant_sing : ∀ x c,
    -    interval_list_invariant ((x, c)::nil) = true ↔ (c ≠ 0)%N.
    +    interval_list_invariant ((x, c)::nil) = true ↔ (c ≠ 0)%N.
    +  Proof.
    +    intros x c.
    +    rewrite interval_list_invariant_cons.
    +    split; tauto.
    +  Qed.

    -  Lemma Ok_cons : ∀ y c s', Ok ((y, c) :: s') ↔
    -    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧ ((c ≠ 0)%N) ∧
    -     is_encoded_elems_list (elementsZ_single y c) ∧ Ok s').
    +  Lemma Ok_cons : ∀ y c s', Ok ((y, c) :: s') ↔
    +    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧ ((c ≠ 0)%N) ∧
    +     is_encoded_elems_list (elementsZ_single y c) ∧ Ok s').
    +  Proof.
    +    intros y c s'.
    +    rewrite /Ok /IsOk interval_list_invariant_cons elementsZ_cons is_encoded_elems_list_app
    +       is_encoded_elems_list_rev.
    +    tauto.
    +  Qed.

      Lemma Nin_elements_greater : ∀ s y,
    -     interval_list_elements_greater y s = true →
    -     interval_list_invariant s = true →
    -     âˆ€ x, x ≤ y → ~(InZ x s).
    - +     interval_list_elements_greater y s = true →
    +     interval_list_invariant s = true →
    +     âˆ€ x, x ≤ y → ~(InZ x s).
    +  Proof.
    +    induction s as [| [z c] s' IH]. {
    +      intros y _ _ x _.
    +      by simpl.
    +    } {
    +      move ⇒ y /interval_list_elements_greater_cons H_y_lt
    +        /interval_list_invariant_cons [H_gr] [H_c] H_s'
    +        x H_x_le.
    +      rewrite InZ_cons In_elementsZ_single.
    +      have H_x_lt : x < z by eapply Z.le_lt_trans; eauto.
    + +
    +      move ⇒ []. {
    +        move ⇒ [H_z_leq] _; contradict H_z_leq.
    +        by apply Z.nle_gt.
    +      } {
    +        eapply IH; eauto.
    +        by apply Z_lt_le_add_r.
    +      }
    +    }
    +  Qed.
    +
      Lemma Nin_elements_greater_equal :
         âˆ€ x s,
    -       interval_list_elements_greater x s = true →
    -       interval_list_invariant s = true →
    -       Â¬ (InZ x s).
    +       interval_list_elements_greater x s = true →
    +       interval_list_invariant s = true →
    +       Â¬ (InZ x s).
    +  Proof.
    +    move ⇒ x s H_inv H_gr.
    +    apply (Nin_elements_greater s x) ⇒ //.
    +    apply Z.le_refl.
    +  Qed.

      Lemma interval_list_elements_greater_alt_def : ∀ s y,
    -     interval_list_invariant s = true →
    -
    -     (interval_list_elements_greater y s = true ↔
    -      (∀ x, x ≤ y → ~(InZ x s))).
    - +     interval_list_invariant s = true →
    +
    +     (interval_list_elements_greater y s = true ↔
    +      (∀ x, x ≤ y → ~(InZ x s))).
    +  Proof.
    +    intros s y H_inv.
    +    split. {
    +      move ⇒ H_gr.
    +      apply Nin_elements_greater ⇒ //.
    +    } {
    +      move : H_inv.
    +      case s as [| [x2 c] s'] ⇒ //.
    +      rewrite interval_list_invariant_cons interval_list_elements_greater_cons.
    +      move ⇒ [_] [H_c_neq] _ H.
    +      apply Z.nle_gt ⇒ H_ge.
    +      apply (H x2) ⇒ //.
    +      rewrite InZ_cons; left.
    +      apply In_elementsZ_single_hd ⇒ //.
    +    }
    +  Qed.
    +
      Lemma interval_list_elements_greater_alt2_def : ∀ s y,
    -     interval_list_invariant s = true →
    +     interval_list_invariant s = true →

    -     (interval_list_elements_greater y s = true ↔
    -      (∀ x, InZ x s → y < x)).
    +     (interval_list_elements_greater y s = true ↔
    +      (∀ x, InZ x s → y < x)).
    +  Proof.
    +    intros s y H.
    +    rewrite interval_list_elements_greater_alt_def //.
    +    firstorder.
    +    apply Z.nle_gt.
    +    move ⇒ H_lt.
    +    eapply H0; eauto.
    +  Qed.

      Lemma interval_list_elements_greater_intro : ∀ s y,
    -     interval_list_invariant s = true →
    -     (∀ x, InZ x s → y < x) →
    -     interval_list_elements_greater y s = true.
    +     interval_list_invariant s = true →
    +     (∀ x, InZ x s → y < x) →
    +     interval_list_elements_greater y s = true.
    +  Proof.
    +    intros s y H1 H2.
    +    rewrite interval_list_elements_greater_alt2_def //.
    +  Qed.

      Lemma interval_list_elements_greater_app_elim_1 : ∀ s1 s2 y,
    -    interval_list_elements_greater y (s1 ++ s2) = true →
    -    interval_list_elements_greater y s1 = true.
    +    interval_list_elements_greater y (s1 ++ s2) = true →
    +    interval_list_elements_greater y s1 = true.
    +  Proof.
    +    intros s1 s2 y.
    +    case s1 ⇒ //.
    +  Qed.

      Lemma interval_list_invariant_app_intro : ∀ s1 s2,
    -      interval_list_invariant s1 = true →
    -      interval_list_invariant s2 = true →
    -      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    -      interval_list_invariant (s1 ++ s2) = true.
    - +      interval_list_invariant s1 = true →
    +      interval_list_invariant s2 = true →
    +      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    +      interval_list_invariant (s1 ++ s2) = true.
    +  Proof.
    +    induction s1 as [| [y1 c1] s1' IH]. {
    +      move ⇒ s2 _ //.
    +    } {
    +      move ⇒ s2.
    +      rewrite -app_comm_cons !interval_list_invariant_cons.
    +      move ⇒ [H_gr] [H_c1_neq] H_inv_s1' H_inv_s2 H_inz_s2.
    +      split; last split. {
    +        move : H_gr H_inz_s2.
    +        case s1' as [| [y1' c1'] s1'']; last done.
    +        move ⇒ _ H_inz_s2.
    +        rewrite app_nil_l.
    +        apply interval_list_elements_greater_intro ⇒ //.
    +        move ⇒ x H_x_in_s2.
    +        suff H_inz : InZ (Z.pred (y1 + Z.of_N c1)) ((y1, c1) :: nil). {
    +          move : (H_inz_s2 _ _ H_inz H_x_in_s2).
    +          by rewrite Z.succ_pred.
    +        }
    +        rewrite InZ_cons In_elementsZ_single -Z.lt_le_pred; left.
    +        split. {
    +          by apply Z_lt_add_r.
    +        } {
    +          apply Z.lt_pred_l.
    +        }
    +      } {
    +        assumption.
    +      } {
    +        apply IH ⇒ //.
    +        intros x1 x2 H_in_x1 H_in_x2.
    +        apply H_inz_s2 ⇒ //.
    +        rewrite InZ_cons; by right.
    +      }
    +    }
    +  Qed.
    +
      Lemma interval_list_invariant_app_elim : ∀ s1 s2,
    -      interval_list_invariant (s1 ++ s2) = true →
    -      interval_list_invariant s1 = true ∧
    -      interval_list_invariant s2 = true ∧
    -      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2).
    - +      interval_list_invariant (s1 ++ s2) = true →
    +      interval_list_invariant s1 = true ∧
    +      interval_list_invariant s2 = true ∧
    +      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2).
    +  Proof.
    +    move ⇒ s1 s2.
    +    induction s1 as [| [y1 c1] s1' IH]; first done.
    +    rewrite -app_comm_cons !interval_list_invariant_cons.
    + +
    +    move ⇒ [H_gr] [H_c1_neq_0] /IH [H_inv_s1'] [H_inv_s2] H_in_s1'_s2.
    +    repeat split; try assumption. {
    +      move : H_gr.
    +      case s1'; first done.
    +      move ⇒ [y2 c2] s1''.
    +      rewrite interval_list_elements_greater_cons //.
    +    } {
    +      move ⇒ x1 x2.
    +      rewrite InZ_cons In_elementsZ_single.
    +      move ⇒ []; last by apply H_in_s1'_s2.
    +      move ⇒ [] H_y1_le H_x1_lt H_x2_in.
    +      move : H_gr.
    +      rewrite interval_list_elements_greater_alt2_def; last first. {
    +          by apply interval_list_invariant_app_intro.
    +      }
    +      move ⇒ H_in_s12'.
    +      have : (y1 + Z.of_N c1 < x2). {
    +        apply H_in_s12'.
    +        rewrite InZ_app.
    +        by right.
    +      }
    +      move ⇒ H_lt_x2.
    +      apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +      by apply Zlt_le_succ.
    +    }
    +  Qed.
    +
      Lemma interval_list_invariant_app_iff : ∀ s1 s2,
    -      interval_list_invariant (s1 ++ s2) = true ↔
    -      (interval_list_invariant s1 = true ∧
    -      interval_list_invariant s2 = true ∧
    -      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2)).
    - +      interval_list_invariant (s1 ++ s2) = true ↔
    +      (interval_list_invariant s1 = true ∧
    +      interval_list_invariant s2 = true ∧
    +      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2)).
    +  Proof.
    +    intros s1 s2.
    +    split. {
    +      by apply interval_list_invariant_app_elim.
    +    } {
    +      move ⇒ [] H_inv_s1 [].
    +      by apply interval_list_invariant_app_intro.
    +    }
    +  Qed.
    +
      Lemma interval_list_invariant_snoc_intro : ∀ s1 y2 c2,
    -      interval_list_invariant s1 = true →
    -      (c2 ≠ 0)%N →
    -      (∀ x, InZ x s1 → Z.succ x < y2) →
    -      interval_list_invariant (s1 ++ ((y2, c2)::nil)) = true.
    - +      interval_list_invariant s1 = true →
    +      (c2 ≠ 0)%N →
    +      (∀ x, InZ x s1 → Z.succ x < y2) →
    +      interval_list_invariant (s1 ++ ((y2, c2)::nil)) = true.
    +  Proof.
    +    intros s1 y2 c2 H_inv_s1 H_c2_neq H_in_s1.
    +    apply interval_list_invariant_app_intro ⇒ //. {
    +      rewrite interval_list_invariant_cons; done.
    +    } {
    +      intros x1 x2 H_in_x1.
    +      rewrite InZ_cons.
    +      move ⇒ [] //.
    +      rewrite In_elementsZ_single.
    +      move ⇒ [H_y2_le] _.
    +      eapply Z.lt_le_trans; eauto.
    +    }
    +  Qed.
    +

    @@ -1300,20 +1977,85 @@

    Library MSetsExtra.MSetIntervals


    -  Lemma encode_decode_eq : ∀ x s, Ok s → InZ x s →
    -    (Enc.encode (Enc.decode x) = x).
    - -
    -  Lemma In_alt_def : ∀ x s, Ok s →
    -    (In x s ↔ List.In x (elements s)).
    - -
    -  Lemma In_InZ : ∀ x s, Ok s →
    -    (In x s ↔ InZ (Enc.encode x) s).
    - -
    -  Lemma InZ_In : ∀ x s, Ok s →
    -    (InZ x s → In (Enc.decode x) s).
    +  Lemma encode_decode_eq : ∀ x s, Ok s → InZ x s →
    +    (Enc.encode (Enc.decode x) = x).
    +  Proof.
    +    intros x s.
    +    rewrite /Ok /IsOk /InZ.
    +    move ⇒ [_] H_enc H_in_x.
    +    move : (H_enc _ H_in_x) ⇒ [x'] <-.
    +    rewrite Enc.decode_encode_ok //.
    +  Qed.
    + +
    +  Lemma In_alt_def : ∀ x s, Ok s →
    +    (In x s ↔ List.In x (elements s)).
    +  Proof.
    +    intros x s H_ok.
    +    rewrite /In InA_alt /elements rev_map_alt_def.
    +    split. {
    +      move ⇒ [y] [H_y_eq].
    +      rewrite -!in_rev !in_map_iff.
    +      move ⇒ [x'] [H_y_eq'] H_x'_in.
    +      suff H_x'_eq : (Enc.encode x = x'). {
    +        âˆƒ x'.
    +        split ⇒ //.
    +        rewrite -H_x'_eq Enc.decode_encode_ok //.
    +      }
    +      have H_enc_list : is_encoded_elems_list (elementsZ s). {
    +        move : H_ok.
    +        rewrite /Ok /IsOk ⇒ [] [] //.
    +      }
    +      move : (H_enc_list _ H_x'_in) ⇒ [x''] H_x'_eq.
    +      move : H_y_eq'.
    +      rewrite -!H_x'_eq Enc.decode_encode_ok ⇒ H_y_eq'.
    +      subst.
    +      suff → : Z.eq (Enc.encode x) (Enc.encode y) by done.
    +      by rewrite Enc.encode_eq.
    +    } {
    +      move ⇒ H_enc_in.
    +      âˆƒ x.
    +      split ⇒ //.
    +      apply Enc.E.eq_equiv.
    +    }
    +  Qed.
    + +
    +  Lemma In_InZ : ∀ x s, Ok s →
    +    (In x s ↔ InZ (Enc.encode x) s).
    +  Proof.
    +    intros x s H_ok.
    +    rewrite /InZ In_alt_def /elements rev_map_alt_def -in_rev in_map_iff.
    +    split; last first. {
    +      âˆƒ (Enc.encode x).
    +      by rewrite Enc.decode_encode_ok.
    +    }
    +    move ⇒ [y] [<-] H_y_in.
    +    suff : ∃ z, (Enc.encode z = y). {
    +      move ⇒ [z] H_y_eq.
    +      move : H_y_in.
    +      by rewrite -!H_y_eq Enc.decode_encode_ok.
    +    }
    +    suff H_enc_list : is_encoded_elems_list (elementsZ s). {
    +      by apply H_enc_list.
    +    }
    +    apply H_ok.
    +  Qed.
    + +
    +  Lemma InZ_In : ∀ x s, Ok s →
    +    (InZ x s → In (Enc.decode x) s).
    +  Proof.
    +    intros x s H_ok.
    +    rewrite In_InZ /InZ.
    +    move : H_ok.
    +    rewrite /Ok /IsOk /is_encoded_elems_list.
    +    move ⇒ [_] H_enc.
    +    move ⇒ H_in.
    +    move : (H_enc _ H_in) ⇒ [e] H_x.
    +    subst.
    +    by rewrite Enc.decode_encode_ok.
    +  Qed.

    @@ -1328,16 +2070,67 @@

    Library MSetsExtra.MSetIntervals


      Lemma memZ_spec :
    -    âˆ€ (s : t) (x : Z) (Hs : Ok s), memZ x s = true ↔ InZ x s.
    - +    âˆ€ (s : t) (x : Z) (Hs : Ok s), memZ x s = true ↔ InZ x s.
    +  Proof.
    +    induction s as [| [y c] s' IH]. {
    +      intros x _.
    +      rewrite /InZ elementsZ_nil //.
    +    } {
    +      move ⇒ x /Ok_cons [H_inf] [H_c] [H_is_enc] H_s'.
    +      rewrite /InZ /memZ elementsZ_cons -/memZ.
    +      rewrite in_app_iff -!in_rev In_elementsZ_single.
    + +
    +      case_eq (x <? y). {
    +        move ⇒ /Z.ltb_lt H_x_lt.
    +        split; first done.
    +        move ⇒ []. {
    +          move ⇒ H_x_in; contradict H_x_in.
    +          apply Nin_elements_greater with (y := (y + Z.of_N c)) ⇒ //. {
    +            apply H_s'.
    +          } {
    +            apply Z_lt_le_add_r ⇒ //.
    +          }
    +        } {
    +          move ⇒ [H_y_le]; contradict H_y_le.
    +          by apply Z.nle_gt.
    +        }
    +      } {
    +        move ⇒ /Z.ltb_ge H_y_le.
    +        case_eq (x <? y + Z.of_N c). {
    +          move ⇒ /Z.ltb_lt H_x_lt.
    +          split; last done.
    +          move ⇒ _.
    +          by right.
    +        } {
    +          move ⇒ /Z.ltb_ge H_yc_le.
    +          rewrite IH.
    +          split; first tauto.
    +          move ⇒ [] //.
    +          move ⇒ [_] H_x_lt; contradict H_x_lt.
    +          by apply Z.nlt_ge.
    +        }
    +      }
    +    }
    +  Qed.
    +
      Lemma mem_spec :
    -   âˆ€ (s : t) (x : elt) (Hs : Ok s), mem x s = true ↔ In x s.
    +   âˆ€ (s : t) (x : elt) (Hs : Ok s), mem x s = true ↔ In x s.
    +  Proof.
    +    intros s x Hs.
    +    rewrite /mem memZ_spec In_InZ //.
    +  Qed.

      Lemma merge_interval_size_neq_0 : ∀ x1 c1 x2 c2,
    -     (c1 ≠ 0%N) →
    -     (merge_interval_size x1 c1 x2 c2 ≠ 0)%N.
    +     (c1 ≠ 0%N) →
    +     (merge_interval_size x1 c1 x2 c2 ≠ 0)%N.
    +  Proof.
    +    intros x1 c1 x2 c2.
    +    rewrite /merge_interval_size !N.neq_0_lt_0 N.max_lt_iff.
    +    by left.
    +  Qed.

    @@ -1350,32 +2143,67 @@

    Library MSetsExtra.MSetIntervals


      Lemma interval_list_invariant_insert_intervalZ_guarded : ∀ x c s,
    -    interval_list_invariant s = true →
    -    interval_list_elements_greater (x + Z.of_N c) s = true →
    -    interval_list_invariant (insert_intervalZ_guarded x c s) = true.
    +    interval_list_invariant s = true →
    +    interval_list_elements_greater (x + Z.of_N c) s = true →
    +    interval_list_invariant (insert_intervalZ_guarded x c s) = true.
    +  Proof.
    +    intros x c s.
    +    rewrite /insert_intervalZ_guarded.
    +    case_eq (c =? 0)%N ⇒ //.
    +    move ⇒ /N.eqb_neq.
    +    rewrite interval_list_invariant_cons.
    +    tauto.
    +  Qed.

      Lemma interval_list_elements_greater_insert_intervalZ_guarded : ∀ x c y s,
    -    interval_list_elements_greater y (insert_intervalZ_guarded x c s) = true ↔
    -    (if (c =? 0)%N then (interval_list_elements_greater y s = true) else (y < x)).
    +    interval_list_elements_greater y (insert_intervalZ_guarded x c s) = true ↔
    +    (if (c =? 0)%N then (interval_list_elements_greater y s = true) else (y < x)).
    +  Proof.
    +    intros x c y s.
    +    rewrite /insert_intervalZ_guarded.
    +    case (c =? 0)%N ⇒ //.
    +    rewrite /interval_list_elements_greater Z.ltb_lt //.
    +  Qed.

      Lemma insert_intervalZ_guarded_app : ∀ x c s1 s2,
    -    (insert_intervalZ_guarded x c s1) ++ s2 =
    -    insert_intervalZ_guarded x c (s1 ++ s2).
    +    (insert_intervalZ_guarded x c s1) ++ s2 =
    +    insert_intervalZ_guarded x c (s1 ++ s2).
    +  Proof.
    +    intros x c s1 s2.
    +    rewrite /insert_intervalZ_guarded.
    +    case (N.eqb c 0) ⇒ //.
    +  Qed.

      Lemma insert_intervalZ_guarded_rev_nil_app : ∀ x c s,
    -    rev (insert_intervalZ_guarded x c nil) ++ s =
    +    rev (insert_intervalZ_guarded x c nil) ++ s =
        insert_intervalZ_guarded x c s.
    +  Proof.
    +    intros x c s.
    +    rewrite /insert_intervalZ_guarded.
    +    case (N.eqb c 0) ⇒ //.
    +  Qed.

    Lemma elementsZ_insert_intervalZ_guarded : ∀ x c s,
    -    elementsZ (insert_intervalZ_guarded x c s) = elementsZ ((x, c) :: s).
    +    elementsZ (insert_intervalZ_guarded x c s) = elementsZ ((x, c) :: s).
    +  Proof.
    +    intros x c s.
    +    rewrite /insert_intervalZ_guarded.
    +    case_eq (c =? 0)%N ⇒ //.
    +    move ⇒ /N.eqb_eq →.
    +    rewrite elementsZ_cons elementsZ_single_base /= app_nil_r //.
    +  Qed.

      Lemma InZ_insert_intervalZ_guarded : ∀ y x c s,
    -    InZ y (insert_intervalZ_guarded x c s) = InZ y ((x, c) :: s).
    +    InZ y (insert_intervalZ_guarded x c s) = InZ y ((x, c) :: s).
    +  Proof.
    +    intros y x c s.
    +    rewrite /InZ elementsZ_insert_intervalZ_guarded //.
    +  Qed.

    @@ -1388,41 +2216,164 @@

    Library MSetsExtra.MSetIntervals


      Lemma merge_interval_size_add : ∀ x c1 c2,
    -     (merge_interval_size x c1 (x + Z.of_N c1) c2 = (c1 + c2))%N.
    +     (merge_interval_size x c1 (x + Z.of_N c1) c2 = (c1 + c2))%N.
    +  Proof.
    +    intros x c1 c2.
    +    rewrite /merge_interval_size.
    +    replace (x + Z.of_N c1 + Z.of_N c2 - x) with
    +            (Z.of_N c1 + Z.of_N c2) by omega.
    +    rewrite -N2Z.inj_add N2Z.id.
    +    apply N.max_r, N.le_add_r.
    +  Qed.

      Lemma merge_interval_size_eq_max : ∀ y1 c1 y2 c2,
    -     y1 ≤ y2 + Z.of_N c2 →
    -     y1 + Z.of_N (merge_interval_size y1 c1 y2 c2) =
    -     Z.max (y1 + Z.of_N c1) (y2 + Z.of_N c2).
    +     y1 ≤ y2 + Z.of_N c2 →
    +     y1 + Z.of_N (merge_interval_size y1 c1 y2 c2) =
    +     Z.max (y1 + Z.of_N c1) (y2 + Z.of_N c2).
    +  Proof.
    +    intros y1 c1 y2 c2 H_y1_le.
    +    rewrite /merge_interval_size N2Z.inj_max Z2N.id; last first. {
    +      by apply Zle_minus_le_0.
    +    }
    +    rewrite -Z.add_max_distr_l.
    +    replace (y1 + (y2 + Z.of_N c2 - y1)) with (y2 + Z.of_N c2) by omega.
    +    done.
    +  Qed.

      Lemma merge_interval_size_invariant : ∀ y1 c1 y2 c2 z s,
    -    interval_list_invariant s = true →
    -    y1 + Z.of_N c1 ≤ y2 + Z.of_N c2 →
    -    y2 + Z.of_N c2 ≤ z →
    -    interval_list_elements_greater z s = true →
    -    (c1 ≠ 0)%N →
    -    interval_list_invariant ((y1, merge_interval_size y1 c1 y2 c2) :: s) =
    -   true.
    - +    interval_list_invariant s = true →
    +    y1 + Z.of_N c1 ≤ y2 + Z.of_N c2 →
    +    y2 + Z.of_N c2 ≤ z →
    +    interval_list_elements_greater z s = true →
    +    (c1 ≠ 0)%N →
    +    interval_list_invariant ((y1, merge_interval_size y1 c1 y2 c2) :: s) =
    +   true.
    +  Proof.
    +    intros y1 c1 y2 c2 z s H_inv H_le H_le_z H_gr H_c1_neq_0.
    +    rewrite interval_list_invariant_cons.
    +    split; last split. {
    +      rewrite merge_interval_size_eq_max; last first. {
    +        eapply Z.le_trans; last apply H_le.
    +        apply Z_le_add_r.
    +      } {
    +        rewrite Z.max_r ⇒ //.
    +        eapply interval_list_elements_greater_impl; first apply H_le_z.
    +        done.
    +      }
    +    } {
    +      apply merge_interval_size_neq_0.
    +      assumption.
    +    } {
    +      assumption.
    +    }
    +  Qed.
    +
      Lemma In_merge_interval : ∀ x1 c1 x2 c2 y,
    -    x1 ≤ x2 →
    -    x2 ≤ x1 + Z.of_N c1 → (
    -    List.In y (elementsZ_single x1 (merge_interval_size x1 c1 x2 c2)) ↔
    -    List.In y (elementsZ_single x1 c1) ∨ List.In y (elementsZ_single x2 c2)).
    - +    x1 ≤ x2 →
    +    x2 ≤ x1 + Z.of_N c1 → (
    +    List.In y (elementsZ_single x1 (merge_interval_size x1 c1 x2 c2)) ↔
    +    List.In y (elementsZ_single x1 c1) ∨ List.In y (elementsZ_single x2 c2)).
    +  Proof.
    +    intros x1 c1 x2 c2 y H_x1_le H_x2_le.
    +    rewrite !In_elementsZ_single merge_interval_size_eq_max;
    +      last first. {
    +      eapply Z.le_trans; eauto.
    +      by apply Z_le_add_r.
    +    }
    +    rewrite Z.max_lt_iff.
    +    split. {
    +      move ⇒ [H_x_le] [] H_y_lt. {
    +        by left.
    +      } {
    +        case_eq (Z.leb x2 y). {
    +          move ⇒ /Z.leb_le H_y'_le.
    +          by right.
    +        } {
    +          move ⇒ /Z.leb_gt H_y_lt_x2.
    +          left.
    +          split ⇒ //.
    +          eapply Z.lt_le_trans; eauto.
    +        }
    +      }
    +    } {
    +      move ⇒ []. {
    +        tauto.
    +      } {
    +        move ⇒ [H_x2_le'] H_y_lt.
    +        split. {
    +          eapply Z.le_trans; eauto.
    +        } {
    +          by right.
    +        }
    +      }
    +    }
    +  Qed.
    +
      Lemma insert_interval_begin_spec : ∀ y s x c,
    -     interval_list_invariant s = true →
    -     interval_list_elements_greater x s = true →
    -     (c ≠ 0)%N →
    -      (
    -     interval_list_invariant (insert_interval_begin x c s) = true ∧
    -     (InZ y (insert_interval_begin x c s) ↔
    -     (List.In y (elementsZ_single x c) ∨ InZ y s))).
    - +     interval_list_invariant s = true →
    +     interval_list_elements_greater x s = true →
    +     (c ≠ 0)%N →
    +      (
    +     interval_list_invariant (insert_interval_begin x c s) = true ∧
    +     (InZ y (insert_interval_begin x c s) ↔
    +     (List.In y (elementsZ_single x c) ∨ InZ y s))).
    +  Proof.
    +    intros y.
    +    induction s as [| [y' c'] s' IH]. {
    +      intros x c _ H_c_neq H_z_lt.
    +      rewrite /insert_interval_begin InZ_cons interval_list_invariant_cons //.
    +    } {
    +      intros x c.
    +      rewrite interval_list_invariant_cons
    +       interval_list_elements_greater_cons.
    +      move ⇒ [H_gr] [H_c'_neq_0] H_inv_s' H_x_lt H_c_neq_0.
    +      unfold insert_interval_begin.
    +      Z_named_compare_cases H_y'; fold insert_interval_begin. {
    +        subst.
    +        rewrite !InZ_cons elementsZ_single_add in_app_iff.
    +        split; last tauto.
    +        rewrite interval_list_invariant_cons N2Z.inj_add
    +          Z.add_assoc N.eq_add_0.
    +        tauto.
    +      } {
    +        rewrite !InZ_cons !interval_list_invariant_cons
    +          interval_list_elements_greater_cons.
    +        repeat split ⇒ //.
    +      } {
    +        set c'' := merge_interval_size x c y' c'.
    +        have H_x_lt' : x < y' + Z.of_N c'. {
    +          eapply Z.lt_le_trans with (m := y') ⇒ //.
    +          by apply Z_le_add_r.
    +        }
    +
    +        have H_pre : interval_list_elements_greater x s' = true. {
    +          eapply interval_list_elements_greater_impl; eauto.
    +          by apply Z.lt_le_incl.
    +        }
    +        have H_pre2 : c'' ≠ 0%N. {
    +          by apply merge_interval_size_neq_0.
    +        }
    +        move : (IH x c'' H_inv_s' H_pre H_pre2) ⇒ {IH} {H_pre} {H_pre2} [->] →.
    + +
    +        split; first reflexivity.
    +        unfold c''; clear c''.
    +        rewrite In_merge_interval. {
    +          rewrite InZ_cons.
    +          tauto.
    +        } {
    +          by apply Z.lt_le_incl.
    +        } {
    +          by apply Z.lt_le_incl.
    +        }
    +      }
    +    }
    +  Qed.
    +
    @@ -1432,23 +2383,153 @@

    Library MSetsExtra.MSetIntervals

      Lemma addZ_InZ :
    -   âˆ€ (s : t) (x y : Z),
    -    interval_list_invariant s = true →
    -    (InZ y (addZ x s) ↔ x = y ∨ InZ y s).
    - +   âˆ€ (s : t) (x y : Z),
    +    interval_list_invariant s = true →
    +    (InZ y (addZ x s) ↔ x = y ∨ InZ y s).
    +  Proof.
    +    move ⇒ s x y.
    +    induction s as [| [z c] s' IH]. {
    +      move ⇒ _.
    +      rewrite /InZ addZ_alt_def
    +              elementsZ_cons elementsZ_nil app_nil_l -in_rev
    +              In_elementsZ_single1 /=.
    +      firstorder.
    +    } {
    +      move ⇒ /interval_list_invariant_cons [H_greater] [H_c_neq_0] H_inv_c'.
    +      move : (IH H_inv_c') ⇒ {IH} IH.
    +      rewrite addZ_alt_def.
    +      have H_succ : ∀ z, z + Z.of_N 1 = Z.succ z by done.
    +      move : (interval_1_compare_elim x z c).
    +      case (interval_1_compare x (z, c));
    +        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
    +        move ⇒ →.
    +        rewrite elementsZ_single_succ_front /=.
    +        tauto.
    +      } {
    +        move ⇒ [] // H_x_in.
    +        split; first tauto.
    +        move ⇒ [] // <-.
    +        left.
    +        by rewrite In_elementsZ_single.
    +      } {
    +        rewrite IH.
    +        tauto.
    +      } {
    +        move ⇒ H_x_eq.
    +        have → : (InZ y (insert_interval_begin z (N.succ c) s') ↔
    +                   List.In y (elementsZ_single z (N.succ c)) ∨ InZ y s'). {
    +          eapply insert_interval_begin_spec. {
    +            by apply H_inv_c'.
    +          } {
    +            eapply interval_list_elements_greater_impl; eauto.
    +            apply Z_le_add_r.
    +          } {
    +            by apply N.neq_succ_0.
    +          }
    +        }
    +        rewrite -H_x_eq elementsZ_single_succ in_app_iff /=.
    +        tauto.
    +      }
    +    }
    +  Qed.
    +
      Lemma addZ_invariant : ∀ s x,
    -    interval_list_invariant s = true →
    -    interval_list_invariant (addZ x s) = true.
    - +    interval_list_invariant s = true →
    +    interval_list_invariant (addZ x s) = true.
    +  Proof.
    +    move ⇒ s x.
    +    induction s as [| [z c] s' IH]. {
    +      move ⇒ _.
    +      by simpl.
    +    } {
    +      move ⇒ /interval_list_invariant_cons [H_greater] [H_c_neq_0]
    +              H_inv_c'.
    +      move : (IH H_inv_c') ⇒ {IH} IH.
    +      rewrite addZ_alt_def.
    +      have H_succ : ∀ z, z + Z.of_N 1 = Z.succ z by done.
    +      move : (interval_1_compare_elim x z c).
    +      case_eq (interval_1_compare x (z, c)) ⇒ H_comp;
    +        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
    +        move ⇒ H_z_gt.
    +        rewrite interval_list_invariant_cons /= !andb_true_iff !H_succ.
    +        repeat split ⇒ //. {
    +          by apply Z.ltb_lt.
    +        } {
    +          apply negb_true_iff, N.eqb_neq ⇒ //.
    +        }
    +      } {
    +        move ⇒ ?; subst.
    +        rewrite /= !andb_true_iff.
    +        repeat split ⇒ //. {
    +          move : H_greater.
    +          rewrite Z.add_succ_l -Z.add_succ_r N2Z.inj_succ //.
    +        } {
    +          apply negb_true_iff, N.eqb_neq ⇒ //.
    +          apply N.neq_succ_0.
    +        }
    +      } {
    +        move ⇒ [] // _.
    +        rewrite interval_list_invariant_cons /=.
    +        tauto.
    +      } {
    +        rewrite interval_list_invariant_cons.
    +        move ⇒ H_lt_x.
    +        repeat split ⇒ //.
    +        apply interval_list_elements_greater_intro ⇒ //.
    +        move ⇒ xx.
    +        rewrite addZ_InZ ⇒ //.
    +        move ⇒ [<- //|].
    +        apply interval_list_elements_greater_alt2_def ⇒ //.
    +      } {
    +        move ⇒ H_x_eq.
    +        apply insert_interval_begin_spec ⇒ //. {
    +          eapply interval_list_elements_greater_impl; eauto.
    +          apply Z_le_add_r.
    +        } {
    +          by apply N.neq_succ_0.
    +        }
    +      }
    +    }
    +  Qed.
    +
      Global Instance add_ok s x : ∀ `(Ok s), Ok (add x s).
    - +  Proof.
    +    move ⇒ H_ok_s.
    +    move : (H_ok_s).
    +    rewrite /Ok /IsOk /is_encoded_elems_list /add.
    +    move ⇒ [H_isok_s] H_pre.
    +    split. {
    +      apply addZ_invariant ⇒ //.
    +    } {
    +      intros y.
    +      move : (addZ_InZ s (Enc.encode x) y H_isok_s).
    +      rewrite /InZ ⇒ →.
    +      move ⇒ []. {
    +        move ⇒ <-.
    +        by ∃ x.
    +      } {
    +        move ⇒ /H_pre //.
    +      }
    +    }
    +  Qed.
    +
      Lemma add_spec :
       âˆ€ (s : t) (x y : elt) (Hs : Ok s),
    -     In y (add x s) ↔ Enc.E.eq y x ∨ In y s.
    - +     In y (add x s) ↔ Enc.E.eq y x ∨ In y s.
    +  Proof.
    +    intros s x y Hs.
    +    have Hs' := (add_ok s x Hs).
    +    rewrite !In_InZ.
    +    rewrite /add addZ_InZ. {
    +      rewrite -Enc.encode_eq; firstorder.
    +    } {
    +      apply Hs.
    +    }
    +  Qed.
    +
    @@ -1460,12 +2541,23 @@

    Library MSetsExtra.MSetIntervals


      Global Instance empty_ok : Ok empty.
    +  Proof.
    +    rewrite /empty Ok_nil //.
    +  Qed.

    -  Lemma empty_spec' : ∀ x, (In x empty ↔ False).
    +  Lemma empty_spec' : ∀ x, (In x empty ↔ False).
    +    rewrite /Empty /empty /In elements_nil.
    +    intros a.
    +    rewrite InA_nil //.
    +  Qed.

      Lemma empty_spec : Empty empty.
    +  Proof.
    +    rewrite /Empty ⇒ a.
    +    rewrite empty_spec' //.
    +  Qed.

    @@ -1477,8 +2569,28 @@

    Library MSetsExtra.MSetIntervals


    -  Lemma is_empty_spec : ∀ (s : t) (Hs : Ok s), is_empty s = true ↔ Empty s.
    - +  Lemma is_empty_spec : ∀ (s : t) (Hs : Ok s), is_empty s = true ↔ Empty s.
    +  Proof.
    +    intros [ | [x c] s]. {
    +      split ⇒ // _.
    +      apply empty_spec.
    +    } {
    +      rewrite /= /Empty Ok_cons.
    +      move ⇒ [_] [H_c_neq] [H_enc] _.
    +      split ⇒ //.
    +      move ⇒ H.
    +      contradiction (H (Enc.decode x)) ⇒ {H}.
    +      rewrite /In InA_alt elements_cons.
    +      âˆƒ (Enc.decode x).
    +      split; first by apply Enc.E.eq_equiv.
    +      rewrite in_app_iff; left.
    +      rewrite /elements_single in_map_iff.
    +      âˆƒ x.
    +      split ⇒ //.
    +      apply In_elementsZ_single_hd ⇒ //.
    +    }
    +  Qed.
    +
    @@ -1490,10 +2602,25 @@

    Library MSetsExtra.MSetIntervals


      Global Instance singleton_ok x : Ok (singleton x).
    +  Proof.
    +    rewrite singleton_alt_def.
    +    apply add_ok.
    +    apply empty_ok.
    +  Qed.

    -  Lemma singleton_spec : ∀ x y : elt, In y (singleton x) ↔ Enc.E.eq y x.
    - +  Lemma singleton_spec : ∀ x y : elt, In y (singleton x) ↔ Enc.E.eq y x.
    +  Proof.
    +    intros x y.
    +    rewrite singleton_alt_def.
    +    rewrite (add_spec empty x y) /empty /In elements_nil InA_nil.
    +    split. {
    +      move ⇒ [] //.
    +    } {
    +      by left.
    +    }
    +  Qed.
    +
    @@ -1504,12 +2631,33 @@

    Library MSetsExtra.MSetIntervals


    -  Lemma add_list_ok : ∀ l s, Ok s → Ok (add_list l s).
    - +  Lemma add_list_ok : ∀ l s, Ok s → Ok (add_list l s).
    +  Proof.
    +    induction l as [| x l' IH]. {
    +      done.
    +    } {
    +      move ⇒ s H_s_ok /=.
    +      apply IH.
    +      by apply add_ok.
    +    }
    +  Qed.
    +
    -  Lemma add_list_spec : ∀ x l s, Ok s →
    -     (In x (add_list l s) ↔ (SetoidList.InA Enc.E.eq x l) ∨ In x s).
    - +  Lemma add_list_spec : ∀ x l s, Ok s →
    +     (In x (add_list l s) ↔ (SetoidList.InA Enc.E.eq x l) ∨ In x s).
    +  Proof.
    +    move ⇒ x.
    +    induction l as [| y l' IH]. {
    +      intros s H.
    +      rewrite /= InA_nil.
    +      tauto.
    +    } {
    +      move ⇒ s H_ok /=.
    +      rewrite IH add_spec InA_cons.
    +      tauto.
    +    }
    +  Qed.
    +
    @@ -1521,19 +2669,19 @@

    Library MSetsExtra.MSetIntervals


      Lemma union_aux_flatten_alt_def : ∀ (s1 s2 : t) acc,
    -    union_aux s1 s2 acc =
    -    match (s1, s2) with
    -    | (nil, _) ⇒ List.rev_append acc s2
    -    | (_, nil) ⇒ List.rev_append acc s1
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
    +    union_aux s1 s2 acc =
    +    match (s1, s2) with
    +    | (nil, _) ⇒ List.rev_append acc s2
    +    | (_, nil) ⇒ List.rev_append acc s1
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
              | ICR_before_touch ⇒
                  union_aux l1 (
    -                insert_interval_begin y1 ((c1+c2)%N) l2) acc
    -          | ICR_after ⇒ union_aux s1 l2 ((y2, c2)::acc)
    +                insert_interval_begin y1 ((c1+c2)%N) l2) acc
    +          | ICR_after ⇒ union_aux s1 l2 ((y2, c2)::acc)
              | ICR_after_touch ⇒ union_aux l1 (
    -              insert_interval_begin y2 ((c1+c2)%N) l2) acc
    +              insert_interval_begin y2 ((c1+c2)%N) l2) acc
              | ICR_overlap_before ⇒
                  union_aux l1 (
                    insert_interval_begin y1
    @@ -1547,26 +2695,48 @@

    Library MSetsExtra.MSetIntervals

              | ICR_subsume_2 ⇒ union_aux s1 l2 acc
            end
        end.
    +  Proof.
    +    intros s1 s2 acc.
    +    case s1, s2 ⇒ //.
    +  Qed.

      Lemma union_aux_alt_def : ∀ (s1 s2 : t) acc,
    -    union_aux s1 s2 acc =
    -    List.rev_append acc (union s1 s2).
    +    union_aux s1 s2 acc =
    +    List.rev_append acc (union s1 s2).
    +  Proof.
    +    rewrite /union.
    +    intros s1 s2 acc.
    +    move : acc s2.
    +    induction s1 as [| [y1 c1] l1 IH1]. {
    +      intros acc s2.
    +      rewrite !union_aux_flatten_alt_def.
    +      rewrite !rev_append_rev //.
    +    }
    +    intros acc s2; move : acc.
    +    induction s2 as [| [y2 c2] l2 IH2]; first by simpl.
    +    move ⇒ acc.
    +    rewrite !(union_aux_flatten_alt_def ((y1, c1)::l1) ((y2, c2)::l2)).
    +    case (interval_compare (y1, c1) (y2, c2));
    +      rewrite ?(IH1 ((y1, c1) :: acc)) ?(IH1 ((y1, c1) :: nil))
    +              ?(IH2 ((y2, c2) :: acc)) ?(IH2 ((y2, c2) :: nil))
    +              ?(IH1 acc) //.
    +  Qed.

      Lemma union_alt_def : ∀ (s1 s2 : t),
    -    union s1 s2 =
    -    match (s1, s2) with
    -    | (nil, _) ⇒ s2
    -    | (_, nil) ⇒ s1
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ (y1, c1) :: (union l1 s2)
    +    union s1 s2 =
    +    match (s1, s2) with
    +    | (nil, _) ⇒ s2
    +    | (_, nil) ⇒ s1
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ (y1, c1) :: (union l1 s2)
              | ICR_before_touch ⇒
    -              union l1 (insert_interval_begin y1 ((c1+c2)%N) l2)
    -          | ICR_after ⇒ (y2, c2) :: union s1 l2
    +              union l1 (insert_interval_begin y1 ((c1+c2)%N) l2)
    +          | ICR_after ⇒ (y2, c2) :: union s1 l2
              | ICR_after_touch ⇒ union l1
    -              (insert_interval_begin y2 ((c1+c2)%N) l2)
    +              (insert_interval_begin y2 ((c1+c2)%N) l2)
              | ICR_overlap_before ⇒
                  union l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2)
              | ICR_overlap_after ⇒
    @@ -1576,31 +2746,340 @@

    Library MSetsExtra.MSetIntervals

              | ICR_subsume_2 ⇒ union s1 l2
            end
         end.
    +  Proof.
    +    intros s1 s2.
    +    rewrite /union union_aux_flatten_alt_def.
    +    case s1 as [| [y1 c1] l1] ⇒ //.
    +    case s2 as [| [y2 c2] l2] ⇒ //.
    +    case (interval_compare (y1, c1) (y2, c2));
    +      rewrite union_aux_alt_def //.
    +  Qed.

      Lemma union_InZ :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    âˆ€ y, (InZ y (union s1 s2) ↔ InZ y s1 ∨ InZ y s2).
    - +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    âˆ€ y, (InZ y (union s1 s2) ↔ InZ y s1 ∨ InZ y s2).
    +  Proof.
    +    intro s1.
    +    induction s1 as [| [y1 c1] l1 IH1]. {
    +      intros s2 _ _ y.
    +      rewrite union_alt_def /InZ /=.
    +      tauto.
    +    } {
    +      induction s2 as [| [y2 c2] l2 IH2]. {
    +        intros _ _ y.
    +        rewrite union_alt_def /InZ /=.
    +        tauto.
    +      } {
    +        move ⇒ H_inv_s1 H_inv_s2.
    +        move : (H_inv_s1) (H_inv_s2).
    +        rewrite !interval_list_invariant_cons.
    +        move ⇒ [H_gr_l1] [H_c1_neq_0] H_inv_l1.
    +        move ⇒ [H_gr_l2] [H_c2_neq_0] H_inv_l2.
    +        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {IH2} IH2.
    +        have : ∀ s2 : t,
    +          interval_list_invariant s2 = true →
    +          âˆ€ y : Z, InZ y (union l1 s2) ↔ InZ y l1 ∨ InZ y s2. {
    +          intros. by apply IH1.
    +        }
    +        move ⇒ {IH1} IH1 y.
    +        rewrite union_alt_def.
    +        move : (interval_compare_elim y1 c1 y2 c2).
    +        case (interval_compare (y1, c1) (y2, c2)). {
    +          rewrite !InZ_cons IH1 // InZ_cons.
    +          tauto.
    +        } {
    +          move ⇒ H_y2_eq.
    +          replace (c1 + c2)%N with (merge_interval_size y1 c1 y2 c2);
    +            last first. {
    +            rewrite -H_y2_eq merge_interval_size_add //.
    +          }
    +          set c'' := merge_interval_size y1 c1 y2 c2.
    +          have [H_inv_insert H_InZ_insert] :
    +               interval_list_invariant (insert_interval_begin y1 c'' l2) = true ∧
    +                (InZ y (insert_interval_begin y1 c'' l2) ↔
    +                List.In y (elementsZ_single y1 c'') ∨ InZ y l2). {
    +            apply insert_interval_begin_spec ⇒ //. {
    +              eapply interval_list_elements_greater_impl; eauto.
    +              rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
    +              apply Z_le_add_r.
    +            } {
    +              by apply merge_interval_size_neq_0.
    +            }
    +          }
    +  
    +          rewrite IH1 ⇒ //.
    +          rewrite H_InZ_insert !InZ_cons /c''.
    +          rewrite -H_y2_eq In_merge_interval. {
    +            tauto.
    +          } {
    +            apply Z_le_add_r.
    +          } {
    +            by apply Z.le_refl.
    +          }
    +        } {
    +          move ⇒ [H_y1_lt] [H_y2_lt] H_y1_c1_lt.
    +          set c'' := merge_interval_size y1 c1 y2 c2.
    +          have [H_inv_insert H_InZ_insert] :
    +               interval_list_invariant (insert_interval_begin y1 c'' l2) = true ∧
    +                (InZ y (insert_interval_begin y1 c'' l2) ↔
    +                List.In y (elementsZ_single y1 c'') ∨ InZ y l2). {
    +            apply insert_interval_begin_spec ⇒ //. {
    +              eapply interval_list_elements_greater_impl; eauto.
    +              apply Z_lt_le_add_r ⇒ //.
    +            } {
    +              by apply merge_interval_size_neq_0.
    +            }
    +          }
    +          rewrite IH1 ⇒ //.
    +          rewrite H_InZ_insert !InZ_cons /c''.
    +          rewrite In_merge_interval. {
    +            tauto.
    +          } {
    +            by apply Z.lt_le_incl.
    +          } {
    +            by apply Z.lt_le_incl.
    +          }
    +        } {
    +          move ⇒ [H_y2_lt] [H_y1_lt] H_y2_c_lt.
    +          set c'' := merge_interval_size y2 c2 y1 c1.
    +          have [H_inv_insert H_InZ_insert] :
    +               interval_list_invariant (insert_interval_begin y2 c'' l2) = true ∧
    +                (InZ y (insert_interval_begin y2 c'' l2) ↔
    +                List.In y (elementsZ_single y2 c'') ∨ InZ y l2). {
    +            apply insert_interval_begin_spec ⇒ //. {
    +              eapply interval_list_elements_greater_impl; eauto.
    +              apply Z_le_add_r.
    +            } {
    +              by apply merge_interval_size_neq_0.
    +            }
    +          }
    +  
    +          rewrite IH1 ⇒ //.
    +          rewrite H_InZ_insert !InZ_cons /c''.
    +          rewrite In_merge_interval. {
    +            tauto.
    +          } {
    +            by apply Z.lt_le_incl.
    +          } {
    +            by apply Z.lt_le_incl.
    +          }
    +        } {
    +          move ⇒ [? ?]; subst.
    +          rewrite IH1 ⇒ //.
    +          rewrite !InZ_cons.
    +          tauto.
    +        } {
    +          move ⇒ [H_y2_le] [H_y1_c1_le] _.
    +          rewrite IH1 ⇒ //.
    +          rewrite !InZ_cons.
    +          suff : (List.In y (elementsZ_single y1 c1) →
    +                  List.In y (elementsZ_single y2 c2)). {
    +            tauto.
    +          }
    +          rewrite !In_elementsZ_single.
    +          move ⇒ [H_y1_le H_y_lt].
    +          split; omega.
    +        } {
    +          move ⇒ [H_y1_le] [H_y2_c2_le] _.
    +          rewrite IH2.
    +          rewrite !InZ_cons.
    +          suff : (List.In y (elementsZ_single y2 c2) →
    +                  List.In y (elementsZ_single y1 c1)). {
    +            tauto.
    +          }
    +          rewrite !In_elementsZ_single.
    +          move ⇒ [H_y2_le H_y_lt].
    +          split; omega.
    +        } {
    +          rewrite !InZ_cons IH2 !InZ_cons.
    +          tauto.
    +        } {
    +          move ⇒ H_y1_eq.
    +          replace (c1 + c2)%N with (merge_interval_size y2 c2 y1 c1);
    +            last first. {
    +            rewrite -H_y1_eq merge_interval_size_add N.add_comm //.
    +          }
    +          set c'' := merge_interval_size y2 c2 y1 c1.
    +          have [H_inv_insert H_InZ_insert] :
    +               interval_list_invariant (insert_interval_begin y2 c'' l2) = true ∧
    +                (InZ y (insert_interval_begin y2 c'' l2) ↔
    +                List.In y (elementsZ_single y2 c'') ∨ InZ y l2). {
    +            apply insert_interval_begin_spec ⇒ //. {
    +              eapply interval_list_elements_greater_impl; eauto.
    +              apply Z_le_add_r.
    +            } {
    +              by apply merge_interval_size_neq_0.
    +            }
    +          }
    +  
    +          rewrite IH1 ⇒ //.
    +          rewrite H_InZ_insert !InZ_cons /c''.
    +          rewrite -H_y1_eq In_merge_interval. {
    +            tauto.
    +          } {
    +            apply Z_le_add_r.
    +          } {
    +            by apply Z.le_refl.
    +          }
    +        }
    +      }
    +    }
    +  Qed.
    +
      Lemma union_invariant :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    interval_list_invariant (union s1 s2) = true.
    - +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    interval_list_invariant (union s1 s2) = true.
    +  Proof.
    +    intro s1.
    +    induction s1 as [| [y1 c1] l1 IH1]. {
    +      intros s2 _ H_inv_s2.
    +      rewrite union_alt_def /InZ //.
    +    } {
    +      induction s2 as [| [y2 c2] l2 IH2]. {
    +        intros H_inv_s1 _.
    +        rewrite union_alt_def /InZ //.
    +      } {
    +        move ⇒ H_inv_s1 H_inv_s2.
    +        move : (H_inv_s1) (H_inv_s2).
    +        rewrite !interval_list_invariant_cons.
    +        move ⇒ [H_gr_l1] [H_c1_neq_0] H_inv_l1.
    +        move ⇒ [H_gr_l2] [H_c2_neq_0] H_inv_l2.
    +        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {IH2} IH2.
    +        have : ∀ s2 : t,
    +          interval_list_invariant s2 = true →
    +          interval_list_invariant (union l1 s2) = true. {
    +          intros. by apply IH1.
    +        }
    +        move ⇒ {IH1} IH1.
    + +
    +        rewrite union_alt_def.
    +        move : (interval_compare_elim y1 c1 y2 c2).
    +        case (interval_compare (y1, c1) (y2, c2)). {
    +          move ⇒ H_lt_y2.
    +          have H_inv' : interval_list_invariant (union l1 ((y2, c2) :: l2)) = true. {
    +            by apply IH1.
    +          }
    +
    +          rewrite interval_list_invariant_cons.
    +          repeat split ⇒ //.
    +          apply interval_list_elements_greater_intro ⇒ // x.
    +          rewrite union_InZ ⇒ //.
    +          move ⇒ []. {
    +            apply interval_list_elements_greater_alt2_def ⇒ //.
    +          } {
    +            apply interval_list_elements_greater_alt2_def ⇒ //.
    +            rewrite interval_list_elements_greater_cons //.
    +          }
    +        } {
    +          move ⇒ H_y2_eq.
    +          apply IH1.
    +          apply insert_interval_begin_spec ⇒ //. {
    +            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
    +            rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
    +            apply Z_le_add_r.
    +          } {
    +            rewrite N.eq_add_0.
    +            tauto.
    +          }
    +        } {
    +          move ⇒ [H_y1_lt] _.
    +          apply IH1.
    +          apply insert_interval_begin_spec ⇒ //. {
    +            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
    +            apply Z_lt_le_add_r ⇒ //.
    +          } {
    +            apply merge_interval_size_neq_0 ⇒ //.
    +          }
    +        } {
    +          move ⇒ [H_y2_lt] _.
    +          apply IH1.
    +          apply insert_interval_begin_spec ⇒ //. {
    +            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
    +            apply Z_le_add_r ⇒ //.
    +          } {
    +            apply merge_interval_size_neq_0 ⇒ //.
    +          }
    +        } {
    +          move ⇒ [? ?]; subst.
    +          apply IH1 ⇒ //.
    +        } {
    +          move ⇒ _.
    +          apply IH1 ⇒ //.
    +        } {
    +          move ⇒ _.
    +          apply IH2 ⇒ //.
    +        } {
    +          move ⇒ H_lt_y1.
    +          rewrite interval_list_invariant_cons ⇒ //.
    +          repeat split ⇒ //.
    +          apply interval_list_elements_greater_intro ⇒ // x.
    +          rewrite union_InZ ⇒ //.
    +          move ⇒ []. {
    +            apply interval_list_elements_greater_alt2_def ⇒ //.
    +            rewrite interval_list_elements_greater_cons //.
    +          } {
    +            apply interval_list_elements_greater_alt2_def ⇒ //.
    +          }
    +        } {
    +          move ⇒ H_y1_eq.
    +          apply IH1 ⇒ //.
    +          apply insert_interval_begin_spec ⇒ //. {
    +            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
    +            apply Z_le_add_r.
    +          } {
    +            rewrite N.eq_add_0.
    +            tauto.
    +          }
    +        }
    +      }
    +    }
    +  Qed.
    +

      Global Instance union_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (union s1 s2).
    - +  Proof.
    +    move ⇒ H_ok_s1 H_ok_s2.
    +    move : (H_ok_s1) (H_ok_s2).
    +    rewrite /Ok /IsOk /is_encoded_elems_list /add.
    +    move ⇒ [H_inv_s1] H_pre1.
    +    move ⇒ [H_inv_s2] H_pre2.
    +    split. {
    +      apply union_invariant ⇒ //.
    +    } {
    +      intros y.
    +      move : (union_InZ s1 s2 H_inv_s1 H_inv_s2).
    +      rewrite /InZ ⇒ →.
    +      move ⇒ []. {
    +        apply H_pre1.
    +      } {
    +        apply H_pre2.
    +      }
    +    }
    +  Qed.
    +
      Lemma union_spec :
       âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    -   In x (union s s') ↔ In x s ∨ In x s'.
    - +   In x (union s s') ↔ In x s ∨ In x s'.
    +  Proof.
    +    intros s s' x H_ok H_ok'.
    +    rewrite !In_InZ.
    +    rewrite union_InZ ⇒ //. {
    +      apply H_ok.
    +    } {
    +      apply H_ok'.
    +    }
    +  Qed.
    +
    @@ -1612,64 +3091,579 @@

    Library MSetsExtra.MSetIntervals


      Lemma inter_aux_alt_def :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      inter_aux y2 c2 acc s = match inter_aux y2 c2 nil s with
    -                               (acc', s') ⇒ (acc' ++ acc, s')
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      inter_aux y2 c2 acc s = match inter_aux y2 c2 nil s with
    +                               (acc', s') ⇒ (acc' ++ acc, s')
                                 end.
    - +  Proof.
    +    intros y2 c2.
    +    induction s as [| [y1 c1] s' IH] ⇒ acc. {
    +      rewrite /inter_aux app_nil_l //.
    +    } {
    +      simpl.
    +      case_eq (inter_aux y2 c2 nil s') ⇒ acc'' s'' H_eq.
    +      case (interval_compare (y1, c1) (y2, c2));
    +        rewrite ?(IH acc)
    +                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: acc))
    +                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: nil))
    +                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: acc))
    +                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: nil))
    +                ?(IH ((y1, c1) :: acc))
    +                ?(IH ((y1, c1) :: nil))
    +                
    +                ?H_eq -?app_assoc -?app_comm_cons //.
    +    }
    +  Qed.
    +
      Lemma inter_aux_props :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      interval_list_invariant (rev acc) = true →
    -      interval_list_invariant s = true →
    -      (∀ x1 x2, InZ x1 acc → InZ x2 s →
    -                     List.In x2 (elementsZ_single y2 c2) →
    -                     Z.succ x1 < x2) →
    -      (c2 ≠ 0%N) →
    -      match (inter_aux y2 c2 acc s) with (acc', s') ⇒
    -        (∀ y, (InZ y acc' ↔
    -                   (InZ y acc ∨ (InZ y s ∧ (List.In y (elementsZ_single y2 c2)))))) ∧
    -        (∀ y, InZ y s' → InZ y s) ∧
    -        (∀ y, InZ y s → y2 + Z.of_N c2 < y → InZ y s') ∧
    -        interval_list_invariant (rev acc') = true ∧
    -        interval_list_invariant s' = true
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      interval_list_invariant (rev acc) = true →
    +      interval_list_invariant s = true →
    +      (∀ x1 x2, InZ x1 acc → InZ x2 s →
    +                     List.In x2 (elementsZ_single y2 c2) →
    +                     Z.succ x1 < x2) →
    +      (c2 ≠ 0%N) →
    +      match (inter_aux y2 c2 acc s) with (acc', s') ⇒
    +        (∀ y, (InZ y acc' ↔
    +                   (InZ y acc ∨ (InZ y s ∧ (List.In y (elementsZ_single y2 c2)))))) ∧
    +        (∀ y, InZ y s' → InZ y s) ∧
    +        (∀ y, InZ y s → y2 + Z.of_N c2 < y → InZ y s') ∧
    +        interval_list_invariant (rev acc') = true ∧
    +        interval_list_invariant s' = true
          end.
    - +  Proof.
    +    intros y2 c2.
    +    induction s as [| [y1 c1] s1' IH] ⇒ acc. {
    +      rewrite /inter_aux.
    +      move ⇒ H_inv_acc _ _ _.
    +      split; last split; try done.
    +      move ⇒ y. rewrite InZ_nil.
    +      tauto.
    +    } {
    +      rewrite interval_list_invariant_cons.
    +      move ⇒ H_inv_acc [H_gr_s1'] [H_c1_neq_0] H_inv_s1'.
    +      move ⇒ H_in_acc_lt H_c2_neq_0.
    + +
    +      rewrite inter_aux_alt_def.
    +      case_eq (inter_aux y2 c2 nil ((y1, c1) :: s1')).
    +      move ⇒ acc' s' H_inter_aux_eq.
    + +
    +      set P1 := ∀ y : Z,
    +        (InZ y acc' ↔
    +        ((InZ y ((y1, c1) :: s1') ∧ List.In y (elementsZ_single y2 c2)))).
    +      set P2 := (∀ y,
    +                  (InZ y s' → InZ y ((y1, c1) :: s1')) ∧
    +                  (InZ y ((y1, c1) :: s1') →
    +                     y2 + Z.of_N c2 < y → InZ y s')).
    +      set P3 := interval_list_invariant (rev acc') = true.
    +      set P4 := interval_list_invariant s' = true.
    + +
    +      suff : (P1 ∧ P2 ∧ P3 ∧ P4). {
    +        move ⇒ [H_P1] [H_P2] [H_P3] H_P4.
    +        split; last split; last split; last split. {
    +          move ⇒ y.
    +          move : (H_P1 y).
    +          rewrite !InZ_app InZ_cons !In_elementsZ_single.
    +          move ⇒ <-.
    +          tauto.
    +        } {
    +          move ⇒ y H_y_in.
    +          by apply H_P2.
    +        } {
    +          move ⇒ y H_y_in.
    +          by apply H_P2.
    +        } {
    +          rewrite rev_app_distr.
    +          apply interval_list_invariant_app_intro ⇒ //.
    +          move ⇒ x1 x2.
    +          rewrite !InZ_rev.
    +          move ⇒ H_x1_in /H_P1 [H_x2_in1] H_x2_in2.
    +          apply H_in_acc_lt ⇒ //.
    +        } {
    +          apply H_P4.
    +        }
    +      }
    +
    +      move : (H_gr_s1').
    +      rewrite interval_list_elements_greater_alt2_def ⇒ // ⇒ H_gr_s1'_alt.
    + +
    +      have : ∀ (acc : list (Z × N)),
    +        interval_list_invariant (rev acc) = true →
    +        (∀ y, InZ y acc ↔ (
    +           y1 ≤ y < y1 + Z.of_N c1 ∧
    +           y2 ≤ y < y2 + Z.of_N c2)) →
    +        (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) →
    +        (inter_aux y2 c2 acc s1' = (acc', s')) →
    +        P1 ∧ P2 ∧ P3 ∧ P4. {
    +        
    +        intros acc0 H_inv_acc0 H_in_acc0 H_y2c_lt H_inter_aux_eq0.
    +        have H_in_acc0_lt : (∀ x1 x2 : Z,
    +          InZ x1 acc0 →
    +          InZ x2 s1' →
    +          List.In x2 (elementsZ_single y2 c2) →
    +          Z.succ x1 < x2). {
    +
    +          intros x1 x2 H_x1_in_acc0 H_x2_in_s1' H_x2_in_yc2.
    + +
    +          suff H_yc1_lt_x2 : Z.succ x1 ≤ y1 + Z.of_N c1. {
    +            apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +            by apply H_gr_s1'_alt.
    +          }
    +          move : (H_x1_in_acc0).
    +          rewrite H_in_acc0 Z.le_succ_l.
    +          tauto.
    +        }
    +        
    +        move : (IH acc0 H_inv_acc0 H_inv_s1' H_in_acc0_lt H_c2_neq_0).
    +        rewrite H_inter_aux_eq0.
    +        move ⇒ [H1] [H2] [H3] [H4] H5.
    +        split; last split ⇒ //. {
    +          move ⇒ y.
    +          rewrite (H1 y).
    +          rewrite InZ_cons !In_elementsZ_single
    +                  H_in_acc0.
    +          tauto.
    +        } {
    +          move ⇒ y.
    +          split. {
    +            move ⇒ /H2.
    +            rewrite InZ_cons.
    +            by right.
    +          } {
    +            rewrite InZ_cons In_elementsZ_single.
    +            move ⇒ []. {
    +              move ⇒ [_] H_y_lt H_lt_y.
    +              exfalso.
    +              suff : (y < y) by apply Z.lt_irrefl.
    +              apply Z.lt_le_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +              apply Z.le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    + +
    +              by apply Z.lt_le_incl.
    +            } {
    +              apply H3.
    +            }
    +          }
    +        }
    +      }
    +      move ⇒ {IH} IH.
    + +
    +      clear H_inv_acc H_in_acc_lt acc.
    +      move : (interval_compare_elim y1 c1 y2 c2) H_inter_aux_eq.
    +      unfold inter_aux.
    +      case_eq (interval_compare (y1, c1) (y2, c2)) ⇒ H_comp;
    +         fold inter_aux. {
    +        move ⇒ H_lt_y2.
    +        apply IH. {
    +          done.
    +        } {
    +          move ⇒ x.
    +          rewrite InZ_nil.
    +          split ⇒ //.
    +          omega.
    +        } {
    +          apply Z.le_trans with (m := y2). {
    +            by apply Z.lt_le_incl.
    +          } {
    +            apply Z_le_add_r.
    +          }
    +        }
    +      } {
    +        move ⇒ H_eq_y2.
    +        apply IH. {
    +          done.
    +        } {
    +          move ⇒ x.
    +          rewrite InZ_nil.
    +          split ⇒ //.
    +          omega.
    +        } {
    +          rewrite H_eq_y2.
    +          apply Z_le_add_r.
    +        }
    +      } {
    +        move ⇒ [H_y1_lt_y2] [H_y2_lt_yc1] H_yc1_lt_yc2.
    +        apply IH. {
    +          rewrite interval_list_invariant_sing.
    +          by apply Z_to_N_minus_neq_0.
    +        } {
    +          move ⇒ x.
    +          rewrite InZ_cons InZ_nil In_elementsZ_single Z2N.id; last omega.
    +          replace (y1 + (y2 - y1)) with y2 by omega.
    +          split; omega.
    +        } {
    +          by apply Z.lt_le_incl.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3 /P4.
    +        move ⇒ [H_y2_lt] [H_y1_lt] H_yc1_lt.
    +        move ⇒ [] H_acc' H_s'.
    +        clear IH P1 P2 P3 P4 H_comp.
    +        subst s' acc'.
    +        split; last split; last split. {
    +          move ⇒ y.
    +          have H_yc2_intro : y1 + Z.of_N (Z.to_N (y2 + Z.of_N c2 - y1)) =
    +                             y2 + Z.of_N c2. {
    +            rewrite Z2N.id; omega.
    +          }
    +          
    +          rewrite !InZ_cons !In_elementsZ_single InZ_nil H_yc2_intro.
    +          split. {
    +            move ⇒ [] //.
    +            move ⇒ [H_y1_le] H_y_lt.
    +            split; last by omega.
    +            left. omega.
    +          } {
    +            move ⇒ [H_in_s] [H_y2_le] H_y_lt.
    +            left.
    +            split; last assumption.
    +            move : H_in_s ⇒ []. {
    +              tauto.
    +            } {
    +              move ⇒ /H_gr_s1'_alt H_lt_y.
    +              apply Z.le_trans with (m := y1 + Z.of_N c1). {
    +                by apply Z_le_add_r.
    +              } {
    +                by apply Z.lt_le_incl.
    +              }
    +            }
    +          }
    +        } {
    +          move ⇒ y.
    +          split; done.
    +        } {
    +          rewrite interval_list_invariant_sing.
    +          by apply Z_to_N_minus_neq_0.
    +        } {
    +          by rewrite interval_list_invariant_cons.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3 /P4.
    +        move ⇒ [H_y12_eq] H_c12_eq [] H_acc' H_s'.
    +        clear IH P1 P2 P3 P4 H_comp.
    +        subst.
    +        split; last split; last split. {
    +          move ⇒ y.
    +          rewrite !InZ_cons InZ_nil In_elementsZ_single.
    +          split; last by tauto. {
    +            move ⇒ [] //.
    +            tauto.
    +          }
    +        } {
    +          move ⇒ y.
    +          rewrite InZ_cons In_elementsZ_single.
    +          split; first by right.
    +          move ⇒ [] //.
    +          move ⇒ [_] H_y_lt H_lt_y.
    +          exfalso.
    +          suff : (y2 + Z.of_N c2 < y2 + Z.of_N c2) by
    +              apply Z.lt_irrefl.
    +          apply Z.lt_trans with (m := y) ⇒ //.
    +        } {
    +          rewrite interval_list_invariant_sing //.
    +        } {
    +          assumption.
    +        }
    +      } {
    +        move ⇒ [H_y2_le_y1] [H_yc1_le_yc2] _.
    +        apply IH. {
    +          by rewrite interval_list_invariant_sing.
    +        } {
    +          move ⇒ y.
    +          rewrite InZ_cons InZ_nil In_elementsZ_single.
    +          split. {
    +            move ⇒ [] //.
    +            move ⇒ [H_y1_le] H_y_lt.
    +            split; first done.
    +            split; omega.
    +          } {
    +            move ⇒ [?] _.
    +            by left.
    +          }
    +        } {
    +          assumption.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3 /P4.
    +        move ⇒ [H_y1_le] [H_yc2_le] _.
    +        move ⇒ [] H_acc' H_s'.
    +        clear IH P1 P2 P3 P4 H_comp.
    +        subst.
    +        split; last split; last split. {
    +          move ⇒ y.
    +          rewrite !InZ_cons !In_elementsZ_single InZ_nil.
    +          split. {
    +            move ⇒ [] //.
    +            move ⇒ [H_y2_le] H_y_lt.
    +            split; last by omega.
    +            left. omega.
    +          } {
    +            move ⇒ [H_in_s] [H_y2_le] H_y_lt.
    +            by left.
    +          }
    +        } {
    +          tauto.
    +        } {
    +          by rewrite interval_list_invariant_sing.
    +        } {
    +          by rewrite interval_list_invariant_cons.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3 /P4.
    +        move ⇒ H_yc2_lt [] H_acc' H_s'.
    +        clear IH P1 P2 P3 P4 H_comp.
    +        subst.
    +        split; last split; last split. {
    +          move ⇒ y.
    +          rewrite InZ_cons !In_elementsZ_single InZ_nil.
    +          split; first done.
    +          move ⇒ [] []. {
    +            move ⇒ [H_y1_le_y] H_y_lt_yc1.
    +            move ⇒ [H_y2_le_y] H_y_lt_yc2.
    +            omega.
    +          } {
    +            move ⇒ /H_gr_s1'_alt H_lt_y [_] H_y_lt.
    +            suff : (y1 + Z.of_N c1 < y1). {
    +              apply Z.nlt_ge.
    +              apply Z_le_add_r.
    +            }
    +            omega.
    +          }
    +        } {
    +          tauto.
    +        } {
    +          done.
    +        } {
    +          by rewrite interval_list_invariant_cons.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3 /P4.
    +        move ⇒ H_y1_eq [] H_acc' H_s'.
    +        clear IH P1 P2 P3 P4 H_comp.
    +        subst acc' s'.
    +        split; last split; last split. {
    +          move ⇒ y.
    +          rewrite InZ_cons !In_elementsZ_single InZ_nil.
    +          split; first done.
    +          move ⇒ [] []. {
    +            move ⇒ [H_y1_le_y] H_y_lt_yc1.
    +            move ⇒ [H_y2_le_y] H_y_lt_yc2.
    +            omega.
    +          } {
    +            move ⇒ /H_gr_s1'_alt H_lt_y [_] H_y_lt.
    +            suff : (y1 + Z.of_N c1 < y1). {
    +              apply Z.nlt_ge.
    +              apply Z_le_add_r.
    +            }
    +            omega.
    +          }
    +        } {
    +          tauto.
    +        } {
    +          done.
    +        } {
    +          by rewrite interval_list_invariant_cons.
    +        }
    +      }
    +    }
    +  Qed.
    +
      Lemma inter_aux2_props :
       âˆ€ (s2 s1 acc : t),
    -    interval_list_invariant (rev acc) = true →
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    (∀ x1 x2, InZ x1 acc → InZ x2 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    -    ((∀ y, (InZ y (inter_aux2 acc s1 s2) ↔
    -               (InZ y acc) ∨ ((InZ y s1) ∧ InZ y s2))) ∧
    -    (interval_list_invariant (inter_aux2 acc s1 s2) = true)).
    - +    interval_list_invariant (rev acc) = true →
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    (∀ x1 x2, InZ x1 acc → InZ x2 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    +    ((∀ y, (InZ y (inter_aux2 acc s1 s2) ↔
    +               (InZ y acc) ∨ ((InZ y s1) ∧ InZ y s2))) ∧
    +    (interval_list_invariant (inter_aux2 acc s1 s2) = true)).
    +  Proof.
    +    induction s2 as [| [y2 c2] s2' IH]. {
    +      move ⇒ s1 acc.
    +      move ⇒ H_inv_acc _ _ _.
    +      unfold inter_aux2.
    +      replace (match s1 with
    +        | nil ⇒ rev' acc
    +        | _ :: _ ⇒ rev' acc
    +               end) with (rev' acc); last by case s1.
    +      rewrite /rev' rev_append_rev app_nil_r.
    +      split; last done.
    +      move ⇒ y.
    +      rewrite InZ_nil InZ_rev.
    +      tauto.
    +    } {
    +      intros s1 acc H_inv_acc H_inv_s1.
    +      rewrite interval_list_invariant_cons.
    +      move ⇒ [H_gr_s2'] [H_c2_neq_0] H_inv_s2'.
    +      move ⇒ H_acc_s12.
    + +
    +      move : H_gr_s2'.
    +      rewrite interval_list_elements_greater_alt2_def //.
    +      move ⇒ H_gr_s2'.
    + +
    +      rewrite /inter_aux2; fold inter_aux2.
    +      case_eq s1. {
    +        move ⇒ H_s1_eq.
    +        split. {
    +          move ⇒ y.
    +          rewrite /rev' rev_append_rev app_nil_r InZ_nil
    +                  InZ_rev.
    +          tauto.
    +        } {
    +          rewrite /rev' rev_append_rev app_nil_r //.
    +        }
    +      } {
    +        move ⇒ [_ _] _ <-.
    +        case_eq (inter_aux y2 c2 acc s1).
    +        move ⇒ acc' s1' H_inter_aux_eq.
    + +
    +        have H_acc_s1_yc2 : ∀ x1 x2 : Z,
    +          InZ x1 acc →
    +          InZ x2 s1 →
    +          List.In x2 (elementsZ_single y2 c2) →
    +          Z.succ x1 < x2. {
    +          intros x1 x2 H_x1_in H_x2_in1 H_x2_in2.
    +          apply H_acc_s12 ⇒ //.
    +          rewrite InZ_cons; by left.
    +        }
    +
    +        move : (inter_aux_props y2 c2 s1 acc H_inv_acc H_inv_s1 H_acc_s1_yc2 H_c2_neq_0).
    +        rewrite H_inter_aux_eq.
    +        move ⇒ [H_in_acc'] [H_in_s1'_elim] [H_in_s1'_intro]
    +                [H_inv_acc'] H_inv_s1'.
    + +
    +        have H_in_acc'_s2' : ∀ x1 x2 : Z,
    +            InZ x1 acc' → InZ x2 s1' → InZ x2 s2' → Z.succ x1 < x2. {
    +          move ⇒ x1 x2 /H_in_acc'.
    +          move ⇒ []. {
    +            move ⇒ H_in_acc H_in_s1' H_in_s2'.
    +            apply H_acc_s12 ⇒ //. {
    +              by apply H_in_s1'_elim.
    +            } {
    +              rewrite InZ_cons; by right.
    +            }
    +          } {
    +            rewrite In_elementsZ_single.
    +            move ⇒ [H_in_s1] [_] H_x1_lt _.
    +            move ⇒ /H_gr_s2' H_lt_x2.
    +            apply Z.le_lt_trans with (m := y2 + Z.of_N c2) ⇒ //.
    +            by apply Z.le_succ_l.
    +          }
    +        }
    +       
    +        move : (IH s1' acc' H_inv_acc' H_inv_s1' H_inv_s2' H_in_acc'_s2').
    +        move ⇒ [H_inZ_res] H_inv_res.
    +        split; last assumption.
    +        move ⇒ y.
    +        rewrite H_inZ_res H_in_acc' InZ_cons
    +                In_elementsZ_single.
    +        split. {
    +          move ⇒ []; first by tauto.
    +          move ⇒ [H_y_in_s1' H_y_in_s2'].
    +          right.
    +          split; last by right.
    +          by apply H_in_s1'_elim.
    +        } {
    +          move ⇒ []. {
    +            move ⇒ H_y_in_acc.
    +            by left; left.
    +          } {
    +            move ⇒ [H_y_in_s1].
    +            move ⇒ []. {
    +              move ⇒ H_in_yc2.
    +              by left; right.
    +            } {
    +              right.
    +              split; last assumption.
    +              apply H_in_s1'_intro ⇒ //.
    +              by apply H_gr_s2'.
    +            }
    +          }
    +        }
    +      }
    +    }
    +  Qed.
    +
      Lemma inter_InZ :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    âˆ€ y, (InZ y (inter s1 s2) ↔ InZ y s1 ∧ InZ y s2).
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    âˆ€ y, (InZ y (inter s1 s2) ↔ InZ y s1 ∧ InZ y s2).
    +  Proof.
    +    intros s1 s2 H_inv_s1 H_inv_s2 y.
    +    rewrite /inter.
    + +
    +    move : (inter_aux2_props s2 s1 nil).
    +    move ⇒ [] //.
    +    move ⇒ H_in_inter _.
    +    rewrite H_in_inter InZ_nil.
    +    tauto.
    +  Qed.

      Lemma inter_invariant :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    interval_list_invariant (inter s1 s2) = true.
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    interval_list_invariant (inter s1 s2) = true.
    +  Proof.
    +    intros s1 s2 H_inv_s1 H_inv_s2.
    +    rewrite /inter.
    + +
    +    move : (inter_aux2_props s2 s1 nil).
    +    move ⇒ [] //.
    +  Qed.


      Global Instance inter_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (inter s1 s2).
    - +  Proof.
    +    move ⇒ H_ok_s1 H_ok_s2.
    +    move : (H_ok_s1) (H_ok_s2).
    +    rewrite /Ok /IsOk /is_encoded_elems_list /add.
    +    move ⇒ [H_inv_s1] H_pre1.
    +    move ⇒ [H_inv_s2] H_pre2.
    +    split. {
    +      apply inter_invariant ⇒ //.
    +    } {
    +      intros y.
    +      move : (inter_InZ s1 s2 H_inv_s1 H_inv_s2).
    +      rewrite /InZ ⇒ →.
    +      move ⇒ [].
    +      move ⇒ /H_pre1 //.
    +    }
    +  Qed.
    +
      Lemma inter_spec :
       âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    -   In x (inter s s') ↔ In x s ∧ In x s'.
    - +   In x (inter s s') ↔ In x s ∧ In x s'.
    +  Proof.
    +    intros s s' x H_ok H_ok'.
    +    rewrite !In_InZ.
    +    rewrite inter_InZ ⇒ //. {
    +      apply H_ok.
    +    } {
    +      apply H_ok'.
    +    }
    +  Qed.
    +

    @@ -1683,62 +3677,632 @@

    Library MSetsExtra.MSetIntervals


      Lemma diff_aux_alt_def :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      diff_aux y2 c2 acc s = match diff_aux y2 c2 nil s with
    -                               (acc', s') ⇒ (acc' ++ acc, s')
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      diff_aux y2 c2 acc s = match diff_aux y2 c2 nil s with
    +                               (acc', s') ⇒ (acc' ++ acc, s')
                                 end.
    - +  Proof.
    +    intros y2 c2.
    +    induction s as [| [y1 c1] acc' IH] ⇒ acc. {
    +      rewrite /diff_aux app_nil_l //.
    +    } {
    +      simpl.
    +      case_eq (diff_aux y2 c2 nil acc') ⇒ acc'' s'' H_eq.
    +      case (interval_compare (y1, c1) (y2, c2));
    +        rewrite ?(IH ((y1, c1)::acc)) ?(IH ((y1, c1)::nil))
    +                ?(IH acc) ?(IH ((y1, Z.to_N (y2 - y1)) :: acc))
    +                ?(IH ((y1, Z.to_N (y2 - y1)) :: nil)) ?H_eq;
    +        rewrite ?insert_intervalZ_guarded_app -?app_assoc -?app_comm_cons //.
    +    }
    +  Qed.
    +

      Lemma diff_aux_props :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      interval_list_invariant (List.rev acc) = true →
    -      interval_list_invariant s = true →
    -      (∀ x1 x2, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    -      (∀ x, InZ x acc → x < y2) →
    -      (c2 ≠ 0%N) →
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      interval_list_invariant (List.rev acc) = true →
    +      interval_list_invariant s = true →
    +      (∀ x1 x2, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    +      (∀ x, InZ x acc → x < y2) →
    +      (c2 ≠ 0%N) →
          match (diff_aux y2 c2 acc s) with
    -        (acc', s') ⇒ (∀ y, InZ y (List.rev_append acc' s') ↔
    -                                 InZ y (List.rev_append acc s) ∧ ~(List.In y (elementsZ_single y2 c2))) ∧
    -                      (interval_list_invariant (List.rev_append acc' s') = true) ∧
    -                      (∀ x, InZ x acc' → x < y2 + Z.of_N c2)
    +        (acc', s') ⇒ (∀ y, InZ y (List.rev_append acc' s') ↔
    +                                 InZ y (List.rev_append acc s) ∧ ~(List.In y (elementsZ_single y2 c2))) ∧
    +                      (interval_list_invariant (List.rev_append acc' s') = true) ∧
    +                      (∀ x, InZ x acc' → x < y2 + Z.of_N c2)
          end.
    - +  Proof.
    +    intros y2 c2.
    +    induction s as [| [y1 c1] s1' IH] ⇒ acc. {
    +      rewrite /diff_aux -rev_alt.
    +      move ⇒ H_inv_acc _ _ H_in_acc H_c2_neq.
    +      split; last split. {
    +        move ⇒ y; split; last by move ⇒ [] //.
    +        rewrite InZ_rev.
    +        move ⇒ H_in. split ⇒ //.
    +        move ⇒ /In_elementsZ_single ⇒ [] [] /Z.nlt_ge H_neq.
    +        contradict H_neq.
    +        by apply H_in_acc.
    +      } {
    +        assumption.
    +      } {
    +        intros x H_in_acc'.
    +        apply Z.lt_le_trans with (m := y2). {
    +          by apply H_in_acc.
    +        } {
    +          by apply Z_le_add_r.
    +        }
    +      }
    +    } {
    +      rewrite interval_list_invariant_cons.
    +      move ⇒ H_inv_acc [H_gr_s1'] [H_c1_neq_0] H_inv_s1'.
    +      move ⇒ H_in_s1 H_in_acc H_c2_neq_0.
    + +
    +      rewrite diff_aux_alt_def.
    +      case_eq (diff_aux y2 c2 nil ((y1, c1) :: s1')).
    +      move ⇒ acc' s' H_diff_aux_eq.
    + +
    +      set P1 := ∀ y : Z,
    +        (InZ y acc' ∨ InZ y s') ↔
    +        InZ y ((y1, c1) :: s1') ∧ ¬ List.In y (elementsZ_single y2 c2).
    +      set P2 := interval_list_invariant (rev acc' ++ s') = true.
    +      set P3 := ∀ x : Z, InZ x acc' → (x < y2 + Z.of_N c2).
    + +
    +      suff : (P1 ∧ P2 ∧ P3). {
    +        move ⇒ [H_P1] [H_P2] H_P3.
    +        split; last split. {
    +          move ⇒ y.
    +          move : (H_P1 y).
    +          rewrite !rev_append_rev rev_app_distr !InZ_app
    +                  !InZ_rev In_elementsZ_single.
    +          suff : (InZ y acc → ¬ y2 ≤ y < y2 + Z.of_N c2). {
    +            tauto.
    +          }
    +          move ⇒ /H_in_acc H_y_lt [H_y_ge] _.
    +          contradict H_y_ge.
    +          by apply Zlt_not_le.
    +        } {
    +          rewrite rev_append_rev rev_app_distr -app_assoc.
    +          apply interval_list_invariant_app_intro ⇒ //.
    +          move ⇒ x1 x2.
    +          rewrite InZ_app !InZ_rev.
    +          move ⇒ H_in_acc' H_x2_in_s'.
    +          suff : (InZ x2 ((y1, c1)::s1')). {
    +            by apply H_in_s1.
    +          }
    +          move : (H_P1 x2).
    +          tauto.
    +        } {
    +          move ⇒ x.
    +          rewrite InZ_app.
    +          move ⇒ []. {
    +            apply H_P3.
    +          } {
    +            move ⇒ /H_in_acc H_x_lt.
    +            eapply Z.lt_trans; eauto.
    +            by apply Z_lt_add_r.
    +          }
    +        }
    +      }
    +
    +      move : (H_gr_s1').
    +      rewrite interval_list_elements_greater_alt2_def ⇒ // ⇒ H_gr_s1'_alt.
    + +
    +      have : ∀ (acc : list (Z × N)),
    +        interval_list_invariant (rev acc) = true →
    +        (∀ x : Z,
    +            InZ x acc ↔
    +            ((y1 ≤ x < y1 + Z.of_N c1) ∧ (x < y2))) →
    +        (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) →
    +        (diff_aux y2 c2 acc s1' = (acc', s')) →
    +        P1 ∧ P2 ∧ P3. {
    +        
    +        intros acc0 H_inv_acc0 H_in_acc0 H_c1_before H_diff_aux_eq0.
    +        have H_in_s1' : (∀ x1 x2 : Z,
    +                            InZ x1 acc0 → InZ x2 s1' → Z.succ x1 < x2). {
    +          intros x1 x2 H_x1_in_acc0.
    +          move ⇒ /H_gr_s1'_alt.
    + +
    +          eapply Z.le_lt_trans.
    +          move : H_x1_in_acc0.
    +          rewrite Z.le_succ_l H_in_acc0.
    +          tauto.
    +        }
    +        have H_in_acc0' : (∀ x : Z, InZ x acc0 → x < y2). {
    +          move ⇒ x.
    +          rewrite H_in_acc0.
    +          move ⇒ [_] //.
    +        }
    +        move : (IH acc0 H_inv_acc0 H_inv_s1' H_in_s1' H_in_acc0' H_c2_neq_0).
    +        rewrite H_diff_aux_eq0 !rev_append_rev.
    +        move ⇒ [H1] [H2] H3.
    +        split; last split ⇒ //. {
    +          move ⇒ y.
    +          move : (H1 y).
    +          rewrite !InZ_app !InZ_rev In_elementsZ_single.
    +          move ⇒ →.
    +          rewrite InZ_cons In_elementsZ_single.
    +          split. {
    +            rewrite H_in_acc0 -(Z.nle_gt y2 y).
    +            tauto.
    +          } {
    +            rewrite H_in_acc0 -(Z.nle_gt y2 y).
    +            move ⇒ [] H_in H_nin_i2.
    +            split; last by assumption.
    +            move : H_in ⇒ [] H_in; last by right.
    +            left.
    +            omega.
    +          }
    +        }
    +      }
    +      move ⇒ {IH} IH.
    + +
    +      clear H_inv_acc H_in_s1 H_in_acc acc.
    +      move : (interval_compare_elim y1 c1 y2 c2) H_diff_aux_eq.
    +      unfold diff_aux.
    +      case_eq (interval_compare (y1, c1) (y2, c2)) ⇒ H_comp;
    +                                                        fold diff_aux. {
    +        move ⇒ H_lt_y2.
    +        apply IH. {
    +          by rewrite interval_list_invariant_sing.
    +        } {
    +          move ⇒ x.
    +          rewrite InZ_cons In_elementsZ_single.
    +          split; last by tauto.
    +          move ⇒ []; last done.
    +          move ⇒ [H_y1_le H_x_lt].
    +          split; first done.
    +          eapply Z.lt_trans; eauto.
    +        } {
    +          apply Z.le_trans with (m := y2).
    +            - by apply Z.lt_le_incl.
    +            - by apply Z_le_add_r.
    +        }
    +      } {
    +        move ⇒ H_eq_y2.
    +        apply IH. {
    +          by rewrite interval_list_invariant_sing.
    +        } {
    +          move ⇒ x.
    +          rewrite InZ_cons In_elementsZ_single -H_eq_y2.
    +          split; last by tauto.
    +          move ⇒ []; last done.
    +          move ⇒ []. done.
    +        } {
    +          rewrite H_eq_y2.
    +          by apply Z_le_add_r.
    +        }
    +      } {
    +        move ⇒ [H_y1_lt_y2] [H_y2_lt_yc1] H_yc1_lt_yc2.
    +        apply IH. {
    +          rewrite interval_list_invariant_sing.
    +          by apply Z_to_N_minus_neq_0.
    +        } {
    +          move ⇒ x.
    +          rewrite InZ_cons In_elementsZ_single Z2N.id; last omega.
    +          replace (y1 + (y2 - y1)) with y2 by omega.
    +          split; last tauto.
    +          move ⇒ [] //.
    +          move ⇒ [H_y1_le] H_x_lt.
    +          repeat split ⇒ //.
    +          apply Z.lt_trans with (m := y2) ⇒ //.
    +        } {
    +          by apply Z.lt_le_incl.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3.
    +        move ⇒ [H_y2_lt] [H_y1_lt] H_yc1_lt [H_acc'] H_s'.
    +        clear IH P1 P2 P3 H_comp.
    +        subst.
    +        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
    +          rewrite Z2N.id; omega.
    +        }
    +        have H_nin_yc2 : ∀ y,
    +            InZ y s1' → ¬ y2 ≤ y < y2 + Z.of_N c2. {
    +          move ⇒ y /H_gr_s1'_alt H_lt_y.
    +          move ⇒ [H_y2_le_y].
    +          apply Z.le_ngt, Z.lt_le_incl.
    +          by apply Z.lt_trans with (m := y1 + Z.of_N c1).
    +        }
    +        split; last split. {
    +          move ⇒ y.
    +          rewrite !InZ_cons !In_elementsZ_single H_yc1_intro.
    +          split. {
    +            move ⇒ [] //.
    +            move ⇒ []. {
    +              move ⇒ [H_le_y] H_y_lt.
    +              split. {
    +                left; omega.
    +              } {
    +                move ⇒ [_].
    +                by apply Z.nlt_ge.
    +              }
    +            } {
    +              move : (H_nin_yc2 y). tauto.
    +            }
    +          } {
    +            move ⇒ [] []; last by right; right.
    +            move ⇒ [H_y_ge] H_y_lt_yc1 H_nin_yc2'.
    +            right; left. omega.
    +          }
    +        } {
    +          rewrite interval_list_invariant_cons H_yc1_intro.
    +          split ⇒ //.
    +          split ⇒ //.
    +          by apply Z_to_N_minus_neq_0.
    +        } {
    +          move ⇒ [] //.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3.
    +        move ⇒ [H_y12_eq] H_c12_eq [] H_acc' H_s'.
    +        clear IH P1 P2 P3 H_comp.
    +        subst.
    +        split; last split. {
    +          move ⇒ y.
    +          rewrite InZ_cons In_elementsZ_single.
    +          split; last by tauto. {
    +            move ⇒ [] //.
    +            move ⇒ H_y_in.
    +            split; first by right.
    +            move ⇒ [] _.
    +            by apply Z.nlt_ge, Z.lt_le_incl, H_gr_s1'_alt.
    +          }
    +        } {
    +          apply H_inv_s1'.
    +        } {
    +          move ⇒ x [] //.
    +        }
    +      } {
    +        move ⇒ [H_y2_le_y1] [H_yc1_le_yc2] _.
    +        apply IH. {
    +          done.
    +        } {
    +          move ⇒ x.
    +          split; first done.
    +          omega.
    +        } {
    +          assumption.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3.
    +        move ⇒ [H_y1_le] [H_yc2_le_yc1] _ [] H_acc' H_s'.
    +        clear IH P1 P2 P3 H_comp.
    +        subst.
    +        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
    +          rewrite Z2N.id; omega.
    +        }
    +        have H_y1_intro : y1 + Z.of_N (Z.to_N (y2 - y1)) = y2. {
    +          rewrite Z2N.id; omega.
    +        }
    +        split; last split. {
    +          move ⇒ y.
    +          rewrite !InZ_insert_intervalZ_guarded
    +                  !InZ_cons !In_elementsZ_single
    +                  H_yc1_intro H_y1_intro InZ_nil.
    +          split. {
    +            rewrite -!or_assoc.
    +            move ⇒ [[[]|]|] //. {
    +              move ⇒ [H_y1_le_y] H_y_lt.
    +              split. {
    +                left.
    +                split ⇒ //.
    +                apply Z.lt_le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    +                apply Z.lt_trans with (m := y2) ⇒ //.
    +                by apply Z_lt_add_r.
    +              } {
    +                move ⇒ [] /Z.le_ngt //.
    +              }
    +            } {
    +              move ⇒ [H_y2c_le_y] H_y_lt_yc1.
    +              split. {
    +                left.
    +                split ⇒ //.
    +                apply Z.le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    +                apply Z.le_trans with (m := y2) ⇒ //.
    +                apply Z_le_add_r.
    +              } {
    +                move ⇒ [] _ /Z.lt_nge //.
    +              }
    +            } {
    +              move ⇒ H_y_in_s1'.
    +              split; first by right.
    +              suff H_suff : y2 + Z.of_N c2 ≤ y. {
    +                move ⇒ [] _ /Z.lt_nge //.
    +              }
    +              apply Z.le_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +              apply Z.lt_le_incl.
    +              by apply H_gr_s1'_alt.
    +            }
    +          } {
    +            move ⇒ [] []; last by tauto.
    +            move ⇒ [H_y1_le_y] H_y_lt H_neq_y2.
    +            apply not_and in H_neq_y2; last by apply Z.le_decidable.
    +            case H_neq_y2. {
    +              move ⇒ /Z.nle_gt H_y_lt'.
    +              left; left; done.
    +            } {
    +              move ⇒ /Z.nlt_ge H_le_y.
    +              right; left; done.
    +            }
    +          }
    +        } {
    +          rewrite insert_intervalZ_guarded_rev_nil_app.
    +          rewrite !interval_list_invariant_insert_intervalZ_guarded ⇒ //. {
    +            rewrite H_yc1_intro ⇒ //.
    +          } {
    +            rewrite /insert_intervalZ_guarded.
    +            case_eq ((Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2)) =? 0)%N). {
    +              rewrite H_y1_intro.
    +              move ⇒ /N.eqb_eq /N2Z.inj_iff.
    +              rewrite Z2N.id; last first. {
    +                by apply Z.le_0_sub.
    +              }
    +              move ⇒ /Zminus_eq H_yc1_eq.
    +              eapply interval_list_elements_greater_impl;
    +                last apply H_gr_s1'.
    +              rewrite H_yc1_eq.
    +              apply Z_le_add_r.
    +            } {
    +              move ⇒ _.
    +              rewrite interval_list_elements_greater_cons
    +                      H_y1_intro.
    +              by apply Z_lt_add_r.
    +            }
    +          }
    +        } {
    +          move ⇒ x.
    +          rewrite InZ_insert_intervalZ_guarded InZ_cons In_elementsZ_single H_y1_intro InZ_nil.
    +          move ⇒ [] //.
    +          move ⇒ [_] H_x_lt.
    +          apply Z.lt_le_trans with (m := y2) ⇒ //.
    +          apply Z_le_add_r.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3.
    +        move ⇒ H_yc2_lt [] H_acc' H_s'.
    +        clear IH P1 P2 P3 H_comp.
    +        subst.
    +        split; last split. {
    +          move ⇒ y.
    +          rewrite InZ_cons In_elementsZ_single.
    +          split; last by tauto. {
    +            move ⇒ [] //.
    +            move ⇒ H_y_in.
    +            split; first assumption.
    +            rewrite In_elementsZ_single.
    +            move ⇒ [] H_y2_le H_y_lt.
    +            case H_y_in; first by omega.
    +            move ⇒ /H_gr_s1'_alt H_lt_y.
    +            suff : y2 + Z.of_N c2 < y. {
    +              move ⇒ ?. omega.
    +            }
    +            apply Z.lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +            apply Z.lt_le_trans with (m := y1) ⇒ //.
    +            apply Z_le_add_r.
    +          }
    +        } {
    +          by rewrite interval_list_invariant_cons.
    +        } {
    +          done.
    +        }
    +      } {
    +        rewrite /P1 /P2 /P3.
    +        move ⇒ H_yc2_eq [] H_acc' H_s'.
    +        clear IH P1 P2 P3 H_comp.
    +        subst.
    +        split; last split. {
    +          move ⇒ y.
    +          rewrite InZ_cons In_elementsZ_single.
    +          split; last by tauto. {
    +            move ⇒ [] //.
    +            move ⇒ H_y_in.
    +            split; first assumption.
    +            rewrite In_elementsZ_single.
    +            move ⇒ [] H_y2_le H_y_lt.
    +            case H_y_in; first by omega.
    +            move ⇒ /H_gr_s1'_alt H_lt_y.
    +            suff : y2 + Z.of_N c2 < y. {
    +              move ⇒ ?. omega.
    +            }
    +            apply Z.lt_trans with (m := (y2 + Z.of_N c2) + Z.of_N c1) ⇒ //.
    +            by apply Z_lt_add_r.
    +          }
    +        } {
    +          by rewrite interval_list_invariant_cons.
    +        } {
    +          done.
    +        }
    +      }
    +    }
    +  Qed.
    +
      Lemma diff_aux2_props :
       âˆ€ (s2 s1 acc : t),
    -    interval_list_invariant (rev_append acc s1) = true →
    -    interval_list_invariant s2 = true →
    -    (∀ x1 x2, InZ x1 acc → InZ x2 s2 → Z.succ x1 < x2) →
    -    ((∀ y, (InZ y (diff_aux2 acc s1 s2) ↔
    -               ((InZ y acc) ∨ (InZ y s1)) ∧ ¬InZ y s2)) ∧
    -    (interval_list_invariant (diff_aux2 acc s1 s2) = true)).
    - +    interval_list_invariant (rev_append acc s1) = true →
    +    interval_list_invariant s2 = true →
    +    (∀ x1 x2, InZ x1 acc → InZ x2 s2 → Z.succ x1 < x2) →
    +    ((∀ y, (InZ y (diff_aux2 acc s1 s2) ↔
    +               ((InZ y acc) ∨ (InZ y s1)) ∧ ¬InZ y s2)) ∧
    +    (interval_list_invariant (diff_aux2 acc s1 s2) = true)).
    +  Proof.
    +    induction s2 as [| [y2 c2] s2' IH]. {
    +      move ⇒ s1 acc H_inv_acc_s1 _ _.
    +      rewrite /diff_aux2.
    +      replace (match s1 with
    +        | nil ⇒ rev_append acc s1
    +        | _ :: _ ⇒ rev_append acc s1
    +       end) with (rev_append acc s1); last by case s1.
    +      split. {
    +        move ⇒ y.
    +        rewrite rev_append_rev InZ_app InZ_rev InZ_nil.
    +        tauto.
    +      } {
    +        assumption.
    +      }
    +    } {
    +      intros s1 acc H_inv_acc_s1.
    +      rewrite interval_list_invariant_cons.
    +      move ⇒ [H_gr_s2'] [H_c2_neq_0] H_inv_s2'.
    +      move ⇒ H_acc_s2.
    +      rewrite /diff_aux2; fold diff_aux2.
    +      case_eq s1. {
    +        move ⇒ H_s1_eq.
    +        split. {
    +          move ⇒ y.
    +          rewrite rev_append_rev InZ_app InZ_nil InZ_rev.
    +          split; last tauto.
    +          move ⇒ [] // H_y_in.
    +          split; first by left.
    +          move ⇒ H_y_in'.
    +          move : (H_acc_s2 _ _ H_y_in H_y_in').
    +          apply Z.nlt_succ_diag_l.
    +        } {
    +          move : H_inv_acc_s1.
    +           by rewrite H_s1_eq.
    +        }
    +      } {
    +        move ⇒ [_ _] _ <-.
    +        case_eq (diff_aux y2 c2 acc s1).
    +        move ⇒ acc' s1' H_diff_aux_eq.
    + +
    +        have H_acc_lt_y2 : (∀ x : Z, InZ x acc → x < y2). {
    +          move ⇒ x H_x_in.
    +          have H_y2_in: (InZ y2 ((y2,c2) :: s2')). {
    +            rewrite InZ_cons.
    +            left.
    +            by apply In_elementsZ_single_hd.
    +          }
    +          move : (H_acc_s2 _ _ H_x_in H_y2_in).
    +          apply Z.lt_trans, Z.lt_succ_diag_r.
    +        }
    +
    +        have [H_inv_acc [H_inv_s1 H_acc_s1]] :
    +          interval_list_invariant (rev acc) = true ∧
    +          interval_list_invariant s1 = true ∧
    +          (∀ x1 x2 : Z,
    +             InZ x1 acc → InZ x2 s1 → Z.succ x1 < x2). {
    +
    +          move : H_inv_acc_s1.
    +          rewrite rev_append_rev.
    +          move ⇒ /interval_list_invariant_app_elim.
    +          move ⇒ [?] [?] H_x.
    +          split; first assumption.
    +          split; first assumption.
    +          move ⇒ x1 x2 H_in_x1.
    +          apply H_x.
    +          by rewrite InZ_rev.
    +        }
    +
    +        move : (diff_aux_props y2 c2 s1 acc H_inv_acc H_inv_s1 H_acc_s1 H_acc_lt_y2 H_c2_neq_0).
    +        rewrite !H_diff_aux_eq.
    +        move ⇒ [H_inZ_res] [H_inv_res] H_inZ_acc'.
    + +
    +        have H_acc'_s2' : (∀ x1 x2 : Z,
    +                              InZ x1 acc' → InZ x2 s2' → Z.succ x1 < x2). {
    +          move ⇒ x1 x2 H_x1_in H_x2_in.
    +          apply Z.le_lt_trans with (m := y2 + Z.of_N c2). {
    +            apply Z.le_succ_l.
    +            by apply H_inZ_acc'.
    +          } {
    +            move : H_gr_s2'.
    +            rewrite interval_list_elements_greater_alt2_def //.
    +            move ⇒ H_gr_s2'.
    +            by apply H_gr_s2'.
    +          }
    +        }
    +
    +        move : (IH s1' acc' H_inv_res H_inv_s2' H_acc'_s2').
    +        move ⇒ [] H_inZ_diff_res →.
    +        split; last done.
    +        move ⇒ y.
    +        rewrite H_inZ_diff_res.
    +        move : (H_inZ_res y).
    +        rewrite !rev_append_rev !InZ_app !InZ_rev InZ_cons.
    +        move ⇒ →.
    +        tauto.
    +      }
    +    }
    +  Qed.
    +
      Lemma diff_InZ :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    âˆ€ y, (InZ y (diff s1 s2) ↔ InZ y s1 ∧ ¬InZ y s2).
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    âˆ€ y, (InZ y (diff s1 s2) ↔ InZ y s1 ∧ ¬InZ y s2).
    +  Proof.
    +    intros s1 s2 H_inv_s1 H_inv_s2 y.
    +    rewrite /diff.
    + +
    +    move : (diff_aux2_props s2 s1 nil).
    +    move ⇒ [] //.
    +    move ⇒ H_in_diff _.
    +    rewrite H_in_diff InZ_nil.
    +    tauto.
    +  Qed.

      Lemma diff_invariant :
       âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    interval_list_invariant (diff s1 s2) = true.
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    interval_list_invariant (diff s1 s2) = true.
    +  Proof.
    +    intros s1 s2 H_inv_s1 H_inv_s2.
    +    rewrite /diff.
    + +
    +    move : (diff_aux2_props s2 s1 nil).
    +    move ⇒ [] //.
    +  Qed.


      Global Instance diff_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (diff s1 s2).
    - +  Proof.
    +    move ⇒ H_ok_s1 H_ok_s2.
    +    move : (H_ok_s1) (H_ok_s2).
    +    rewrite /Ok /IsOk /is_encoded_elems_list /add.
    +    move ⇒ [H_inv_s1] H_pre1.
    +    move ⇒ [H_inv_s2] H_pre2.
    +    split. {
    +      apply diff_invariant ⇒ //.
    +    } {
    +      intros y.
    +      move : (diff_InZ s1 s2 H_inv_s1 H_inv_s2).
    +      rewrite /InZ ⇒ →.
    +      move ⇒ [].
    +      move ⇒ /H_pre1 //.
    +    }
    +  Qed.
    +
      Lemma diff_spec :
       âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    -   In x (diff s s') ↔ In x s ∧ ¬In x s'.
    - +   In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +  Proof.
    +    intros s s' x H_ok H_ok'.
    +    rewrite !In_InZ.
    +    rewrite diff_InZ ⇒ //. {
    +      apply H_ok.
    +    } {
    +      apply H_ok'.
    +    }
    +  Qed.
    +
    @@ -1750,27 +4314,214 @@

    Library MSetsExtra.MSetIntervals


      Lemma removeZ_alt_def : ∀ x s acc,
    -    interval_list_invariant s = true →
    -    removeZ_aux acc x s = match diff_aux x (1%N) acc s with
    -                    (acc', s') ⇒ rev_append acc' s'
    +    interval_list_invariant s = true →
    +    removeZ_aux acc x s = match diff_aux x (1%N) acc s with
    +                    (acc', s') ⇒ rev_append acc' s'
                      end.
    - -
    -  Lemma removeZ_interval_list_invariant : ∀ s x, interval_list_invariant s = true → interval_list_invariant (removeZ x s) = true.
    +  Proof.
    +    intros y2.
    +    induction s as [| [y1 c1] s' IH]; first done.
    + +
    +    move ⇒ acc.
    +    rewrite interval_list_invariant_cons /=.
    +    move ⇒ [H_gr] [H_c1_neq_0] H_inv_s'.
    +    move : (interval_1_compare_elim y2 y1 c1).
    +    rewrite interval_1_compare_alt_def.
    +    rewrite !(interval_compare_swap y1 c1 y2); last first. {
    +      right. done.
    +    }
    +    move : (interval_compare_elim y1 c1 y2 (1%N)).
    +    case_eq (interval_compare (y1, c1) (y2, (1%N))) ⇒ H_eq. {
    +      move ⇒ H_lt_y2 _.
    +      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
    +        by apply Z.nlt_ge, Z.lt_le_incl.
    +      }
    +      have H_y2_nlt : ~(y2 < y1). {
    +        move ⇒ H_y2_y1.
    +        apply H_yc2_nlt.
    +        apply Z.lt_le_trans with (m := y1) ⇒ //.
    +        apply Z_le_add_r.
    +      }
    +      move : (H_y2_nlt) (H_yc2_nlt) ⇒ /Z.ltb_nlt → /Z.ltb_nlt →.
    +      rewrite IH //.
    +    } {
    +      move ⇒ H_lt_y2 _.
    +      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
    +        apply Z.nlt_ge.
    +        rewrite H_lt_y2.
    +        apply Z.le_refl.
    +      }
    +      have H_y2_nlt : ~(y2 < y1). {
    +        move ⇒ H_y2_y1.
    +        apply H_yc2_nlt.
    +        apply Z.lt_le_trans with (m := y1) ⇒ //.
    +        apply Z_le_add_r.
    +      }
    +      move : (H_y2_nlt) (H_yc2_nlt) ⇒ /Z.ltb_nlt → /Z.ltb_nlt →.
    +      rewrite IH //.
    +    } {
    +      done.
    +    } {
    +      done.
    +    } {
    +      move ⇒ [H_y1_eq] H_c1_eq.
    +      move ⇒ [] // .
    +      move ⇒ [H_lt_y2] H_y2_lt.
    +      have H_y2_nlt : ~(y2 < y1). {
    +        apply Z.nlt_ge ⇒ //.
    +      }
    +      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    +      rewrite /insert_intervalZ_guarded.
    + +
    +      have → : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
    +        rewrite H_y1_eq H_c1_eq Z.add_1_r Z.sub_diag //.
    +      }
    +
    +      have → : (Z.to_N (y2 - y1) =? 0 = true)%N. {
    +        rewrite H_y1_eq Z.sub_diag //.
    +      }
    +      done.
    +    } {
    +      move ⇒ [H_y2_le] [H_yc1_le] _.
    +      move ⇒ [] //.
    +      move ⇒ [H_y1_le] H_y2_lt.
    +      have H_y2_nlt : ~(y2 < y1). {
    +        apply Z.nlt_ge ⇒ //.
    +      }
    +      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    + +
    +      have H_y1_eq : (y1 = y2) by omega.
    +      have H_yc1_eq : (y1 + Z.of_N c1 = Z.succ y2). {
    +        apply Z.le_antisymm. {
    +          move : H_yc1_le.
    +          rewrite Z.add_1_r //.
    +        } {
    +          by apply Z.le_succ_l.
    +        }
    +      }
    +
    +      rewrite /insert_intervalZ_guarded.
    +      have → : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
    +        rewrite H_yc1_eq Z.sub_diag //.
    +      }
    +
    +      have → : (Z.to_N (y2 - y1) =? 0 = true)%N. {
    +        rewrite H_y1_eq Z.sub_diag //.
    +      }
    +
    +      suff → : diff_aux y2 (1%N) acc s' = (acc, s') by done.
    + +
    +      move : H_gr.
    +      rewrite H_yc1_eq.
    +      case s' as [| [y' c'] s'']. {
    +        done.
    +      } {
    +        rewrite interval_list_elements_greater_cons /=
    +                /interval_compare.
    +        move ⇒ H_lt_y'.
    +        have → : y2 + Z.of_N 1 ?= y' = Lt. {
    +          apply Z.compare_lt_iff.
    +          by rewrite Z.add_1_r.
    +        }
    +        done.
    +      }
    +    } {
    +      move ⇒ [H_y1_le] [H_yc2_le] _.
    +      move ⇒ [] //.
    +      move ⇒ [_] H_y2_lt.
    +      have H_y2_nlt : ~(y2 < y1). {
    +        apply Z.nlt_ge ⇒ //.
    +      }
    +      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    +      rewrite !rev_append_rev /insert_intervalZ_guarded Z.add_1_r.
    +      case ((Z.to_N (y2 - y1) =? 0)%N), (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0)%N. {
    +        reflexivity.
    +      } {
    +        rewrite /= -!app_assoc //.
    +      } {
    +        reflexivity.
    +      } {
    +        rewrite /= -!app_assoc //.
    +      }
    +    } {
    +      move ⇒ _ H_y2_lt'.
    +      have H_y2_lt : (y2 < y1). {
    +        apply Z.lt_trans with (m := Z.succ y2) ⇒ //.
    +        apply Z.lt_succ_diag_r.
    +      }
    +      move : (H_y2_lt) ⇒ /Z.ltb_lt → //.
    +    } {
    +      move ⇒ _ H_y1_eq.
    +      have H_y2_lt : (y2 < y1). {
    +        rewrite H_y1_eq.
    +        apply Z.lt_succ_diag_r.
    +      }
    +      move : (H_y2_lt) ⇒ /Z.ltb_lt → //.
    +    }
    +  Qed.
    + +
    +  Lemma removeZ_interval_list_invariant : ∀ s x, interval_list_invariant s = true → interval_list_invariant (removeZ x s) = true.
    +  Proof.
    +    intros s x H_inv.
    +    rewrite /removeZ removeZ_alt_def //.
    +    move : (diff_aux_props x (1%N) s nil).
    +    case_eq (diff_aux x 1%N nil s).
    +    move ⇒ acc' s' H_eq [] //.
    +    move ⇒ _ [] //.
    +  Qed.

      Lemma removeZ_spec :
    -   âˆ€ (s : t) (x y : Z) (Hs : interval_list_invariant s = true),
    -    InZ y (removeZ x s) ↔ InZ y s ∧ ¬Z.eq y x.
    +   âˆ€ (s : t) (x y : Z) (Hs : interval_list_invariant s = true),
    +    InZ y (removeZ x s) ↔ InZ y s ∧ ¬Z.eq y x.
    +  Proof.
    +    intros s x y H_inv.
    +    rewrite /removeZ removeZ_alt_def //.
    +    move : (diff_aux_props x (1%N) s nil).
    +    case_eq (diff_aux x 1%N nil s).
    +    move ⇒ acc' s' H_eq [] //.
    +    move ⇒ → _.
    +    rewrite rev_append_rev InZ_app InZ_rev InZ_nil
    +            In_elementsZ_single1.
    +    split; move ⇒ [H1 H2]; split ⇒ //;
    +      move ⇒ H3; apply H2; by rewrite H3.
    +  Qed.

      Global Instance remove_ok s x : ∀ `(Ok s), Ok (remove x s).
    - +  Proof.
    +    rewrite /Ok /interval_list_invariant /remove.
    +    move ⇒ [H_is_ok_s H_enc_s].
    +    split. {
    +      by apply removeZ_interval_list_invariant.
    +    } {
    +      rewrite /is_encoded_elems_list ⇒ y.
    +      move : (removeZ_spec s (Enc.encode x) y H_is_ok_s).
    +      rewrite /InZ ⇒ → [] H_y_in _.
    +      apply H_enc_s ⇒ //.
    +    }
    +  Qed.
    +
      Lemma remove_spec :
       âˆ€ (s : t) (x y : elt) (Hs : Ok s),
    -    In y (remove x s) ↔ In y s ∧ ¬Enc.E.eq y x.
    - +    In y (remove x s) ↔ In y s ∧ ¬Enc.E.eq y x.
    +  Proof.
    +    intros s x y Hs.
    +    have H_rs := (remove_ok s x Hs).
    +    rewrite /remove !In_InZ.
    +    rewrite removeZ_spec. {
    +      rewrite Enc.encode_eq //.
    +    } {
    +      apply Hs.
    +    }
    +  Qed.
    +
    @@ -1781,12 +4532,33 @@

    Library MSetsExtra.MSetIntervals


    -  Lemma remove_list_ok : ∀ l s, Ok s → Ok (remove_list l s).
    - +  Lemma remove_list_ok : ∀ l s, Ok s → Ok (remove_list l s).
    +  Proof.
    +    induction l as [| x l' IH]. {
    +      done.
    +    } {
    +      move ⇒ s H_s_ok /=.
    +      apply IH.
    +      by apply remove_ok.
    +    }
    +  Qed.
    +
    -  Lemma remove_list_spec : ∀ x l s, Ok s →
    -     (In x (remove_list l s) ↔ ~(InA Enc.E.eq x l) ∧ In x s).
    - +  Lemma remove_list_spec : ∀ x l s, Ok s →
    +     (In x (remove_list l s) ↔ ~(InA Enc.E.eq x l) ∧ In x s).
    +  Proof.
    +    move ⇒ x.
    +    induction l as [| y l' IH]. {
    +      intros s H.
    +      rewrite /= InA_nil.
    +      tauto.
    +    } {
    +      move ⇒ s H_ok /=.
    +      rewrite IH remove_spec InA_cons.
    +      tauto.
    +    }
    +  Qed.
    +

    @@ -1800,54 +4572,271 @@

    Library MSetsExtra.MSetIntervals


      Lemma subset_flatten_alt_def : ∀ (s1 s2 : t),
    -    subset s1 s2 =
    -    match (s1, s2) with
    -    | (nil, _) ⇒ true
    -    | (_ :: _, nil) ⇒ false
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ false
    -          | ICR_before_touch ⇒ false
    +    subset s1 s2 =
    +    match (s1, s2) with
    +    | (nil, _) ⇒ true
    +    | (_ :: _, nil) ⇒ false
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ false
    +          | ICR_before_touch ⇒ false
              | ICR_after ⇒ subset s1 l2
    -          | ICR_after_touch ⇒ false
    -          | ICR_overlap_before ⇒ false
    -          | ICR_overlap_after ⇒ false
    +          | ICR_after_touch ⇒ false
    +          | ICR_overlap_before ⇒ false
    +          | ICR_overlap_after ⇒ false
              | ICR_equal ⇒ subset l1 l2
              | ICR_subsume_1 ⇒ subset l1 s2
    -          | ICR_subsume_2 ⇒ false
    +          | ICR_subsume_2 ⇒ false
            end
        end.
    +  Proof.
    +    intros s1 s2.
    +    case s1, s2 ⇒ //.
    +  Qed.

      Lemma subset_props_aux : ∀ y1 c1 l1 y2 c2 l2,
    -    (∃ y, InZ y ((y1, c1) :: l1) ∧ ¬InZ y ((y2, c2) :: l2)) →
    -    (false = true ↔
    -    (∀ y : Z,
    -        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
    +    (∃ y, InZ y ((y1, c1) :: l1) ∧ ¬InZ y ((y2, c2) :: l2)) →
    +    (false = true ↔
    +    (∀ y : Z,
    +        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
    +  Proof.
    +    intros y1 c1 l1 y2 c2 l2.
    +    move ⇒ [y] [H_y_in H_y_nin].
    +    split; first done.
    +    move ⇒ H.
    +    contradict H_y_nin.
    +    by apply H.
    +  Qed.

      Lemma subset_props_aux_before : ∀ y1 c1 l1 y2 c2 l2,
    -    (c1 ≠ 0%N) →
    -    interval_list_invariant ((y2, c2) :: l2) = true →
    -    (y1 < y2) →
    -    (false = true ↔
    -    (∀ y : Z,
    -        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
    - +    (c1 ≠ 0%N) →
    +    interval_list_invariant ((y2, c2) :: l2) = true →
    +    (y1 < y2) →
    +    (false = true ↔
    +    (∀ y : Z,
    +        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
    +  Proof.
    +    intros y1 c1 l1 y2 c2 l2.
    +    rewrite interval_list_invariant_cons.
    +    move ⇒ H_c1_neq_0 [H_gr] [H_inv_l2] H_c2_neq_0 H_y1_lt.
    +    apply subset_props_aux.
    +    âˆƒ y1.
    +    split. {
    +      rewrite InZ_cons.
    +      left.
    +      by apply In_elementsZ_single_hd.
    +    } {
    +      rewrite InZ_cons.
    +      suff : ¬ (List.In y1 (elementsZ_single y2 c2)) ∧ ¬ InZ y1 l2 by tauto.
    +      split. {
    +        rewrite In_elementsZ_single.
    +        move ⇒ [] /Z.le_ngt //.
    +      } {
    +        eapply Nin_elements_greater; eauto.
    +        apply Z.le_trans with (m := y2). {
    +          by apply Z.lt_le_incl.
    +        } {
    +          apply Z_le_add_r.
    +        }
    +      }
    +    }
    +  Qed.
    +

      Lemma subset_props : ∀ s1 s2 : t,
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    (subset s1 s2 = true ↔
    -     (∀ y, InZ y s1 → InZ y s2)).
    - +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    (subset s1 s2 = true ↔
    +     (∀ y, InZ y s1 → InZ y s2)).
    +  Proof.
    +    induction s1 as [| [y1 c1] l1 IH1]. {
    +      move ⇒ s2 _ _.
    +      rewrite subset_flatten_alt_def.
    +      split; done.
    +    } {
    +      induction s2 as [| [y2 c2] l2 IH2]. {
    +        rewrite interval_list_invariant_cons
    +                subset_flatten_alt_def.
    +        move ⇒ [_] [H_c1_neq_0] _ _.
    +        split ⇒ //.
    +        move ⇒ H; move : (H y1).
    +        rewrite InZ_nil ⇒ {H} H.
    +        contradict H.
    +        rewrite InZ_cons; left.
    +        by apply In_elementsZ_single_hd.
    +      } {
    +        move ⇒ H_inv_s1 H_inv_s2.
    +        move : (H_inv_s1) (H_inv_s2).
    +        rewrite !interval_list_invariant_cons.
    +        move ⇒ [H_gr_l1] [H_c1_neq_0] H_inv_l1.
    +        move ⇒ [H_gr_l2] [H_c2_neq_0] H_inv_l2.
    +        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {IH2} IH2.
    +        have : ∀ s2 : t,
    +          interval_list_invariant s2 = true →
    +          (subset l1 s2 = true ↔
    +          (∀ y : Z, InZ y l1 → InZ y s2)). {
    +          intros. by apply IH1.
    +        }
    +        move ⇒ {IH1} IH1.
    + +
    +        have H_yc2_nin : ¬ InZ (y2 + Z.of_N c2) ((y2, c2) :: l2). {
    +          rewrite !InZ_cons !In_elementsZ_single.
    +          move ⇒ []. {
    +            move ⇒ [_] /Z.lt_irrefl //.
    +          } {
    +            eapply Nin_elements_greater; eauto.
    +            apply Z.le_refl.
    +          }
    +        }
    +          
    +        rewrite subset_flatten_alt_def.
    +        move : (interval_compare_elim y1 c1 y2 c2).
    +        case (interval_compare (y1, c1) (y2, c2)). {
    +          move ⇒ H_lt_y2.
    +          apply subset_props_aux_before ⇒ //.
    +          apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +          apply Z_le_add_r.
    +        } {
    +          move ⇒ H_y2_eq.
    +          apply subset_props_aux_before ⇒ //.
    +          rewrite -H_y2_eq.
    +          by apply Z_lt_add_r.
    +        } {
    +          move ⇒ [H_y1_lt] _.
    +          apply subset_props_aux_before ⇒ //.
    +        } {
    +          move ⇒ [H_y2_lt] [H_y1_lt] H_yc2_lt.
    +          apply subset_props_aux.
    +          âˆƒ (y2 + Z.of_N c2).
    +          split ⇒ //.
    +          rewrite !InZ_cons !In_elementsZ_single.
    +          left.
    +          split ⇒ //.
    +          by apply Z.lt_le_incl.
    +        } {
    +          move ⇒ [H_y1_eq] H_c1_eq; subst.
    +          rewrite IH1 ⇒ //.
    +          split; move ⇒ H_pre y; move : (H_pre y) ⇒ {H_pre};
    +            rewrite !InZ_cons. {
    +            tauto.
    +          } {
    +            move ⇒ H_pre H_y_in_l1.
    +            suff : ~(List.In y (elementsZ_single y2 c2)). {
    +              tauto.
    +            }
    +            move : H_gr_l1.
    +            rewrite interval_list_elements_greater_alt2_def
    +              // In_elementsZ_single.
    +            move ⇒ H; move : (H y H_y_in_l1) ⇒ {H}.
    +            move ⇒ /Z.lt_ngt H_neq [_] //.
    +          }
    +        } {
    +          move ⇒ [H_y2_lt_y1] [H_yc1_le] _.
    +          rewrite IH1.
    +          split; move ⇒ H_pre y; move : (H_pre y) ⇒ {H_pre};
    +            rewrite !InZ_cons. {
    +            move ⇒ H []; last apply H. move ⇒ {H}.
    +            rewrite !In_elementsZ_single.
    +            move ⇒ [H_y1_le] H_y_lt.
    +            left.
    +            omega.
    +          } {
    +            move ⇒ H_pre H_y_in_l1.
    +            apply H_pre.
    +            by right.
    +          } {
    +            assumption.
    +          }
    +        } {
    +          move ⇒ [H_y1_le] [_] []. {
    +            apply subset_props_aux_before ⇒ //.
    +          } {
    +            move ⇒ H_yc2_lt.
    +            apply subset_props_aux.
    +            âˆƒ (y2 + Z.of_N c2).
    +            split ⇒ //.
    +            rewrite !InZ_cons !In_elementsZ_single.
    +            left.
    +            split ⇒ //.
    +            apply Z.le_trans with (m := y2) ⇒ //.
    +            apply Z_le_add_r.
    +          }
    +        } {
    +          move ⇒ H_yc2_lt_y1.
    +          rewrite IH2.
    +          split; move ⇒ H_pre y; move : (H_pre y) ⇒ {H_pre};
    +                                                        rewrite !InZ_cons. {
    +            tauto.
    +          } {
    +            rewrite !In_elementsZ_single.
    +            move ⇒ H_pre H_y_in.
    +            suff : ~(y2 ≤ y < y2 + Z.of_N c2). {
    +              tauto.
    +            }
    +            move ⇒ [H_y2_le H_y_lt].
    +            move : H_y_in ⇒ []. {
    +              move ⇒ [H_y1_le] H_y_lt'.
    +              omega.
    +            } {
    +              eapply Nin_elements_greater; eauto.
    +              apply Z.le_trans with (m := y1); last first. {
    +                apply Z_le_add_r.
    +              }
    +              apply Z.lt_le_incl.
    +              apply Z.lt_trans with (m := y2 + Z.of_N c2) ⇒ //.
    +            }
    +          }
    +        } {
    +          move ⇒ H_y1_eq.
    +          apply subset_props_aux.
    +          âˆƒ y1.
    +          rewrite !InZ_cons.
    +          split. {
    +            left.
    +            by apply In_elementsZ_single_hd.
    +          } {
    +            rewrite !In_elementsZ_single H_y1_eq.
    +            move ⇒ []. {
    +              move ⇒ [_] /Z.lt_irrefl //.
    +            } {
    +              eapply Nin_elements_greater; eauto.
    +              rewrite H_y1_eq.
    +              apply Z.le_refl.
    +            }
    +          }
    +        }
    +      }
    +    }
    +  Qed.
    +
      Lemma subset_spec :
       âˆ€ (s s' : t) (Hs : Ok s) (Hs' : Ok s'),
    -   subset s s' = true ↔ Subset s s'.
    - +   subset s s' = true ↔ Subset s s'.
    +  Proof.
    +    intros s s' Hs Hs'.
    +    move : (Hs) (Hs').
    +    rewrite /Ok /IsOk.
    +    move ⇒ [H_inv_s H_enc_s] [H_inv_s' H_enc_s'].
    +    rewrite (subset_props s s' H_inv_s H_inv_s').
    +    rewrite /Subset.
    +    split. {
    +      move ⇒ H_pre enc_y.
    +      rewrite !In_InZ.
    +      apply H_pre.
    +    } {
    +      move ⇒ H_pre y H_y_in.
    +      move : (H_enc_s _ H_y_in) ⇒ [e] H_e. subst.
    +      move : (H_pre e) H_y_in.
    +      rewrite !In_InZ //.
    +    }
    +  Qed.
    +
    @@ -1858,18 +4847,105 @@

    Library MSetsExtra.MSetIntervals


    -  Lemma elements_spec1 : ∀ (s : t) (x : elt) (Hs : Ok s), List.In x (elements s) ↔ In x s.
    +  Lemma elements_spec1 : ∀ (s : t) (x : elt) (Hs : Ok s), List.In x (elements s) ↔ In x s.
    +  Proof.
    +    intros s x Hs.
    +    by rewrite In_alt_def.
    +  Qed.

      Lemma NoDupA_elementsZ_single: ∀ c x,
    -    NoDupA Z.eq (elementsZ_single x c).
    - -
    -  Lemma elementsZ_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Z.eq (elementsZ s).
    - -
    -  Lemma elements_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Enc.E.eq (elements s).
    - +    NoDupA Z.eq (elementsZ_single x c).
    +  Proof.
    +    induction c as [| c' IH] using N.peano_ind. {
    +      intros x.
    +      rewrite elementsZ_single_base //.
    +    } {
    +      intros x.
    +      rewrite elementsZ_single_succ.
    +      apply NoDupA_app. {
    +        apply Z.eq_equiv.
    +      } {
    +        apply IH.
    +      } {
    +        apply NoDupA_singleton.
    +      } {
    +        move ⇒ y.
    +        rewrite !InA_alt.
    +        move ⇒ [_] [<-] H_y_in.
    +        move ⇒ [_] [<-] H_y_in'.
    +        move : H_y_in H_y_in'.
    +        rewrite In_elementsZ_single /=.
    +        move ⇒ [H_x_le] H_y_lt [] // H_y_eq.
    +        contradict H_y_lt.
    +        rewrite H_y_eq.
    +        apply Z.lt_irrefl.
    +      }
    +    }
    +  Qed.
    + +
    +  Lemma elementsZ_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Z.eq (elementsZ s).
    +  Proof.
    +    induction s as [| [x c] s' IH]. {
    +      move ⇒ _.
    +      rewrite elementsZ_nil.
    +      apply NoDupA_nil.
    +    } {
    +      move ⇒ H_ok_s.
    +      move : (H_ok_s) ⇒ /Ok_cons [H_interval_list_elements_greater] [H_c] [H_enc] H_s'.
    +      rewrite elementsZ_cons.
    +      apply NoDupA_app. {
    +        apply Z.eq_equiv.
    +      } {
    +        by apply IH.
    +      } {
    +        apply NoDupA_rev. {
    +          apply Z.eq_equiv.
    +        } {
    +          apply NoDupA_elementsZ_single.
    +        }
    +      } {
    +        move ⇒ y.
    +        rewrite !InA_alt.
    +        move ⇒ [_] [<-] H_y_in.
    +        move ⇒ [_] [<-] H_y_in'.
    +        move : H_y_in'.
    +        rewrite -in_rev In_elementsZ_single /=.
    +        move ⇒ [H_x_le] H_y_lt.
    +        eapply (Nin_elements_greater s' (x + Z.of_N c)) ⇒ //. {
    +          apply H_s'.
    +        } {
    +          apply Z.lt_le_incl, H_y_lt.
    +        } {
    +          apply H_y_in.
    +        }
    +      }
    +    }
    +  Qed.
    + +
    +  Lemma elements_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Enc.E.eq (elements s).
    +  Proof.
    +    intros s Hs.
    +    rewrite /elements rev_map_alt_def.
    +    apply NoDupA_rev. {
    +      apply Enc.E.eq_equiv.
    +    } {
    +      eapply NoDupA_map; first by apply elementsZ_spec2w.
    + +
    +      intros x1 x2 H_x1_in H_x2_in H_dec_eq.
    +      have H_is_enc: is_encoded_elems_list (elementsZ s). {
    +        apply Hs.
    +      }
    +      move : (H_is_enc _ H_x1_in) ⇒ [y1 H_x1_eq].
    +      move : (H_is_enc _ H_x2_in) ⇒ [y2 H_x2_eq].
    +      move : H_dec_eq.
    +      rewrite -H_x1_eq -H_x2_eq !Enc.decode_encode_ok Enc.encode_eq //.
    +    }
    +  Qed.
    +
    @@ -1881,18 +4957,94 @@

    Library MSetsExtra.MSetIntervals


      Lemma equal_alt_def : ∀ s1 s2,
    -    equal s1 s2 = true ↔ (s1 = s2).
    - +    equal s1 s2 = true ↔ (s1 = s2).
    +  Proof.
    +    induction s1 as [| [x cx] xs IH]. {
    +      move ⇒ [] //.
    +    } {
    +      move ⇒ [] //=.
    +      move ⇒ [y cy] ys.
    +      rewrite !andb_true_iff IH N.eqb_eq Z.eqb_eq.
    +      split. {
    +        move ⇒ [->] [->] → //.
    +      } {
    +        move ⇒ [->] → → //.
    +      }
    +    }
    +  Qed.
    +
      Lemma equal_elementsZ :
        âˆ€ (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
    -    (∀ x, (InZ x s ↔ InZ x s')) → (s = s').
    +    (∀ x, (InZ x s ↔ InZ x s')) → (s = s').
    +  Proof.
    +    intros s s'.
    +    move ⇒ H_ok_s H_ok_s' H_InZ_eq.
    +    have [] : ((subset s s' = true) ∧ (subset s' s = true)). {
    +      rewrite !subset_spec /Subset.
    +      split ⇒ x; rewrite !In_InZ H_InZ_eq //.
    +    }
    +    have : interval_list_invariant s' = true by apply H_ok_s'.
    +    have : interval_list_invariant s = true by apply H_ok_s.
    +    clear H_ok_s H_ok_s' H_InZ_eq.
    +    move : s s'.
    +    induction s as [| [x1 c1] s1 IH];
    +      case s' as [| [x2 c2] s2] ⇒ //.
    +    rewrite !interval_list_invariant_cons.
    +    move ⇒ [H_gr_s1] [H_c1_neq_0] H_inv_s1.
    +    move ⇒ [H_gr_s2] [H_c2_neq_0] H_inv_s2.
    +    rewrite subset_flatten_alt_def
    +            (subset_flatten_alt_def ((x2, c2)::s2)).
    +    rewrite (interval_compare_swap x1 c1); last by left.
    +    move : (interval_compare_elim x1 c1 x2 c2).
    +    case (interval_compare (x1, c1) (x2, c2)) ⇒ //.
    +    move ⇒ [->] → H_sub_s12 H_sub_s21.
    + +
    +    suff → : s1 = s2 by done.
    +    by apply IH.
    +  Qed.

      Lemma equal_spec :
        âˆ€ (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
    -    equal s s' = true ↔ Equal s s'.
    - +    equal s s' = true ↔ Equal s s'.
    +  Proof.
    +    intros s s' Hs Hs'.
    +    rewrite equal_alt_def /Equal.
    +    split. {
    +      move ⇒ → //.
    +    } {
    +      move ⇒ H.
    +      apply equal_elementsZ ⇒ // x.
    +      move : (H (Enc.decode x)).
    +      rewrite !In_InZ.
    + +
    +      suff H_ex : (∀ s'', Ok s'' → InZ x s'' → (∃ z, Enc.encode z = x)). {
    +        move ⇒ HH.
    +        split. {
    +          move ⇒ H3.
    +          move : HH (H3).
    +          move : (H_ex s Hs H3) ⇒ [z] <-.
    +          rewrite Enc.decode_encode_ok ⇒ <- //.
    +        } {
    +          move ⇒ H3.
    +          move : HH (H3).
    +          move : (H_ex s' Hs' H3) ⇒ [z] <-.
    +          rewrite Enc.decode_encode_ok ⇒ <- //.
    +        }
    +      }
    +      clear.
    +      intros s'' H_ok H_in_x.
    +      have H_enc : is_encoded_elems_list (elementsZ s''). {
    +        apply H_ok.
    +      }
    +      apply H_enc.
    +      apply H_in_x.
    +    }
    +  Qed.
    +
    @@ -1901,42 +5053,203 @@

    Library MSetsExtra.MSetIntervals

    -  Definition lt (s1 s2 : t) : Prop := (compare s1 s2 = Lt).
    +  Definition lt (s1 s2 : t) : Prop := (compare s1 s2 = Lt).

      Lemma compare_eq_Eq : ∀ s1 s2,
    -    (compare s1 s2 = Eq ↔ equal s1 s2 = true).
    - +    (compare s1 s2 = Eq ↔ equal s1 s2 = true).
    +  Proof.
    +    induction s1 as [| [y1 c1] s1' IH];
    +      case s2 as [| [y2 c2] s2'] ⇒ //.
    +    rewrite /= !andb_true_iff -IH Z.eqb_eq N.eqb_eq.
    +    move : (Z.compare_eq_iff y1 y2).
    +    case (Z.compare y1 y2). {
    +      move ⇒ H.
    +      have → : y1 = y2. by apply H.
    +      clear H.
    + +
    +      move : (N.compare_eq_iff c1 c2).
    +      case (N.compare c1 c2). {
    +        move ⇒ H.
    +        have → : c1 = c2. by apply H.
    +        tauto.
    +      } {
    +        move ⇒ H.
    +        have H_neq : ~(c1 = c2). by rewrite -H ⇒ {H}.
    +        tauto.
    +      } {
    +        move ⇒ H.
    +        have H_neq : ~(c1 = c2). by rewrite -H ⇒ {H}.
    +        tauto.
    +      }
    +    } {
    +      move ⇒ H.
    +      have H_neq : ~(y1 = y2). by rewrite -H ⇒ {H}.
    +      tauto.
    +    } {
    +      move ⇒ H.
    +      have H_neq : ~(y1 = y2). by rewrite -H ⇒ {H}.
    +      tauto.
    +    }
    +  Qed.
    +
      Lemma compare_eq_Lt_nil_l : ∀ s,
    -    compare nil s = Lt ↔ s ≠ nil.
    +    compare nil s = Lt ↔ s ≠ nil.
    +  Proof.
    +    intros s.
    +    case s ⇒ //=.
    +  Qed.

      Lemma compare_eq_Lt_nil_r : ∀ s,
    -    ~(compare s nil = Lt).
    +    ~(compare s nil = Lt).
    +  Proof.
    +    intros s.
    +    case s as [| [y1 c1] s'] ⇒ //=.
    +  Qed.

      Lemma compare_eq_Lt_cons : ∀ y1 y2 c1 c2 s1 s2,
    -    compare ((y1, c1)::s1) ((y2, c2)::s2) = Lt ↔
    -    (y1 < y2) ∨ ((y1 = y2) ∧ (c1 < c2)%N) ∨
    -    ((y1 = y2) ∧ (c1 = c2) ∧ compare s1 s2 = Lt).
    - +    compare ((y1, c1)::s1) ((y2, c2)::s2) = Lt ↔
    +    (y1 < y2) ∨ ((y1 = y2) ∧ (c1 < c2)%N) ∨
    +    ((y1 = y2) ∧ (c1 = c2) ∧ compare s1 s2 = Lt).
    +  Proof.
    +    intros y1 y2 c1 c2 s1 s2.
    +    rewrite /=.
    +    case_eq (Z.compare y1 y2). {
    +      move ⇒ /Z.compare_eq_iff →.
    +      case_eq (N.compare c1 c2). {
    +        move ⇒ /N.compare_eq_iff →.
    +        split. {
    +          move ⇒ H.
    +          right; right.
    +          done.
    +        } {
    +          move ⇒ [| []]. {
    +            move ⇒ /Z.lt_irrefl //.
    +          } {
    +            move ⇒ [_] /N.lt_irrefl //.
    +          } {
    +            move ⇒ [_] [_] → //.
    +          }
    +        }
    +      } {
    +        move ⇒ /N.compare_lt_iff H_c1_lt.
    +        split ⇒ //.
    +        move ⇒ _.
    +        right; left. done.
    +      } {
    +        move ⇒ /N.compare_gt_iff H_c2_lt.
    +        split ⇒ //.
    +        move ⇒ [| []]. {
    +          move ⇒ /Z.lt_irrefl //.
    +        } {
    +          move ⇒ [_] /N.lt_asymm //.
    +        } {
    +          move ⇒ [_] [] H_c1_eq.
    +          contradict H_c2_lt.
    +          subst c1.
    +          by apply N.lt_irrefl.
    +        }
    +      }
    +    } {
    +      move ⇒ /Z.compare_lt_iff.
    +      tauto.
    +    } {
    +      move ⇒ /Z.compare_gt_iff H_y2_lt.
    +      split ⇒ //.
    +      move ⇒ [| []]. {
    +        move ⇒ /Z.lt_asymm //.
    +      } {
    +        move ⇒ [] H_y1_eq.
    +        exfalso. omega.
    +      } {
    +        move ⇒ [] H_y1_eq.
    +        exfalso. omega.
    +      }
    +    }
    +  Qed.
    +

      Lemma compare_antisym: ∀ (s1 s2 : t),
    -    (compare s1 s2) = CompOpp (compare s2 s1).
    +    (compare s1 s2) = CompOpp (compare s2 s1).
    +  Proof.
    +    induction s1 as [| [y1 c1] s1' IH];
    +      case s2 as [| [y2 c2] s2'] ⇒ //.
    +    rewrite /= (Z.compare_antisym y1 y2) (N.compare_antisym c1 c2).
    +    case (Z.compare y1 y2) ⇒ //=.
    +    case (N.compare c1 c2) ⇒ //=.
    +  Qed.

      Lemma compare_spec : ∀ s1 s2,
    -    CompSpec eq lt s1 s2 (compare s1 s2).
    - -
    -  Lemma lt_Irreflexive : Irreflexive lt.
    +    CompSpec eq lt s1 s2 (compare s1 s2).
    +  Proof.
    +    intros s1 s2.
    +    rewrite /CompSpec /lt (compare_antisym s2 s1).
    +    case_eq (compare s1 s2). {
    +      rewrite compare_eq_Eq equal_alt_def ⇒ →.
    +      by apply CompEq.
    +    } {
    +      move ⇒ _.
    +      by apply CompLt.
    +    } {
    +      move ⇒ _.
    +      by apply CompGt.
    +    }
    +  Qed.
    + +
    +  Lemma lt_Irreflexive : Irreflexive lt.
    +  Proof.
    +    rewrite /Irreflexive /Reflexive /complement /lt.
    +    intros x.
    +    suff → : compare x x = Eq by done.
    +    rewrite compare_eq_Eq equal_alt_def //.
    +  Qed.
    + +
    +  Lemma lt_Transitive : Transitive lt.
    +  Proof.
    +    rewrite /Transitive /lt.
    +    induction x as [| [y1 c1] s1' IH];
    +      case y as [| [y2 c2] s2'];
    +      case z as [| [y3 c3] s3'] ⇒ //.
    + +
    +    rewrite !compare_eq_Lt_cons.
    +    move ⇒ [H_y1_lt | [[->] H_c1_lt | [->] [->] H_comp]]
    + [H_y2_lt | [[<-] H_c2_lt | [<-] [<-] H_comp']]. {
    +      left.
    +      by apply Z.lt_trans with (m := y2).
    +    } {
    +      by left.
    +    } {
    +      by left.
    +    } {
    +      by left.
    +    } {
    +      right; left.
    +      split ⇒ //.
    +      by apply N.lt_trans with (m := c2).
    +    } {
    +      by right; left.
    +    } {
    +      by left.
    +    } {
    +      by right; left.
    +    } {
    +      right; right.
    +      split ⇒ //.
    +      split ⇒ //.
    +      by apply (IH s2').
    +    }
    +  Qed.
    -
    -  Lemma lt_Transitive : Transitive lt.
    -
    @@ -1948,20 +5261,112 @@

    Library MSetsExtra.MSetIntervals


      Lemma elementsZ_single_sorted : ∀ c x,
    -    sort Z.lt (elementsZ_single x c).
    - +    sort Z.lt (elementsZ_single x c).
    +  Proof.
    +    induction c as [| c' IH] using N.peano_ind. {
    +      intro x.
    +      rewrite elementsZ_single_base.
    +      apply Sorted_nil.
    +    } {
    +      intro x.
    +      rewrite elementsZ_single_succ_front.
    +      apply Sorted_cons. {
    +        apply IH.
    +      } {
    +        case (N.zero_or_succ c'). {
    +          move ⇒ →.
    +          rewrite elementsZ_single_base //.
    +        } {
    +          move ⇒ [c''] →.
    +          rewrite elementsZ_single_succ_front.
    +          constructor.
    +          apply Z.lt_succ_diag_r.
    +        }
    +      }
    +    }
    +  Qed.
    +
      Lemma elementsZ_sorted : ∀ s,
    -    interval_list_invariant s = true →
    -    sort Z.lt (rev (elementsZ s)).
    - +    interval_list_invariant s = true →
    +    sort Z.lt (rev (elementsZ s)).
    +  Proof.
    +    induction s as [| [y c] s' IH]. {
    +      move ⇒ _.
    +      rewrite elementsZ_nil.
    +      apply Sorted_nil.
    +    } {
    +      rewrite interval_list_invariant_cons elementsZ_cons
    +              rev_app_distr rev_involutive.
    +      move ⇒ [H_gr] [H_c_neq_0] H_inv_s'.
    +      apply SortA_app with (eqA := Logic.eq). {
    +        apply eq_equivalence.
    +      } {
    +        apply elementsZ_single_sorted.
    +      } {
    +        by apply IH.
    +      } {
    +        intros x1 x2.
    +        move ⇒ /InA_alt [_] [<-] /In_elementsZ_single [_ H_x1_lt].
    +        move ⇒ /InA_alt [_] [<-].
    +        rewrite -In_rev ⇒ H_x2_in.
    + +
    +        apply Z.lt_trans with (m := (y + Z.of_N c)) ⇒ //.
    +        eapply interval_list_elements_greater_alt2_def;
    +          eauto.
    +      }
    +    }
    +  Qed.
    +

      Lemma elements_sorted : ∀ s,
    -    Ok s →
    -    sort Enc.E.lt (elements s).
    - +    Ok s →
    +    sort Enc.E.lt (elements s).
    +  Proof.
    +    move ⇒ s [H_inv] H_enc.
    +    rewrite /elements rev_map_alt_def -map_rev.
    +    have : (∀ x : Z, List.In x (rev (elementsZ s)) →
    +           âˆƒ e : Enc.E.t, Enc.encode e = x). {
    +      move ⇒ x.
    +      move : (H_enc x).
    +      rewrite In_rev //.
    +    }
    +    move : (elementsZ_sorted s H_inv) ⇒ {H_enc}.
    +    generalize (rev (elementsZ s)).
    +    induction l as [| x xs IH]. {
    +      rewrite /= ⇒ _ _.
    +      apply Sorted_nil.
    +    } {
    +      move ⇒ H_sort H_enc.
    +      apply Sorted_inv in H_sort as [H_sort H_hd_rel].
    +      simpl.
    +      apply Sorted_cons. {
    +        apply IH ⇒ //.
    +        move ⇒ xx H_xx_in.
    +        apply H_enc.
    +        by apply in_cons.
    +      } {
    +        move : H_hd_rel H_enc.
    +        case xs ⇒ //=.
    +        move ⇒ x' xs' H_hd_rel H_enc.
    +        apply HdRel_inv in H_hd_rel.
    +        apply HdRel_cons.
    +        rewrite -Enc.encode_lt.
    +        have [y H_y] : (∃ y, Enc.encode y = x). {
    +          apply H_enc. by left.
    +        }
    +        have [y' H_y'] : (∃ y', Enc.encode y' = x'). {
    +          apply H_enc. by right; left.
    +        }
    +        move : H_hd_rel.
    +        rewrite -!H_y -!H_y' !Enc.decode_encode_ok //.
    +      }
    +    }
    +  Qed.
    +

    @@ -1975,79 +5380,254 @@

    Library MSetsExtra.MSetIntervals


      Definition min_eltZ_spec1 :
    -    âˆ€ (s : t) (x : Z),
    -      interval_list_invariant s = true →
    -      min_eltZ s = Some x → InZ x s.
    - +    âˆ€ (s : t) (x : Z),
    +      interval_list_invariant s = true →
    +      min_eltZ s = Some x → InZ x s.
    +  Proof.
    +    intros s x.
    +    case s as [| [x' c] s']. {
    +      rewrite /min_eltZ //.
    +    } {
    +      rewrite /min_eltZ InZ_cons interval_list_invariant_cons.
    +      move ⇒ [_] [H_c_neq] _ [->].
    +      left.
    +      by apply In_elementsZ_single_hd.
    +    }
    +  Qed.
    +
      Lemma min_eltZ_spec2 :
    -    âˆ€ (s : t) (x y : Z) (Hs : Ok s),
    -    min_eltZ s = Some x → InZ y s → ¬ Z.lt y x.
    - +    âˆ€ (s : t) (x y : Z) (Hs : Ok s),
    +    min_eltZ s = Some x → InZ y s → ¬ Z.lt y x.
    +  Proof.
    +    intros s x y H_ok H_min H_in H_y_lt_x.
    +    eapply (Nin_elements_greater s (Z.pred x)) ⇒ //; last apply H_in. {
    +      move : H_ok H_min.
    +      case s ⇒ //.
    +      move ⇒ [z c] s' _ [<-].
    +      rewrite interval_list_elements_greater_cons.
    +      apply Z.lt_pred_l.
    +    } {
    +      apply H_ok.
    +    } {
    +      by apply Z.lt_le_pred.
    +    }
    +  Qed.
    +
      Definition min_eltZ_spec3 :
        âˆ€ (s : t),
    -      min_eltZ s = None → ∀ x, ¬InZ x s.
    - +      min_eltZ s = None → ∀ x, ¬InZ x s.
    +  Proof.
    +    intros s.
    +    case s as [| [x' c] s'];
    +      rewrite /min_eltZ //.
    +    move ⇒ _ x //.
    +  Qed.
    +
      Definition min_elt_spec1 :
    -    âˆ€ (s : t) (x : elt) (Hs : Ok s), min_elt s = Some x → In x s.
    - +    âˆ€ (s : t) (x : elt) (Hs : Ok s), min_elt s = Some x → In x s.
    +  Proof.
    +    rewrite /min_elt.
    +    move ⇒ s x H_ok.
    +    case_eq (min_eltZ s) ⇒ //.
    +    move ⇒ z H_min_elt [<-].
    +    apply InZ_In ⇒ //.
    +    apply min_eltZ_spec1 ⇒ //.
    +    apply H_ok.
    +  Qed.
    +
      Definition min_elt_spec2 :
    -    âˆ€ (s : t) (x y : elt) (Hs : Ok s), min_elt s = Some x → In y s → ~(Enc.E.lt y x).
    - +    âˆ€ (s : t) (x y : elt) (Hs : Ok s), min_elt s = Some x → In y s → ~(Enc.E.lt y x).
    +  Proof.
    +    rewrite /min_elt.
    +    move ⇒ s x y H_ok.
    +    case_eq (min_eltZ s) ⇒ //.
    +    move ⇒ z H_min_elt [<-].
    +    rewrite In_InZ ⇒ H_inZ.
    +    have H_y_eq : y = Enc.decode (Enc.encode y). {
    +      by rewrite Enc.decode_encode_ok.
    +    }
    +    rewrite H_y_eq -Enc.encode_lt.
    +    apply (min_eltZ_spec2 _ _ _ H_ok); last first. {
    +      by rewrite Enc.decode_encode_ok.
    +    }
    +    suff → : Enc.encode (Enc.decode z) = z by assumption.
    +    apply encode_decode_eq with (s := s) ⇒ //.
    +    apply min_eltZ_spec1 ⇒ //.
    +    apply H_ok.
    +  Qed.
    +
      Definition min_elt_spec3 :
    -    âˆ€ s : t, min_elt s = None → Empty s.
    - +    âˆ€ s : t, min_elt s = None → Empty s.
    +  Proof.
    +    rewrite /min_elt /min_eltZ /Empty /In.
    +    case s as [| [x' c] s'] ⇒ //.
    +    move ⇒ _ e.
    +    rewrite elements_nil InA_nil //.
    +  Qed.
    +
      Definition choose_spec1 :
    -    âˆ€ (s : t) (x : elt) (Hs : Ok s), choose s = Some x → In x s.
    - +    âˆ€ (s : t) (x : elt) (Hs : Ok s), choose s = Some x → In x s.
    +  Proof.
    +    rewrite /choose.
    +    apply min_elt_spec1.
    +  Qed.
    +
      Definition choose_spec2 :
    -    âˆ€ s : t, choose s = None → Empty s.
    - +    âˆ€ s : t, choose s = None → Empty s.
    +  Proof.
    +    rewrite /choose.
    +    apply min_elt_spec3.
    +  Qed.
    +
    -  Lemma choose_spec3: ∀ s s' x x', Ok s → Ok s' →
    -    choose s = Some x → choose s' = Some x' → Equal s s' → x = x'.
    +  Lemma choose_spec3: ∀ s s' x x', Ok s → Ok s' →
    +    choose s = Some x → choose s' = Some x' → Equal s s' → x = x'.
    +  Proof.
    +    intros s s' x x' Hs Hs' Hx Hx'.
    +    rewrite -equal_spec equal_alt_def ⇒ H_s_eq.
    +    move : Hx Hx'.
    +    rewrite H_s_eq ⇒ → [] //.
    +  Qed.


      Definition max_eltZ_spec1 :
    -    âˆ€ (s : t) (x : Z),
    -      interval_list_invariant s = true →
    -      max_eltZ s = Some x → InZ x s.
    - +    âˆ€ (s : t) (x : Z),
    +      interval_list_invariant s = true →
    +      max_eltZ s = Some x → InZ x s.
    +  Proof.
    +    intros s x.
    +    induction s as [| [x' c] s' IH]. {
    +      rewrite /max_eltZ //.
    +    } {
    +      rewrite InZ_cons interval_list_invariant_cons /=.
    +      move ⇒ [_] [H_c_neq].
    +      case s' as [| [y' c'] s'']. {
    +        move ⇒ _ [<-].
    +        left. {
    +          rewrite In_elementsZ_single.
    +          split. {
    +            rewrite -Z.lt_le_pred.
    +            by apply Z_lt_add_r.
    +          } {
    +            apply Z.lt_pred_l.
    +          }
    +        }
    +      } {
    +        move ⇒ H_inv H_max_eq.
    +        right.
    +        by apply IH.
    +      }
    +    }
    +  Qed.
    +
      Lemma max_eltZ_spec2 :
    -    âˆ€ (s : t) (x y : Z),
    -    interval_list_invariant s = true →
    -    max_eltZ s = Some x → InZ y s → ¬ Z.lt x y.
    - +    âˆ€ (s : t) (x y : Z),
    +    interval_list_invariant s = true →
    +    max_eltZ s = Some x → InZ y s → ¬ Z.lt x y.
    +  Proof.
    +    induction s as [| [y c] s' IH]. {
    +      done.
    +    } {
    +      move ⇒ x x'.
    +      rewrite interval_list_invariant_cons.
    +      move ⇒ [H_gr] [H_c_neq_0] H_inv_s'.
    +      have H_gr' : (∀ xx : Z, InZ xx (s') → y + Z.of_N c < xx). {
    +        apply interval_list_elements_greater_alt2_def ⇒ //.
    +      }
    +
    +      case s' as [| [y' c'] s'']. {
    +        move ⇒ [<-].
    +        rewrite InZ_cons InZ_nil In_elementsZ_single.
    +        omega.
    +      } {
    +        move ⇒ H_max_eq.
    +        rewrite InZ_cons.
    +        move ⇒ []; last by apply IH.
    +        rewrite In_elementsZ_single.
    +        move ⇒ [_] H_x'_lt H_lt_x'.
    + +
    +        have H_x_in : InZ x ((y', c')::s''). {
    +          by apply max_eltZ_spec1.
    +        }
    +        
    +        move : (H_gr' _ H_x_in).
    +        apply Z.nlt_ge, Z.lt_le_incl.
    +        by apply Z.lt_trans with (m := x').
    +      }
    +    }
    +  Qed.
    +
      Lemma max_eltZ_eq_None :
        âˆ€ (s : t),
    -      max_eltZ s = None → s = nil.
    +      max_eltZ s = None → s = nil.
    +  Proof.
    +    induction s as [| [x' c] s' IH] ⇒ //.
    +    move : IH.
    +    case s' as [| [y' c'] s''] ⇒ //=.
    +    move ⇒ H H_pre.
    +    move : (H H_pre) ⇒ //.
    +  Qed.

      Definition max_eltZ_spec3 :
        âˆ€ (s : t),
    -      max_eltZ s = None → ∀ x, ¬InZ x s.
    +      max_eltZ s = None → ∀ x, ¬InZ x s.
    +  Proof.
    +    move ⇒ s /max_eltZ_eq_None → x /InZ_nil //.
    +  Qed.

      Definition max_elt_spec1 :
    -    âˆ€ (s : t) (x : elt) (Hs : Ok s), max_elt s = Some x → In x s.
    - +    âˆ€ (s : t) (x : elt) (Hs : Ok s), max_elt s = Some x → In x s.
    +  Proof.
    +    rewrite /max_elt.
    +    move ⇒ s x H_ok.
    +    case_eq (max_eltZ s) ⇒ //.
    +    move ⇒ z H_max_elt [<-].
    +    apply InZ_In ⇒ //.
    +    apply max_eltZ_spec1 ⇒ //.
    +    apply H_ok.
    +  Qed.
    +
      Definition max_elt_spec2 :
    -    âˆ€ (s : t) (x y : elt) (Hs : Ok s), max_elt s = Some x → In y s → ~(Enc.E.lt x y).
    - +    âˆ€ (s : t) (x y : elt) (Hs : Ok s), max_elt s = Some x → In y s → ~(Enc.E.lt x y).
    +  Proof.
    +    rewrite /max_elt.
    +    move ⇒ s x y H_ok.
    +    move : (H_ok) ⇒ [H_inv] _.
    +    case_eq (max_eltZ s) ⇒ //.
    +    move ⇒ z H_max_elt [<-].
    +    rewrite In_InZ ⇒ H_inZ.
    +    rewrite -Enc.encode_lt.
    +    apply (max_eltZ_spec2 _ _ _ H_inv) ⇒ //.
    +    suff → : Enc.encode (Enc.decode z) = z ⇒ //.
    +    apply encode_decode_eq with (s := s) ⇒ //.
    +    apply max_eltZ_spec1 ⇒ //.
    +  Qed.
    +
      Definition max_elt_spec3 :
    -    âˆ€ s : t, max_elt s = None → Empty s.
    +    âˆ€ s : t, max_elt s = None → Empty s.
    +  Proof.
    +    intro s.
    +    rewrite /max_elt /Empty.
    +    case_eq (max_eltZ s) ⇒ //.
    +    move ⇒ /max_eltZ_eq_None → _ x.
    +    rewrite /In elements_nil InA_nil //.
    +  Qed.

    @@ -2060,9 +5640,22 @@

    Library MSetsExtra.MSetIntervals


      Lemma fold_spec :
    -   âˆ€ (s : t) (A : Type) (i : A) (f : elt → A → A),
    -   fold f s i = fold_left (flip f) (elements s) i.
    - +   âˆ€ (s : t) (A : Type) (i : A) (f : elt → A → A),
    +   fold f s i = fold_left (flip f) (elements s) i.
    +  Proof.
    +    intros s A i f.
    +    rewrite /fold fold_elementsZ_alt_def /elements
    +            rev_map_alt_def -map_rev.
    +    move : i.
    +    generalize (rev (elementsZ s)).
    +    induction l as [| x xs IH]. {
    +      done.
    +    } {
    +      move ⇒ i.
    +      rewrite /= IH //.
    +    }
    +  Qed.
    +

    @@ -2075,13 +5668,30 @@

    Library MSetsExtra.MSetIntervals


    -  Lemma cardinalN_spec : ∀ (s : t) (c : N),
    -    cardinalN c s = (c + N.of_nat (length (elements s)))%N.
    - +  Lemma cardinalN_spec : ∀ (s : t) (c : N),
    +    cardinalN c s = (c + N.of_nat (length (elements s)))%N.
    +  Proof.
    +    induction s as [| [x cx] xs IH]. {
    +      intros c.
    +      rewrite elements_nil /= N.add_0_r //.
    +    } {
    +      intros c.
    +      rewrite /= IH.
    +      rewrite /elements !rev_map_alt_def !rev_length !map_length.
    +      rewrite elementsZ_cons app_length Nat2N.inj_add rev_length.
    +      rewrite length_elementsZ_single N2Nat.id.
    +      ring.
    +    }
    +  Qed.
    +
      Lemma cardinal_spec :
       âˆ€ (s : t),
    -   cardinal s = length (elements s).
    +   cardinal s = length (elements s).
    +  Proof.
    +    intros s.
    +    rewrite /cardinal cardinalN_spec N.add_0_l Nat2N.id //.
    +  Qed.

    @@ -2094,10 +5704,50 @@

    Library MSetsExtra.MSetIntervals


      Lemma for_all_spec :
    -   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    -   Proper (Enc.E.eq==>eq) f →
    -   (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    - +   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    +   Proper (Enc.E.eq==>eq) f →
    +   (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +  Proof.
    +    intros s f Hs H.
    +    rewrite /for_all /For_all /In fold_elementsZ_alt_def
    +            /elements rev_map_alt_def -map_rev.
    +    generalize (rev (elementsZ s)).
    +    induction l as [| x xs IH]. {
    +      split ⇒ // _ x /= /InA_nil //.
    +    } {
    +      rewrite /=.
    +      case_eq (f (Enc.decode x)) ⇒ H_f_eq. {
    +        rewrite IH.
    +        split. {
    +          move ⇒ HH x' /InA_cons []. {
    +            by move ⇒ →.
    +          } {
    +            apply HH.
    +          }
    +        } {
    +          move ⇒ HH x' H_in.
    +          apply HH.
    +          apply InA_cons.
    +          by right.
    +        }
    +      } {
    +        split; move ⇒ HH. {
    +          contradict HH.
    +          case xs ⇒ //.
    +        } {
    +          exfalso.
    +          have H_in: (InA Enc.E.eq (Enc.decode x) (Enc.decode x :: map Enc.decode xs)). {
    +            apply InA_cons.
    +            left.
    +            apply Enc.E.eq_equiv.
    +          }
    +          move : (HH _ H_in).
    +          rewrite H_f_eq ⇒ //.
    +        }
    +      }
    +    }
    +  Qed.
    +
    @@ -2109,10 +5759,49 @@

    Library MSetsExtra.MSetIntervals


      Lemma exists_spec :
    -   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    -   Proper (Enc.E.eq==>eq) f →
    -   (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    - +   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    +   Proper (Enc.E.eq==>eq) f →
    +   (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +  Proof.
    +    intros s f Hs H.
    +    rewrite /exists_ /Exists /In fold_elementsZ_alt_def
    +            /elements rev_map_alt_def -map_rev.
    +    generalize (rev (elementsZ s)).
    +    induction l as [| x xs IH]. {
    +      split ⇒ //.
    +      move ⇒ [x] /= [] /InA_nil //.
    +    } {
    +      rewrite /=.
    +      case_eq (f (Enc.decode x)) ⇒ H_f_eq. {
    +        split ⇒ _. {
    +          âˆƒ (Enc.decode x).
    +          split ⇒ //.
    +          apply InA_cons.
    +          left.
    +          apply Enc.E.eq_equiv.
    +        } {
    +          case xs ⇒ //.
    +        }
    +      } {
    +        rewrite IH.
    +        split. {
    +          move ⇒ [x0] [H_in] H_f_x0.
    +          âˆƒ x0.
    +          split ⇒ //.
    +          apply InA_cons.
    +          by right.
    +        } {
    +          move ⇒ [x0] [] /InA_cons H_in H_f_x0.
    +          âˆƒ x0.
    +          split ⇒ //.
    +          move : H_in ⇒ [] // H_in.
    +          contradict H_f_x0.
    +          rewrite H_in H_f_eq //.
    +        }
    +      }
    +    }
    +  Qed.
    +
    @@ -2123,110 +5812,411 @@

    Library MSetsExtra.MSetIntervals


    -  Definition partitionZ_aux_invariant (x : Z) acc c :=
    -    interval_list_invariant (List.rev (partitionZ_fold_skip acc c)) = true ∧
    +  Definition partitionZ_aux_invariant (x : Z) acc c :=
    +    interval_list_invariant (List.rev (partitionZ_fold_skip acc c)) = true ∧
        match c with
    -      None ⇒ (∀ y', InZ y' acc → Z.succ y' < x)
    -    | Some (y, c') ⇒ (x = y + Z.of_N c')
    +      None ⇒ (∀ y', InZ y' acc → Z.succ y' < x)
    +    | Some (y, c') ⇒ (x = y + Z.of_N c')
        end.

      Lemma partitionZ_aux_invariant_insert : ∀ x acc c,
    -    partitionZ_aux_invariant x acc c →
    -    partitionZ_aux_invariant (Z.succ x) acc
    -      (Some (partitionZ_fold_insert c x)).
    - +    partitionZ_aux_invariant x acc c →
    +    partitionZ_aux_invariant (Z.succ x) acc
    +      (Some (partitionZ_fold_insert c x)).
    +  Proof.
    +    intros x acc c.
    +    rewrite /partitionZ_fold_insert /partitionZ_aux_invariant
    +            /partitionZ_fold_skip.
    +    case c; last first. {
    +      move ⇒ [H_inv] H_in.
    +      rewrite /= interval_list_invariant_app_iff Z.add_1_r.
    +      split; last done.
    +      split; first done.
    +      split; first done.
    +      move ⇒ x1 x2.
    +      rewrite InZ_rev InZ_cons InZ_nil In_elementsZ_single1.
    +      move ⇒ H_x1_in [] // <-.
    +      by apply H_in.
    +    } {
    +      move ⇒ [y c'].
    +      rewrite /= !interval_list_invariant_app_iff
    +        N2Z.inj_succ Z.add_succ_r .
    +      rewrite !interval_list_invariant_cons !interval_list_invariant_nil.
    +      move ⇒ [] [H_inv_acc] [] [] _ [H_c_neq_0] _
    +        H_in_c →.
    +      split; last done.
    +      split; first done.
    +      split. {
    +        split; first done.
    +        split; last done.
    +        apply N.neq_succ_0.
    +      } {
    +        move ⇒ x1 x2.
    +        rewrite InZ_cons InZ_nil In_elementsZ_single.
    +        move ⇒ H_x1_in [] // [H_y_le] H_x2_lt.
    +        apply Z.lt_le_trans with (m := y) ⇒ //.
    +        apply H_in_c ⇒ //.
    +        rewrite InZ_cons In_elementsZ_single.
    +        left.
    +        split. {
    +          apply Z.le_refl.
    +        } {
    +          by apply Z_lt_add_r.
    +        }
    +      }
    +    }
    +  Qed.
    +
      Lemma partitionZ_aux_invariant_skip : ∀ x acc c,
    -    partitionZ_aux_invariant x acc c →
    -    partitionZ_aux_invariant (Z.succ x) (partitionZ_fold_skip acc c) None.
    - -
    -  Definition partitionZ_fold_current (c : option (Z × N)) :=
    +    partitionZ_aux_invariant x acc c →
    +    partitionZ_aux_invariant (Z.succ x) (partitionZ_fold_skip acc c) None.
    +  Proof.
    +    intros x acc c.
    +    rewrite /partitionZ_fold_skip /partitionZ_aux_invariant
    +            /partitionZ_fold_skip.
    +    case c; last first. {
    +      move ⇒ [H_inv] H_in.
    +      split; first done.
    +      move ⇒ y' H_y'_in.
    +      apply Z.lt_trans with (m := x). {
    +        by apply H_in.
    +      } {
    +        apply Z.lt_succ_diag_r.
    +      }
    +    } {
    +      move ⇒ [y c'] [H_inv] →.
    +      split ⇒ //.
    +      move ⇒ y'.
    +      rewrite InZ_cons In_elementsZ_single.
    +      move ⇒ []. {
    +        move ⇒ [_].
    +        rewrite -Z.succ_lt_mono //.
    +      } {
    +        move : H_inv.
    +        rewrite /= !interval_list_invariant_app_iff interval_list_invariant_cons.
    +        move ⇒ [_] [] [_] [H_c'_neq] _ H_pre H_y'_in.
    +        apply Z.lt_trans with (m := y). {
    +          apply H_pre. {
    +            by rewrite InZ_rev.
    +          } {
    +            rewrite InZ_cons.
    +            left.
    +            by apply In_elementsZ_single_hd.
    +          }
    +        }
    +        apply Z.lt_succ_r, Z_le_add_r.
    +      }
    +    }
    +  Qed.
    + +
    +  Definition partitionZ_fold_current (c : option (Z × N)) :=
        match c with
    -       None ⇒ nil
    -     | Some yc ⇒ yc::nil
    +       None ⇒ nil
    +     | Some yc ⇒ yc::nil
        end.

      Lemma InZ_partitionZ_fold_current_Some : ∀ yc y,
    -     InZ y (partitionZ_fold_current (Some yc)) ↔
    -     InZ y (yc :: nil).
    - +     InZ y (partitionZ_fold_current (Some yc)) ↔
    +     InZ y (yc :: nil).
    +  Proof. done. Qed.
    +
      Lemma InZ_partitionZ_fold_insert : ∀ c x y l,
       match c with
    -   | Some (y, c') ⇒ x = y + Z.of_N c'
    -   | None ⇒ True
    -   end → (
    -   InZ y (partitionZ_fold_insert c x :: l) ↔
    -    ((x = y) ∨ InZ y (partitionZ_fold_current c) ∨
    -       InZ y l)).
    - +   | Some (y, c') ⇒ x = y + Z.of_N c'
    +   | None ⇒ True
    +   end → (
    +   InZ y (partitionZ_fold_insert c x :: l) ↔
    +    ((x = y) ∨ InZ y (partitionZ_fold_current c) ∨
    +       InZ y l)).
    +  Proof.
    +    intros c x y l.
    +    rewrite /partitionZ_fold_insert /partitionZ_fold_current
    +            /partitionZ_fold_skip.
    +    case c. {
    +      move ⇒ [y' c'] →.
    +      rewrite !InZ_cons elementsZ_single_succ in_app_iff
    +              InZ_nil /=.
    +      tauto.
    +    } {
    +      rewrite InZ_cons InZ_nil In_elementsZ_single1.
    +      tauto.
    +    }
    +  Qed.
    +
      Lemma InZ_partitionZ_fold_skip : ∀ c acc y,
    -    InZ y (partitionZ_fold_skip acc c) ↔
    -    (InZ y (partitionZ_fold_current c) ∨ InZ y acc).
    - +    InZ y (partitionZ_fold_skip acc c) ↔
    +    (InZ y (partitionZ_fold_current c) ∨ InZ y acc).
    +  Proof.
    +    intros c acc y.
    +    rewrite /partitionZ_fold_skip /partitionZ_fold_current
    +            /partitionZ_fold_skip.
    +    case c. {
    +      move ⇒ [y' c'].
    +      rewrite !InZ_cons InZ_nil /=.
    +      tauto.
    +    } {
    +      rewrite InZ_nil.
    +      tauto.
    +    }
    +  Qed.
    +
      Lemma filterZ_single_aux_props :
        âˆ€ f c x acc cur,
    -      partitionZ_aux_invariant x acc cur →
    -      match (filterZ_single_aux f (acc, cur) x c) with
    -        (acc', c') ⇒
    -        let r := partitionZ_fold_skip acc' c' in
    -        interval_list_invariant (List.rev r) = true ∧
    -        (∀ y', InZ y' r ↔ (InZ y' (partitionZ_fold_skip acc cur) ∨
    -                                    (f y' = true ∧ List.In y' (elementsZ_single x c))))
    +      partitionZ_aux_invariant x acc cur →
    +      match (filterZ_single_aux f (acc, cur) x c) with
    +        (acc', c') ⇒
    +        let r := partitionZ_fold_skip acc' c' in
    +        interval_list_invariant (List.rev r) = true ∧
    +        (∀ y', InZ y' r ↔ (InZ y' (partitionZ_fold_skip acc cur) ∨
    +                                    (f y' = true ∧ List.In y' (elementsZ_single x c))))

          end.
    - +  Proof.
    +    intro f.
    +    induction c as [| c' IH] using N.peano_ind. {
    +      intros x acc cur.
    +      rewrite /partitionZ_aux_invariant.
    +      move ⇒ [H_inv] _.
    +      rewrite /filterZ_single_aux fold_elementsZ_single_zero /=.
    +      tauto.
    +    }
    +    intros x acc cur H_inv.
    +    have → : filterZ_single_aux f (acc, cur) x (N.succ c') =
    +               filterZ_single_aux f (filterZ_fold_fun f (acc, cur) x) (Z.succ x) c'. {
    +        by rewrite /filterZ_single_aux fold_elementsZ_single_succ.
    +    }
    +    case_eq (filterZ_fold_fun f (acc, cur) x).
    +    move ⇒ acc' cur' H_fold_eq.
    + +
    +    case_eq (filterZ_single_aux f (acc', cur') (Z.succ x) c').
    +    move ⇒ acc'' cur'' H_succ_eq.
    + +
    +    have H_inv' : partitionZ_aux_invariant (Z.succ x) acc' cur'. {
    +      move : H_fold_eq H_inv.
    +      rewrite /filterZ_fold_fun.
    +      case (f x); move ⇒ [<-] <-. {
    +        apply partitionZ_aux_invariant_insert.
    +      } {
    +        apply partitionZ_aux_invariant_skip.
    +      }
    +    }
    +   
    +    move : (IH (Z.succ x) acc' cur' H_inv') ⇒ {IH}.
    +    rewrite H_succ_eq /=.
    +    set r := partitionZ_fold_skip acc'' cur''.
    +    move ⇒ [H_inv_r] H_in_r.
    +    split; first assumption.
    + +
    +    move ⇒ y'.
    +    move : H_fold_eq.
    +    rewrite H_in_r /filterZ_fold_fun.
    +    case_eq (f x) ⇒ H_fx [<-] <-. {
    +      rewrite InZ_partitionZ_fold_skip InZ_partitionZ_fold_current_Some InZ_partitionZ_fold_skip elementsZ_single_succ_front.
    +      rewrite InZ_partitionZ_fold_insert; last first. {
    +        move : H_inv.
    +        rewrite /partitionZ_aux_invariant ⇒ [[_]].
    +        case cur ⇒ //.
    +      }
    +      rewrite InZ_nil /=.
    +      split; last by tauto.
    +      move ⇒ []; last by tauto.
    +      move ⇒ []; last by tauto.
    +      move ⇒ []. {
    +        move ⇒ <-.
    +        tauto.
    +      } {
    +        tauto.
    +      }
    +    } {
    +      rewrite InZ_partitionZ_fold_skip /partitionZ_fold_current InZ_partitionZ_fold_skip elementsZ_single_succ_front !InZ_nil /=.
    +      split; first by tauto.
    +      move ⇒ []; first by tauto.
    +      move ⇒ [] H_fy' []. {
    +        move ⇒ H_x_eq; subst y'.
    +        contradict H_fy'.
    +        by rewrite H_fx.
    +      } {
    +        tauto.
    +      }
    +    }
    +  Qed.
    +
      Lemma filterZ_single_props :
        âˆ€ f c x acc,
    -      interval_list_invariant (rev acc) = true →
    -      (∀ y' : Z, InZ y' acc → Z.succ y' < x) →
    +      interval_list_invariant (rev acc) = true →
    +      (∀ y' : Z, InZ y' acc → Z.succ y' < x) →
          match (filterZ_single f acc x c) with
            r ⇒
    -        interval_list_invariant (List.rev r) = true ∧
    -        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    -                                 (f y' = true ∧ List.In y' (elementsZ_single x c))))
    +        interval_list_invariant (List.rev r) = true ∧
    +        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    +                                 (f y' = true ∧ List.In y' (elementsZ_single x c))))

          end.
    +  Proof.
    +    intros f c x acc.
    +    move ⇒ H_inv H_acc.
    +    rewrite /filterZ_single.
    + +
    +    have H_inv' : partitionZ_aux_invariant x acc None. {
    +      by rewrite /partitionZ_aux_invariant /=.
    +    }
    +    move : (filterZ_single_aux_props f c x acc None H_inv').
    +    case_eq (filterZ_single_aux f (acc, None) x c).
    +    move ⇒ acc' cur' /= H_res.
    +    tauto.
    +  Qed.


      Lemma filterZ_aux_props :
        âˆ€ f s acc,
    -      interval_list_invariant s = true →
    -      interval_list_invariant (rev acc) = true →
    -      (∀ x1 x2 : Z, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    +      interval_list_invariant s = true →
    +      interval_list_invariant (rev acc) = true →
    +      (∀ x1 x2 : Z, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
          match (filterZ_aux acc f s) with
            r ⇒
    -        interval_list_invariant r = true ∧
    -        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    -                                 (f y' = true ∧ InZ y' s)))
    +        interval_list_invariant r = true ∧
    +        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    +                                 (f y' = true ∧ InZ y' s)))
          end.
    - +  Proof.
    +    intro f.
    +    induction s as [| [y c] s' IH]. {
    +      intros acc.
    +      move ⇒ _ H_inv _.
    +      rewrite /filterZ_aux.
    +      split; first assumption.
    +      move ⇒ y'; rewrite InZ_rev InZ_nil; tauto.
    +    } {
    +      intros acc.
    +      rewrite interval_list_invariant_cons.
    +      move ⇒ [H_gr] [H_c_neq_0] H_inv_s' H_inv H_in_acc /=.
    +      move : H_gr.
    +      rewrite interval_list_elements_greater_alt2_def ⇒ // H_gr.
    + +
    +      have H_pre : (∀ y' : Z, InZ y' acc → Z.succ y' < y). {
    +        move ⇒ x1 H_x1_in.
    +        apply H_in_acc ⇒ //.
    +        rewrite InZ_cons.
    +        by left; apply In_elementsZ_single_hd.
    +      }
    +      
    +      move : (filterZ_single_props f c y acc H_inv H_pre) ⇒ {H_pre}.
    +      set acc' := filterZ_single f acc y c.
    +      move ⇒ [H_inv'] H_in_acc'.
    + +
    +      have H_pre : (∀ x1 x2 : Z,
    +                      InZ x1 acc' → InZ x2 s' → Z.succ x1 < x2). {
    +        move ⇒ x1 x2.
    +        rewrite H_in_acc' In_elementsZ_single.
    +        move ⇒ []. {
    +          move ⇒ H_x1_in H_x2_in.
    +          apply H_in_acc ⇒ //.
    +          rewrite InZ_cons.
    +          by right.
    +        } {
    +          move ⇒ [_] [_] H_x1_lt H_x2_in.
    +          apply Z.le_lt_trans with (m := y + Z.of_N c).
    +            - by apply Z.le_succ_l.
    +            - by apply H_gr.
    +        }
    +      }
    +      move : (IH acc' H_inv_s' H_inv' H_pre) ⇒ {H_pre}.
    + +
    +      move ⇒ [H_inv_r] H_in_r.
    +      split; first assumption.
    +      move ⇒ y'.
    +      rewrite H_in_r H_in_acc' InZ_cons.
    +      tauto.
    +    }
    +  Qed.
    +
      Lemma filterZ_props :
        âˆ€ f s,
    -      interval_list_invariant s = true →
    +      interval_list_invariant s = true →
          match (filterZ f s) with r ⇒
    -        interval_list_invariant r = true ∧
    -        (∀ y', InZ y' r ↔ (f y' = true ∧ InZ y' s))
    +        interval_list_invariant r = true ∧
    +        (∀ y', InZ y' r ↔ (f y' = true ∧ InZ y' s))
          end.
    +  Proof.
    +    intros f s H_inv_s.
    +    rewrite /filterZ.
    + +
    +    have H_pre_1 : interval_list_invariant (rev nil) = true by done.
    +    have H_pre_2 : (∀ x1 x2 : Z, InZ x1 nil → InZ x2 s → Z.succ x1 < x2) by done.
    + +
    +    move : (filterZ_aux_props f s nil H_inv_s H_pre_1 H_pre_2) ⇒ {H_pre_1} {H_pre_2}.
    +    move ⇒ [H_inv'] H_in_r.
    +    split; first assumption.
    +    move ⇒ y'.
    +    rewrite H_in_r InZ_nil.
    +    tauto.
    +  Qed.

      Global Instance filter_ok s f : ∀ `(Ok s), Ok (filter f s).
    +  Proof.
    +    move ⇒ [H_inv H_enc].
    +    rewrite /filter.
    +    set f' := (fun z : Z ⇒ f (Enc.decode z)).
    +    move : (filterZ_props f' s H_inv).
    +    move ⇒ [H_inv'] H_in_r.
    +    rewrite /Ok /IsOk /is_encoded_elems_list.
    +    split; first assumption.
    +    move ⇒ x /H_in_r [_] H_x_in.
    +    by apply H_enc.
    +  Qed.

      Lemma filter_spec :
    -   âˆ€ (s : t) (x : elt) (f : elt → bool),
    -   Ok s →
    -   (In x (filter f s) ↔ In x s ∧ f x = true).
    - +   âˆ€ (s : t) (x : elt) (f : elt → bool),
    +   Ok s →
    +   (In x (filter f s) ↔ In x s ∧ f x = true).
    +  Proof.
    +    move ⇒ s x f H_ok.
    +    suff H_suff :
    +      (∀ x, (InZ x (filter f s)) ↔
    +                 InZ x s ∧ f (Enc.decode x) = true). {
    +      rewrite !In_alt_def /elements !rev_map_alt_def
    +              -!in_rev !in_map_iff.
    +      setoid_rewrite H_suff.
    +      rewrite /InZ.
    +      split. {
    +        move ⇒ [y] [<-] [?] ?.
    +        split ⇒ //.
    +        by ∃ y.
    +      } {
    +        move ⇒ [] [y] [<-] ? ?.
    +        by ∃ y.
    +      }
    +    }
    +    rewrite /filter.
    +    set f' := (fun z : Z ⇒ f (Enc.decode z)).
    +    move : (H_ok) ⇒ [H_inv _].
    +    move : (filterZ_props f' s H_inv).
    +    move ⇒ [H_inv'] H_in_r.
    +    move ⇒ y; rewrite H_in_r; tauto.
    +  Qed.
    +
    @@ -2238,42 +6228,102 @@

    Library MSetsExtra.MSetIntervals


      Lemma partitionZ_single_aux_alt_def : ∀ f c y acc_t c_t acc_f c_f,
    -    partitionZ_single_aux f ((acc_t, c_t), (acc_f, c_f)) y c =
    -    (filterZ_single_aux f (acc_t, c_t) y c,
    -     filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y c).
    +    partitionZ_single_aux f ((acc_t, c_t), (acc_f, c_f)) y c =
    +    (filterZ_single_aux f (acc_t, c_t) y c,
    +     filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y c).
    +  Proof.
    +    intros f.
    +    rewrite /partitionZ_single_aux /filterZ_single_aux.
    +    induction c as [| c' IH] using N.peano_ind. {
    +      intros y acc_t c_t acc_f c_f.
    +      rewrite !fold_elementsZ_single_zero //. } {
    +      intros y acc_t c_t acc_f c_f.
    +      rewrite !fold_elementsZ_single_succ.
    +      case_eq (partitionZ_fold_fun f (acc_t, c_t, (acc_f, c_f)) y) ⇒ [] [acc_t' c_t'] [acc_f' c_f'] H_fold_eq.
    +      rewrite IH ⇒ {IH}.
    +      suff : (filterZ_fold_fun f (acc_t, c_t) y = (acc_t', c_t')) ∧
    +             (filterZ_fold_fun (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y = (acc_f', c_f')). {
    +        move ⇒ [->] → //.
    +      }
    +      move : H_fold_eq.
    +      rewrite /partitionZ_fold_fun /filterZ_fold_fun.
    +      case (f y); move ⇒ [<-] <- <- <- //.
    +    }
    +  Qed.

      Lemma partitionZ_aux_alt_def : ∀ f s acc_t acc_f,
    -   partitionZ_aux acc_t acc_f f s =
    -   (filterZ_aux acc_t f s,
    -    filterZ_aux acc_f (fun x : Z ⇒ negb (f x)) s).
    - +   partitionZ_aux acc_t acc_f f s =
    +   (filterZ_aux acc_t f s,
    +    filterZ_aux acc_f (fun x : Z ⇒ negb (f x)) s).
    +  Proof.
    +    intros f.
    +    induction s as [| [y c] s' IH]. {
    +      done.
    +    } {
    +      intros acc_t acc_f.
    +      rewrite /= /partitionZ_single /filterZ_single
    +              partitionZ_single_aux_alt_def.
    +      case (filterZ_single_aux f (acc_t, None) y c) ⇒ acc_t' c_t'.
    +      case (filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, None) y c) ⇒ acc_f' c_f'.
    +      rewrite IH //.
    +    }
    +  Qed.
    +
      Lemma partitionZ_alt_def : ∀ f s,
    -    partitionZ f s = (filterZ f s,
    -                      filterZ (fun x ⇒ negb (f x)) s).
    +    partitionZ f s = (filterZ f s,
    +                      filterZ (fun x ⇒ negb (f x)) s).
    +  Proof.
    +    intros f s.
    +    rewrite /partitionZ /filterZ
    +            partitionZ_aux_alt_def //.
    +  Qed.

      Lemma partition_alt_def : ∀ f s,
    -    partition f s = (filter f s,
    -                     filter (fun x ⇒ negb (f x)) s).
    +    partition f s = (filter f s,
    +                     filter (fun x ⇒ negb (f x)) s).
    +  Proof.
    +    intros f s.
    +    rewrite /partition /filter partitionZ_alt_def.
    +    done.
    +  Qed.

    -  Global Instance partition_ok1 s f : ∀ `(Ok s), Ok (fst (partition f s)).
    +  Global Instance partition_ok1 s f : ∀ `(Ok s), Ok (fst (partition f s)).
    +  Proof.
    +    move ⇒ H_ok.
    +    rewrite partition_alt_def /fst.
    +    by apply filter_ok.
    +  Qed.

    -  Global Instance partition_ok2 s f : ∀ `(Ok s), Ok (snd (partition f s)).
    +  Global Instance partition_ok2 s f : ∀ `(Ok s), Ok (snd (partition f s)).
    +  Proof.
    +    move ⇒ H_ok.
    +    rewrite partition_alt_def /snd.
    +    by apply filter_ok.
    +  Qed.

      Lemma partition_spec1 :
    -   âˆ€ (s : t) (f : elt → bool),
    -   Equal (fst (partition f s)) (filter f s).
    +   âˆ€ (s : t) (f : elt → bool),
    +   Equal (fst (partition f s)) (filter f s).
    +  Proof.
    +    intros s f.
    +    rewrite partition_alt_def /fst /Equal //.
    +  Qed.

      Lemma partition_spec2 :
    -   âˆ€ (s : t) (f : elt → bool),
    -   Ok s →
    -   Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
    +   âˆ€ (s : t) (f : elt → bool),
    +   Ok s →
    +   Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
    +  Proof.
    +    intros s f.
    +    rewrite partition_alt_def /snd /Equal //.
    +  Qed.

    End Raw.
    @@ -2293,123 +6343,217 @@

    Library MSetsExtra.MSetIntervals


    -Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E.
    +Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E.
      Module E := Enc.E.
      Module Raw := Raw Enc.

    + Local Unset Elimination Schemes.
    + Local Unset Case Analysis Schemes.

    - Definition elt := Raw.elt.
    - Record t_ := Mkt {this :> Raw.t; is_ok : Raw.Ok this}.
    + Definition elt := Raw.elt.
    + Record t_ := Mkt {this :> Raw.t; is_ok : Raw.Ok this}.
     Definition t := t_.
    + Arguments Mkt this {is_ok}.
     Hint Resolve is_ok : typeclass_instances.

    - Definition In (x : elt)(s : t) := Raw.In x s.(this).
    - Definition Equal (s s' : t) := ∀ a : elt, In a s ↔ In a s'.
    - Definition Subset (s s' : t) := ∀ a : elt, In a s → In a s'.
    - Definition Empty (s : t) := ∀ a : elt, ¬ In a s.
    - Definition For_all (P : elt → Prop)(s : t) := ∀ x, In x s → P x.
    - Definition Exists (P : elt → Prop)(s : t) := ∃ x, In x s ∧ P x.
    - -
    - Definition mem (x : elt)(s : t) := Raw.mem x s.(this).
    - Definition add (x : elt)(s : t) : t := Mkt (Raw.add x s.(this)).
    - Definition remove (x : elt)(s : t) : t := Mkt (Raw.remove x s.(this)).
    - Definition singleton (x : elt) : t := Mkt (Raw.singleton x).
    - Definition union (s s' : t) : t := Mkt (Raw.union s s').
    - Definition inter (s s' : t) : t := Mkt (Raw.inter s s').
    - Definition diff (s s' : t) : t := Mkt (Raw.diff s s').
    - Definition equal (s s' : t) := Raw.equal s s'.
    - Definition subset (s s' : t) := Raw.subset s s'.(this).
    - Definition empty : t := Mkt Raw.empty.
    - Definition is_empty (s : t) := Raw.is_empty s.
    - Definition elements (s : t) : list elt := Raw.elements s.
    - Definition min_elt (s : t) : option elt := Raw.min_elt s.
    - Definition max_elt (s : t) : option elt := Raw.max_elt s.
    - Definition choose (s : t) : option elt := Raw.choose s.
    - Definition compare (s1 s2 : t) : comparison := Raw.compare s1 s2.
    - Definition fold {A : Type}(f : elt → A → A)(s : t) : A → A := Raw.fold f s.
    - Definition cardinal (s : t) := Raw.cardinal s.
    - Definition filter (f : elt → bool)(s : t) : t := Mkt (Raw.filter f s).
    - Definition for_all (f : elt → bool)(s : t) := Raw.for_all f s.
    - Definition exists_ (f : elt → bool)(s : t) := Raw.exists_ f s.
    - Definition partition (f : elt → bool)(s : t) : t × t :=
    -   let p := Raw.partition f s in (Mkt (fst p), Mkt (snd p)).
    - -
    - Instance In_compat : Proper (E.eq==>eq==>iff) In.
    - -
    - Definition eq : t → t → Prop := Equal.
    - -
    - Instance eq_equiv : Equivalence eq.
    - -
    - Definition eq_dec : ∀ (s s':t), { eq s s' }+{ ¬eq s s' }.
    - -
    - Definition lt : t → t → Prop := Raw.lt.
    - -
    - Instance lt_strorder : StrictOrder lt.
    - -
    - Instance lt_compat : Proper (eq==>eq==>iff) lt.
    + Definition In (x : elt)(s : t) := Raw.In x s.(this).
    + Definition Equal (s s' : t) := ∀ a : elt, In a s ↔ In a s'.
    + Definition Subset (s s' : t) := ∀ a : elt, In a s → In a s'.
    + Definition Empty (s : t) := ∀ a : elt, ¬ In a s.
    + Definition For_all (P : elt → Prop)(s : t) := ∀ x, In x s → P x.
    + Definition Exists (P : elt → Prop)(s : t) := ∃ x, In x s ∧ P x.
    + +
    + Definition mem (x : elt)(s : t) := Raw.mem x s.(this).
    + Definition add (x : elt)(s : t) : t := Mkt (Raw.add x s.(this)).
    + Definition remove (x : elt)(s : t) : t := Mkt (Raw.remove x s.(this)).
    + Definition singleton (x : elt) : t := Mkt (Raw.singleton x).
    + Definition union (s s' : t) : t := Mkt (Raw.union s s').
    + Definition inter (s s' : t) : t := Mkt (Raw.inter s s').
    + Definition diff (s s' : t) : t := Mkt (Raw.diff s s').
    + Definition equal (s s' : t) := Raw.equal s s'.
    + Definition subset (s s' : t) := Raw.subset s s'.(this).
    + Definition empty : t := Mkt Raw.empty.
    + Definition is_empty (s : t) := Raw.is_empty s.
    + Definition elements (s : t) : list elt := Raw.elements s.
    + Definition min_elt (s : t) : option elt := Raw.min_elt s.
    + Definition max_elt (s : t) : option elt := Raw.max_elt s.
    + Definition choose (s : t) : option elt := Raw.choose s.
    + Definition compare (s1 s2 : t) : comparison := Raw.compare s1 s2.
    + Definition fold {A : Type}(f : elt → A → A)(s : t) : A → A := Raw.fold f s.
    + Definition cardinal (s : t) := Raw.cardinal s.
    + Definition filter (f : elt → bool)(s : t) : t := Mkt (Raw.filter f s).
    + Definition for_all (f : elt → bool)(s : t) := Raw.for_all f s.
    + Definition exists_ (f : elt → bool)(s : t) := Raw.exists_ f s.
    + Definition partition (f : elt → bool)(s : t) : t × t :=
    +   let p := Raw.partition f s in (Mkt (fst p), Mkt (snd p)).
    + +
    + Instance In_compat : Proper (E.eq==>eq==>iff) In.
    + Proof.
    +   repeat red.
    +   move ⇒ x y H_eq_xy x' y' →.
    +   rewrite /In /Raw.In.
    +   setoid_rewrite H_eq_xy.
    +   done.
    + Qed.
    + +
    + Definition eq : t → t → Prop := Equal.
    + +
    + Instance eq_equiv : Equivalence eq.
    + Proof. firstorder. Qed.
    + +
    + Definition eq_dec : ∀ (s s':t), { eq s s' }+{ ¬eq s s' }.
    + Proof.
    +  intros (s,Hs) (s',Hs').
    +  change ({Raw.Equal s s'}+{¬Raw.Equal s s'}).
    +  destruct (Raw.equal s s') eqn:H; [left|right];
    +   rewrite <- Raw.equal_spec; congruence.
    + Defined.
    + +
    + Definition lt : t → t → Prop := Raw.lt.
    + +
    + Instance lt_strorder : StrictOrder lt.
    + Proof.
    +   unfold lt.
    +   constructor. {
    +     move : Raw.lt_Irreflexive.
    +     rewrite /Irreflexive /complement /Reflexive.
    +     move ⇒ H x.
    +     apply H.
    +   } {
    +     move : Raw.lt_Transitive.
    +     rewrite /Transitive.
    +     move ⇒ H x y z.
    +     apply H.
    +   }
    + Qed.
    + +
    + Instance lt_compat : Proper (eq==>eq==>iff) lt.
    + Proof.
    +   repeat red.
    +   move ⇒ [x1 H_x1_ok] [y1 H_y1_ok] H_eq.
    +   move ⇒ [x2 H_x2_ok] [y2 H_y2_ok].
    +   move : H_eq.
    +   rewrite /eq /lt /Equal /In /=.
    +   replace (∀ a : elt, Raw.In a x1 ↔ Raw.In a y1) with
    +     (Raw.Equal x1 y1) by done.
    +   replace (∀ a : elt, Raw.In a x2 ↔ Raw.In a y2) with
    +     (Raw.Equal x2 y2) by done.
    +   rewrite -!Raw.equal_spec !Raw.equal_alt_def.
    +   move ⇒ → → //.
    + Qed.

     Section Spec.
      Variable s s' : t.
      Variable x y : elt.
    -  Variable f : elt → bool.
    -  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    - -
    -  Lemma mem_spec : mem x s = true ↔ In x s.
    -   Lemma equal_spec : equal s s' = true ↔ Equal s s'.
    -   Lemma subset_spec : subset s s' = true ↔ Subset s s'.
    -   Lemma empty_spec : Empty empty.
    -   Lemma is_empty_spec : is_empty s = true ↔ Empty s.
    -   Lemma add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    -   Lemma remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    -   Lemma singleton_spec : In y (singleton x) ↔ E.eq y x.
    -   Lemma union_spec : In x (union s s') ↔ In x s ∨ In x s'.
    -   Lemma inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
    -   Lemma diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
    -   Lemma fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    -      fold f s i = fold_left (fun a e ⇒ f e a) (elements s) i.
    -   Lemma cardinal_spec : cardinal s = length (elements s).
    -   Lemma filter_spec : compatb f →
    -    (In x (filter f s) ↔ In x s ∧ f x = true).
    -   Lemma for_all_spec : compatb f →
    -    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    -   Lemma exists_spec : compatb f →
    -    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    -   Lemma partition_spec1 : compatb f → Equal (fst (partition f s)) (filter f s).
    -   Lemma partition_spec2 : compatb f →
    -      Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
    -   Lemma elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    -   Lemma elements_spec2w : NoDupA E.eq (elements s).
    -   Lemma elements_spec2 : sort E.lt (elements s).
    -   Lemma choose_spec1 : choose s = Some x → In x s.
    -   Lemma choose_spec2 : choose s = None → Empty s.
    -   Lemma choose_spec3 : choose s = Some x → choose s' = Some y →
    -    Equal s s' → E.eq x y.
    - -
    -  Lemma min_elt_spec1 : choose s = Some x → In x s.
    -   Lemma min_elt_spec2 : min_elt s = Some x → In y s → ¬ E.lt y x.
    -   Lemma min_elt_spec3 : choose s = None → Empty s.
    - -
    -  Lemma max_elt_spec1 : max_elt s = Some x → In x s.
    -   Lemma max_elt_spec2 : max_elt s = Some x → In y s → ¬ E.lt x y.
    -   Lemma max_elt_spec3 : max_elt s = None → Empty s.
    - -
    -  Lemma compare_spec : CompSpec eq lt s s' (compare s s').
    +  Variable f : elt → bool.
    +  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    + +
    +  Lemma mem_spec : mem x s = true ↔ In x s.
    +  Proof. exact (Raw.mem_spec _ _ _). Qed.
    +  Lemma equal_spec : equal s s' = true ↔ Equal s s'.
    +  Proof. rewrite Raw.equal_spec //. Qed.
    +  Lemma subset_spec : subset s s' = true ↔ Subset s s'.
    +  Proof. exact (Raw.subset_spec _ _ _ _). Qed.
    +  Lemma empty_spec : Empty empty.
    +  Proof. exact Raw.empty_spec. Qed.
    +  Lemma is_empty_spec : is_empty s = true ↔ Empty s.
    +  Proof. rewrite Raw.is_empty_spec //. Qed.
    +  Lemma add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    +  Proof. exact (Raw.add_spec _ _ _ _). Qed.
    +  Lemma remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    +  Proof. exact (Raw.remove_spec _ _ _ _). Qed.
    +  Lemma singleton_spec : In y (singleton x) ↔ E.eq y x.
    +  Proof. exact (Raw.singleton_spec _ _). Qed.
    +  Lemma union_spec : In x (union s s') ↔ In x s ∨ In x s'.
    +  Proof. exact (Raw.union_spec _ _ _ _ _). Qed.
    +  Lemma inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
    +  Proof. exact (Raw.inter_spec _ _ _ _ _). Qed.
    +  Lemma diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +  Proof. exact (Raw.diff_spec _ _ _ _ _). Qed.
    +  Lemma fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    +      fold f s i = fold_left (fun a e ⇒ f e a) (elements s) i.
    +  Proof. exact (@Raw.fold_spec _). Qed.
    +  Lemma cardinal_spec : cardinal s = length (elements s).
    +  Proof. exact (@Raw.cardinal_spec s). Qed.
    +  Lemma filter_spec : compatb f →
    +    (In x (filter f s) ↔ In x s ∧ f x = true).
    +  Proof. move ⇒ _; exact (@Raw.filter_spec _ _ _ _). Qed.
    +  Lemma for_all_spec : compatb f →
    +    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +  Proof. exact (@Raw.for_all_spec _ _ _). Qed.
    +  Lemma exists_spec : compatb f →
    +    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +  Proof. exact (@Raw.exists_spec _ _ _). Qed.
    +  Lemma partition_spec1 : compatb f → Equal (fst (partition f s)) (filter f s).
    +  Proof. move ⇒ _; exact (@Raw.partition_spec1 _ _). Qed.
    +  Lemma partition_spec2 : compatb f →
    +      Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
    +  Proof. move ⇒ _; exact (@Raw.partition_spec2 _ _ _). Qed.
    +  Lemma elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    +  Proof. rewrite /In /Raw.In /elements //. Qed.
    +  Lemma elements_spec2w : NoDupA E.eq (elements s).
    +  Proof. exact (Raw.elements_spec2w _ _). Qed.
    +  Lemma elements_spec2 : sort E.lt (elements s).
    +  Proof. exact (Raw.elements_sorted _ _). Qed.
    +  Lemma choose_spec1 : choose s = Some x → In x s.
    +  Proof. exact (Raw.choose_spec1 _ _ _). Qed.
    +  Lemma choose_spec2 : choose s = None → Empty s.
    +  Proof. exact (Raw.choose_spec2 _). Qed.
    +  Lemma choose_spec3 : choose s = Some x → choose s' = Some y →
    +    Equal s s' → E.eq x y.
    +  Proof.
    +    intros H1 H2 H3.
    +    suff → : x = y. {
    +      apply E.eq_equiv.
    +    }
    +    move : H1 H2 H3.
    +    exact (Raw.choose_spec3 _ _ _ _ _ _).
    +  Qed.
    + +
    +  Lemma min_elt_spec1 : choose s = Some x → In x s.
    +  Proof. exact (Raw.min_elt_spec1 _ _ _). Qed.
    +  Lemma min_elt_spec2 : min_elt s = Some x → In y s → ¬ E.lt y x.
    +  Proof. exact (Raw.min_elt_spec2 _ _ _ _). Qed.
    +  Lemma min_elt_spec3 : choose s = None → Empty s.
    +  Proof. exact (Raw.min_elt_spec3 _). Qed.
    + +
    +  Lemma max_elt_spec1 : max_elt s = Some x → In x s.
    +  Proof. exact (Raw.max_elt_spec1 _ _ _). Qed.
    +  Lemma max_elt_spec2 : max_elt s = Some x → In y s → ¬ E.lt x y.
    +  Proof. exact (Raw.max_elt_spec2 _ _ _ _). Qed.
    +  Lemma max_elt_spec3 : max_elt s = None → Empty s.
    +  Proof. exact (Raw.max_elt_spec3 _). Qed.
    + +
    +  Lemma compare_spec : CompSpec eq lt s s' (compare s s').
    +  Proof.
    +    generalize s s'.
    +    move ⇒ [s1 H_ok_s1] [s2 H_ok_s2].
    +    move : (Raw.compare_spec s1 s2).
    +    rewrite /CompSpec /eq /Equal /In /lt /compare /=.
    +    replace (∀ a : elt, Raw.In a s1 ↔ Raw.In a s2) with
    +    (Raw.Equal s1 s2) by done.
    +    suff H_eq : (Raw.Equal s1 s2) ↔ (s1 = s2). {
    +      move ⇒ [] H; constructor ⇒ //.
    +      by rewrite H_eq.
    +    }
    +    rewrite -Raw.equal_spec Raw.equal_alt_def //.
    +  Qed.

     End Spec.
    @@ -2436,29 +6580,32 @@

    Library MSetsExtra.MSetIntervals


    Module ElementEncodeZ <: ElementEncode.
    -  Module E := Z.
    +  Module E := Z.

    -  Definition encode (z : Z) := z.
    -  Definition decode (z : Z) := z.
    +  Definition encode (z : Z) := z.
    +  Definition decode (z : Z) := z.

    -  Lemma decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    - +  Lemma decode_encode_ok: ∀ (e : E.t),
    +    decode (encode e) = e.
    +  Proof. by []. Qed.
    +
    -  Lemma encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    - +  Lemma encode_eq : ∀ (e1 e2 : E.t),
    +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +  Proof. by []. Qed.
    +
    -  Lemma encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    - +  Lemma encode_lt : ∀ (e1 e2 : E.t),
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Proof. by []. Qed.
    +
    End ElementEncodeZ.

    -Module MSetIntervalsZ <: SetsOn Z := MSetIntervals ElementEncodeZ.
    +Module MSetIntervalsZ <: SetsOn Z := MSetIntervals ElementEncodeZ.

    @@ -2471,29 +6618,41 @@

    Library MSetsExtra.MSetIntervals


    Module ElementEncodeN <: ElementEncode.
    -  Module E := N.
    +  Module E := N.

    -  Definition encode (n : N) := Z.of_N n.
    -  Definition decode (z : Z) := Z.to_N z.
    +  Definition encode (n : N) := Z.of_N n.
    +  Definition decode (z : Z) := Z.to_N z.

    -  Lemma decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    - +  Lemma decode_encode_ok: ∀ (e : E.t),
    +    decode (encode e) = e.
    +  Proof.
    +    intros e.
    +    rewrite /encode /decode N2Z.id //.
    +  Qed.
    +
    -  Lemma encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    - +  Lemma encode_eq : ∀ (e1 e2 : E.t),
    +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +  Proof.
    +    intros e1 e2.
    +    rewrite /encode /Z.eq N2Z.inj_iff /E.eq //.
    +  Qed.
    +
    -  Lemma encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Lemma encode_lt : ∀ (e1 e2 : E.t),
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Proof.
    +    intros e1 e2.
    +    rewrite /encode -N2Z.inj_lt //.
    +  Qed.

    End ElementEncodeN.

    -Module MSetIntervalsN <: SetsOn N := MSetIntervals ElementEncodeN.
    +Module MSetIntervalsN <: SetsOn N := MSetIntervals ElementEncodeN.

    @@ -2504,29 +6663,41 @@

    Library MSetsExtra.MSetIntervals

    Module ElementEncodeNat <: ElementEncode.
    -  Module E := Nat.
    +  Module E := Nat.

    -  Definition encode (n : nat) := Z.of_nat n.
    -  Definition decode (z : Z) := Z.to_nat z.
    +  Definition encode (n : nat) := Z.of_nat n.
    +  Definition decode (z : Z) := Z.to_nat z.

    -  Lemma decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    - +  Lemma decode_encode_ok: ∀ (e : E.t),
    +    decode (encode e) = e.
    +  Proof.
    +    intros e.
    +    rewrite /encode /decode Nat2Z.id //.
    +  Qed.
    +
    -  Lemma encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    - +  Lemma encode_eq : ∀ (e1 e2 : E.t),
    +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +  Proof.
    +    intros e1 e2.
    +    rewrite /encode /Z.eq Nat2Z.inj_iff /E.eq //.
    +  Qed.
    +
    -  Lemma encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Lemma encode_lt : ∀ (e1 e2 : E.t),
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Proof.
    +    intros e1 e2.
    +    rewrite /encode -Nat2Z.inj_lt //.
    +  Qed.

    End ElementEncodeNat.

    -Module MSetIntervalsNat <: SetsOn Nat := MSetIntervals ElementEncodeNat.
    +Module MSetIntervalsNat <: SetsOn Nat := MSetIntervals ElementEncodeNat.

    diff --git a/html/MSetsExtra.MSetListWithDups.html b/html/MSetsExtra.MSetListWithDups.html index 7ec5cc9..26bd69f 100644 --- a/html/MSetsExtra.MSetListWithDups.html +++ b/html/MSetsExtra.MSetListWithDups.html @@ -45,10 +45,10 @@

    Library MSetsExtra.MSetListWithDups


    -Require Export MSetInterface.
    +Require Export MSetInterface.
    Require Import mathcomp.ssreflect.ssreflect.
    -Require Import List OrdersFacts OrdersLists.
    -Require Import Sorting Permutation.
    +Require Import List OrdersFacts OrdersLists.
    +Require Import Sorting Permutation.
    Require Import MSetWithDups.

    @@ -72,7 +72,7 @@

    Library MSetsExtra.MSetListWithDups

    duplicates.
    -Module RemoveDupsFromSorted (Import X:OrderedType).
    +Module RemoveDupsFromSorted (Import X:OrderedType).

    @@ -81,30 +81,41 @@

    Library MSetsExtra.MSetListWithDups

    First, we need some infrastructure for our ordered type
    -  Module Import MX := OrderedTypeFacts X.
    +  Module Import MX := OrderedTypeFacts X.

    -  Module Import XTotalLeBool <: TotalLeBool.
    +  Module Import XTotalLeBool <: TotalLeBool.
        Definition t := X.t.
        Definition leb x y :=
          match X.compare x y with
    -        | Lt ⇒ true
    -        | Eq ⇒ true
    -        | Gt ⇒ false
    +        | Lt ⇒ true
    +        | Eq ⇒ true
    +        | Gt ⇒ false
          end.

    -    Infix "<=?" := leb (at level 35).
    +    Infix "<=?" := leb (at level 35).

    -    Theorem leb_total : ∀ (a1 a2 : t), (a1 <=? a2 = true) ∨ (a2 <=? a1 = true).
    +    Theorem leb_total : ∀ (a1 a2 : t), (a1 <=? a2 = true) ∨ (a2 <=? a1 = true).
    +    Proof.
    +      intros a1 a2.
    +      unfold leb.
    +      rewrite (compare_antisym a1 a2).
    +      case (X.compare a1 a2); rewrite /=; tauto.
    +    Qed.

    -    Definition le x y := (leb x y = true).
    +    Definition le x y := (leb x y = true).
      End XTotalLeBool.

    -  Lemma eqb_eq_alt : ∀ x y, eqb x y = true ↔ eq x y.
    +  Lemma eqb_eq_alt : ∀ x y, eqb x y = true ↔ eq x y.
    +  Proof.
    +    intros x y.
    +    rewrite eqb_alt -compare_eq_iff.
    +    case (compare x y) ⇒ //.
    +  Qed.

    @@ -113,22 +124,22 @@

    Library MSetsExtra.MSetListWithDups

    Now we can define our main function
    -  Fixpoint remove_dups_from_sortedA_aux (acc : list t) (l : list t) : list t :=
    +  Fixpoint remove_dups_from_sortedA_aux (acc : list t) (l : list t) : list t :=
        match l with
    -    | nil ⇒ List.rev' acc
    -    | x :: xs ⇒
    +    | nil ⇒ List.rev' acc
    +    | x :: xs ⇒
           match xs with
    -       | nil ⇒ List.rev' (x :: acc)
    -       | y :: ys ⇒
    -           if eqb x y then
    +       | nil ⇒ List.rev' (x :: acc)
    +       | y :: ys ⇒
    +           if eqb x y then
                 remove_dups_from_sortedA_aux acc xs
    -           else
    -             remove_dups_from_sortedA_aux (x::acc) xs
    +           else
    +             remove_dups_from_sortedA_aux (x::acc) xs
           end
        end.

    -  Definition remove_dups_from_sortedA := remove_dups_from_sortedA_aux (nil : list t).
    +  Definition remove_dups_from_sortedA := remove_dups_from_sortedA_aux (nil : list t).

    @@ -137,34 +148,87 @@

    Library MSetsExtra.MSetListWithDups

    We can prove some technical lemmata
    -  Lemma remove_dups_from_sortedA_aux_alt : ∀ (l : list X.t) acc,
    -    remove_dups_from_sortedA_aux acc l =
    -    List.rev acc ++ (remove_dups_from_sortedA l).
    - +  Lemma remove_dups_from_sortedA_aux_alt : ∀ (l : list X.t) acc,
    +    remove_dups_from_sortedA_aux acc l =
    +    List.rev acc ++ (remove_dups_from_sortedA l).
    +  Proof.
    +    unfold remove_dups_from_sortedA.
    +    induction l as [| x xs IH] ⇒ acc. {
    +      rewrite /remove_dups_from_sortedA_aux /rev' -!rev_alt /= app_nil_r //.
    +    } {
    +      rewrite /=.
    +      case_eq xs. {
    +        rewrite /rev' -!rev_alt //.
    +      } {
    +        move ⇒ y ys H_xs_eq.
    +        rewrite -!H_xs_eq !(IH acc) !(IH (x :: acc)) (IH (x::nil)).
    +        case (eqb x y) ⇒ //.
    +        rewrite /= -app_assoc //.
    +      }
    +    }
    +  Qed.
    +
      Lemma remove_dups_from_sortedA_alt :
    -    âˆ€ (l : list t),
    -    remove_dups_from_sortedA l =
    +    âˆ€ (l : list t),
    +    remove_dups_from_sortedA l =
        match l with
    -    | nil ⇒ nil
    -    | x :: xs ⇒
    +    | nil ⇒ nil
    +    | x :: xs ⇒
           match xs with
    -       | nil ⇒ l
    -       | y :: ys ⇒
    -           if eqb x y then
    +       | nil ⇒ l
    +       | y :: ys ⇒
    +           if eqb x y then
                 remove_dups_from_sortedA xs
    -           else
    -             x :: remove_dups_from_sortedA xs
    +           else
    +             x :: remove_dups_from_sortedA xs
           end
        end.
    - +  Proof.
    +    case. {
    +      done.
    +    } {
    +      intros x xs.
    +      rewrite /remove_dups_from_sortedA /= /rev' /=.
    +      case xs ⇒ //.
    +      move ⇒ y ys.
    +      rewrite !remove_dups_from_sortedA_aux_alt /= //.
    +    }
    +  Qed.
    +
      Lemma remove_dups_from_sortedA_hd :
          âˆ€ x xs,
    -      âˆƒ (x':t) xs',
    -        remove_dups_from_sortedA (x :: xs) =
    -        (x' :: xs') ∧ (eqb x x' = true).
    - +      âˆƒ (x':t) xs',
    +        remove_dups_from_sortedA (x :: xs) =
    +        (x' :: xs') ∧ (eqb x x' = true).
    +  Proof.
    +    intros x xs.
    +    move : x;
    +    induction xs as [| y ys IH] ⇒ x. {
    +      rewrite remove_dups_from_sortedA_alt.
    +      âˆƒ x, nil.
    +      split; first reflexivity.
    +      rewrite eqb_alt compare_refl //.
    +    } {
    +      rewrite remove_dups_from_sortedA_alt.
    +      case_eq (eqb x y); last first. {
    +        move ⇒ _.
    +        âˆƒ x, (remove_dups_from_sortedA (y :: ys)).
    +        split; first reflexivity.
    +        rewrite eqb_alt compare_refl //.
    +      } {
    +        move ⇒ H_eqb_xy.
    +        move : (IH y) ⇒ {IH} [x'] [xs'] [->] H_eqb_yx'.
    +        âˆƒ x', xs'.
    +        split; first done.
    +        move : H_eqb_xy H_eqb_yx'.
    +        rewrite !eqb_eq_alt.
    +        apply MX.eq_trans.
    +      }
    +    }
    +  Qed.
    +
    @@ -173,14 +237,102 @@

    Library MSetsExtra.MSetListWithDups

      Lemma remove_dups_from_sortedA_spec :
    -    âˆ€ (l : list t),
    -      Sorted le l →
    -      let l' := remove_dups_from_sortedA l in (
    +    âˆ€ (l : list t),
    +      Sorted le l →
    +      let l' := remove_dups_from_sortedA l in (
        
    -      Sorted lt l' ∧
    -      NoDupA eq l' ∧
    -      (∀ x, InA eq x l ↔ InA eq x l')).
    - +      Sorted lt l' ∧
    +      NoDupA eq l' ∧
    +      (∀ x, InA eq x l ↔ InA eq x l')).
    +  Proof.
    +    simpl.
    +    induction l as [| x xs IH]. {
    +      rewrite remove_dups_from_sortedA_alt.
    +      done.
    +    } {
    +      rewrite remove_dups_from_sortedA_alt.
    +      move : IH.
    +      case xs ⇒ {xs}. {
    +        move ⇒ _.
    +        split; last split. {
    +          apply Sorted_cons ⇒ //.
    +        } {
    +          apply NoDupA_singleton.
    +        } {
    +          done.
    +        }
    +      } {
    +        move ⇒ y ys IH H_sorted_x_y_ys.
    +        apply Sorted_inv in H_sorted_x_y_ys as [H_sorted_y_ys H_hd_rel].
    +        apply HdRel_inv in H_hd_rel.
    + +
    +        have : ∃ y' ys',
    +          remove_dups_from_sortedA (y :: ys) = y' :: ys' ∧
    +          eqb y y' = true. {
    +          apply remove_dups_from_sortedA_hd ⇒ //.
    +        }
    +        move ⇒ [y'] [ys'] [H_yys'_intro] /eqb_eq_alt H_eq_y_y'.
    + +
    +        move : (IH H_sorted_y_ys).
    +        rewrite !H_yys'_intro.
    +        move ⇒ {IH} [IH1] [IH2] IH3.
    + +
    +        case_eq (eqb x y). {
    +          rewrite eqb_eq_alt ⇒ H_eq_x_y.
    +          split ⇒ //.
    +          split ⇒ //.
    +          move ⇒ x'.
    +          rewrite InA_cons IH3.
    +          split; last tauto.
    +          move ⇒ [] //.
    +          move ⇒ H_eq_x'_x.
    +          apply InA_cons_hd.
    +          apply eq_trans with (y := x) ⇒ //.
    +          apply eq_trans with (y := y) ⇒ //.
    +        }
    +        move ⇒ H_neqb_x_y.
    + +
    +        have H_sorted : Sorted lt (x :: y' :: ys'). {
    +          apply Sorted_cons ⇒ //.
    +          apply HdRel_cons.
    +          rewrite -compare_lt_iff.
    +          suff : (compare x y = Lt). {
    +            setoid_rewrite compare_compat; eauto;
    +              apply eq_refl.
    +          }
    +          move : H_hd_rel H_neqb_x_y.
    +          rewrite eqb_alt /le /leb.
    +          case (compare x y) ⇒ //.
    +        }
    +        split; last split. {
    +          assumption.
    +        } {
    +          apply NoDupA_cons ⇒ //.
    + +
    +          move ⇒ /InA_alt [x'] [H_eq_xx'] H_in_x'.
    +          have : Forall (lt x) (y' :: ys'). {
    +            apply Sorted_extends ⇒ //.
    +            rewrite /Relations_1.Transitive.
    +            by apply lt_trans.
    +          }
    +          rewrite Forall_forall ⇒ H_forall.
    +          move : (H_forall _ H_in_x') ⇒ {H_forall}.
    +          move : H_eq_xx'.
    +          rewrite -compare_lt_iff -compare_eq_iff.
    +          move ⇒ → //.
    +        } {
    +          move ⇒ x0.
    +          rewrite !(InA_cons eq x0 x) IH3 //.
    +        }
    +      }
    +    }
    +  Qed.
    +
    @@ -188,21 +340,57 @@

    Library MSetsExtra.MSetListWithDups

    Next, we combine it with sorting
    -  Module Import XSort := Sort XTotalLeBool.
    +  Module Import XSort := Sort XTotalLeBool.

    -  Definition remove_dups_by_sortingA (l : list t) : list t :=
    -    remove_dups_from_sortedA (XSort.sort l).
    +  Definition remove_dups_by_sortingA (l : list t) : list t :=
    +    remove_dups_from_sortedA (XSort.sort l).

      Lemma remove_dups_by_sortingA_spec :
    -    âˆ€ (l : list t),
    -      let l' := remove_dups_by_sortingA l in (
    +    âˆ€ (l : list t),
    +      let l' := remove_dups_by_sortingA l in (
        
    -      Sorted lt l' ∧
    -      NoDupA eq l' ∧
    -      (∀ x, InA eq x l ↔ InA eq x l')).
    - +      Sorted lt l' ∧
    +      NoDupA eq l' ∧
    +      (∀ x, InA eq x l ↔ InA eq x l')).
    +  Proof.
    +    intro l.
    + +
    +    suff : (∀ x : X.t, InA eq x (sort l) ↔ InA eq x l) ∧
    +           Sorted le (sort l). {
    +      
    +      unfold remove_dups_by_sortingA.
    +      move : (remove_dups_from_sortedA_spec (sort l)).
    +      simpl.
    +      move ⇒ H_spec [H_in_sort H_sorted_sort].
    +      move : (H_spec H_sorted_sort).
    +      move ⇒ [H1] [H2] H3.
    +      split ⇒ //.
    +      split ⇒ //.
    +      move ⇒ x.
    +      rewrite -H_in_sort H3 //.
    +    }
    +
    +    split. {
    +      have H_in_sort : ∀ x, List.In x (XSort.sort l) ↔ List.In x l. {
    +        intros x.
    +        have H_perm := (XSort.Permuted_sort l).
    +        split; apply Permutation_in ⇒ //.
    +        by apply Permutation_sym.
    +      }
    +
    +      intros x.
    +      rewrite !InA_alt.
    +      setoid_rewrite H_in_sort ⇒ //.
    +    } {
    +
    +      move : (LocallySorted_sort l).
    +      rewrite /is_true /le /leb //.
    +    }
    +  Qed.
    +
    End RemoveDupsFromSorted.
    @@ -222,113 +410,113 @@

    Library MSetsExtra.MSetListWithDups


    -Module Ops (X:OrderedType) <: WOps X.
    +Module Ops (X:OrderedType) <: WOps X.

      Module RDFS := RemoveDupsFromSorted X.
    -  Module Import MX := OrderedTypeFacts X.
    +  Module Import MX := OrderedTypeFacts X.

      Definition elt := X.t.
    -  Definition t := list elt.
    +  Definition t := list elt.

    -  Definition empty : t := nil.
    +  Definition empty : t := nil.

    -  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.
    -  Fixpoint mem (x : elt) (s : t) : bool :=
    +  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.
    +  Fixpoint mem (x : elt) (s : t) : bool :=
        match s with
    -    | nil ⇒ false
    -    | y :: l ⇒
    +    | nil ⇒ false
    +    | y :: l ⇒
               match X.compare x y with
    -               Eq ⇒ true
    +               Eq ⇒ true
                 | _ ⇒ mem x l
               end
        end.

    -  Definition add x (s : t) := x :: s.
    -  Definition singleton (x : elt) := x :: nil.
    +  Definition add x (s : t) := x :: s.
    +  Definition singleton (x : elt) := x :: nil.

    -  Fixpoint rev_filter_aux acc (f : elt → bool) s :=
    +  Fixpoint rev_filter_aux acc (f : elt → bool) s :=
        match s with
    -       nil ⇒ acc
    -     | (x :: xs) ⇒ rev_filter_aux (if (f x) then (x :: acc) else acc) f xs
    +       nil ⇒ acc
    +     | (x :: xs) ⇒ rev_filter_aux (if (f x) then (x :: acc) else acc) f xs
        end.
    -  Definition rev_filter := rev_filter_aux nil.
    +  Definition rev_filter := rev_filter_aux nil.

    -  Definition filter (f : elt → bool) (s : t) : t := rev_filter f s.
    +  Definition filter (f : elt → bool) (s : t) : t := rev_filter f s.

      Definition remove x s :=
    -    rev_filter (fun y ⇒ match X.compare x y with Eq ⇒ false | _ ⇒ true end) s.
    +    rev_filter (fun y ⇒ match X.compare x y with Eq ⇒ false | _ ⇒ true end) s.

      Definition union (s1 s2 : t) : t :=
    -    List.rev_append s2 s1.
    +    List.rev_append s2 s1.

      Definition inter (s1 s2 : t) : t :=
        rev_filter (fun y ⇒ mem y s2) s1.

    -  Definition elements (x : t) : list elt := x.
    +  Definition elements (x : t) : list elt := x.

    -  Definition elements_dist (x : t) : list elt :=
    -    RDFS.remove_dups_by_sortingA x.
    +  Definition elements_dist (x : t) : list elt :=
    +    RDFS.remove_dups_by_sortingA x.

    -  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    -    fold_left (flip f) (elements s) i.
    +  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    +    fold_left (flip f) (elements s) i.

      Definition diff (s s' : t) : t := fold remove s' s.

    -  Definition subset (s s' : t) : bool :=
    -    List.forallb (fun x ⇒ mem x s') s.
    +  Definition subset (s s' : t) : bool :=
    +    List.forallb (fun x ⇒ mem x s') s.

    -  Definition equal (s s' : t) : bool := andb (subset s s') (subset s' s).
    +  Definition equal (s s' : t) : bool := andb (subset s s') (subset s' s).

    -  Fixpoint for_all (f : elt → bool) (s : t) : bool :=
    +  Fixpoint for_all (f : elt → bool) (s : t) : bool :=
        match s with
    -    | nil ⇒ true
    -    | x :: l ⇒ if f x then for_all f l else false
    +    | nil ⇒ true
    +    | x :: l ⇒ if f x then for_all f l else false
        end.

    -  Fixpoint exists_ (f : elt → bool) (s : t) : bool :=
    +  Fixpoint exists_ (f : elt → bool) (s : t) : bool :=
        match s with
    -    | nil ⇒ false
    -    | x :: l ⇒ if f x then true else exists_ f l
    +    | nil ⇒ false
    +    | x :: l ⇒ if f x then true else exists_ f l
        end.

    -  Fixpoint partition_aux (a1 a2 : t) (f : elt → bool) (s : t) : t × t :=
    +  Fixpoint partition_aux (a1 a2 : t) (f : elt → bool) (s : t) : t × t :=
        match s with
    -    | nil ⇒ (a1, a2)
    -    | x :: l ⇒
    -        if f x then partition_aux (x :: a1) a2 f l else
    -                    partition_aux a1 (x :: a2) f l
    +    | nil ⇒ (a1, a2)
    +    | x :: l ⇒
    +        if f x then partition_aux (x :: a1) a2 f l else
    +                    partition_aux a1 (x :: a2) f l
        end.

    -  Definition partition := partition_aux nil nil.
    +  Definition partition := partition_aux nil nil.

    -  Definition cardinal (s : t) : nat := length (elements_dist s).
    +  Definition cardinal (s : t) : nat := length (elements_dist s).

    -  Definition choose (s : t) : option elt :=
    +  Definition choose (s : t) : option elt :=
         match s with
    -      | nil ⇒ None
    -      | x::_ ⇒ Some x
    +      | nil ⇒ None
    +      | x::_ ⇒ Some x
         end.

    @@ -352,7 +540,7 @@

    Library MSetsExtra.MSetListWithDups

    -Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
    +Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
      Include Ops E.
      Import MX.
    @@ -364,26 +552,36 @@

    Library MSetsExtra.MSetListWithDups

    Logical predicates
    -  Definition In x (s : t) := SetoidList.InA E.eq x s.
    +  Definition In x (s : t) := SetoidList.InA E.eq x s.

    -  Instance In_compat : Proper (E.eq==>eq==>iff) In.
    - +  Instance In_compat : Proper (E.eq==>eq==>iff) In.
    +  Proof. repeat red. intros. rewrite H H0. auto. Qed.
    +
    -  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    -  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    -  Definition Empty s := ∀ a : elt, ¬ In a s.
    -  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    -  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.
    +  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    +  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    +  Definition Empty s := ∀ a : elt, ¬ In a s.
    +  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    +  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.

    -  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
    -  Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).
    +  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
    +  Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).

    -  Definition eq : t → t → Prop := Equal.
    -  Lemma eq_equiv : Equivalence eq.
    - +  Definition eq : t → t → Prop := Equal.
    +  Lemma eq_equiv : Equivalence eq.
    +  Proof.
    +    constructor. {
    +      done.
    +    } {
    +      by constructor; rewrite H.
    +    } {
    +      by constructor; rewrite H H0.
    +    }
    +  Qed.
    +

    @@ -395,107 +593,376 @@

    Library MSetsExtra.MSetListWithDups


    -  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    - -
    -  Lemma mem_spec : ∀ s x, mem x s = true ↔ In x s.
    - -
    -  Lemma subset_spec : ∀ s s', subset s s' = true ↔ s[<=]s'.
    - -
    -  Lemma equal_spec : ∀ s s', equal s s' = true ↔ s[=]s'.
    - -
    -  Lemma eq_dec : ∀ x y : t, {eq x y} + {¬ eq x y}.
    +  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    + +
    +  Lemma mem_spec : ∀ s x, mem x s = true ↔ In x s.
    +  Proof.
    +    induction s as [| y s' IH]. {
    +      move ⇒ x.
    +      rewrite /= /In InA_nil.
    +      split ⇒ //.
    +    } {
    +      move ⇒ x.
    +      rewrite /= /In InA_cons.
    +      move : (MX.compare_eq_iff x y).
    +      case (E.compare x y). {
    +        tauto.
    +      } {
    +        rewrite IH; intuition; inversion H.
    +      } {
    +        rewrite IH; intuition; inversion H.
    +      }
    +    }
    + Qed.
    + +
    +  Lemma subset_spec : ∀ s s', subset s s' = true ↔ s[<=]s'.
    +  Proof.
    +    intros s s'.
    +    rewrite /subset forallb_forall /Subset /In.
    +    split. {
    +      move ⇒ H z /InA_alt [] x [H_z_eq] H_in.
    +      move : (H _ H_in).
    +      rewrite mem_spec.
    +      setoid_replace z with x ⇒ //.
    +    } {
    +      move ⇒ H z H_in.
    +      rewrite mem_spec.
    +      apply H, In_InA ⇒ //.
    +      apply E.eq_equiv.
    +    }
    +  Qed.
    + +
    +  Lemma equal_spec : ∀ s s', equal s s' = true ↔ s[=]s'.
    +  Proof.
    +    intros s s'.
    +    rewrite /Equal /equal Bool.andb_true_iff !subset_spec /Subset.
    +    split. {
    +      move ⇒ [H1 H2] a.
    +      split.
    +        - by apply H1.
    +        - by apply H2.
    +    } {
    +      move ⇒ H.
    +      split; move ⇒ a; rewrite H //.
    +    }
    +  Qed.
    + +
    +  Lemma eq_dec : ∀ x y : t, {eq x y} + {¬ eq x y}.
    +  Proof.
    +    intros x y.
    +    change ({Equal x y}+{¬Equal x y}).
    +    destruct (equal x y) eqn:H; [left|right];
    +     rewrite <- equal_spec; congruence.
    +  Qed.

      Lemma empty_spec : Empty empty.
    - -
    -  Lemma is_empty_spec : ∀ s, is_empty s = true ↔ Empty s.
    - -
    -  Lemma add_spec : ∀ s x y, In y (add x s) ↔ E.eq y x ∨ In y s.
    - -
    -  Lemma singleton_spec : ∀ x y, In y (singleton x) ↔ E.eq y x.
    - -
    -  Hint Resolve (@Equivalence_Reflexive _ _ E.eq_equiv).
    -  Hint Immediate (@Equivalence_Symmetric _ _ E.eq_equiv).
    -  Hint Resolve (@Equivalence_Transitive _ _ E.eq_equiv).
    +  Proof. rewrite /Empty /empty /In. move ⇒ a /InA_nil //. Qed.
    + +
    +  Lemma is_empty_spec : ∀ s, is_empty s = true ↔ Empty s.
    +  Proof.
    +    rewrite /is_empty /Empty /In.
    +    case; split ⇒ //. {
    +      move ⇒ _ a.
    +      rewrite InA_nil //.
    +    } {
    +      move ⇒ H; contradiction (H a).
    +      apply InA_cons_hd.
    +      apply Equivalence_Reflexive.
    +    }
    +  Qed.
    + +
    +  Lemma add_spec : ∀ s x y, In y (add x s) ↔ E.eq y x ∨ In y s.
    +  Proof.
    +    intros s x y.
    +    rewrite /add /In InA_cons //.
    +  Qed.
    + +
    +  Lemma singleton_spec : ∀ x y, In y (singleton x) ↔ E.eq y x.
    +  Proof.
    +    intros x y.
    +    rewrite /singleton /In InA_cons.
    +    split. {
    +      move ⇒ [] // /InA_nil //.
    +    } {
    +      by left.
    +    }
    +  Qed.
    + +
    +  Hint Resolve (@Equivalence_Reflexive _ _ E.eq_equiv).
    +  Hint Immediate (@Equivalence_Symmetric _ _ E.eq_equiv).
    +  Hint Resolve (@Equivalence_Transitive _ _ E.eq_equiv).
    + +
    +  Lemma rev_filter_aux_spec : ∀ s acc x f, compatb f →
    +    (In x (rev_filter_aux acc f s) ↔ (In x s ∧ f x = true) ∨ (In x acc)).
    +  Proof.
    +    intros s acc x f H_compat.
    +    move : x acc.
    +    induction s as [| y s' IH]. {
    +      intros x acc.
    +      rewrite /rev_filter_aux /In InA_nil.
    +      tauto.
    +    } {
    +      intros x acc.
    +      rewrite /= IH /In.
    +      case_eq (f y) ⇒ H_fy; rewrite !InA_cons; intuition. {
    +        left.
    +        split; first by left.
    +        setoid_replace x with y ⇒ //.
    +      } {
    +        contradict H1.
    +        setoid_replace x with y ⇒ //.
    +        by rewrite H_fy.
    +      }
    +    }
    +  Qed.
    + +
    +  Lemma filter_spec : ∀ s x f, compatb f →
    +    (In x (filter f s) ↔ In x s ∧ f x = true).
    +  Proof.
    +    intros s x f H_compat.
    +    rewrite /filter /rev_filter rev_filter_aux_spec /In InA_nil.
    +    tauto.
    +  Qed.
    + +
    +  Lemma remove_spec : ∀ s x y, In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    +  Proof.
    +    intros s x y.
    +    rewrite /remove /rev_filter.
    +    have H_compat : compatb ((fun y0 : elt ⇒
    +         match E.compare x y0 with
    +         | Eq ⇒ false
    +         | _ ⇒ true
    +         end)). {
    +      repeat red; intros.
    +      setoid_replace x0 with y0 ⇒ //.
    +    }
    +    rewrite rev_filter_aux_spec /In InA_nil.
    +    have → : (E.eq y x ↔ E.eq x y). {
    +      split; move ⇒ ?; by apply Equivalence_Symmetric.
    +    }
    +    rewrite -compare_eq_iff.
    +    case (E.compare x y). {
    +      intuition.
    +    } {
    +      intuition.
    +      inversion H0.
    +    } {
    +      intuition.
    +      inversion H0.
    +    }
    +  Qed.
    + +
    +  Lemma union_spec : ∀ s s' x, In x (union s s') ↔ In x s ∨ In x s'.
    +  Proof.
    +    intros s s' x.
    +    rewrite /union /In rev_append_rev InA_app_iff InA_rev; tauto.
    +  Qed.
    + +
    +  Lemma inter_spec : ∀ s s' x, In x (inter s s') ↔ In x s ∧ In x s'.
    +  Proof.
    +    intros s s' x.
    +    have H_compat : compatb (fun y : elt ⇒ mem y s'). {
    +      repeat red; intros.
    +      suff : ( mem x0 s' = true ↔ mem y s' = true). {
    +        case (mem y s'), (mem x0 s'); intuition.
    +      }
    +      rewrite !mem_spec /In.
    +      setoid_replace x0 with y ⇒ //.
    +    }
    +    rewrite /inter rev_filter_aux_spec mem_spec /In InA_nil.
    +    tauto.
    +  Qed.
    + +
    +  Lemma fold_spec : ∀ s (A : Type) (i : A) (f : elt → A → A),
    +    fold f s i = fold_left (flip f) (elements s) i.
    +  Proof. done. Qed.
    + +
    +  Lemma elements_spec1 : ∀ s x, InA E.eq x (elements s) ↔ In x s.
    +  Proof.
    +    intros s x.
    +    rewrite /elements /In //.
    +  Qed.
    + +
    +  Lemma diff_spec : ∀ s s' x, In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +  Proof.
    +    intros s s' x.
    +    rewrite /diff fold_spec -(elements_spec1 s').
    +    move : s.
    +    induction (elements s') as [| y ys IH] ⇒ s. {
    +      rewrite InA_nil /=; tauto.
    +    } {
    +      rewrite /= IH InA_cons /flip remove_spec.
    +      tauto.
    +    }
    +  Qed.
    + +
    +  Lemma cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    +  Proof. rewrite /cardinal //. Qed.
    + +
    +  Lemma for_all_spec : ∀ s f, compatb f →
    +    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +  Proof.
    +    intros s f H_compat.
    +    rewrite /For_all.
    +    induction s as [| x xs IH]. {
    +      rewrite /= /In.
    +      split ⇒ //.
    +      move ⇒ _ x /InA_nil //.
    +    } {
    +      rewrite /=.
    +      case_eq (f x) ⇒ H_fx. {
    +        rewrite IH.
    +        split. {
    +          move ⇒ H x' /InA_cons []. {
    +            move ⇒ → //.
    +          } {
    +            apply H.
    +          }
    +        } {
    +          move ⇒ H x' H_in.
    +          apply H.
    +          apply InA_cons.
    +          by right.
    +        }
    +      } {
    +        split ⇒ //.
    +        move ⇒ H.
    +        suff : f x = true. {
    +          rewrite H_fx //.
    +        }
    +        apply H.
    +        apply InA_cons_hd.
    +        done.
    +      }
    +    }
    +  Qed.
    + +
    +  Lemma exists_spec : ∀ s f, compatb f →
    +    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +  Proof.
    +    intros s f H_compat.
    +    rewrite /Exists.
    +    induction s as [| x xs IH]. {
    +      rewrite /= /In.
    +      split ⇒ //.
    +      move ⇒ [x] [] /InA_nil //.
    +    } {
    +      rewrite /=.
    +      case_eq (f x) ⇒ H_fx. {
    +        split ⇒ // _.
    +        âˆƒ x.
    +        split ⇒ //.
    +        apply InA_cons_hd.
    +        done.
    +      } {
    +        rewrite IH.
    +        split. {
    +          move ⇒ [x'] [H_in] H_fx'.
    +          âˆƒ x'.
    +          split ⇒ //.
    +          apply InA_cons.
    +          by right.
    +        } {
    +          move ⇒ [x'] [] /InA_cons []. {
    +            move ⇒ →.
    +            rewrite H_fx //.
    +          } {
    +            by ∃ x'.
    +          }
    +        }
    +      }
    +    }
    +  Qed.

    -  Lemma rev_filter_aux_spec : ∀ s acc x f, compatb f →
    -    (In x (rev_filter_aux acc f s) ↔ (In x s ∧ f x = true) ∨ (In x acc)).
    - -
    -  Lemma filter_spec : ∀ s x f, compatb f →
    -    (In x (filter f s) ↔ In x s ∧ f x = true).
    - -
    -  Lemma remove_spec : ∀ s x y, In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    - -
    -  Lemma union_spec : ∀ s s' x, In x (union s s') ↔ In x s ∨ In x s'.
    +  Lemma partition_aux_spec : ∀ a1 a2 s f,
    +    (partition_aux a1 a2 f s = (rev_filter_aux a1 f s, rev_filter_aux a2 (fun x ⇒ negb (f x)) s)).
    +  Proof.
    +    move ⇒ a1 a2 s f.
    +    move : a1 a2.
    +    induction s as [| x xs IH]. {
    +      rewrite /partition_aux /rev_filter_aux //.
    +    } {
    +      intros a1 a2.
    +      rewrite /= IH.
    +      case (f x) ⇒ //.

    -  Lemma inter_spec : ∀ s s' x, In x (inter s s') ↔ In x s ∧ In x s'.
    +    }
    +  Qed.

    -  Lemma fold_spec : ∀ s (A : Type) (i : A) (f : elt → A → A),
    -    fold f s i = fold_left (flip f) (elements s) i.
    - -
    -  Lemma elements_spec1 : ∀ s x, InA E.eq x (elements s) ↔ In x s.
    +  Lemma partition_spec1 : ∀ s f, compatb f →
    +    fst (partition f s) [=] filter f s.
    +  Proof.
    +    move ⇒ s f _.
    +    rewrite /partition partition_aux_spec /fst /filter /rev_filter //.
    +  Qed.

    -  Lemma diff_spec : ∀ s s' x, In x (diff s s') ↔ In x s ∧ ¬In x s'.
    - -
    -  Lemma cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    - -
    -  Lemma for_all_spec : ∀ s f, compatb f →
    -    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    - -
    -  Lemma exists_spec : ∀ s f, compatb f →
    -    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    - -
    -  Lemma partition_aux_spec : ∀ a1 a2 s f,
    -    (partition_aux a1 a2 f s = (rev_filter_aux a1 f s, rev_filter_aux a2 (fun x ⇒ negb (f x)) s)).
    - -
    -  Lemma partition_spec1 : ∀ s f, compatb f →
    -    fst (partition f s) [=] filter f s.
    - -
    -  Lemma partition_spec2 : ∀ s f, compatb f →
    -    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    - +  Lemma partition_spec2 : ∀ s f, compatb f →
    +    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    +  Proof.
    +    move ⇒ s f _.
    +    rewrite /partition partition_aux_spec /snd /filter /rev_filter //.
    +  Qed.
    +
    -  Lemma choose_spec1 : ∀ s x, choose s = Some x → In x s.
    - +  Lemma choose_spec1 : ∀ s x, choose s = Some x → In x s.
    +  Proof.
    +   move ⇒ [] // y s' x [->].
    +   rewrite /In.
    +   apply InA_cons_hd.
    +   apply Equivalence_Reflexive.
    +  Qed.
    +
    -  Lemma choose_spec2 : ∀ s, choose s = None → Empty s.
    - +  Lemma choose_spec2 : ∀ s, choose s = None → Empty s.
    +  Proof. move ⇒ [] // _ a. rewrite /In InA_nil //. Qed.
    +
      Lemma elements_dist_spec_full :
        âˆ€ s,
    -      Sorted E.lt (elements_dist s) ∧
    -      NoDupA E.eq (elements_dist s) ∧
    -      (∀ x, InA E.eq x (elements_dist s) ↔ InA E.eq x (elements s)).
    +      Sorted E.lt (elements_dist s) ∧
    +      NoDupA E.eq (elements_dist s) ∧
    +      (∀ x, InA E.eq x (elements_dist s) ↔ InA E.eq x (elements s)).
    +  Proof.
    +    move ⇒ s.
    +    rewrite /elements_dist /elements.
    +    move : (RDFS.remove_dups_by_sortingA_spec s).
    +    simpl.
    +    firstorder.
    +  Qed.

    -  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    -                                          InA E.eq x (elements s).
    - +  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    +                                          InA E.eq x (elements s).
    +  Proof. intros; apply elements_dist_spec_full. Qed.
    +
    -  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    - +  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    +  Proof. intros; apply elements_dist_spec_full. Qed.
    +
    End Make.
    diff --git a/html/MSetsExtra.MSetWithDups.html b/html/MSetsExtra.MSetWithDups.html index f4d3ac0..9ba7eed 100644 --- a/html/MSetsExtra.MSetWithDups.html +++ b/html/MSetsExtra.MSetWithDups.html @@ -66,7 +66,7 @@

    Library MSetsExtra.MSetWithDups


    -Require Import Coq.MSets.MSetInterface.
    +Require Import Coq.MSets.MSetInterface.
    Require Import mathcomp.ssreflect.ssreflect.

    @@ -91,68 +91,68 @@

    Library MSetsExtra.MSetWithDups

    -Module Type WSetsOnWithDups (E : DecidableType).
    -  Include WOps E.
    +Module Type WSetsOnWithDups (E : DecidableType).
    +  Include WOps E.

    -  Parameter In : elt → t → Prop.
    -  Declare Instance In_compat : Proper (E.eq==>eq==>iff) In.
    +  Parameter In : elt → t → Prop.
    +  Declare Instance In_compat : Proper (E.eq==>eq==>iff) In.

    -  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    -  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    -  Definition Empty s := ∀ a : elt, ¬ In a s.
    -  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    -  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.
    +  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    +  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    +  Definition Empty s := ∀ a : elt, ¬ In a s.
    +  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    +  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.

    -  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
    -  Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).
    +  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
    +  Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).

    -  Definition eq : t → t → Prop := Equal.
    -  Include IsEq.
    +  Definition eq : t → t → Prop := Equal.
    +  Include IsEq.
    eq is obviously an equivalence, for subtyping only
    -  Include HasEqDec.
    +  Include HasEqDec.

      Section Spec.
      Variable s s': t.
      Variable x y : elt.
    -  Variable f : elt → bool.
    -  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    +  Variable f : elt → bool.
    +  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).

    -  Parameter mem_spec : mem x s = true ↔ In x s.
    -  Parameter equal_spec : equal s s' = true ↔ s[=]s'.
    -  Parameter subset_spec : subset s s' = true ↔ s[<=]s'.
    +  Parameter mem_spec : mem x s = true ↔ In x s.
    +  Parameter equal_spec : equal s s' = true ↔ s[=]s'.
    +  Parameter subset_spec : subset s s' = true ↔ s[<=]s'.
      Parameter empty_spec : Empty empty.
    -  Parameter is_empty_spec : is_empty s = true ↔ Empty s.
    -  Parameter add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    -  Parameter remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    -  Parameter singleton_spec : In y (singleton x) ↔ E.eq y x.
    -  Parameter union_spec : In x (union s s') ↔ In x s ∨ In x s'.
    -  Parameter inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
    -  Parameter diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
    -  Parameter fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    -    fold f s i = fold_left (flip f) (elements s) i.
    -  Parameter filter_spec : compatb f →
    -    (In x (filter f s) ↔ In x s ∧ f x = true).
    -  Parameter for_all_spec : compatb f →
    -    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    -  Parameter exists_spec : compatb f →
    -    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    -  Parameter partition_spec1 : compatb f →
    -    fst (partition f s) [=] filter f s.
    -  Parameter partition_spec2 : compatb f →
    -    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    -  Parameter elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    -  Parameter choose_spec1 : choose s = Some x → In x s.
    -  Parameter choose_spec2 : choose s = None → Empty s.
    +  Parameter is_empty_spec : is_empty s = true ↔ Empty s.
    +  Parameter add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    +  Parameter remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    +  Parameter singleton_spec : In y (singleton x) ↔ E.eq y x.
    +  Parameter union_spec : In x (union s s') ↔ In x s ∨ In x s'.
    +  Parameter inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
    +  Parameter diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +  Parameter fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    +    fold f s i = fold_left (flip f) (elements s) i.
    +  Parameter filter_spec : compatb f →
    +    (In x (filter f s) ↔ In x s ∧ f x = true).
    +  Parameter for_all_spec : compatb f →
    +    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +  Parameter exists_spec : compatb f →
    +    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +  Parameter partition_spec1 : compatb f →
    +    fst (partition f s) [=] filter f s.
    +  Parameter partition_spec2 : compatb f →
    +    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    +  Parameter elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    +  Parameter choose_spec1 : choose s = Some x → In x s.
    +  Parameter choose_spec2 : choose s = None → Empty s.

      End Spec.
    @@ -174,7 +174,7 @@

    Library MSetsExtra.MSetWithDups

    the original behavior of elements.
    -Module Type WSetsOnWithDupsExtra (E : DecidableType).
    +Module Type WSetsOnWithDupsExtra (E : DecidableType).
      Include WSetsOnWithDups E.

    @@ -184,14 +184,14 @@

    Library MSetsExtra.MSetWithDups

    An operation for getting an elements list without duplicates
    -  Parameter elements_dist : t → list elt.
    +  Parameter elements_dist : t → list elt.

    -  Parameter elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    -                                              InA E.eq x (elements s).
    +  Parameter elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    +                                              InA E.eq x (elements s).

    -  Parameter elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    +  Parameter elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).

    @@ -200,7 +200,7 @@

    Library MSetsExtra.MSetWithDups

    Cardinality can then be specified with respect to elements_dist.
    -  Parameter cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    +  Parameter cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    End WSetsOnWithDupsExtra.

    @@ -219,7 +219,7 @@

    Library MSetsExtra.MSetWithDups


    -Module WSetsOn_TO_WSetsOnWithDupsExtra (E : DecidableType) (W : WSetsOn E) <:
    +Module WSetsOn_TO_WSetsOnWithDupsExtra (E : DecidableType) (W : WSetsOn E) <:
      WSetsOnWithDupsExtra E.

    @@ -229,12 +229,14 @@

    Library MSetsExtra.MSetWithDups

      Definition elements_dist := W.elements.

    -  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    -                                          InA E.eq x (elements s).
    - +  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    +                                          InA E.eq x (elements s).
    +  Proof. done. Qed.
    +
    -  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    - +  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    +  Proof. apply elements_spec2w. Qed.
    +
    End WSetsOn_TO_WSetsOnWithDupsExtra.
    diff --git a/html/index.html b/html/index.html index 51c254e..eb94472 100644 --- a/html/index.html +++ b/html/index.html @@ -627,8 +627,8 @@

    Global Index

    Make.Subset [definition, in MSetsExtra.MSetListWithDups]
    Make.subset_spec [lemma, in MSetsExtra.MSetListWithDups]
    Make.union_spec [lemma, in MSetsExtra.MSetListWithDups]
    -_ [<=] _ [notation, in MSetsExtra.MSetListWithDups]
    -_ [=] _ [notation, in MSetsExtra.MSetListWithDups]
    +_ [<=] _ [notation, in MSetsExtra.MSetListWithDups]
    +_ [=] _ [notation, in MSetsExtra.MSetListWithDups]
    MSetFoldWithAbort [library]
    MSetIntervals [module, in MSetsExtra.MSetIntervals]
    MSetIntervals [library]
    @@ -1042,7 +1042,7 @@

    Global Index

    RemoveDupsFromSorted.XTotalLeBool.leb [definition, in MSetsExtra.MSetListWithDups]
    RemoveDupsFromSorted.XTotalLeBool.leb_total [lemma, in MSetsExtra.MSetListWithDups]
    RemoveDupsFromSorted.XTotalLeBool.t [definition, in MSetsExtra.MSetListWithDups]
    -_ <=? _ [notation, in MSetsExtra.MSetListWithDups]
    +_ <=? _ [notation, in MSetsExtra.MSetListWithDups]
    rev_map_alt_def [lemma, in MSetsExtra.MSetIntervals]
    rev_map_aux_alt_def [lemma, in MSetsExtra.MSetIntervals]
    rev_map [definition, in MSetsExtra.MSetIntervals]
    @@ -1092,8 +1092,8 @@

    Global Index

    WSetsOnWithDups.Subset [definition, in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.subset_spec [axiom, in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.union_spec [axiom, in MSetsExtra.MSetWithDups]
    -_ [<=] _ [notation, in MSetsExtra.MSetWithDups]
    -_ [=] _ [notation, in MSetsExtra.MSetWithDups]
    +_ [<=] _ [notation, in MSetsExtra.MSetWithDups]
    +_ [=] _ [notation, in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w [lemma, in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1 [lemma, in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist [definition, in MSetsExtra.MSetWithDups]
    @@ -1110,13 +1110,13 @@

    Global Index



    Notation Index

    M

    -_ [<=] _ [in MSetsExtra.MSetListWithDups]
    -_ [=] _ [in MSetsExtra.MSetListWithDups]
    +_ [<=] _ [in MSetsExtra.MSetListWithDups]
    +_ [=] _ [in MSetsExtra.MSetListWithDups]


    R

    -_ <=? _ [in MSetsExtra.MSetListWithDups]
    +_ <=? _ [in MSetsExtra.MSetListWithDups]


    W

    -_ [<=] _ [in MSetsExtra.MSetWithDups]
    -_ [=] _ [in MSetsExtra.MSetWithDups]
    +_ [<=] _ [in MSetsExtra.MSetWithDups]
    +_ [=] _ [in MSetsExtra.MSetWithDups]


    Module Index

    E

    From 27b0ba5dd086b83cd176a801086c599a48d536ee Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Thomas=20T=C3=BCrk?= Date: Fri, 30 Jun 2023 13:46:51 +0200 Subject: [PATCH 7/7] adapt to Coq 8.16.1 --- MSetFoldWithAbort.v | 8 +- MSetIntervals.v | 139 +- MSetListWithDups.v | 17 +- MSetWithDups.v | 2 +- Makefile | 441 ++- all.pdf | Bin 1106488 -> 1241338 bytes html/MSetsExtra.MSetFoldWithAbort.html | 986 +++--- html/MSetsExtra.MSetIntervals.html | 3900 ++++++++++++------------ html/MSetsExtra.MSetListWithDups.html | 528 ++-- html/MSetsExtra.MSetWithDups.html | 130 +- html/coqdoc.css | 9 + html/index.html | 3353 +++++++++++++++++++- html/toc.html | 8 +- 13 files changed, 6397 insertions(+), 3124 deletions(-) diff --git a/MSetFoldWithAbort.v b/MSetFoldWithAbort.v index f9c212c..145290f 100644 --- a/MSetFoldWithAbort.v +++ b/MSetFoldWithAbort.v @@ -861,7 +861,7 @@ End SetsWithFoldA. (** *** GenTree implementation Finally, provide such a fold with abort operation for generic trees. *) Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp) - (Import Raw:Ops E I) + (Import Raw: MSetGenTree.Ops E I) (M : MSetGenTree.Props E I Raw). Fixpoint foldWithAbort_Raw {A B: Type} (f_pre : E.t -> B) f_lt (f: E.t -> B -> A -> A) f_gt t (base: A) : A := @@ -926,7 +926,7 @@ Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp) M.bst l /\ M.bst r /\ M.lt_tree e l /\ M.gt_tree e r. { inversion H_bst. done. } - + have H_sub_l : (forall e0 : E.t, M.In e0 l -> M.In e0 s /\ E.lt e0 e). { intros e0 H_in_l. split; last by apply H_lt_tree_l. @@ -934,7 +934,7 @@ Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp) rewrite /M.In M.In_node_iff. tauto. } - move : (IHl H_bst_l) => {IHl} IHl {H_bst_l} {H_lt_tree_l}. + move : (IHl H_bst_l) => {} IHl {H_bst_l} {H_lt_tree_l}. have H_sub_r : (forall e0 : E.t, M.In e0 r -> M.In e0 s /\ E.lt e e0). { intros e0 H_in_r. split; last by apply H_gt_tree_r. @@ -942,7 +942,7 @@ Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp) rewrite /M.In M.In_node_iff. tauto. } - move : (IHr H_bst_r) => {IHr} IHr {H_bst_r} {H_gt_tree_r}. + move : (IHr H_bst_r) => {} IHr {H_bst_r} {H_gt_tree_r}. have H_in_e : M.In e s. { eapply H_sub. rewrite /M.In M.In_node_iff. diff --git a/MSetIntervals.v b/MSetIntervals.v index 2828068..4a28c26 100644 --- a/MSetIntervals.v +++ b/MSetIntervals.v @@ -46,6 +46,7 @@ Require Import mathcomp.ssreflect.ssreflect. Require Import NArith. Require Import ZArith. Require Import NOrder. +Require Import Lia. Require Import DecidableTypeEx. Module Import NOP := NOrderProp N. Open Scope Z_scope. @@ -715,8 +716,6 @@ Module Raw (Enc : ElementEncode). Section ForNotations. Class Ok (s:t) : Prop := ok : IsOk s. - Hint Resolve @ok. - Hint Unfold Ok. Instance IsOk_Ok s `(Hs : IsOk s) : Ok s := { ok := Hs }. Definition In x s := (SetoidList.InA Enc.E.eq x (elements s)). @@ -792,14 +791,14 @@ Module Raw (Enc : ElementEncode). induction c as [| c' IH] using N.peano_ind. { intros y x. rewrite elementsZ_single_base Z.add_0_r /=. - omega. + lia. } { intros y x. rewrite elementsZ_single_succ in_app_iff IH /= N2Z.inj_succ Z.add_succ_r Z.lt_succ_r. split. { move => [ | []] //. { move => [H_x_le H_y_le]. - omega. + lia. } { move => <-. split. @@ -808,7 +807,7 @@ Module Raw (Enc : ElementEncode). } } { move => [H_x_le] H_y_lt. - omega. + lia. } } Qed. @@ -819,7 +818,7 @@ Module Raw (Enc : ElementEncode). Proof. intros y x. rewrite In_elementsZ_single /= Z.add_1_r Z.lt_succ_r. - omega. + lia. Qed. Lemma length_elementsZ_single : forall cx x, @@ -1490,11 +1489,18 @@ Module Raw (Enc : ElementEncode). (forall x, InZ x s -> y < x)). Proof. intros s y H. + rewrite interval_list_elements_greater_alt_def //. - firstorder. - apply Z.nle_gt. - move => H_lt. - eapply H0; eauto. + split. { + move => H_notInZ x H_inZ. + apply Z.nle_gt. + move => H_lt. + move : (H_notInZ x H_lt H_inZ) => //. + } { + move => H_lt x H_le H_inZ. + move : (H_lt x H_inZ). + lia. + } Qed. Lemma interval_list_elements_greater_intro : forall s y, @@ -1838,7 +1844,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, intros x c1 c2. rewrite /merge_interval_size. replace (x + Z.of_N c1 + Z.of_N c2 - x) with - (Z.of_N c1 + Z.of_N c2) by omega. + (Z.of_N c1 + Z.of_N c2) by lia. rewrite -N2Z.inj_add N2Z.id. apply N.max_r, N.le_add_r. Qed. @@ -1853,7 +1859,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, by apply Zle_minus_le_0. } rewrite -Z.add_max_distr_l. - replace (y1 + (y2 + Z.of_N c2 - y1)) with (y2 + Z.of_N c2) by omega. + replace (y1 + (y2 + Z.of_N c2 - y1)) with (y2 + Z.of_N c2) by lia. done. Qed. @@ -2003,7 +2009,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, firstorder. } { move => /interval_list_invariant_cons [H_greater] [H_c_neq_0] H_inv_c'. - move : (IH H_inv_c') => {IH} IH. + move : (IH H_inv_c') => {} IH. rewrite addZ_alt_def. have H_succ : forall z, z + Z.of_N 1 = Z.succ z by done. move : (interval_1_compare_elim x z c). @@ -2051,7 +2057,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, } { move => /interval_list_invariant_cons [H_greater] [H_c_neq_0] H_inv_c'. - move : (IH H_inv_c') => {IH} IH. + move : (IH H_inv_c') => {} IH. rewrite addZ_alt_def. have H_succ : forall z, z + Z.of_N 1 = Z.succ z by done. move : (interval_1_compare_elim x z c). @@ -2128,7 +2134,8 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, have Hs' := (add_ok s x Hs). rewrite !In_InZ. rewrite /add addZ_InZ. { - rewrite -Enc.encode_eq; firstorder. + rewrite -Enc.encode_eq /Z.eq. + firstorder. } { apply Hs. } @@ -2337,13 +2344,13 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, rewrite !interval_list_invariant_cons. move => [H_gr_l1] [H_c1_neq_0] H_inv_l1. move => [H_gr_l2] [H_c2_neq_0] H_inv_l2. - move : (IH2 H_inv_s1 H_inv_l2) => {IH2} IH2. + move : (IH2 H_inv_s1 H_inv_l2) => {} IH2. have : forall s2 : t, interval_list_invariant s2 = true -> forall y : Z, InZ y (union l1 s2) <-> InZ y l1 \/ InZ y s2. { intros. by apply IH1. } - move => {IH1} IH1 y. + move => {} IH1 y. rewrite union_alt_def. move : (interval_compare_elim y1 c1 y2 c2). case (interval_compare (y1, c1) (y2, c2)). { @@ -2440,7 +2447,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, } rewrite !In_elementsZ_single. move => [H_y1_le H_y_lt]. - split; omega. + split; lia. } { move => [H_y1_le] [H_y2_c2_le] _. rewrite IH2. @@ -2451,7 +2458,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, } rewrite !In_elementsZ_single. move => [H_y2_le H_y_lt]. - split; omega. + split; lia. } { rewrite !InZ_cons IH2 !InZ_cons. tauto. @@ -2508,13 +2515,13 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, rewrite !interval_list_invariant_cons. move => [H_gr_l1] [H_c1_neq_0] H_inv_l1. move => [H_gr_l2] [H_c2_neq_0] H_inv_l2. - move : (IH2 H_inv_s1 H_inv_l2) => {IH2} IH2. + move : (IH2 H_inv_s1 H_inv_l2) => {} IH2. have : forall s2 : t, interval_list_invariant s2 = true -> interval_list_invariant (union l1 s2) = true. { intros. by apply IH1. } - move => {IH1} IH1. + move => {} IH1. rewrite union_alt_def. move : (interval_compare_elim y1 c1 y2 c2). @@ -2792,8 +2799,8 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, } } } - move => {IH} IH. - + move => {} IH. + clear H_inv_acc H_in_acc_lt acc. move : (interval_compare_elim y1 c1 y2 c2) H_inter_aux_eq. unfold inter_aux. @@ -2806,7 +2813,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => x. rewrite InZ_nil. split => //. - omega. + lia. } { apply Z.le_trans with (m := y2). { by apply Z.lt_le_incl. @@ -2822,7 +2829,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => x. rewrite InZ_nil. split => //. - omega. + lia. } { rewrite H_eq_y2. apply Z_le_add_r. @@ -2834,9 +2841,9 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, by apply Z_to_N_minus_neq_0. } { move => x. - rewrite InZ_cons InZ_nil In_elementsZ_single Z2N.id; last omega. - replace (y1 + (y2 - y1)) with y2 by omega. - split; omega. + rewrite InZ_cons InZ_nil In_elementsZ_single Z2N.id; last lia. + replace (y1 + (y2 - y1)) with y2 by lia. + split; lia. } { by apply Z.lt_le_incl. } @@ -2850,15 +2857,15 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => y. have H_yc2_intro : y1 + Z.of_N (Z.to_N (y2 + Z.of_N c2 - y1)) = y2 + Z.of_N c2. { - rewrite Z2N.id; omega. + rewrite Z2N.id; lia. } rewrite !InZ_cons !In_elementsZ_single InZ_nil H_yc2_intro. split. { move => [] //. move => [H_y1_le] H_y_lt. - split; last by omega. - left. omega. + split; last by lia. + left. lia. } { move => [H_in_s] [H_y2_le] H_y_lt. left. @@ -2921,7 +2928,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => [] //. move => [H_y1_le] H_y_lt. split; first done. - split; omega. + split; lia. } { move => [?] _. by left. @@ -2941,8 +2948,8 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, split. { move => [] //. move => [H_y2_le] H_y_lt. - split; last by omega. - left. omega. + split; last by lia. + left. lia. } { move => [H_in_s] [H_y2_le] H_y_lt. by left. @@ -2966,14 +2973,14 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => [] []. { move => [H_y1_le_y] H_y_lt_yc1. move => [H_y2_le_y] H_y_lt_yc2. - omega. + lia. } { move => /H_gr_s1'_alt H_lt_y [_] H_y_lt. suff : (y1 + Z.of_N c1 < y1). { apply Z.nlt_ge. apply Z_le_add_r. } - omega. + lia. } } { tauto. @@ -2994,14 +3001,14 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => [] []. { move => [H_y1_le_y] H_y_lt_yc1. move => [H_y2_le_y] H_y_lt_yc2. - omega. + lia. } { move => /H_gr_s1'_alt H_lt_y [_] H_y_lt. suff : (y1 + Z.of_N c1 < y1). { apply Z.nlt_ge. apply Z_le_add_r. } - omega. + lia. } } { tauto. @@ -3348,11 +3355,11 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, split; last by assumption. move : H_in => [] H_in; last by right. left. - omega. + lia. } } } - move => {IH} IH. + move => {} IH. clear H_inv_acc H_in_s1 H_in_acc acc. move : (interval_compare_elim y1 c1 y2 c2) H_diff_aux_eq. @@ -3396,8 +3403,8 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, by apply Z_to_N_minus_neq_0. } { move => x. - rewrite InZ_cons In_elementsZ_single Z2N.id; last omega. - replace (y1 + (y2 - y1)) with y2 by omega. + rewrite InZ_cons In_elementsZ_single Z2N.id; last lia. + replace (y1 + (y2 - y1)) with y2 by lia. split; last tauto. move => [] //. move => [H_y1_le] H_x_lt. @@ -3412,7 +3419,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, clear IH P1 P2 P3 H_comp. subst. have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. { - rewrite Z2N.id; omega. + rewrite Z2N.id; lia. } have H_nin_yc2 : forall y, InZ y s1' -> ~ y2 <= y < y2 + Z.of_N c2. { @@ -3429,7 +3436,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => []. { move => [H_le_y] H_y_lt. split. { - left; omega. + left; lia. } { move => [_]. by apply Z.nlt_ge. @@ -3440,7 +3447,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, } { move => [] []; last by right; right. move => [H_y_ge] H_y_lt_yc1 H_nin_yc2'. - right; left. omega. + right; left. lia. } } { rewrite interval_list_invariant_cons H_yc1_intro. @@ -3477,7 +3484,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, } { move => x. split; first done. - omega. + lia. } { assumption. } @@ -3487,10 +3494,10 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, clear IH P1 P2 P3 H_comp. subst. have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. { - rewrite Z2N.id; omega. + rewrite Z2N.id; lia. } have H_y1_intro : y1 + Z.of_N (Z.to_N (y2 - y1)) = y2. { - rewrite Z2N.id; omega. + rewrite Z2N.id; lia. } split; last split. { move => y. @@ -3589,10 +3596,10 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, split; first assumption. rewrite In_elementsZ_single. move => [] H_y2_le H_y_lt. - case H_y_in; first by omega. + case H_y_in; first by lia. move => /H_gr_s1'_alt H_lt_y. suff : y2 + Z.of_N c2 < y. { - move => ?. omega. + move => ?. lia. } apply Z.lt_trans with (m := y1 + Z.of_N c1) => //. apply Z.lt_le_trans with (m := y1) => //. @@ -3617,10 +3624,10 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, split; first assumption. rewrite In_elementsZ_single. move => [] H_y2_le H_y_lt. - case H_y_in; first by omega. + case H_y_in; first by lia. move => /H_gr_s1'_alt H_lt_y. suff : y2 + Z.of_N c2 < y. { - move => ?. omega. + move => ?. lia. } apply Z.lt_trans with (m := (y2 + Z.of_N c2) + Z.of_N c1) => //. by apply Z_lt_add_r. @@ -3885,7 +3892,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, } move : (H_y2_nlt) (H_y2_lt) => /Z.ltb_nlt -> /Z.ltb_lt ->. - have H_y1_eq : (y1 = y2) by omega. + have H_y1_eq : (y1 = y2) by lia. have H_yc1_eq : (y1 + Z.of_N c1 = Z.succ y2). { apply Z.le_antisymm. { move : H_yc1_le. @@ -4130,7 +4137,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => [_] [H_c1_neq_0] _ _. split => //. move => H; move : (H y1). - rewrite InZ_nil => {H} H. + rewrite InZ_nil => {} H. contradict H. rewrite InZ_cons; left. by apply In_elementsZ_single_hd. @@ -4140,14 +4147,14 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, rewrite !interval_list_invariant_cons. move => [H_gr_l1] [H_c1_neq_0] H_inv_l1. move => [H_gr_l2] [H_c2_neq_0] H_inv_l2. - move : (IH2 H_inv_s1 H_inv_l2) => {IH2} IH2. + move : (IH2 H_inv_s1 H_inv_l2) => {} IH2. have : forall s2 : t, interval_list_invariant s2 = true -> (subset l1 s2 = true <-> (forall y : Z, InZ y l1 -> InZ y s2)). { intros. by apply IH1. } - move => {IH1} IH1. + move => {} IH1. have H_yc2_nin : ~ InZ (y2 + Z.of_N c2) ((y2, c2) :: l2). { rewrite !InZ_cons !In_elementsZ_single. @@ -4209,7 +4216,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, rewrite !In_elementsZ_single. move => [H_y1_le] H_y_lt. left. - omega. + lia. } { move => H_pre H_y_in_l1. apply H_pre. @@ -4246,7 +4253,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => [H_y2_le H_y_lt]. move : H_y_in => []. { move => [H_y1_le] H_y_lt'. - omega. + lia. } { eapply Nin_elements_greater; eauto. apply Z.le_trans with (m := y1); last first. { @@ -4598,10 +4605,10 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, move => /Z.lt_asymm //. } { move => [] H_y1_eq. - exfalso. omega. + exfalso. lia. } { move => [] H_y1_eq. - exfalso. omega. + exfalso. lia. } } Qed. @@ -4946,7 +4953,7 @@ Lemma elementsZ_insert_intervalZ_guarded : forall x c s, case s' as [| [y' c'] s'']. { move => [<-]. rewrite InZ_cons InZ_nil In_elementsZ_single. - omega. + lia. } { move => H_max_eq. rewrite InZ_cons. @@ -5665,7 +5672,7 @@ Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E. Record t_ := Mkt {this :> Raw.t; is_ok : Raw.Ok this}. Definition t := t_. Arguments Mkt this {is_ok}. - Hint Resolve is_ok : typeclass_instances. + #[local] Hint Resolve is_ok : typeclass_instances. Definition In (x : elt)(s : t) := Raw.In x s.(this). Definition Equal (s s' : t) := forall a : elt, In a s <-> In a s'. @@ -5698,7 +5705,7 @@ Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E. Definition partition (f : elt -> bool)(s : t) : t * t := let p := Raw.partition f s in (Mkt (fst p), Mkt (snd p)). - Instance In_compat : Proper (E.eq==>eq==>iff) In. + #[local] Instance In_compat : Proper (E.eq==>eq==>iff) In. Proof. repeat red. move => x y H_eq_xy x' y' ->. @@ -5709,7 +5716,7 @@ Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E. Definition eq : t -> t -> Prop := Equal. - Instance eq_equiv : Equivalence eq. + #[local] Instance eq_equiv : Equivalence eq. Proof. firstorder. Qed. Definition eq_dec : forall (s s':t), { eq s s' }+{ ~eq s s' }. @@ -5722,7 +5729,7 @@ Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E. Definition lt : t -> t -> Prop := Raw.lt. - Instance lt_strorder : StrictOrder lt. + #[local] Instance lt_strorder : StrictOrder lt. Proof. unfold lt. constructor. { @@ -5738,7 +5745,7 @@ Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E. } Qed. - Instance lt_compat : Proper (eq==>eq==>iff) lt. + #[local] Instance lt_compat : Proper (eq==>eq==>iff) lt. Proof. repeat red. move => [x1 H_x1_ok] [y1 H_y1_ok] H_eq. diff --git a/MSetListWithDups.v b/MSetListWithDups.v index c5e888b..fefef58 100644 --- a/MSetListWithDups.v +++ b/MSetListWithDups.v @@ -325,8 +325,7 @@ Module RemoveDupsFromSorted (Import X:OrderedType). rewrite !InA_alt. setoid_rewrite H_in_sort => //. } { - - move : (LocallySorted_sort l). + move : (Sorted_sort l). rewrite /is_true /le /leb //. } Qed. @@ -447,7 +446,7 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E. (** Logical predicates *) Definition In x (s : t) := SetoidList.InA E.eq x s. - Instance In_compat : Proper (E.eq==>eq==>iff) In. + #[local] Instance In_compat : Proper (E.eq==>eq==>iff) In. Proof. repeat red. intros. rewrite H H0. auto. Qed. Definition Equal s s' := forall a : elt, In a s <-> In a s'. @@ -489,9 +488,9 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E. case (E.compare x y). { tauto. } { - rewrite IH; intuition; inversion H. + rewrite IH; intuition; inversion H1. } { - rewrite IH; intuition; inversion H. + rewrite IH; intuition; inversion H1. } } Qed. @@ -570,10 +569,6 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E. } Qed. - Hint Resolve (@Equivalence_Reflexive _ _ E.eq_equiv). - Hint Immediate (@Equivalence_Symmetric _ _ E.eq_equiv). - Hint Resolve (@Equivalence_Transitive _ _ E.eq_equiv). - Lemma rev_filter_aux_spec : forall s acc x f, compatb f -> (In x (rev_filter_aux acc f s) <-> (In x s /\ f x = true) \/ (In x acc)). Proof. @@ -717,7 +712,7 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E. } apply H. apply InA_cons_hd. - done. + apply (@Equivalence_Reflexive _ _ E.eq_equiv). } } Qed. @@ -738,7 +733,7 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E. exists x. split => //. apply InA_cons_hd. - done. + apply (@Equivalence_Reflexive _ _ E.eq_equiv). } { rewrite IH. split. { diff --git a/MSetWithDups.v b/MSetWithDups.v index 2c4a0c6..87d7d4b 100644 --- a/MSetWithDups.v +++ b/MSetWithDups.v @@ -67,7 +67,7 @@ Module Type WSetsOnWithDups (E : DecidableType). Include WOps E. Parameter In : elt -> t -> Prop. - Declare Instance In_compat : Proper (E.eq==>eq==>iff) In. + #[local] Declare Instance In_compat : Proper (E.eq==>eq==>iff) In. Definition Equal s s' := forall a : elt, In a s <-> In a s'. Definition Subset s s' := forall a : elt, In a s -> In a s'. diff --git a/Makefile b/Makefile index 5e064e4..b10449a 100644 --- a/Makefile +++ b/Makefile @@ -1,10 +1,13 @@ -############################################################################### -## v # The Coq Proof Assistant ## -## /dev/null 2>/dev/null; echo $$?)) +ifeq (0,$(shell command time -f "" true >/dev/null 2>/dev/null; echo $$?)) STDTIME?=command time -f $(TIMEFMT) else -ifeq (0,$(shell gtime -f $(TIMEFMT) true >/dev/null 2>/dev/null; echo $$?)) +ifeq (0,$(shell gtime -f "" true >/dev/null 2>/dev/null; echo $$?)) STDTIME?=gtime -f $(TIMEFMT) else STDTIME?=command time @@ -81,31 +91,37 @@ else STDTIME?=command time -f $(TIMEFMT) endif +COQBIN?= +ifneq (,$(COQBIN)) +# add an ending / +COQBIN:=$(COQBIN)/ +endif + # Coq binaries COQC ?= "$(COQBIN)coqc" COQTOP ?= "$(COQBIN)coqtop" COQCHK ?= "$(COQBIN)coqchk" +COQNATIVE ?= "$(COQBIN)coqnative" COQDEP ?= "$(COQBIN)coqdep" COQDOC ?= "$(COQBIN)coqdoc" +COQPP ?= "$(COQBIN)coqpp" COQMKFILE ?= "$(COQBIN)coq_makefile" +OCAMLLIBDEP ?= "$(COQBIN)ocamllibdep" # Timing scripts -COQMAKE_ONE_TIME_FILE ?= "$(COQLIB)/tools/make-one-time-file.py" -COQMAKE_BOTH_TIME_FILES ?= "$(COQLIB)/tools/make-both-time-files.py" -COQMAKE_BOTH_SINGLE_TIMING_FILES ?= "$(COQLIB)/tools/make-both-single-timing-files.py" +COQMAKE_ONE_TIME_FILE ?= "$(COQCORELIB)/tools/make-one-time-file.py" +COQMAKE_BOTH_TIME_FILES ?= "$(COQCORELIB)/tools/make-both-time-files.py" +COQMAKE_BOTH_SINGLE_TIMING_FILES ?= "$(COQCORELIB)/tools/make-both-single-timing-files.py" BEFORE ?= AFTER ?= -# FIXME this should be generated by Coq (modules already linked by Coq) -CAMLDONTLINK=camlp5.gramlib,unix,str - # OCaml binaries CAMLC ?= "$(OCAMLFIND)" ocamlc -c CAMLOPTC ?= "$(OCAMLFIND)" opt -c -CAMLLINK ?= "$(OCAMLFIND)" ocamlc -linkpkg -dontlink $(CAMLDONTLINK) -CAMLOPTLINK ?= "$(OCAMLFIND)" opt -linkpkg -dontlink $(CAMLDONTLINK) +CAMLLINK ?= "$(OCAMLFIND)" ocamlc -linkall +CAMLOPTLINK ?= "$(OCAMLFIND)" opt -linkall CAMLDOC ?= "$(OCAMLFIND)" ocamldoc -CAMLDEP ?= "$(OCAMLFIND)" ocamldep -slash -ml-synonym .ml4 -ml-synonym .mlpack +CAMLDEP ?= "$(OCAMLFIND)" ocamldep -slash -ml-synonym .mlpack # DESTDIR is prepended to all installation paths DESTDIR ?= @@ -116,11 +132,20 @@ COQDEBUG ?= # Extra packages to be linked in (as in findlib -package) CAMLPKGS ?= +FINDLIBPKGS = -package coq-core.plugins.ltac $(CAMLPKGS) # Option for making timing files TIMING?= # Option for changing sorting of timing output file TIMING_SORT_BY ?= auto +# Option for changing the fuzz parameter on the output file +TIMING_FUZZ ?= 0 +# Option for changing whether to use real or user time for timing tables +TIMING_REAL?= +# Option for including the memory column(s) +TIMING_INCLUDE_MEM?= +# Option for sorting by the memory column +TIMING_SORT_BY_MEM?= # Output file names for timed builds TIME_OF_BUILD_FILE ?= time-of-build.log TIME_OF_BUILD_BEFORE_FILE ?= time-of-build-before.log @@ -129,6 +154,45 @@ TIME_OF_PRETTY_BUILD_FILE ?= time-of-build-pretty.log TIME_OF_PRETTY_BOTH_BUILD_FILE ?= time-of-build-both.log TIME_OF_PRETTY_BUILD_EXTRA_FILES ?= - # also output to the command line +TGTS ?= + +# Retro compatibility (DESTDIR is standard on Unix, DSTROOT is not) +ifdef DSTROOT +DESTDIR := $(DSTROOT) +endif + +# Substitution of the path by appending $(DESTDIR) if needed. +# The variable $(COQMF_WINDRIVE) can be needed for Cygwin environments. +windrive_path = $(if $(COQMF_WINDRIVE),$(subst $(COQMF_WINDRIVE),/,$(1)),$(1)) +destination_path = $(if $(DESTDIR),$(DESTDIR)/$(call windrive_path,$(1)),$(1)) + +# Installation paths of libraries and documentation. +COQLIBINSTALL ?= $(call destination_path,$(COQLIB)/user-contrib) +COQDOCINSTALL ?= $(call destination_path,$(DOCDIR)/coq/user-contrib) +COQPLUGININSTALL ?= $(call destination_path,$(COQCORELIB)/..) +COQTOPINSTALL ?= $(call destination_path,$(COQLIB)/toploop) # FIXME: Unused variable? + +# findlib files installation +FINDLIBPREINST= mkdir -p "$(COQPLUGININSTALL)/" +FINDLIBDESTDIR= -destdir "$(COQPLUGININSTALL)/" + +# we need to move out of sight $(METAFILE) otherwise findlib thinks the +# package is already installed +findlib_install = \ + $(HIDE)if [ "$(METAFILE)" ]; then \ + $(FINDLIBPREINST) && \ + mv "$(METAFILE)" "$(METAFILE).skip" ; \ + "$(OCAMLFIND)" install $(2) $(FINDLIBDESTDIR) $(FINDLIBPACKAGE) $(1); \ + rc=$$?; \ + mv "$(METAFILE).skip" "$(METAFILE)"; \ + exit $$rc; \ + fi +findlib_remove = \ + $(HIDE)if [ ! -z "$(METAFILE)" ]; then\ + "$(OCAMLFIND)" remove $(FINDLIBDESTDIR) $(FINDLIBPACKAGE); \ + fi + + ########## End of parameters ################################################## # What follows may be relevant to you only if you need to # extend this Makefile. If so, look for 'Extension point' here and @@ -171,13 +235,41 @@ DYNOBJ:=.cmxs DYNLIB:=.cmxs endif -# these variables are meant to be overriden if you want to add *extra* flags +# these variables are meant to be overridden if you want to add *extra* flags COQEXTRAFLAGS?= COQCHKEXTRAFLAGS?= COQDOCEXTRAFLAGS?= +# Find the last argument of the form "-native-compiler FLAG" +COQUSERNATIVEFLAG:=$(strip \ +$(subst -native-compiler-,,\ +$(lastword \ +$(filter -native-compiler-%,\ +$(subst -native-compiler ,-native-compiler-,\ +$(strip $(COQEXTRAFLAGS))))))) + +COQFILTEREDEXTRAFLAGS:=$(strip \ +$(filter-out -native-compiler-%,\ +$(subst -native-compiler ,-native-compiler-,\ +$(strip $(COQEXTRAFLAGS))))) + +COQACTUALNATIVEFLAG:=$(lastword $(COQMF_COQ_NATIVE_COMPILER_DEFAULT) $(COQMF_COQPROJECTNATIVEFLAG) $(COQUSERNATIVEFLAG)) + +ifeq '$(COQACTUALNATIVEFLAG)' 'yes' + COQNATIVEFLAG="-w" "-deprecated-native-compiler-option" "-native-compiler" "ondemand" + COQDONATIVE="yes" +else +ifeq '$(COQACTUALNATIVEFLAG)' 'ondemand' + COQNATIVEFLAG="-w" "-deprecated-native-compiler-option" "-native-compiler" "ondemand" + COQDONATIVE="no" +else + COQNATIVEFLAG="-w" "-deprecated-native-compiler-option" "-native-compiler" "no" + COQDONATIVE="no" +endif +endif + # these flags do NOT contain the libraries, to make them easier to overwrite -COQFLAGS?=-q $(OPT) $(OTHERFLAGS) $(COQEXTRAFLAGS) +COQFLAGS?=-q $(OTHERFLAGS) $(COQFILTEREDEXTRAFLAGS) $(COQNATIVEFLAG) COQCHKFLAGS?=-silent -o $(COQCHKEXTRAFLAGS) COQDOCFLAGS?=-interpolate -utf8 $(COQDOCEXTRAFLAGS) @@ -186,25 +278,19 @@ COQDOCLIBS?=$(COQLIBS_NOML) # The version of Coq being run and the version of coq_makefile that # generated this makefile COQ_VERSION:=$(shell $(COQC) --print-version | cut -d " " -f 1) -COQMAKEFILE_VERSION:=8.9.0 - -COQSRCLIBS?= $(foreach d,$(COQ_SRC_SUBDIRS), -I "$(COQLIB)$(d)") +COQMAKEFILE_VERSION:=8.16.1 -CAMLFLAGS+=$(OCAMLLIBS) $(COQSRCLIBS) -I $(CAMLP5LIB) +# COQ_SRC_SUBDIRS is for user-overriding, usually to add +# `user-contrib/Foo` to the includes, we keep COQCORE_SRC_SUBDIRS for +# Coq's own core libraries, which should be replaced by ocamlfind +# options at some point. +COQ_SRC_SUBDIRS?= +COQSRCLIBS?= $(foreach d,$(COQ_SRC_SUBDIRS), -I "$(COQLIB)/$(d)") +CAMLFLAGS+=$(OCAMLLIBS) $(COQSRCLIBS) # ocamldoc fails with unknown argument otherwise -CAMLDOCFLAGS=$(filter-out -annot, $(filter-out -bin-annot, $(CAMLFLAGS))) - -# FIXME This should be generated by Coq -GRAMMARS:=grammar.cma -CAMLP5EXTEND=pa_extend.cmo q_MLast.cmo pa_macro.cmo - -CAMLLIB:=$(shell "$(OCAMLFIND)" printconf stdlib 2> /dev/null) -ifeq (,$(CAMLLIB)) -PP=$(error "Cannot find the 'ocamlfind' binary used to build Coq ($(OCAMLFIND)). Pre-compiled binary packages of Coq do not support compiling plugins this way. Please download the sources of Coq and run the Windows build script.") -else -PP:=-pp '$(CAMLP5O) -I $(CAMLLIB) -I "$(COQLIB)/grammar" $(CAMLP5EXTEND) $(GRAMMARS) $(CAMLP5OPTIONS) -impl' -endif +CAMLDOCFLAGS:=$(filter-out -annot, $(filter-out -bin-annot, $(CAMLFLAGS))) +CAMLFLAGS+=$(OCAMLWARN) ifneq (,$(TIMING)) TIMING_ARG=-time @@ -221,26 +307,15 @@ else TIMING_ARG= endif -# Retro compatibility (DESTDIR is standard on Unix, DSTROOT is not) -ifdef DSTROOT -DESTDIR := $(DSTROOT) -endif - -concat_path = $(if $(1),$(1)/$(if $(COQMF_WINDRIVE),$(subst $(COQMF_WINDRIVE),/,$(2)),$(2)),$(2)) - -COQLIBINSTALL = $(call concat_path,$(DESTDIR),$(COQLIB)user-contrib) -COQDOCINSTALL = $(call concat_path,$(DESTDIR),$(DOCDIR)user-contrib) -COQTOPINSTALL = $(call concat_path,$(DESTDIR),$(COQLIB)toploop) - # Files ####################################################################### # # We here define a bunch of variables about the files being part of the # Coq project in order to ease the writing of build target and build rules -VDFILE := .coqdeps +VDFILE := .Makefile.d ALLSRCFILES := \ - $(ML4FILES) \ + $(MLGFILES) \ $(MLFILES) \ $(MLPACKFILES) \ $(MLLIBFILES) \ @@ -251,7 +326,13 @@ vo_to_obj = $(addsuffix .o,\ $(filter-out Warning: Error:,\ $(shell $(COQTOP) -q -noinit -batch -quiet -print-mod-uid $(1)))) strip_dotslash = $(patsubst ./%,%,$(1)) + +# without this we get undefined variables in the expansion for the +# targets of the [deprecated,use-mllib-or-mlpack] rule +with_undef = $(if $(filter-out undefined, $(origin $(1))),$($(1))) + VO = vo +VOS = vos VOFILES = $(VFILES:.v=.$(VO)) GLOBFILES = $(VFILES:.v=.glob) @@ -261,7 +342,7 @@ BEAUTYFILES = $(addsuffix .beautified,$(VFILES)) TEXFILES = $(VFILES:.v=.tex) GTEXFILES = $(VFILES:.v=.g.tex) CMOFILES = \ - $(ML4FILES:.ml4=.cmo) \ + $(MLGFILES:.mlg=.cmo) \ $(MLFILES:.ml=.cmo) \ $(MLPACKFILES:.mlpack=.cmo) CMXFILES = $(CMOFILES:.cmo=.cmx) @@ -272,25 +353,25 @@ CMIFILES = \ $(CMOFILES:.cmo=.cmi) \ $(MLIFILES:.mli=.cmi) # the /if/ is because old _CoqProject did not list a .ml(pack|lib) but just -# a .ml4 file +# a .mlg file CMXSFILES = \ $(MLPACKFILES:.mlpack=.cmxs) \ $(CMXAFILES:.cmxa=.cmxs) \ $(if $(MLPACKFILES)$(CMXAFILES),,\ - $(ML4FILES:.ml4=.cmxs) $(MLFILES:.ml=.cmxs)) + $(MLGFILES:.mlg=.cmxs) $(MLFILES:.ml=.cmxs)) # files that are packed into a plugin (no extension) PACKEDFILES = \ $(call strip_dotslash, \ $(foreach lib, \ - $(call strip_dotslash, \ - $(MLPACKFILES:.mlpack=_MLPACK_DEPENDENCIES)),$($(lib)))) + $(call strip_dotslash, \ + $(MLPACKFILES:.mlpack=_MLPACK_DEPENDENCIES)),$(call with_undef,$(lib)))) # files that are archived into a .cma (mllib) LIBEDFILES = \ $(call strip_dotslash, \ $(foreach lib, \ - $(call strip_dotslash, \ - $(MLLIBFILES:.mllib=_MLLIB_DEPENDENCIES)),$($(lib)))) + $(call strip_dotslash, \ + $(MLLIBFILES:.mllib=_MLLIB_DEPENDENCIES)),$(call with_undef,$(lib)))) CMIFILESTOINSTALL = $(filter-out $(addsuffix .cmi,$(PACKEDFILES)),$(CMIFILES)) CMOFILESTOINSTALL = $(filter-out $(addsuffix .cmo,$(PACKEDFILES)),$(CMOFILES)) OBJFILES = $(call vo_to_obj,$(VOFILES)) @@ -298,6 +379,8 @@ ALLNATIVEFILES = \ $(OBJFILES:.o=.cmi) \ $(OBJFILES:.o=.cmx) \ $(OBJFILES:.o=.cmxs) +FINDLIBPACKAGE=$(patsubst .%,%,$(suffix $(METAFILE))) + # trick: wildcard filters out non-existing files, so that `install` doesn't show # warnings and `clean` doesn't pass to rm a list of files that is too long for # the shell. @@ -307,18 +390,17 @@ FILESTOINSTALL = \ $(VFILES) \ $(GLOBFILES) \ $(NATIVEFILES) \ + $(CMXSFILES) # to be removed when we remove legacy loading +FINDLIBFILESTOINSTALL = \ $(CMIFILESTOINSTALL) -BYTEFILESTOINSTALL = \ - $(CMOFILESTOINSTALL) \ - $(CMAFILES) ifeq '$(HASNATDYNLINK)' 'true' DO_NATDYNLINK = yes -FILESTOINSTALL += $(CMXSFILES) $(CMXAFILES) $(CMOFILESTOINSTALL:.cmo=.cmx) +FINDLIBFILESTOINSTALL += $(CMXSFILES) $(CMXAFILES) $(CMOFILESTOINSTALL:.cmo=.cmx) else DO_NATDYNLINK = endif -ALLDFILES = $(addsuffix .d,$(ALLSRCFILES) $(VDFILE)) +ALLDFILES = $(addsuffix .d,$(ALLSRCFILES)) $(VDFILE) # Compilation targets ######################################################### @@ -334,6 +416,31 @@ all.timing.diff: $(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" post-all .PHONY: all.timing.diff +ifeq (0,$(TIMING_REAL)) +TIMING_REAL_ARG := +TIMING_USER_ARG := --user +else +ifeq (1,$(TIMING_REAL)) +TIMING_REAL_ARG := --real +TIMING_USER_ARG := +else +TIMING_REAL_ARG := +TIMING_USER_ARG := +endif +endif + +ifeq (0,$(TIMING_INCLUDE_MEM)) +TIMING_INCLUDE_MEM_ARG := --no-include-mem +else +TIMING_INCLUDE_MEM_ARG := +endif + +ifeq (1,$(TIMING_SORT_BY_MEM)) +TIMING_SORT_BY_MEM_ARG := --sort-by-mem +else +TIMING_SORT_BY_MEM_ARG := +endif + make-pretty-timed-before:: TIME_OF_BUILD_FILE=$(TIME_OF_BUILD_BEFORE_FILE) make-pretty-timed-after:: TIME_OF_BUILD_FILE=$(TIME_OF_BUILD_AFTER_FILE) make-pretty-timed make-pretty-timed-before make-pretty-timed-after:: @@ -341,9 +448,9 @@ make-pretty-timed make-pretty-timed-before make-pretty-timed-after:: $(HIDE)($(MAKE) --no-print-directory -f "$(PARENT)" $(TGTS) TIMED=1 2>&1 && touch pretty-timed-success.ok) | tee -a $(TIME_OF_BUILD_FILE) $(HIDE)rm pretty-timed-success.ok # must not be -f; must fail if the touch failed print-pretty-timed:: - $(HIDE)$(COQMAKE_ONE_TIME_FILE) $(TIME_OF_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_EXTRA_FILES) + $(HIDE)$(COQMAKE_ONE_TIME_FILE) $(TIMING_INCLUDE_MEM_ARG) $(TIMING_SORT_BY_MEM_ARG) $(TIMING_REAL_ARG) $(TIME_OF_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_EXTRA_FILES) print-pretty-timed-diff:: - $(HIDE)$(COQMAKE_BOTH_TIME_FILES) --sort-by=$(TIMING_SORT_BY) $(TIME_OF_BUILD_AFTER_FILE) $(TIME_OF_BUILD_BEFORE_FILE) $(TIME_OF_PRETTY_BOTH_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_EXTRA_FILES) + $(HIDE)$(COQMAKE_BOTH_TIME_FILES) --sort-by=$(TIMING_SORT_BY) $(TIMING_INCLUDE_MEM_ARG) $(TIMING_SORT_BY_MEM_ARG) $(TIMING_REAL_ARG) $(TIME_OF_BUILD_AFTER_FILE) $(TIME_OF_BUILD_BEFORE_FILE) $(TIME_OF_PRETTY_BOTH_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_EXTRA_FILES) ifeq (,$(BEFORE)) print-pretty-single-time-diff:: @echo 'Error: Usage: $(MAKE) print-pretty-single-time-diff AFTER=path/to/file.v.after-timing BEFORE=path/to/file.v.before-timing' @@ -355,7 +462,7 @@ print-pretty-single-time-diff:: $(HIDE)false else print-pretty-single-time-diff:: - $(HIDE)$(COQMAKE_BOTH_SINGLE_TIMING_FILES) --sort-by=$(TIMING_SORT_BY) $(AFTER) $(BEFORE) $(TIME_OF_PRETTY_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_EXTRA_FILES) + $(HIDE)$(COQMAKE_BOTH_SINGLE_TIMING_FILES) --fuzz=$(TIMING_FUZZ) --sort-by=$(TIMING_SORT_BY) $(TIMING_USER_ARG) $(AFTER) $(BEFORE) $(TIME_OF_PRETTY_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_EXTRA_FILES) endif endif pretty-timed: @@ -389,7 +496,11 @@ optfiles: $(if $(DO_NATDYNLINK),$(CMXSFILES)) .PHONY: optfiles # FIXME, see Ralf's bugreport -quick: $(VOFILES:.vo=.vio) +# quick is deprecated, now renamed vio +vio: $(VOFILES:.vo=.vio) +.PHONY: vio +quick: vio + $(warning "'make quick' is deprecated, use 'make vio' or consider using 'vos' files") .PHONY: quick vio2vo: @@ -397,8 +508,9 @@ vio2vo: -schedule-vio2vo $(J) $(VOFILES:%.vo=%.vio) .PHONY: vio2vo +# quick2vo is undocumented quick2vo: - $(HIDE)make -j $(J) quick + $(HIDE)make -j $(J) vio $(HIDE)VIOFILES=$$(for vofile in $(VOFILES); do \ viofile="$$(echo "$$vofile" | sed "s/\.vo$$/.vio/")"; \ if [ "$$vofile" -ot "$$viofile" -o ! -e "$$vofile" ]; then printf "$$viofile "; fi; \ @@ -414,8 +526,14 @@ checkproofs: -schedule-vio-checking $(J) $(VOFILES:%.vo=%.vio) .PHONY: checkproofs +vos: $(VOFILES:%.vo=%.vos) +.PHONY: vos + +vok: $(VOFILES:%.vo=%.vok) +.PHONY: vok + validate: $(VOFILES) - $(TIMER) $(COQCHK) $(COQCHKFLAGS) $(COQLIBS) $^ + $(TIMER) $(COQCHK) $(COQCHKFLAGS) $(COQLIBS_NOML) $^ .PHONY: validate only: $(TGTS) @@ -433,24 +551,24 @@ mlihtml: $(MLIFILES:.mli=.cmi) $(SHOW)'CAMLDOC -d $@' $(HIDE)mkdir $@ || rm -rf $@/* $(HIDE)$(CAMLDOC) -html \ - -d $@ -m A $(CAMLDEBUG) $(CAMLDOCFLAGS) $(MLIFILES) + -d $@ -m A $(CAMLDEBUG) $(CAMLDOCFLAGS) $(MLIFILES) $(FINDLIBPKGS) all-mli.tex: $(MLIFILES:.mli=.cmi) $(SHOW)'CAMLDOC -latex $@' $(HIDE)$(CAMLDOC) -latex \ - -o $@ -m A $(CAMLDEBUG) $(CAMLDOCFLAGS) $(MLIFILES) + -o $@ -m A $(CAMLDEBUG) $(CAMLDOCFLAGS) $(MLIFILES) $(FINDLIBPKGS) all.ps: $(VFILES) $(SHOW)'COQDOC -ps $(GAL)' $(HIDE)$(COQDOC) \ -toc $(COQDOCFLAGS) -ps $(GAL) $(COQDOCLIBS) \ - -o $@ `$(COQDEP) -sort -suffix .v $(VFILES)` + -o $@ `$(COQDEP) -sort $(VFILES)` all.pdf: $(VFILES) $(SHOW)'COQDOC -pdf $(GAL)' $(HIDE)$(COQDOC) \ -toc $(COQDOCFLAGS) -pdf $(GAL) $(COQDOCLIBS) \ - -o $@ `$(COQDEP) -sort -suffix .v $(VFILES)` + -o $@ `$(COQDEP) -sort $(VFILES)` # FIXME: not quite right, since the output name is different gallinahtml: GAL=-g @@ -474,7 +592,12 @@ beautify: $(BEAUTYFILES) # There rules can be extended in Makefile.local # Extensions can't assume when they run. -install: +# findlib needs the package to not be installed, so we remove it before +# installing it (see the call to findlib_remove) +install: META + $(HIDE)code=0; for f in $(FILESTOINSTALL); do\ + if ! [ -f "$$f" ]; then >&2 echo $$f does not exist; code=1; fi \ + done; exit $$code $(HIDE)for f in $(FILESTOINSTALL); do\ df="`$(COQMKFILE) -destination-of "$$f" $(COQLIBS)`";\ if [ "$$?" != "0" -o -z "$$df" ]; then\ @@ -485,22 +608,20 @@ install: echo INSTALL "$$f" "$(COQLIBINSTALL)/$$df";\ fi;\ done + $(call findlib_remove) + $(call findlib_install, META $(FINDLIBFILESTOINSTALL)) $(HIDE)$(MAKE) install-extra -f "$(SELF)" install-extra:: @# Extension point .PHONY: install install-extra +META: $(METAFILE) + $(HIDE)if [ "$(METAFILE)" ]; then \ + cat "$(METAFILE)" | grep -v 'directory.*=.*' > META; \ + fi + install-byte: - $(HIDE)for f in $(BYTEFILESTOINSTALL); do\ - df="`$(COQMKFILE) -destination-of "$$f" $(COQLIBS)`";\ - if [ "$$?" != "0" -o -z "$$df" ]; then\ - echo SKIP "$$f" since it has no logical path;\ - else\ - install -d "$(COQLIBINSTALL)/$$df" &&\ - install -m 0644 "$$f" "$(COQLIBINSTALL)/$$df" &&\ - echo INSTALL "$$f" "$(COQLIBINSTALL)/$$df";\ - fi;\ - done + $(call findlib_install, $(CMAFILES) $(CMOFILESTOINSTALL), -add) install-doc:: html mlihtml @# Extension point @@ -521,13 +642,19 @@ install-doc:: html mlihtml uninstall:: @# Extension point + $(call findlib_remove) $(HIDE)for f in $(FILESTOINSTALL); do \ df="`$(COQMKFILE) -destination-of "$$f" $(COQLIBS)`" &&\ instf="$(COQLIBINSTALL)/$$df/`basename $$f`" &&\ rm -f "$$instf" &&\ - echo RM "$$instf" &&\ - (rmdir "$(call concat_path,,$(COQLIBINSTALL)/$$df/)" 2>/dev/null || true); \ + echo RM "$$instf" ;\ + done + $(HIDE)for f in $(FILESTOINSTALL); do \ + df="`$(COQMKFILE) -destination-of "$$f" $(COQLIBS)`" &&\ + echo RMDIR "$(COQLIBINSTALL)/$$df/" &&\ + (rmdir "$(COQLIBINSTALL)/$$df/" 2>/dev/null || true); \ done + .PHONY: uninstall uninstall-doc:: @@ -555,17 +682,21 @@ clean:: $(HIDE)rm -f $(CMXSFILES) $(HIDE)rm -f $(CMOFILES:.cmo=.o) $(HIDE)rm -f $(CMXAFILES:.cmxa=.a) + $(HIDE)rm -f $(MLGFILES:.mlg=.ml) $(HIDE)rm -f $(ALLDFILES) $(HIDE)rm -f $(NATIVEFILES) $(HIDE)find . -name .coq-native -type d -empty -delete $(HIDE)rm -f $(VOFILES) $(HIDE)rm -f $(VOFILES:.vo=.vio) + $(HIDE)rm -f $(VOFILES:.vo=.vos) + $(HIDE)rm -f $(VOFILES:.vo=.vok) $(HIDE)rm -f $(BEAUTYFILES) $(VFILES:=.old) $(HIDE)rm -f all.ps all-gal.ps all.pdf all-gal.pdf all.glob all-mli.tex $(HIDE)rm -f $(VFILES:.v=.glob) $(HIDE)rm -f $(VFILES:.v=.tex) $(HIDE)rm -f $(VFILES:.v=.g.tex) $(HIDE)rm -f pretty-timed-success.ok + $(HIDE)rm -f META $(HIDE)rm -rf html mlihtml .PHONY: clean @@ -578,6 +709,7 @@ cleanall:: clean $(HIDE)rm -f $(VOFILES:.vo=.v.before-timing) $(HIDE)rm -f $(VOFILES:.vo=.v.after-timing) $(HIDE)rm -f $(VOFILES:.vo=.v.timing.diff) + $(HIDE)rm -f .lia.cache .nia.cache .PHONY: cleanall archclean:: @@ -592,64 +724,62 @@ archclean:: $(MLIFILES:.mli=.cmi): %.cmi: %.mli $(SHOW)'CAMLC -c $<' - $(HIDE)$(CAMLC) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) $< - -$(ML4FILES:.ml4=.cmo): %.cmo: %.ml4 - $(SHOW)'CAMLC -pp -c $<' - $(HIDE)$(CAMLC) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) $(PP) -impl $< + $(HIDE)$(TIMER) $(CAMLC) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) $< -$(ML4FILES:.ml4=.cmx): %.cmx: %.ml4 - $(SHOW)'CAMLOPT -pp -c $(FOR_PACK) $<' - $(HIDE)$(CAMLOPTC) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) $(PP) $(FOR_PACK) -impl $< +$(MLGFILES:.mlg=.ml): %.ml: %.mlg + $(SHOW)'COQPP $<' + $(HIDE)$(COQPP) $< -$(MLFILES:.ml=.cmo): %.cmo: %.ml +# Stupid hack around a deficient syntax: we cannot concatenate two expansions +$(filter %.cmo, $(MLFILES:.ml=.cmo) $(MLGFILES:.mlg=.cmo)): %.cmo: %.ml $(SHOW)'CAMLC -c $<' - $(HIDE)$(CAMLC) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) $< + $(HIDE)$(TIMER) $(CAMLC) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) $< -$(MLFILES:.ml=.cmx): %.cmx: %.ml +# Same hack +$(filter %.cmx, $(MLFILES:.ml=.cmx) $(MLGFILES:.mlg=.cmx)): %.cmx: %.ml $(SHOW)'CAMLOPT -c $(FOR_PACK) $<' - $(HIDE)$(CAMLOPTC) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) $(FOR_PACK) $< + $(HIDE)$(TIMER) $(CAMLOPTC) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) $(FOR_PACK) $< $(MLLIBFILES:.mllib=.cmxs): %.cmxs: %.cmxa $(SHOW)'CAMLOPT -shared -o $@' - $(HIDE)$(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) \ - -linkall -shared -o $@ $< + $(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) \ + -shared -o $@ $< $(MLLIBFILES:.mllib=.cma): %.cma: | %.mllib $(SHOW)'CAMLC -a -o $@' - $(HIDE)$(CAMLLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) -a -o $@ $^ + $(HIDE)$(TIMER) $(CAMLLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -a -o $@ $^ $(MLLIBFILES:.mllib=.cmxa): %.cmxa: | %.mllib $(SHOW)'CAMLOPT -a -o $@' - $(HIDE)$(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) -a -o $@ $^ + $(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -a -o $@ $^ $(MLPACKFILES:.mlpack=.cmxs): %.cmxs: %.cmxa $(SHOW)'CAMLOPT -shared -o $@' - $(HIDE)$(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) \ - -shared -linkall -o $@ $< + $(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) \ + -shared -o $@ $< -$(MLPACKFILES:.mlpack=.cmxa): %.cmxa: %.cmx +$(MLPACKFILES:.mlpack=.cmxa): %.cmxa: %.cmx | %.mlpack $(SHOW)'CAMLOPT -a -o $@' - $(HIDE)$(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) -a -o $@ $< + $(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -a -o $@ $< $(MLPACKFILES:.mlpack=.cma): %.cma: %.cmo | %.mlpack $(SHOW)'CAMLC -a -o $@' - $(HIDE)$(CAMLLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) -a -o $@ $^ + $(HIDE)$(TIMER) $(CAMLLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -a -o $@ $^ $(MLPACKFILES:.mlpack=.cmo): %.cmo: | %.mlpack $(SHOW)'CAMLC -pack -o $@' - $(HIDE)$(CAMLLINK) $(CAMLDEBUG) $(CAMLFLAGS) -pack -o $@ $^ + $(HIDE)$(TIMER) $(CAMLLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -pack -o $@ $^ $(MLPACKFILES:.mlpack=.cmx): %.cmx: | %.mlpack $(SHOW)'CAMLOPT -pack -o $@' - $(HIDE)$(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) -pack -o $@ $^ + $(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -pack -o $@ $^ # This rule is for _CoqProject with no .mllib nor .mlpack -$(filter-out $(MLLIBFILES:.mllib=.cmxs) $(MLPACKFILES:.mlpack=.cmxs) $(addsuffix .cmxs,$(PACKEDFILES)) $(addsuffix .cmxs,$(LIBEDFILES)),$(MLFILES:.ml=.cmxs) $(ML4FILES:.ml4=.cmxs)): %.cmxs: %.cmx +$(filter-out $(MLLIBFILES:.mllib=.cmxs) $(MLPACKFILES:.mlpack=.cmxs) $(addsuffix .cmxs,$(PACKEDFILES)) $(addsuffix .cmxs,$(LIBEDFILES)),$(MLFILES:.ml=.cmxs) $(MLGFILES:.mlg=.cmxs)): %.cmxs: %.cmx $(SHOW)'[deprecated,use-mllib-or-mlpack] CAMLOPT -shared -o $@' - $(HIDE)$(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(CAMLPKGS) \ + $(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) \ -shared -o $@ $< ifneq (,$(TIMING)) @@ -658,20 +788,32 @@ else TIMING_EXTRA = endif -$(VOFILES): %.vo: %.v +$(VOFILES): %.vo: %.v | $(VDFILE) $(SHOW)COQC $< $(HIDE)$(TIMER) $(COQC) $(COQDEBUG) $(TIMING_ARG) $(COQFLAGS) $(COQLIBS) $< $(TIMING_EXTRA) +ifeq ($(COQDONATIVE), "yes") + $(SHOW)COQNATIVE $@ + $(HIDE)$(COQNATIVE) $(COQLIBS) $@ +endif # FIXME ?merge with .vo / .vio ? $(GLOBFILES): %.glob: %.v $(TIMER) $(COQC) $(COQDEBUG) $(COQFLAGS) $(COQLIBS) $< $(VFILES:.v=.vio): %.vio: %.v - $(SHOW)COQC -quick $< - $(HIDE)$(TIMER) $(COQC) -quick $(COQDEBUG) $(COQFLAGS) $(COQLIBS) $< + $(SHOW)COQC -vio $< + $(HIDE)$(TIMER) $(COQC) -vio $(COQDEBUG) $(COQFLAGS) $(COQLIBS) $< + +$(VFILES:.v=.vos): %.vos: %.v + $(SHOW)COQC -vos $< + $(HIDE)$(TIMER) $(COQC) -vos $(COQDEBUG) $(COQFLAGS) $(COQLIBS) $< + +$(VFILES:.v=.vok): %.vok: %.v + $(SHOW)COQC -vok $< + $(HIDE)$(TIMER) $(COQC) -vok $(COQDEBUG) $(COQFLAGS) $(COQLIBS) $< $(addsuffix .timing.diff,$(VFILES)): %.timing.diff : %.before-timing %.after-timing - $(SHOW)PYTHON TIMING-DIFF $< + $(SHOW)PYTHON TIMING-DIFF $*.{before,after}-timing $(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" print-pretty-single-time-diff BEFORE=$*.before-timing AFTER=$*.after-timing TIME_OF_PRETTY_BUILD_FILE="$@" $(BEAUTYFILES): %.v.beautified: %.v @@ -696,11 +838,11 @@ $(GHTMLFILES): %.g.html: %.v %.glob # Dependency files ############################################################ -ifneq ($(filter-out archclean clean cleanall printenv make-pretty-timed make-pretty-timed-before make-pretty-timed-after print-pretty-timed print-pretty-timed-diff print-pretty-single-time-diff,$(MAKECMDGOALS)),) - -include $(ALLDFILES) -else - ifeq ($(MAKECMDGOALS),) +ifndef MAKECMDGOALS -include $(ALLDFILES) +else + ifneq ($(filter-out archclean clean cleanall printenv make-pretty-timed make-pretty-timed-before make-pretty-timed-after print-pretty-timed print-pretty-timed-diff print-pretty-single-time-diff,$(MAKECMDGOALS)),) + -include $(ALLDFILES) endif endif @@ -708,34 +850,37 @@ endif redir_if_ok = > "$@" || ( RV=$$?; rm -f "$@"; exit $$RV ) +GENMLFILES:=$(MLGFILES:.mlg=.ml) +$(addsuffix .d,$(ALLSRCFILES)): $(GENMLFILES) + $(addsuffix .d,$(MLIFILES)): %.mli.d: %.mli $(SHOW)'CAMLDEP $<' $(HIDE)$(CAMLDEP) $(OCAMLLIBS) "$<" $(redir_if_ok) -$(addsuffix .d,$(ML4FILES)): %.ml4.d: %.ml4 - $(SHOW)'CAMLDEP -pp $<' - $(HIDE)$(CAMLDEP) $(OCAMLLIBS) $(PP) -impl "$<" $(redir_if_ok) +$(addsuffix .d,$(MLGFILES)): %.mlg.d: %.ml + $(SHOW)'CAMLDEP $<' + $(HIDE)$(CAMLDEP) $(OCAMLLIBS) "$<" $(redir_if_ok) $(addsuffix .d,$(MLFILES)): %.ml.d: %.ml $(SHOW)'CAMLDEP $<' $(HIDE)$(CAMLDEP) $(OCAMLLIBS) "$<" $(redir_if_ok) $(addsuffix .d,$(MLLIBFILES)): %.mllib.d: %.mllib - $(SHOW)'COQDEP $<' - $(HIDE)$(COQDEP) $(OCAMLLIBS) -c "$<" $(redir_if_ok) + $(SHOW)'OCAMLLIBDEP $<' + $(HIDE)$(OCAMLLIBDEP) -c $(OCAMLLIBS) "$<" $(redir_if_ok) $(addsuffix .d,$(MLPACKFILES)): %.mlpack.d: %.mlpack - $(SHOW)'COQDEP $<' - $(HIDE)$(COQDEP) $(OCAMLLIBS) -c "$<" $(redir_if_ok) + $(SHOW)'OCAMLLIBDEP $<' + $(HIDE)$(OCAMLLIBDEP) -c $(OCAMLLIBS) "$<" $(redir_if_ok) # If this makefile is created using a _CoqProject we have coqdep get # options from it. This avoids argument length limits for pathological # projects. Note that extra options might be on the command line. VDFILE_FLAGS:=$(if _CoqProject,-f _CoqProject,) $(CMDLINE_COQLIBS) $(CMDLINE_VFILES) -$(VDFILE).d: $(VFILES) +$(VDFILE): _CoqProject $(VFILES) $(SHOW)'COQDEP VFILES' - $(HIDE)$(COQDEP) -dyndep var $(VDFILE_FLAGS) $(redir_if_ok) + $(HIDE)$(COQDEP) $(if $(strip $(METAFILE)),-m "$(METAFILE)") -vos -dyndep var $(VDFILE_FLAGS) $(redir_if_ok) # Misc ######################################################################## @@ -752,17 +897,14 @@ opt: printenv:: $(warning printenv is deprecated) $(warning write extensions in Makefile.local or include Makefile.conf) - @echo 'LOCAL = $(LOCAL)' @echo 'COQLIB = $(COQLIB)' + @echo 'COQCORELIB = $(COQCORELIB)' @echo 'DOCDIR = $(DOCDIR)' @echo 'OCAMLFIND = $(OCAMLFIND)' - @echo 'CAMLP5O = $(CAMLP5O)' - @echo 'CAMLP5BIN = $(CAMLP5BIN)' - @echo 'CAMLP5LIB = $(CAMLP5LIB)' - @echo 'CAMLP5OPTIONS = $(CAMLP5OPTIONS)' @echo 'HASNATDYNLINK = $(HASNATDYNLINK)' @echo 'SRC_SUBDIRS = $(SRC_SUBDIRS)' @echo 'COQ_SRC_SUBDIRS = $(COQ_SRC_SUBDIRS)' + @echo 'COQCORE_SRC_SUBDIRS = $(COQCORE_SRC_SUBDIRS)' @echo 'OCAMLFIND = $(OCAMLFIND)' @echo 'PP = $(PP)' @echo 'COQFLAGS = $(COQFLAGS)' @@ -776,10 +918,10 @@ printenv:: .merlin: $(SHOW)'FILL .merlin' $(HIDE)echo 'FLG $(COQMF_CAMLFLAGS)' > .merlin - $(HIDE)echo 'B $(COQLIB)' >> .merlin - $(HIDE)echo 'S $(COQLIB)' >> .merlin - $(HIDE)$(foreach d,$(COQ_SRC_SUBDIRS), \ - echo 'B $(COQLIB)$(d)' >> .merlin;) + $(HIDE)echo 'B $(COQCORELIB)' >> .merlin + $(HIDE)echo 'S $(COQCORELIB)' >> .merlin + $(HIDE)$(foreach d,$(COQCORE_SRC_SUBDIRS), \ + echo 'B $(COQCORELIB)$(d)' >> .merlin;) $(HIDE)$(foreach d,$(COQ_SRC_SUBDIRS), \ echo 'S $(COQLIB)$(d)' >> .merlin;) $(HIDE)$(foreach d,$(SRC_SUBDIRS), echo 'B $(d)' >> .merlin;) @@ -800,3 +942,12 @@ debug: .PHONY: debug .DEFAULT_GOAL := all + +# Users can create Makefile.local-late to hook into double-colon rules +# or add other needed Makefile code, using defined +# variables if necessary. +-include Makefile.local-late + +# Local Variables: +# mode: makefile-gmake +# End: diff --git a/all.pdf b/all.pdf index f76df38d1c3355e02d9d423bcad1d3787b5cf4f8..44427998f7409baaf498d064f8196d44e32de010 100644 GIT binary patch literal 1241338 zcmce-bFA>}*6urP+qP|6&$MmZwr$(CZJW=uZF|oBeqX-qtaH}R&idozbTrM~t4-Ua z&B*l|<1VSZh!`y+9V-;+{PM_OC`M*h0tNy*LrW+g9w>Sl6I(N9a{@*NHWq^aUZCj3 zEUcYP90};ftPPw^L`;nAj7^~U_@JDe9Zd{upxigEGjyDi1W-b6KdIZb>H5Ea+N(FX zTsA6wIer3-|{tN!+MT?s+z2ga<5^bpbDYvl5t^3sE!ZF|gnxq)oIJinc- zQ10WE7Q_R29BRqw#~EQFN%h<>cY)b8E{{Q>G;Gs+&mF!#KYtkR6QWR!)^}id0-nBb z9f3nR+UIT^Dg4Z6m&m2gByqwcnto(Hd_^f})5!Ird7xzGsYyEgy0-lNauaDlIM2I5 zYxiNY$;FYE5(;QQC%{tF{KLGAEQ&zHAa%1?u72Erk1#!8le&o?$A?A`+nde{nCh8o z`^A@$F8u21hL@3J=W1q{4=FDqX$f9hHA0giBHH~Uu;bES08)U@0O9vs`YbNNv?`>G zKf6BC5Dp#mNm6(s+h38T<)E74L8#x;hZlA?G` z5_>xFpaC?IOBmmdHcs6nA`8)hCd3Ov#VB(UrNI+db1|aCjnNUr%)4Ep^CZrDX%i_y z@CmH2+cs$3u6You=^K5T0eBx@?Do=t4Q~}CojZd&O5@fYe7Ia%a26%4(H0+^FqH%$sefC5@F?}FgO6@&fQVfr?k3+zcu3eIbPBvzV+yIynf0QQqz2ig@ 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      Library MSetsExtra.MSetFoldWithAbort

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    -

    Fold with abort for sets

    +

    Fold with abort for sets

    @@ -38,18 +38,18 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Require Export MSetInterface.
    +Require Export MSetInterface.
    Require Import mathcomp.ssreflect.ssreflect.
    Require Import MSetWithDups.
    -Require Import Int.
    -Require Import MSetGenTree MSetAVL MSetRBT.
    -Require Import MSetList MSetWeakList.
    +Require Import Int.
    +Require Import MSetGenTree MSetAVL MSetRBT.
    +Require Import MSetList MSetWeakList.

    -

    Fold With Abort Operations

    +

    Fold With Abort Operations

    @@ -62,60 +62,59 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Definition foldWithAbortType
    -    elt
    +Definition foldWithAbortType
    +    elt
    element type of set
    -    t
    +    t
    type of set
    -    A
    +    A
    -return type -
    +return type
    :=
    -    (elt → A → A) →
    +    (elt → A → A) →
    f
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_abort
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value
    -    A.
    +    A.

    -Definition foldWithAbortSpecPred {elt t : Type}
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    -    (foldWithAbort : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=
    +Definition foldWithAbortSpecPred {elt t : Type}
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (foldWithAbort : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=

        âˆ€
    -      (A : Type)
    +      (A : Type)
          
    @@ -123,7 +122,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (i i' : A)
    +      (i i' : A)
          
    @@ -131,7 +130,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → A → A) (f' : elt → A → A)
    +      (f : elt → A → A) (f' : elt → A → A)
          
    @@ -139,7 +138,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_abort : elt → A → bool)
    +      (f_abort : elt → A → bool)
          
    @@ -147,28 +146,27 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (s : t)
    +      (s : t)
    sets to fold over

    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    -equivalence relation on results -
    +equivalence relation on results
    ,

        
    -P is an equivalence relation +P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →
          
        
    @@ -176,28 +174,28 @@

    Library MSetsExtra.MSetFoldWithAbort

    f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →
    +    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →

        
    -f and f' agree for the elements of s +f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e st) (f' e st))) →
    +    (∀ e st, In e s → (P (f e st) (f' e st))) →

        
    -f_abort is OK, i.e. all other elements can be skipped without +f_abort is OK, i.e. all other elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_abort e1 st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → e2 ≠ e1 →
    -                        P st (f e2 st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_abort e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → e2 ≠ e1 →
    +                        P st (f e2 st'))) →

        
    @@ -205,7 +203,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i' →
    +    P i i' →

        
    @@ -213,7 +211,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The results are in equivalence relation
    -    P (foldWithAbort f f_abort s i) (fold f' s i').
    +    P (foldWithAbort f f_abort s i) (fold f' s i').

    @@ -228,61 +226,60 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Definition foldWithAbortGtType
    -    elt
    +Definition foldWithAbortGtType
    +    elt
    element type of set
    -    t
    +    t
    type of set
    -    A
    +    A
    -return type -
    +return type
    :=
    -    (elt → A → A) →
    +    (elt → A → A) →
    f
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_gt
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value
    -    A.
    +    A.

    -Definition foldWithAbortGtSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    -    (foldWithAbortGt : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=
    +Definition foldWithAbortGtSpecPred {elt t : Type}
    +    (lt : elt → elt → Prop)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (foldWithAbortGt : ∀ {A : Type}, foldWithAbortType elt t A) : Prop :=

        âˆ€
    -      (A : Type)
    +      (A : Type)
          
    @@ -290,7 +287,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (i i' : A)
    +      (i i' : A)
          
    @@ -298,7 +295,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → A → A) (f' : elt → A → A)
    +      (f : elt → A → A) (f' : elt → A → A)
          
    @@ -306,7 +303,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_gt : elt → A → bool)
    +      (f_gt : elt → A → bool)
          
    @@ -314,18 +311,17 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (s : t)
    +      (s : t)
    sets to fold over

    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    -equivalence relation on results -
    +equivalence relation on results
    ,

    @@ -333,10 +329,10 @@

    Library MSetsExtra.MSetFoldWithAbort

        
    -P is an equivalence relation +P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →

        
    @@ -344,28 +340,28 @@

    Library MSetsExtra.MSetFoldWithAbort

    f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →
    +    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →

        
    -f and f' agree for the elements of s +f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e st) (f' e st))) →
    +    (∀ e st, In e s → (P (f e st) (f' e st))) →

        
    -f_abort is OK, i.e. all other elements can be skipped without +f_abort is OK, i.e. all other elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_gt e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e1 e2 →
    +                        P st (f e2 st'))) →

        
    @@ -373,7 +369,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i' →
    +    P i i' →

        
    @@ -381,7 +377,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The results are in equivalence relation
    -    P (foldWithAbortGt f f_gt s i) (fold f' s i').
    +    P (foldWithAbortGt f f_gt s i) (fold f' s i').

    @@ -404,67 +400,66 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Definition foldWithAbortGtLtType
    -    elt
    +Definition foldWithAbortGtLtType
    +    elt
    element type of set
    -    t
    +    t
    type of set
    -    A
    +    A
    -return type -
    +return type
    :=
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_lt
    -    (elt → A → A) →
    +    (elt → A → A) →
    f
    -    (elt → A → bool) →
    +    (elt → A → bool) →
    f_gt
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value
    -    A.
    +    A.

    -Definition foldWithAbortGtLtSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    -    (foldWithAbortGtLt : ∀ {A : Type}, foldWithAbortGtLtType elt t A) : Prop :=
    +Definition foldWithAbortGtLtSpecPred {elt t : Type}
    +    (lt : elt → elt → Prop)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (foldWithAbortGtLt : ∀ {A : Type}, foldWithAbortGtLtType elt t A) : Prop :=

        âˆ€
    -      (A : Type)
    +      (A : Type)
          
    @@ -472,7 +467,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (i i' : A)
    +      (i i' : A)
          
    @@ -480,7 +475,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → A → A) (f' : elt → A → A)
    +      (f : elt → A → A) (f' : elt → A → A)
          
    @@ -488,7 +483,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_lt f_gt : elt → A → bool)
    +      (f_lt f_gt : elt → A → bool)
          
    @@ -496,18 +491,17 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (s : t)
    +      (s : t)
    sets to fold over

    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    -equivalence relation on results -
    +equivalence relation on results
    ,

    @@ -515,10 +509,10 @@

    Library MSetsExtra.MSetFoldWithAbort

        
    -P is an equivalence relation +P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →

        
    @@ -526,41 +520,41 @@

    Library MSetsExtra.MSetFoldWithAbort

    f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →
    +    (∀ st st' e, In e s → P st st' → P (f e st) (f e st')) →

        
    -f and f' agree for the elements of s +f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e st) (f' e st))) →
    +    (∀ e st, In e s → (P (f e st) (f' e st))) →

        
    -f_lt is OK, i.e. smaller elements can be skipped without +f_lt is OK, i.e. smaller elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_lt e1 st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e2 e1 →
    -                        P st (f e2 st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_lt e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e2 e1 →
    +                        P st (f e2 st'))) →

        
    -f_gt is OK, i.e. greater elements can be skipped without +f_gt is OK, i.e. greater elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_gt e1 st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e1 e2 →
    +                        P st (f e2 st'))) →

        
    @@ -568,7 +562,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i' →
    +    P i i' →

        
    @@ -576,7 +570,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The results are in equivalence relation
    -    P (foldWithAbortGtLt f_lt f f_gt s i) (fold f' s i').
    +    P (foldWithAbortGtLt f_lt f f_gt s i) (fold f' s i').

    @@ -584,101 +578,100 @@

    Library MSetsExtra.MSetFoldWithAbort

    We are interested in folding with abort mainly for runtime performance reasons of extracted code. The argument functions - f_lt, f_gt and f of foldWithAbortGtLt often share a large, + f_lt, f_gt and f of foldWithAbortGtLt often share a large, comparably expensive part of their computation.
    In order to further improve runtime performance, therefore another - version foldWithAbortPrecompute f_precompute f_lt f f_gt that - uses an extra function f_precompute to allows to compute the + version foldWithAbortPrecompute f_precompute f_lt f f_gt that + uses an extra function f_precompute to allows to compute the commonly used parts of these functions only once. This leads to the following definitions.

    -Definition foldWithAbortPrecomputeType
    -    elt
    +Definition foldWithAbortPrecomputeType
    +    elt
    element type of set
    -    t
    +    t
    type of set
    -    A
    +    A
    return type
    -    B
    +    B
    -type of precomputed results -
    +type of precomputed results
    :=

    -    (elt → B) →
    +    (elt → B) →
    f_precompute
    -    (elt → B → A → bool) →
    +    (elt → B → A → bool) →
    f_lt
    -    (elt → B → A → A) →
    +    (elt → B → A → A) →
    f
    -    (elt → B → A → bool) →
    +    (elt → B → A → bool) →
    f_gt
    -    t →
    +    t →
    input set
    -    A →
    +    A →
    base value
    -    A.
    +    A.

    The specification is similar to the one without precompute, - but uses f_precompute so avoid doing computations multiple times + but uses f_precompute so avoid doing computations multiple times
    -Definition foldWithAbortPrecomputeSpecPred {elt t : Type}
    -    (lt : elt → elt → Prop)
    -    (In : elt → t → Prop)
    -    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    -    (foldWithAbortPrecompute : ∀ {A B : Type}, foldWithAbortPrecomputeType elt t A B) : Prop :=
    +Definition foldWithAbortPrecomputeSpecPred {elt t : Type}
    +    (lt : elt → elt → Prop)
    +    (In : elt → t → Prop)
    +    (fold : ∀ {A : Type}, (elt → A → A) → t → A → A)
    +    (foldWithAbortPrecompute : ∀ {A B : Type}, foldWithAbortPrecomputeType elt t A B) : Prop :=

        âˆ€
    -      (A B : Type)
    +      (A B : Type)
          
    @@ -686,7 +679,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (i i' : A)
    +      (i i' : A)
          
    @@ -694,7 +687,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_precompute : elt → B)
    +      (f_precompute : elt → B)
          
    @@ -702,7 +695,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f : elt → B → A → A) (f' : elt → A → A)
    +      (f : elt → B → A → A) (f' : elt → A → A)
          
    @@ -710,7 +703,7 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (f_lt f_gt : elt → B → A → bool)
    +      (f_lt f_gt : elt → B → A → bool)
          
    @@ -718,18 +711,17 @@

    Library MSetsExtra.MSetFoldWithAbort


    -      (s : t)
    +      (s : t)
    sets to fold over

    -      (P : A → A → Prop)
    +      (P : A → A → Prop)
    -equivalence relation on results -
    +equivalence relation on results
    ,

    @@ -737,10 +729,10 @@

    Library MSetsExtra.MSetFoldWithAbort

        
    -P is an equivalence relation +P is an equivalence relation
    -    Equivalence P →
    +    Equivalence P →

        
    @@ -748,41 +740,41 @@

    Library MSetsExtra.MSetFoldWithAbort

    f is for the elements of s compatible with the equivalence relation P
    -    (∀ st st' e, In e s → P st st' → P (f e (f_precompute e) st) (f e (f_precompute e) st')) →
    +    (∀ st st' e, In e s → P st st' → P (f e (f_precompute e) st) (f e (f_precompute e) st')) →

        
    -f and f' agree for the elements of s +f and f' agree for the elements of s
    -    (∀ e st, In e s → (P (f e (f_precompute e) st) (f' e st))) →
    +    (∀ e st, In e s → (P (f e (f_precompute e) st) (f' e st))) →

        
    -f_lt is OK, i.e. smaller elements can be skipped without +f_lt is OK, i.e. smaller elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_lt e1 (f_precompute e1) st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e2 e1 →
    -                        P st (f e2 (f_precompute e2) st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_lt e1 (f_precompute e1) st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e2 e1 →
    +                        P st (f e2 (f_precompute e2) st'))) →

        
    -f_gt is OK, i.e. greater elements can be skipped without +f_gt is OK, i.e. greater elements can be skipped without leaving the equivalence relation.
    -    (∀ e1 st,
    -        In e1 s → f_gt e1 (f_precompute e1) st = true →
    -        (∀ st' e2, P st st' →
    -                        In e2 s → lt e1 e2 →
    -                        P st (f e2 (f_precompute e2) st'))) →
    +    (∀ e1 st,
    +        In e1 s → f_gt e1 (f_precompute e1) st = true →
    +        (∀ st' e2, P st st' →
    +                        In e2 s → lt e1 e2 →
    +                        P st (f e2 (f_precompute e2) st'))) →

        
    @@ -790,7 +782,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    The base values are in equivalence relation
    -    P i i' →
    +    P i i' →

        
    @@ -798,13 +790,13 @@

    Library MSetsExtra.MSetFoldWithAbort

    The results are in equivalence relation
    -    P (foldWithAbortPrecompute f_precompute f_lt f f_gt s i) (fold f' s i').
    +    P (foldWithAbortPrecompute f_precompute f_lt f f_gt s i) (fold f' s i').

    -

    Module Types

    +

    Module Types

    @@ -815,14 +807,14 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Module Type HasFoldWithAbort (E : OrderedType) (Import C : WSetsOnWithDups E).
    +Module Type HasFoldWithAbort (E : OrderedType) (Import C : WSetsOnWithDups E).

    -  Parameter foldWithAbortPrecompute : ∀ {A B : Type},
    -    foldWithAbortPrecomputeType elt t A B.
    +  Parameter foldWithAbortPrecompute : ∀ {A B : Type},
    +    foldWithAbortPrecomputeType elt t A B.

    -  Parameter foldWithAbortPrecomputeSpec :
    +  Parameter foldWithAbortPrecomputeSpec :
         foldWithAbortPrecomputeSpecPred E.lt In (@fold) (@foldWithAbortPrecompute).

    @@ -832,7 +824,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Derived operations

    +

    Derived operations

    @@ -844,29 +836,29 @@

    Library MSetsExtra.MSetFoldWithAbort


    -Module HasFoldWithAbortOps (E : OrderedType) (C : WSetsOnWithDups E)
    +Module HasFoldWithAbortOps (E : OrderedType) (C : WSetsOnWithDups E)
                               (FT : HasFoldWithAbort E C).

    -  Import FT.
    -  Import C.
    +  Import FT.
    +  Import C.

    -

    First lets define the other folding with abort variants

    +

    First lets define the other folding with abort variants


    -  Definition foldWithAbortGtLt {A} f_lt (f : (elt → A → A)) f_gt :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_lt e st)
    -      (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).
    +  Definition foldWithAbortGtLt {A} f_lt (f : (elt → A → A)) f_gt :=
    +    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_lt e st)
    +      (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).

    -  Lemma foldWithAbortGtLtSpec :
    +  Lemma foldWithAbortGtLtSpec :
         foldWithAbortGtLtSpecPred E.lt In (@fold) (@foldWithAbortGtLt).
      Proof.
        rewrite /foldWithAbortGtLt /foldWithAbortGtLtSpecPred.
    @@ -876,12 +868,12 @@

    Library MSetsExtra.MSetFoldWithAbort

      Qed.

    -  Definition foldWithAbortGt {A} (f : (elt → A → A)) f_gt :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun _ _ _ ⇒ false)
    -      (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).
    +  Definition foldWithAbortGt {A} (f : (elt → A → A)) f_gt :=
    +    foldWithAbortPrecompute (fun _ ⇒ tt) (fun _ _ _ ⇒ false)
    +      (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_gt e st).

    -  Lemma foldWithAbortGtSpec :
    +  Lemma foldWithAbortGtSpec :
         foldWithAbortGtSpecPred E.lt In (@fold) (@foldWithAbortGt).
      Proof.
        rewrite /foldWithAbortGt /foldWithAbortGtSpecPred.
    @@ -891,22 +883,22 @@

    Library MSetsExtra.MSetFoldWithAbort

      Qed.

    -  Definition foldWithAbort {A} (f : (elt → A → A)) f_abort :=
    -    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_abort e st)
    -      (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_abort e st).
    +  Definition foldWithAbort {A} (f : (elt → A → A)) f_abort :=
    +    foldWithAbortPrecompute (fun _ ⇒ tt) (fun e _ st ⇒ f_abort e st)
    +      (fun e _ st ⇒ f e st) (fun e _ st ⇒ f_abort e st).

    -  Lemma foldWithAbortSpec :
    +  Lemma foldWithAbortSpec :
         foldWithAbortSpecPred In (@fold) (@foldWithAbort).
      Proof.
        rewrite /foldWithAbort /foldWithAbortGtSpecPred.
        intros A i i' f f' f_abort s P.
        move ⇒ H_equiv_P H_f_compat H_ff' H_abort H_ii'.
    -    have H_lt_neq: (∀ e1 e2, E.lt e1 e2 → e1 ≠ e2). {
    +    have H_lt_neq: (∀ e1 e2, E.lt e1 e2 → e1 ≠ e2). {
          move ⇒ e1 e2 H_lt H_e12_eq.
          rewrite H_e12_eq in H_lt.
    -      have : ( Irreflexive E.lt) by apply StrictOrder_Irreflexive.
    -      rewrite /Irreflexive /Reflexive /complement ⇒ H.
    +      have : ( Irreflexive E.lt) by apply StrictOrder_Irreflexive.
    +      rewrite /Irreflexive /Reflexive /complement ⇒ H.
          eapply H, H_lt.
        }
        apply foldWithAbortPrecomputeSpec ⇒ //; (
    @@ -920,7 +912,7 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Specialisations for equality

    +

    Specialisations for equality

    @@ -928,30 +920,30 @@

    Library MSetsExtra.MSetFoldWithAbort

    of an arbitrary equivalence relation on results.
    -  Lemma foldWithAbortPrecomputeSpec_Equal : ∀ (A B : Type) (i : A) (f_pre : elt → B)
    -      (f : elt → B → A → A) (f' : elt → A → A) (f_lt f_gt : elt → B → A → bool) (s : t),
    +  Lemma foldWithAbortPrecomputeSpec_Equal : ∀ (A B : Type) (i : A) (f_pre : elt → B)
    +      (f : elt → B → A → A) (f' : elt → A → A) (f_lt f_gt : elt → B → A → bool) (s : t),

    -      (∀ e st, In e s → (f e (f_pre e) st = f' e st)) →
    +      (∀ e st, In e s → (f e (f_pre e) st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_lt e1 (f_pre e1) st = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 →
    -                      (f e2 (f_pre e2) st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_lt e1 (f_pre e1) st = true →
    +          (∀ e2, In e2 s → E.lt e2 e1 →
    +                      (f e2 (f_pre e2) st = st))) →

          
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 (f_pre e1) st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 (f_pre e2) st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_gt e1 (f_pre e1) st = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 →
    +                      (f e2 (f_pre e2) st = st))) →

    -      (foldWithAbortPrecompute f_pre f_lt f f_gt s i) = (fold f' s i).
    +      (foldWithAbortPrecompute f_pre f_lt f f_gt s i) = (fold f' s i).
      Proof.
        intros A B i f_pre f f' f_lt f_gt s H_f' H_lt H_gt.

         eapply (foldWithAbortPrecomputeSpec A B i i f_pre f f'); eauto. {
    -       apply eq_equivalence.
    +       apply eq_equivalence.
         } {
           move ⇒ st1 st2 e H_in → //.
         } {
    @@ -966,30 +958,30 @@

    Library MSetsExtra.MSetFoldWithAbort

      Qed.

    -  Lemma foldWithAbortGtLtSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f' : elt → A → A) (f_lt f_gt : elt → A → bool) (s : t),
    +  Lemma foldWithAbortGtLtSpec_Equal : ∀ (A : Type) (i : A)
    +      (f : elt → A → A) (f' : elt → A → A) (f_lt f_gt : elt → A → bool) (s : t),

    -      (∀ e st, In e s → (f e st = f' e st)) →
    +      (∀ e st, In e s → (f e st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_lt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_lt e1 st = true →
    +          (∀ e2, In e2 s → E.lt e2 e1 →
    +                      (f e2 st = st))) →

          
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_gt e1 st = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 →
    +                      (f e2 st = st))) →

    -      (foldWithAbortGtLt f_lt f f_gt s i) = (fold f' s i).
    +      (foldWithAbortGtLt f_lt f f_gt s i) = (fold f' s i).
      Proof.
        intros A i f f' f_lt f_gt s H_f' H_lt H_gt.

         eapply (foldWithAbortGtLtSpec A i i f f'); eauto. {
    -       apply eq_equivalence.
    +       apply eq_equivalence.
         } {
           move ⇒ st1 st2 e H_in → //.
         } {
    @@ -1004,24 +996,24 @@

    Library MSetsExtra.MSetFoldWithAbort

      Qed.

    -  Lemma foldWithAbortGtSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f' : elt → A → A) (f_gt : elt → A → bool) (s : t),
    +  Lemma foldWithAbortGtSpec_Equal : ∀ (A : Type) (i : A)
    +      (f : elt → A → A) (f' : elt → A → A) (f_gt : elt → A → bool) (s : t),

    -      (∀ e st, In e s → (f e st = f' e st)) →
    +      (∀ e st, In e s → (f e st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_gt e1 st = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_gt e1 st = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 →
    +                      (f e2 st = st))) →

    -      (foldWithAbortGt f f_gt s i) = (fold f' s i).
    +      (foldWithAbortGt f f_gt s i) = (fold f' s i).
      Proof.
        intros A i f f' f_gt s H_f' H_gt.

         eapply (foldWithAbortGtSpec A i i f f'); eauto. {
    -       apply eq_equivalence.
    +       apply eq_equivalence.
         } {
           move ⇒ st1 st2 e H_in → //.
         } {
    @@ -1032,24 +1024,24 @@

    Library MSetsExtra.MSetFoldWithAbort

      Qed.

    -  Lemma foldWithAbortSpec_Equal : ∀ (A : Type) (i : A)
    -      (f : elt → A → A) (f' : elt → A → A) (f_abort : elt → A → bool) (s : t),
    +  Lemma foldWithAbortSpec_Equal : ∀ (A : Type) (i : A)
    +      (f : elt → A → A) (f' : elt → A → A) (f_abort : elt → A → bool) (s : t),

    -      (∀ e st, In e s → (f e st = f' e st)) →
    +      (∀ e st, In e s → (f e st = f' e st)) →

          
    -      (∀ e1 st,
    -          In e1 s → f_abort e1 st = true →
    -          (∀ e2, In e2 s → e1 ≠ e2 →
    -                      (f e2 st = st))) →
    +      (∀ e1 st,
    +          In e1 s → f_abort e1 st = true →
    +          (∀ e2, In e2 s → e1 ≠ e2 →
    +                      (f e2 st = st))) →

    -      (foldWithAbort f f_abort s i) = (fold f' s i).
    +      (foldWithAbort f f_abort s i) = (fold f' s i).
      Proof.
         intros A i f f' f_abort s H_f' H_abort.

         eapply (foldWithAbortSpec A i i f f'); eauto. {
    -       apply eq_equivalence.
    +       apply eq_equivalence.
         } {
           move ⇒ st1 st2 e H_in → //.
         } {
    @@ -1063,44 +1055,44 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    FoldWithAbortSpecArgs

    +

    FoldWithAbortSpecArgs

    While folding, we are often interested in skipping elements that do not - satisfy a certain property P. This needs expressing in terms of + satisfy a certain property P. This needs expressing in terms of skips of smaller of larger elements in order to be done efficiently by our folding functions. Formally, this leads to the definition of foldWithAbortSpecForPred.
    - Given a FoldWithAbortSpecArg for a predicate P and a + Given a FoldWithAbortSpecArg for a predicate P and a set s, many operations can be implemented efficiently. Below we will provide efficient versions of filter, choose, ∃, ∀ and more.
    -  Record FoldWithAbortSpecArg {B} := {
    -     fwasa_f_pre : (elt → B);
    +  Record FoldWithAbortSpecArg {B} := {
    +     fwasa_f_pre : (elt → B);
    The precompute function
    -     fwasa_f_lt : (elt → B → bool);
    +     fwasa_f_lt : (elt → B → bool);
    f_lt without state argument
    -     fwasa_f_gt : (elt → B → bool);
    +     fwasa_f_gt : (elt → B → bool);
    f_gt without state argument
    -     fwasa_P' : (elt → B → bool)
    +     fwasa_P' : (elt → B → bool)
    the predicate P @@ -1112,20 +1104,20 @@

    Library MSetsExtra.MSetFoldWithAbort

    -foldWithAbortSpecForPred s P fwasa holds, if - the argument fwasa fits the predicate P for set s. +foldWithAbortSpecForPred s P fwasa holds, if + the argument fwasa fits the predicate P for set s.
    -  Definition foldWithAbortSpecArgsForPred {A : Type}
    -     (s : t) (P : elt → bool) (fwasa : @FoldWithAbortSpecArg A) :=
    +  Definition foldWithAbortSpecArgsForPred {A : Type}
    +     (s : t) (P : elt → bool) (fwasa : @FoldWithAbortSpecArg A) :=

         
    -the predicate P' coincides for s and the given precomputation with P +the predicate P' coincides for s and the given precomputation with P
    -     (∀ e, In e s → (fwasa_P' fwasa e (fwasa_f_pre fwasa e) = P e)) ∧
    +     (∀ e, In e s → (fwasa_P' fwasa e (fwasa_f_pre fwasa e) = P e)) ∧

         
    @@ -1134,9 +1126,9 @@

    Library MSetsExtra.MSetFoldWithAbort

    don't satisfy predicate P.
    -     (∀ e1,
    -          In e1 s → fwasa_f_lt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    -          (∀ e2, In e2 s → E.lt e2 e1 → (P e2 = false))) ∧
    +     (∀ e1,
    +          In e1 s → fwasa_f_lt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    +          (∀ e2, In e2 s → E.lt e2 e1 → (P e2 = false))) ∧

         
    @@ -1145,9 +1137,9 @@

    Library MSetsExtra.MSetFoldWithAbort

    don't satisfy predicate P.
    -     (∀ e1,
    -          In e1 s → fwasa_f_gt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    -          (∀ e2, In e2 s → E.lt e1 e2 → (P e2 = false))).
    +     (∀ e1,
    +          In e1 s → fwasa_f_gt fwasa e1 (fwasa_f_pre fwasa e1) = true →
    +          (∀ e2, In e2 s → E.lt e1 e2 → (P e2 = false))).

    @@ -1155,21 +1147,21 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Filter with abort

    +

    Filter with abort

    -  Definition filter_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
    -     @foldWithAbortPrecompute t B (fwasa_f_pre fwasa) (fun e p _ ⇒ fwasa_f_lt fwasa e p)
    -       (fun e e_pre s ⇒ if fwasa_P' fwasa e e_pre then add e s else s)
    -       (fun e p _ ⇒ fwasa_f_gt fwasa e p) s empty.
    +  Definition filter_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
    +     @foldWithAbortPrecompute t B (fwasa_f_pre fwasa) (fun e p _ ⇒ fwasa_f_lt fwasa e p)
    +       (fun e e_pre s ⇒ if fwasa_P' fwasa e e_pre then add e s else s)
    +       (fun e p _ ⇒ fwasa_f_gt fwasa e p) s empty.

    -  Lemma filter_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    Equal (filter_with_abort fwasa s)
    -          (filter P s).
    +  Lemma filter_with_abort_spec {B} : ∀ fwasa P s,
    +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    Equal (filter_with_abort fwasa s)
    +          (filter P s).
      Proof.
        unfold foldWithAbortSpecArgsForPred.
        move ⇒ [] f_pre f_lt f_gt P' P s /=.
    @@ -1177,22 +1169,22 @@

    Library MSetsExtra.MSetFoldWithAbort

        rewrite /filter_with_abort /=.

    -    have → : (foldWithAbortPrecompute f_pre (fun e p _ ⇒ f_lt e p)
    -     (fun (e : elt) (e_pre : B) (s0 : t) ⇒
    -      if P' e e_pre then add e s0 else s0) (fun e p _ ⇒ f_gt e p) s empty =
    -     (fold (fun e s0 ⇒ if P e then add e s0 else s0) s empty)). {
    +    have → : (foldWithAbortPrecompute f_pre (fun e p _ ⇒ f_lt e p)
    +     (fun (e : elt) (e_pre : B) (s0 : t) ⇒
    +      if P' e e_pre then add e s0 else s0) (fun e p _ ⇒ f_gt e p) s empty =
    +     (fold (fun e s0 ⇒ if P e then add e s0 else s0) s empty)). {
          apply foldWithAbortPrecomputeSpec_Equal. {
            intros e st H_e_in.
            rewrite H_f' //.
          } {
            intros e1 st H_e1_in H_f_lt_eq e2 H_e2_in H_lt_e2_e1.
            rewrite (H_f' _ H_e2_in).
    -        suff → : (P e2 = false) by done.
    +        suff → : (P e2 = false) by done.
            apply (H_lt e1); eauto.
          } {
            intros e1 st H_e1_in H_f_gt_eq e2 H_e2_in H_gt_e2_e1.
            rewrite (H_f' _ H_e2_in).
    -        suff → : (P e2 = false) by done.
    +        suff → : (P e2 = false) by done.
            apply (H_gt e1); eauto.
          }
        }
    @@ -1202,18 +1194,18 @@

    Library MSetsExtra.MSetFoldWithAbort

        setoid_rewrite filter_spec ⇒ //.

    -    suff → : ∀ acc, In e
    -      (fold_left
    -         (flip (fun (e0 : elt) (s0 : t) ⇒ if P e0 then add e0 s0 else s0))
    -         (elements s) acc) ↔ (InA E.eq e (elements s) ∧ P e = true) ∨ (In e acc). {
    +    suff → : ∀ acc, In e
    +      (fold_left
    +         (flip (fun (e0 : elt) (s0 : t) ⇒ if P e0 then add e0 s0 else s0))
    +         (elements s) acc) ↔ (InA E.eq e (elements s) ∧ P e = true) ∨ (In e acc). {
          rewrite elements_spec1.
    -      suff : (¬In e empty) by tauto.
    +      suff : (¬In e empty) by tauto.
          apply empty_spec.
        }
        induction (elements s) as [| x xs IH] ⇒ acc. {
    -      rewrite /= InA_nil. tauto.
    +      rewrite /= InA_nil. tauto.
        } {
    -      rewrite /= /flip IH InA_cons.
    +      rewrite /= /flip IH InA_cons.
          case_eq (P x). {
            rewrite add_spec.
            intuition.
    @@ -1221,7 +1213,7 @@

    Library MSetsExtra.MSetFoldWithAbort

            rewrite H0.
            split ⇒ //.
            left.
    -        apply Equivalence_Reflexive.
    +        apply Equivalence_Reflexive.
          } {
            intuition.
            contradict H2.
    @@ -1235,27 +1227,27 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Choose with abort

    +

    Choose with abort

    -  Definition choose_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
    -     foldWithAbortPrecompute (fwasa_f_pre fwasa)
    -       (fun e p st ⇒ match st with None ⇒ (fwasa_f_lt fwasa e p) | _ ⇒ true end)
    -       (fun e e_pre st ⇒ match st with None ⇒
    -          if (fwasa_P' fwasa e e_pre) then Some e else None | _ ⇒ st end)
    +  Definition choose_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
    +     foldWithAbortPrecompute (fwasa_f_pre fwasa)
    +       (fun e p st ⇒ match st with None ⇒ (fwasa_f_lt fwasa e p) | _ ⇒ true end)
    +       (fun e e_pre st ⇒ match st with None ⇒
    +          if (fwasa_P' fwasa e e_pre) then Some e else None | _ ⇒ st end)

    -       (fun e p st ⇒ match st with None ⇒ (fwasa_f_gt fwasa e p) | _ ⇒ true end)
    -       s None.
    +       (fun e p st ⇒ match st with None ⇒ (fwasa_f_gt fwasa e p) | _ ⇒ true end)
    +       s None.

    -  Lemma choose_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (match (choose_with_abort fwasa s) with
    -       | None ⇒ (∀ e, In e s → P e = false)
    -       | Some e ⇒ In e s ∧ (P e = true)
    -     end).
    +  Lemma choose_with_abort_spec {B} : ∀ fwasa P s,
    +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    (match (choose_with_abort fwasa s) with
    +       | None ⇒ (∀ e, In e s → P e = false)
    +       | Some e ⇒ In e s ∧ (P e = true)
    +     end).
      Proof.
        rewrite /foldWithAbortSpecArgsForPred.
        move ⇒ [] f_pre f_lt f_gt P' P s /=.
    @@ -1263,15 +1255,15 @@

    Library MSetsExtra.MSetFoldWithAbort


        set fwasa := {|
    -       fwasa_f_pre := f_pre;
    -       fwasa_f_lt := f_lt;
    -       fwasa_f_gt := f_gt;
    -       fwasa_P' := P' |}.
    +       fwasa_f_pre := f_pre;
    +       fwasa_f_lt := f_lt;
    +       fwasa_f_gt := f_gt;
    +       fwasa_P' := P' |}.

        suff : (match (choose_with_abort fwasa s) with
    -       | None ⇒ (∀ e, InA E.eq e (elements s) → P e = false)
    -       | Some e ⇒ InA E.eq e (elements s) ∧ (P e = true)
    +       | None ⇒ (∀ e, InA E.eq e (elements s) → P e = false)
    +       | Some e ⇒ InA E.eq e (elements s) ∧ (P e = true)
         end). {
           case (choose_with_abort fwasa s). {
             move ⇒ e.
    @@ -1283,11 +1275,11 @@

    Library MSetsExtra.MSetFoldWithAbort

           }
        }

    -    have → : (choose_with_abort fwasa s =
    -      (fold (fun e st ⇒
    -        match st with
    -          | None ⇒ if P e then Some e else None
    -          | _ ⇒ st end) s None)). {
    +    have → : (choose_with_abort fwasa s =
    +      (fold (fun e st ⇒
    +        match st with
    +          | None ⇒ if P e then Some e else None
    +          | _ ⇒ st end) s None)). {
          apply foldWithAbortPrecomputeSpec_Equal. {
            intros e st H_e_in.
            case st ⇒ //=.
    @@ -1297,44 +1289,44 @@

    Library MSetsExtra.MSetFoldWithAbort

            rewrite (H_f' _ H_e2_in).
            case_eq (P e2) ⇒ // H_P_e2.
            contradict H_P_e2.
    -        apply not_true_iff_false, (H_lt e1); auto.
    +        apply not_true_iff_false, (H_lt e1); auto.
          } {
            move ⇒ e1 [] //= H_e1_in H_f_gt_eq e2 H_e2_in H_gt_e2_e1.
            rewrite (H_f' _ H_e2_in).
            case_eq (P e2) ⇒ // H_P_e2.
            contradict H_P_e2.
    -        apply not_true_iff_false, (H_gt e1); auto.
    +        apply not_true_iff_false, (H_gt e1); auto.
          }
        }

    -    rewrite fold_spec /flip.
    +    rewrite fold_spec /flip.
        induction (elements s) as [| x xs IH]. {
          rewrite /=.
    -      move ⇒ e /InA_nil //.
    +      move ⇒ e /InA_nil //.
        } {
          case_eq (P x) ⇒ H_Px; rewrite /= H_Px. {
    -        have → : ∀ xs, fold_left (fun (x0 : option elt) (y : elt) ⇒
    -                    match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    -                    end) xs (Some x) = Some x. {
    +        have → : ∀ xs, fold_left (fun (x0 : option elt) (y : elt) ⇒
    +                    match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    +                    end) xs (Some x) = Some x. {
              move ⇒ ys.
              induction ys ⇒ //.
            }
            split; last assumption.
    -        apply InA_cons_hd.
    +        apply InA_cons_hd.
            apply E.eq_equiv.
          } {
            move : IH.
    -        case (fold_left
    -          (fun (x0 : option elt) (y : elt) ⇒
    -             match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    -             end) xs None). {
    +        case (fold_left
    +          (fun (x0 : option elt) (y : elt) ⇒
    +             match x0 with | Some _ ⇒ x0 | None ⇒ if P y then Some y else None
    +             end) xs None). {

                 move ⇒ e [H_e_in] H_Pe.
                 split; last assumption.
    -             apply InA_cons_tl ⇒ //.
    +             apply InA_cons_tl ⇒ //.
            } {
              move ⇒ H_e_nin e H_e_in.
    -          have : (InA E.eq e xs ∨ (E.eq e x)). {
    +          have : (InA E.eq e xs ∨ (E.eq e x)). {
                inversion H_e_in; tauto.
              }
              move ⇒ []. {
    @@ -1351,25 +1343,25 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Exists and Forall with abort

    +

    Exists and Forall with abort

    -  Definition exists_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
    -    match choose_with_abort fwasa s with
    -      | None ⇒ false
    -      | Some _ ⇒ true
    +  Definition exists_with_abort {B} (fwasa : @FoldWithAbortSpecArg B) s :=
    +    match choose_with_abort fwasa s with
    +      | None ⇒ false
    +      | Some _ ⇒ true
        end.

    -  Lemma exists_with_abort_spec {B} : ∀ fwasa P s,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (exists_with_abort fwasa s =
    -     exists_ P s).
    +  Lemma exists_with_abort_spec {B} : ∀ fwasa P s,
    +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    (exists_with_abort fwasa s =
    +     exists_ P s).
      Proof.
        intros fwasa P s H_fwasa H_proper.
    -    apply Logic.eq_sym.
    +    apply Logic.eq_sym.
        rewrite /exists_with_abort.
        move : (choose_with_abort_spec _ _ _ H_fwasa H_proper).
        case (choose_with_abort fwasa s). {
    @@ -1378,7 +1370,7 @@

    Library MSetsExtra.MSetFoldWithAbort

          by ∃ e.
        } {
          move ⇒ H_not_ex.
    -      apply not_true_iff_false.
    +      apply not_true_iff_false.
          rewrite exists_spec /Exists.
          move ⇒ [e] [H_in] H_pe.
          move : (H_not_ex e H_in).
    @@ -1393,21 +1385,21 @@

    Library MSetsExtra.MSetFoldWithAbort

    Negation leads to forall.
    -  Definition forall_with_abort {B} fwasa s :=
    -     negb (@exists_with_abort B fwasa s).
    +  Definition forall_with_abort {B} fwasa s :=
    +     negb (@exists_with_abort B fwasa s).

    -  Lemma forall_with_abort_spec {B} : ∀ fwasa s P,
    -    @foldWithAbortSpecArgsForPred B s P fwasa →
    -    Proper (E.eq ==> Logic.eq) P →
    -    (forall_with_abort fwasa s =
    -     for_all (fun e ⇒ negb (P e)) s).
    +  Lemma forall_with_abort_spec {B} : ∀ fwasa s P,
    +    @foldWithAbortSpecArgsForPred B s P fwasa →
    +    Proper (E.eq ==> Logic.eq) P →
    +    (forall_with_abort fwasa s =
    +     for_all (fun e ⇒ negb (P e)) s).
      Proof.
        intros fwasa s P H_ok H_proper.
        rewrite /forall_with_abort exists_with_abort_spec; auto.

    -    rewrite eq_iff_eq_true negb_true_iff -not_true_iff_false.
    +    rewrite eq_iff_eq_true negb_true_iff -not_true_iff_false.
        rewrite exists_spec.
        setoid_rewrite for_all_spec; last solve_proper.
    @@ -1415,7 +1407,7 @@

    Library MSetsExtra.MSetFoldWithAbort

        rewrite /Exists /For_all.
        split. {
          move ⇒ H_pre x H_x_in.
    -      rewrite negb_true_iff -not_true_iff_false ⇒ H_Px.
    +      rewrite negb_true_iff -not_true_iff_false ⇒ H_Px.
          apply H_pre.
          by ∃ x.
        } {
    @@ -1433,29 +1425,29 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Modules Types For Sets with Fold with Abort

    +

    Modules Types For Sets with Fold with Abort

    -Module Type WSetsWithDupsFoldA.
    -  Declare Module E : OrderedType.
    +Module Type WSetsWithDupsFoldA.
    +  Declare Module E : OrderedType.
      Include WSetsOnWithDups E.
      Include HasFoldWithAbort E.
      Include HasFoldWithAbortOps E.
    End WSetsWithDupsFoldA.

    -Module Type WSetsWithFoldA <: WSets.
    -  Declare Module E : OrderedType.
    -  Include WSetsOn E.
    +Module Type WSetsWithFoldA <: WSets.
    +  Declare Module E : OrderedType.
    +  Include WSetsOn E.
      Include HasFoldWithAbort E.
      Include HasFoldWithAbortOps E.
    End WSetsWithFoldA.

    -Module Type SetsWithFoldA <: Sets.
    -  Declare Module E : OrderedType.
    -  Include SetsOn E.
    +Module Type SetsWithFoldA <: Sets.
    +  Declare Module E : OrderedType.
    +  Include SetsOn E.
      Include HasFoldWithAbort E.
      Include HasFoldWithAbortOps E.
    End SetsWithFoldA.
    @@ -1464,57 +1456,57 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Implementations

    +

    Implementations

    -

    GenTree implementation

    +

    GenTree implementation

    Finally, provide such a fold with abort operation for generic trees.
    -Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
    -  (Import Raw:Ops E I)
    -  (M : MSetGenTree.Props E I Raw).
    +Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
    +  (Import Raw: MSetGenTree.Ops E I)
    +  (M : MSetGenTree.Props E I Raw).

    -  Fixpoint foldWithAbort_Raw {A B: Type} (f_pre : E.t → B) f_lt (f: E.t → B → A → A) f_gt t (base: A) : A :=
    -    match t with
    -    | Raw.Leaf ⇒ base
    +  Fixpoint foldWithAbort_Raw {A B: Type} (f_pre : E.t → B) f_lt (f: E.t → B → A → A) f_gt t (base: A) : A :=
    +    match t with
    +    | Raw.Leaf ⇒ base
        | Raw.Node _ l x r ⇒
    -        let x_pre := f_pre x in
    -        let st0 := if f_lt x x_pre base then base else foldWithAbort_Raw f_pre f_lt f f_gt l base in
    -        let st1 := f x x_pre st0 in
    -        let st2 := if f_gt x x_pre st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1 in
    -        st2
    +        let x_pre := f_pre x in
    +        let st0 := if f_lt x x_pre base then base else foldWithAbort_Raw f_pre f_lt f f_gt l base in
    +        let st1 := f x x_pre st0 in
    +        let st2 := if f_gt x x_pre st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1 in
    +        st2
        end.

    -  Lemma foldWithAbort_RawSpec : ∀ (A B : Type) (i i' : A) (f_pre : E.t → B)
    -      (f : E.t → B → A → A) (f' : E.t → A → A) (f_lt f_gt : E.t → B → A → bool) (s : Raw.tree)
    -      (P : A → A → Prop),
    +  Lemma foldWithAbort_RawSpec : ∀ (A B : Type) (i i' : A) (f_pre : E.t → B)
    +      (f : E.t → B → A → A) (f' : E.t → A → A) (f_lt f_gt : E.t → B → A → bool) (s : Raw.tree)
    +      (P : A → A → Prop),

    -      (M.bst s) →
    -      Equivalence P →
    -      (∀ st st' e, M.In e s → P st st' → P (f e (f_pre e) st) (f e (f_pre e) st')) →
    -      (∀ e st, M.In e s → P (f e (f_pre e) st) (f' e st)) →
    +      (M.bst s) →
    +      Equivalence P →
    +      (∀ st st' e, M.In e s → P st st' → P (f e (f_pre e) st) (f e (f_pre e) st')) →
    +      (∀ e st, M.In e s → P (f e (f_pre e) st) (f' e st)) →

          
    -      (∀ e1 st,
    -          M.In e1 s → f_lt e1 (f_pre e1) st = true →
    -          (∀ st' e2, P st st' →
    -                          M.In e2 s → E.lt e2 e1 →
    -                          P st (f e2 (f_pre e2) st'))) →
    +      (∀ e1 st,
    +          M.In e1 s → f_lt e1 (f_pre e1) st = true →
    +          (∀ st' e2, P st st' →
    +                          M.In e2 s → E.lt e2 e1 →
    +                          P st (f e2 (f_pre e2) st'))) →

          
    -      (∀ e1 st,
    -          M.In e1 s → f_gt e1 (f_pre e1) st = true →
    -          (∀ st' e2, P st st' →
    -                          M.In e2 s → E.lt e1 e2 →
    -                          P st (f e2 (f_pre e2) st'))) →
    -
    -      P i i' →
    -      P (foldWithAbort_Raw f_pre f_lt f f_gt s i) (fold f' s i').
    +      (∀ e1 st,
    +          M.In e1 s → f_gt e1 (f_pre e1) st = true →
    +          (∀ st' e2, P st st' →
    +                          M.In e2 s → E.lt e1 e2 →
    +                          P st (f e2 (f_pre e2) st'))) →
    +
    +      P i i' →
    +      P (foldWithAbort_Raw f_pre f_lt f f_gt s i) (fold f' s i').
       Proof.
         intros A B i i' f_pre f f' f_lt f_gt s P.
         move ⇒ H_bst H_equiv_P H_P_f H_f' H_RL H_RG.
    @@ -1524,8 +1516,8 @@

    Library MSetsExtra.MSetFoldWithAbort


         move : i i'.
    -     have : (∀ e, M.In e base → M.In e s). {
    -       rewrite /In /base //.
    +     have : (∀ e, M.In e base → M.In e s). {
    +       rewrite /In /base //.
         }
         have : M.bst base. {
           apply H_bst.
    @@ -1542,41 +1534,41 @@

    Library MSetsExtra.MSetFoldWithAbort


           have [H_bst_l [H_bst_r [H_lt_tree_l H_gt_tree_r]]]:
    -         M.bst l ∧ M.bst r ∧ M.lt_tree e l ∧ M.gt_tree e r. {
    +         M.bst l ∧ M.bst r ∧ M.lt_tree e l ∧ M.gt_tree e r. {
             inversion H_bst. done.
           }
    -       
    -       have H_sub_l : (∀ e0 : E.t, M.In e0 l → M.In e0 s ∧ E.lt e0 e). {
    +
    +       have H_sub_l : (∀ e0 : E.t, M.In e0 l → M.In e0 s ∧ E.lt e0 e). {
             intros e0 H_in_l.
             split; last by apply H_lt_tree_l.
             eapply H_sub.
             rewrite /M.In M.In_node_iff.
             tauto.
           }
    -       move : (IHl H_bst_l) ⇒ {IHl} IHl {H_bst_l} {H_lt_tree_l}.
    -       have H_sub_r : (∀ e0 : E.t, M.In e0 r → M.In e0 s ∧ E.lt e e0). {
    +       move : (IHl H_bst_l) ⇒ {} IHl {H_bst_l} {H_lt_tree_l}.
    +       have H_sub_r : (∀ e0 : E.t, M.In e0 r → M.In e0 s ∧ E.lt e e0). {
             intros e0 H_in_r.
             split; last by apply H_gt_tree_r.
             eapply H_sub.
             rewrite /M.In M.In_node_iff.
             tauto.
           }
    -       move : (IHr H_bst_r) ⇒ {IHr} IHr {H_bst_r} {H_gt_tree_r}.
    +       move : (IHr H_bst_r) ⇒ {} IHr {H_bst_r} {H_gt_tree_r}.
           have H_in_e : M.In e s. {
             eapply H_sub.
             rewrite /M.In M.In_node_iff.
             right; left.
    -         apply Equivalence_Reflexive.
    +         apply Equivalence_Reflexive.
           }
           move ⇒ {H_sub}.

           rewrite /=.
    -       set st0 := if f_lt e (f_pre e) i then i else foldWithAbort_Raw f_pre f_lt f f_gt l i.
    +       set st0 := if f_lt e (f_pre e) i then i else foldWithAbort_Raw f_pre f_lt f f_gt l i.
           set st0' := Raw.fold f' l i'.
           set st1 := f e (f_pre e) st0.
           set st1' := f' e st0'.
    -       set st2 := if f_gt e (f_pre e) st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1.
    +       set st2 := if f_gt e (f_pre e) st1 then st1 else foldWithAbort_Raw f_pre f_lt f f_gt r st1.
           set st2' := Raw.fold f' r st1'.

    @@ -1599,15 +1591,15 @@

    Library MSetsExtra.MSetFoldWithAbort

                   apply H_sub_l.
                   rewrite /M.In M.In_node_iff. tauto.
                 }
    -             have [] : (M.In e' s ∧ E.lt e' e). {
    +             have [] : (M.In e' s ∧ E.lt e' e). {
                   apply H_sub_l.
                   rewrite /M.In M.In_node_iff.
                   right; left.
    -               apply Equivalence_Reflexive.
    +               apply Equivalence_Reflexive.
                 }
                 move ⇒ H_e'_in H_lt_in.
                 suff H_P_i : (P i (f e' (f_pre e') (fold f' l i'))). {
    -               eapply Equivalence_Transitive; first apply H_P_i.
    +               eapply Equivalence_Transitive; first apply H_P_i.
                   by apply H_f'.
                 }
                 eapply H_RL ⇒ //.
    @@ -1649,15 +1641,15 @@

    Library MSetsExtra.MSetFoldWithAbort

                   apply H_sub_r.
                   rewrite /M.In M.In_node_iff. tauto.
                 }
    -             have [] : (M.In e' s ∧ E.lt e e'). {
    +             have [] : (M.In e' s ∧ E.lt e e'). {
                   apply H_sub_r.
                   rewrite /M.In M.In_node_iff.
                   right; left.
    -               apply Equivalence_Reflexive.
    +               apply Equivalence_Reflexive.
                 }
                 move ⇒ H_e'_in H_lt_ee'.
                 suff H_P_st1_aux : (P st1 (f e' (f_pre e') (fold f' l st1'))). {
    -               eapply Equivalence_Transitive; first apply H_P_st1_aux.
    +               eapply Equivalence_Transitive; first apply H_P_st1_aux.
                   by apply H_f'.
                 }
                 eapply H_RG ⇒ //.
    @@ -1683,23 +1675,23 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    AVL implementation

    +

    AVL implementation

    The generic tree implementation naturally leads to an AVL one.

    -Module MakeAVLSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetAVL.Make X.
    -  Include MakeGenTreeFoldA X Z_as_Int Raw Raw.
    +Module MakeAVLSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    +  Include MSetAVL.Make X.
    +  Include MakeGenTreeFoldA X Z_as_Int Raw Raw.

    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    -    foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.
    +  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    +    foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.

    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
      Proof.
        intros A B i i' f_pre f f' f_lt f_gt s P.
        move ⇒ H_P_f H_f' H_RL H_RG H_P_ii'.
    @@ -1720,21 +1712,21 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    RBT implementation

    +

    RBT implementation

    The generic tree implementation naturally leads to an RBT one.
    -Module MakeRBTSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetRBT.Make X.
    -  Include MakeGenTreeFoldA X Color Raw Raw.
    +Module MakeRBTSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    +  Include MSetRBT.Make X.
    +  Include MakeGenTreeFoldA X Color Raw Raw.

    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    -    foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.
    +  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt (f: elt → B → A → A) f_gt t (base: A) : A :=
    +    foldWithAbort_Raw f_pre f_lt f f_gt (t.(this)) base.

    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
      Proof.
        intros A B i i' f_pre f f' f_lt f_gt s P.
        move ⇒ H_P_f H_f' H_RL H_RG H_P_ii'.
    @@ -1755,34 +1747,34 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Sorted Lists Implementation

    +

    Sorted Lists Implementation

    -Module MakeListSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    -  Include MSetList.Make X.
    +Module MakeListSetsWithFoldA (X : OrderedType) <: SetsWithFoldA with Module E := X.
    +  Include MSetList.Make X.

    -  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    -    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    -  match t with
    -    | nil ⇒ acc
    -    | x :: xs ⇒ (
    -        let pre_x := f_pre x in
    -        let acc := f x (pre_x) acc in
    -        if (f_gt x pre_x acc) then
    -          acc
    -        else
    -          foldWithAbortRaw f_pre f_lt f f_gt xs acc
    +  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    +    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    +  match t with
    +    | nil ⇒ acc
    +    | x :: xs ⇒ (
    +        let pre_x := f_pre x in
    +        let acc := f x (pre_x) acc in
    +        if (f_gt x pre_x acc) then
    +          acc
    +        else
    +          foldWithAbortRaw f_pre f_lt f f_gt xs acc
          )
      end.

    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt f f_gt t acc :=
    -    @foldWithAbortRaw A B f_pre f_lt f f_gt t.(this) acc.
    +  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt f f_gt t acc :=
    +    @foldWithAbortRaw A B f_pre f_lt f f_gt t.(this) acc.

    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
      Proof.
        intros A B i i' f_pre f f' f_lt f_gt.
        move ⇒ [] l H_is_ok_l P H_equiv_P.
    @@ -1792,10 +1784,10 @@

    Library MSetsExtra.MSetFoldWithAbort


        set base := l.
        move : i i'.
    -    have : (∀ e, InA X.eq e base → InA X.eq e l). {
    +    have : (∀ e, InA X.eq e base → InA X.eq e l). {
          rewrite /base //.
        }
    -    have : sort X.lt base. {
    +    have : sort X.lt base. {
          rewrite Raw.isok_iff /base //.
        }
        clear H_is_ok_l.
    @@ -1807,7 +1799,7 @@

    Library MSetsExtra.MSetFoldWithAbort

        move ⇒ H_sort H_in_xxs i i' Pii' /=.

    -    have [H_sort_xs H_hd_rel {H_sort}] : Sorted X.lt xs ∧ HdRel X.lt x xs. {
    +    have [H_sort_xs H_hd_rel {H_sort}] : Sorted X.lt xs ∧ HdRel X.lt x xs. {
          by inversion H_sort.
        }

    @@ -1815,18 +1807,18 @@

    Library MSetsExtra.MSetFoldWithAbort

        rewrite (Raw.ML.Inf_alt x H_sort_xs) ⇒ H_lt_xs.

    -    have H_x_in_l : InA X.eq x l. {
    +    have H_x_in_l : InA X.eq x l. {
          apply H_in_xxs.
    -      apply InA_cons_hd.
    +      apply InA_cons_hd.
          apply X.eq_equiv.
        }
    -    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
    +    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
          intros e H_in.
    -      apply H_in_xxs, InA_cons_tl ⇒ //.
    +      apply H_in_xxs, InA_cons_tl ⇒ //.
        }
       
    -    have H_P_next : P (f x (f_pre x) i) (flip f' i' x). {
    -      rewrite /flip -H_f' //.
    +    have H_P_next : P (f x (f_pre x) i) (flip f' i' x). {
    +      rewrite /flip -H_f' //.
          apply H_P_f ⇒ //.
        }

    @@ -1835,8 +1827,8 @@

    Library MSetsExtra.MSetFoldWithAbort

          apply IH ⇒ //.
        } {
          move ⇒ H_gt.
    -      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    -         P (f x (f_pre x) i) (fold_left (flip f') xs st)). {
    +      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    +         P (f x (f_pre x) i) (fold_left (flip f') xs st)). {
             apply H_suff ⇒ //.
          }
          move : H_in_xs H_lt_xs.
    @@ -1849,21 +1841,21 @@

    Library MSetsExtra.MSetFoldWithAbort

          } {
            intros H_in_xs H_lt_xs st H_P_st.
            rewrite /=.
    -        have H_x'_in_l : InA X.eq x' l. {
    +        have H_x'_in_l : InA X.eq x' l. {
              apply H_in_xs.
    -          apply InA_cons_hd, X.eq_equiv.
    +          apply InA_cons_hd, X.eq_equiv.
            }
            apply IH'. {
              intros e H.
    -          apply H_in_xs, InA_cons_tl ⇒ //.
    +          apply H_in_xs, InA_cons_tl ⇒ //.
            } {
              intros e H.
    -          apply H_lt_xs, InA_cons_tl ⇒ //.
    +          apply H_lt_xs, InA_cons_tl ⇒ //.
            } {
    -          rewrite /flip -H_f' //.
    +          rewrite /flip -H_f' //.
              apply H_RG_x ⇒ //.
              apply H_lt_xs.
    -          apply InA_cons_hd, X.eq_equiv.
    +          apply InA_cons_hd, X.eq_equiv.
            }
          }
        }
    @@ -1879,36 +1871,36 @@

    Library MSetsExtra.MSetFoldWithAbort

    -

    Unsorted Lists without Dups Implementation

    +

    Unsorted Lists without Dups Implementation

    -Module MakeWeakListSetsWithFoldA (X : OrderedType) <: WSetsWithFoldA with Module E := X.
    -  Module Raw := MSetWeakList.MakeRaw X.
    -  Module E := X.
    -  Include WRaw2SetsOn E Raw.
    +Module MakeWeakListSetsWithFoldA (X : OrderedType) <: WSetsWithFoldA with Module E := X.
    +  Module Raw := MSetWeakList.MakeRaw X.
    +  Module E := X.
    +  Include WRaw2SetsOn E Raw.

    -  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    -    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    -  match t with
    -    | nil ⇒ acc
    -    | x :: xs ⇒ (
    -        let pre_x := f_pre x in
    -        let acc := f x (pre_x) acc in
    -        if (f_gt x pre_x acc) && (f_lt x pre_x acc) then
    -          acc
    -        else
    -          foldWithAbortRaw f_pre f_lt f f_gt xs acc
    +  Fixpoint foldWithAbortRaw {A B: Type} (f_pre : X.t → B) (f_lt : X.t → B → A → bool)
    +    (f: X.t → B → A → A) (f_gt : X.t → B → A → bool) (t : list X.t) (acc : A) : A :=
    +  match t with
    +    | nil ⇒ acc
    +    | x :: xs ⇒ (
    +        let pre_x := f_pre x in
    +        let acc := f x (pre_x) acc in
    +        if (f_gt x pre_x acc) && (f_lt x pre_x acc) then
    +          acc
    +        else
    +          foldWithAbortRaw f_pre f_lt f f_gt xs acc
          )
      end.

    -  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt f f_gt t acc :=
    -    @foldWithAbortRaw A B f_pre f_lt f f_gt t.(this) acc.
    +  Definition foldWithAbortPrecompute {A B: Type} f_pre f_lt f f_gt t acc :=
    +    @foldWithAbortRaw A B f_pre f_lt f f_gt t.(this) acc.

    -  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
    +  Lemma foldWithAbortPrecomputeSpec : foldWithAbortPrecomputeSpecPred X.lt In fold (@foldWithAbortPrecompute).
      Proof.
        intros A B i i' f_pre f f' f_lt f_gt.
        move ⇒ [] l H_is_ok_l P H_P_equiv.
    @@ -1918,10 +1910,10 @@

    Library MSetsExtra.MSetFoldWithAbort


        set base := l.
        move : i i'.
    -    have : (∀ e, InA X.eq e base → InA X.eq e l). {
    +    have : (∀ e, InA X.eq e base → InA X.eq e l). {
          rewrite /base //.
        }
    -    have : NoDupA X.eq base. {
    +    have : NoDupA X.eq base. {
          apply H_is_ok_l.
        }
        clear H_is_ok_l.
    @@ -1933,44 +1925,44 @@

    Library MSetsExtra.MSetFoldWithAbort

        move ⇒ H_nodup_xxs H_in_xxs i i' Pii' /=.

    -    have [H_nin_x_xs H_nodup_xs {H_nodup_xxs}] : ¬ InA X.eq x xs ∧ NoDupA X.eq xs. {
    +    have [H_nin_x_xs H_nodup_xs {H_nodup_xxs}] : ¬ InA X.eq x xs ∧ NoDupA X.eq xs. {
          by inversion H_nodup_xxs.
        }

    -    have H_x_in_l : InA X.eq x l. {
    +    have H_x_in_l : InA X.eq x l. {
          apply H_in_xxs.
    -      apply InA_cons_hd.
    +      apply InA_cons_hd.
          apply X.eq_equiv.
        }
    -    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
    +    have H_in_xs : (∀ e : X.t, InA X.eq e xs → InA X.eq e l). {
          intros e H_in.
    -      apply H_in_xxs, InA_cons_tl ⇒ //.
    +      apply H_in_xxs, InA_cons_tl ⇒ //.
        }
       
    -    have H_P_next : P (f x (f_pre x) i) (flip f' i' x). {
    -      rewrite /flip -H_f' //.
    +    have H_P_next : P (f x (f_pre x) i) (flip f' i' x). {
    +      rewrite /flip -H_f' //.
          apply H_P_f ⇒ //.
        }

    -    case_eq (f_gt x (f_pre x) (f x (f_pre x) i) &&
    +    case_eq (f_gt x (f_pre x) (f x (f_pre x) i) &&
                 f_lt x (f_pre x) (f x (f_pre x) i)); last first. {
          move ⇒ _.
          apply IH ⇒ //.
        } {
    -      move ⇒ /andb_true_iff [H_gt H_lt].
    -      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    -         P (f x (f_pre x) i) (fold_left (flip f') xs st)). {
    +      move ⇒ /andb_true_iff [H_gt H_lt].
    +      suff H_suff : (∀ st, P (f x (f_pre x) i) st →
    +         P (f x (f_pre x) i) (fold_left (flip f') xs st)). {
             apply H_suff ⇒ //.
          }

    -      have H_neq_xs : ∀ e, List.In e xs → X.lt x e ∨ X.lt e x. {
    +      have H_neq_xs : ∀ e, List.In e xs → X.lt x e ∨ X.lt e x. {
            intros e H_in.

            move : (X.compare_spec x e).
            case (X.compare x e) ⇒ H_cmp; inversion H_cmp. {
              contradict H_nin_x_xs.
    -          rewrite InA_alt.
    +          rewrite InA_alt.
              by ∃ e.
            } {
              by left.
    @@ -1989,24 +1981,24 @@

    Library MSetsExtra.MSetFoldWithAbort

          } {
            intros H_in_xs H_neq_xs st H_P_st.
            rewrite /=.
    -        have H_x'_in_xxs' : List.In x' (x' :: xs'). {
    +        have H_x'_in_xxs' : List.In x' (x' :: xs'). {
              simpl; by left.
            }
    -        have H_x'_in_l : InA X.eq x' l. {
    +        have H_x'_in_l : InA X.eq x' l. {
              apply H_in_xs.
    -          apply InA_cons_hd, X.eq_equiv.
    +          apply InA_cons_hd, X.eq_equiv.
            }
            apply IH'. {
              intros H.
    -          apply H_nin_x_xs, InA_cons_tl ⇒ //.
    +          apply H_nin_x_xs, InA_cons_tl ⇒ //.
            } {
              intros e H.
    -          apply H_in_xs, InA_cons_tl ⇒ //.
    +          apply H_in_xs, InA_cons_tl ⇒ //.
            } {
              intros e H.
    -          apply H_neq_xs, in_cons ⇒ //.
    +          apply H_neq_xs, in_cons ⇒ //.
            } {
    -          rewrite /flip -H_f' //.
    +          rewrite /flip -H_f' //.
              move : (H_neq_xs x' H_x'_in_xxs') ⇒ [] H_cmp. {
                apply H_RG_x ⇒ //.
              } {
    diff --git a/html/MSetsExtra.MSetIntervals.html b/html/MSetsExtra.MSetIntervals.html index 649c296..d66b966 100644 --- a/html/MSetsExtra.MSetIntervals.html +++ b/html/MSetsExtra.MSetIntervals.html @@ -26,7 +26,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Weak sets implemented by interval lists

    +

    Weak sets implemented by interval lists

    @@ -47,113 +47,114 @@

    Library MSetsExtra.MSetIntervals


    -Require Import MSetInterface OrdersFacts OrdersLists.
    -Require Import BinNat.
    +Require Import MSetInterface OrdersFacts OrdersLists.
    +Require Import BinNat.
    Require Import mathcomp.ssreflect.ssreflect.
    -Require Import NArith.
    -Require Import ZArith.
    -Require Import NOrder.
    -Require Import DecidableTypeEx.
    -Module Import NOP := NOrderProp N.
    +Require Import NArith.
    +Require Import ZArith.
    +Require Import NOrder.
    +Require Import Lia.
    +Require Import DecidableTypeEx.
    +Module Import NOP := NOrderProp N.
    Open Scope Z_scope.

    -

    Auxiliary stuff

    +

    Auxiliary stuff

    Simple auxiliary lemmata
    -Lemma Z_le_add_r : ∀ (z : Z) (n : N),
    -  z ≤ z + Z.of_N n.
    +Lemma Z_le_add_r : ∀ (z : Z) (n : N),
    +  z ≤ z + Z.of_N n.
    Proof.
      intros z n.
    -  suff : (z + 0 ≤ z + Z.of_N n). {
    -    rewrite Z.add_0_r //.
    +  suff : (z + 0 ≤ z + Z.of_N n). {
    +    rewrite Z.add_0_r //.
      }
    -  apply Zplus_le_compat_l.
    -  apply N2Z.is_nonneg.
    +  apply Zplus_le_compat_l.
    +  apply N2Z.is_nonneg.
    Qed.

    -Lemma Z_lt_add_r : ∀ (z : Z) (n : N),
    -  (n ≠ 0)%N →
    -  z < z + Z.of_N n.
    +Lemma Z_lt_add_r : ∀ (z : Z) (n : N),
    +  (n ≠ 0)%N →
    +  z < z + Z.of_N n.
    Proof.
      move ⇒ z n H_neq_0.
    -  suff : (z + Z.of_N 0 < z + Z.of_N n). {
    -    rewrite Z.add_0_r //.
    +  suff : (z + Z.of_N 0 < z + Z.of_N n). {
    +    rewrite Z.add_0_r //.
      }
    -  apply Z.add_lt_mono_l, N2Z.inj_lt.
    -  by apply N.neq_0_lt_0.
    +  apply Z.add_lt_mono_l, N2Z.inj_lt.
    +  by apply N.neq_0_lt_0.
    Qed.

    -Lemma Z_lt_le_add_r : ∀ y1 y2 c,
    -  y1 < y2 →
    -  y1 ≤ y2 + Z.of_N c.
    +Lemma Z_lt_le_add_r : ∀ y1 y2 c,
    +  y1 < y2 →
    +  y1 ≤ y2 + Z.of_N c.
    Proof.
      intros y1 y2 c H.
    -  apply Z.le_trans with (m := y2). {
    -    by apply Z.lt_le_incl.
    +  apply Z.le_trans with (m := y2). {
    +    by apply Z.lt_le_incl.
      } {
        apply Z_le_add_r.
      }
    Qed.

    -Lemma Z_to_N_minus_neq_0 : ∀ (x y : Z),
    -    y < x →
    -    Z.to_N (x - y) ≠ 0%N.
    +Lemma Z_to_N_minus_neq_0 : ∀ (x y : Z),
    +    y < x →
    +    Z.to_N (x - y) ≠ 0%N.
    Proof.
      intros x y H_y_lt.
    -  apply N.neq_0_lt_0.
    -  apply N2Z.inj_lt.
    -  suff H : 0 < x - y. {
    -    rewrite Z2N.id ⇒ //.
    -    by apply Z.lt_le_incl.
    +  apply N.neq_0_lt_0.
    +  apply N2Z.inj_lt.
    +  suff H : 0 < x - y. {
    +    rewrite Z2N.id ⇒ //.
    +    by apply Z.lt_le_incl.
      }
    -  by apply Z.lt_0_sub.
    +  by apply Z.lt_0_sub.
    Qed.

    -Lemma add_add_sub_eq : ∀ (x y : Z), (x + (y - x) = y).
    +Lemma add_add_sub_eq : ∀ (x y : Z), (x + (y - x) = y).
    Proof.
      intros x y.
    -  rewrite Z.add_sub_assoc ⇒ //.
    -  rewrite Z.add_sub_swap Z.sub_diag Z.add_0_l //.
    +  rewrite Z.add_sub_assoc ⇒ //.
    +  rewrite Z.add_sub_swap Z.sub_diag Z.add_0_l //.
    Qed.

    -Lemma NoDupA_map {A B} : ∀ (eqA : A → A → Prop) (eqB : B → B → Prop) (f : A → B) l,
    -  NoDupA eqA l →
    -  (∀ x1 x2, List.In x1 l → List.In x2 l →
    -                  eqB (f x1) (f x2) → eqA x1 x2) →
    -  NoDupA eqB (map f l).
    +Lemma NoDupA_map {A B} : ∀ (eqA : A → A → Prop) (eqB : B → B → Prop) (f : A → B) l,
    +  NoDupA eqA l →
    +  (∀ x1 x2, List.In x1 l → List.In x2 l →
    +                  eqB (f x1) (f x2) → eqA x1 x2) →
    +  NoDupA eqB (map f l).
    Proof.
      intros eqA eqB f.
      induction l as [| x xs IH]. {
        move ⇒ _ _; rewrite /=.
    -    apply NoDupA_nil.
    +    apply NoDupA_nil.
      } {
        move ⇒ H_pre H_eqA_impl.
    -    have [H_nin_x H_no_dup_xs] : ¬ InA eqA x xs ∧ NoDupA eqA xs. {
    +    have [H_nin_x H_no_dup_xs] : ¬ InA eqA x xs ∧ NoDupA eqA xs. {
          by inversion_clear H_pre.
        }
        simpl.
    -    apply NoDupA_cons; last first. {
    +    apply NoDupA_cons; last first. {
          apply IH ⇒ //.
          intros x1 x2 H_in_x1 H_in_x2 H_eqB.
          apply H_eqA_impl ⇒ //=; by right.
        }
        move ⇒ H_in_map; apply H_nin_x.
        move : H_in_map.
    -    rewrite !InA_alt ⇒ [[y] [H_eqB_y]].
    -    rewrite in_map_iff ⇒ [[y'] [H_y_eq] H_y'_in].
    +    rewrite !InA_alt ⇒ [[y] [H_eqB_y]].
    +    rewrite in_map_iff ⇒ [[y'] [H_y_eq] H_y'_in].
        subst.
        âˆƒ y'.
        split ⇒ //.
    @@ -169,7 +170,7 @@

    Library MSetsExtra.MSetIntervals

    -

    rev_map

    +

    rev_map

    @@ -177,14 +178,14 @@

    Library MSetsExtra.MSetIntervals

    rev_map is used for efficiency.
    -Fixpoint rev_map_aux {A B} (f : A → B) (acc : list B) (l : list A) :=
    -  match l with
    -   | nil ⇒ acc
    -   | x :: xs ⇒ rev_map_aux f ((f x)::acc) xs
    +Fixpoint rev_map_aux {A B} (f : A → B) (acc : list B) (l : list A) :=
    +  match l with
    +   | nil ⇒ acc
    +   | x :: xs ⇒ rev_map_aux f ((f x)::acc) xs
      end.

    -Definition rev_map {A B} (f : A → B) (l : list A) : list B := rev_map_aux f nil l.
    +Definition rev_map {A B} (f : A → B) (l : list A) : list B := rev_map_aux f nil l.

    @@ -193,8 +194,8 @@

    Library MSetsExtra.MSetIntervals

    Lemmata about rev_map
    -Lemma rev_map_aux_alt_def {A B} : ∀ (f : A → B) l acc,
    -  rev_map_aux f acc l = List.rev_append (List.map f l) acc.
    +Lemma rev_map_aux_alt_def {A B} : ∀ (f : A → B) l acc,
    +  rev_map_aux f acc l = List.rev_append (List.map f l) acc.
    Proof.
      intro f.
      induction l as [| x xs IH]. {
    @@ -207,18 +208,18 @@

    Library MSetsExtra.MSetIntervals

    Qed.

    -Lemma rev_map_alt_def {A B} : ∀ (f : A → B) l,
    -  rev_map f l = List.rev (List.map f l).
    +Lemma rev_map_alt_def {A B} : ∀ (f : A → B) l,
    +  rev_map f l = List.rev (List.map f l).
    Proof.
      intros f l.
    -  rewrite /rev_map rev_map_aux_alt_def -rev_alt //.
    +  rewrite /rev_map rev_map_aux_alt_def -rev_alt //.
    Qed.

    -

    Encoding Elements

    +

    Encoding Elements

    @@ -230,12 +231,12 @@

    Library MSetsExtra.MSetIntervals


    -Module Type ElementEncode.
    -  Declare Module E : OrderedType.
    +Module Type ElementEncode.
    +  Declare Module E : OrderedType.

    -  Parameter encode : E.t → Z.
    -  Parameter decode : Z → E.t.
    +  Parameter encode : E.t → Z.
    +  Parameter decode : Z → E.t.

    @@ -243,12 +244,12 @@

    Library MSetsExtra.MSetIntervals

    Decoding is the inverse of encoding. Notice that the reverse is not demanded. This means that - we do need to provide for all integers z an - element e with encode v = z. + we do need to provide for all integers z an + element e with encode v = z.
    -  Axiom decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    +  Axiom decode_encode_ok: ∀ (e : E.t),
    +    decode (encode e) = e.

    @@ -257,8 +258,8 @@

    Library MSetsExtra.MSetIntervals

    Encoding is compatible with the equality of elements.
    -  Axiom encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +  Axiom encode_eq : ∀ (e1 e2 : E.t),
    +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.

    @@ -267,8 +268,8 @@

    Library MSetsExtra.MSetIntervals

    Encoding is compatible with the order of elements.
    -  Axiom encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Axiom encode_lt : ∀ (e1 e2 : E.t),
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.

    End ElementEncode.
    @@ -277,7 +278,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Set Operations

    +

    Set Operations

    @@ -301,8 +302,8 @@

    Library MSetsExtra.MSetIntervals

    - Intervals we represent by tuples (Z, N). The tuple (z, c) represents the - interval z-(z+c). + Intervals we represent by tuples (Z, N). The tuple (z, c) represents the + interval z-(z+c).
    @@ -312,9 +313,9 @@

    Library MSetsExtra.MSetIntervals

    -Module Ops (Enc : ElementEncode) <: Ops Enc.E.
    -  Definition elt := Enc.E.t.
    -  Definition t := list (Z × N).
    +Module Ops (Enc : ElementEncode) <: Ops Enc.E.
    +  Definition elt := Enc.E.t.
    +  Definition t := list (Z × N).

    @@ -323,8 +324,8 @@

    Library MSetsExtra.MSetIntervals

    The empty list is trivial to define and check for.
    -  Definition empty : t := nil.
    -  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.
    +  Definition empty : t := nil.
    +  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.

    @@ -336,46 +337,46 @@

    Library MSetsExtra.MSetIntervals

    if it needs to be executable.
    -  Lemma acc_pred : ∀ n p, n = Npos p → Acc N.lt n → Acc N.lt (N.pred n).
    +  Lemma acc_pred : ∀ n p, n = Npos p → Acc N.lt n → Acc N.lt (N.pred n).
      Proof.
        intros n p H0 H1.
        apply H1.
        rewrite H0.
    -    apply N.lt_pred_l.
    +    apply N.lt_pred_l.
        discriminate.
      Defined.

    -  Fixpoint fold_elementsZ_aux {A} (f : A → Z → option A) (acc : A) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : (bool × A) :=
    -    match c as c0 return c = c0 → (bool × A) with
    -    | N0 ⇒ fun _ ⇒ (false, acc)
    -    | c ⇒ fun Heq ⇒ match (f acc x) with
    -        | None ⇒ (true, acc)
    -        | Some acc' ⇒
    -          fold_elementsZ_aux f acc' (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end
    -    end (refl_equal _).
    +  Fixpoint fold_elementsZ_aux {A} (f : A → Z → option A) (acc : A) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : (bool × A) :=
    +    match c as c0 return c = c0 → (bool × A) with
    +    | N0 ⇒ fun _ ⇒ (false, acc)
    +    | c ⇒ fun Heq ⇒ match (f acc x) with
    +        | None ⇒ (true, acc)
    +        | Some acc' ⇒
    +          fold_elementsZ_aux f acc' (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end
    +    end (refl_equal _).

    -  Definition fold_elementsZ_single {A} f (acc : A) x c := fold_elementsZ_aux f acc x c (lt_wf_0 _).
    +  Definition fold_elementsZ_single {A} f (acc : A) x c := fold_elementsZ_aux f acc x c (lt_wf_0 _).

    -  Fixpoint fold_elementsZ {A} f (acc : A) (s : t) : (bool × A) :=
    -    match s with
    -    | nil ⇒ (false, acc)
    -    | (x, c) :: s' ⇒
    -      match fold_elementsZ_single f acc x c with
    -          (false, acc') ⇒ fold_elementsZ f acc' s'
    -        | (true, acc') ⇒ (true, acc')
    +  Fixpoint fold_elementsZ {A} f (acc : A) (s : t) : (bool × A) :=
    +    match s with
    +    | nil ⇒ (false, acc)
    +    | (x, c) :: s' ⇒
    +      match fold_elementsZ_single f acc x c with
    +          (false, acc') ⇒ fold_elementsZ f acc' s'
    +        | (true, acc') ⇒ (true, acc')
          end
        end.

    -  Definition elementsZ (s : t) : list Z :=
    -    snd (fold_elementsZ (fun l x ⇒ Some (x :: l)) nil s).
    +  Definition elementsZ (s : t) : list Z :=
    +    snd (fold_elementsZ (fun l x ⇒ Some (x :: l)) nil s).

    -  Definition elements (s : t) : list elt :=
    -    rev_map Enc.decode (elementsZ s).
    +  Definition elements (s : t) : list elt :=
    +    rev_map Enc.decode (elementsZ s).

    @@ -384,17 +385,17 @@

    Library MSetsExtra.MSetIntervals

    membership is easily defined
    -  Fixpoint memZ (x : Z) (s : t) :=
    -    match s with
    -    | nil ⇒ false
    -    | (y, c) :: l ⇒
    -        if (Z.ltb x y) then false else
    -        if (Z.ltb x (y+Z.of_N c)) then true else
    -        memZ x l
    +  Fixpoint memZ (x : Z) (s : t) :=
    +    match s with
    +    | nil ⇒ false
    +    | (y, c) :: l ⇒
    +        if (Z.ltb x y) then false else
    +        if (Z.ltb x (y+Z.of_N c)) then true else
    +        memZ x l
        end.

    -  Definition mem (x : elt) (s : t) := memZ (Enc.encode x) s.
    +  Definition mem (x : elt) (s : t) := memZ (Enc.encode x) s.

    @@ -403,71 +404,71 @@

    Library MSetsExtra.MSetIntervals

    Comparing intervals
    -  Inductive interval_compare_result :=
    -      ICR_before
    -    | ICR_before_touch
    -    | ICR_overlap_before
    -    | ICR_overlap_after
    -    | ICR_equal
    -    | ICR_subsume_1
    -    | ICR_subsume_2
    -    | ICR_after
    -    | ICR_after_touch.
    - -
    -  Definition interval_compare (i1 i2 : (Z × N)) : interval_compare_result :=
    -    match (i1, i2) with ((y1, c1), (y2, c2)) ⇒
    -      let yc2 := (y2+Z.of_N c2) in
    -      match (Z.compare yc2 y1) with
    -        | Lt ⇒ ICR_after
    -        | Eq ⇒ ICR_after_touch
    -        | Gt ⇒ let yc1 := (y1+Z.of_N c1) in
    -                match (Z.compare yc1 y2) with
    -                | Lt ⇒ ICR_before
    -                | Eq ⇒ ICR_before_touch
    -                | Gt ⇒
    -                        match (Z.compare y1 y2, Z.compare yc1 yc2) with
    -                        | (Lt, Lt) ⇒ ICR_overlap_before
    -                        | (Lt, _) ⇒ ICR_subsume_2
    -                        | (Eq, Lt) ⇒ ICR_subsume_1
    -                        | (Eq, Gt) ⇒ ICR_subsume_2
    -                        | (Eq, Eq) ⇒ ICR_equal
    -                        | (Gt, Gt) ⇒ ICR_overlap_after
    -                        | (Gt, _) ⇒ ICR_subsume_1
    +  Inductive interval_compare_result :=
    +      ICR_before
    +    | ICR_before_touch
    +    | ICR_overlap_before
    +    | ICR_overlap_after
    +    | ICR_equal
    +    | ICR_subsume_1
    +    | ICR_subsume_2
    +    | ICR_after
    +    | ICR_after_touch.
    + +
    +  Definition interval_compare (i1 i2 : (Z × N)) : interval_compare_result :=
    +    match (i1, i2) with ((y1, c1), (y2, c2)) ⇒
    +      let yc2 := (y2+Z.of_N c2) in
    +      match (Z.compare yc2 y1) with
    +        | Lt ⇒ ICR_after
    +        | Eq ⇒ ICR_after_touch
    +        | Gt ⇒ let yc1 := (y1+Z.of_N c1) in
    +                match (Z.compare yc1 y2) with
    +                | Lt ⇒ ICR_before
    +                | Eq ⇒ ICR_before_touch
    +                | Gt ⇒
    +                        match (Z.compare y1 y2, Z.compare yc1 yc2) with
    +                        | (Lt, Lt) ⇒ ICR_overlap_before
    +                        | (Lt, _) ⇒ ICR_subsume_2
    +                        | (Eq, Lt) ⇒ ICR_subsume_1
    +                        | (Eq, Gt) ⇒ ICR_subsume_2
    +                        | (Eq, Eq) ⇒ ICR_equal
    +                        | (Gt, Gt) ⇒ ICR_overlap_after
    +                        | (Gt, _) ⇒ ICR_subsume_1
                            end
                    end
          end
        end.

    -  Definition interval_1_compare (y1 : Z) (i : (Z × N)) : interval_compare_result :=
    -    match i with (y2, c2) ⇒
    -      let yc2 := (y2+Z.of_N c2) in
    -      match (Z.compare yc2 y1) with
    -        | Lt ⇒ ICR_after
    -        | Eq ⇒ ICR_after_touch
    -        | Gt ⇒ match (Z.compare (Z.succ y1) y2) with
    -                | Lt ⇒ ICR_before
    -                | Eq ⇒ ICR_before_touch
    -                | Gt ⇒ ICR_subsume_1
    +  Definition interval_1_compare (y1 : Z) (i : (Z × N)) : interval_compare_result :=
    +    match i with (y2, c2) ⇒
    +      let yc2 := (y2+Z.of_N c2) in
    +      match (Z.compare yc2 y1) with
    +        | Lt ⇒ ICR_after
    +        | Eq ⇒ ICR_after_touch
    +        | Gt ⇒ match (Z.compare (Z.succ y1) y2) with
    +                | Lt ⇒ ICR_before
    +                | Eq ⇒ ICR_before_touch
    +                | Gt ⇒ ICR_subsume_1
                    end
          end
        end.

    -  Fixpoint compare (s1 s2 : t) :=
    -    match (s1, s2) with
    -      | (nil, nil) ⇒ Eq
    -      | (nil, _ :: _) ⇒ Lt
    -      | (_ :: _, nil) ⇒ Gt
    -      | ((y1, c1)::s1', (y2, c2)::s2') ⇒
    -        match (Z.compare y1 y2) with
    -          | Lt ⇒ Lt
    -          | Gt ⇒ Gt
    -          | Eq ⇒ match N.compare c1 c2 with
    -                    | Lt ⇒ Lt
    -                    | Gt ⇒ Gt
    -                    | Eq ⇒ compare s1' s2'
    +  Fixpoint compare (s1 s2 : t) :=
    +    match (s1, s2) with
    +      | (nil, nil) ⇒ Eq
    +      | (nil, _ :: _) ⇒ Lt
    +      | (_ :: _, nil) ⇒ Gt
    +      | ((y1, c1)::s1', (y2, c2)::s2') ⇒
    +        match (Z.compare y1 y2) with
    +          | Lt ⇒ Lt
    +          | Gt ⇒ Gt
    +          | Eq ⇒ match N.compare c1 c2 with
    +                    | Lt ⇒ Lt
    +                    | Gt ⇒ Gt
    +                    | Eq ⇒ compare s1' s2'
                      end
            end
        end.
    @@ -479,18 +480,18 @@

    Library MSetsExtra.MSetIntervals

    Auxiliary functions for inserting at front and merging intervals
    -  Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N :=
    -    (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))).
    - -
    -  Fixpoint insert_interval_begin (x : Z) (c : N) (l : t) :=
    -    match l with
    -    | nil ⇒ (x,c)::nil
    -    | (y, c')::l' ⇒
    -         match (Z.compare (x + Z.of_N c) y) with
    -         | Lt ⇒ (x, c) :: l
    -         | Eq ⇒ (x, (c+c')%N) :: l'
    -         | Gt ⇒ insert_interval_begin x (merge_interval_size x c y c') l'
    +  Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N :=
    +    (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))).
    + +
    +  Fixpoint insert_interval_begin (x : Z) (c : N) (l : t) :=
    +    match l with
    +    | nil ⇒ (x,c)::nil
    +    | (y, c')::l' ⇒
    +         match (Z.compare (x + Z.of_N c) y) with
    +         | Lt ⇒ (x, c) :: l
    +         | Eq ⇒ (x, (c+c')%N) :: l'
    +         | Gt ⇒ insert_interval_begin x (merge_interval_size x c y c') l'
             end
        end.
    @@ -502,22 +503,22 @@

    Library MSetsExtra.MSetIntervals

    generate efficient code
    -  Fixpoint addZ_aux (acc : list (Z × N)) (x : Z) (s : t) :=
    -    match s with
    -    | nil ⇒ List.rev' ((x, (1%N))::acc)
    -    | (y, c) :: l ⇒
    -        match (interval_1_compare x (y,c)) with
    -          | ICR_before ⇒ List.rev_append ((x, (1%N))::acc) s
    -          | ICR_before_touch ⇒ List.rev_append ((x, N.succ c)::acc) l
    -          | ICR_after ⇒ addZ_aux ((y,c) :: acc) x l
    -          | ICR_after_touch ⇒ List.rev_append acc (insert_interval_begin y (N.succ c) l)
    -          | _ ⇒ List.rev_append ((y, c)::acc) l
    +  Fixpoint addZ_aux (acc : list (Z × N)) (x : Z) (s : t) :=
    +    match s with
    +    | nil ⇒ List.rev' ((x, (1%N))::acc)
    +    | (y, c) :: l ⇒
    +        match (interval_1_compare x (y,c)) with
    +          | ICR_before ⇒ List.rev_append ((x, (1%N))::acc) s
    +          | ICR_before_touch ⇒ List.rev_append ((x, N.succ c)::acc) l
    +          | ICR_after ⇒ addZ_aux ((y,c) :: acc) x l
    +          | ICR_after_touch ⇒ List.rev_append acc (insert_interval_begin y (N.succ c) l)
    +          | _ ⇒ List.rev_append ((y, c)::acc) l
            end
        end.

    -  Definition addZ x s := addZ_aux nil x s.
    -  Definition add x s := addZ (Enc.encode x) s.
    +  Definition addZ x s := addZ_aux nil x s.
    +  Definition add x s := addZ (Enc.encode x) s.

    @@ -527,11 +528,11 @@

    Library MSetsExtra.MSetIntervals

    This is used to define the function from_elements.
    -  Definition add_list (l : list elt) (s : t) : t :=
    -     List.fold_left (fun s x ⇒ add x s) l s.
    +  Definition add_list (l : list elt) (s : t) : t :=
    +     List.fold_left (fun s x ⇒ add x s) l s.

    -  Definition from_elements (l : list elt) : t := add_list l empty.
    +  Definition from_elements (l : list elt) : t := add_list l empty.

    @@ -540,10 +541,10 @@

    Library MSetsExtra.MSetIntervals

    singleton is trivial to define
    -  Definition singleton (x : elt) : t := (Enc.encode x, 1%N) :: nil.
    +  Definition singleton (x : elt) : t := (Enc.encode x, 1%N) :: nil.

    -  Lemma singleton_alt_def : ∀ x, singleton x = add x empty.
    +  Lemma singleton_alt_def : ∀ x, singleton x = add x empty.
      Proof. by []. Qed.

    @@ -553,29 +554,29 @@

    Library MSetsExtra.MSetIntervals

    removing needs to be done with code extraction in mind again.
    -  Definition insert_intervalZ_guarded (x : Z) (c : N) s :=
    -     if (N.eqb c 0) then s else (x, c) :: s.
    - -
    -  Fixpoint removeZ_aux (acc : list (Z × N)) (x : Z) (s : t) : t :=
    -    match s with
    -    | nil ⇒ List.rev' acc
    -    | (y, c) :: l ⇒
    -        if (Z.ltb x y) then List.rev_append acc s else
    -        if (Z.ltb x (y+Z.of_N c)) then (
    -           List.rev_append (insert_intervalZ_guarded (Z.succ x)
    -              (Z.to_N ((y+Z.of_N c)- (Z.succ x)))
    -             (insert_intervalZ_guarded y (Z.to_N (x-y)) acc)) l
    -        ) else removeZ_aux ((y,c)::acc) x l
    +  Definition insert_intervalZ_guarded (x : Z) (c : N) s :=
    +     if (N.eqb c 0) then s else (x, c) :: s.
    + +
    +  Fixpoint removeZ_aux (acc : list (Z × N)) (x : Z) (s : t) : t :=
    +    match s with
    +    | nil ⇒ List.rev' acc
    +    | (y, c) :: l ⇒
    +        if (Z.ltb x y) then List.rev_append acc s else
    +        if (Z.ltb x (y+Z.of_N c)) then (
    +           List.rev_append (insert_intervalZ_guarded (Z.succ x)
    +              (Z.to_N ((y+Z.of_N c)- (Z.succ x)))
    +             (insert_intervalZ_guarded y (Z.to_N (x-y)) acc)) l
    +        ) else removeZ_aux ((y,c)::acc) x l
        end.

    -  Definition removeZ (x : Z) (s : t) : t := removeZ_aux nil x s.
    -  Definition remove (x : elt) (s : t) : t := removeZ (Enc.encode x) s.
    +  Definition removeZ (x : Z) (s : t) : t := removeZ_aux nil x s.
    +  Definition remove (x : elt) (s : t) : t := removeZ (Enc.encode x) s.

    -  Definition remove_list (l : list elt) (s : t) : t :=
    -     List.fold_left (fun s x ⇒ remove x s) l s.
    +  Definition remove_list (l : list elt) (s : t) : t :=
    +     List.fold_left (fun s x ⇒ remove x s) l s.

    @@ -584,32 +585,32 @@

    Library MSetsExtra.MSetIntervals

    union
    -  Fixpoint union_aux (s1 : t) :=
    -    fix aux (s2 : t) (acc : list (Z × N)) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ List.rev_append acc s2
    -    | (_, nil) ⇒ List.rev_append acc s1
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
    +  Fixpoint union_aux (s1 : t) :=
    +    fix aux (s2 : t) (acc : list (Z × N)) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ List.rev_append acc s2
    +    | (_, nil) ⇒ List.rev_append acc s1
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
              | ICR_before_touch ⇒
    -              union_aux l1 (
    -               insert_interval_begin y1 ((c1+c2)%N) l2) acc
    -          | ICR_after ⇒ aux l2 ((y2, c2)::acc)
    -          | ICR_after_touch ⇒ union_aux l1 (
    -              insert_interval_begin y2 ((c1+c2)%N) l2) acc
    +              union_aux l1 (
    +               insert_interval_begin y1 ((c1+c2)%N) l2) acc
    +          | ICR_after ⇒ aux l2 ((y2, c2)::acc)
    +          | ICR_after_touch ⇒ union_aux l1 (
    +              insert_interval_begin y2 ((c1+c2)%N) l2) acc
              | ICR_overlap_before ⇒
    -              union_aux l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2) acc
    +              union_aux l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2) acc
              | ICR_overlap_after ⇒
    -              union_aux l1 (insert_interval_begin y2 (merge_interval_size y2 c2 y1 c1) l2) acc
    -          | ICR_equal ⇒ union_aux l1 s2 acc
    -          | ICR_subsume_1 ⇒ union_aux l1 s2 acc
    -          | ICR_subsume_2 ⇒ aux l2 acc
    +              union_aux l1 (insert_interval_begin y2 (merge_interval_size y2 c2 y1 c1) l2) acc
    +          | ICR_equal ⇒ union_aux l1 s2 acc
    +          | ICR_subsume_1 ⇒ union_aux l1 s2 acc
    +          | ICR_subsume_2 ⇒ aux l2 acc
            end
        end.

    -  Definition union s1 s2 := union_aux s1 s2 nil.
    +  Definition union s1 s2 := union_aux s1 s2 nil.

    @@ -620,38 +621,38 @@

    Library MSetsExtra.MSetIntervals


    -  Fixpoint diff_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
    -    match s with
    -    | nil ⇒ (acc, nil)
    -    | ((y1, c1) :: l1) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    -          | ICR_before_touch ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    -          | ICR_after ⇒ (acc, s)
    -          | ICR_after_touch ⇒ (acc, s)
    -          | ICR_overlap_before ⇒ diff_aux y2 c2 ((y1, Z.to_N (y2 - y1))::acc) l1
    -          | ICR_overlap_after ⇒ (acc, (y2+Z.of_N c2, Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) :: l1)
    -          | ICR_equal ⇒ (acc, l1)
    -          | ICR_subsume_1 ⇒ diff_aux y2 c2 acc l1
    -          | ICR_subsume_2 ⇒ ((insert_intervalZ_guarded y1
    -                (Z.to_N (y2 - y1)) acc),
    -              insert_intervalZ_guarded (y2+Z.of_N c2) (Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) l1)
    +  Fixpoint diff_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
    +    match s with
    +    | nil ⇒ (acc, nil)
    +    | ((y1, c1) :: l1) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    +          | ICR_before_touch ⇒ diff_aux y2 c2 ((y1, c1)::acc) l1
    +          | ICR_after ⇒ (acc, s)
    +          | ICR_after_touch ⇒ (acc, s)
    +          | ICR_overlap_before ⇒ diff_aux y2 c2 ((y1, Z.to_N (y2 - y1))::acc) l1
    +          | ICR_overlap_after ⇒ (acc, (y2+Z.of_N c2, Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) :: l1)
    +          | ICR_equal ⇒ (acc, l1)
    +          | ICR_subsume_1 ⇒ diff_aux y2 c2 acc l1
    +          | ICR_subsume_2 ⇒ ((insert_intervalZ_guarded y1
    +                (Z.to_N (y2 - y1)) acc),
    +              insert_intervalZ_guarded (y2+Z.of_N c2) (Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) l1)
            end
        end.

    -  Fixpoint diff_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ rev_append acc s1
    -    | (_, nil) ⇒ rev_append acc s1
    -    | (_, (y2, c2) :: l2) ⇒
    -      match diff_aux y2 c2 acc s1 with
    -        (acc', s1') ⇒ diff_aux2 acc' s1' l2
    +  Fixpoint diff_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ rev_append acc s1
    +    | (_, nil) ⇒ rev_append acc s1
    +    | (_, (y2, c2) :: l2) ⇒
    +      match diff_aux y2 c2 acc s1 with
    +        (acc', s1') ⇒ diff_aux2 acc' s1' l2
          end
        end.

    -  Definition diff s1 s2 := diff_aux2 nil s1 s2.
    +  Definition diff s1 s2 := diff_aux2 nil s1 s2.

    @@ -660,22 +661,22 @@

    Library MSetsExtra.MSetIntervals

    subset
    -  Fixpoint subset (s1 : t) :=
    -    fix aux (s2 : t) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ true
    -    | (_ :: _, nil) ⇒ false
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ false
    -          | ICR_before_touch ⇒ false
    -          | ICR_after ⇒ aux l2
    -          | ICR_after_touch ⇒ false
    -          | ICR_overlap_before ⇒ false
    -          | ICR_overlap_after ⇒ false
    -          | ICR_equal ⇒ subset l1 l2
    -          | ICR_subsume_1 ⇒ subset l1 s2
    -          | ICR_subsume_2 ⇒ false
    +  Fixpoint subset (s1 : t) :=
    +    fix aux (s2 : t) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ true
    +    | (_ :: _, nil) ⇒ false
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ false
    +          | ICR_before_touch ⇒ false
    +          | ICR_after ⇒ aux l2
    +          | ICR_after_touch ⇒ false
    +          | ICR_overlap_before ⇒ false
    +          | ICR_overlap_after ⇒ false
    +          | ICR_equal ⇒ subset l1 l2
    +          | ICR_subsume_1 ⇒ subset l1 s2
    +          | ICR_subsume_2 ⇒ false
            end
        end.
    @@ -686,10 +687,10 @@

    Library MSetsExtra.MSetIntervals

    equal
    -  Fixpoint equal (s s' : t) : bool := match s, s' with
    -    | nil, nil ⇒ true
    -    | ((x,cx)::xs), ((y,cy)::ys) ⇒ andb (Z.eqb x y) (andb (N.eqb cx cy) (equal xs ys))
    -    | _, _ ⇒ false
    +  Fixpoint equal (s s' : t) : bool := match s, s' with
    +    | nil, nil ⇒ true
    +    | ((x,cx)::xs), ((y,cy)::ys) ⇒ andb (Z.eqb x y) (andb (N.eqb cx cy) (equal xs ys))
    +    | _, _ ⇒ false
      end.

    @@ -699,36 +700,36 @@

    Library MSetsExtra.MSetIntervals

    inter
    -  Fixpoint inter_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
    -    match s with
    -    | nil ⇒ (acc, nil)
    -    | ((y1, c1) :: l1) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ inter_aux y2 c2 acc l1
    -          | ICR_before_touch ⇒ inter_aux y2 c2 acc l1
    -          | ICR_after ⇒ (acc, s)
    -          | ICR_after_touch ⇒ (acc, s)
    -          | ICR_overlap_before ⇒ inter_aux y2 c2 ((y2, Z.to_N (y1 + Z.of_N c1 - y2))::acc) l1
    -          | ICR_overlap_after ⇒ ((y1, Z.to_N (y2 + Z.of_N c2 - y1))::acc, s)
    -          | ICR_equal ⇒ ((y1,c1)::acc, l1)
    -          | ICR_subsume_1 ⇒ inter_aux y2 c2 ((y1, c1)::acc) l1
    -          | ICR_subsume_2 ⇒ ((y2, c2)::acc, s)
    +  Fixpoint inter_aux (y2 : Z) (c2 : N) (acc : list (Z × N)) (s : t) : (list (Z × N) × t) :=
    +    match s with
    +    | nil ⇒ (acc, nil)
    +    | ((y1, c1) :: l1) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ inter_aux y2 c2 acc l1
    +          | ICR_before_touch ⇒ inter_aux y2 c2 acc l1
    +          | ICR_after ⇒ (acc, s)
    +          | ICR_after_touch ⇒ (acc, s)
    +          | ICR_overlap_before ⇒ inter_aux y2 c2 ((y2, Z.to_N (y1 + Z.of_N c1 - y2))::acc) l1
    +          | ICR_overlap_after ⇒ ((y1, Z.to_N (y2 + Z.of_N c2 - y1))::acc, s)
    +          | ICR_equal ⇒ ((y1,c1)::acc, l1)
    +          | ICR_subsume_1 ⇒ inter_aux y2 c2 ((y1, c1)::acc) l1
    +          | ICR_subsume_2 ⇒ ((y2, c2)::acc, s)
            end
        end.

    -  Fixpoint inter_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    -    match (s1, s2) with
    -    | (nil, _) ⇒ List.rev' acc
    -    | (_, nil) ⇒ List.rev' acc
    -    | (_, (y2, c2) :: l2) ⇒
    -      match inter_aux y2 c2 acc s1 with
    -        (acc', s1') ⇒ inter_aux2 acc' s1' l2
    +  Fixpoint inter_aux2 (acc : list (Z × N)) (s1 s2 : t) : (list (Z × N)) :=
    +    match (s1, s2) with
    +    | (nil, _) ⇒ List.rev' acc
    +    | (_, nil) ⇒ List.rev' acc
    +    | (_, (y2, c2) :: l2) ⇒
    +      match inter_aux y2 c2 acc s1 with
    +        (acc', s1') ⇒ inter_aux2 acc' s1' l2
          end
        end.

    -  Definition inter s1 s2 := inter_aux2 nil s1 s2.
    +  Definition inter s1 s2 := inter_aux2 nil s1 s2.

    @@ -739,95 +740,95 @@

    Library MSetsExtra.MSetIntervals


    -  Definition partitionZ_fold_insert
    -             (cur : option (Z × N)) (x : Z) :=
    -    match cur with
    -       None ⇒ (x, 1%N)
    -     | Some (y, c) ⇒ (y, N.succ c)
    +  Definition partitionZ_fold_insert
    +             (cur : option (Z × N)) (x : Z) :=
    +    match cur with
    +       None ⇒ (x, 1%N)
    +     | Some (y, c) ⇒ (y, N.succ c)
        end.

    -  Definition partitionZ_fold_skip (acc : list (Z × N))
    -             (cur : option (Z × N)) : (list (Z × N)) :=
    -    match cur with
    -       None ⇒ acc
    -     | Some yc ⇒ yc::acc
    +  Definition partitionZ_fold_skip (acc : list (Z × N))
    +             (cur : option (Z × N)) : (list (Z × N)) :=
    +    match cur with
    +       None ⇒ acc
    +     | Some yc ⇒ yc::acc
        end.

    -  Definition partitionZ_fold_fun f st (x : Z) :=
    -    match st with ((acc_t, c_t), (acc_f, c_f)) ⇒
    -      if (f x) then
    -        ((acc_t, Some (partitionZ_fold_insert c_t x)),
    -         (partitionZ_fold_skip acc_f c_f, None))
    -      else
    -        ((partitionZ_fold_skip acc_t c_t, None),
    -         (acc_f, Some (partitionZ_fold_insert c_f x)))
    +  Definition partitionZ_fold_fun f st (x : Z) :=
    +    match st with ((acc_t, c_t), (acc_f, c_f)) ⇒
    +      if (f x) then
    +        ((acc_t, Some (partitionZ_fold_insert c_t x)),
    +         (partitionZ_fold_skip acc_f c_f, None))
    +      else
    +        ((partitionZ_fold_skip acc_t c_t, None),
    +         (acc_f, Some (partitionZ_fold_insert c_f x)))
        end.

    -  Definition partitionZ_single_aux f st (x : Z) (c : N) :=
    -    snd (fold_elementsZ_single (fun st x ⇒ Some (partitionZ_fold_fun f st x)) st x c).
    +  Definition partitionZ_single_aux f st (x : Z) (c : N) :=
    +    snd (fold_elementsZ_single (fun st x ⇒ Some (partitionZ_fold_fun f st x)) st x c).

    -  Definition partitionZ_single f acc_t acc_f x c :=
    -    match partitionZ_single_aux f ((acc_t, None), (acc_f, None)) x c with
    -    | ((acc_t, c_t), (acc_f, c_f)) ⇒
    -        (partitionZ_fold_skip acc_t c_t,
    -         partitionZ_fold_skip acc_f c_f)
    +  Definition partitionZ_single f acc_t acc_f x c :=
    +    match partitionZ_single_aux f ((acc_t, None), (acc_f, None)) x c with
    +    | ((acc_t, c_t), (acc_f, c_f)) ⇒
    +        (partitionZ_fold_skip acc_t c_t,
    +         partitionZ_fold_skip acc_f c_f)
        end.

    -  Fixpoint partitionZ_aux acc_t acc_f f s :=
    -    match s with
    -    | nil ⇒ (List.rev acc_t, List.rev acc_f)
    -    | (y, c) :: s' ⇒
    -      match partitionZ_single f acc_t acc_f y c with
    -      | (acc_t', acc_f') ⇒ partitionZ_aux acc_t' acc_f' f s'
    +  Fixpoint partitionZ_aux acc_t acc_f f s :=
    +    match s with
    +    | nil ⇒ (List.rev acc_t, List.rev acc_f)
    +    | (y, c) :: s' ⇒
    +      match partitionZ_single f acc_t acc_f y c with
    +      | (acc_t', acc_f') ⇒ partitionZ_aux acc_t' acc_f' f s'
          end
        end.

    -  Definition partitionZ := partitionZ_aux nil nil.
    +  Definition partitionZ := partitionZ_aux nil nil.

    -  Definition partition (f : elt → bool) : t → (t × t) :=
    -    partitionZ (fun z ⇒ f (Enc.decode z)).
    +  Definition partition (f : elt → bool) : t → (t × t) :=
    +    partitionZ (fun z ⇒ f (Enc.decode z)).

    -  Definition filterZ_fold_fun f st (x : Z) :=
    -    match st with (acc_t, c_t) ⇒
    -      if (f x) then
    -        (acc_t, Some (partitionZ_fold_insert c_t x))
    -      else
    -        (partitionZ_fold_skip acc_t c_t, None)
    +  Definition filterZ_fold_fun f st (x : Z) :=
    +    match st with (acc_t, c_t) ⇒
    +      if (f x) then
    +        (acc_t, Some (partitionZ_fold_insert c_t x))
    +      else
    +        (partitionZ_fold_skip acc_t c_t, None)
        end.

    -  Definition filterZ_single_aux f st (x : Z) (c : N) :=
    -    snd (fold_elementsZ_single (fun st x ⇒ Some (filterZ_fold_fun f st x)) st x c).
    +  Definition filterZ_single_aux f st (x : Z) (c : N) :=
    +    snd (fold_elementsZ_single (fun st x ⇒ Some (filterZ_fold_fun f st x)) st x c).

    -  Definition filterZ_single f acc x c :=
    -    match filterZ_single_aux f (acc, None) x c with
    -    | (acc, c) ⇒
    -        (partitionZ_fold_skip acc c)
    +  Definition filterZ_single f acc x c :=
    +    match filterZ_single_aux f (acc, None) x c with
    +    | (acc, c) ⇒
    +        (partitionZ_fold_skip acc c)
        end.

    -  Fixpoint filterZ_aux acc f s :=
    -    match s with
    -    | nil ⇒ (List.rev acc)
    -    | (y, c) :: s' ⇒
    -      filterZ_aux (filterZ_single f acc y c) f s'
    +  Fixpoint filterZ_aux acc f s :=
    +    match s with
    +    | nil ⇒ (List.rev acc)
    +    | (y, c) :: s' ⇒
    +      filterZ_aux (filterZ_single f acc y c) f s'
        end.

    -  Definition filterZ := filterZ_aux nil.
    +  Definition filterZ := filterZ_aux nil.

    -  Definition filter (f : elt → bool) : t → t :=
    -    filterZ (fun z ⇒ f (Enc.decode z)).
    +  Definition filter (f : elt → bool) : t → t :=
    +    filterZ (fun z ⇒ f (Enc.decode z)).

    @@ -840,62 +841,62 @@

    Library MSetsExtra.MSetIntervals


    -  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    -    snd (fold_elementsZ (fun b z ⇒ Some (f (Enc.decode z) b)) i s).
    +  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    +    snd (fold_elementsZ (fun b z ⇒ Some (f (Enc.decode z) b)) i s).

    -  Definition for_all (f : elt → bool) (s : t) : bool :=
    -    snd (fold_elementsZ (fun b z ⇒
    -      if b then
    -        Some (f (Enc.decode z))
    -      else None) true s).
    +  Definition for_all (f : elt → bool) (s : t) : bool :=
    +    snd (fold_elementsZ (fun b z ⇒
    +      if b then
    +        Some (f (Enc.decode z))
    +      else None) true s).

    -  Definition exists_ (f : elt → bool) (s : t) : bool :=
    -    snd (fold_elementsZ (fun b z ⇒
    -      if b then
    -        None
    -      else Some (f (Enc.decode z))) false s).
    +  Definition exists_ (f : elt → bool) (s : t) : bool :=
    +    snd (fold_elementsZ (fun b z ⇒
    +      if b then
    +        None
    +      else Some (f (Enc.decode z))) false s).

    -  Fixpoint cardinalN c (s : t) : N := match s with
    -    | nil ⇒ c
    -    | (_,cx)::xs ⇒ cardinalN (c + cx)%N xs
    +  Fixpoint cardinalN c (s : t) : N := match s with
    +    | nil ⇒ c
    +    | (_,cx)::xs ⇒ cardinalN (c + cx)%N xs
      end.

    -  Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s).
    +  Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s).

    -  Definition min_eltZ (s : t) : option Z :=
    -    match s with
    -    | nil ⇒ None
    -    | (x, _) :: _ ⇒ Some x
    +  Definition min_eltZ (s : t) : option Z :=
    +    match s with
    +    | nil ⇒ None
    +    | (x, _) :: _ ⇒ Some x
        end.

    -  Definition min_elt (s : t) : option elt :=
    -    match (min_eltZ s) with
    -    | None ⇒ None
    -    | Some x ⇒ Some (Enc.decode x)
    +  Definition min_elt (s : t) : option elt :=
    +    match (min_eltZ s) with
    +    | None ⇒ None
    +    | Some x ⇒ Some (Enc.decode x)
        end.

    -  Definition choose := min_elt.
    +  Definition choose := min_elt.

    -  Fixpoint max_eltZ (s : t) : option Z :=
    -    match s with
    -    | nil ⇒ None
    -    | (x, c) :: nil ⇒ Some (Z.pred (x + Z.of_N c))
    -    | (x, _) :: s' ⇒ max_eltZ s'
    +  Fixpoint max_eltZ (s : t) : option Z :=
    +    match s with
    +    | nil ⇒ None
    +    | (x, c) :: nil ⇒ Some (Z.pred (x + Z.of_N c))
    +    | (x, _) :: s' ⇒ max_eltZ s'
        end.

    -  Definition max_elt (s : t) : option elt :=
    -    match (max_eltZ s) with
    -    | None ⇒ None
    -    | Some x ⇒ Some (Enc.decode x)
    +  Definition max_elt (s : t) : option elt :=
    +    match (max_eltZ s) with
    +    | None ⇒ None
    +    | Some x ⇒ Some (Enc.decode x)
        end.

    @@ -905,7 +906,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Raw Module

    +

    Raw Module

    @@ -917,62 +918,60 @@

    Library MSetsExtra.MSetIntervals

    the WSetsOn interface.
    -Module Raw (Enc : ElementEncode).
    +Module Raw (Enc : ElementEncode).
      Include (Ops Enc).

    -

    Defining invariant IsOk

    +

    Defining invariant IsOk

    -  Definition is_encoded_elems_list (l : list Z) : Prop :=
    -    (∀ x, List.In x l → ∃ e, Enc.encode e = x).
    +  Definition is_encoded_elems_list (l : list Z) : Prop :=
    +    (∀ x, List.In x l → ∃ e, Enc.encode e = x).

    -  Definition interval_list_elements_greater (x : Z) (l : t) : bool :=
    -    match l with
    -      | nil ⇒ true
    -      | (y, _)::_ ⇒ Z.ltb x y
    +  Definition interval_list_elements_greater (x : Z) (l : t) : bool :=
    +    match l with
    +      | nil ⇒ true
    +      | (y, _)::_ ⇒ Z.ltb x y
        end.

    -  Fixpoint interval_list_invariant (l : t) :=
    -   match l with
    -   | nil ⇒ true
    -   | (x, c) :: l' ⇒
    -       interval_list_elements_greater (x + (Z.of_N c)) l' && negb (N.eqb c 0) && interval_list_invariant l'
    +  Fixpoint interval_list_invariant (l : t) :=
    +   match l with
    +   | nil ⇒ true
    +   | (x, c) :: l' ⇒
    +       interval_list_elements_greater (x + (Z.of_N c)) l' && negb (N.eqb c 0) && interval_list_invariant l'
       end.

    -  Definition IsOk s := (interval_list_invariant s = true ∧ is_encoded_elems_list (elementsZ s)).
    +  Definition IsOk s := (interval_list_invariant s = true ∧ is_encoded_elems_list (elementsZ s)).

    -

    Defining notations

    +

    Defining notations

    -  Section ForNotations.
    +  Section ForNotations.

    -    Class Ok (s:t) : Prop := ok : IsOk s.
    -    Hint Resolve @ok.
    -    Hint Unfold Ok.
    -    Instance IsOk_Ok s `(Hs : IsOk s) : Ok s := { ok := Hs }.
    +    Class Ok (s:t) : Prop := ok : IsOk s.
    +    Instance IsOk_Ok s `(Hs : IsOk s) : Ok s := { ok := Hs }.

    -    Definition In x s := (SetoidList.InA Enc.E.eq x (elements s)).
    -    Definition InZ x s := (List.In x (elementsZ s)).
    -    Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    -    Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    -    Definition Empty s := ∀ a : elt, ¬ In a s.
    -    Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    -    Definition Exists (P : elt → Prop) (s : t) := ∃ x, In x s ∧ P x.
    +    Definition In x s := (SetoidList.InA Enc.E.eq x (elements s)).
    +    Definition InZ x s := (List.In x (elementsZ s)).
    +    Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    +    Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    +    Definition Empty s := ∀ a : elt, ¬ In a s.
    +    Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    +    Definition Exists (P : elt → Prop) (s : t) := ∃ x, In x s ∧ P x.

      End ForNotations.
    @@ -981,7 +980,7 @@

    Library MSetsExtra.MSetIntervals

    -

    elements list properties

    +

    elements list properties

    @@ -994,122 +993,122 @@

    Library MSetsExtra.MSetIntervals


    -  Lemma elementsZ_nil : (elementsZ (nil : t) = nil).
    +  Lemma elementsZ_nil : (elementsZ (nil : t) = nil).
      Proof. done. Qed.

    -  Lemma elements_nil : (elements (nil : t) = nil).
    +  Lemma elements_nil : (elements (nil : t) = nil).
      Proof. done. Qed.

    -  Definition elementsZ_single (x:Z) (c:N) :=
    -      List.rev' (N.peano_rec (fun _ ⇒ list Z)
    -                  nil (fun n ls ⇒ (x+Z.of_N n)%Z :: ls) c).
    +  Definition elementsZ_single (x:Z) (c:N) :=
    +      List.rev' (N.peano_rec (fun _ ⇒ list Z)
    +                  nil (fun n ls ⇒ (x+Z.of_N n)%Z :: ls) c).

    -  Definition elements_single x c :=
    -    List.map Enc.decode (elementsZ_single x c).
    +  Definition elements_single x c :=
    +    List.map Enc.decode (elementsZ_single x c).

    -  Lemma elementsZ_single_base : ∀ x,
    -    elementsZ_single x (0%N) = nil.
    +  Lemma elementsZ_single_base : ∀ x,
    +    elementsZ_single x (0%N) = nil.
      Proof. done. Qed.

    -  Lemma elementsZ_single_succ : ∀ x c,
    -    elementsZ_single x (N.succ c) =
    -    elementsZ_single x c ++ (x+Z.of_N c)::nil.
    +  Lemma elementsZ_single_succ : ∀ x c,
    +    elementsZ_single x (N.succ c) =
    +    elementsZ_single x c ++ (x+Z.of_N c)::nil.
      Proof.
        intros x c.
    -    rewrite /elementsZ_single N.peano_rec_succ /rev' -!rev_alt //.
    +    rewrite /elementsZ_single N.peano_rec_succ /rev' -!rev_alt //.
      Qed.

    -  Lemma elementsZ_single_add : ∀ x c2 c1,
    -    elementsZ_single x (c1 + c2)%N =
    -    elementsZ_single x c1 ++ elementsZ_single (x+Z.of_N c1) c2.
    +  Lemma elementsZ_single_add : ∀ x c2 c1,
    +    elementsZ_single x (c1 + c2)%N =
    +    elementsZ_single x c1 ++ elementsZ_single (x+Z.of_N c1) c2.
      Proof.
        intros x.
    -    induction c2 as [| c2' IH] using N.peano_ind. {
    +    induction c2 as [| c2' IH] using N.peano_ind. {
          move ⇒ c1.
    -      rewrite elementsZ_single_base /= app_nil_r N.add_0_r //.
    +      rewrite elementsZ_single_base /= app_nil_r N.add_0_r //.
        } {
          move ⇒ c1.
    -      rewrite N.add_succ_r !elementsZ_single_succ IH app_assoc N2Z.inj_add Z.add_assoc //.
    +      rewrite N.add_succ_r !elementsZ_single_succ IH app_assoc N2Z.inj_add Z.add_assoc //.
        }
      Qed.

    -  Lemma elementsZ_single_succ_front : ∀ x c,
    -    elementsZ_single x (N.succ c) =
    -    x :: elementsZ_single (Z.succ x) c.
    +  Lemma elementsZ_single_succ_front : ∀ x c,
    +    elementsZ_single x (N.succ c) =
    +    x :: elementsZ_single (Z.succ x) c.
      Proof.
        intros x c.
    -    rewrite -N.add_1_l elementsZ_single_add.
    -    rewrite N.one_succ elementsZ_single_succ elementsZ_single_base /= Z.add_0_r.
    -    by rewrite Z.add_1_r.
    +    rewrite -N.add_1_l elementsZ_single_add.
    +    rewrite N.one_succ elementsZ_single_succ elementsZ_single_base /= Z.add_0_r.
    +    by rewrite Z.add_1_r.
      Qed.

    -  Lemma In_elementsZ_single : ∀ c y x,
    -    List.In y (elementsZ_single x c) ↔
    -    (x ≤ y) ∧ (y < (x+Z.of_N c)).
    +  Lemma In_elementsZ_single : ∀ c y x,
    +    List.In y (elementsZ_single x c) ↔
    +    (x ≤ y) ∧ (y < (x+Z.of_N c)).
      Proof.
    -    induction c as [| c' IH] using N.peano_ind. {
    +    induction c as [| c' IH] using N.peano_ind. {
          intros y x.
    -      rewrite elementsZ_single_base Z.add_0_r /=.
    -      omega.
    +      rewrite elementsZ_single_base Z.add_0_r /=.
    +      lia.
        } {
          intros y x.
    -      rewrite elementsZ_single_succ in_app_iff IH /= N2Z.inj_succ Z.add_succ_r Z.lt_succ_r.
    +      rewrite elementsZ_single_succ in_app_iff IH /= N2Z.inj_succ Z.add_succ_r Z.lt_succ_r.
          split. {
            move ⇒ [ | []] //. {
              move ⇒ [H_x_le H_y_le].
    -          omega.
    +          lia.
            } {
              move ⇒ <-.
              split.
                - by apply Z_le_add_r.
    -            - by apply Z.le_refl.
    +            - by apply Z.le_refl.
            }
          } {
            move ⇒ [H_x_le] H_y_lt.
    -        omega.
    +        lia.
          }
        }
      Qed.

    -  Lemma In_elementsZ_single1 : ∀ y x,
    -    List.In y (elementsZ_single x (1%N)) ↔
    -    (x = y).
    +  Lemma In_elementsZ_single1 : ∀ y x,
    +    List.In y (elementsZ_single x (1%N)) ↔
    +    (x = y).
      Proof.
        intros y x.
    -    rewrite In_elementsZ_single /= Z.add_1_r Z.lt_succ_r.
    -    omega.
    +    rewrite In_elementsZ_single /= Z.add_1_r Z.lt_succ_r.
    +    lia.
      Qed.

    -  Lemma length_elementsZ_single : ∀ cx x,
    -    length (elementsZ_single x cx) = N.to_nat cx.
    +  Lemma length_elementsZ_single : ∀ cx x,
    +    length (elementsZ_single x cx) = N.to_nat cx.
      Proof.
    -    induction cx as [| cx' IH] using N.peano_ind. {
    +    induction cx as [| cx' IH] using N.peano_ind. {
          by simpl.
        } {
          intros x.
          rewrite elementsZ_single_succ_front /=.
    -      rewrite IH N2Nat.inj_succ //.
    +      rewrite IH N2Nat.inj_succ //.
        }
      Qed.

    -  Lemma fold_elementsZ_aux_irrel {A} :
    -    âˆ€ f c (acc : A) x H1 H2,
    -      fold_elementsZ_aux f acc x c H1 =
    -      fold_elementsZ_aux f acc x c H2.
    +  Lemma fold_elementsZ_aux_irrel {A} :
    +    âˆ€ f c (acc : A) x H1 H2,
    +      fold_elementsZ_aux f acc x c H1 =
    +      fold_elementsZ_aux f acc x c H2.
      Proof.
        intros f c.
    -    induction c as [c IH] using (well_founded_ind lt_wf_0).
    +    induction c as [c IH] using (well_founded_ind lt_wf_0).
        case_eq c. {
          intros H_c acc x; case; intro H_H1; case; intro H_H2.
          reflexivity.
    @@ -1119,19 +1118,19 @@

    Library MSetsExtra.MSetIntervals

          case (f acc x) ⇒ // acc'.
          apply IH.
          rewrite H_c.
    -      apply N.lt_pred_l.
    +      apply N.lt_pred_l.
          discriminate.
        }
      Qed.

    -  Lemma fold_elementsZ_single_pos {A} : ∀ f (acc : A) x p,
    -    fold_elementsZ_single f acc x (N.pos p) =
    -    match f acc x with
    -    | Some acc' ⇒
    -        fold_elementsZ_single f acc' (Z.succ x)
    -         (N.pred (N.pos p))
    -    | None ⇒ (true, acc)
    +  Lemma fold_elementsZ_single_pos {A} : ∀ f (acc : A) x p,
    +    fold_elementsZ_single f acc x (N.pos p) =
    +    match f acc x with
    +    | Some acc' ⇒
    +        fold_elementsZ_single f acc' (Z.succ x)
    +         (N.pred (N.pos p))
    +    | None ⇒ (true, acc)
        end.
      Proof.
        intros f acc x p.
    @@ -1145,8 +1144,8 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma fold_elementsZ_single_zero {A} : ∀ f (acc : A) x,
    -      fold_elementsZ_single f acc x (0%N) = (false, acc).
    +  Lemma fold_elementsZ_single_zero {A} : ∀ f (acc : A) x,
    +      fold_elementsZ_single f acc x (0%N) = (false, acc).
      Proof.
        intros f acc x.
        unfold fold_elementsZ_single.
    @@ -1156,12 +1155,12 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma fold_elementsZ_single_succ {A} : ∀ f (acc : A) x c,
    -    fold_elementsZ_single f acc x (N.succ c) =
    -    match f acc x with
    -      | Some acc' ⇒
    -          fold_elementsZ_single f acc' (Z.succ x) c
    -      | None ⇒ (true, acc)
    +  Lemma fold_elementsZ_single_succ {A} : ∀ f (acc : A) x c,
    +    fold_elementsZ_single f acc x (N.succ c) =
    +    match f acc x with
    +      | Some acc' ⇒
    +          fold_elementsZ_single f acc' (Z.succ x) c
    +      | None ⇒ (true, acc)
        end.
      Proof.
        intros f acc x c.
    @@ -1171,43 +1170,43 @@

    Library MSetsExtra.MSetIntervals

          intro p; simpl.
          rewrite fold_elementsZ_single_pos.
          case (f acc x) ⇒ // acc' /=.
    -      by rewrite Pos.pred_N_succ.
    +      by rewrite Pos.pred_N_succ.
        }
      Qed.

    -  Fixpoint fold_opt {A B} f (acc : A) (bs : list B) : (bool × A) :=
    -    match bs with
    -      | nil ⇒ (false, acc)
    -      | (b :: bs') ⇒
    -        match f acc b with
    -        | Some acc' ⇒ fold_opt f acc' bs'
    -        | None ⇒ (true, acc)
    +  Fixpoint fold_opt {A B} f (acc : A) (bs : list B) : (bool × A) :=
    +    match bs with
    +      | nil ⇒ (false, acc)
    +      | (b :: bs') ⇒
    +        match f acc b with
    +        | Some acc' ⇒ fold_opt f acc' bs'
    +        | None ⇒ (true, acc)
            end
        end.

    -  Lemma fold_opt_list_cons : ∀ {A} (bs : list A) (acc : list A),
    -    fold_opt (fun l x ⇒ Some (x :: l)) acc bs =
    -    (false, List.rev bs ++ acc).
    +  Lemma fold_opt_list_cons : ∀ {A} (bs : list A) (acc : list A),
    +    fold_opt (fun l x ⇒ Some (x :: l)) acc bs =
    +    (false, List.rev bs ++ acc).
      Proof.
        induction bs as [| b bs' IH] ⇒ acc. {
          by simpl.
        } {
    -      rewrite /= IH -app_assoc //.
    +      rewrite /= IH -app_assoc //.
        }
      Qed.

    -  Lemma fold_opt_app {A B} : ∀ f (acc : A) (l1 l2 : list B),
    -    fold_opt f acc (l1 ++ l2) =
    -    (let (ab, acc') := fold_opt f acc l1 in
    -     if ab then (true, acc') else fold_opt f acc' l2).
    +  Lemma fold_opt_app {A B} : ∀ f (acc : A) (l1 l2 : list B),
    +    fold_opt f acc (l1 ++ l2) =
    +    (let (ab, acc') := fold_opt f acc l1 in
    +     if ab then (true, acc') else fold_opt f acc' l2).
      Proof.
        intros f acc l1 l2.
        move : acc.
        induction l1 as [| b l1' IH] ⇒ acc. {
    -      rewrite app_nil_l //.
    +      rewrite app_nil_l //.
        } {
          rewrite /=.
          case (f acc b); last done.
    @@ -1219,12 +1218,12 @@

    Library MSetsExtra.MSetIntervals



    -  Lemma fold_elementsZ_single_alt_def {A} : ∀ f c (acc : A) x,
    -     fold_elementsZ_single f acc x c =
    -     fold_opt f acc (elementsZ_single x c).
    +  Lemma fold_elementsZ_single_alt_def {A} : ∀ f c (acc : A) x,
    +     fold_elementsZ_single f acc x c =
    +     fold_opt f acc (elementsZ_single x c).
      Proof.
        intro f.
    -    induction c as [| c' IH] using N.peano_ind. {
    +    induction c as [| c' IH] using N.peano_ind. {
          intros acc x.
          rewrite fold_elementsZ_single_zero
                  elementsZ_single_base /fold_opt //.
    @@ -1239,42 +1238,42 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma fold_elementsZ_nil {A} : ∀ f (acc : A),
    -     fold_elementsZ f acc nil = (false, acc).
    +  Lemma fold_elementsZ_nil {A} : ∀ f (acc : A),
    +     fold_elementsZ f acc nil = (false, acc).
      Proof. done. Qed.

    -  Lemma fold_elementsZ_cons {A} : ∀ f (acc : A) y c s,
    -    fold_elementsZ f acc ((y, c)::s) =
    -    (let (ab, acc') := fold_elementsZ_single f acc y c in
    -     if ab then (true, acc') else fold_elementsZ f acc' s).
    +  Lemma fold_elementsZ_cons {A} : ∀ f (acc : A) y c s,
    +    fold_elementsZ f acc ((y, c)::s) =
    +    (let (ab, acc') := fold_elementsZ_single f acc y c in
    +     if ab then (true, acc') else fold_elementsZ f acc' s).
      Proof.
        intros f acc y c s.
        done.
      Qed.

    -  Lemma fold_elementsZ_alt_def_aux : ∀ (s : t) base,
    -    (snd (fold_elementsZ
    -      (fun (l : list Z) (x : Z) ⇒ Some (x :: l)) base s)) =
    -    elementsZ s ++ base.
    +  Lemma fold_elementsZ_alt_def_aux : ∀ (s : t) base,
    +    (snd (fold_elementsZ
    +      (fun (l : list Z) (x : Z) ⇒ Some (x :: l)) base s)) =
    +    elementsZ s ++ base.
      Proof.
        induction s as [| [y1 c1] s' IH] ⇒ base. {
    -      rewrite elementsZ_nil /fold_elementsZ /fold_opt /snd
    -        app_nil_l //.
    +      rewrite elementsZ_nil /fold_elementsZ /fold_opt /snd
    +        app_nil_l //.
        } {
          rewrite /elementsZ !fold_elementsZ_cons.
          rewrite !fold_elementsZ_single_alt_def !fold_opt_list_cons.
    -      rewrite !IH app_nil_r app_assoc //.
    +      rewrite !IH app_nil_r app_assoc //.
        }
      Qed.


    -  Lemma fold_elementsZ_alt_def {A} : ∀ f s (acc : A),
    -     fold_elementsZ f acc s =
    -     fold_opt f acc (rev (elementsZ s)).
    +  Lemma fold_elementsZ_alt_def {A} : ∀ f s (acc : A),
    +     fold_elementsZ f acc s =
    +     fold_opt f acc (rev (elementsZ s)).
      Proof.
        intro f.
        induction s as [| [y1 c1] s' IH] ⇒ acc. {
    @@ -1282,105 +1281,105 @@

    Library MSetsExtra.MSetIntervals

        } {
          rewrite /elementsZ !fold_elementsZ_cons.
          rewrite !fold_elementsZ_single_alt_def
    -              fold_opt_list_cons app_nil_r
    -              fold_elementsZ_alt_def_aux rev_app_distr
    -              rev_involutive fold_opt_app.
    +              fold_opt_list_cons app_nil_r
    +              fold_elementsZ_alt_def_aux rev_app_distr
    +              rev_involutive fold_opt_app.
          case (fold_opt f acc (elementsZ_single y1 c1)).
          move ⇒ [] //.
        }
      Qed.

    -  Lemma elementsZ_cons : ∀ x c s, elementsZ (((x, c) :: s) : t) =
    -     ((elementsZ s) ++ (List.rev (elementsZ_single x c))).
    +  Lemma elementsZ_cons : ∀ x c s, elementsZ (((x, c) :: s) : t) =
    +     ((elementsZ s) ++ (List.rev (elementsZ_single x c))).
      Proof.
        intros x c s.
        rewrite /elementsZ fold_elementsZ_cons
                !fold_elementsZ_alt_def
                fold_elementsZ_single_alt_def.
        rewrite !fold_opt_list_cons.
    -    rewrite !app_nil_r !rev_involutive /=.
    +    rewrite !app_nil_r !rev_involutive /=.
        rewrite fold_elementsZ_alt_def_aux //.
      Qed.

    -  Lemma elements_cons : ∀ x c s, elements (((x, c) :: s) : t) =
    -     ((elements_single x c) ++ elements s).
    +  Lemma elements_cons : ∀ x c s, elements (((x, c) :: s) : t) =
    +     ((elements_single x c) ++ elements s).
      Proof.
        intros x c s.
        rewrite /elements /elements_single elementsZ_cons.
    -    rewrite !rev_map_alt_def map_app rev_app_distr map_rev rev_involutive //.
    +    rewrite !rev_map_alt_def map_app rev_app_distr map_rev rev_involutive //.
      Qed.

    -  Lemma elementsZ_app : ∀ (s1 s2 : t), elementsZ (s1 ++ s2) =
    -     ((elementsZ s2) ++ (elementsZ s1)).
    +  Lemma elementsZ_app : ∀ (s1 s2 : t), elementsZ (s1 ++ s2) =
    +     ((elementsZ s2) ++ (elementsZ s1)).
      Proof.
        induction s1 as [| [x1 c1] s1 IH1]. {
          move ⇒ s2.
    -      rewrite elementsZ_nil app_nil_r //.
    +      rewrite elementsZ_nil app_nil_r //.
        }
        move ⇒ s2.
    -    rewrite -app_comm_cons !elementsZ_cons IH1 -app_assoc //.
    +    rewrite -app_comm_cons !elementsZ_cons IH1 -app_assoc //.
      Qed.

    -  Lemma InZ_nil : ∀ y, InZ y nil ↔ False.
    +  Lemma InZ_nil : ∀ y, InZ y nil ↔ False.
      Proof.
        intro y.
        done.
      Qed.

    -  Lemma InZ_cons : ∀ y x c s, InZ y (((x, c) :: s) : t) ↔
    -     List.In y (elementsZ_single x c) ∨ InZ y s.
    +  Lemma InZ_cons : ∀ y x c s, InZ y (((x, c) :: s) : t) ↔
    +     List.In y (elementsZ_single x c) ∨ InZ y s.
      Proof.
        intros y x c s.
    -    rewrite /InZ elementsZ_cons in_app_iff -in_rev.
    +    rewrite /InZ elementsZ_cons in_app_iff -in_rev.
        firstorder.
      Qed.

    -  Lemma InZ_app : ∀ s1 s2 y,
    -     InZ y (s1 ++ s2) ↔ InZ y s1 ∨ InZ y s2.
    +  Lemma InZ_app : ∀ s1 s2 y,
    +     InZ y (s1 ++ s2) ↔ InZ y s1 ∨ InZ y s2.
      Proof.
        intros s1 s2 y.
    -    rewrite /InZ elementsZ_app in_app_iff.
    +    rewrite /InZ elementsZ_app in_app_iff.
        tauto.
      Qed.

    -  Lemma InZ_rev : ∀ s y,
    -     InZ y (List.rev s) ↔ InZ y s.
    +  Lemma InZ_rev : ∀ s y,
    +     InZ y (List.rev s) ↔ InZ y s.
      Proof.
        intros s x.
        rewrite /InZ.
        induction s as [| [y c] s' IH]; first done.
        simpl.
    -    rewrite elementsZ_app in_app_iff IH.
    -    rewrite !elementsZ_cons !in_app_iff elementsZ_nil
    -            -!in_rev /=.
    +    rewrite elementsZ_app in_app_iff IH.
    +    rewrite !elementsZ_cons !in_app_iff elementsZ_nil
    +            -!in_rev /=.
        tauto.
      Qed.

    -  Lemma In_elementsZ_single_dec : ∀ y x c,
    -    {List.In y (elementsZ_single x c)} +
    -    {¬ List.In y (elementsZ_single x c)}.
    +  Lemma In_elementsZ_single_dec : ∀ y x c,
    +    {List.In y (elementsZ_single x c)} +
    +    {¬ List.In y (elementsZ_single x c)}.
      Proof.
        intros y x c.
    -    case (Z_le_dec x y); last first. {
    +    case (Z_le_dec x y); last first. {
          right; rewrite In_elementsZ_single; tauto.
        }
    -    case (Z_lt_dec y (x + Z.of_N c)); last first. {
    +    case (Z_lt_dec y (x + Z.of_N c)); last first. {
          right; rewrite In_elementsZ_single; tauto.
        }
        left; rewrite In_elementsZ_single; tauto.
      Qed.

    -  Lemma InZ_dec : ∀ y s,
    -     {InZ y s} + {¬InZ y s}.
    +  Lemma InZ_dec : ∀ y s,
    +     {InZ y s} + {¬InZ y s}.
      Proof.
        intros y.
        induction s as [| [x c] s IH]. {
    @@ -1397,17 +1396,17 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma In_elementsZ_single_hd : ∀ (c : N) x, (c ≠ 0)%N → List.In x (elementsZ_single x c).
    +  Lemma In_elementsZ_single_hd : ∀ (c : N) x, (c ≠ 0)%N → List.In x (elementsZ_single x c).
      Proof.
        intros c x H_c_neq.
        rewrite In_elementsZ_single.
        split. {
    -      apply Z.le_refl.
    +      apply Z.le_refl.
        } {
    -      apply Z.lt_add_pos_r.
    -      have → : 0 = Z.of_N (0%N) by [].
    -      apply N2Z.inj_lt.
    -      by apply N.neq_0_lt_0.
    +      apply Z.lt_add_pos_r.
    +      have → : 0 = Z.of_N (0%N) by [].
    +      apply N2Z.inj_lt.
    +      by apply N.neq_0_lt_0.
        }
      Qed.
    @@ -1417,51 +1416,51 @@

    Library MSetsExtra.MSetIntervals

    -

    comparing intervals

    +

    comparing intervals


      Ltac Z_named_compare_cases H := match goal with
    -    | [ |- context [Z.compare ?z1 ?z2] ] ⇒
    -      case_eq (Z.compare z1 z2); [move ⇒ /Z.compare_eq_iff | move ⇒ /Z.compare_lt_iff | move ⇒ /Z.compare_gt_iff]; move ⇒ H //
    +    | [ |- context [Z.compare ?z1 ?z2] ] ⇒
    +      case_eq (Z.compare z1 z2); [move ⇒ /Z.compare_eq_iff | move ⇒ /Z.compare_lt_iff | move ⇒ /Z.compare_gt_iff]; move ⇒ H //
      end.

      Ltac Z_compare_cases := let H := fresh "H" in Z_named_compare_cases H.

    -  Lemma interval_compare_elim : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    -    match (interval_compare (y1, c1) (y2, c2)) with
    -      | ICR_before ⇒ (y1 + Z.of_N c1) < y2
    -      | ICR_before_touch ⇒ (y1 + Z.of_N c1) = y2
    -      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    -      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    -      | ICR_equal ⇒ (y1 = y2) ∧ (c1 = c2)
    -      | ICR_overlap_before ⇒ (y1 < y2) ∧ (y2 < y1 + Z.of_N c1) ∧ (y1 + Z.of_N c1 < y2 + Z.of_N c2)
    -      | ICR_overlap_after ⇒ (y2 < y1) ∧ (y1 < y2 + Z.of_N c2) ∧ (y2 + Z.of_N c2 < y1 + Z.of_N c1)
    -      | ICR_subsume_1 ⇒ (y2 ≤ y1) ∧ (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) ∧ (y2 < y1 ∨ y1 + Z.of_N c1 < y2 + Z.of_N c2)
    -      | ICR_subsume_2 ⇒ (y1 ≤ y2) ∧ (y2 + Z.of_N c2 ≤ y1 + Z.of_N c1) ∧ (y1 < y2 ∨ y2 + Z.of_N c2 < y1 + Z.of_N c1)
    +  Lemma interval_compare_elim : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    +    match (interval_compare (y1, c1) (y2, c2)) with
    +      | ICR_before ⇒ (y1 + Z.of_N c1) < y2
    +      | ICR_before_touch ⇒ (y1 + Z.of_N c1) = y2
    +      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    +      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    +      | ICR_equal ⇒ (y1 = y2) ∧ (c1 = c2)
    +      | ICR_overlap_before ⇒ (y1 < y2) ∧ (y2 < y1 + Z.of_N c1) ∧ (y1 + Z.of_N c1 < y2 + Z.of_N c2)
    +      | ICR_overlap_after ⇒ (y2 < y1) ∧ (y1 < y2 + Z.of_N c2) ∧ (y2 + Z.of_N c2 < y1 + Z.of_N c1)
    +      | ICR_subsume_1 ⇒ (y2 ≤ y1) ∧ (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) ∧ (y2 < y1 ∨ y1 + Z.of_N c1 < y2 + Z.of_N c2)
    +      | ICR_subsume_2 ⇒ (y1 ≤ y2) ∧ (y2 + Z.of_N c2 ≤ y1 + Z.of_N c1) ∧ (y1 < y2 ∨ y2 + Z.of_N c2 < y1 + Z.of_N c1)
        end.
      Proof.
        intros y1 c1 y2 c2.
        rewrite /interval_compare.
        (repeat Z_compare_cases); subst; repeat split;
    -       try (by apply Z.eq_le_incl);
    -       try (by apply Z.lt_le_incl);
    +       try (by apply Z.eq_le_incl);
    +       try (by apply Z.lt_le_incl);
           try (by left); try (by right).

    -    apply Z.add_reg_l in H2.
    -    by apply N2Z.inj.
    +    apply Z.add_reg_l in H2.
    +    by apply N2Z.inj.
      Qed.

    -  Lemma interval_compare_swap : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    -    (c1 ≠ 0%N) ∨ (c2 ≠ 0%N) →
    -    interval_compare (y2, c2) (y1, c1) =
    -    match (interval_compare (y1, c1) (y2, c2)) with
    +  Lemma interval_compare_swap : ∀ (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
    +    (c1 ≠ 0%N) ∨ (c2 ≠ 0%N) →
    +    interval_compare (y2, c2) (y1, c1) =
    +    match (interval_compare (y1, c1) (y2, c2)) with
          | ICR_before ⇒ ICR_after
          | ICR_before_touch ⇒ ICR_after_touch
          | ICR_after ⇒ ICR_before
    @@ -1475,50 +1474,50 @@

    Library MSetsExtra.MSetIntervals

      Proof.
        intros y1 c1 y2 c2 H_c12_neq_0.
        rewrite /interval_compare.
    -    move : (Z.compare_antisym y1 y2) ⇒ →.
    -    move : (Z.compare_antisym (y1 + Z.of_N c1) (y2 + Z.of_N c2)) ⇒ →.
    -    have H_suff : y1 + Z.of_N c1 ≤ y2 → y2 + Z.of_N c2 ≤ y1 → False. {
    +    move : (Z.compare_antisym y1 y2) ⇒ →.
    +    move : (Z.compare_antisym (y1 + Z.of_N c1) (y2 + Z.of_N c2)) ⇒ →.
    +    have H_suff : y1 + Z.of_N c1 ≤ y2 → y2 + Z.of_N c2 ≤ y1 → False. {
          move ⇒ H1 H2.
          case H_c12_neq_0 ⇒ H_c_neq_0. {
    -        suff : (y1 + Z.of_N c1 ≤ y1). {
    -          apply Z.nle_gt.
    +        suff : (y1 + Z.of_N c1 ≤ y1). {
    +          apply Z.nle_gt.
              by apply Z_lt_add_r.
            }
    -        eapply Z.le_trans; eauto.
    -        eapply Z.le_trans; eauto.
    +        eapply Z.le_trans; eauto.
    +        eapply Z.le_trans; eauto.
            apply Z_le_add_r.
          } {
    -        suff : (y2 + Z.of_N c2 ≤ y2). {
    -          apply Z.nle_gt.
    +        suff : (y2 + Z.of_N c2 ≤ y2). {
    +          apply Z.nle_gt.
              by apply Z_lt_add_r.
            }
    -        eapply Z.le_trans; eauto.
    -        eapply Z.le_trans; eauto.
    +        eapply Z.le_trans; eauto.
    +        eapply Z.le_trans; eauto.
            apply Z_le_add_r.
          }
        }
        repeat Z_compare_cases. {
          exfalso; apply H_suff.
    -        - by rewrite H; apply Z.le_refl.
    -        - by rewrite H0; apply Z.le_refl.
    +        - by rewrite H; apply Z.le_refl.
    +        - by rewrite H0; apply Z.le_refl.
        } {
          exfalso; apply H_suff.
    -        - by rewrite H; apply Z.le_refl.
    -        - by apply Z.lt_le_incl.
    +        - by rewrite H; apply Z.le_refl.
    +        - by apply Z.lt_le_incl.
        } {
          exfalso; apply H_suff.
    -        - by apply Z.lt_le_incl.
    -        - by rewrite H0; apply Z.le_refl.
    +        - by apply Z.lt_le_incl.
    +        - by rewrite H0; apply Z.le_refl.
        } {
          exfalso; apply H_suff.
    -        - by apply Z.lt_le_incl.
    -        - by apply Z.lt_le_incl.
    +        - by apply Z.lt_le_incl.
    +        - by apply Z.lt_le_incl.
        }
      Qed.

    -  Lemma interval_1_compare_alt_def : ∀ (y : Z) (i : (Z × N)),
    -    interval_1_compare y i = match (interval_compare (y, (1%N)) i) with
    +  Lemma interval_1_compare_alt_def : ∀ (y : Z) (i : (Z × N)),
    +    interval_1_compare y i = match (interval_compare (y, (1%N)) i) with
          | ICR_equal ⇒ ICR_subsume_1
          | ICR_subsume_1 ⇒ ICR_subsume_1
          | ICR_subsume_2 ⇒ ICR_subsume_1
    @@ -1527,68 +1526,68 @@

    Library MSetsExtra.MSetIntervals

      Proof.
        move ⇒ y1 [y2 c2].
        rewrite /interval_1_compare /interval_compare.
    -    replace (y1 + Z.of_N 1) with (Z.succ y1); last done.
    +    replace (y1 + Z.of_N 1) with (Z.succ y1); last done.
        repeat Z_compare_cases. {
          contradict H1.
    -      by apply Zle_not_lt, Zlt_succ_le.
    +      by apply Zle_not_lt, Zlt_succ_le.
        } {
          contradict H.
    -      by apply Zle_not_lt, Zlt_succ_le.
    +      by apply Zle_not_lt, Zlt_succ_le.
        }
      Qed.

    -  Lemma interval_1_compare_elim : ∀ (y1 : Z) (y2 : Z) (c2 : N),
    -    match (interval_1_compare y1 (y2, c2)) with
    -      | ICR_before ⇒ Z.succ y1 < y2
    -      | ICR_before_touch ⇒ y2 = Z.succ y1
    -      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    -      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    -      | ICR_equal ⇒ False
    -      | ICR_overlap_before ⇒ False
    -      | ICR_overlap_after ⇒ False
    -      | ICR_subsume_1 ⇒ (c2 = 0%N) ∨ ((y2 ≤ y1) ∧ (y1 < y2 + Z.of_N c2))
    -      | ICR_subsume_2 ⇒ False
    +  Lemma interval_1_compare_elim : ∀ (y1 : Z) (y2 : Z) (c2 : N),
    +    match (interval_1_compare y1 (y2, c2)) with
    +      | ICR_before ⇒ Z.succ y1 < y2
    +      | ICR_before_touch ⇒ y2 = Z.succ y1
    +      | ICR_after ⇒ (y2 + Z.of_N c2) < y1
    +      | ICR_after_touch ⇒ (y2 + Z.of_N c2) = y1
    +      | ICR_equal ⇒ False
    +      | ICR_overlap_before ⇒ False
    +      | ICR_overlap_after ⇒ False
    +      | ICR_subsume_1 ⇒ (c2 = 0%N) ∨ ((y2 ≤ y1) ∧ (y1 < y2 + Z.of_N c2))
    +      | ICR_subsume_2 ⇒ False
        end.
      Proof.
        intros y1 y2 c2.
        move : (interval_compare_elim y1 (1%N) y2 c2).
        rewrite interval_1_compare_alt_def.
    -    have H_succ: ∀ z, z + Z.of_N 1 = Z.succ z by done.
    +    have H_succ: ∀ z, z + Z.of_N 1 = Z.succ z by done.

    -    case_eq (interval_compare (y1, 1%N) (y2, c2)) ⇒ H_comp;
    -      rewrite ?H_succ ?Z.lt_succ_r ?Z.le_succ_l //. {
    +    case_eq (interval_compare (y1, 1%N) (y2, c2)) ⇒ H_comp;
    +      rewrite ?H_succ ?Z.lt_succ_r ?Z.le_succ_l //. {
          move ⇒ [H_lt] [H_le] _.
          contradict H_lt.
    -      by apply Zle_not_lt.
    +      by apply Zle_not_lt.
        } {
          move ⇒ [_] [H_lt] H_le.
          contradict H_lt.
    -      by apply Zle_not_lt.
    +      by apply Zle_not_lt.
        } {
          move ⇒ [->] <-.
    -      rewrite ?Z.lt_succ_r.
    +      rewrite ?Z.lt_succ_r.
          right.
    -      split; apply Z.le_refl.
    +      split; apply Z.le_refl.
        } {
          tauto.
        } {
    -      case (N.zero_or_succ c2). {
    +      case (N.zero_or_succ c2). {
            move ⇒ → _; by left.
          } {
            move ⇒ [c2'] →.
    -        rewrite !N2Z.inj_succ Z.add_succ_r -Z.succ_le_mono Z.le_succ_l.
    +        rewrite !N2Z.inj_succ Z.add_succ_r -Z.succ_le_mono Z.le_succ_l.
            move ⇒ [H_y1_le] [H_le_y1].
    -        suff → : y1 = y2. {
    +        suff → : y1 = y2. {
              move ⇒ [] H_pre; contradict H_pre. {
    -            apply Z.lt_irrefl.
    +            apply Z.lt_irrefl.
              } {
    -            apply Zle_not_lt, Z_le_add_r.
    +            apply Zle_not_lt, Z_le_add_r.
              }
            }
    -        apply Z.le_antisymm ⇒ //.
    -        eapply Z.le_trans; last apply H_le_y1.
    +        apply Z.le_antisymm ⇒ //.
    +        eapply Z.le_trans; last apply H_le_y1.
            apply Z_le_add_r.
          }
        }
    @@ -1598,38 +1597,38 @@

    Library MSetsExtra.MSetIntervals

    -

    Alternative definition of addZ

    +

    Alternative definition of addZ

    -  Lemma addZ_aux_alt_def : ∀ x s acc,
    -    addZ_aux acc x s = (List.rev acc) ++ addZ x s.
    +  Lemma addZ_aux_alt_def : ∀ x s acc,
    +    addZ_aux acc x s = (List.rev acc) ++ addZ x s.
      Proof.
        intros y1 s.
        unfold addZ.
        induction s as [| [y2 c2] s' IH] ⇒ acc. {
    -      rewrite /addZ_aux /addZ /= /rev' !rev_append_rev /= app_nil_r //.
    +      rewrite /addZ_aux /addZ /= /rev' !rev_append_rev /= app_nil_r //.
        } {
          unfold addZ_aux.
    -      case (interval_1_compare y1 (y2, c2)); fold addZ_aux;
    -        rewrite ?rev_append_rev /= ?app_assoc_reverse //.
    -      rewrite (IH ((y2,c2)::acc)) (IH ((y2,c2)::nil)).
    -      rewrite /= app_assoc_reverse //.
    +      case (interval_1_compare y1 (y2, c2)); fold addZ_aux;
    +        rewrite ?rev_append_rev /= ?app_assoc_reverse //.
    +      rewrite (IH ((y2,c2)::acc)) (IH ((y2,c2)::nil)).
    +      rewrite /= app_assoc_reverse //.
        }
      Qed.

    -  Lemma addZ_alt_def : ∀ x s,
    -    addZ x s =
    -    match s with
    -    | nil ⇒ (x, (1%N))::nil
    -    | (y, c) :: l ⇒
    -        match (interval_1_compare x (y,c)) with
    -          | ICR_before ⇒ (x, (1%N))::s
    -          | ICR_before_touch ⇒ (x, N.succ c)::l
    -          | ICR_after ⇒ (y,c)::(addZ x l)
    -          | ICR_after_touch ⇒ insert_interval_begin y (N.succ c) l
    -          | _ ⇒ (y, c)::l
    +  Lemma addZ_alt_def : ∀ x s,
    +    addZ x s =
    +    match s with
    +    | nil ⇒ (x, (1%N))::nil
    +    | (y, c) :: l ⇒
    +        match (interval_1_compare x (y,c)) with
    +          | ICR_before ⇒ (x, (1%N))::s
    +          | ICR_before_touch ⇒ (x, N.succ c)::l
    +          | ICR_after ⇒ (y,c)::(addZ x l)
    +          | ICR_after_touch ⇒ insert_interval_begin y (N.succ c) l
    +          | _ ⇒ (y, c)::l
            end
        end.
      Proof.
    @@ -1638,8 +1637,8 @@

    Library MSetsExtra.MSetIntervals

        case s ⇒ //.
        move ⇒ [y c] s'.
        unfold addZ_aux.
    -    case (interval_1_compare x (y, c)); fold addZ_aux;
    -      rewrite ?rev_append_rev /= ?app_assoc_reverse //.
    +    case (interval_1_compare x (y, c)); fold addZ_aux;
    +      rewrite ?rev_append_rev /= ?app_assoc_reverse //.
        rewrite addZ_aux_alt_def //.
      Qed.
    @@ -1649,63 +1648,63 @@

    Library MSetsExtra.MSetIntervals

    -

    Auxiliary Lemmata about Invariant

    +

    Auxiliary Lemmata about Invariant


    -  Lemma interval_list_elements_greater_cons : ∀ z x c s,
    -    interval_list_elements_greater z ((x, c) :: s) = true ↔
    -    (z < x).
    +  Lemma interval_list_elements_greater_cons : ∀ z x c s,
    +    interval_list_elements_greater z ((x, c) :: s) = true ↔
    +    (z < x).
      Proof.
        intros z x c s.
        rewrite /=.
    -    apply Z.ltb_lt.
    +    apply Z.ltb_lt.
      Qed.

    -  Lemma interval_list_elements_greater_impl : ∀ x y s,
    -    (y ≤ x) →
    -    interval_list_elements_greater x s = true →
    -    interval_list_elements_greater y s = true.
    +  Lemma interval_list_elements_greater_impl : ∀ x y s,
    +    (y ≤ x) →
    +    interval_list_elements_greater x s = true →
    +    interval_list_elements_greater y s = true.
      Proof.
        intros x y s.
        case s ⇒ //.
        move ⇒ [z c] s'.
        rewrite /interval_list_elements_greater.
    -    move ⇒ H_y_leq /Z.ltb_lt H_x_lt.
    -    apply Z.ltb_lt.
    -    eapply Z.le_lt_trans; eauto.
    +    move ⇒ H_y_leq /Z.ltb_lt H_x_lt.
    +    apply Z.ltb_lt.
    +    eapply Z.le_lt_trans; eauto.
      Qed.

    -  Lemma interval_list_invariant_nil : interval_list_invariant nil = true.
    +  Lemma interval_list_invariant_nil : interval_list_invariant nil = true.
      Proof.
        by [].
      Qed.

    -  Lemma Ok_nil : Ok nil ↔ True.
    +  Lemma Ok_nil : Ok nil ↔ True.
      Proof.
        rewrite /Ok /IsOk /interval_list_invariant /is_encoded_elems_list //.
      Qed.

    -  Lemma is_encoded_elems_list_app : ∀ l1 l2,
    -    is_encoded_elems_list (l1 ++ l2) ↔
    -    (is_encoded_elems_list l1 ∧ is_encoded_elems_list l2).
    +  Lemma is_encoded_elems_list_app : ∀ l1 l2,
    +    is_encoded_elems_list (l1 ++ l2) ↔
    +    (is_encoded_elems_list l1 ∧ is_encoded_elems_list l2).
      Proof.
        intros l1 l2.
        rewrite /is_encoded_elems_list.
    -    setoid_rewrite in_app_iff.
    +    setoid_rewrite in_app_iff.
        split; firstorder.
      Qed.

    -  Lemma is_encoded_elems_list_rev : ∀ l,
    -    is_encoded_elems_list (List.rev l) ↔
    -    is_encoded_elems_list l.
    +  Lemma is_encoded_elems_list_rev : ∀ l,
    +    is_encoded_elems_list (List.rev l) ↔
    +    is_encoded_elems_list l.
      Proof.
        intros l.
        rewrite /is_encoded_elems_list.
    @@ -1713,29 +1712,29 @@

    Library MSetsExtra.MSetIntervals

          move ⇒ H x H_in;
          apply H;
          move : H_in;
    -      rewrite -in_rev ⇒ //
    +      rewrite -in_rev ⇒ //
        ).
      Qed.

    -  Lemma interval_list_invariant_cons : ∀ y c s',
    -    interval_list_invariant ((y, c) :: s') = true ↔
    -    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧
    -      ((c ≠ 0)%N) ∧ interval_list_invariant s' = true).
    +  Lemma interval_list_invariant_cons : ∀ y c s',
    +    interval_list_invariant ((y, c) :: s') = true ↔
    +    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧
    +      ((c ≠ 0)%N) ∧ interval_list_invariant s' = true).
      Proof.
        rewrite /interval_list_invariant -/interval_list_invariant.
        intros y c s'.
    -    rewrite !Bool.andb_true_iff negb_true_iff.
    +    rewrite !Bool.andb_true_iff negb_true_iff.
        split. {
    -      move ⇒ [] [H_inf] /N.eqb_neq H_c H_s'. tauto.
    +      move ⇒ [] [H_inf] /N.eqb_neq H_c H_s'. tauto.
        } {
    -      move ⇒ [H_inf] [/N.eqb_neq H_c] H_s'. tauto.
    +      move ⇒ [H_inf] [/N.eqb_neq H_c] H_s'. tauto.
        }
      Qed.

    -  Lemma interval_list_invariant_sing : ∀ x c,
    -    interval_list_invariant ((x, c)::nil) = true ↔ (c ≠ 0)%N.
    +  Lemma interval_list_invariant_sing : ∀ x c,
    +    interval_list_invariant ((x, c)::nil) = true ↔ (c ≠ 0)%N.
      Proof.
        intros x c.
        rewrite interval_list_invariant_cons.
    @@ -1743,9 +1742,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma Ok_cons : ∀ y c s', Ok ((y, c) :: s') ↔
    -    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧ ((c ≠ 0)%N) ∧
    -     is_encoded_elems_list (elementsZ_single y c) ∧ Ok s').
    +  Lemma Ok_cons : ∀ y c s', Ok ((y, c) :: s') ↔
    +    (interval_list_elements_greater (y+Z.of_N c) s' = true ∧ ((c ≠ 0)%N) ∧
    +     is_encoded_elems_list (elementsZ_single y c) ∧ Ok s').
      Proof.
        intros y c s'.
        rewrite /Ok /IsOk interval_list_invariant_cons elementsZ_cons is_encoded_elems_list_app
    @@ -1754,10 +1753,10 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma Nin_elements_greater : ∀ s y,
    -     interval_list_elements_greater y s = true →
    -     interval_list_invariant s = true →
    -     âˆ€ x, x ≤ y → ~(InZ x s).
    +  Lemma Nin_elements_greater : ∀ s y,
    +     interval_list_elements_greater y s = true →
    +     interval_list_invariant s = true →
    +     âˆ€ x, x ≤ y → ~(InZ x s).
      Proof.
        induction s as [| [z c] s' IH]. {
          intros y _ _ x _.
    @@ -1767,12 +1766,12 @@

    Library MSetsExtra.MSetIntervals

            /interval_list_invariant_cons [H_gr] [H_c] H_s'
            x H_x_le.
          rewrite InZ_cons In_elementsZ_single.
    -      have H_x_lt : x < z by eapply Z.le_lt_trans; eauto.
    +      have H_x_lt : x < z by eapply Z.le_lt_trans; eauto.

          move ⇒ []. {
            move ⇒ [H_z_leq] _; contradict H_z_leq.
    -        by apply Z.nle_gt.
    +        by apply Z.nle_gt.
          } {
            eapply IH; eauto.
            by apply Z_lt_le_add_r.
    @@ -1781,23 +1780,23 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma Nin_elements_greater_equal :
    -     âˆ€ x s,
    -       interval_list_elements_greater x s = true →
    -       interval_list_invariant s = true →
    -       Â¬ (InZ x s).
    +  Lemma Nin_elements_greater_equal :
    +     âˆ€ x s,
    +       interval_list_elements_greater x s = true →
    +       interval_list_invariant s = true →
    +       Â¬ (InZ x s).
      Proof.
        move ⇒ x s H_inv H_gr.
        apply (Nin_elements_greater s x) ⇒ //.
    -    apply Z.le_refl.
    +    apply Z.le_refl.
      Qed.

    -  Lemma interval_list_elements_greater_alt_def : ∀ s y,
    -     interval_list_invariant s = true →
    +  Lemma interval_list_elements_greater_alt_def : ∀ s y,
    +     interval_list_invariant s = true →

    -     (interval_list_elements_greater y s = true ↔
    -      (∀ x, x ≤ y → ~(InZ x s))).
    +     (interval_list_elements_greater y s = true ↔
    +      (∀ x, x ≤ y → ~(InZ x s))).
      Proof.
        intros s y H_inv.
        split. {
    @@ -1808,7 +1807,7 @@

    Library MSetsExtra.MSetIntervals

          case s as [| [x2 c] s'] ⇒ //.
          rewrite interval_list_invariant_cons interval_list_elements_greater_cons.
          move ⇒ [_] [H_c_neq] _ H.
    -      apply Z.nle_gt ⇒ H_ge.
    +      apply Z.nle_gt ⇒ H_ge.
          apply (H x2) ⇒ //.
          rewrite InZ_cons; left.
          apply In_elementsZ_single_hd ⇒ //.
    @@ -1816,68 +1815,76 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma interval_list_elements_greater_alt2_def : ∀ s y,
    -     interval_list_invariant s = true →
    +  Lemma interval_list_elements_greater_alt2_def : ∀ s y,
    +     interval_list_invariant s = true →

    -     (interval_list_elements_greater y s = true ↔
    -      (∀ x, InZ x s → y < x)).
    +     (interval_list_elements_greater y s = true ↔
    +      (∀ x, InZ x s → y < x)).
      Proof.
        intros s y H.
    + +
        rewrite interval_list_elements_greater_alt_def //.
    -    firstorder.
    -    apply Z.nle_gt.
    -    move ⇒ H_lt.
    -    eapply H0; eauto.
    +    split. {
    +      move ⇒ H_notInZ x H_inZ.
    +      apply Z.nle_gt.
    +      move ⇒ H_lt.
    +      move : (H_notInZ x H_lt H_inZ) ⇒ //.
    +    } {
    +      move ⇒ H_lt x H_le H_inZ.
    +      move : (H_lt x H_inZ).
    +      lia.
    +    }
      Qed.

    -  Lemma interval_list_elements_greater_intro : ∀ s y,
    -     interval_list_invariant s = true →
    -     (∀ x, InZ x s → y < x) →
    -     interval_list_elements_greater y s = true.
    +  Lemma interval_list_elements_greater_intro : ∀ s y,
    +     interval_list_invariant s = true →
    +     (∀ x, InZ x s → y < x) →
    +     interval_list_elements_greater y s = true.
      Proof.
        intros s y H1 H2.
        rewrite interval_list_elements_greater_alt2_def //.
      Qed.

    -  Lemma interval_list_elements_greater_app_elim_1 : ∀ s1 s2 y,
    -    interval_list_elements_greater y (s1 ++ s2) = true →
    -    interval_list_elements_greater y s1 = true.
    +  Lemma interval_list_elements_greater_app_elim_1 : ∀ s1 s2 y,
    +    interval_list_elements_greater y (s1 ++ s2) = true →
    +    interval_list_elements_greater y s1 = true.
      Proof.
        intros s1 s2 y.
        case s1 ⇒ //.
      Qed.

    -  Lemma interval_list_invariant_app_intro : ∀ s1 s2,
    -      interval_list_invariant s1 = true →
    -      interval_list_invariant s2 = true →
    -      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    -      interval_list_invariant (s1 ++ s2) = true.
    +  Lemma interval_list_invariant_app_intro : ∀ s1 s2,
    +      interval_list_invariant s1 = true →
    +      interval_list_invariant s2 = true →
    +      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    +      interval_list_invariant (s1 ++ s2) = true.
      Proof.
        induction s1 as [| [y1 c1] s1' IH]. {
          move ⇒ s2 _ //.
        } {
          move ⇒ s2.
    -      rewrite -app_comm_cons !interval_list_invariant_cons.
    +      rewrite -app_comm_cons !interval_list_invariant_cons.
          move ⇒ [H_gr] [H_c1_neq] H_inv_s1' H_inv_s2 H_inz_s2.
          split; last split. {
            move : H_gr H_inz_s2.
            case s1' as [| [y1' c1'] s1'']; last done.
            move ⇒ _ H_inz_s2.
    -        rewrite app_nil_l.
    +        rewrite app_nil_l.
            apply interval_list_elements_greater_intro ⇒ //.
            move ⇒ x H_x_in_s2.
    -        suff H_inz : InZ (Z.pred (y1 + Z.of_N c1)) ((y1, c1) :: nil). {
    +        suff H_inz : InZ (Z.pred (y1 + Z.of_N c1)) ((y1, c1) :: nil). {
              move : (H_inz_s2 _ _ H_inz H_x_in_s2).
    -          by rewrite Z.succ_pred.
    +          by rewrite Z.succ_pred.
            }
    -        rewrite InZ_cons In_elementsZ_single -Z.lt_le_pred; left.
    +        rewrite InZ_cons In_elementsZ_single -Z.lt_le_pred; left.
            split. {
              by apply Z_lt_add_r.
            } {
    -          apply Z.lt_pred_l.
    +          apply Z.lt_pred_l.
            }
          } {
            assumption.
    @@ -1891,15 +1898,15 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma interval_list_invariant_app_elim : ∀ s1 s2,
    -      interval_list_invariant (s1 ++ s2) = true →
    -      interval_list_invariant s1 = true ∧
    -      interval_list_invariant s2 = true ∧
    -      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2).
    +  Lemma interval_list_invariant_app_elim : ∀ s1 s2,
    +      interval_list_invariant (s1 ++ s2) = true →
    +      interval_list_invariant s1 = true ∧
    +      interval_list_invariant s2 = true ∧
    +      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2).
      Proof.
        move ⇒ s1 s2.
        induction s1 as [| [y1 c1] s1' IH]; first done.
    -    rewrite -app_comm_cons !interval_list_invariant_cons.
    +    rewrite -app_comm_cons !interval_list_invariant_cons.

        move ⇒ [H_gr] [H_c1_neq_0] /IH [H_inv_s1'] [H_inv_s2] H_in_s1'_s2.
    @@ -1918,23 +1925,23 @@

    Library MSetsExtra.MSetIntervals

              by apply interval_list_invariant_app_intro.
          }
          move ⇒ H_in_s12'.
    -      have : (y1 + Z.of_N c1 < x2). {
    +      have : (y1 + Z.of_N c1 < x2). {
            apply H_in_s12'.
            rewrite InZ_app.
            by right.
          }
          move ⇒ H_lt_x2.
    -      apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -      by apply Zlt_le_succ.
    +      apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +      by apply Zlt_le_succ.
        }
      Qed.

    -  Lemma interval_list_invariant_app_iff : ∀ s1 s2,
    -      interval_list_invariant (s1 ++ s2) = true ↔
    -      (interval_list_invariant s1 = true ∧
    -      interval_list_invariant s2 = true ∧
    -      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2)).
    +  Lemma interval_list_invariant_app_iff : ∀ s1 s2,
    +      interval_list_invariant (s1 ++ s2) = true ↔
    +      (interval_list_invariant s1 = true ∧
    +      interval_list_invariant s2 = true ∧
    +      (∀ (x1 x2 : Z), InZ x1 s1 → InZ x2 s2 → Z.succ x1 < x2)).
      Proof.
        intros s1 s2.
        split. {
    @@ -1946,11 +1953,11 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma interval_list_invariant_snoc_intro : ∀ s1 y2 c2,
    -      interval_list_invariant s1 = true →
    -      (c2 ≠ 0)%N →
    -      (∀ x, InZ x s1 → Z.succ x < y2) →
    -      interval_list_invariant (s1 ++ ((y2, c2)::nil)) = true.
    +  Lemma interval_list_invariant_snoc_intro : ∀ s1 y2 c2,
    +      interval_list_invariant s1 = true →
    +      (c2 ≠ 0)%N →
    +      (∀ x, InZ x s1 → Z.succ x < y2) →
    +      interval_list_invariant (s1 ++ ((y2, c2)::nil)) = true.
      Proof.
        intros s1 y2 c2 H_inv_s1 H_c2_neq H_in_s1.
        apply interval_list_invariant_app_intro ⇒ //. {
    @@ -1961,7 +1968,7 @@

    Library MSetsExtra.MSetIntervals

          move ⇒ [] //.
          rewrite In_elementsZ_single.
          move ⇒ [H_y2_le] _.
    -      eapply Z.lt_le_trans; eauto.
    +      eapply Z.lt_le_trans; eauto.
        }
      Qed.
    @@ -1971,14 +1978,14 @@

    Library MSetsExtra.MSetIntervals

    -

    Properties of In and InZ

    +

    Properties of In and InZ


    -  Lemma encode_decode_eq : ∀ x s, Ok s → InZ x s →
    -    (Enc.encode (Enc.decode x) = x).
    +  Lemma encode_decode_eq : ∀ x s, Ok s → InZ x s →
    +    (Enc.encode (Enc.decode x) = x).
      Proof.
        intros x s.
        rewrite /Ok /IsOk /InZ.
    @@ -1988,16 +1995,16 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma In_alt_def : ∀ x s, Ok s →
    -    (In x s ↔ List.In x (elements s)).
    +  Lemma In_alt_def : ∀ x s, Ok s →
    +    (In x s ↔ List.In x (elements s)).
      Proof.
        intros x s H_ok.
    -    rewrite /In InA_alt /elements rev_map_alt_def.
    +    rewrite /In InA_alt /elements rev_map_alt_def.
        split. {
          move ⇒ [y] [H_y_eq].
    -      rewrite -!in_rev !in_map_iff.
    +      rewrite -!in_rev !in_map_iff.
          move ⇒ [x'] [H_y_eq'] H_x'_in.
    -      suff H_x'_eq : (Enc.encode x = x'). {
    +      suff H_x'_eq : (Enc.encode x = x'). {
            âˆƒ x'.
            split ⇒ //.
            rewrite -H_x'_eq Enc.decode_encode_ok //.
    @@ -2010,7 +2017,7 @@

    Library MSetsExtra.MSetIntervals

          move : H_y_eq'.
          rewrite -!H_x'_eq Enc.decode_encode_ok ⇒ H_y_eq'.
          subst.
    -      suff → : Z.eq (Enc.encode x) (Enc.encode y) by done.
    +      suff → : Z.eq (Enc.encode x) (Enc.encode y) by done.
          by rewrite Enc.encode_eq.
        } {
          move ⇒ H_enc_in.
    @@ -2021,17 +2028,17 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma In_InZ : ∀ x s, Ok s →
    -    (In x s ↔ InZ (Enc.encode x) s).
    +  Lemma In_InZ : ∀ x s, Ok s →
    +    (In x s ↔ InZ (Enc.encode x) s).
      Proof.
        intros x s H_ok.
    -    rewrite /InZ In_alt_def /elements rev_map_alt_def -in_rev in_map_iff.
    +    rewrite /InZ In_alt_def /elements rev_map_alt_def -in_rev in_map_iff.
        split; last first. {
          âˆƒ (Enc.encode x).
          by rewrite Enc.decode_encode_ok.
        }
        move ⇒ [y] [<-] H_y_in.
    -    suff : ∃ z, (Enc.encode z = y). {
    +    suff : ∃ z, (Enc.encode z = y). {
          move ⇒ [z] H_y_eq.
          move : H_y_in.
          by rewrite -!H_y_eq Enc.decode_encode_ok.
    @@ -2043,8 +2050,8 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma InZ_In : ∀ x s, Ok s →
    -    (InZ x s → In (Enc.decode x) s).
    +  Lemma InZ_In : ∀ x s, Ok s →
    +    (InZ x s → In (Enc.decode x) s).
      Proof.
        intros x s H_ok.
        rewrite In_InZ /InZ.
    @@ -2063,14 +2070,14 @@

    Library MSetsExtra.MSetIntervals

    -

    Membership specification

    +

    Membership specification


    -  Lemma memZ_spec :
    -    âˆ€ (s : t) (x : Z) (Hs : Ok s), memZ x s = true ↔ InZ x s.
    +  Lemma memZ_spec :
    +    âˆ€ (s : t) (x : Z) (Hs : Ok s), memZ x s = true ↔ InZ x s.
      Proof.
        induction s as [| [y c] s' IH]. {
          intros x _.
    @@ -2078,57 +2085,57 @@

    Library MSetsExtra.MSetIntervals

        } {
          move ⇒ x /Ok_cons [H_inf] [H_c] [H_is_enc] H_s'.
          rewrite /InZ /memZ elementsZ_cons -/memZ.
    -      rewrite in_app_iff -!in_rev In_elementsZ_single.
    +      rewrite in_app_iff -!in_rev In_elementsZ_single.

    -      case_eq (x <? y). {
    -        move ⇒ /Z.ltb_lt H_x_lt.
    +      case_eq (x <? y). {
    +        move ⇒ /Z.ltb_lt H_x_lt.
            split; first done.
            move ⇒ []. {
              move ⇒ H_x_in; contradict H_x_in.
    -          apply Nin_elements_greater with (y := (y + Z.of_N c)) ⇒ //. {
    +          apply Nin_elements_greater with (y := (y + Z.of_N c)) ⇒ //. {
                apply H_s'.
              } {
                apply Z_lt_le_add_r ⇒ //.
              }
            } {
              move ⇒ [H_y_le]; contradict H_y_le.
    -          by apply Z.nle_gt.
    +          by apply Z.nle_gt.
            }
          } {
    -        move ⇒ /Z.ltb_ge H_y_le.
    -        case_eq (x <? y + Z.of_N c). {
    -          move ⇒ /Z.ltb_lt H_x_lt.
    +        move ⇒ /Z.ltb_ge H_y_le.
    +        case_eq (x <? y + Z.of_N c). {
    +          move ⇒ /Z.ltb_lt H_x_lt.
              split; last done.
              move ⇒ _.
              by right.
            } {
    -          move ⇒ /Z.ltb_ge H_yc_le.
    +          move ⇒ /Z.ltb_ge H_yc_le.
              rewrite IH.
              split; first tauto.
              move ⇒ [] //.
              move ⇒ [_] H_x_lt; contradict H_x_lt.
    -          by apply Z.nlt_ge.
    +          by apply Z.nlt_ge.
            }
          }
        }
      Qed.

    -  Lemma mem_spec :
    -   âˆ€ (s : t) (x : elt) (Hs : Ok s), mem x s = true ↔ In x s.
    +  Lemma mem_spec :
    +   âˆ€ (s : t) (x : elt) (Hs : Ok s), mem x s = true ↔ In x s.
      Proof.
        intros s x Hs.
        rewrite /mem memZ_spec In_InZ //.
      Qed.

    -  Lemma merge_interval_size_neq_0 : ∀ x1 c1 x2 c2,
    -     (c1 ≠ 0%N) →
    -     (merge_interval_size x1 c1 x2 c2 ≠ 0)%N.
    +  Lemma merge_interval_size_neq_0 : ∀ x1 c1 x2 c2,
    +     (c1 ≠ 0%N) →
    +     (merge_interval_size x1 c1 x2 c2 ≠ 0)%N.
      Proof.
        intros x1 c1 x2 c2.
    -    rewrite /merge_interval_size !N.neq_0_lt_0 N.max_lt_iff.
    +    rewrite /merge_interval_size !N.neq_0_lt_0 N.max_lt_iff.
        by left.
      Qed.
    @@ -2136,70 +2143,70 @@

    Library MSetsExtra.MSetIntervals

    -

    insert if length not 0

    +

    insert if length not 0


    -  Lemma interval_list_invariant_insert_intervalZ_guarded : ∀ x c s,
    -    interval_list_invariant s = true →
    -    interval_list_elements_greater (x + Z.of_N c) s = true →
    -    interval_list_invariant (insert_intervalZ_guarded x c s) = true.
    +  Lemma interval_list_invariant_insert_intervalZ_guarded : ∀ x c s,
    +    interval_list_invariant s = true →
    +    interval_list_elements_greater (x + Z.of_N c) s = true →
    +    interval_list_invariant (insert_intervalZ_guarded x c s) = true.
      Proof.
        intros x c s.
        rewrite /insert_intervalZ_guarded.
    -    case_eq (c =? 0)%N ⇒ //.
    -    move ⇒ /N.eqb_neq.
    +    case_eq (c =? 0)%N ⇒ //.
    +    move ⇒ /N.eqb_neq.
        rewrite interval_list_invariant_cons.
        tauto.
      Qed.

    -  Lemma interval_list_elements_greater_insert_intervalZ_guarded : ∀ x c y s,
    -    interval_list_elements_greater y (insert_intervalZ_guarded x c s) = true ↔
    -    (if (c =? 0)%N then (interval_list_elements_greater y s = true) else (y < x)).
    +  Lemma interval_list_elements_greater_insert_intervalZ_guarded : ∀ x c y s,
    +    interval_list_elements_greater y (insert_intervalZ_guarded x c s) = true ↔
    +    (if (c =? 0)%N then (interval_list_elements_greater y s = true) else (y < x)).
      Proof.
        intros x c y s.
        rewrite /insert_intervalZ_guarded.
    -    case (c =? 0)%N ⇒ //.
    -    rewrite /interval_list_elements_greater Z.ltb_lt //.
    +    case (c =? 0)%N ⇒ //.
    +    rewrite /interval_list_elements_greater Z.ltb_lt //.
      Qed.

    -  Lemma insert_intervalZ_guarded_app : ∀ x c s1 s2,
    -    (insert_intervalZ_guarded x c s1) ++ s2 =
    -    insert_intervalZ_guarded x c (s1 ++ s2).
    +  Lemma insert_intervalZ_guarded_app : ∀ x c s1 s2,
    +    (insert_intervalZ_guarded x c s1) ++ s2 =
    +    insert_intervalZ_guarded x c (s1 ++ s2).
      Proof.
        intros x c s1 s2.
        rewrite /insert_intervalZ_guarded.
    -    case (N.eqb c 0) ⇒ //.
    +    case (N.eqb c 0) ⇒ //.
      Qed.

    -  Lemma insert_intervalZ_guarded_rev_nil_app : ∀ x c s,
    -    rev (insert_intervalZ_guarded x c nil) ++ s =
    -    insert_intervalZ_guarded x c s.
    +  Lemma insert_intervalZ_guarded_rev_nil_app : ∀ x c s,
    +    rev (insert_intervalZ_guarded x c nil) ++ s =
    +    insert_intervalZ_guarded x c s.
      Proof.
        intros x c s.
        rewrite /insert_intervalZ_guarded.
    -    case (N.eqb c 0) ⇒ //.
    +    case (N.eqb c 0) ⇒ //.
      Qed.

    -Lemma elementsZ_insert_intervalZ_guarded : ∀ x c s,
    -    elementsZ (insert_intervalZ_guarded x c s) = elementsZ ((x, c) :: s).
    +Lemma elementsZ_insert_intervalZ_guarded : ∀ x c s,
    +    elementsZ (insert_intervalZ_guarded x c s) = elementsZ ((x, c) :: s).
      Proof.
        intros x c s.
        rewrite /insert_intervalZ_guarded.
    -    case_eq (c =? 0)%N ⇒ //.
    -    move ⇒ /N.eqb_eq →.
    -    rewrite elementsZ_cons elementsZ_single_base /= app_nil_r //.
    +    case_eq (c =? 0)%N ⇒ //.
    +    move ⇒ /N.eqb_eq →.
    +    rewrite elementsZ_cons elementsZ_single_base /= app_nil_r //.
      Qed.

    -  Lemma InZ_insert_intervalZ_guarded : ∀ y x c s,
    -    InZ y (insert_intervalZ_guarded x c s) = InZ y ((x, c) :: s).
    +  Lemma InZ_insert_intervalZ_guarded : ∀ y x c s,
    +    InZ y (insert_intervalZ_guarded x c s) = InZ y ((x, c) :: s).
      Proof.
        intros y x c s.
        rewrite /InZ elementsZ_insert_intervalZ_guarded //.
    @@ -2209,56 +2216,56 @@

    Library MSetsExtra.MSetIntervals

    -

    Merging intervals

    +

    Merging intervals


    -  Lemma merge_interval_size_add : ∀ x c1 c2,
    -     (merge_interval_size x c1 (x + Z.of_N c1) c2 = (c1 + c2))%N.
    +  Lemma merge_interval_size_add : ∀ x c1 c2,
    +     (merge_interval_size x c1 (x + Z.of_N c1) c2 = (c1 + c2))%N.
      Proof.
        intros x c1 c2.
        rewrite /merge_interval_size.
    -    replace (x + Z.of_N c1 + Z.of_N c2 - x) with
    -            (Z.of_N c1 + Z.of_N c2) by omega.
    -    rewrite -N2Z.inj_add N2Z.id.
    -    apply N.max_r, N.le_add_r.
    +    replace (x + Z.of_N c1 + Z.of_N c2 - x) with
    +            (Z.of_N c1 + Z.of_N c2) by lia.
    +    rewrite -N2Z.inj_add N2Z.id.
    +    apply N.max_r, N.le_add_r.
      Qed.

    -  Lemma merge_interval_size_eq_max : ∀ y1 c1 y2 c2,
    -     y1 ≤ y2 + Z.of_N c2 →
    -     y1 + Z.of_N (merge_interval_size y1 c1 y2 c2) =
    -     Z.max (y1 + Z.of_N c1) (y2 + Z.of_N c2).
    +  Lemma merge_interval_size_eq_max : ∀ y1 c1 y2 c2,
    +     y1 ≤ y2 + Z.of_N c2 →
    +     y1 + Z.of_N (merge_interval_size y1 c1 y2 c2) =
    +     Z.max (y1 + Z.of_N c1) (y2 + Z.of_N c2).
      Proof.
        intros y1 c1 y2 c2 H_y1_le.
    -    rewrite /merge_interval_size N2Z.inj_max Z2N.id; last first. {
    -      by apply Zle_minus_le_0.
    +    rewrite /merge_interval_size N2Z.inj_max Z2N.id; last first. {
    +      by apply Zle_minus_le_0.
        }
    -    rewrite -Z.add_max_distr_l.
    -    replace (y1 + (y2 + Z.of_N c2 - y1)) with (y2 + Z.of_N c2) by omega.
    +    rewrite -Z.add_max_distr_l.
    +    replace (y1 + (y2 + Z.of_N c2 - y1)) with (y2 + Z.of_N c2) by lia.
        done.
      Qed.

    -  Lemma merge_interval_size_invariant : ∀ y1 c1 y2 c2 z s,
    -    interval_list_invariant s = true →
    -    y1 + Z.of_N c1 ≤ y2 + Z.of_N c2 →
    -    y2 + Z.of_N c2 ≤ z →
    -    interval_list_elements_greater z s = true →
    -    (c1 ≠ 0)%N →
    -    interval_list_invariant ((y1, merge_interval_size y1 c1 y2 c2) :: s) =
    -   true.
    +  Lemma merge_interval_size_invariant : ∀ y1 c1 y2 c2 z s,
    +    interval_list_invariant s = true →
    +    y1 + Z.of_N c1 ≤ y2 + Z.of_N c2 →
    +    y2 + Z.of_N c2 ≤ z →
    +    interval_list_elements_greater z s = true →
    +    (c1 ≠ 0)%N →
    +    interval_list_invariant ((y1, merge_interval_size y1 c1 y2 c2) :: s) =
    +   true.
      Proof.
        intros y1 c1 y2 c2 z s H_inv H_le H_le_z H_gr H_c1_neq_0.
        rewrite interval_list_invariant_cons.
        split; last split. {
          rewrite merge_interval_size_eq_max; last first. {
    -        eapply Z.le_trans; last apply H_le.
    +        eapply Z.le_trans; last apply H_le.
            apply Z_le_add_r.
          } {
    -        rewrite Z.max_r ⇒ //.
    +        rewrite Z.max_r ⇒ //.
            eapply interval_list_elements_greater_impl; first apply H_le_z.
            done.
          }
    @@ -2271,31 +2278,31 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma In_merge_interval : ∀ x1 c1 x2 c2 y,
    -    x1 ≤ x2 →
    -    x2 ≤ x1 + Z.of_N c1 → (
    -    List.In y (elementsZ_single x1 (merge_interval_size x1 c1 x2 c2)) ↔
    -    List.In y (elementsZ_single x1 c1) ∨ List.In y (elementsZ_single x2 c2)).
    +  Lemma In_merge_interval : ∀ x1 c1 x2 c2 y,
    +    x1 ≤ x2 →
    +    x2 ≤ x1 + Z.of_N c1 → (
    +    List.In y (elementsZ_single x1 (merge_interval_size x1 c1 x2 c2)) ↔
    +    List.In y (elementsZ_single x1 c1) ∨ List.In y (elementsZ_single x2 c2)).
      Proof.
        intros x1 c1 x2 c2 y H_x1_le H_x2_le.
        rewrite !In_elementsZ_single merge_interval_size_eq_max;
          last first. {
    -      eapply Z.le_trans; eauto.
    +      eapply Z.le_trans; eauto.
          by apply Z_le_add_r.
        }
    -    rewrite Z.max_lt_iff.
    +    rewrite Z.max_lt_iff.
        split. {
          move ⇒ [H_x_le] [] H_y_lt. {
            by left.
          } {
    -        case_eq (Z.leb x2 y). {
    -          move ⇒ /Z.leb_le H_y'_le.
    +        case_eq (Z.leb x2 y). {
    +          move ⇒ /Z.leb_le H_y'_le.
              by right.
            } {
    -          move ⇒ /Z.leb_gt H_y_lt_x2.
    +          move ⇒ /Z.leb_gt H_y_lt_x2.
              left.
              split ⇒ //.
    -          eapply Z.lt_le_trans; eauto.
    +          eapply Z.lt_le_trans; eauto.
            }
          }
        } {
    @@ -2304,7 +2311,7 @@

    Library MSetsExtra.MSetIntervals

          } {
            move ⇒ [H_x2_le'] H_y_lt.
            split. {
    -          eapply Z.le_trans; eauto.
    +          eapply Z.le_trans; eauto.
            } {
              by right.
            }
    @@ -2313,14 +2320,14 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma insert_interval_begin_spec : ∀ y s x c,
    -     interval_list_invariant s = true →
    -     interval_list_elements_greater x s = true →
    -     (c ≠ 0)%N →
    -      (
    -     interval_list_invariant (insert_interval_begin x c s) = true ∧
    -     (InZ y (insert_interval_begin x c s) ↔
    -     (List.In y (elementsZ_single x c) ∨ InZ y s))).
    +  Lemma insert_interval_begin_spec : ∀ y s x c,
    +     interval_list_invariant s = true →
    +     interval_list_elements_greater x s = true →
    +     (c ≠ 0)%N →
    +      (
    +     interval_list_invariant (insert_interval_begin x c s) = true ∧
    +     (InZ y (insert_interval_begin x c s) ↔
    +     (List.In y (elementsZ_single x c) ∨ InZ y s))).
      Proof.
        intros y.
        induction s as [| [y' c'] s' IH]. {
    @@ -2334,10 +2341,10 @@

    Library MSetsExtra.MSetIntervals

          unfold insert_interval_begin.
          Z_named_compare_cases H_y'; fold insert_interval_begin. {
            subst.
    -        rewrite !InZ_cons elementsZ_single_add in_app_iff.
    +        rewrite !InZ_cons elementsZ_single_add in_app_iff.
            split; last tauto.
    -        rewrite interval_list_invariant_cons N2Z.inj_add
    -          Z.add_assoc N.eq_add_0.
    +        rewrite interval_list_invariant_cons N2Z.inj_add
    +          Z.add_assoc N.eq_add_0.
            tauto.
          } {
            rewrite !InZ_cons !interval_list_invariant_cons
    @@ -2345,16 +2352,16 @@

    Library MSetsExtra.MSetIntervals

            repeat split ⇒ //.
          } {
            set c'' := merge_interval_size x c y' c'.
    -        have H_x_lt' : x < y' + Z.of_N c'. {
    -          eapply Z.lt_le_trans with (m := y') ⇒ //.
    +        have H_x_lt' : x < y' + Z.of_N c'. {
    +          eapply Z.lt_le_trans with (m := y') ⇒ //.
              by apply Z_le_add_r.
            }

    -        have H_pre : interval_list_elements_greater x s' = true. {
    +        have H_pre : interval_list_elements_greater x s' = true. {
              eapply interval_list_elements_greater_impl; eauto.
    -          by apply Z.lt_le_incl.
    +          by apply Z.lt_le_incl.
            }
    -        have H_pre2 : c'' ≠ 0%N. {
    +        have H_pre2 : c'' ≠ 0%N. {
              by apply merge_interval_size_neq_0.
            }
            move : (IH x c'' H_inv_s' H_pre H_pre2) ⇒ {IH} {H_pre} {H_pre2} [->] →.
    @@ -2366,9 +2373,9 @@

    Library MSetsExtra.MSetIntervals

              rewrite InZ_cons.
              tauto.
            } {
    -          by apply Z.lt_le_incl.
    +          by apply Z.lt_le_incl.
            } {
    -          by apply Z.lt_le_incl.
    +          by apply Z.lt_le_incl.
            }
          }
        }
    @@ -2378,30 +2385,30 @@

    Library MSetsExtra.MSetIntervals

    -

    add specification

    +

    add specification

    -  Lemma addZ_InZ :
    -   âˆ€ (s : t) (x y : Z),
    -    interval_list_invariant s = true →
    -    (InZ y (addZ x s) ↔ x = y ∨ InZ y s).
    +  Lemma addZ_InZ :
    +   âˆ€ (s : t) (x y : Z),
    +    interval_list_invariant s = true →
    +    (InZ y (addZ x s) ↔ x = y ∨ InZ y s).
      Proof.
        move ⇒ s x y.
        induction s as [| [z c] s' IH]. {
          move ⇒ _.
          rewrite /InZ addZ_alt_def
    -              elementsZ_cons elementsZ_nil app_nil_l -in_rev
    +              elementsZ_cons elementsZ_nil app_nil_l -in_rev
                  In_elementsZ_single1 /=.
          firstorder.
        } {
          move ⇒ /interval_list_invariant_cons [H_greater] [H_c_neq_0] H_inv_c'.
    -      move : (IH H_inv_c') ⇒ {IH} IH.
    +      move : (IH H_inv_c') ⇒ {} IH.
          rewrite addZ_alt_def.
    -      have H_succ : ∀ z, z + Z.of_N 1 = Z.succ z by done.
    +      have H_succ : ∀ z, z + Z.of_N 1 = Z.succ z by done.
          move : (interval_1_compare_elim x z c).
    -      case (interval_1_compare x (z, c));
    -        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
    +      case (interval_1_compare x (z, c));
    +        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
            move ⇒ →.
            rewrite elementsZ_single_succ_front /=.
            tauto.
    @@ -2416,27 +2423,27 @@

    Library MSetsExtra.MSetIntervals

            tauto.
          } {
            move ⇒ H_x_eq.
    -        have → : (InZ y (insert_interval_begin z (N.succ c) s') ↔
    -                   List.In y (elementsZ_single z (N.succ c)) ∨ InZ y s'). {
    +        have → : (InZ y (insert_interval_begin z (N.succ c) s') ↔
    +                   List.In y (elementsZ_single z (N.succ c)) ∨ InZ y s'). {
              eapply insert_interval_begin_spec. {
                by apply H_inv_c'.
              } {
                eapply interval_list_elements_greater_impl; eauto.
                apply Z_le_add_r.
              } {
    -            by apply N.neq_succ_0.
    +            by apply N.neq_succ_0.
              }
            }
    -        rewrite -H_x_eq elementsZ_single_succ in_app_iff /=.
    +        rewrite -H_x_eq elementsZ_single_succ in_app_iff /=.
            tauto.
          }
        }
      Qed.

    -  Lemma addZ_invariant : ∀ s x,
    -    interval_list_invariant s = true →
    -    interval_list_invariant (addZ x s) = true.
    +  Lemma addZ_invariant : ∀ s x,
    +    interval_list_invariant s = true →
    +    interval_list_invariant (addZ x s) = true.
      Proof.
        move ⇒ s x.
        induction s as [| [z c] s' IH]. {
    @@ -2445,28 +2452,28 @@

    Library MSetsExtra.MSetIntervals

        } {
          move ⇒ /interval_list_invariant_cons [H_greater] [H_c_neq_0]
                  H_inv_c'.
    -      move : (IH H_inv_c') ⇒ {IH} IH.
    +      move : (IH H_inv_c') ⇒ {} IH.
          rewrite addZ_alt_def.
    -      have H_succ : ∀ z, z + Z.of_N 1 = Z.succ z by done.
    +      have H_succ : ∀ z, z + Z.of_N 1 = Z.succ z by done.
          move : (interval_1_compare_elim x z c).
    -      case_eq (interval_1_compare x (z, c)) ⇒ H_comp;
    -        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
    +      case_eq (interval_1_compare x (z, c)) ⇒ H_comp;
    +        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
            move ⇒ H_z_gt.
    -        rewrite interval_list_invariant_cons /= !andb_true_iff !H_succ.
    +        rewrite interval_list_invariant_cons /= !andb_true_iff !H_succ.
            repeat split ⇒ //. {
    -          by apply Z.ltb_lt.
    +          by apply Z.ltb_lt.
            } {
    -          apply negb_true_iff, N.eqb_neq ⇒ //.
    +          apply negb_true_iff, N.eqb_neq ⇒ //.
            }
          } {
            move ⇒ ?; subst.
    -        rewrite /= !andb_true_iff.
    +        rewrite /= !andb_true_iff.
            repeat split ⇒ //. {
              move : H_greater.
    -          rewrite Z.add_succ_l -Z.add_succ_r N2Z.inj_succ //.
    +          rewrite Z.add_succ_l -Z.add_succ_r N2Z.inj_succ //.
            } {
    -          apply negb_true_iff, N.eqb_neq ⇒ //.
    -          apply N.neq_succ_0.
    +          apply negb_true_iff, N.eqb_neq ⇒ //.
    +          apply N.neq_succ_0.
            }
          } {
            move ⇒ [] // _.
    @@ -2487,14 +2494,14 @@

    Library MSetsExtra.MSetIntervals

              eapply interval_list_elements_greater_impl; eauto.
              apply Z_le_add_r.
            } {
    -          by apply N.neq_succ_0.
    +          by apply N.neq_succ_0.
            }
          }
        }
      Qed.

    -  Global Instance add_ok s x : ∀ `(Ok s), Ok (add x s).
    +  Global Instance add_ok s x : ∀ `(Ok s), Ok (add x s).
      Proof.
        move ⇒ H_ok_s.
        move : (H_ok_s).
    @@ -2516,15 +2523,16 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma add_spec :
    -   âˆ€ (s : t) (x y : elt) (Hs : Ok s),
    -     In y (add x s) ↔ Enc.E.eq y x ∨ In y s.
    +  Lemma add_spec :
    +   âˆ€ (s : t) (x y : elt) (Hs : Ok s),
    +     In y (add x s) ↔ Enc.E.eq y x ∨ In y s.
      Proof.
        intros s x y Hs.
        have Hs' := (add_ok s x Hs).
        rewrite !In_InZ.
        rewrite /add addZ_InZ. {
    -      rewrite -Enc.encode_eq; firstorder.
    +      rewrite -Enc.encode_eq /Z.eq.
    +      firstorder.
        } {
          apply Hs.
        }
    @@ -2534,26 +2542,26 @@

    Library MSetsExtra.MSetIntervals

    -

    empty specification

    +

    empty specification


    -  Global Instance empty_ok : Ok empty.
    +  Global Instance empty_ok : Ok empty.
      Proof.
        rewrite /empty Ok_nil //.
      Qed.

    -  Lemma empty_spec' : ∀ x, (In x empty ↔ False).
    +  Lemma empty_spec' : ∀ x, (In x empty ↔ False).
        rewrite /Empty /empty /In elements_nil.
        intros a.
    -    rewrite InA_nil //.
    +    rewrite InA_nil //.
      Qed.

    -  Lemma empty_spec : Empty empty.
    +  Lemma empty_spec : Empty empty.
      Proof.
        rewrite /Empty ⇒ a.
        rewrite empty_spec' //.
    @@ -2563,13 +2571,13 @@

    Library MSetsExtra.MSetIntervals

    -

    is_empty specification

    +

    is_empty specification


    -  Lemma is_empty_spec : ∀ (s : t) (Hs : Ok s), is_empty s = true ↔ Empty s.
    +  Lemma is_empty_spec : ∀ (s : t) (Hs : Ok s), is_empty s = true ↔ Empty s.
      Proof.
        intros [ | [x c] s]. {
          split ⇒ // _.
    @@ -2580,11 +2588,11 @@

    Library MSetsExtra.MSetIntervals

          split ⇒ //.
          move ⇒ H.
          contradiction (H (Enc.decode x)) ⇒ {H}.
    -      rewrite /In InA_alt elements_cons.
    +      rewrite /In InA_alt elements_cons.
          âˆƒ (Enc.decode x).
          split; first by apply Enc.E.eq_equiv.
    -      rewrite in_app_iff; left.
    -      rewrite /elements_single in_map_iff.
    +      rewrite in_app_iff; left.
    +      rewrite /elements_single in_map_iff.
          âˆƒ x.
          split ⇒ //.
          apply In_elementsZ_single_hd ⇒ //.
    @@ -2595,13 +2603,13 @@

    Library MSetsExtra.MSetIntervals

    -

    singleton specification

    +

    singleton specification


    -  Global Instance singleton_ok x : Ok (singleton x).
    +  Global Instance singleton_ok x : Ok (singleton x).
      Proof.
        rewrite singleton_alt_def.
        apply add_ok.
    @@ -2609,11 +2617,11 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma singleton_spec : ∀ x y : elt, In y (singleton x) ↔ Enc.E.eq y x.
    +  Lemma singleton_spec : ∀ x y : elt, In y (singleton x) ↔ Enc.E.eq y x.
      Proof.
        intros x y.
        rewrite singleton_alt_def.
    -    rewrite (add_spec empty x y) /empty /In elements_nil InA_nil.
    +    rewrite (add_spec empty x y) /empty /In elements_nil InA_nil.
        split. {
          move ⇒ [] //.
        } {
    @@ -2625,13 +2633,13 @@

    Library MSetsExtra.MSetIntervals

    -

    add_list specification

    +

    add_list specification


    -  Lemma add_list_ok : ∀ l s, Ok s → Ok (add_list l s).
    +  Lemma add_list_ok : ∀ l s, Ok s → Ok (add_list l s).
      Proof.
        induction l as [| x l' IH]. {
          done.
    @@ -2643,17 +2651,17 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma add_list_spec : ∀ x l s, Ok s →
    -     (In x (add_list l s) ↔ (SetoidList.InA Enc.E.eq x l) ∨ In x s).
    +  Lemma add_list_spec : ∀ x l s, Ok s →
    +     (In x (add_list l s) ↔ (SetoidList.InA Enc.E.eq x l) ∨ In x s).
      Proof.
        move ⇒ x.
        induction l as [| y l' IH]. {
          intros s H.
    -      rewrite /= InA_nil.
    +      rewrite /= InA_nil.
          tauto.
        } {
          move ⇒ s H_ok /=.
    -      rewrite IH add_spec InA_cons.
    +      rewrite IH add_spec InA_cons.
          tauto.
        }
      Qed.
    @@ -2662,37 +2670,37 @@

    Library MSetsExtra.MSetIntervals

    -

    union specification

    +

    union specification


    -  Lemma union_aux_flatten_alt_def : ∀ (s1 s2 : t) acc,
    -    union_aux s1 s2 acc =
    -    match (s1, s2) with
    -    | (nil, _) ⇒ List.rev_append acc s2
    -    | (_, nil) ⇒ List.rev_append acc s1
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
    +  Lemma union_aux_flatten_alt_def : ∀ (s1 s2 : t) acc,
    +    union_aux s1 s2 acc =
    +    match (s1, s2) with
    +    | (nil, _) ⇒ List.rev_append acc s2
    +    | (_, nil) ⇒ List.rev_append acc s1
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ union_aux l1 s2 ((y1, c1)::acc)
              | ICR_before_touch ⇒
                  union_aux l1 (
    -                insert_interval_begin y1 ((c1+c2)%N) l2) acc
    -          | ICR_after ⇒ union_aux s1 l2 ((y2, c2)::acc)
    +                insert_interval_begin y1 ((c1+c2)%N) l2) acc
    +          | ICR_after ⇒ union_aux s1 l2 ((y2, c2)::acc)
              | ICR_after_touch ⇒ union_aux l1 (
    -              insert_interval_begin y2 ((c1+c2)%N) l2) acc
    +              insert_interval_begin y2 ((c1+c2)%N) l2) acc
              | ICR_overlap_before ⇒
                  union_aux l1 (
                    insert_interval_begin y1
    -                  (merge_interval_size y1 c1 y2 c2) l2) acc
    +                  (merge_interval_size y1 c1 y2 c2) l2) acc
              | ICR_overlap_after ⇒
                  union_aux l1 (
                    insert_interval_begin y2
    -                  (merge_interval_size y2 c2 y1 c1) l2) acc
    -          | ICR_equal ⇒ union_aux l1 s2 acc
    -          | ICR_subsume_1 ⇒ union_aux l1 s2 acc
    -          | ICR_subsume_2 ⇒ union_aux s1 l2 acc
    +                  (merge_interval_size y2 c2 y1 c1) l2) acc
    +          | ICR_equal ⇒ union_aux l1 s2 acc
    +          | ICR_subsume_1 ⇒ union_aux l1 s2 acc
    +          | ICR_subsume_2 ⇒ union_aux s1 l2 acc
            end
        end.
      Proof.
    @@ -2701,9 +2709,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma union_aux_alt_def : ∀ (s1 s2 : t) acc,
    -    union_aux s1 s2 acc =
    -    List.rev_append acc (union s1 s2).
    +  Lemma union_aux_alt_def : ∀ (s1 s2 : t) acc,
    +    union_aux s1 s2 acc =
    +    List.rev_append acc (union s1 s2).
      Proof.
        rewrite /union.
        intros s1 s2 acc.
    @@ -2711,39 +2719,39 @@

    Library MSetsExtra.MSetIntervals

        induction s1 as [| [y1 c1] l1 IH1]. {
          intros acc s2.
          rewrite !union_aux_flatten_alt_def.
    -      rewrite !rev_append_rev //.
    +      rewrite !rev_append_rev //.
        }
        intros acc s2; move : acc.
        induction s2 as [| [y2 c2] l2 IH2]; first by simpl.
        move ⇒ acc.
    -    rewrite !(union_aux_flatten_alt_def ((y1, c1)::l1) ((y2, c2)::l2)).
    -    case (interval_compare (y1, c1) (y2, c2));
    -      rewrite ?(IH1 ((y1, c1) :: acc)) ?(IH1 ((y1, c1) :: nil))
    -              ?(IH2 ((y2, c2) :: acc)) ?(IH2 ((y2, c2) :: nil))
    +    rewrite !(union_aux_flatten_alt_def ((y1, c1)::l1) ((y2, c2)::l2)).
    +    case (interval_compare (y1, c1) (y2, c2));
    +      rewrite ?(IH1 ((y1, c1) :: acc)) ?(IH1 ((y1, c1) :: nil))
    +              ?(IH2 ((y2, c2) :: acc)) ?(IH2 ((y2, c2) :: nil))
                  ?(IH1 acc) //.
      Qed.

    -  Lemma union_alt_def : ∀ (s1 s2 : t),
    -    union s1 s2 =
    -    match (s1, s2) with
    -    | (nil, _) ⇒ s2
    -    | (_, nil) ⇒ s1
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ (y1, c1) :: (union l1 s2)
    +  Lemma union_alt_def : ∀ (s1 s2 : t),
    +    union s1 s2 =
    +    match (s1, s2) with
    +    | (nil, _) ⇒ s2
    +    | (_, nil) ⇒ s1
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ (y1, c1) :: (union l1 s2)
              | ICR_before_touch ⇒
    -              union l1 (insert_interval_begin y1 ((c1+c2)%N) l2)
    -          | ICR_after ⇒ (y2, c2) :: union s1 l2
    +              union l1 (insert_interval_begin y1 ((c1+c2)%N) l2)
    +          | ICR_after ⇒ (y2, c2) :: union s1 l2
              | ICR_after_touch ⇒ union l1
    -              (insert_interval_begin y2 ((c1+c2)%N) l2)
    +              (insert_interval_begin y2 ((c1+c2)%N) l2)
              | ICR_overlap_before ⇒
                  union l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2)
              | ICR_overlap_after ⇒
                  union l1 (insert_interval_begin y2 (merge_interval_size y2 c2 y1 c1) l2)
    -          | ICR_equal ⇒ union l1 s2
    -          | ICR_subsume_1 ⇒ union l1 s2
    -          | ICR_subsume_2 ⇒ union s1 l2
    +          | ICR_equal ⇒ union l1 s2
    +          | ICR_subsume_1 ⇒ union l1 s2
    +          | ICR_subsume_2 ⇒ union s1 l2
            end
         end.
      Proof.
    @@ -2751,16 +2759,16 @@

    Library MSetsExtra.MSetIntervals

        rewrite /union union_aux_flatten_alt_def.
        case s1 as [| [y1 c1] l1] ⇒ //.
        case s2 as [| [y2 c2] l2] ⇒ //.
    -    case (interval_compare (y1, c1) (y2, c2));
    +    case (interval_compare (y1, c1) (y2, c2));
          rewrite union_aux_alt_def //.
      Qed.

    -  Lemma union_InZ :
    -   âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    âˆ€ y, (InZ y (union s1 s2) ↔ InZ y s1 ∨ InZ y s2).
    +  Lemma union_InZ :
    +   âˆ€ (s1 s2 : t),
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    âˆ€ y, (InZ y (union s1 s2) ↔ InZ y s1 ∨ InZ y s2).
      Proof.
        intro s1.
        induction s1 as [| [y1 c1] l1 IH1]. {
    @@ -2778,32 +2786,32 @@

    Library MSetsExtra.MSetIntervals

            rewrite !interval_list_invariant_cons.
            move ⇒ [H_gr_l1] [H_c1_neq_0] H_inv_l1.
            move ⇒ [H_gr_l2] [H_c2_neq_0] H_inv_l2.
    -        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {IH2} IH2.
    -        have : ∀ s2 : t,
    -          interval_list_invariant s2 = true →
    -          âˆ€ y : Z, InZ y (union l1 s2) ↔ InZ y l1 ∨ InZ y s2. {
    +        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {} IH2.
    +        have : ∀ s2 : t,
    +          interval_list_invariant s2 = true →
    +          âˆ€ y : Z, InZ y (union l1 s2) ↔ InZ y l1 ∨ InZ y s2. {
              intros. by apply IH1.
            }
    -        move ⇒ {IH1} IH1 y.
    +        move ⇒ {} IH1 y.
            rewrite union_alt_def.
            move : (interval_compare_elim y1 c1 y2 c2).
    -        case (interval_compare (y1, c1) (y2, c2)). {
    +        case (interval_compare (y1, c1) (y2, c2)). {
              rewrite !InZ_cons IH1 // InZ_cons.
              tauto.
            } {
              move ⇒ H_y2_eq.
    -          replace (c1 + c2)%N with (merge_interval_size y1 c1 y2 c2);
    +          replace (c1 + c2)%N with (merge_interval_size y1 c1 y2 c2);
                last first. {
                rewrite -H_y2_eq merge_interval_size_add //.
              }
              set c'' := merge_interval_size y1 c1 y2 c2.
              have [H_inv_insert H_InZ_insert] :
    -               interval_list_invariant (insert_interval_begin y1 c'' l2) = true ∧
    -                (InZ y (insert_interval_begin y1 c'' l2) ↔
    -                List.In y (elementsZ_single y1 c'') ∨ InZ y l2). {
    +               interval_list_invariant (insert_interval_begin y1 c'' l2) = true ∧
    +                (InZ y (insert_interval_begin y1 c'' l2) ↔
    +                List.In y (elementsZ_single y1 c'') ∨ InZ y l2). {
                apply insert_interval_begin_spec ⇒ //. {
                  eapply interval_list_elements_greater_impl; eauto.
    -              rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
    +              rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
                  apply Z_le_add_r.
                } {
                  by apply merge_interval_size_neq_0.
    @@ -2817,15 +2825,15 @@

    Library MSetsExtra.MSetIntervals

              } {
                apply Z_le_add_r.
              } {
    -            by apply Z.le_refl.
    +            by apply Z.le_refl.
              }
            } {
              move ⇒ [H_y1_lt] [H_y2_lt] H_y1_c1_lt.
              set c'' := merge_interval_size y1 c1 y2 c2.
              have [H_inv_insert H_InZ_insert] :
    -               interval_list_invariant (insert_interval_begin y1 c'' l2) = true ∧
    -                (InZ y (insert_interval_begin y1 c'' l2) ↔
    -                List.In y (elementsZ_single y1 c'') ∨ InZ y l2). {
    +               interval_list_invariant (insert_interval_begin y1 c'' l2) = true ∧
    +                (InZ y (insert_interval_begin y1 c'' l2) ↔
    +                List.In y (elementsZ_single y1 c'') ∨ InZ y l2). {
                apply insert_interval_begin_spec ⇒ //. {
                  eapply interval_list_elements_greater_impl; eauto.
                  apply Z_lt_le_add_r ⇒ //.
    @@ -2838,17 +2846,17 @@

    Library MSetsExtra.MSetIntervals

              rewrite In_merge_interval. {
                tauto.
              } {
    -            by apply Z.lt_le_incl.
    +            by apply Z.lt_le_incl.
              } {
    -            by apply Z.lt_le_incl.
    +            by apply Z.lt_le_incl.
              }
            } {
              move ⇒ [H_y2_lt] [H_y1_lt] H_y2_c_lt.
              set c'' := merge_interval_size y2 c2 y1 c1.
              have [H_inv_insert H_InZ_insert] :
    -               interval_list_invariant (insert_interval_begin y2 c'' l2) = true ∧
    -                (InZ y (insert_interval_begin y2 c'' l2) ↔
    -                List.In y (elementsZ_single y2 c'') ∨ InZ y l2). {
    +               interval_list_invariant (insert_interval_begin y2 c'' l2) = true ∧
    +                (InZ y (insert_interval_begin y2 c'' l2) ↔
    +                List.In y (elementsZ_single y2 c'') ∨ InZ y l2). {
                apply insert_interval_begin_spec ⇒ //. {
                  eapply interval_list_elements_greater_impl; eauto.
                  apply Z_le_add_r.
    @@ -2862,9 +2870,9 @@

    Library MSetsExtra.MSetIntervals

              rewrite In_merge_interval. {
                tauto.
              } {
    -            by apply Z.lt_le_incl.
    +            by apply Z.lt_le_incl.
              } {
    -            by apply Z.lt_le_incl.
    +            by apply Z.lt_le_incl.
              }
            } {
              move ⇒ [? ?]; subst.
    @@ -2875,38 +2883,38 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ [H_y2_le] [H_y1_c1_le] _.
              rewrite IH1 ⇒ //.
              rewrite !InZ_cons.
    -          suff : (List.In y (elementsZ_single y1 c1) →
    -                  List.In y (elementsZ_single y2 c2)). {
    +          suff : (List.In y (elementsZ_single y1 c1) →
    +                  List.In y (elementsZ_single y2 c2)). {
                tauto.
              }
              rewrite !In_elementsZ_single.
              move ⇒ [H_y1_le H_y_lt].
    -          split; omega.
    +          split; lia.
            } {
              move ⇒ [H_y1_le] [H_y2_c2_le] _.
              rewrite IH2.
              rewrite !InZ_cons.
    -          suff : (List.In y (elementsZ_single y2 c2) →
    -                  List.In y (elementsZ_single y1 c1)). {
    +          suff : (List.In y (elementsZ_single y2 c2) →
    +                  List.In y (elementsZ_single y1 c1)). {
                tauto.
              }
              rewrite !In_elementsZ_single.
              move ⇒ [H_y2_le H_y_lt].
    -          split; omega.
    +          split; lia.
            } {
              rewrite !InZ_cons IH2 !InZ_cons.
              tauto.
            } {
              move ⇒ H_y1_eq.
    -          replace (c1 + c2)%N with (merge_interval_size y2 c2 y1 c1);
    +          replace (c1 + c2)%N with (merge_interval_size y2 c2 y1 c1);
                last first. {
    -            rewrite -H_y1_eq merge_interval_size_add N.add_comm //.
    +            rewrite -H_y1_eq merge_interval_size_add N.add_comm //.
              }
              set c'' := merge_interval_size y2 c2 y1 c1.
              have [H_inv_insert H_InZ_insert] :
    -               interval_list_invariant (insert_interval_begin y2 c'' l2) = true ∧
    -                (InZ y (insert_interval_begin y2 c'' l2) ↔
    -                List.In y (elementsZ_single y2 c'') ∨ InZ y l2). {
    +               interval_list_invariant (insert_interval_begin y2 c'' l2) = true ∧
    +                (InZ y (insert_interval_begin y2 c'' l2) ↔
    +                List.In y (elementsZ_single y2 c'') ∨ InZ y l2). {
                apply insert_interval_begin_spec ⇒ //. {
                  eapply interval_list_elements_greater_impl; eauto.
                  apply Z_le_add_r.
    @@ -2922,7 +2930,7 @@

    Library MSetsExtra.MSetIntervals

              } {
                apply Z_le_add_r.
              } {
    -            by apply Z.le_refl.
    +            by apply Z.le_refl.
              }
            }
          }
    @@ -2930,11 +2938,11 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma union_invariant :
    -   âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    interval_list_invariant (union s1 s2) = true.
    +  Lemma union_invariant :
    +   âˆ€ (s1 s2 : t),
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    interval_list_invariant (union s1 s2) = true.
      Proof.
        intro s1.
        induction s1 as [| [y1 c1] l1 IH1]. {
    @@ -2950,20 +2958,20 @@

    Library MSetsExtra.MSetIntervals

            rewrite !interval_list_invariant_cons.
            move ⇒ [H_gr_l1] [H_c1_neq_0] H_inv_l1.
            move ⇒ [H_gr_l2] [H_c2_neq_0] H_inv_l2.
    -        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {IH2} IH2.
    -        have : ∀ s2 : t,
    -          interval_list_invariant s2 = true →
    -          interval_list_invariant (union l1 s2) = true. {
    +        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {} IH2.
    +        have : ∀ s2 : t,
    +          interval_list_invariant s2 = true →
    +          interval_list_invariant (union l1 s2) = true. {
              intros. by apply IH1.
            }
    -        move ⇒ {IH1} IH1.
    +        move ⇒ {} IH1.

            rewrite union_alt_def.
            move : (interval_compare_elim y1 c1 y2 c2).
    -        case (interval_compare (y1, c1) (y2, c2)). {
    +        case (interval_compare (y1, c1) (y2, c2)). {
              move ⇒ H_lt_y2.
    -          have H_inv' : interval_list_invariant (union l1 ((y2, c2) :: l2)) = true. {
    +          have H_inv' : interval_list_invariant (union l1 ((y2, c2) :: l2)) = true. {
                by apply IH1.
              }

    @@ -2982,10 +2990,10 @@

    Library MSetsExtra.MSetIntervals

              apply IH1.
              apply insert_interval_begin_spec ⇒ //. {
                eapply interval_list_elements_greater_impl; last apply H_gr_l2.
    -            rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
    +            rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
                apply Z_le_add_r.
              } {
    -            rewrite N.eq_add_0.
    +            rewrite N.eq_add_0.
                tauto.
              }
            } {
    @@ -3034,7 +3042,7 @@

    Library MSetsExtra.MSetIntervals

                eapply interval_list_elements_greater_impl; last apply H_gr_l2.
                apply Z_le_add_r.
              } {
    -            rewrite N.eq_add_0.
    +            rewrite N.eq_add_0.
                tauto.
              }
            }
    @@ -3045,7 +3053,7 @@

    Library MSetsExtra.MSetIntervals



    -  Global Instance union_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (union s1 s2).
    +  Global Instance union_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (union s1 s2).
      Proof.
        move ⇒ H_ok_s1 H_ok_s2.
        move : (H_ok_s1) (H_ok_s2).
    @@ -3067,9 +3075,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma union_spec :
    -   âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    -   In x (union s s') ↔ In x s ∨ In x s'.
    +  Lemma union_spec :
    +   âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    +   In x (union s s') ↔ In x s ∨ In x s'.
      Proof.
        intros s s' x H_ok H_ok'.
        rewrite !In_InZ.
    @@ -3084,53 +3092,53 @@

    Library MSetsExtra.MSetIntervals

    -

    inter specification

    +

    inter specification


    -  Lemma inter_aux_alt_def :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      inter_aux y2 c2 acc s = match inter_aux y2 c2 nil s with
    -                               (acc', s') ⇒ (acc' ++ acc, s')
    +  Lemma inter_aux_alt_def :
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      inter_aux y2 c2 acc s = match inter_aux y2 c2 nil s with
    +                               (acc', s') ⇒ (acc' ++ acc, s')
                                 end.
      Proof.
        intros y2 c2.
        induction s as [| [y1 c1] s' IH] ⇒ acc. {
    -      rewrite /inter_aux app_nil_l //.
    +      rewrite /inter_aux app_nil_l //.
        } {
          simpl.
    -      case_eq (inter_aux y2 c2 nil s') ⇒ acc'' s'' H_eq.
    -      case (interval_compare (y1, c1) (y2, c2));
    +      case_eq (inter_aux y2 c2 nil s') ⇒ acc'' s'' H_eq.
    +      case (interval_compare (y1, c1) (y2, c2));
            rewrite ?(IH acc)
    -                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: acc))
    -                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: nil))
    -                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: acc))
    -                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: nil))
    -                ?(IH ((y1, c1) :: acc))
    -                ?(IH ((y1, c1) :: nil))
    +                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: acc))
    +                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: nil))
    +                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: acc))
    +                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: nil))
    +                ?(IH ((y1, c1) :: acc))
    +                ?(IH ((y1, c1) :: nil))
                    
    -                ?H_eq -?app_assoc -?app_comm_cons //.
    +                ?H_eq -?app_assoc -?app_comm_cons //.
        }
      Qed.

    -  Lemma inter_aux_props :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      interval_list_invariant (rev acc) = true →
    -      interval_list_invariant s = true →
    -      (∀ x1 x2, InZ x1 acc → InZ x2 s →
    -                     List.In x2 (elementsZ_single y2 c2) →
    -                     Z.succ x1 < x2) →
    -      (c2 ≠ 0%N) →
    -      match (inter_aux y2 c2 acc s) with (acc', s') ⇒
    -        (∀ y, (InZ y acc' ↔
    -                   (InZ y acc ∨ (InZ y s ∧ (List.In y (elementsZ_single y2 c2)))))) ∧
    -        (∀ y, InZ y s' → InZ y s) ∧
    -        (∀ y, InZ y s → y2 + Z.of_N c2 < y → InZ y s') ∧
    -        interval_list_invariant (rev acc') = true ∧
    -        interval_list_invariant s' = true
    +  Lemma inter_aux_props :
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      interval_list_invariant (rev acc) = true →
    +      interval_list_invariant s = true →
    +      (∀ x1 x2, InZ x1 acc → InZ x2 s →
    +                     List.In x2 (elementsZ_single y2 c2) →
    +                     Z.succ x1 < x2) →
    +      (c2 ≠ 0%N) →
    +      match (inter_aux y2 c2 acc s) with (acc', s') ⇒
    +        (∀ y, (InZ y acc' ↔
    +                   (InZ y acc ∨ (InZ y s ∧ (List.In y (elementsZ_single y2 c2)))))) ∧
    +        (∀ y, InZ y s' → InZ y s) ∧
    +        (∀ y, InZ y s → y2 + Z.of_N c2 < y → InZ y s') ∧
    +        interval_list_invariant (rev acc') = true ∧
    +        interval_list_invariant s' = true
          end.
      Proof.
        intros y2 c2.
    @@ -3147,22 +3155,22 @@

    Library MSetsExtra.MSetIntervals


          rewrite inter_aux_alt_def.
    -      case_eq (inter_aux y2 c2 nil ((y1, c1) :: s1')).
    +      case_eq (inter_aux y2 c2 nil ((y1, c1) :: s1')).
          move ⇒ acc' s' H_inter_aux_eq.

    -      set P1 := ∀ y : Z,
    -        (InZ y acc' ↔
    -        ((InZ y ((y1, c1) :: s1') ∧ List.In y (elementsZ_single y2 c2)))).
    -      set P2 := (∀ y,
    -                  (InZ y s' → InZ y ((y1, c1) :: s1')) ∧
    -                  (InZ y ((y1, c1) :: s1') →
    -                     y2 + Z.of_N c2 < y → InZ y s')).
    -      set P3 := interval_list_invariant (rev acc') = true.
    -      set P4 := interval_list_invariant s' = true.
    +      set P1 := ∀ y : Z,
    +        (InZ y acc' ↔
    +        ((InZ y ((y1, c1) :: s1') ∧ List.In y (elementsZ_single y2 c2)))).
    +      set P2 := (∀ y,
    +                  (InZ y s' → InZ y ((y1, c1) :: s1')) ∧
    +                  (InZ y ((y1, c1) :: s1') →
    +                     y2 + Z.of_N c2 < y → InZ y s')).
    +      set P3 := interval_list_invariant (rev acc') = true.
    +      set P4 := interval_list_invariant s' = true.

    -      suff : (P1 ∧ P2 ∧ P3 ∧ P4). {
    +      suff : (P1 ∧ P2 ∧ P3 ∧ P4). {
            move ⇒ [H_P1] [H_P2] [H_P3] H_P4.
            split; last split; last split; last split. {
              move ⇒ y.
    @@ -3177,7 +3185,7 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ y H_y_in.
              by apply H_P2.
            } {
    -          rewrite rev_app_distr.
    +          rewrite rev_app_distr.
              apply interval_list_invariant_app_intro ⇒ //.
              move ⇒ x1 x2.
              rewrite !InZ_rev.
    @@ -3192,31 +3200,31 @@

    Library MSetsExtra.MSetIntervals

          rewrite interval_list_elements_greater_alt2_def ⇒ // ⇒ H_gr_s1'_alt.

    -      have : ∀ (acc : list (Z × N)),
    -        interval_list_invariant (rev acc) = true →
    -        (∀ y, InZ y acc ↔ (
    -           y1 ≤ y < y1 + Z.of_N c1 ∧
    -           y2 ≤ y < y2 + Z.of_N c2)) →
    -        (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) →
    -        (inter_aux y2 c2 acc s1' = (acc', s')) →
    -        P1 ∧ P2 ∧ P3 ∧ P4. {
    +      have : ∀ (acc : list (Z × N)),
    +        interval_list_invariant (rev acc) = true →
    +        (∀ y, InZ y acc ↔ (
    +           y1 ≤ y < y1 + Z.of_N c1 ∧
    +           y2 ≤ y < y2 + Z.of_N c2)) →
    +        (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) →
    +        (inter_aux y2 c2 acc s1' = (acc', s')) →
    +        P1 ∧ P2 ∧ P3 ∧ P4. {
            
            intros acc0 H_inv_acc0 H_in_acc0 H_y2c_lt H_inter_aux_eq0.
    -        have H_in_acc0_lt : (∀ x1 x2 : Z,
    -          InZ x1 acc0 →
    -          InZ x2 s1' →
    -          List.In x2 (elementsZ_single y2 c2) →
    -          Z.succ x1 < x2). {
    +        have H_in_acc0_lt : (∀ x1 x2 : Z,
    +          InZ x1 acc0 →
    +          InZ x2 s1' →
    +          List.In x2 (elementsZ_single y2 c2) →
    +          Z.succ x1 < x2). {

              intros x1 x2 H_x1_in_acc0 H_x2_in_s1' H_x2_in_yc2.

    -          suff H_yc1_lt_x2 : Z.succ x1 ≤ y1 + Z.of_N c1. {
    -            apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +          suff H_yc1_lt_x2 : Z.succ x1 ≤ y1 + Z.of_N c1. {
    +            apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
                by apply H_gr_s1'_alt.
              }
              move : (H_x1_in_acc0).
    -          rewrite H_in_acc0 Z.le_succ_l.
    +          rewrite H_in_acc0 Z.le_succ_l.
              tauto.
            }
            
    @@ -3240,25 +3248,25 @@

    Library MSetsExtra.MSetIntervals

                move ⇒ []. {
                  move ⇒ [_] H_y_lt H_lt_y.
                  exfalso.
    -              suff : (y < y) by apply Z.lt_irrefl.
    -              apply Z.lt_le_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -              apply Z.le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    +              suff : (y < y) by apply Z.lt_irrefl.
    +              apply Z.lt_le_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +              apply Z.le_trans with (m := y2 + Z.of_N c2) ⇒ //.

    -              by apply Z.lt_le_incl.
    +              by apply Z.lt_le_incl.
                } {
                  apply H3.
                }
              }
            }
          }
    -      move ⇒ {IH} IH.
    +      move ⇒ {} IH.

          clear H_inv_acc H_in_acc_lt acc.
          move : (interval_compare_elim y1 c1 y2 c2) H_inter_aux_eq.
          unfold inter_aux.
    -      case_eq (interval_compare (y1, c1) (y2, c2)) ⇒ H_comp;
    +      case_eq (interval_compare (y1, c1) (y2, c2)) ⇒ H_comp;
             fold inter_aux. {
            move ⇒ H_lt_y2.
            apply IH. {
    @@ -3267,10 +3275,10 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ x.
              rewrite InZ_nil.
              split ⇒ //.
    -          omega.
    +          lia.
            } {
    -          apply Z.le_trans with (m := y2). {
    -            by apply Z.lt_le_incl.
    +          apply Z.le_trans with (m := y2). {
    +            by apply Z.lt_le_incl.
              } {
                apply Z_le_add_r.
              }
    @@ -3283,7 +3291,7 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ x.
              rewrite InZ_nil.
              split ⇒ //.
    -          omega.
    +          lia.
            } {
              rewrite H_eq_y2.
              apply Z_le_add_r.
    @@ -3295,11 +3303,11 @@

    Library MSetsExtra.MSetIntervals

              by apply Z_to_N_minus_neq_0.
            } {
              move ⇒ x.
    -          rewrite InZ_cons InZ_nil In_elementsZ_single Z2N.id; last omega.
    -          replace (y1 + (y2 - y1)) with y2 by omega.
    -          split; omega.
    +          rewrite InZ_cons InZ_nil In_elementsZ_single Z2N.id; last lia.
    +          replace (y1 + (y2 - y1)) with y2 by lia.
    +          split; lia.
            } {
    -          by apply Z.lt_le_incl.
    +          by apply Z.lt_le_incl.
            }
          } {
            rewrite /P1 /P2 /P3 /P4.
    @@ -3309,17 +3317,17 @@

    Library MSetsExtra.MSetIntervals

            subst s' acc'.
            split; last split; last split. {
              move ⇒ y.
    -          have H_yc2_intro : y1 + Z.of_N (Z.to_N (y2 + Z.of_N c2 - y1)) =
    -                             y2 + Z.of_N c2. {
    -            rewrite Z2N.id; omega.
    +          have H_yc2_intro : y1 + Z.of_N (Z.to_N (y2 + Z.of_N c2 - y1)) =
    +                             y2 + Z.of_N c2. {
    +            rewrite Z2N.id; lia.
              }
              
              rewrite !InZ_cons !In_elementsZ_single InZ_nil H_yc2_intro.
              split. {
                move ⇒ [] //.
                move ⇒ [H_y1_le] H_y_lt.
    -            split; last by omega.
    -            left. omega.
    +            split; last by lia.
    +            left. lia.
              } {
                move ⇒ [H_in_s] [H_y2_le] H_y_lt.
                left.
    @@ -3328,10 +3336,10 @@

    Library MSetsExtra.MSetIntervals

                  tauto.
                } {
                  move ⇒ /H_gr_s1'_alt H_lt_y.
    -              apply Z.le_trans with (m := y1 + Z.of_N c1). {
    +              apply Z.le_trans with (m := y1 + Z.of_N c1). {
                    by apply Z_le_add_r.
                  } {
    -                by apply Z.lt_le_incl.
    +                by apply Z.lt_le_incl.
                  }
                }
              }
    @@ -3363,9 +3371,9 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ [] //.
              move ⇒ [_] H_y_lt H_lt_y.
              exfalso.
    -          suff : (y2 + Z.of_N c2 < y2 + Z.of_N c2) by
    -              apply Z.lt_irrefl.
    -          apply Z.lt_trans with (m := y) ⇒ //.
    +          suff : (y2 + Z.of_N c2 < y2 + Z.of_N c2) by
    +              apply Z.lt_irrefl.
    +          apply Z.lt_trans with (m := y) ⇒ //.
            } {
              rewrite interval_list_invariant_sing //.
            } {
    @@ -3382,7 +3390,7 @@

    Library MSetsExtra.MSetIntervals

                move ⇒ [] //.
                move ⇒ [H_y1_le] H_y_lt.
                split; first done.
    -            split; omega.
    +            split; lia.
              } {
                move ⇒ [?] _.
                by left.
    @@ -3402,8 +3410,8 @@

    Library MSetsExtra.MSetIntervals

              split. {
                move ⇒ [] //.
                move ⇒ [H_y2_le] H_y_lt.
    -            split; last by omega.
    -            left. omega.
    +            split; last by lia.
    +            left. lia.
              } {
                move ⇒ [H_in_s] [H_y2_le] H_y_lt.
                by left.
    @@ -3427,14 +3435,14 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ [] []. {
                move ⇒ [H_y1_le_y] H_y_lt_yc1.
                move ⇒ [H_y2_le_y] H_y_lt_yc2.
    -            omega.
    +            lia.
              } {
                move ⇒ /H_gr_s1'_alt H_lt_y [_] H_y_lt.
    -            suff : (y1 + Z.of_N c1 < y1). {
    -              apply Z.nlt_ge.
    +            suff : (y1 + Z.of_N c1 < y1). {
    +              apply Z.nlt_ge.
                  apply Z_le_add_r.
                }
    -            omega.
    +            lia.
              }
            } {
              tauto.
    @@ -3455,14 +3463,14 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ [] []. {
                move ⇒ [H_y1_le_y] H_y_lt_yc1.
                move ⇒ [H_y2_le_y] H_y_lt_yc2.
    -            omega.
    +            lia.
              } {
                move ⇒ /H_gr_s1'_alt H_lt_y [_] H_y_lt.
    -            suff : (y1 + Z.of_N c1 < y1). {
    -              apply Z.nlt_ge.
    +            suff : (y1 + Z.of_N c1 < y1). {
    +              apply Z.nlt_ge.
                  apply Z_le_add_r.
                }
    -            omega.
    +            lia.
              }
            } {
              tauto.
    @@ -3476,25 +3484,25 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma inter_aux2_props :
    -   âˆ€ (s2 s1 acc : t),
    -    interval_list_invariant (rev acc) = true →
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    (∀ x1 x2, InZ x1 acc → InZ x2 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    -    ((∀ y, (InZ y (inter_aux2 acc s1 s2) ↔
    -               (InZ y acc) ∨ ((InZ y s1) ∧ InZ y s2))) ∧
    -    (interval_list_invariant (inter_aux2 acc s1 s2) = true)).
    +  Lemma inter_aux2_props :
    +   âˆ€ (s2 s1 acc : t),
    +    interval_list_invariant (rev acc) = true →
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    (∀ x1 x2, InZ x1 acc → InZ x2 s1 → InZ x2 s2 → Z.succ x1 < x2) →
    +    ((∀ y, (InZ y (inter_aux2 acc s1 s2) ↔
    +               (InZ y acc) ∨ ((InZ y s1) ∧ InZ y s2))) ∧
    +    (interval_list_invariant (inter_aux2 acc s1 s2) = true)).
      Proof.
        induction s2 as [| [y2 c2] s2' IH]. {
          move ⇒ s1 acc.
          move ⇒ H_inv_acc _ _ _.
          unfold inter_aux2.
          replace (match s1 with
    -        | nil ⇒ rev' acc
    -        | _ :: _ ⇒ rev' acc
    -               end) with (rev' acc); last by case s1.
    -      rewrite /rev' rev_append_rev app_nil_r.
    +        | nil ⇒ rev' acc
    +        | _ :: _ ⇒ rev' acc
    +               end) with (rev' acc); last by case s1.
    +      rewrite /rev' rev_append_rev app_nil_r.
          split; last done.
          move ⇒ y.
          rewrite InZ_nil InZ_rev.
    @@ -3516,11 +3524,11 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ H_s1_eq.
            split. {
              move ⇒ y.
    -          rewrite /rev' rev_append_rev app_nil_r InZ_nil
    +          rewrite /rev' rev_append_rev app_nil_r InZ_nil
                      InZ_rev.
              tauto.
            } {
    -          rewrite /rev' rev_append_rev app_nil_r //.
    +          rewrite /rev' rev_append_rev app_nil_r //.
            }
          } {
            move ⇒ [_ _] _ <-.
    @@ -3528,11 +3536,11 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ acc' s1' H_inter_aux_eq.

    -        have H_acc_s1_yc2 : ∀ x1 x2 : Z,
    -          InZ x1 acc →
    -          InZ x2 s1 →
    -          List.In x2 (elementsZ_single y2 c2) →
    -          Z.succ x1 < x2. {
    +        have H_acc_s1_yc2 : ∀ x1 x2 : Z,
    +          InZ x1 acc →
    +          InZ x2 s1 →
    +          List.In x2 (elementsZ_single y2 c2) →
    +          Z.succ x1 < x2. {
              intros x1 x2 H_x1_in H_x2_in1 H_x2_in2.
              apply H_acc_s12 ⇒ //.
              rewrite InZ_cons; by left.
    @@ -3544,8 +3552,8 @@

    Library MSetsExtra.MSetIntervals

                    [H_inv_acc'] H_inv_s1'.

    -        have H_in_acc'_s2' : ∀ x1 x2 : Z,
    -            InZ x1 acc' → InZ x2 s1' → InZ x2 s2' → Z.succ x1 < x2. {
    +        have H_in_acc'_s2' : ∀ x1 x2 : Z,
    +            InZ x1 acc' → InZ x2 s1' → InZ x2 s2' → Z.succ x1 < x2. {
              move ⇒ x1 x2 /H_in_acc'.
              move ⇒ []. {
                move ⇒ H_in_acc H_in_s1' H_in_s2'.
    @@ -3558,8 +3566,8 @@

    Library MSetsExtra.MSetIntervals

                rewrite In_elementsZ_single.
                move ⇒ [H_in_s1] [_] H_x1_lt _.
                move ⇒ /H_gr_s2' H_lt_x2.
    -            apply Z.le_lt_trans with (m := y2 + Z.of_N c2) ⇒ //.
    -            by apply Z.le_succ_l.
    +            apply Z.le_lt_trans with (m := y2 + Z.of_N c2) ⇒ //.
    +            by apply Z.le_succ_l.
              }
            }
           
    @@ -3597,17 +3605,17 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma inter_InZ :
    -   âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    âˆ€ y, (InZ y (inter s1 s2) ↔ InZ y s1 ∧ InZ y s2).
    +  Lemma inter_InZ :
    +   âˆ€ (s1 s2 : t),
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    âˆ€ y, (InZ y (inter s1 s2) ↔ InZ y s1 ∧ InZ y s2).
      Proof.
        intros s1 s2 H_inv_s1 H_inv_s2 y.
        rewrite /inter.

    -    move : (inter_aux2_props s2 s1 nil).
    +    move : (inter_aux2_props s2 s1 nil).
        move ⇒ [] //.
        move ⇒ H_in_inter _.
        rewrite H_in_inter InZ_nil.
    @@ -3615,24 +3623,24 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma inter_invariant :
    -   âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    interval_list_invariant (inter s1 s2) = true.
    +  Lemma inter_invariant :
    +   âˆ€ (s1 s2 : t),
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    interval_list_invariant (inter s1 s2) = true.
      Proof.
        intros s1 s2 H_inv_s1 H_inv_s2.
        rewrite /inter.

    -    move : (inter_aux2_props s2 s1 nil).
    +    move : (inter_aux2_props s2 s1 nil).
        move ⇒ [] //.
      Qed.


    -  Global Instance inter_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (inter s1 s2).
    +  Global Instance inter_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (inter s1 s2).
      Proof.
        move ⇒ H_ok_s1 H_ok_s2.
        move : (H_ok_s1) (H_ok_s2).
    @@ -3651,9 +3659,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma inter_spec :
    -   âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    -   In x (inter s s') ↔ In x s ∧ In x s'.
    +  Lemma inter_spec :
    +   âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    +   In x (inter s s') ↔ In x s ∧ In x s'.
      Proof.
        intros s s' x H_ok H_ok'.
        rewrite !In_InZ.
    @@ -3670,65 +3678,65 @@

    Library MSetsExtra.MSetIntervals

    -

    diff specification

    +

    diff specification


    -  Lemma diff_aux_alt_def :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      diff_aux y2 c2 acc s = match diff_aux y2 c2 nil s with
    -                               (acc', s') ⇒ (acc' ++ acc, s')
    +  Lemma diff_aux_alt_def :
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      diff_aux y2 c2 acc s = match diff_aux y2 c2 nil s with
    +                               (acc', s') ⇒ (acc' ++ acc, s')
                                 end.
      Proof.
        intros y2 c2.
        induction s as [| [y1 c1] acc' IH] ⇒ acc. {
    -      rewrite /diff_aux app_nil_l //.
    +      rewrite /diff_aux app_nil_l //.
        } {
          simpl.
    -      case_eq (diff_aux y2 c2 nil acc') ⇒ acc'' s'' H_eq.
    -      case (interval_compare (y1, c1) (y2, c2));
    -        rewrite ?(IH ((y1, c1)::acc)) ?(IH ((y1, c1)::nil))
    -                ?(IH acc) ?(IH ((y1, Z.to_N (y2 - y1)) :: acc))
    -                ?(IH ((y1, Z.to_N (y2 - y1)) :: nil)) ?H_eq;
    -        rewrite ?insert_intervalZ_guarded_app -?app_assoc -?app_comm_cons //.
    +      case_eq (diff_aux y2 c2 nil acc') ⇒ acc'' s'' H_eq.
    +      case (interval_compare (y1, c1) (y2, c2));
    +        rewrite ?(IH ((y1, c1)::acc)) ?(IH ((y1, c1)::nil))
    +                ?(IH acc) ?(IH ((y1, Z.to_N (y2 - y1)) :: acc))
    +                ?(IH ((y1, Z.to_N (y2 - y1)) :: nil)) ?H_eq;
    +        rewrite ?insert_intervalZ_guarded_app -?app_assoc -?app_comm_cons //.
        }
      Qed.


    -  Lemma diff_aux_props :
    -    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    -      interval_list_invariant (List.rev acc) = true →
    -      interval_list_invariant s = true →
    -      (∀ x1 x2, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    -      (∀ x, InZ x acc → x < y2) →
    -      (c2 ≠ 0%N) →
    -      match (diff_aux y2 c2 acc s) with
    -        (acc', s') ⇒ (∀ y, InZ y (List.rev_append acc' s') ↔
    -                                 InZ y (List.rev_append acc s) ∧ ~(List.In y (elementsZ_single y2 c2))) ∧
    -                      (interval_list_invariant (List.rev_append acc' s') = true) ∧
    -                      (∀ x, InZ x acc' → x < y2 + Z.of_N c2)
    +  Lemma diff_aux_props :
    +    âˆ€ (y2 : Z) (c2 : N) (s : t) acc,
    +      interval_list_invariant (List.rev acc) = true →
    +      interval_list_invariant s = true →
    +      (∀ x1 x2, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    +      (∀ x, InZ x acc → x < y2) →
    +      (c2 ≠ 0%N) →
    +      match (diff_aux y2 c2 acc s) with
    +        (acc', s') ⇒ (∀ y, InZ y (List.rev_append acc' s') ↔
    +                                 InZ y (List.rev_append acc s) ∧ ~(List.In y (elementsZ_single y2 c2))) ∧
    +                      (interval_list_invariant (List.rev_append acc' s') = true) ∧
    +                      (∀ x, InZ x acc' → x < y2 + Z.of_N c2)
          end.
      Proof.
        intros y2 c2.
        induction s as [| [y1 c1] s1' IH] ⇒ acc. {
    -      rewrite /diff_aux -rev_alt.
    +      rewrite /diff_aux -rev_alt.
          move ⇒ H_inv_acc _ _ H_in_acc H_c2_neq.
          split; last split. {
            move ⇒ y; split; last by move ⇒ [] //.
            rewrite InZ_rev.
            move ⇒ H_in. split ⇒ //.
    -        move ⇒ /In_elementsZ_single ⇒ [] [] /Z.nlt_ge H_neq.
    +        move ⇒ /In_elementsZ_single ⇒ [] [] /Z.nlt_ge H_neq.
            contradict H_neq.
            by apply H_in_acc.
          } {
            assumption.
          } {
            intros x H_in_acc'.
    -        apply Z.lt_le_trans with (m := y2). {
    +        apply Z.lt_le_trans with (m := y2). {
              by apply H_in_acc.
            } {
              by apply Z_le_add_r.
    @@ -3741,37 +3749,37 @@

    Library MSetsExtra.MSetIntervals


          rewrite diff_aux_alt_def.
    -      case_eq (diff_aux y2 c2 nil ((y1, c1) :: s1')).
    +      case_eq (diff_aux y2 c2 nil ((y1, c1) :: s1')).
          move ⇒ acc' s' H_diff_aux_eq.

    -      set P1 := ∀ y : Z,
    -        (InZ y acc' ∨ InZ y s') ↔
    -        InZ y ((y1, c1) :: s1') ∧ ¬ List.In y (elementsZ_single y2 c2).
    -      set P2 := interval_list_invariant (rev acc' ++ s') = true.
    -      set P3 := ∀ x : Z, InZ x acc' → (x < y2 + Z.of_N c2).
    +      set P1 := ∀ y : Z,
    +        (InZ y acc' ∨ InZ y s') ↔
    +        InZ y ((y1, c1) :: s1') ∧ ¬ List.In y (elementsZ_single y2 c2).
    +      set P2 := interval_list_invariant (rev acc' ++ s') = true.
    +      set P3 := ∀ x : Z, InZ x acc' → (x < y2 + Z.of_N c2).

    -      suff : (P1 ∧ P2 ∧ P3). {
    +      suff : (P1 ∧ P2 ∧ P3). {
            move ⇒ [H_P1] [H_P2] H_P3.
            split; last split. {
              move ⇒ y.
              move : (H_P1 y).
    -          rewrite !rev_append_rev rev_app_distr !InZ_app
    +          rewrite !rev_append_rev rev_app_distr !InZ_app
                      !InZ_rev In_elementsZ_single.
    -          suff : (InZ y acc → ¬ y2 ≤ y < y2 + Z.of_N c2). {
    +          suff : (InZ y acc → ¬ y2 ≤ y < y2 + Z.of_N c2). {
                tauto.
              }
              move ⇒ /H_in_acc H_y_lt [H_y_ge] _.
              contradict H_y_ge.
    -          by apply Zlt_not_le.
    +          by apply Zlt_not_le.
            } {
    -          rewrite rev_append_rev rev_app_distr -app_assoc.
    +          rewrite rev_append_rev rev_app_distr -app_assoc.
              apply interval_list_invariant_app_intro ⇒ //.
              move ⇒ x1 x2.
              rewrite InZ_app !InZ_rev.
              move ⇒ H_in_acc' H_x2_in_s'.
    -          suff : (InZ x2 ((y1, c1)::s1')). {
    +          suff : (InZ x2 ((y1, c1)::s1')). {
                by apply H_in_s1.
              }
              move : (H_P1 x2).
    @@ -3783,7 +3791,7 @@

    Library MSetsExtra.MSetIntervals

                apply H_P3.
              } {
                move ⇒ /H_in_acc H_x_lt.
    -            eapply Z.lt_trans; eauto.
    +            eapply Z.lt_trans; eauto.
                by apply Z_lt_add_r.
              }
            }
    @@ -3793,34 +3801,34 @@

    Library MSetsExtra.MSetIntervals

          rewrite interval_list_elements_greater_alt2_def ⇒ // ⇒ H_gr_s1'_alt.

    -      have : ∀ (acc : list (Z × N)),
    -        interval_list_invariant (rev acc) = true →
    -        (∀ x : Z,
    -            InZ x acc ↔
    -            ((y1 ≤ x < y1 + Z.of_N c1) ∧ (x < y2))) →
    -        (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) →
    -        (diff_aux y2 c2 acc s1' = (acc', s')) →
    -        P1 ∧ P2 ∧ P3. {
    +      have : ∀ (acc : list (Z × N)),
    +        interval_list_invariant (rev acc) = true →
    +        (∀ x : Z,
    +            InZ x acc ↔
    +            ((y1 ≤ x < y1 + Z.of_N c1) ∧ (x < y2))) →
    +        (y1 + Z.of_N c1 ≤ y2 + Z.of_N c2) →
    +        (diff_aux y2 c2 acc s1' = (acc', s')) →
    +        P1 ∧ P2 ∧ P3. {
            
            intros acc0 H_inv_acc0 H_in_acc0 H_c1_before H_diff_aux_eq0.
    -        have H_in_s1' : (∀ x1 x2 : Z,
    -                            InZ x1 acc0 → InZ x2 s1' → Z.succ x1 < x2). {
    +        have H_in_s1' : (∀ x1 x2 : Z,
    +                            InZ x1 acc0 → InZ x2 s1' → Z.succ x1 < x2). {
              intros x1 x2 H_x1_in_acc0.
              move ⇒ /H_gr_s1'_alt.

    -          eapply Z.le_lt_trans.
    +          eapply Z.le_lt_trans.
              move : H_x1_in_acc0.
    -          rewrite Z.le_succ_l H_in_acc0.
    +          rewrite Z.le_succ_l H_in_acc0.
              tauto.
            }
    -        have H_in_acc0' : (∀ x : Z, InZ x acc0 → x < y2). {
    +        have H_in_acc0' : (∀ x : Z, InZ x acc0 → x < y2). {
              move ⇒ x.
              rewrite H_in_acc0.
              move ⇒ [_] //.
            }
            move : (IH acc0 H_inv_acc0 H_inv_s1' H_in_s1' H_in_acc0' H_c2_neq_0).
    -        rewrite H_diff_aux_eq0 !rev_append_rev.
    +        rewrite H_diff_aux_eq0 !rev_append_rev.
            move ⇒ [H1] [H2] H3.
            split; last split ⇒ //. {
              move ⇒ y.
    @@ -3829,25 +3837,25 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ →.
              rewrite InZ_cons In_elementsZ_single.
              split. {
    -            rewrite H_in_acc0 -(Z.nle_gt y2 y).
    +            rewrite H_in_acc0 -(Z.nle_gt y2 y).
                tauto.
              } {
    -            rewrite H_in_acc0 -(Z.nle_gt y2 y).
    +            rewrite H_in_acc0 -(Z.nle_gt y2 y).
                move ⇒ [] H_in H_nin_i2.
                split; last by assumption.
                move : H_in ⇒ [] H_in; last by right.
                left.
    -            omega.
    +            lia.
              }
            }
          }
    -      move ⇒ {IH} IH.
    +      move ⇒ {} IH.

          clear H_inv_acc H_in_s1 H_in_acc acc.
          move : (interval_compare_elim y1 c1 y2 c2) H_diff_aux_eq.
          unfold diff_aux.
    -      case_eq (interval_compare (y1, c1) (y2, c2)) ⇒ H_comp;
    +      case_eq (interval_compare (y1, c1) (y2, c2)) ⇒ H_comp;
                                                            fold diff_aux. {
            move ⇒ H_lt_y2.
            apply IH. {
    @@ -3859,10 +3867,10 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ []; last done.
              move ⇒ [H_y1_le H_x_lt].
              split; first done.
    -          eapply Z.lt_trans; eauto.
    +          eapply Z.lt_trans; eauto.
            } {
    -          apply Z.le_trans with (m := y2).
    -            - by apply Z.lt_le_incl.
    +          apply Z.le_trans with (m := y2).
    +            - by apply Z.lt_le_incl.
                - by apply Z_le_add_r.
            }
          } {
    @@ -3886,30 +3894,30 @@

    Library MSetsExtra.MSetIntervals

              by apply Z_to_N_minus_neq_0.
            } {
              move ⇒ x.
    -          rewrite InZ_cons In_elementsZ_single Z2N.id; last omega.
    -          replace (y1 + (y2 - y1)) with y2 by omega.
    +          rewrite InZ_cons In_elementsZ_single Z2N.id; last lia.
    +          replace (y1 + (y2 - y1)) with y2 by lia.
              split; last tauto.
              move ⇒ [] //.
              move ⇒ [H_y1_le] H_x_lt.
              repeat split ⇒ //.
    -          apply Z.lt_trans with (m := y2) ⇒ //.
    +          apply Z.lt_trans with (m := y2) ⇒ //.
            } {
    -          by apply Z.lt_le_incl.
    +          by apply Z.lt_le_incl.
            }
          } {
            rewrite /P1 /P2 /P3.
            move ⇒ [H_y2_lt] [H_y1_lt] H_yc1_lt [H_acc'] H_s'.
            clear IH P1 P2 P3 H_comp.
            subst.
    -        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
    -          rewrite Z2N.id; omega.
    +        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
    +          rewrite Z2N.id; lia.
            }
    -        have H_nin_yc2 : ∀ y,
    -            InZ y s1' → ¬ y2 ≤ y < y2 + Z.of_N c2. {
    +        have H_nin_yc2 : ∀ y,
    +            InZ y s1' → ¬ y2 ≤ y < y2 + Z.of_N c2. {
              move ⇒ y /H_gr_s1'_alt H_lt_y.
              move ⇒ [H_y2_le_y].
    -          apply Z.le_ngt, Z.lt_le_incl.
    -          by apply Z.lt_trans with (m := y1 + Z.of_N c1).
    +          apply Z.le_ngt, Z.lt_le_incl.
    +          by apply Z.lt_trans with (m := y1 + Z.of_N c1).
            }
            split; last split. {
              move ⇒ y.
    @@ -3919,10 +3927,10 @@

    Library MSetsExtra.MSetIntervals

                move ⇒ []. {
                  move ⇒ [H_le_y] H_y_lt.
                  split. {
    -                left; omega.
    +                left; lia.
                  } {
                    move ⇒ [_].
    -                by apply Z.nlt_ge.
    +                by apply Z.nlt_ge.
                  }
                } {
                  move : (H_nin_yc2 y). tauto.
    @@ -3930,7 +3938,7 @@

    Library MSetsExtra.MSetIntervals

              } {
                move ⇒ [] []; last by right; right.
                move ⇒ [H_y_ge] H_y_lt_yc1 H_nin_yc2'.
    -            right; left. omega.
    +            right; left. lia.
              }
            } {
              rewrite interval_list_invariant_cons H_yc1_intro.
    @@ -3953,7 +3961,7 @@

    Library MSetsExtra.MSetIntervals

                move ⇒ H_y_in.
                split; first by right.
                move ⇒ [] _.
    -            by apply Z.nlt_ge, Z.lt_le_incl, H_gr_s1'_alt.
    +            by apply Z.nlt_ge, Z.lt_le_incl, H_gr_s1'_alt.
              }
            } {
              apply H_inv_s1'.
    @@ -3967,7 +3975,7 @@

    Library MSetsExtra.MSetIntervals

            } {
              move ⇒ x.
              split; first done.
    -          omega.
    +          lia.
            } {
              assumption.
            }
    @@ -3976,11 +3984,11 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ [H_y1_le] [H_yc2_le_yc1] _ [] H_acc' H_s'.
            clear IH P1 P2 P3 H_comp.
            subst.
    -        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
    -          rewrite Z2N.id; omega.
    +        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
    +          rewrite Z2N.id; lia.
            }
    -        have H_y1_intro : y1 + Z.of_N (Z.to_N (y2 - y1)) = y2. {
    -          rewrite Z2N.id; omega.
    +        have H_y1_intro : y1 + Z.of_N (Z.to_N (y2 - y1)) = y2. {
    +          rewrite Z2N.id; lia.
            }
            split; last split. {
              move ⇒ y.
    @@ -3988,48 +3996,48 @@

    Library MSetsExtra.MSetIntervals

                      !InZ_cons !In_elementsZ_single
                      H_yc1_intro H_y1_intro InZ_nil.
              split. {
    -            rewrite -!or_assoc.
    +            rewrite -!or_assoc.
                move ⇒ [[[]|]|] //. {
                  move ⇒ [H_y1_le_y] H_y_lt.
                  split. {
                    left.
                    split ⇒ //.
    -                apply Z.lt_le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    -                apply Z.lt_trans with (m := y2) ⇒ //.
    +                apply Z.lt_le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    +                apply Z.lt_trans with (m := y2) ⇒ //.
                    by apply Z_lt_add_r.
                  } {
    -                move ⇒ [] /Z.le_ngt //.
    +                move ⇒ [] /Z.le_ngt //.
                  }
                } {
                  move ⇒ [H_y2c_le_y] H_y_lt_yc1.
                  split. {
                    left.
                    split ⇒ //.
    -                apply Z.le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    -                apply Z.le_trans with (m := y2) ⇒ //.
    +                apply Z.le_trans with (m := y2 + Z.of_N c2) ⇒ //.
    +                apply Z.le_trans with (m := y2) ⇒ //.
                    apply Z_le_add_r.
                  } {
    -                move ⇒ [] _ /Z.lt_nge //.
    +                move ⇒ [] _ /Z.lt_nge //.
                  }
                } {
                  move ⇒ H_y_in_s1'.
                  split; first by right.
    -              suff H_suff : y2 + Z.of_N c2 ≤ y. {
    -                move ⇒ [] _ /Z.lt_nge //.
    +              suff H_suff : y2 + Z.of_N c2 ≤ y. {
    +                move ⇒ [] _ /Z.lt_nge //.
                  }
    -              apply Z.le_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -              apply Z.lt_le_incl.
    +              apply Z.le_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +              apply Z.lt_le_incl.
                  by apply H_gr_s1'_alt.
                }
              } {
                move ⇒ [] []; last by tauto.
                move ⇒ [H_y1_le_y] H_y_lt H_neq_y2.
    -            apply not_and in H_neq_y2; last by apply Z.le_decidable.
    +            apply not_and in H_neq_y2; last by apply Z.le_decidable.
                case H_neq_y2. {
    -              move ⇒ /Z.nle_gt H_y_lt'.
    +              move ⇒ /Z.nle_gt H_y_lt'.
                  left; left; done.
                } {
    -              move ⇒ /Z.nlt_ge H_le_y.
    +              move ⇒ /Z.nlt_ge H_le_y.
                  right; left; done.
                }
              }
    @@ -4039,13 +4047,13 @@

    Library MSetsExtra.MSetIntervals

                rewrite H_yc1_intro ⇒ //.
              } {
                rewrite /insert_intervalZ_guarded.
    -            case_eq ((Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2)) =? 0)%N). {
    +            case_eq ((Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2)) =? 0)%N). {
                  rewrite H_y1_intro.
    -              move ⇒ /N.eqb_eq /N2Z.inj_iff.
    -              rewrite Z2N.id; last first. {
    -                by apply Z.le_0_sub.
    +              move ⇒ /N.eqb_eq /N2Z.inj_iff.
    +              rewrite Z2N.id; last first. {
    +                by apply Z.le_0_sub.
                  }
    -              move ⇒ /Zminus_eq H_yc1_eq.
    +              move ⇒ /Zminus_eq H_yc1_eq.
                  eapply interval_list_elements_greater_impl;
                    last apply H_gr_s1'.
                  rewrite H_yc1_eq.
    @@ -4062,7 +4070,7 @@

    Library MSetsExtra.MSetIntervals

              rewrite InZ_insert_intervalZ_guarded InZ_cons In_elementsZ_single H_y1_intro InZ_nil.
              move ⇒ [] //.
              move ⇒ [_] H_x_lt.
    -          apply Z.lt_le_trans with (m := y2) ⇒ //.
    +          apply Z.lt_le_trans with (m := y2) ⇒ //.
              apply Z_le_add_r.
            }
          } {
    @@ -4079,13 +4087,13 @@

    Library MSetsExtra.MSetIntervals

                split; first assumption.
                rewrite In_elementsZ_single.
                move ⇒ [] H_y2_le H_y_lt.
    -            case H_y_in; first by omega.
    +            case H_y_in; first by lia.
                move ⇒ /H_gr_s1'_alt H_lt_y.
    -            suff : y2 + Z.of_N c2 < y. {
    -              move ⇒ ?. omega.
    +            suff : y2 + Z.of_N c2 < y. {
    +              move ⇒ ?. lia.
                }
    -            apply Z.lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    -            apply Z.lt_le_trans with (m := y1) ⇒ //.
    +            apply Z.lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +            apply Z.lt_le_trans with (m := y1) ⇒ //.
                apply Z_le_add_r.
              }
            } {
    @@ -4107,12 +4115,12 @@

    Library MSetsExtra.MSetIntervals

                split; first assumption.
                rewrite In_elementsZ_single.
                move ⇒ [] H_y2_le H_y_lt.
    -            case H_y_in; first by omega.
    +            case H_y_in; first by lia.
                move ⇒ /H_gr_s1'_alt H_lt_y.
    -            suff : y2 + Z.of_N c2 < y. {
    -              move ⇒ ?. omega.
    +            suff : y2 + Z.of_N c2 < y. {
    +              move ⇒ ?. lia.
                }
    -            apply Z.lt_trans with (m := (y2 + Z.of_N c2) + Z.of_N c1) ⇒ //.
    +            apply Z.lt_trans with (m := (y2 + Z.of_N c2) + Z.of_N c1) ⇒ //.
                by apply Z_lt_add_r.
              }
            } {
    @@ -4125,25 +4133,25 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma diff_aux2_props :
    -   âˆ€ (s2 s1 acc : t),
    -    interval_list_invariant (rev_append acc s1) = true →
    -    interval_list_invariant s2 = true →
    -    (∀ x1 x2, InZ x1 acc → InZ x2 s2 → Z.succ x1 < x2) →
    -    ((∀ y, (InZ y (diff_aux2 acc s1 s2) ↔
    -               ((InZ y acc) ∨ (InZ y s1)) ∧ ¬InZ y s2)) ∧
    -    (interval_list_invariant (diff_aux2 acc s1 s2) = true)).
    +  Lemma diff_aux2_props :
    +   âˆ€ (s2 s1 acc : t),
    +    interval_list_invariant (rev_append acc s1) = true →
    +    interval_list_invariant s2 = true →
    +    (∀ x1 x2, InZ x1 acc → InZ x2 s2 → Z.succ x1 < x2) →
    +    ((∀ y, (InZ y (diff_aux2 acc s1 s2) ↔
    +               ((InZ y acc) ∨ (InZ y s1)) ∧ ¬InZ y s2)) ∧
    +    (interval_list_invariant (diff_aux2 acc s1 s2) = true)).
      Proof.
        induction s2 as [| [y2 c2] s2' IH]. {
          move ⇒ s1 acc H_inv_acc_s1 _ _.
          rewrite /diff_aux2.
          replace (match s1 with
    -        | nil ⇒ rev_append acc s1
    -        | _ :: _ ⇒ rev_append acc s1
    -       end) with (rev_append acc s1); last by case s1.
    +        | nil ⇒ rev_append acc s1
    +        | _ :: _ ⇒ rev_append acc s1
    +       end) with (rev_append acc s1); last by case s1.
          split. {
            move ⇒ y.
    -        rewrite rev_append_rev InZ_app InZ_rev InZ_nil.
    +        rewrite rev_append_rev InZ_app InZ_rev InZ_nil.
            tauto.
          } {
            assumption.
    @@ -4158,13 +4166,13 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ H_s1_eq.
            split. {
              move ⇒ y.
    -          rewrite rev_append_rev InZ_app InZ_nil InZ_rev.
    +          rewrite rev_append_rev InZ_app InZ_nil InZ_rev.
              split; last tauto.
              move ⇒ [] // H_y_in.
              split; first by left.
              move ⇒ H_y_in'.
              move : (H_acc_s2 _ _ H_y_in H_y_in').
    -          apply Z.nlt_succ_diag_l.
    +          apply Z.nlt_succ_diag_l.
            } {
              move : H_inv_acc_s1.
               by rewrite H_s1_eq.
    @@ -4175,25 +4183,25 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ acc' s1' H_diff_aux_eq.

    -        have H_acc_lt_y2 : (∀ x : Z, InZ x acc → x < y2). {
    +        have H_acc_lt_y2 : (∀ x : Z, InZ x acc → x < y2). {
              move ⇒ x H_x_in.
    -          have H_y2_in: (InZ y2 ((y2,c2) :: s2')). {
    +          have H_y2_in: (InZ y2 ((y2,c2) :: s2')). {
                rewrite InZ_cons.
                left.
                by apply In_elementsZ_single_hd.
              }
              move : (H_acc_s2 _ _ H_x_in H_y2_in).
    -          apply Z.lt_trans, Z.lt_succ_diag_r.
    +          apply Z.lt_trans, Z.lt_succ_diag_r.
            }

            have [H_inv_acc [H_inv_s1 H_acc_s1]] :
    -          interval_list_invariant (rev acc) = true ∧
    -          interval_list_invariant s1 = true ∧
    -          (∀ x1 x2 : Z,
    -             InZ x1 acc → InZ x2 s1 → Z.succ x1 < x2). {
    +          interval_list_invariant (rev acc) = true ∧
    +          interval_list_invariant s1 = true ∧
    +          (∀ x1 x2 : Z,
    +             InZ x1 acc → InZ x2 s1 → Z.succ x1 < x2). {

              move : H_inv_acc_s1.
    -          rewrite rev_append_rev.
    +          rewrite rev_append_rev.
              move ⇒ /interval_list_invariant_app_elim.
              move ⇒ [?] [?] H_x.
              split; first assumption.
    @@ -4208,11 +4216,11 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ [H_inZ_res] [H_inv_res] H_inZ_acc'.

    -        have H_acc'_s2' : (∀ x1 x2 : Z,
    -                              InZ x1 acc' → InZ x2 s2' → Z.succ x1 < x2). {
    +        have H_acc'_s2' : (∀ x1 x2 : Z,
    +                              InZ x1 acc' → InZ x2 s2' → Z.succ x1 < x2). {
              move ⇒ x1 x2 H_x1_in H_x2_in.
    -          apply Z.le_lt_trans with (m := y2 + Z.of_N c2). {
    -            apply Z.le_succ_l.
    +          apply Z.le_lt_trans with (m := y2 + Z.of_N c2). {
    +            apply Z.le_succ_l.
                by apply H_inZ_acc'.
              } {
                move : H_gr_s2'.
    @@ -4228,7 +4236,7 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ y.
            rewrite H_inZ_diff_res.
            move : (H_inZ_res y).
    -        rewrite !rev_append_rev !InZ_app !InZ_rev InZ_cons.
    +        rewrite !rev_append_rev !InZ_app !InZ_rev InZ_cons.
            move ⇒ →.
            tauto.
          }
    @@ -4236,17 +4244,17 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma diff_InZ :
    -   âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    âˆ€ y, (InZ y (diff s1 s2) ↔ InZ y s1 ∧ ¬InZ y s2).
    +  Lemma diff_InZ :
    +   âˆ€ (s1 s2 : t),
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    âˆ€ y, (InZ y (diff s1 s2) ↔ InZ y s1 ∧ ¬InZ y s2).
      Proof.
        intros s1 s2 H_inv_s1 H_inv_s2 y.
        rewrite /diff.

    -    move : (diff_aux2_props s2 s1 nil).
    +    move : (diff_aux2_props s2 s1 nil).
        move ⇒ [] //.
        move ⇒ H_in_diff _.
        rewrite H_in_diff InZ_nil.
    @@ -4254,24 +4262,24 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma diff_invariant :
    -   âˆ€ (s1 s2 : t),
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    interval_list_invariant (diff s1 s2) = true.
    +  Lemma diff_invariant :
    +   âˆ€ (s1 s2 : t),
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    interval_list_invariant (diff s1 s2) = true.
      Proof.
        intros s1 s2 H_inv_s1 H_inv_s2.
        rewrite /diff.

    -    move : (diff_aux2_props s2 s1 nil).
    +    move : (diff_aux2_props s2 s1 nil).
        move ⇒ [] //.
      Qed.


    -  Global Instance diff_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (diff s1 s2).
    +  Global Instance diff_ok s1 s2 : ∀ `(Ok s1, Ok s2), Ok (diff s1 s2).
      Proof.
        move ⇒ H_ok_s1 H_ok_s2.
        move : (H_ok_s1) (H_ok_s2).
    @@ -4290,9 +4298,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma diff_spec :
    -   âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    -   In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +  Lemma diff_spec :
    +   âˆ€ (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    +   In x (diff s s') ↔ In x s ∧ ¬In x s'.
      Proof.
        intros s s' x H_ok H_ok'.
        rewrite !In_InZ.
    @@ -4307,16 +4315,16 @@

    Library MSetsExtra.MSetIntervals

    -

    remove specification

    +

    remove specification


    -  Lemma removeZ_alt_def : ∀ x s acc,
    -    interval_list_invariant s = true →
    -    removeZ_aux acc x s = match diff_aux x (1%N) acc s with
    -                    (acc', s') ⇒ rev_append acc' s'
    +  Lemma removeZ_alt_def : ∀ x s acc,
    +    interval_list_invariant s = true →
    +    removeZ_aux acc x s = match diff_aux x (1%N) acc s with
    +                    (acc', s') ⇒ rev_append acc' s'
                      end.
      Proof.
        intros y2.
    @@ -4332,33 +4340,33 @@

    Library MSetsExtra.MSetIntervals

          right. done.
        }
        move : (interval_compare_elim y1 c1 y2 (1%N)).
    -    case_eq (interval_compare (y1, c1) (y2, (1%N))) ⇒ H_eq. {
    +    case_eq (interval_compare (y1, c1) (y2, (1%N))) ⇒ H_eq. {
          move ⇒ H_lt_y2 _.
    -      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
    -        by apply Z.nlt_ge, Z.lt_le_incl.
    +      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
    +        by apply Z.nlt_ge, Z.lt_le_incl.
          }
    -      have H_y2_nlt : ~(y2 < y1). {
    +      have H_y2_nlt : ~(y2 < y1). {
            move ⇒ H_y2_y1.
            apply H_yc2_nlt.
    -        apply Z.lt_le_trans with (m := y1) ⇒ //.
    +        apply Z.lt_le_trans with (m := y1) ⇒ //.
            apply Z_le_add_r.
          }
    -      move : (H_y2_nlt) (H_yc2_nlt) ⇒ /Z.ltb_nlt → /Z.ltb_nlt →.
    +      move : (H_y2_nlt) (H_yc2_nlt) ⇒ /Z.ltb_nlt → /Z.ltb_nlt →.
          rewrite IH //.
        } {
          move ⇒ H_lt_y2 _.
    -      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
    -        apply Z.nlt_ge.
    +      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
    +        apply Z.nlt_ge.
            rewrite H_lt_y2.
    -        apply Z.le_refl.
    +        apply Z.le_refl.
          }
    -      have H_y2_nlt : ~(y2 < y1). {
    +      have H_y2_nlt : ~(y2 < y1). {
            move ⇒ H_y2_y1.
            apply H_yc2_nlt.
    -        apply Z.lt_le_trans with (m := y1) ⇒ //.
    +        apply Z.lt_le_trans with (m := y1) ⇒ //.
            apply Z_le_add_r.
          }
    -      move : (H_y2_nlt) (H_yc2_nlt) ⇒ /Z.ltb_nlt → /Z.ltb_nlt →.
    +      move : (H_y2_nlt) (H_yc2_nlt) ⇒ /Z.ltb_nlt → /Z.ltb_nlt →.
          rewrite IH //.
        } {
          done.
    @@ -4368,51 +4376,51 @@

    Library MSetsExtra.MSetIntervals

          move ⇒ [H_y1_eq] H_c1_eq.
          move ⇒ [] // .
          move ⇒ [H_lt_y2] H_y2_lt.
    -      have H_y2_nlt : ~(y2 < y1). {
    -        apply Z.nlt_ge ⇒ //.
    +      have H_y2_nlt : ~(y2 < y1). {
    +        apply Z.nlt_ge ⇒ //.
          }
    -      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    +      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
          rewrite /insert_intervalZ_guarded.

    -      have → : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
    -        rewrite H_y1_eq H_c1_eq Z.add_1_r Z.sub_diag //.
    +      have → : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
    +        rewrite H_y1_eq H_c1_eq Z.add_1_r Z.sub_diag //.
          }

    -      have → : (Z.to_N (y2 - y1) =? 0 = true)%N. {
    -        rewrite H_y1_eq Z.sub_diag //.
    +      have → : (Z.to_N (y2 - y1) =? 0 = true)%N. {
    +        rewrite H_y1_eq Z.sub_diag //.
          }
          done.
        } {
          move ⇒ [H_y2_le] [H_yc1_le] _.
          move ⇒ [] //.
          move ⇒ [H_y1_le] H_y2_lt.
    -      have H_y2_nlt : ~(y2 < y1). {
    -        apply Z.nlt_ge ⇒ //.
    +      have H_y2_nlt : ~(y2 < y1). {
    +        apply Z.nlt_ge ⇒ //.
          }
    -      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    +      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.

    -      have H_y1_eq : (y1 = y2) by omega.
    -      have H_yc1_eq : (y1 + Z.of_N c1 = Z.succ y2). {
    -        apply Z.le_antisymm. {
    +      have H_y1_eq : (y1 = y2) by lia.
    +      have H_yc1_eq : (y1 + Z.of_N c1 = Z.succ y2). {
    +        apply Z.le_antisymm. {
              move : H_yc1_le.
    -          rewrite Z.add_1_r //.
    +          rewrite Z.add_1_r //.
            } {
    -          by apply Z.le_succ_l.
    +          by apply Z.le_succ_l.
            }
          }

          rewrite /insert_intervalZ_guarded.
    -      have → : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
    -        rewrite H_yc1_eq Z.sub_diag //.
    +      have → : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
    +        rewrite H_yc1_eq Z.sub_diag //.
          }

    -      have → : (Z.to_N (y2 - y1) =? 0 = true)%N. {
    -        rewrite H_y1_eq Z.sub_diag //.
    +      have → : (Z.to_N (y2 - y1) =? 0 = true)%N. {
    +        rewrite H_y1_eq Z.sub_diag //.
          }

    -      suff → : diff_aux y2 (1%N) acc s' = (acc, s') by done.
    +      suff → : diff_aux y2 (1%N) acc s' = (acc, s') by done.

          move : H_gr.
    @@ -4423,9 +4431,9 @@

    Library MSetsExtra.MSetIntervals

            rewrite interval_list_elements_greater_cons /=
                    /interval_compare.
            move ⇒ H_lt_y'.
    -        have → : y2 + Z.of_N 1 ?= y' = Lt. {
    -          apply Z.compare_lt_iff.
    -          by rewrite Z.add_1_r.
    +        have → : y2 + Z.of_N 1 ?= y' = Lt. {
    +          apply Z.compare_lt_iff.
    +          by rewrite Z.add_1_r.
            }
            done.
          }
    @@ -4433,67 +4441,67 @@

    Library MSetsExtra.MSetIntervals

          move ⇒ [H_y1_le] [H_yc2_le] _.
          move ⇒ [] //.
          move ⇒ [_] H_y2_lt.
    -      have H_y2_nlt : ~(y2 < y1). {
    -        apply Z.nlt_ge ⇒ //.
    +      have H_y2_nlt : ~(y2 < y1). {
    +        apply Z.nlt_ge ⇒ //.
          }
    -      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    -      rewrite !rev_append_rev /insert_intervalZ_guarded Z.add_1_r.
    -      case ((Z.to_N (y2 - y1) =? 0)%N), (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0)%N. {
    +      move : (H_y2_nlt) (H_y2_lt) ⇒ /Z.ltb_nlt → /Z.ltb_lt →.
    +      rewrite !rev_append_rev /insert_intervalZ_guarded Z.add_1_r.
    +      case ((Z.to_N (y2 - y1) =? 0)%N), (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0)%N. {
            reflexivity.
          } {
    -        rewrite /= -!app_assoc //.
    +        rewrite /= -!app_assoc //.
          } {
            reflexivity.
          } {
    -        rewrite /= -!app_assoc //.
    +        rewrite /= -!app_assoc //.
          }
        } {
          move ⇒ _ H_y2_lt'.
    -      have H_y2_lt : (y2 < y1). {
    -        apply Z.lt_trans with (m := Z.succ y2) ⇒ //.
    -        apply Z.lt_succ_diag_r.
    +      have H_y2_lt : (y2 < y1). {
    +        apply Z.lt_trans with (m := Z.succ y2) ⇒ //.
    +        apply Z.lt_succ_diag_r.
          }
    -      move : (H_y2_lt) ⇒ /Z.ltb_lt → //.
    +      move : (H_y2_lt) ⇒ /Z.ltb_lt → //.
        } {
          move ⇒ _ H_y1_eq.
    -      have H_y2_lt : (y2 < y1). {
    +      have H_y2_lt : (y2 < y1). {
            rewrite H_y1_eq.
    -        apply Z.lt_succ_diag_r.
    +        apply Z.lt_succ_diag_r.
          }
    -      move : (H_y2_lt) ⇒ /Z.ltb_lt → //.
    +      move : (H_y2_lt) ⇒ /Z.ltb_lt → //.
        }
      Qed.

    -  Lemma removeZ_interval_list_invariant : ∀ s x, interval_list_invariant s = true → interval_list_invariant (removeZ x s) = true.
    +  Lemma removeZ_interval_list_invariant : ∀ s x, interval_list_invariant s = true → interval_list_invariant (removeZ x s) = true.
      Proof.
        intros s x H_inv.
        rewrite /removeZ removeZ_alt_def //.
    -    move : (diff_aux_props x (1%N) s nil).
    -    case_eq (diff_aux x 1%N nil s).
    +    move : (diff_aux_props x (1%N) s nil).
    +    case_eq (diff_aux x 1%N nil s).
        move ⇒ acc' s' H_eq [] //.
        move ⇒ _ [] //.
      Qed.

    -  Lemma removeZ_spec :
    -   âˆ€ (s : t) (x y : Z) (Hs : interval_list_invariant s = true),
    -    InZ y (removeZ x s) ↔ InZ y s ∧ ¬Z.eq y x.
    +  Lemma removeZ_spec :
    +   âˆ€ (s : t) (x y : Z) (Hs : interval_list_invariant s = true),
    +    InZ y (removeZ x s) ↔ InZ y s ∧ ¬Z.eq y x.
      Proof.
        intros s x y H_inv.
        rewrite /removeZ removeZ_alt_def //.
    -    move : (diff_aux_props x (1%N) s nil).
    -    case_eq (diff_aux x 1%N nil s).
    +    move : (diff_aux_props x (1%N) s nil).
    +    case_eq (diff_aux x 1%N nil s).
        move ⇒ acc' s' H_eq [] //.
        move ⇒ → _.
    -    rewrite rev_append_rev InZ_app InZ_rev InZ_nil
    +    rewrite rev_append_rev InZ_app InZ_rev InZ_nil
                In_elementsZ_single1.
        split; move ⇒ [H1 H2]; split ⇒ //;
          move ⇒ H3; apply H2; by rewrite H3.
      Qed.

    -  Global Instance remove_ok s x : ∀ `(Ok s), Ok (remove x s).
    +  Global Instance remove_ok s x : ∀ `(Ok s), Ok (remove x s).
      Proof.
        rewrite /Ok /interval_list_invariant /remove.
        move ⇒ [H_is_ok_s H_enc_s].
    @@ -4508,9 +4516,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma remove_spec :
    -   âˆ€ (s : t) (x y : elt) (Hs : Ok s),
    -    In y (remove x s) ↔ In y s ∧ ¬Enc.E.eq y x.
    +  Lemma remove_spec :
    +   âˆ€ (s : t) (x y : elt) (Hs : Ok s),
    +    In y (remove x s) ↔ In y s ∧ ¬Enc.E.eq y x.
      Proof.
        intros s x y Hs.
        have H_rs := (remove_ok s x Hs).
    @@ -4526,13 +4534,13 @@

    Library MSetsExtra.MSetIntervals

    -

    remove_list specification

    +

    remove_list specification


    -  Lemma remove_list_ok : ∀ l s, Ok s → Ok (remove_list l s).
    +  Lemma remove_list_ok : ∀ l s, Ok s → Ok (remove_list l s).
      Proof.
        induction l as [| x l' IH]. {
          done.
    @@ -4544,17 +4552,17 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma remove_list_spec : ∀ x l s, Ok s →
    -     (In x (remove_list l s) ↔ ~(InA Enc.E.eq x l) ∧ In x s).
    +  Lemma remove_list_spec : ∀ x l s, Ok s →
    +     (In x (remove_list l s) ↔ ~(InA Enc.E.eq x l) ∧ In x s).
      Proof.
        move ⇒ x.
        induction l as [| y l' IH]. {
          intros s H.
    -      rewrite /= InA_nil.
    +      rewrite /= InA_nil.
          tauto.
        } {
          move ⇒ s H_ok /=.
    -      rewrite IH remove_spec InA_cons.
    +      rewrite IH remove_spec InA_cons.
          tauto.
        }
      Qed.
    @@ -4565,28 +4573,28 @@

    Library MSetsExtra.MSetIntervals

    -

    subset specification

    +

    subset specification


    -  Lemma subset_flatten_alt_def : ∀ (s1 s2 : t),
    -    subset s1 s2 =
    -    match (s1, s2) with
    -    | (nil, _) ⇒ true
    -    | (_ :: _, nil) ⇒ false
    -    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    -        match (interval_compare (y1, c1) (y2,c2)) with
    -          | ICR_before ⇒ false
    -          | ICR_before_touch ⇒ false
    -          | ICR_after ⇒ subset s1 l2
    -          | ICR_after_touch ⇒ false
    -          | ICR_overlap_before ⇒ false
    -          | ICR_overlap_after ⇒ false
    +  Lemma subset_flatten_alt_def : ∀ (s1 s2 : t),
    +    subset s1 s2 =
    +    match (s1, s2) with
    +    | (nil, _) ⇒ true
    +    | (_ :: _, nil) ⇒ false
    +    | ((y1, c1) :: l1, (y2, c2) :: l2) ⇒
    +        match (interval_compare (y1, c1) (y2,c2)) with
    +          | ICR_before ⇒ false
    +          | ICR_before_touch ⇒ false
    +          | ICR_after ⇒ subset s1 l2
    +          | ICR_after_touch ⇒ false
    +          | ICR_overlap_before ⇒ false
    +          | ICR_overlap_after ⇒ false
              | ICR_equal ⇒ subset l1 l2
    -          | ICR_subsume_1 ⇒ subset l1 s2
    -          | ICR_subsume_2 ⇒ false
    +          | ICR_subsume_1 ⇒ subset l1 s2
    +          | ICR_subsume_2 ⇒ false
            end
        end.
      Proof.
    @@ -4595,11 +4603,11 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma subset_props_aux : ∀ y1 c1 l1 y2 c2 l2,
    -    (∃ y, InZ y ((y1, c1) :: l1) ∧ ¬InZ y ((y2, c2) :: l2)) →
    -    (false = true ↔
    -    (∀ y : Z,
    -        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
    +  Lemma subset_props_aux : ∀ y1 c1 l1 y2 c2 l2,
    +    (∃ y, InZ y ((y1, c1) :: l1) ∧ ¬InZ y ((y2, c2) :: l2)) →
    +    (false = true ↔
    +    (∀ y : Z,
    +        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
      Proof.
        intros y1 c1 l1 y2 c2 l2.
        move ⇒ [y] [H_y_in H_y_nin].
    @@ -4610,13 +4618,13 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma subset_props_aux_before : ∀ y1 c1 l1 y2 c2 l2,
    -    (c1 ≠ 0%N) →
    -    interval_list_invariant ((y2, c2) :: l2) = true →
    -    (y1 < y2) →
    -    (false = true ↔
    -    (∀ y : Z,
    -        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
    +  Lemma subset_props_aux_before : ∀ y1 c1 l1 y2 c2 l2,
    +    (c1 ≠ 0%N) →
    +    interval_list_invariant ((y2, c2) :: l2) = true →
    +    (y1 < y2) →
    +    (false = true ↔
    +    (∀ y : Z,
    +        InZ y ((y1, c1) :: l1) → InZ y ((y2, c2) :: l2))).
      Proof.
        intros y1 c1 l1 y2 c2 l2.
        rewrite interval_list_invariant_cons.
    @@ -4629,14 +4637,14 @@

    Library MSetsExtra.MSetIntervals

          by apply In_elementsZ_single_hd.
        } {
          rewrite InZ_cons.
    -      suff : ¬ (List.In y1 (elementsZ_single y2 c2)) ∧ ¬ InZ y1 l2 by tauto.
    +      suff : ¬ (List.In y1 (elementsZ_single y2 c2)) ∧ ¬ InZ y1 l2 by tauto.
          split. {
            rewrite In_elementsZ_single.
    -        move ⇒ [] /Z.le_ngt //.
    +        move ⇒ [] /Z.le_ngt //.
          } {
            eapply Nin_elements_greater; eauto.
    -        apply Z.le_trans with (m := y2). {
    -          by apply Z.lt_le_incl.
    +        apply Z.le_trans with (m := y2). {
    +          by apply Z.lt_le_incl.
            } {
              apply Z_le_add_r.
            }
    @@ -4647,11 +4655,11 @@

    Library MSetsExtra.MSetIntervals



    -  Lemma subset_props : ∀ s1 s2 : t,
    -    interval_list_invariant s1 = true →
    -    interval_list_invariant s2 = true →
    -    (subset s1 s2 = true ↔
    -     (∀ y, InZ y s1 → InZ y s2)).
    +  Lemma subset_props : ∀ s1 s2 : t,
    +    interval_list_invariant s1 = true →
    +    interval_list_invariant s2 = true →
    +    (subset s1 s2 = true ↔
    +     (∀ y, InZ y s1 → InZ y s2)).
      Proof.
        induction s1 as [| [y1 c1] l1 IH1]. {
          move ⇒ s2 _ _.
    @@ -4664,7 +4672,7 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ [_] [H_c1_neq_0] _ _.
            split ⇒ //.
            move ⇒ H; move : (H y1).
    -        rewrite InZ_nil ⇒ {H} H.
    +        rewrite InZ_nil ⇒ {} H.
            contradict H.
            rewrite InZ_cons; left.
            by apply In_elementsZ_single_hd.
    @@ -4674,32 +4682,32 @@

    Library MSetsExtra.MSetIntervals

            rewrite !interval_list_invariant_cons.
            move ⇒ [H_gr_l1] [H_c1_neq_0] H_inv_l1.
            move ⇒ [H_gr_l2] [H_c2_neq_0] H_inv_l2.
    -        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {IH2} IH2.
    -        have : ∀ s2 : t,
    -          interval_list_invariant s2 = true →
    -          (subset l1 s2 = true ↔
    -          (∀ y : Z, InZ y l1 → InZ y s2)). {
    +        move : (IH2 H_inv_s1 H_inv_l2) ⇒ {} IH2.
    +        have : ∀ s2 : t,
    +          interval_list_invariant s2 = true →
    +          (subset l1 s2 = true ↔
    +          (∀ y : Z, InZ y l1 → InZ y s2)). {
              intros. by apply IH1.
            }
    -        move ⇒ {IH1} IH1.
    +        move ⇒ {} IH1.

    -        have H_yc2_nin : ¬ InZ (y2 + Z.of_N c2) ((y2, c2) :: l2). {
    +        have H_yc2_nin : ¬ InZ (y2 + Z.of_N c2) ((y2, c2) :: l2). {
              rewrite !InZ_cons !In_elementsZ_single.
              move ⇒ []. {
    -            move ⇒ [_] /Z.lt_irrefl //.
    +            move ⇒ [_] /Z.lt_irrefl //.
              } {
                eapply Nin_elements_greater; eauto.
    -            apply Z.le_refl.
    +            apply Z.le_refl.
              }
            }
              
            rewrite subset_flatten_alt_def.
            move : (interval_compare_elim y1 c1 y2 c2).
    -        case (interval_compare (y1, c1) (y2, c2)). {
    +        case (interval_compare (y1, c1) (y2, c2)). {
              move ⇒ H_lt_y2.
              apply subset_props_aux_before ⇒ //.
    -          apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
    +          apply Z.le_lt_trans with (m := y1 + Z.of_N c1) ⇒ //.
              apply Z_le_add_r.
            } {
              move ⇒ H_y2_eq.
    @@ -4712,12 +4720,12 @@

    Library MSetsExtra.MSetIntervals

            } {
              move ⇒ [H_y2_lt] [H_y1_lt] H_yc2_lt.
              apply subset_props_aux.
    -          âˆƒ (y2 + Z.of_N c2).
    +          âˆƒ (y2 + Z.of_N c2).
              split ⇒ //.
              rewrite !InZ_cons !In_elementsZ_single.
              left.
              split ⇒ //.
    -          by apply Z.lt_le_incl.
    +          by apply Z.lt_le_incl.
            } {
              move ⇒ [H_y1_eq] H_c1_eq; subst.
              rewrite IH1 ⇒ //.
    @@ -4726,14 +4734,14 @@

    Library MSetsExtra.MSetIntervals

                tauto.
              } {
                move ⇒ H_pre H_y_in_l1.
    -            suff : ~(List.In y (elementsZ_single y2 c2)). {
    +            suff : ~(List.In y (elementsZ_single y2 c2)). {
                  tauto.
                }
                move : H_gr_l1.
                rewrite interval_list_elements_greater_alt2_def
                  // In_elementsZ_single.
                move ⇒ H; move : (H y H_y_in_l1) ⇒ {H}.
    -            move ⇒ /Z.lt_ngt H_neq [_] //.
    +            move ⇒ /Z.lt_ngt H_neq [_] //.
              }
            } {
              move ⇒ [H_y2_lt_y1] [H_yc1_le] _.
    @@ -4744,7 +4752,7 @@

    Library MSetsExtra.MSetIntervals

                rewrite !In_elementsZ_single.
                move ⇒ [H_y1_le] H_y_lt.
                left.
    -            omega.
    +            lia.
              } {
                move ⇒ H_pre H_y_in_l1.
                apply H_pre.
    @@ -4758,12 +4766,12 @@

    Library MSetsExtra.MSetIntervals

              } {
                move ⇒ H_yc2_lt.
                apply subset_props_aux.
    -            âˆƒ (y2 + Z.of_N c2).
    +            âˆƒ (y2 + Z.of_N c2).
                split ⇒ //.
                rewrite !InZ_cons !In_elementsZ_single.
                left.
                split ⇒ //.
    -            apply Z.le_trans with (m := y2) ⇒ //.
    +            apply Z.le_trans with (m := y2) ⇒ //.
                apply Z_le_add_r.
              }
            } {
    @@ -4775,20 +4783,20 @@

    Library MSetsExtra.MSetIntervals

              } {
                rewrite !In_elementsZ_single.
                move ⇒ H_pre H_y_in.
    -            suff : ~(y2 ≤ y < y2 + Z.of_N c2). {
    +            suff : ~(y2 ≤ y < y2 + Z.of_N c2). {
                  tauto.
                }
                move ⇒ [H_y2_le H_y_lt].
                move : H_y_in ⇒ []. {
                  move ⇒ [H_y1_le] H_y_lt'.
    -              omega.
    +              lia.
                } {
                  eapply Nin_elements_greater; eauto.
    -              apply Z.le_trans with (m := y1); last first. {
    +              apply Z.le_trans with (m := y1); last first. {
                    apply Z_le_add_r.
                  }
    -              apply Z.lt_le_incl.
    -              apply Z.lt_trans with (m := y2 + Z.of_N c2) ⇒ //.
    +              apply Z.lt_le_incl.
    +              apply Z.lt_trans with (m := y2 + Z.of_N c2) ⇒ //.
                }
              }
            } {
    @@ -4802,11 +4810,11 @@

    Library MSetsExtra.MSetIntervals

              } {
                rewrite !In_elementsZ_single H_y1_eq.
                move ⇒ []. {
    -              move ⇒ [_] /Z.lt_irrefl //.
    +              move ⇒ [_] /Z.lt_irrefl //.
                } {
                  eapply Nin_elements_greater; eauto.
                  rewrite H_y1_eq.
    -              apply Z.le_refl.
    +              apply Z.le_refl.
                }
              }
            }
    @@ -4815,9 +4823,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma subset_spec :
    -   âˆ€ (s s' : t) (Hs : Ok s) (Hs' : Ok s'),
    -   subset s s' = true ↔ Subset s s'.
    +  Lemma subset_spec :
    +   âˆ€ (s s' : t) (Hs : Ok s) (Hs' : Ok s'),
    +   subset s s' = true ↔ Subset s s'.
      Proof.
        intros s s' Hs Hs'.
        move : (Hs) (Hs').
    @@ -4841,37 +4849,37 @@

    Library MSetsExtra.MSetIntervals

    -

    elements and elementsZ specification

    +

    elements and elementsZ specification


    -  Lemma elements_spec1 : ∀ (s : t) (x : elt) (Hs : Ok s), List.In x (elements s) ↔ In x s.
    +  Lemma elements_spec1 : ∀ (s : t) (x : elt) (Hs : Ok s), List.In x (elements s) ↔ In x s.
      Proof.
        intros s x Hs.
        by rewrite In_alt_def.
      Qed.

    -  Lemma NoDupA_elementsZ_single: ∀ c x,
    -    NoDupA Z.eq (elementsZ_single x c).
    +  Lemma NoDupA_elementsZ_single: ∀ c x,
    +    NoDupA Z.eq (elementsZ_single x c).
      Proof.
    -    induction c as [| c' IH] using N.peano_ind. {
    +    induction c as [| c' IH] using N.peano_ind. {
          intros x.
          rewrite elementsZ_single_base //.
        } {
          intros x.
          rewrite elementsZ_single_succ.
    -      apply NoDupA_app. {
    -        apply Z.eq_equiv.
    +      apply NoDupA_app. {
    +        apply Z.eq_equiv.
          } {
            apply IH.
          } {
    -        apply NoDupA_singleton.
    +        apply NoDupA_singleton.
          } {
            move ⇒ y.
    -        rewrite !InA_alt.
    +        rewrite !InA_alt.
            move ⇒ [_] [<-] H_y_in.
            move ⇒ [_] [<-] H_y_in'.
            move : H_y_in H_y_in'.
    @@ -4879,44 +4887,44 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ [H_x_le] H_y_lt [] // H_y_eq.
            contradict H_y_lt.
            rewrite H_y_eq.
    -        apply Z.lt_irrefl.
    +        apply Z.lt_irrefl.
          }
        }
      Qed.

    -  Lemma elementsZ_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Z.eq (elementsZ s).
    +  Lemma elementsZ_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Z.eq (elementsZ s).
      Proof.
        induction s as [| [x c] s' IH]. {
          move ⇒ _.
          rewrite elementsZ_nil.
    -      apply NoDupA_nil.
    +      apply NoDupA_nil.
        } {
          move ⇒ H_ok_s.
          move : (H_ok_s) ⇒ /Ok_cons [H_interval_list_elements_greater] [H_c] [H_enc] H_s'.
          rewrite elementsZ_cons.
    -      apply NoDupA_app. {
    -        apply Z.eq_equiv.
    +      apply NoDupA_app. {
    +        apply Z.eq_equiv.
          } {
            by apply IH.
          } {
    -        apply NoDupA_rev. {
    -          apply Z.eq_equiv.
    +        apply NoDupA_rev. {
    +          apply Z.eq_equiv.
            } {
              apply NoDupA_elementsZ_single.
            }
          } {
            move ⇒ y.
    -        rewrite !InA_alt.
    +        rewrite !InA_alt.
            move ⇒ [_] [<-] H_y_in.
            move ⇒ [_] [<-] H_y_in'.
            move : H_y_in'.
    -        rewrite -in_rev In_elementsZ_single /=.
    +        rewrite -in_rev In_elementsZ_single /=.
            move ⇒ [H_x_le] H_y_lt.
    -        eapply (Nin_elements_greater s' (x + Z.of_N c)) ⇒ //. {
    +        eapply (Nin_elements_greater s' (x + Z.of_N c)) ⇒ //. {
              apply H_s'.
            } {
    -          apply Z.lt_le_incl, H_y_lt.
    +          apply Z.lt_le_incl, H_y_lt.
            } {
              apply H_y_in.
            }
    @@ -4925,11 +4933,11 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma elements_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Enc.E.eq (elements s).
    +  Lemma elements_spec2w : ∀ (s : t) (Hs : Ok s), NoDupA Enc.E.eq (elements s).
      Proof.
        intros s Hs.
        rewrite /elements rev_map_alt_def.
    -    apply NoDupA_rev. {
    +    apply NoDupA_rev. {
          apply Enc.E.eq_equiv.
        } {
          eapply NoDupA_map; first by apply elementsZ_spec2w.
    @@ -4950,21 +4958,21 @@

    Library MSetsExtra.MSetIntervals

    -

    equal specification

    +

    equal specification


    -  Lemma equal_alt_def : ∀ s1 s2,
    -    equal s1 s2 = true ↔ (s1 = s2).
    +  Lemma equal_alt_def : ∀ s1 s2,
    +    equal s1 s2 = true ↔ (s1 = s2).
      Proof.
        induction s1 as [| [x cx] xs IH]. {
          move ⇒ [] //.
        } {
          move ⇒ [] //=.
          move ⇒ [y cy] ys.
    -      rewrite !andb_true_iff IH N.eqb_eq Z.eqb_eq.
    +      rewrite !andb_true_iff IH N.eqb_eq Z.eqb_eq.
          split. {
            move ⇒ [->] [->] → //.
          } {
    @@ -4974,18 +4982,18 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma equal_elementsZ :
    -    âˆ€ (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
    -    (∀ x, (InZ x s ↔ InZ x s')) → (s = s').
    +  Lemma equal_elementsZ :
    +    âˆ€ (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
    +    (∀ x, (InZ x s ↔ InZ x s')) → (s = s').
      Proof.
        intros s s'.
        move ⇒ H_ok_s H_ok_s' H_InZ_eq.
    -    have [] : ((subset s s' = true) ∧ (subset s' s = true)). {
    +    have [] : ((subset s s' = true) ∧ (subset s' s = true)). {
          rewrite !subset_spec /Subset.
          split ⇒ x; rewrite !In_InZ H_InZ_eq //.
        }
    -    have : interval_list_invariant s' = true by apply H_ok_s'.
    -    have : interval_list_invariant s = true by apply H_ok_s.
    +    have : interval_list_invariant s' = true by apply H_ok_s'.
    +    have : interval_list_invariant s = true by apply H_ok_s.
        clear H_ok_s H_ok_s' H_InZ_eq.
        move : s s'.
        induction s as [| [x1 c1] s1 IH];
    @@ -4994,21 +5002,21 @@

    Library MSetsExtra.MSetIntervals

        move ⇒ [H_gr_s1] [H_c1_neq_0] H_inv_s1.
        move ⇒ [H_gr_s2] [H_c2_neq_0] H_inv_s2.
        rewrite subset_flatten_alt_def
    -            (subset_flatten_alt_def ((x2, c2)::s2)).
    +            (subset_flatten_alt_def ((x2, c2)::s2)).
        rewrite (interval_compare_swap x1 c1); last by left.
        move : (interval_compare_elim x1 c1 x2 c2).
    -    case (interval_compare (x1, c1) (x2, c2)) ⇒ //.
    +    case (interval_compare (x1, c1) (x2, c2)) ⇒ //.
        move ⇒ [->] → H_sub_s12 H_sub_s21.

    -    suff → : s1 = s2 by done.
    +    suff → : s1 = s2 by done.
        by apply IH.
      Qed.

    -  Lemma equal_spec :
    -    âˆ€ (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
    -    equal s s' = true ↔ Equal s s'.
    +  Lemma equal_spec :
    +    âˆ€ (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
    +    equal s s' = true ↔ Equal s s'.
      Proof.
        intros s s' Hs Hs'.
        rewrite equal_alt_def /Equal.
    @@ -5021,7 +5029,7 @@

    Library MSetsExtra.MSetIntervals

          rewrite !In_InZ.

    -      suff H_ex : (∀ s'', Ok s'' → InZ x s'' → (∃ z, Enc.encode z = x)). {
    +      suff H_ex : (∀ s'', Ok s'' → InZ x s'' → (∃ z, Enc.encode z = x)). {
            move ⇒ HH.
            split. {
              move ⇒ H3.
    @@ -5049,125 +5057,125 @@

    Library MSetsExtra.MSetIntervals

    -

    compare

    +

    compare

    -  Definition lt (s1 s2 : t) : Prop := (compare s1 s2 = Lt).
    +  Definition lt (s1 s2 : t) : Prop := (compare s1 s2 = Lt).

    -  Lemma compare_eq_Eq : ∀ s1 s2,
    -    (compare s1 s2 = Eq ↔ equal s1 s2 = true).
    +  Lemma compare_eq_Eq : ∀ s1 s2,
    +    (compare s1 s2 = Eq ↔ equal s1 s2 = true).
      Proof.
        induction s1 as [| [y1 c1] s1' IH];
          case s2 as [| [y2 c2] s2'] ⇒ //.
    -    rewrite /= !andb_true_iff -IH Z.eqb_eq N.eqb_eq.
    -    move : (Z.compare_eq_iff y1 y2).
    -    case (Z.compare y1 y2). {
    +    rewrite /= !andb_true_iff -IH Z.eqb_eq N.eqb_eq.
    +    move : (Z.compare_eq_iff y1 y2).
    +    case (Z.compare y1 y2). {
          move ⇒ H.
    -      have → : y1 = y2. by apply H.
    +      have → : y1 = y2. by apply H.
          clear H.

    -      move : (N.compare_eq_iff c1 c2).
    -      case (N.compare c1 c2). {
    +      move : (N.compare_eq_iff c1 c2).
    +      case (N.compare c1 c2). {
            move ⇒ H.
    -        have → : c1 = c2. by apply H.
    +        have → : c1 = c2. by apply H.
            tauto.
          } {
            move ⇒ H.
    -        have H_neq : ~(c1 = c2). by rewrite -H ⇒ {H}.
    +        have H_neq : ~(c1 = c2). by rewrite -H ⇒ {H}.
            tauto.
          } {
            move ⇒ H.
    -        have H_neq : ~(c1 = c2). by rewrite -H ⇒ {H}.
    +        have H_neq : ~(c1 = c2). by rewrite -H ⇒ {H}.
            tauto.
          }
        } {
          move ⇒ H.
    -      have H_neq : ~(y1 = y2). by rewrite -H ⇒ {H}.
    +      have H_neq : ~(y1 = y2). by rewrite -H ⇒ {H}.
          tauto.
        } {
          move ⇒ H.
    -      have H_neq : ~(y1 = y2). by rewrite -H ⇒ {H}.
    +      have H_neq : ~(y1 = y2). by rewrite -H ⇒ {H}.
          tauto.
        }
      Qed.

    -  Lemma compare_eq_Lt_nil_l : ∀ s,
    -    compare nil s = Lt ↔ s ≠ nil.
    +  Lemma compare_eq_Lt_nil_l : ∀ s,
    +    compare nil s = Lt ↔ s ≠ nil.
      Proof.
        intros s.
        case s ⇒ //=.
      Qed.

    -  Lemma compare_eq_Lt_nil_r : ∀ s,
    -    ~(compare s nil = Lt).
    +  Lemma compare_eq_Lt_nil_r : ∀ s,
    +    ~(compare s nil = Lt).
      Proof.
        intros s.
        case s as [| [y1 c1] s'] ⇒ //=.
      Qed.

    -  Lemma compare_eq_Lt_cons : ∀ y1 y2 c1 c2 s1 s2,
    -    compare ((y1, c1)::s1) ((y2, c2)::s2) = Lt ↔
    -    (y1 < y2) ∨ ((y1 = y2) ∧ (c1 < c2)%N) ∨
    -    ((y1 = y2) ∧ (c1 = c2) ∧ compare s1 s2 = Lt).
    +  Lemma compare_eq_Lt_cons : ∀ y1 y2 c1 c2 s1 s2,
    +    compare ((y1, c1)::s1) ((y2, c2)::s2) = Lt ↔
    +    (y1 < y2) ∨ ((y1 = y2) ∧ (c1 < c2)%N) ∨
    +    ((y1 = y2) ∧ (c1 = c2) ∧ compare s1 s2 = Lt).
      Proof.
        intros y1 y2 c1 c2 s1 s2.
        rewrite /=.
    -    case_eq (Z.compare y1 y2). {
    -      move ⇒ /Z.compare_eq_iff →.
    -      case_eq (N.compare c1 c2). {
    -        move ⇒ /N.compare_eq_iff →.
    +    case_eq (Z.compare y1 y2). {
    +      move ⇒ /Z.compare_eq_iff →.
    +      case_eq (N.compare c1 c2). {
    +        move ⇒ /N.compare_eq_iff →.
            split. {
              move ⇒ H.
              right; right.
              done.
            } {
              move ⇒ [| []]. {
    -            move ⇒ /Z.lt_irrefl //.
    +            move ⇒ /Z.lt_irrefl //.
              } {
    -            move ⇒ [_] /N.lt_irrefl //.
    +            move ⇒ [_] /N.lt_irrefl //.
              } {
                move ⇒ [_] [_] → //.
              }
            }
          } {
    -        move ⇒ /N.compare_lt_iff H_c1_lt.
    +        move ⇒ /N.compare_lt_iff H_c1_lt.
            split ⇒ //.
            move ⇒ _.
            right; left. done.
          } {
    -        move ⇒ /N.compare_gt_iff H_c2_lt.
    +        move ⇒ /N.compare_gt_iff H_c2_lt.
            split ⇒ //.
            move ⇒ [| []]. {
    -          move ⇒ /Z.lt_irrefl //.
    +          move ⇒ /Z.lt_irrefl //.
            } {
    -          move ⇒ [_] /N.lt_asymm //.
    +          move ⇒ [_] /N.lt_asymm //.
            } {
              move ⇒ [_] [] H_c1_eq.
              contradict H_c2_lt.
              subst c1.
    -          by apply N.lt_irrefl.
    +          by apply N.lt_irrefl.
            }
          }
        } {
    -      move ⇒ /Z.compare_lt_iff.
    +      move ⇒ /Z.compare_lt_iff.
          tauto.
        } {
    -      move ⇒ /Z.compare_gt_iff H_y2_lt.
    +      move ⇒ /Z.compare_gt_iff H_y2_lt.
          split ⇒ //.
          move ⇒ [| []]. {
    -        move ⇒ /Z.lt_asymm //.
    +        move ⇒ /Z.lt_asymm //.
          } {
            move ⇒ [] H_y1_eq.
    -        exfalso. omega.
    +        exfalso. lia.
          } {
            move ⇒ [] H_y1_eq.
    -        exfalso. omega.
    +        exfalso. lia.
          }
        }
      Qed.
    @@ -5175,47 +5183,47 @@

    Library MSetsExtra.MSetIntervals



    -  Lemma compare_antisym: ∀ (s1 s2 : t),
    -    (compare s1 s2) = CompOpp (compare s2 s1).
    +  Lemma compare_antisym: ∀ (s1 s2 : t),
    +    (compare s1 s2) = CompOpp (compare s2 s1).
      Proof.
        induction s1 as [| [y1 c1] s1' IH];
          case s2 as [| [y2 c2] s2'] ⇒ //.
    -    rewrite /= (Z.compare_antisym y1 y2) (N.compare_antisym c1 c2).
    -    case (Z.compare y1 y2) ⇒ //=.
    -    case (N.compare c1 c2) ⇒ //=.
    +    rewrite /= (Z.compare_antisym y1 y2) (N.compare_antisym c1 c2).
    +    case (Z.compare y1 y2) ⇒ //=.
    +    case (N.compare c1 c2) ⇒ //=.
      Qed.

    -  Lemma compare_spec : ∀ s1 s2,
    -    CompSpec eq lt s1 s2 (compare s1 s2).
    +  Lemma compare_spec : ∀ s1 s2,
    +    CompSpec eq lt s1 s2 (compare s1 s2).
      Proof.
        intros s1 s2.
    -    rewrite /CompSpec /lt (compare_antisym s2 s1).
    +    rewrite /CompSpec /lt (compare_antisym s2 s1).
        case_eq (compare s1 s2). {
          rewrite compare_eq_Eq equal_alt_def ⇒ →.
    -      by apply CompEq.
    +      by apply CompEq.
        } {
          move ⇒ _.
    -      by apply CompLt.
    +      by apply CompLt.
        } {
          move ⇒ _.
    -      by apply CompGt.
    +      by apply CompGt.
        }
      Qed.

    -  Lemma lt_Irreflexive : Irreflexive lt.
    +  Lemma lt_Irreflexive : Irreflexive lt.
      Proof.
    -    rewrite /Irreflexive /Reflexive /complement /lt.
    +    rewrite /Irreflexive /Reflexive /complement /lt.
        intros x.
    -    suff → : compare x x = Eq by done.
    +    suff → : compare x x = Eq by done.
        rewrite compare_eq_Eq equal_alt_def //.
      Qed.

    -  Lemma lt_Transitive : Transitive lt.
    +  Lemma lt_Transitive : Transitive lt.
      Proof.
    -    rewrite /Transitive /lt.
    +    rewrite /Transitive /lt.
        induction x as [| [y1 c1] s1' IH];
          case y as [| [y2 c2] s2'];
          case z as [| [y3 c3] s3'] ⇒ //.
    @@ -5225,7 +5233,7 @@

    Library MSetsExtra.MSetIntervals

        move ⇒ [H_y1_lt | [[->] H_c1_lt | [->] [->] H_comp]]
    [H_y2_lt | [[<-] H_c2_lt | [<-] [<-] H_comp']]. {
          left.
    -      by apply Z.lt_trans with (m := y2).
    +      by apply Z.lt_trans with (m := y2).
        } {
          by left.
        } {
    @@ -5235,7 +5243,7 @@

    Library MSetsExtra.MSetIntervals

        } {
          right; left.
          split ⇒ //.
    -      by apply N.lt_trans with (m := c2).
    +      by apply N.lt_trans with (m := c2).
        } {
          by right; left.
        } {
    @@ -5254,65 +5262,65 @@

    Library MSetsExtra.MSetIntervals

    -

    elements is sorted

    +

    elements is sorted


    -  Lemma elementsZ_single_sorted : ∀ c x,
    -    sort Z.lt (elementsZ_single x c).
    +  Lemma elementsZ_single_sorted : ∀ c x,
    +    sort Z.lt (elementsZ_single x c).
      Proof.
    -    induction c as [| c' IH] using N.peano_ind. {
    +    induction c as [| c' IH] using N.peano_ind. {
          intro x.
          rewrite elementsZ_single_base.
    -      apply Sorted_nil.
    +      apply Sorted_nil.
        } {
          intro x.
          rewrite elementsZ_single_succ_front.
    -      apply Sorted_cons. {
    +      apply Sorted_cons. {
            apply IH.
          } {
    -        case (N.zero_or_succ c'). {
    +        case (N.zero_or_succ c'). {
              move ⇒ →.
              rewrite elementsZ_single_base //.
            } {
              move ⇒ [c''] →.
              rewrite elementsZ_single_succ_front.
              constructor.
    -          apply Z.lt_succ_diag_r.
    +          apply Z.lt_succ_diag_r.
            }
          }
        }
      Qed.

    -  Lemma elementsZ_sorted : ∀ s,
    -    interval_list_invariant s = true →
    -    sort Z.lt (rev (elementsZ s)).
    +  Lemma elementsZ_sorted : ∀ s,
    +    interval_list_invariant s = true →
    +    sort Z.lt (rev (elementsZ s)).
      Proof.
        induction s as [| [y c] s' IH]. {
          move ⇒ _.
          rewrite elementsZ_nil.
    -      apply Sorted_nil.
    +      apply Sorted_nil.
        } {
          rewrite interval_list_invariant_cons elementsZ_cons
    -              rev_app_distr rev_involutive.
    +              rev_app_distr rev_involutive.
          move ⇒ [H_gr] [H_c_neq_0] H_inv_s'.
    -      apply SortA_app with (eqA := Logic.eq). {
    -        apply eq_equivalence.
    +      apply SortA_app with (eqA := Logic.eq). {
    +        apply eq_equivalence.
          } {
            apply elementsZ_single_sorted.
          } {
            by apply IH.
          } {
            intros x1 x2.
    -        move ⇒ /InA_alt [_] [<-] /In_elementsZ_single [_ H_x1_lt].
    -        move ⇒ /InA_alt [_] [<-].
    -        rewrite -In_rev ⇒ H_x2_in.
    +        move ⇒ /InA_alt [_] [<-] /In_elementsZ_single [_ H_x1_lt].
    +        move ⇒ /InA_alt [_] [<-].
    +        rewrite -In_rev ⇒ H_x2_in.

    -        apply Z.lt_trans with (m := (y + Z.of_N c)) ⇒ //.
    +        apply Z.lt_trans with (m := (y + Z.of_N c)) ⇒ //.
            eapply interval_list_elements_greater_alt2_def;
              eauto.
          }
    @@ -5322,43 +5330,43 @@

    Library MSetsExtra.MSetIntervals



    -  Lemma elements_sorted : ∀ s,
    -    Ok s →
    -    sort Enc.E.lt (elements s).
    +  Lemma elements_sorted : ∀ s,
    +    Ok s →
    +    sort Enc.E.lt (elements s).
      Proof.
        move ⇒ s [H_inv] H_enc.
    -    rewrite /elements rev_map_alt_def -map_rev.
    -    have : (∀ x : Z, List.In x (rev (elementsZ s)) →
    -           âˆƒ e : Enc.E.t, Enc.encode e = x). {
    +    rewrite /elements rev_map_alt_def -map_rev.
    +    have : (∀ x : Z, List.In x (rev (elementsZ s)) →
    +           âˆƒ e : Enc.E.t, Enc.encode e = x). {
          move ⇒ x.
          move : (H_enc x).
    -      rewrite In_rev //.
    +      rewrite In_rev //.
        }
        move : (elementsZ_sorted s H_inv) ⇒ {H_enc}.
    -    generalize (rev (elementsZ s)).
    +    generalize (rev (elementsZ s)).
        induction l as [| x xs IH]. {
          rewrite /= ⇒ _ _.
    -      apply Sorted_nil.
    +      apply Sorted_nil.
        } {
          move ⇒ H_sort H_enc.
    -      apply Sorted_inv in H_sort as [H_sort H_hd_rel].
    +      apply Sorted_inv in H_sort as [H_sort H_hd_rel].
          simpl.
    -      apply Sorted_cons. {
    +      apply Sorted_cons. {
            apply IH ⇒ //.
            move ⇒ xx H_xx_in.
            apply H_enc.
    -        by apply in_cons.
    +        by apply in_cons.
          } {
            move : H_hd_rel H_enc.
            case xs ⇒ //=.
            move ⇒ x' xs' H_hd_rel H_enc.
    -        apply HdRel_inv in H_hd_rel.
    -        apply HdRel_cons.
    +        apply HdRel_inv in H_hd_rel.
    +        apply HdRel_cons.
            rewrite -Enc.encode_lt.
    -        have [y H_y] : (∃ y, Enc.encode y = x). {
    +        have [y H_y] : (∃ y, Enc.encode y = x). {
              apply H_enc. by left.
            }
    -        have [y' H_y'] : (∃ y', Enc.encode y' = x'). {
    +        have [y' H_y'] : (∃ y', Enc.encode y' = x'). {
              apply H_enc. by right; left.
            }
            move : H_hd_rel.
    @@ -5373,16 +5381,16 @@

    Library MSetsExtra.MSetIntervals

    -

    choose specification

    +

    choose specification


    -  Definition min_eltZ_spec1 :
    -    âˆ€ (s : t) (x : Z),
    -      interval_list_invariant s = true →
    -      min_eltZ s = Some x → InZ x s.
    +  Definition min_eltZ_spec1 :
    +    âˆ€ (s : t) (x : Z),
    +      interval_list_invariant s = true →
    +      min_eltZ s = Some x → InZ x s.
      Proof.
        intros s x.
        case s as [| [x' c] s']. {
    @@ -5396,28 +5404,28 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma min_eltZ_spec2 :
    -    âˆ€ (s : t) (x y : Z) (Hs : Ok s),
    -    min_eltZ s = Some x → InZ y s → ¬ Z.lt y x.
    +  Lemma min_eltZ_spec2 :
    +    âˆ€ (s : t) (x y : Z) (Hs : Ok s),
    +    min_eltZ s = Some x → InZ y s → ¬ Z.lt y x.
      Proof.
        intros s x y H_ok H_min H_in H_y_lt_x.
    -    eapply (Nin_elements_greater s (Z.pred x)) ⇒ //; last apply H_in. {
    +    eapply (Nin_elements_greater s (Z.pred x)) ⇒ //; last apply H_in. {
          move : H_ok H_min.
          case s ⇒ //.
          move ⇒ [z c] s' _ [<-].
          rewrite interval_list_elements_greater_cons.
    -      apply Z.lt_pred_l.
    +      apply Z.lt_pred_l.
        } {
          apply H_ok.
        } {
    -      by apply Z.lt_le_pred.
    +      by apply Z.lt_le_pred.
        }
      Qed.

    -  Definition min_eltZ_spec3 :
    -    âˆ€ (s : t),
    -      min_eltZ s = None → ∀ x, ¬InZ x s.
    +  Definition min_eltZ_spec3 :
    +    âˆ€ (s : t),
    +      min_eltZ s = None → ∀ x, ¬InZ x s.
      Proof.
        intros s.
        case s as [| [x' c] s'];
    @@ -5426,8 +5434,8 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Definition min_elt_spec1 :
    -    âˆ€ (s : t) (x : elt) (Hs : Ok s), min_elt s = Some x → In x s.
    +  Definition min_elt_spec1 :
    +    âˆ€ (s : t) (x : elt) (Hs : Ok s), min_elt s = Some x → In x s.
      Proof.
        rewrite /min_elt.
        move ⇒ s x H_ok.
    @@ -5439,56 +5447,56 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Definition min_elt_spec2 :
    -    âˆ€ (s : t) (x y : elt) (Hs : Ok s), min_elt s = Some x → In y s → ~(Enc.E.lt y x).
    +  Definition min_elt_spec2 :
    +    âˆ€ (s : t) (x y : elt) (Hs : Ok s), min_elt s = Some x → In y s → ~(Enc.E.lt y x).
      Proof.
        rewrite /min_elt.
        move ⇒ s x y H_ok.
        case_eq (min_eltZ s) ⇒ //.
        move ⇒ z H_min_elt [<-].
        rewrite In_InZ ⇒ H_inZ.
    -    have H_y_eq : y = Enc.decode (Enc.encode y). {
    +    have H_y_eq : y = Enc.decode (Enc.encode y). {
          by rewrite Enc.decode_encode_ok.
        }
        rewrite H_y_eq -Enc.encode_lt.
        apply (min_eltZ_spec2 _ _ _ H_ok); last first. {
          by rewrite Enc.decode_encode_ok.
        }
    -    suff → : Enc.encode (Enc.decode z) = z by assumption.
    +    suff → : Enc.encode (Enc.decode z) = z by assumption.
        apply encode_decode_eq with (s := s) ⇒ //.
        apply min_eltZ_spec1 ⇒ //.
        apply H_ok.
      Qed.

    -  Definition min_elt_spec3 :
    -    âˆ€ s : t, min_elt s = None → Empty s.
    +  Definition min_elt_spec3 :
    +    âˆ€ s : t, min_elt s = None → Empty s.
      Proof.
        rewrite /min_elt /min_eltZ /Empty /In.
        case s as [| [x' c] s'] ⇒ //.
        move ⇒ _ e.
    -    rewrite elements_nil InA_nil //.
    +    rewrite elements_nil InA_nil //.
      Qed.

    -  Definition choose_spec1 :
    -    âˆ€ (s : t) (x : elt) (Hs : Ok s), choose s = Some x → In x s.
    +  Definition choose_spec1 :
    +    âˆ€ (s : t) (x : elt) (Hs : Ok s), choose s = Some x → In x s.
      Proof.
        rewrite /choose.
        apply min_elt_spec1.
      Qed.

    -  Definition choose_spec2 :
    -    âˆ€ s : t, choose s = None → Empty s.
    +  Definition choose_spec2 :
    +    âˆ€ s : t, choose s = None → Empty s.
      Proof.
        rewrite /choose.
        apply min_elt_spec3.
      Qed.

    -  Lemma choose_spec3: ∀ s s' x x', Ok s → Ok s' →
    -    choose s = Some x → choose s' = Some x' → Equal s s' → x = x'.
    +  Lemma choose_spec3: ∀ s s' x x', Ok s → Ok s' →
    +    choose s = Some x → choose s' = Some x' → Equal s s' → x = x'.
      Proof.
        intros s s' x x' Hs Hs' Hx Hx'.
        rewrite -equal_spec equal_alt_def ⇒ H_s_eq.
    @@ -5499,10 +5507,10 @@

    Library MSetsExtra.MSetIntervals



    -  Definition max_eltZ_spec1 :
    -    âˆ€ (s : t) (x : Z),
    -      interval_list_invariant s = true →
    -      max_eltZ s = Some x → InZ x s.
    +  Definition max_eltZ_spec1 :
    +    âˆ€ (s : t) (x : Z),
    +      interval_list_invariant s = true →
    +      max_eltZ s = Some x → InZ x s.
      Proof.
        intros s x.
        induction s as [| [x' c] s' IH]. {
    @@ -5515,10 +5523,10 @@

    Library MSetsExtra.MSetIntervals

            left. {
              rewrite In_elementsZ_single.
              split. {
    -            rewrite -Z.lt_le_pred.
    +            rewrite -Z.lt_le_pred.
                by apply Z_lt_add_r.
              } {
    -            apply Z.lt_pred_l.
    +            apply Z.lt_pred_l.
              }
            }
          } {
    @@ -5530,10 +5538,10 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma max_eltZ_spec2 :
    -    âˆ€ (s : t) (x y : Z),
    -    interval_list_invariant s = true →
    -    max_eltZ s = Some x → InZ y s → ¬ Z.lt x y.
    +  Lemma max_eltZ_spec2 :
    +    âˆ€ (s : t) (x y : Z),
    +    interval_list_invariant s = true →
    +    max_eltZ s = Some x → InZ y s → ¬ Z.lt x y.
      Proof.
        induction s as [| [y c] s' IH]. {
          done.
    @@ -5541,14 +5549,14 @@

    Library MSetsExtra.MSetIntervals

          move ⇒ x x'.
          rewrite interval_list_invariant_cons.
          move ⇒ [H_gr] [H_c_neq_0] H_inv_s'.
    -      have H_gr' : (∀ xx : Z, InZ xx (s') → y + Z.of_N c < xx). {
    +      have H_gr' : (∀ xx : Z, InZ xx (s') → y + Z.of_N c < xx). {
            apply interval_list_elements_greater_alt2_def ⇒ //.
          }

          case s' as [| [y' c'] s'']. {
            move ⇒ [<-].
            rewrite InZ_cons InZ_nil In_elementsZ_single.
    -        omega.
    +        lia.
          } {
            move ⇒ H_max_eq.
            rewrite InZ_cons.
    @@ -5557,21 +5565,21 @@

    Library MSetsExtra.MSetIntervals

            move ⇒ [_] H_x'_lt H_lt_x'.

    -        have H_x_in : InZ x ((y', c')::s''). {
    +        have H_x_in : InZ x ((y', c')::s''). {
              by apply max_eltZ_spec1.
            }
            
            move : (H_gr' _ H_x_in).
    -        apply Z.nlt_ge, Z.lt_le_incl.
    -        by apply Z.lt_trans with (m := x').
    +        apply Z.nlt_ge, Z.lt_le_incl.
    +        by apply Z.lt_trans with (m := x').
          }
        }
      Qed.

    -  Lemma max_eltZ_eq_None :
    -    âˆ€ (s : t),
    -      max_eltZ s = None → s = nil.
    +  Lemma max_eltZ_eq_None :
    +    âˆ€ (s : t),
    +      max_eltZ s = None → s = nil.
      Proof.
        induction s as [| [x' c] s' IH] ⇒ //.
        move : IH.
    @@ -5581,16 +5589,16 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Definition max_eltZ_spec3 :
    -    âˆ€ (s : t),
    -      max_eltZ s = None → ∀ x, ¬InZ x s.
    +  Definition max_eltZ_spec3 :
    +    âˆ€ (s : t),
    +      max_eltZ s = None → ∀ x, ¬InZ x s.
      Proof.
        move ⇒ s /max_eltZ_eq_None → x /InZ_nil //.
      Qed.

    -  Definition max_elt_spec1 :
    -    âˆ€ (s : t) (x : elt) (Hs : Ok s), max_elt s = Some x → In x s.
    +  Definition max_elt_spec1 :
    +    âˆ€ (s : t) (x : elt) (Hs : Ok s), max_elt s = Some x → In x s.
      Proof.
        rewrite /max_elt.
        move ⇒ s x H_ok.
    @@ -5602,8 +5610,8 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Definition max_elt_spec2 :
    -    âˆ€ (s : t) (x y : elt) (Hs : Ok s), max_elt s = Some x → In y s → ~(Enc.E.lt x y).
    +  Definition max_elt_spec2 :
    +    âˆ€ (s : t) (x y : elt) (Hs : Ok s), max_elt s = Some x → In y s → ~(Enc.E.lt x y).
      Proof.
        rewrite /max_elt.
        move ⇒ s x y H_ok.
    @@ -5613,41 +5621,41 @@

    Library MSetsExtra.MSetIntervals

        rewrite In_InZ ⇒ H_inZ.
        rewrite -Enc.encode_lt.
        apply (max_eltZ_spec2 _ _ _ H_inv) ⇒ //.
    -    suff → : Enc.encode (Enc.decode z) = z ⇒ //.
    +    suff → : Enc.encode (Enc.decode z) = z ⇒ //.
        apply encode_decode_eq with (s := s) ⇒ //.
        apply max_eltZ_spec1 ⇒ //.
      Qed.

    -  Definition max_elt_spec3 :
    -    âˆ€ s : t, max_elt s = None → Empty s.
    +  Definition max_elt_spec3 :
    +    âˆ€ s : t, max_elt s = None → Empty s.
      Proof.
        intro s.
        rewrite /max_elt /Empty.
        case_eq (max_eltZ s) ⇒ //.
        move ⇒ /max_eltZ_eq_None → _ x.
    -    rewrite /In elements_nil InA_nil //.
    +    rewrite /In elements_nil InA_nil //.
      Qed.

    -

    fold specification

    +

    fold specification


    -  Lemma fold_spec :
    -   âˆ€ (s : t) (A : Type) (i : A) (f : elt → A → A),
    -   fold f s i = fold_left (flip f) (elements s) i.
    +  Lemma fold_spec :
    +   âˆ€ (s : t) (A : Type) (i : A) (f : elt → A → A),
    +   fold f s i = fold_left (flip f) (elements s) i.
      Proof.
        intros s A i f.
        rewrite /fold fold_elementsZ_alt_def /elements
    -            rev_map_alt_def -map_rev.
    +            rev_map_alt_def -map_rev.
        move : i.
    -    generalize (rev (elementsZ s)).
    +    generalize (rev (elementsZ s)).
        induction l as [| x xs IH]. {
          done.
        } {
    @@ -5662,64 +5670,64 @@

    Library MSetsExtra.MSetIntervals

    -

    cardinal specification

    +

    cardinal specification


    -  Lemma cardinalN_spec : ∀ (s : t) (c : N),
    -    cardinalN c s = (c + N.of_nat (length (elements s)))%N.
    +  Lemma cardinalN_spec : ∀ (s : t) (c : N),
    +    cardinalN c s = (c + N.of_nat (length (elements s)))%N.
      Proof.
        induction s as [| [x cx] xs IH]. {
          intros c.
    -      rewrite elements_nil /= N.add_0_r //.
    +      rewrite elements_nil /= N.add_0_r //.
        } {
          intros c.
          rewrite /= IH.
    -      rewrite /elements !rev_map_alt_def !rev_length !map_length.
    -      rewrite elementsZ_cons app_length Nat2N.inj_add rev_length.
    -      rewrite length_elementsZ_single N2Nat.id.
    +      rewrite /elements !rev_map_alt_def !rev_length !map_length.
    +      rewrite elementsZ_cons app_length Nat2N.inj_add rev_length.
    +      rewrite length_elementsZ_single N2Nat.id.
          ring.
        }
      Qed.

    -  Lemma cardinal_spec :
    -   âˆ€ (s : t),
    -   cardinal s = length (elements s).
    +  Lemma cardinal_spec :
    +   âˆ€ (s : t),
    +   cardinal s = length (elements s).
      Proof.
        intros s.
    -    rewrite /cardinal cardinalN_spec N.add_0_l Nat2N.id //.
    +    rewrite /cardinal cardinalN_spec N.add_0_l Nat2N.id //.
      Qed.

    -

    for_all specification

    +

    for_all specification


    -  Lemma for_all_spec :
    -   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    -   Proper (Enc.E.eq==>eq) f →
    -   (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +  Lemma for_all_spec :
    +   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    +   Proper (Enc.E.eq==>eq) f →
    +   (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
      Proof.
        intros s f Hs H.
        rewrite /for_all /For_all /In fold_elementsZ_alt_def
    -            /elements rev_map_alt_def -map_rev.
    -    generalize (rev (elementsZ s)).
    +            /elements rev_map_alt_def -map_rev.
    +    generalize (rev (elementsZ s)).
        induction l as [| x xs IH]. {
    -      split ⇒ // _ x /= /InA_nil //.
    +      split ⇒ // _ x /= /InA_nil //.
        } {
          rewrite /=.
          case_eq (f (Enc.decode x)) ⇒ H_f_eq. {
            rewrite IH.
            split. {
    -          move ⇒ HH x' /InA_cons []. {
    +          move ⇒ HH x' /InA_cons []. {
                by move ⇒ →.
              } {
                apply HH.
    @@ -5727,7 +5735,7 @@

    Library MSetsExtra.MSetIntervals

            } {
              move ⇒ HH x' H_in.
              apply HH.
    -          apply InA_cons.
    +          apply InA_cons.
              by right.
            }
          } {
    @@ -5736,8 +5744,8 @@

    Library MSetsExtra.MSetIntervals

              case xs ⇒ //.
            } {
              exfalso.
    -          have H_in: (InA Enc.E.eq (Enc.decode x) (Enc.decode x :: map Enc.decode xs)). {
    -            apply InA_cons.
    +          have H_in: (InA Enc.E.eq (Enc.decode x) (Enc.decode x :: map Enc.decode xs)). {
    +            apply InA_cons.
                left.
                apply Enc.E.eq_equiv.
              }
    @@ -5752,31 +5760,31 @@

    Library MSetsExtra.MSetIntervals

    -

    exists specification

    +

    exists specification


    -  Lemma exists_spec :
    -   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    -   Proper (Enc.E.eq==>eq) f →
    -   (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +  Lemma exists_spec :
    +   âˆ€ (s : t) (f : elt → bool) (Hs : Ok s),
    +   Proper (Enc.E.eq==>eq) f →
    +   (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
      Proof.
        intros s f Hs H.
        rewrite /exists_ /Exists /In fold_elementsZ_alt_def
    -            /elements rev_map_alt_def -map_rev.
    -    generalize (rev (elementsZ s)).
    +            /elements rev_map_alt_def -map_rev.
    +    generalize (rev (elementsZ s)).
        induction l as [| x xs IH]. {
          split ⇒ //.
    -      move ⇒ [x] /= [] /InA_nil //.
    +      move ⇒ [x] /= [] /InA_nil //.
        } {
          rewrite /=.
          case_eq (f (Enc.decode x)) ⇒ H_f_eq. {
            split ⇒ _. {
              âˆƒ (Enc.decode x).
              split ⇒ //.
    -          apply InA_cons.
    +          apply InA_cons.
              left.
              apply Enc.E.eq_equiv.
            } {
    @@ -5788,10 +5796,10 @@

    Library MSetsExtra.MSetIntervals

              move ⇒ [x0] [H_in] H_f_x0.
              âˆƒ x0.
              split ⇒ //.
    -          apply InA_cons.
    +          apply InA_cons.
              by right.
            } {
    -          move ⇒ [x0] [] /InA_cons H_in H_f_x0.
    +          move ⇒ [x0] [] /InA_cons H_in H_f_x0.
              âˆƒ x0.
              split ⇒ //.
              move : H_in ⇒ [] // H_in.
    @@ -5806,31 +5814,31 @@

    Library MSetsExtra.MSetIntervals

    -

    filter specification

    +

    filter specification


    -  Definition partitionZ_aux_invariant (x : Z) acc c :=
    -    interval_list_invariant (List.rev (partitionZ_fold_skip acc c)) = true ∧
    -    match c with
    -      None ⇒ (∀ y', InZ y' acc → Z.succ y' < x)
    -    | Some (y, c') ⇒ (x = y + Z.of_N c')
    +  Definition partitionZ_aux_invariant (x : Z) acc c :=
    +    interval_list_invariant (List.rev (partitionZ_fold_skip acc c)) = true ∧
    +    match c with
    +      None ⇒ (∀ y', InZ y' acc → Z.succ y' < x)
    +    | Some (y, c') ⇒ (x = y + Z.of_N c')
        end.

    -  Lemma partitionZ_aux_invariant_insert : ∀ x acc c,
    -    partitionZ_aux_invariant x acc c →
    -    partitionZ_aux_invariant (Z.succ x) acc
    -      (Some (partitionZ_fold_insert c x)).
    +  Lemma partitionZ_aux_invariant_insert : ∀ x acc c,
    +    partitionZ_aux_invariant x acc c →
    +    partitionZ_aux_invariant (Z.succ x) acc
    +      (Some (partitionZ_fold_insert c x)).
      Proof.
        intros x acc c.
        rewrite /partitionZ_fold_insert /partitionZ_aux_invariant
                /partitionZ_fold_skip.
        case c; last first. {
          move ⇒ [H_inv] H_in.
    -      rewrite /= interval_list_invariant_app_iff Z.add_1_r.
    +      rewrite /= interval_list_invariant_app_iff Z.add_1_r.
          split; last done.
          split; first done.
          split; first done.
    @@ -5841,7 +5849,7 @@

    Library MSetsExtra.MSetIntervals

        } {
          move ⇒ [y c'].
          rewrite /= !interval_list_invariant_app_iff
    -        N2Z.inj_succ Z.add_succ_r .
    +        N2Z.inj_succ Z.add_succ_r .
          rewrite !interval_list_invariant_cons !interval_list_invariant_nil.
          move ⇒ [] [H_inv_acc] [] [] _ [H_c_neq_0] _
            H_in_c →.
    @@ -5850,17 +5858,17 @@

    Library MSetsExtra.MSetIntervals

          split. {
            split; first done.
            split; last done.
    -        apply N.neq_succ_0.
    +        apply N.neq_succ_0.
          } {
            move ⇒ x1 x2.
            rewrite InZ_cons InZ_nil In_elementsZ_single.
            move ⇒ H_x1_in [] // [H_y_le] H_x2_lt.
    -        apply Z.lt_le_trans with (m := y) ⇒ //.
    +        apply Z.lt_le_trans with (m := y) ⇒ //.
            apply H_in_c ⇒ //.
            rewrite InZ_cons In_elementsZ_single.
            left.
            split. {
    -          apply Z.le_refl.
    +          apply Z.le_refl.
            } {
              by apply Z_lt_add_r.
            }
    @@ -5869,9 +5877,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma partitionZ_aux_invariant_skip : ∀ x acc c,
    -    partitionZ_aux_invariant x acc c →
    -    partitionZ_aux_invariant (Z.succ x) (partitionZ_fold_skip acc c) None.
    +  Lemma partitionZ_aux_invariant_skip : ∀ x acc c,
    +    partitionZ_aux_invariant x acc c →
    +    partitionZ_aux_invariant (Z.succ x) (partitionZ_fold_skip acc c) None.
      Proof.
        intros x acc c.
        rewrite /partitionZ_fold_skip /partitionZ_aux_invariant
    @@ -5880,10 +5888,10 @@

    Library MSetsExtra.MSetIntervals

          move ⇒ [H_inv] H_in.
          split; first done.
          move ⇒ y' H_y'_in.
    -      apply Z.lt_trans with (m := x). {
    +      apply Z.lt_trans with (m := x). {
            by apply H_in.
          } {
    -        apply Z.lt_succ_diag_r.
    +        apply Z.lt_succ_diag_r.
          }
        } {
          move ⇒ [y c'] [H_inv] →.
    @@ -5892,12 +5900,12 @@

    Library MSetsExtra.MSetIntervals

          rewrite InZ_cons In_elementsZ_single.
          move ⇒ []. {
            move ⇒ [_].
    -        rewrite -Z.succ_lt_mono //.
    +        rewrite -Z.succ_lt_mono //.
          } {
            move : H_inv.
            rewrite /= !interval_list_invariant_app_iff interval_list_invariant_cons.
            move ⇒ [_] [] [_] [H_c'_neq] _ H_pre H_y'_in.
    -        apply Z.lt_trans with (m := y). {
    +        apply Z.lt_trans with (m := y). {
              apply H_pre. {
                by rewrite InZ_rev.
              } {
    @@ -5906,40 +5914,40 @@

    Library MSetsExtra.MSetIntervals

                by apply In_elementsZ_single_hd.
              }
            }
    -        apply Z.lt_succ_r, Z_le_add_r.
    +        apply Z.lt_succ_r, Z_le_add_r.
          }
        }
      Qed.

    -  Definition partitionZ_fold_current (c : option (Z × N)) :=
    -    match c with
    -       None ⇒ nil
    -     | Some yc ⇒ yc::nil
    +  Definition partitionZ_fold_current (c : option (Z × N)) :=
    +    match c with
    +       None ⇒ nil
    +     | Some yc ⇒ yc::nil
        end.

    -  Lemma InZ_partitionZ_fold_current_Some : ∀ yc y,
    -     InZ y (partitionZ_fold_current (Some yc)) ↔
    -     InZ y (yc :: nil).
    +  Lemma InZ_partitionZ_fold_current_Some : ∀ yc y,
    +     InZ y (partitionZ_fold_current (Some yc)) ↔
    +     InZ y (yc :: nil).
      Proof. done. Qed.

    -  Lemma InZ_partitionZ_fold_insert : ∀ c x y l,
    -   match c with
    -   | Some (y, c') ⇒ x = y + Z.of_N c'
    -   | None ⇒ True
    -   end → (
    -   InZ y (partitionZ_fold_insert c x :: l) ↔
    -    ((x = y) ∨ InZ y (partitionZ_fold_current c) ∨
    -       InZ y l)).
    +  Lemma InZ_partitionZ_fold_insert : ∀ c x y l,
    +   match c with
    +   | Some (y, c') ⇒ x = y + Z.of_N c'
    +   | None ⇒ True
    +   end → (
    +   InZ y (partitionZ_fold_insert c x :: l) ↔
    +    ((x = y) ∨ InZ y (partitionZ_fold_current c) ∨
    +       InZ y l)).
      Proof.
        intros c x y l.
        rewrite /partitionZ_fold_insert /partitionZ_fold_current
                /partitionZ_fold_skip.
        case c. {
          move ⇒ [y' c'] →.
    -      rewrite !InZ_cons elementsZ_single_succ in_app_iff
    +      rewrite !InZ_cons elementsZ_single_succ in_app_iff
                  InZ_nil /=.
          tauto.
        } {
    @@ -5949,9 +5957,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma InZ_partitionZ_fold_skip : ∀ c acc y,
    -    InZ y (partitionZ_fold_skip acc c) ↔
    -    (InZ y (partitionZ_fold_current c) ∨ InZ y acc).
    +  Lemma InZ_partitionZ_fold_skip : ∀ c acc y,
    +    InZ y (partitionZ_fold_skip acc c) ↔
    +    (InZ y (partitionZ_fold_current c) ∨ InZ y acc).
      Proof.
        intros c acc y.
        rewrite /partitionZ_fold_skip /partitionZ_fold_current
    @@ -5967,20 +5975,20 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma filterZ_single_aux_props :
    -    âˆ€ f c x acc cur,
    -      partitionZ_aux_invariant x acc cur →
    -      match (filterZ_single_aux f (acc, cur) x c) with
    -        (acc', c') ⇒
    -        let r := partitionZ_fold_skip acc' c' in
    -        interval_list_invariant (List.rev r) = true ∧
    -        (∀ y', InZ y' r ↔ (InZ y' (partitionZ_fold_skip acc cur) ∨
    -                                    (f y' = true ∧ List.In y' (elementsZ_single x c))))
    +  Lemma filterZ_single_aux_props :
    +    âˆ€ f c x acc cur,
    +      partitionZ_aux_invariant x acc cur →
    +      match (filterZ_single_aux f (acc, cur) x c) with
    +        (acc', c') ⇒
    +        let r := partitionZ_fold_skip acc' c' in
    +        interval_list_invariant (List.rev r) = true ∧
    +        (∀ y', InZ y' r ↔ (InZ y' (partitionZ_fold_skip acc cur) ∨
    +                                    (f y' = true ∧ List.In y' (elementsZ_single x c))))

          end.
      Proof.
        intro f.
    -    induction c as [| c' IH] using N.peano_ind. {
    +    induction c as [| c' IH] using N.peano_ind. {
          intros x acc cur.
          rewrite /partitionZ_aux_invariant.
          move ⇒ [H_inv] _.
    @@ -5988,19 +5996,19 @@

    Library MSetsExtra.MSetIntervals

          tauto.
        }
        intros x acc cur H_inv.
    -    have → : filterZ_single_aux f (acc, cur) x (N.succ c') =
    -               filterZ_single_aux f (filterZ_fold_fun f (acc, cur) x) (Z.succ x) c'. {
    +    have → : filterZ_single_aux f (acc, cur) x (N.succ c') =
    +               filterZ_single_aux f (filterZ_fold_fun f (acc, cur) x) (Z.succ x) c'. {
            by rewrite /filterZ_single_aux fold_elementsZ_single_succ.
        }
    -    case_eq (filterZ_fold_fun f (acc, cur) x).
    +    case_eq (filterZ_fold_fun f (acc, cur) x).
        move ⇒ acc' cur' H_fold_eq.

    -    case_eq (filterZ_single_aux f (acc', cur') (Z.succ x) c').
    +    case_eq (filterZ_single_aux f (acc', cur') (Z.succ x) c').
        move ⇒ acc'' cur'' H_succ_eq.

    -    have H_inv' : partitionZ_aux_invariant (Z.succ x) acc' cur'. {
    +    have H_inv' : partitionZ_aux_invariant (Z.succ x) acc' cur'. {
          move : H_fold_eq H_inv.
          rewrite /filterZ_fold_fun.
          case (f x); move ⇒ [<-] <-. {
    @@ -6010,7 +6018,7 @@

    Library MSetsExtra.MSetIntervals

          }
        }
       
    -    move : (IH (Z.succ x) acc' cur' H_inv') ⇒ {IH}.
    +    move : (IH (Z.succ x) acc' cur' H_inv') ⇒ {IH}.
        rewrite H_succ_eq /=.
        set r := partitionZ_fold_skip acc'' cur''.
        move ⇒ [H_inv_r] H_in_r.
    @@ -6052,15 +6060,15 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma filterZ_single_props :
    -    âˆ€ f c x acc,
    -      interval_list_invariant (rev acc) = true →
    -      (∀ y' : Z, InZ y' acc → Z.succ y' < x) →
    -      match (filterZ_single f acc x c) with
    +  Lemma filterZ_single_props :
    +    âˆ€ f c x acc,
    +      interval_list_invariant (rev acc) = true →
    +      (∀ y' : Z, InZ y' acc → Z.succ y' < x) →
    +      match (filterZ_single f acc x c) with
            r ⇒
    -        interval_list_invariant (List.rev r) = true ∧
    -        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    -                                 (f y' = true ∧ List.In y' (elementsZ_single x c))))
    +        interval_list_invariant (List.rev r) = true ∧
    +        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    +                                 (f y' = true ∧ List.In y' (elementsZ_single x c))))

          end.
      Proof.
    @@ -6069,11 +6077,11 @@

    Library MSetsExtra.MSetIntervals

        rewrite /filterZ_single.

    -    have H_inv' : partitionZ_aux_invariant x acc None. {
    +    have H_inv' : partitionZ_aux_invariant x acc None. {
          by rewrite /partitionZ_aux_invariant /=.
        }
    -    move : (filterZ_single_aux_props f c x acc None H_inv').
    -    case_eq (filterZ_single_aux f (acc, None) x c).
    +    move : (filterZ_single_aux_props f c x acc None H_inv').
    +    case_eq (filterZ_single_aux f (acc, None) x c).
        move ⇒ acc' cur' /= H_res.
        tauto.
      Qed.
    @@ -6081,16 +6089,16 @@

    Library MSetsExtra.MSetIntervals



    -  Lemma filterZ_aux_props :
    -    âˆ€ f s acc,
    -      interval_list_invariant s = true →
    -      interval_list_invariant (rev acc) = true →
    -      (∀ x1 x2 : Z, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    -      match (filterZ_aux acc f s) with
    +  Lemma filterZ_aux_props :
    +    âˆ€ f s acc,
    +      interval_list_invariant s = true →
    +      interval_list_invariant (rev acc) = true →
    +      (∀ x1 x2 : Z, InZ x1 acc → InZ x2 s → Z.succ x1 < x2) →
    +      match (filterZ_aux acc f s) with
            r ⇒
    -        interval_list_invariant r = true ∧
    -        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    -                                 (f y' = true ∧ InZ y' s)))
    +        interval_list_invariant r = true ∧
    +        (∀ y', InZ y' r ↔ (InZ y' acc ∨
    +                                 (f y' = true ∧ InZ y' s)))
          end.
      Proof.
        intro f.
    @@ -6108,7 +6116,7 @@

    Library MSetsExtra.MSetIntervals

          rewrite interval_list_elements_greater_alt2_def ⇒ // H_gr.

    -      have H_pre : (∀ y' : Z, InZ y' acc → Z.succ y' < y). {
    +      have H_pre : (∀ y' : Z, InZ y' acc → Z.succ y' < y). {
            move ⇒ x1 H_x1_in.
            apply H_in_acc ⇒ //.
            rewrite InZ_cons.
    @@ -6120,8 +6128,8 @@

    Library MSetsExtra.MSetIntervals

          move ⇒ [H_inv'] H_in_acc'.

    -      have H_pre : (∀ x1 x2 : Z,
    -                      InZ x1 acc' → InZ x2 s' → Z.succ x1 < x2). {
    +      have H_pre : (∀ x1 x2 : Z,
    +                      InZ x1 acc' → InZ x2 s' → Z.succ x1 < x2). {
            move ⇒ x1 x2.
            rewrite H_in_acc' In_elementsZ_single.
            move ⇒ []. {
    @@ -6131,8 +6139,8 @@

    Library MSetsExtra.MSetIntervals

              by right.
            } {
              move ⇒ [_] [_] H_x1_lt H_x2_in.
    -          apply Z.le_lt_trans with (m := y + Z.of_N c).
    -            - by apply Z.le_succ_l.
    +          apply Z.le_lt_trans with (m := y + Z.of_N c).
    +            - by apply Z.le_succ_l.
                - by apply H_gr.
            }
          }
    @@ -6148,23 +6156,23 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma filterZ_props :
    -    âˆ€ f s,
    -      interval_list_invariant s = true →
    -      match (filterZ f s) with r ⇒
    -        interval_list_invariant r = true ∧
    -        (∀ y', InZ y' r ↔ (f y' = true ∧ InZ y' s))
    +  Lemma filterZ_props :
    +    âˆ€ f s,
    +      interval_list_invariant s = true →
    +      match (filterZ f s) with r ⇒
    +        interval_list_invariant r = true ∧
    +        (∀ y', InZ y' r ↔ (f y' = true ∧ InZ y' s))
          end.
      Proof.
        intros f s H_inv_s.
        rewrite /filterZ.

    -    have H_pre_1 : interval_list_invariant (rev nil) = true by done.
    -    have H_pre_2 : (∀ x1 x2 : Z, InZ x1 nil → InZ x2 s → Z.succ x1 < x2) by done.
    +    have H_pre_1 : interval_list_invariant (rev nil) = true by done.
    +    have H_pre_2 : (∀ x1 x2 : Z, InZ x1 nil → InZ x2 s → Z.succ x1 < x2) by done.

    -    move : (filterZ_aux_props f s nil H_inv_s H_pre_1 H_pre_2) ⇒ {H_pre_1} {H_pre_2}.
    +    move : (filterZ_aux_props f s nil H_inv_s H_pre_1 H_pre_2) ⇒ {H_pre_1} {H_pre_2}.
        move ⇒ [H_inv'] H_in_r.
        split; first assumption.
        move ⇒ y'.
    @@ -6173,11 +6181,11 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Global Instance filter_ok s f : ∀ `(Ok s), Ok (filter f s).
    +  Global Instance filter_ok s f : ∀ `(Ok s), Ok (filter f s).
      Proof.
        move ⇒ [H_inv H_enc].
        rewrite /filter.
    -    set f' := (fun z : Z ⇒ f (Enc.decode z)).
    +    set f' := (fun z : Z ⇒ f (Enc.decode z)).
        move : (filterZ_props f' s H_inv).
        move ⇒ [H_inv'] H_in_r.
        rewrite /Ok /IsOk /is_encoded_elems_list.
    @@ -6187,17 +6195,17 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma filter_spec :
    -   âˆ€ (s : t) (x : elt) (f : elt → bool),
    -   Ok s →
    -   (In x (filter f s) ↔ In x s ∧ f x = true).
    +  Lemma filter_spec :
    +   âˆ€ (s : t) (x : elt) (f : elt → bool),
    +   Ok s →
    +   (In x (filter f s) ↔ In x s ∧ f x = true).
      Proof.
        move ⇒ s x f H_ok.
        suff H_suff :
    -      (∀ x, (InZ x (filter f s)) ↔
    -                 InZ x s ∧ f (Enc.decode x) = true). {
    +      (∀ x, (InZ x (filter f s)) ↔
    +                 InZ x s ∧ f (Enc.decode x) = true). {
          rewrite !In_alt_def /elements !rev_map_alt_def
    -              -!in_rev !in_map_iff.
    +              -!in_rev !in_map_iff.
          setoid_rewrite H_suff.
          rewrite /InZ.
          split. {
    @@ -6210,7 +6218,7 @@

    Library MSetsExtra.MSetIntervals

          }
        }
        rewrite /filter.
    -    set f' := (fun z : Z ⇒ f (Enc.decode z)).
    +    set f' := (fun z : Z ⇒ f (Enc.decode z)).
        move : (H_ok) ⇒ [H_inv _].
        move : (filterZ_props f' s H_inv).
        move ⇒ [H_inv'] H_in_r.
    @@ -6221,28 +6229,28 @@

    Library MSetsExtra.MSetIntervals

    -

    partition specification

    +

    partition specification


    -  Lemma partitionZ_single_aux_alt_def : ∀ f c y acc_t c_t acc_f c_f,
    -    partitionZ_single_aux f ((acc_t, c_t), (acc_f, c_f)) y c =
    -    (filterZ_single_aux f (acc_t, c_t) y c,
    -     filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y c).
    +  Lemma partitionZ_single_aux_alt_def : ∀ f c y acc_t c_t acc_f c_f,
    +    partitionZ_single_aux f ((acc_t, c_t), (acc_f, c_f)) y c =
    +    (filterZ_single_aux f (acc_t, c_t) y c,
    +     filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y c).
      Proof.
        intros f.
        rewrite /partitionZ_single_aux /filterZ_single_aux.
    -    induction c as [| c' IH] using N.peano_ind. {
    +    induction c as [| c' IH] using N.peano_ind. {
          intros y acc_t c_t acc_f c_f.
          rewrite !fold_elementsZ_single_zero //. } {
          intros y acc_t c_t acc_f c_f.
          rewrite !fold_elementsZ_single_succ.
    -      case_eq (partitionZ_fold_fun f (acc_t, c_t, (acc_f, c_f)) y) ⇒ [] [acc_t' c_t'] [acc_f' c_f'] H_fold_eq.
    +      case_eq (partitionZ_fold_fun f (acc_t, c_t, (acc_f, c_f)) y) ⇒ [] [acc_t' c_t'] [acc_f' c_f'] H_fold_eq.
          rewrite IH ⇒ {IH}.
    -      suff : (filterZ_fold_fun f (acc_t, c_t) y = (acc_t', c_t')) ∧
    -             (filterZ_fold_fun (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y = (acc_f', c_f')). {
    +      suff : (filterZ_fold_fun f (acc_t, c_t) y = (acc_t', c_t')) ∧
    +             (filterZ_fold_fun (fun x : Z ⇒ negb (f x)) (acc_f, c_f) y = (acc_f', c_f')). {
            move ⇒ [->] → //.
          }
          move : H_fold_eq.
    @@ -6252,10 +6260,10 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma partitionZ_aux_alt_def : ∀ f s acc_t acc_f,
    -   partitionZ_aux acc_t acc_f f s =
    -   (filterZ_aux acc_t f s,
    -    filterZ_aux acc_f (fun x : Z ⇒ negb (f x)) s).
    +  Lemma partitionZ_aux_alt_def : ∀ f s acc_t acc_f,
    +   partitionZ_aux acc_t acc_f f s =
    +   (filterZ_aux acc_t f s,
    +    filterZ_aux acc_f (fun x : Z ⇒ negb (f x)) s).
      Proof.
        intros f.
        induction s as [| [y c] s' IH]. {
    @@ -6264,16 +6272,16 @@

    Library MSetsExtra.MSetIntervals

          intros acc_t acc_f.
          rewrite /= /partitionZ_single /filterZ_single
                  partitionZ_single_aux_alt_def.
    -      case (filterZ_single_aux f (acc_t, None) y c) ⇒ acc_t' c_t'.
    -      case (filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, None) y c) ⇒ acc_f' c_f'.
    +      case (filterZ_single_aux f (acc_t, None) y c) ⇒ acc_t' c_t'.
    +      case (filterZ_single_aux (fun x : Z ⇒ negb (f x)) (acc_f, None) y c) ⇒ acc_f' c_f'.
          rewrite IH //.
        }
      Qed.

    -  Lemma partitionZ_alt_def : ∀ f s,
    -    partitionZ f s = (filterZ f s,
    -                      filterZ (fun x ⇒ negb (f x)) s).
    +  Lemma partitionZ_alt_def : ∀ f s,
    +    partitionZ f s = (filterZ f s,
    +                      filterZ (fun x ⇒ negb (f x)) s).
      Proof.
        intros f s.
        rewrite /partitionZ /filterZ
    @@ -6281,9 +6289,9 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma partition_alt_def : ∀ f s,
    -    partition f s = (filter f s,
    -                     filter (fun x ⇒ negb (f x)) s).
    +  Lemma partition_alt_def : ∀ f s,
    +    partition f s = (filter f s,
    +                     filter (fun x ⇒ negb (f x)) s).
      Proof.
        intros f s.
        rewrite /partition /filter partitionZ_alt_def.
    @@ -6291,38 +6299,38 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Global Instance partition_ok1 s f : ∀ `(Ok s), Ok (fst (partition f s)).
    +  Global Instance partition_ok1 s f : ∀ `(Ok s), Ok (fst (partition f s)).
      Proof.
        move ⇒ H_ok.
    -    rewrite partition_alt_def /fst.
    +    rewrite partition_alt_def /fst.
        by apply filter_ok.
      Qed.

    -  Global Instance partition_ok2 s f : ∀ `(Ok s), Ok (snd (partition f s)).
    +  Global Instance partition_ok2 s f : ∀ `(Ok s), Ok (snd (partition f s)).
      Proof.
        move ⇒ H_ok.
    -    rewrite partition_alt_def /snd.
    +    rewrite partition_alt_def /snd.
        by apply filter_ok.
      Qed.

    -  Lemma partition_spec1 :
    -   âˆ€ (s : t) (f : elt → bool),
    -   Equal (fst (partition f s)) (filter f s).
    +  Lemma partition_spec1 :
    +   âˆ€ (s : t) (f : elt → bool),
    +   Equal (fst (partition f s)) (filter f s).
      Proof.
        intros s f.
    -    rewrite partition_alt_def /fst /Equal //.
    +    rewrite partition_alt_def /fst /Equal //.
      Qed.

    -  Lemma partition_spec2 :
    -   âˆ€ (s : t) (f : elt → bool),
    -   Ok s →
    -   Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
    +  Lemma partition_spec2 :
    +   âˆ€ (s : t) (f : elt → bool),
    +   Ok s →
    +   Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
      Proof.
        intros s f.
    -    rewrite partition_alt_def /snd /Equal //.
    +    rewrite partition_alt_def /snd /Equal //.
      Qed.

    @@ -6332,7 +6340,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Main Module

    +

    Main Module

    @@ -6343,56 +6351,56 @@

    Library MSetsExtra.MSetIntervals


    -Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E.
    -  Module E := Enc.E.
    -  Module Raw := Raw Enc.
    +Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E.
    +  Module E := Enc.E.
    +  Module Raw := Raw Enc.

     Local Unset Elimination Schemes.
     Local Unset Case Analysis Schemes.

    - Definition elt := Raw.elt.
    - Record t_ := Mkt {this :> Raw.t; is_ok : Raw.Ok this}.
    - Definition t := t_.
    - Arguments Mkt this {is_ok}.
    - Hint Resolve is_ok : typeclass_instances.
    - -
    - Definition In (x : elt)(s : t) := Raw.In x s.(this).
    - Definition Equal (s s' : t) := ∀ a : elt, In a s ↔ In a s'.
    - Definition Subset (s s' : t) := ∀ a : elt, In a s → In a s'.
    - Definition Empty (s : t) := ∀ a : elt, ¬ In a s.
    - Definition For_all (P : elt → Prop)(s : t) := ∀ x, In x s → P x.
    - Definition Exists (P : elt → Prop)(s : t) := ∃ x, In x s ∧ P x.
    - -
    - Definition mem (x : elt)(s : t) := Raw.mem x s.(this).
    - Definition add (x : elt)(s : t) : t := Mkt (Raw.add x s.(this)).
    - Definition remove (x : elt)(s : t) : t := Mkt (Raw.remove x s.(this)).
    - Definition singleton (x : elt) : t := Mkt (Raw.singleton x).
    - Definition union (s s' : t) : t := Mkt (Raw.union s s').
    - Definition inter (s s' : t) : t := Mkt (Raw.inter s s').
    - Definition diff (s s' : t) : t := Mkt (Raw.diff s s').
    - Definition equal (s s' : t) := Raw.equal s s'.
    - Definition subset (s s' : t) := Raw.subset s s'.(this).
    - Definition empty : t := Mkt Raw.empty.
    - Definition is_empty (s : t) := Raw.is_empty s.
    - Definition elements (s : t) : list elt := Raw.elements s.
    - Definition min_elt (s : t) : option elt := Raw.min_elt s.
    - Definition max_elt (s : t) : option elt := Raw.max_elt s.
    - Definition choose (s : t) : option elt := Raw.choose s.
    - Definition compare (s1 s2 : t) : comparison := Raw.compare s1 s2.
    - Definition fold {A : Type}(f : elt → A → A)(s : t) : A → A := Raw.fold f s.
    - Definition cardinal (s : t) := Raw.cardinal s.
    - Definition filter (f : elt → bool)(s : t) : t := Mkt (Raw.filter f s).
    - Definition for_all (f : elt → bool)(s : t) := Raw.for_all f s.
    - Definition exists_ (f : elt → bool)(s : t) := Raw.exists_ f s.
    - Definition partition (f : elt → bool)(s : t) : t × t :=
    -   let p := Raw.partition f s in (Mkt (fst p), Mkt (snd p)).
    - -
    - Instance In_compat : Proper (E.eq==>eq==>iff) In.
    + Definition elt := Raw.elt.
    + Record t_ := Mkt {this :> Raw.t; is_ok : Raw.Ok this}.
    + Definition t := t_.
    + Arguments Mkt this {is_ok}.
    + #[local] Hint Resolve is_ok : typeclass_instances.
    + +
    + Definition In (x : elt)(s : t) := Raw.In x s.(this).
    + Definition Equal (s s' : t) := ∀ a : elt, In a s ↔ In a s'.
    + Definition Subset (s s' : t) := ∀ a : elt, In a s → In a s'.
    + Definition Empty (s : t) := ∀ a : elt, ¬ In a s.
    + Definition For_all (P : elt → Prop)(s : t) := ∀ x, In x s → P x.
    + Definition Exists (P : elt → Prop)(s : t) := ∃ x, In x s ∧ P x.
    + +
    + Definition mem (x : elt)(s : t) := Raw.mem x s.(this).
    + Definition add (x : elt)(s : t) : t := Mkt (Raw.add x s.(this)).
    + Definition remove (x : elt)(s : t) : t := Mkt (Raw.remove x s.(this)).
    + Definition singleton (x : elt) : t := Mkt (Raw.singleton x).
    + Definition union (s s' : t) : t := Mkt (Raw.union s s').
    + Definition inter (s s' : t) : t := Mkt (Raw.inter s s').
    + Definition diff (s s' : t) : t := Mkt (Raw.diff s s').
    + Definition equal (s s' : t) := Raw.equal s s'.
    + Definition subset (s s' : t) := Raw.subset s s'.(this).
    + Definition empty : t := Mkt Raw.empty.
    + Definition is_empty (s : t) := Raw.is_empty s.
    + Definition elements (s : t) : list elt := Raw.elements s.
    + Definition min_elt (s : t) : option elt := Raw.min_elt s.
    + Definition max_elt (s : t) : option elt := Raw.max_elt s.
    + Definition choose (s : t) : option elt := Raw.choose s.
    + Definition compare (s1 s2 : t) : comparison := Raw.compare s1 s2.
    + Definition fold {A : Type}(f : elt → A → A)(s : t) : A → A := Raw.fold f s.
    + Definition cardinal (s : t) := Raw.cardinal s.
    + Definition filter (f : elt → bool)(s : t) : t := Mkt (Raw.filter f s).
    + Definition for_all (f : elt → bool)(s : t) := Raw.for_all f s.
    + Definition exists_ (f : elt → bool)(s : t) := Raw.exists_ f s.
    + Definition partition (f : elt → bool)(s : t) : t × t :=
    +   let p := Raw.partition f s in (Mkt (fst p), Mkt (snd p)).
    + +
    + #[local] Instance In_compat : Proper (E.eq==>eq==>iff) In.
     Proof.
       repeat red.
       move ⇒ x y H_eq_xy x' y' →.
    @@ -6402,121 +6410,121 @@

    Library MSetsExtra.MSetIntervals

     Qed.

    - Definition eq : t → t → Prop := Equal.
    + Definition eq : t → t → Prop := Equal.

    - Instance eq_equiv : Equivalence eq.
    + #[local] Instance eq_equiv : Equivalence eq.
     Proof. firstorder. Qed.

    - Definition eq_dec : ∀ (s s':t), { eq s s' }+{ ¬eq s s' }.
    + Definition eq_dec : ∀ (s s':t), { eq s s' }+{ ¬eq s s' }.
     Proof.
      intros (s,Hs) (s',Hs').
    -  change ({Raw.Equal s s'}+{¬Raw.Equal s s'}).
    +  change ({Raw.Equal s s'}+{¬Raw.Equal s s'}).
      destruct (Raw.equal s s') eqn:H; [left|right];
       rewrite <- Raw.equal_spec; congruence.
     Defined.

    - Definition lt : t → t → Prop := Raw.lt.
    + Definition lt : t → t → Prop := Raw.lt.

    - Instance lt_strorder : StrictOrder lt.
    + #[local] Instance lt_strorder : StrictOrder lt.
     Proof.
       unfold lt.
       constructor. {
         move : Raw.lt_Irreflexive.
    -     rewrite /Irreflexive /complement /Reflexive.
    +     rewrite /Irreflexive /complement /Reflexive.
         move ⇒ H x.
         apply H.
       } {
         move : Raw.lt_Transitive.
    -     rewrite /Transitive.
    +     rewrite /Transitive.
         move ⇒ H x y z.
         apply H.
       }
     Qed.

    - Instance lt_compat : Proper (eq==>eq==>iff) lt.
    + #[local] Instance lt_compat : Proper (eq==>eq==>iff) lt.
     Proof.
       repeat red.
       move ⇒ [x1 H_x1_ok] [y1 H_y1_ok] H_eq.
       move ⇒ [x2 H_x2_ok] [y2 H_y2_ok].
       move : H_eq.
       rewrite /eq /lt /Equal /In /=.
    -   replace (∀ a : elt, Raw.In a x1 ↔ Raw.In a y1) with
    +   replace (∀ a : elt, Raw.In a x1 ↔ Raw.In a y1) with
         (Raw.Equal x1 y1) by done.
    -   replace (∀ a : elt, Raw.In a x2 ↔ Raw.In a y2) with
    +   replace (∀ a : elt, Raw.In a x2 ↔ Raw.In a y2) with
         (Raw.Equal x2 y2) by done.
       rewrite -!Raw.equal_spec !Raw.equal_alt_def.
       move ⇒ → → //.
     Qed.

    - Section Spec.
    -  Variable s s' : t.
    -  Variable x y : elt.
    -  Variable f : elt → bool.
    -  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    + Section Spec.
    +  Variable s s' : t.
    +  Variable x y : elt.
    +  Variable f : elt → bool.
    +  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).

    -  Lemma mem_spec : mem x s = true ↔ In x s.
    +  Lemma mem_spec : mem x s = true ↔ In x s.
      Proof. exact (Raw.mem_spec _ _ _). Qed.
    -  Lemma equal_spec : equal s s' = true ↔ Equal s s'.
    +  Lemma equal_spec : equal s s' = true ↔ Equal s s'.
      Proof. rewrite Raw.equal_spec //. Qed.
    -  Lemma subset_spec : subset s s' = true ↔ Subset s s'.
    +  Lemma subset_spec : subset s s' = true ↔ Subset s s'.
      Proof. exact (Raw.subset_spec _ _ _ _). Qed.
    -  Lemma empty_spec : Empty empty.
    +  Lemma empty_spec : Empty empty.
      Proof. exact Raw.empty_spec. Qed.
    -  Lemma is_empty_spec : is_empty s = true ↔ Empty s.
    +  Lemma is_empty_spec : is_empty s = true ↔ Empty s.
      Proof. rewrite Raw.is_empty_spec //. Qed.
    -  Lemma add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    +  Lemma add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
      Proof. exact (Raw.add_spec _ _ _ _). Qed.
    -  Lemma remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    +  Lemma remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
      Proof. exact (Raw.remove_spec _ _ _ _). Qed.
    -  Lemma singleton_spec : In y (singleton x) ↔ E.eq y x.
    +  Lemma singleton_spec : In y (singleton x) ↔ E.eq y x.
      Proof. exact (Raw.singleton_spec _ _). Qed.
    -  Lemma union_spec : In x (union s s') ↔ In x s ∨ In x s'.
    +  Lemma union_spec : In x (union s s') ↔ In x s ∨ In x s'.
      Proof. exact (Raw.union_spec _ _ _ _ _). Qed.
    -  Lemma inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
    +  Lemma inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
      Proof. exact (Raw.inter_spec _ _ _ _ _). Qed.
    -  Lemma diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +  Lemma diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
      Proof. exact (Raw.diff_spec _ _ _ _ _). Qed.
    -  Lemma fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    -      fold f s i = fold_left (fun a e ⇒ f e a) (elements s) i.
    +  Lemma fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    +      fold f s i = fold_left (fun a e ⇒ f e a) (elements s) i.
      Proof. exact (@Raw.fold_spec _). Qed.
    -  Lemma cardinal_spec : cardinal s = length (elements s).
    +  Lemma cardinal_spec : cardinal s = length (elements s).
      Proof. exact (@Raw.cardinal_spec s). Qed.
    -  Lemma filter_spec : compatb f →
    -    (In x (filter f s) ↔ In x s ∧ f x = true).
    +  Lemma filter_spec : compatb f →
    +    (In x (filter f s) ↔ In x s ∧ f x = true).
      Proof. move ⇒ _; exact (@Raw.filter_spec _ _ _ _). Qed.
    -  Lemma for_all_spec : compatb f →
    -    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +  Lemma for_all_spec : compatb f →
    +    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
      Proof. exact (@Raw.for_all_spec _ _ _). Qed.
    -  Lemma exists_spec : compatb f →
    -    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +  Lemma exists_spec : compatb f →
    +    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
      Proof. exact (@Raw.exists_spec _ _ _). Qed.
    -  Lemma partition_spec1 : compatb f → Equal (fst (partition f s)) (filter f s).
    +  Lemma partition_spec1 : compatb f → Equal (fst (partition f s)) (filter f s).
      Proof. move ⇒ _; exact (@Raw.partition_spec1 _ _). Qed.
    -  Lemma partition_spec2 : compatb f →
    -      Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
    +  Lemma partition_spec2 : compatb f →
    +      Equal (snd (partition f s)) (filter (fun x ⇒ negb (f x)) s).
      Proof. move ⇒ _; exact (@Raw.partition_spec2 _ _ _). Qed.
    -  Lemma elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    +  Lemma elements_spec1 : InA E.eq x (elements s) ↔ In x s.
      Proof. rewrite /In /Raw.In /elements //. Qed.
    -  Lemma elements_spec2w : NoDupA E.eq (elements s).
    +  Lemma elements_spec2w : NoDupA E.eq (elements s).
      Proof. exact (Raw.elements_spec2w _ _). Qed.
    -  Lemma elements_spec2 : sort E.lt (elements s).
    +  Lemma elements_spec2 : sort E.lt (elements s).
      Proof. exact (Raw.elements_sorted _ _). Qed.
    -  Lemma choose_spec1 : choose s = Some x → In x s.
    +  Lemma choose_spec1 : choose s = Some x → In x s.
      Proof. exact (Raw.choose_spec1 _ _ _). Qed.
    -  Lemma choose_spec2 : choose s = None → Empty s.
    +  Lemma choose_spec2 : choose s = None → Empty s.
      Proof. exact (Raw.choose_spec2 _). Qed.
    -  Lemma choose_spec3 : choose s = Some x → choose s' = Some y →
    -    Equal s s' → E.eq x y.
    +  Lemma choose_spec3 : choose s = Some x → choose s' = Some y →
    +    Equal s s' → E.eq x y.
      Proof.
        intros H1 H2 H3.
    -    suff → : x = y. {
    +    suff → : x = y. {
          apply E.eq_equiv.
        }
        move : H1 H2 H3.
    @@ -6524,31 +6532,31 @@

    Library MSetsExtra.MSetIntervals

      Qed.

    -  Lemma min_elt_spec1 : choose s = Some x → In x s.
    +  Lemma min_elt_spec1 : choose s = Some x → In x s.
      Proof. exact (Raw.min_elt_spec1 _ _ _). Qed.
    -  Lemma min_elt_spec2 : min_elt s = Some x → In y s → ¬ E.lt y x.
    +  Lemma min_elt_spec2 : min_elt s = Some x → In y s → ¬ E.lt y x.
      Proof. exact (Raw.min_elt_spec2 _ _ _ _). Qed.
    -  Lemma min_elt_spec3 : choose s = None → Empty s.
    +  Lemma min_elt_spec3 : choose s = None → Empty s.
      Proof. exact (Raw.min_elt_spec3 _). Qed.

    -  Lemma max_elt_spec1 : max_elt s = Some x → In x s.
    +  Lemma max_elt_spec1 : max_elt s = Some x → In x s.
      Proof. exact (Raw.max_elt_spec1 _ _ _). Qed.
    -  Lemma max_elt_spec2 : max_elt s = Some x → In y s → ¬ E.lt x y.
    +  Lemma max_elt_spec2 : max_elt s = Some x → In y s → ¬ E.lt x y.
      Proof. exact (Raw.max_elt_spec2 _ _ _ _). Qed.
    -  Lemma max_elt_spec3 : max_elt s = None → Empty s.
    +  Lemma max_elt_spec3 : max_elt s = None → Empty s.
      Proof. exact (Raw.max_elt_spec3 _). Qed.

    -  Lemma compare_spec : CompSpec eq lt s s' (compare s s').
    +  Lemma compare_spec : CompSpec eq lt s s' (compare s s').
      Proof.
        generalize s s'.
        move ⇒ [s1 H_ok_s1] [s2 H_ok_s2].
        move : (Raw.compare_spec s1 s2).
    -    rewrite /CompSpec /eq /Equal /In /lt /compare /=.
    -    replace (∀ a : elt, Raw.In a s1 ↔ Raw.In a s2) with
    +    rewrite /CompSpec /eq /Equal /In /lt /compare /=.
    +    replace (∀ a : elt, Raw.In a s1 ↔ Raw.In a s2) with
        (Raw.Equal s1 s2) by done.
    -    suff H_eq : (Raw.Equal s1 s2) ↔ (s1 = s2). {
    +    suff H_eq : (Raw.Equal s1 s2) ↔ (s1 = s2). {
          move ⇒ [] H; constructor ⇒ //.
          by rewrite H_eq.
        }
    @@ -6565,7 +6573,7 @@

    Library MSetsExtra.MSetIntervals

    -

    Instantiations

    +

    Instantiations

    @@ -6573,131 +6581,131 @@

    Library MSetsExtra.MSetIntervals

    It remains to provide instantiations for commonly used datatypes.
    -

    Z

    +

    Z


    -Module ElementEncodeZ <: ElementEncode.
    -  Module E := Z.
    +Module ElementEncodeZ <: ElementEncode.
    +  Module E := Z.

    -  Definition encode (z : Z) := z.
    -  Definition decode (z : Z) := z.
    +  Definition encode (z : Z) := z.
    +  Definition decode (z : Z) := z.

    -  Lemma decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    +  Lemma decode_encode_ok: ∀ (e : E.t),
    +    decode (encode e) = e.
      Proof. by []. Qed.

    -  Lemma encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +  Lemma encode_eq : ∀ (e1 e2 : E.t),
    +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
      Proof. by []. Qed.

    -  Lemma encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Lemma encode_lt : ∀ (e1 e2 : E.t),
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
      Proof. by []. Qed.

    End ElementEncodeZ.

    -Module MSetIntervalsZ <: SetsOn Z := MSetIntervals ElementEncodeZ.
    +Module MSetIntervalsZ <: SetsOn Z := MSetIntervals ElementEncodeZ.

    -

    N

    +

    N


    -Module ElementEncodeN <: ElementEncode.
    -  Module E := N.
    +Module ElementEncodeN <: ElementEncode.
    +  Module E := N.

    -  Definition encode (n : N) := Z.of_N n.
    -  Definition decode (z : Z) := Z.to_N z.
    +  Definition encode (n : N) := Z.of_N n.
    +  Definition decode (z : Z) := Z.to_N z.

    -  Lemma decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    +  Lemma decode_encode_ok: ∀ (e : E.t),
    +    decode (encode e) = e.
      Proof.
        intros e.
    -    rewrite /encode /decode N2Z.id //.
    +    rewrite /encode /decode N2Z.id //.
      Qed.

    -  Lemma encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +  Lemma encode_eq : ∀ (e1 e2 : E.t),
    +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
      Proof.
        intros e1 e2.
    -    rewrite /encode /Z.eq N2Z.inj_iff /E.eq //.
    +    rewrite /encode /Z.eq N2Z.inj_iff /E.eq //.
      Qed.

    -  Lemma encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Lemma encode_lt : ∀ (e1 e2 : E.t),
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
      Proof.
        intros e1 e2.
    -    rewrite /encode -N2Z.inj_lt //.
    +    rewrite /encode -N2Z.inj_lt //.
      Qed.

    End ElementEncodeN.

    -Module MSetIntervalsN <: SetsOn N := MSetIntervals ElementEncodeN.
    +Module MSetIntervalsN <: SetsOn N := MSetIntervals ElementEncodeN.

    -

    nat

    +

    nat

    -Module ElementEncodeNat <: ElementEncode.
    -  Module E := Nat.
    +Module ElementEncodeNat <: ElementEncode.
    +  Module E := Nat.

    -  Definition encode (n : nat) := Z.of_nat n.
    -  Definition decode (z : Z) := Z.to_nat z.
    +  Definition encode (n : nat) := Z.of_nat n.
    +  Definition decode (z : Z) := Z.to_nat z.

    -  Lemma decode_encode_ok: ∀ (e : E.t),
    -    decode (encode e) = e.
    +  Lemma decode_encode_ok: ∀ (e : E.t),
    +    decode (encode e) = e.
      Proof.
        intros e.
    -    rewrite /encode /decode Nat2Z.id //.
    +    rewrite /encode /decode Nat2Z.id //.
      Qed.

    -  Lemma encode_eq : ∀ (e1 e2 : E.t),
    -    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
    +  Lemma encode_eq : ∀ (e1 e2 : E.t),
    +    (Z.eq (encode e1) (encode e2)) ↔ E.eq e1 e2.
      Proof.
        intros e1 e2.
    -    rewrite /encode /Z.eq Nat2Z.inj_iff /E.eq //.
    +    rewrite /encode /Z.eq Nat2Z.inj_iff /E.eq //.
      Qed.

    -  Lemma encode_lt : ∀ (e1 e2 : E.t),
    -    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
    +  Lemma encode_lt : ∀ (e1 e2 : E.t),
    +    (Z.lt (encode e1) (encode e2)) ↔ E.lt e1 e2.
      Proof.
        intros e1 e2.
    -    rewrite /encode -Nat2Z.inj_lt //.
    +    rewrite /encode -Nat2Z.inj_lt //.
      Qed.

    End ElementEncodeNat.

    -Module MSetIntervalsNat <: SetsOn Nat := MSetIntervals ElementEncodeNat.
    +Module MSetIntervalsNat <: SetsOn Nat := MSetIntervals ElementEncodeNat.

    diff --git a/html/MSetsExtra.MSetListWithDups.html b/html/MSetsExtra.MSetListWithDups.html index 26bd69f..4b0ff38 100644 --- a/html/MSetsExtra.MSetListWithDups.html +++ b/html/MSetsExtra.MSetListWithDups.html @@ -24,7 +24,7 @@

    Library MSetsExtra.MSetListWithDups

    -

    Weak sets implemented as lists with duplicates

    +

    Weak sets implemented as lists with duplicates

    @@ -45,17 +45,17 @@

    Library MSetsExtra.MSetListWithDups


    -Require Export MSetInterface.
    +Require Export MSetInterface.
    Require Import mathcomp.ssreflect.ssreflect.
    -Require Import List OrdersFacts OrdersLists.
    -Require Import Sorting Permutation.
    +Require Import List OrdersFacts OrdersLists.
    +Require Import Sorting Permutation.
    Require Import MSetWithDups.

    -

    Removing duplicates from sorted lists

    +

    Removing duplicates from sorted lists

    @@ -72,7 +72,7 @@

    Library MSetsExtra.MSetListWithDups

    duplicates.
    -Module RemoveDupsFromSorted (Import X:OrderedType).
    +Module RemoveDupsFromSorted (Import X:OrderedType).

    @@ -81,23 +81,23 @@

    Library MSetsExtra.MSetListWithDups

    First, we need some infrastructure for our ordered type
    -  Module Import MX := OrderedTypeFacts X.
    +  Module Import MX := OrderedTypeFacts X.

    -  Module Import XTotalLeBool <: TotalLeBool.
    -    Definition t := X.t.
    -    Definition leb x y :=
    -      match X.compare x y with
    -        | Lt ⇒ true
    -        | Eq ⇒ true
    -        | Gt ⇒ false
    +  Module Import XTotalLeBool <: TotalLeBool.
    +    Definition t := X.t.
    +    Definition leb x y :=
    +      match X.compare x y with
    +        | Lt ⇒ true
    +        | Eq ⇒ true
    +        | Gt ⇒ false
          end.

    -    Infix "<=?" := leb (at level 35).
    +    Infix "<=?" := leb (at level 35).

    -    Theorem leb_total : ∀ (a1 a2 : t), (a1 <=? a2 = true) ∨ (a2 <=? a1 = true).
    +    Theorem leb_total : ∀ (a1 a2 : t), (a1 <=? a2 = true) ∨ (a2 <=? a1 = true).
        Proof.
          intros a1 a2.
          unfold leb.
    @@ -106,11 +106,11 @@

    Library MSetsExtra.MSetListWithDups

        Qed.

    -    Definition le x y := (leb x y = true).
    +    Definition le x y := (leb x y = true).
      End XTotalLeBool.

    -  Lemma eqb_eq_alt : ∀ x y, eqb x y = true ↔ eq x y.
    +  Lemma eqb_eq_alt : ∀ x y, eqb x y = true ↔ eq x y.
      Proof.
        intros x y.
        rewrite eqb_alt -compare_eq_iff.
    @@ -124,22 +124,22 @@

    Library MSetsExtra.MSetListWithDups

    Now we can define our main function
    -  Fixpoint remove_dups_from_sortedA_aux (acc : list t) (l : list t) : list t :=
    -    match l with
    -    | nil ⇒ List.rev' acc
    -    | x :: xs ⇒
    +  Fixpoint remove_dups_from_sortedA_aux (acc : list t) (l : list t) : list t :=
    +    match l with
    +    | nil ⇒ List.rev' acc
    +    | x :: xs ⇒
           match xs with
    -       | nil ⇒ List.rev' (x :: acc)
    -       | y :: ys ⇒
    -           if eqb x y then
    -             remove_dups_from_sortedA_aux acc xs
    -           else
    -             remove_dups_from_sortedA_aux (x::acc) xs
    +       | nil ⇒ List.rev' (x :: acc)
    +       | y :: ys ⇒
    +           if eqb x y then
    +             remove_dups_from_sortedA_aux acc xs
    +           else
    +             remove_dups_from_sortedA_aux (x::acc) xs
           end
        end.

    -  Definition remove_dups_from_sortedA := remove_dups_from_sortedA_aux (nil : list t).
    +  Definition remove_dups_from_sortedA := remove_dups_from_sortedA_aux (nil : list t).

    @@ -148,40 +148,40 @@

    Library MSetsExtra.MSetListWithDups

    We can prove some technical lemmata
    -  Lemma remove_dups_from_sortedA_aux_alt : ∀ (l : list X.t) acc,
    -    remove_dups_from_sortedA_aux acc l =
    -    List.rev acc ++ (remove_dups_from_sortedA l).
    +  Lemma remove_dups_from_sortedA_aux_alt : ∀ (l : list X.t) acc,
    +    remove_dups_from_sortedA_aux acc l =
    +    List.rev acc ++ (remove_dups_from_sortedA l).
      Proof.
        unfold remove_dups_from_sortedA.
        induction l as [| x xs IH] ⇒ acc. {
    -      rewrite /remove_dups_from_sortedA_aux /rev' -!rev_alt /= app_nil_r //.
    +      rewrite /remove_dups_from_sortedA_aux /rev' -!rev_alt /= app_nil_r //.
        } {
          rewrite /=.
          case_eq xs. {
    -        rewrite /rev' -!rev_alt //.
    +        rewrite /rev' -!rev_alt //.
          } {
            move ⇒ y ys H_xs_eq.
    -        rewrite -!H_xs_eq !(IH acc) !(IH (x :: acc)) (IH (x::nil)).
    +        rewrite -!H_xs_eq !(IH acc) !(IH (x :: acc)) (IH (x::nil)).
            case (eqb x y) ⇒ //.
    -        rewrite /= -app_assoc //.
    +        rewrite /= -app_assoc //.
          }
        }
      Qed.

    -  Lemma remove_dups_from_sortedA_alt :
    -    âˆ€ (l : list t),
    -    remove_dups_from_sortedA l =
    -    match l with
    -    | nil ⇒ nil
    -    | x :: xs ⇒
    +  Lemma remove_dups_from_sortedA_alt :
    +    âˆ€ (l : list t),
    +    remove_dups_from_sortedA l =
    +    match l with
    +    | nil ⇒ nil
    +    | x :: xs ⇒
           match xs with
    -       | nil ⇒ l
    -       | y :: ys ⇒
    -           if eqb x y then
    +       | nil ⇒ l
    +       | y :: ys ⇒
    +           if eqb x y then
                 remove_dups_from_sortedA xs
    -           else
    -             x :: remove_dups_from_sortedA xs
    +           else
    +             x :: remove_dups_from_sortedA xs
           end
        end.
      Proof.
    @@ -189,7 +189,7 @@

    Library MSetsExtra.MSetListWithDups

          done.
        } {
          intros x xs.
    -      rewrite /remove_dups_from_sortedA /= /rev' /=.
    +      rewrite /remove_dups_from_sortedA /= /rev' /=.
          case xs ⇒ //.
          move ⇒ y ys.
          rewrite !remove_dups_from_sortedA_aux_alt /= //.
    @@ -197,24 +197,24 @@

    Library MSetsExtra.MSetListWithDups

      Qed.

    -  Lemma remove_dups_from_sortedA_hd :
    -      âˆ€ x xs,
    -      âˆƒ (x':t) xs',
    -        remove_dups_from_sortedA (x :: xs) =
    -        (x' :: xs') ∧ (eqb x x' = true).
    +  Lemma remove_dups_from_sortedA_hd :
    +      âˆ€ x xs,
    +      âˆƒ (x':t) xs',
    +        remove_dups_from_sortedA (x :: xs) =
    +        (x' :: xs') ∧ (eqb x x' = true).
      Proof.
        intros x xs.
        move : x;
        induction xs as [| y ys IH] ⇒ x. {
          rewrite remove_dups_from_sortedA_alt.
    -      âˆƒ x, nil.
    +      âˆƒ x, nil.
          split; first reflexivity.
          rewrite eqb_alt compare_refl //.
        } {
          rewrite remove_dups_from_sortedA_alt.
          case_eq (eqb x y); last first. {
            move ⇒ _.
    -        âˆƒ x, (remove_dups_from_sortedA (y :: ys)).
    +        âˆƒ x, (remove_dups_from_sortedA (y :: ys)).
            split; first reflexivity.
            rewrite eqb_alt compare_refl //.
          } {
    @@ -236,14 +236,14 @@

    Library MSetsExtra.MSetListWithDups

    Finally we get our main result for removing duplicates from sorted lists
    -  Lemma remove_dups_from_sortedA_spec :
    -    âˆ€ (l : list t),
    -      Sorted le l →
    -      let l' := remove_dups_from_sortedA l in (
    +  Lemma remove_dups_from_sortedA_spec :
    +    âˆ€ (l : list t),
    +      Sorted le l →
    +      let l' := remove_dups_from_sortedA l in (
        
    -      Sorted lt l' ∧
    -      NoDupA eq l' ∧
    -      (∀ x, InA eq x l ↔ InA eq x l')).
    +      Sorted lt l' ∧
    +      NoDupA eq l' ∧
    +      (∀ x, InA eq x l ↔ InA eq x l')).
      Proof.
        simpl.
        induction l as [| x xs IH]. {
    @@ -255,21 +255,21 @@

    Library MSetsExtra.MSetListWithDups

          case xs ⇒ {xs}. {
            move ⇒ _.
            split; last split. {
    -          apply Sorted_cons ⇒ //.
    +          apply Sorted_cons ⇒ //.
            } {
    -          apply NoDupA_singleton.
    +          apply NoDupA_singleton.
            } {
              done.
            }
          } {
            move ⇒ y ys IH H_sorted_x_y_ys.
    -        apply Sorted_inv in H_sorted_x_y_ys as [H_sorted_y_ys H_hd_rel].
    -        apply HdRel_inv in H_hd_rel.
    +        apply Sorted_inv in H_sorted_x_y_ys as [H_sorted_y_ys H_hd_rel].
    +        apply HdRel_inv in H_hd_rel.

    -        have : ∃ y' ys',
    -          remove_dups_from_sortedA (y :: ys) = y' :: ys' ∧
    -          eqb y y' = true. {
    +        have : ∃ y' ys',
    +          remove_dups_from_sortedA (y :: ys) = y' :: ys' ∧
    +          eqb y y' = true. {
              apply remove_dups_from_sortedA_hd ⇒ //.
            }
            move ⇒ [y'] [ys'] [H_yys'_intro] /eqb_eq_alt H_eq_y_y'.
    @@ -285,22 +285,22 @@

    Library MSetsExtra.MSetListWithDups

              split ⇒ //.
              split ⇒ //.
              move ⇒ x'.
    -          rewrite InA_cons IH3.
    +          rewrite InA_cons IH3.
              split; last tauto.
              move ⇒ [] //.
              move ⇒ H_eq_x'_x.
    -          apply InA_cons_hd.
    +          apply InA_cons_hd.
              apply eq_trans with (y := x) ⇒ //.
              apply eq_trans with (y := y) ⇒ //.
            }
            move ⇒ H_neqb_x_y.

    -        have H_sorted : Sorted lt (x :: y' :: ys'). {
    -          apply Sorted_cons ⇒ //.
    -          apply HdRel_cons.
    +        have H_sorted : Sorted lt (x :: y' :: ys'). {
    +          apply Sorted_cons ⇒ //.
    +          apply HdRel_cons.
              rewrite -compare_lt_iff.
    -          suff : (compare x y = Lt). {
    +          suff : (compare x y = Lt). {
                setoid_rewrite compare_compat; eauto;
                  apply eq_refl.
              }
    @@ -311,23 +311,23 @@

    Library MSetsExtra.MSetListWithDups

            split; last split. {
              assumption.
            } {
    -          apply NoDupA_cons ⇒ //.
    +          apply NoDupA_cons ⇒ //.

    -          move ⇒ /InA_alt [x'] [H_eq_xx'] H_in_x'.
    -          have : Forall (lt x) (y' :: ys'). {
    -            apply Sorted_extends ⇒ //.
    -            rewrite /Relations_1.Transitive.
    +          move ⇒ /InA_alt [x'] [H_eq_xx'] H_in_x'.
    +          have : Forall (lt x) (y' :: ys'). {
    +            apply Sorted_extends ⇒ //.
    +            rewrite /Relations_1.Transitive.
                by apply lt_trans.
              }
    -          rewrite Forall_forall ⇒ H_forall.
    +          rewrite Forall_forall ⇒ H_forall.
              move : (H_forall _ H_in_x') ⇒ {H_forall}.
              move : H_eq_xx'.
              rewrite -compare_lt_iff -compare_eq_iff.
              move ⇒ → //.
            } {
              move ⇒ x0.
    -          rewrite !(InA_cons eq x0 x) IH3 //.
    +          rewrite !(InA_cons eq x0 x) IH3 //.
            }
          }
        }
    @@ -340,26 +340,26 @@

    Library MSetsExtra.MSetListWithDups

    Next, we combine it with sorting
    -  Module Import XSort := Sort XTotalLeBool.
    +  Module Import XSort := Sort XTotalLeBool.

    -  Definition remove_dups_by_sortingA (l : list t) : list t :=
    -    remove_dups_from_sortedA (XSort.sort l).
    +  Definition remove_dups_by_sortingA (l : list t) : list t :=
    +    remove_dups_from_sortedA (XSort.sort l).

    -  Lemma remove_dups_by_sortingA_spec :
    -    âˆ€ (l : list t),
    -      let l' := remove_dups_by_sortingA l in (
    +  Lemma remove_dups_by_sortingA_spec :
    +    âˆ€ (l : list t),
    +      let l' := remove_dups_by_sortingA l in (
        
    -      Sorted lt l' ∧
    -      NoDupA eq l' ∧
    -      (∀ x, InA eq x l ↔ InA eq x l')).
    +      Sorted lt l' ∧
    +      NoDupA eq l' ∧
    +      (∀ x, InA eq x l ↔ InA eq x l')).
      Proof.
        intro l.

    -    suff : (∀ x : X.t, InA eq x (sort l) ↔ InA eq x l) ∧
    -           Sorted le (sort l). {
    +    suff : (∀ x : X.t, InA eq x (sort l) ↔ InA eq x l) ∧
    +           Sorted le (sort l). {
          
          unfold remove_dups_by_sortingA.
          move : (remove_dups_from_sortedA_spec (sort l)).
    @@ -374,20 +374,19 @@

    Library MSetsExtra.MSetListWithDups

        }

        split. {
    -      have H_in_sort : ∀ x, List.In x (XSort.sort l) ↔ List.In x l. {
    +      have H_in_sort : ∀ x, List.In x (XSort.sort l) ↔ List.In x l. {
            intros x.
            have H_perm := (XSort.Permuted_sort l).
    -        split; apply Permutation_in ⇒ //.
    -        by apply Permutation_sym.
    +        split; apply Permutation_in ⇒ //.
    +        by apply Permutation_sym.
          }

          intros x.
    -      rewrite !InA_alt.
    +      rewrite !InA_alt.
          setoid_rewrite H_in_sort ⇒ //.
        } {
    -
    -      move : (LocallySorted_sort l).
    -      rewrite /is_true /le /leb //.
    +      move : (Sorted_sort l).
    +      rewrite /is_true /le /leb //.
        }
      Qed.
    @@ -398,7 +397,7 @@

    Library MSetsExtra.MSetListWithDups

    -

    Operations Module

    +

    Operations Module

    @@ -410,113 +409,113 @@

    Library MSetsExtra.MSetListWithDups


    -Module Ops (X:OrderedType) <: WOps X.
    +Module Ops (X:OrderedType) <: WOps X.

    -  Module RDFS := RemoveDupsFromSorted X.
    -  Module Import MX := OrderedTypeFacts X.
    +  Module RDFS := RemoveDupsFromSorted X.
    +  Module Import MX := OrderedTypeFacts X.

    -  Definition elt := X.t.
    -  Definition t := list elt.
    +  Definition elt := X.t.
    +  Definition t := list elt.

    -  Definition empty : t := nil.
    +  Definition empty : t := nil.

    -  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.
    -  Fixpoint mem (x : elt) (s : t) : bool :=
    -    match s with
    -    | nil ⇒ false
    -    | y :: l ⇒
    -           match X.compare x y with
    -               Eq ⇒ true
    -             | _ ⇒ mem x l
    +  Definition is_empty (l : t) := match l with nil ⇒ true | _ ⇒ false end.
    +  Fixpoint mem (x : elt) (s : t) : bool :=
    +    match s with
    +    | nil ⇒ false
    +    | y :: l ⇒
    +           match X.compare x y with
    +               Eq ⇒ true
    +             | _ ⇒ mem x l
               end
        end.

    -  Definition add x (s : t) := x :: s.
    -  Definition singleton (x : elt) := x :: nil.
    +  Definition add x (s : t) := x :: s.
    +  Definition singleton (x : elt) := x :: nil.

    -  Fixpoint rev_filter_aux acc (f : elt → bool) s :=
    -    match s with
    -       nil ⇒ acc
    -     | (x :: xs) ⇒ rev_filter_aux (if (f x) then (x :: acc) else acc) f xs
    +  Fixpoint rev_filter_aux acc (f : elt → bool) s :=
    +    match s with
    +       nil ⇒ acc
    +     | (x :: xs) ⇒ rev_filter_aux (if (f x) then (x :: acc) else acc) f xs
        end.
    -  Definition rev_filter := rev_filter_aux nil.
    +  Definition rev_filter := rev_filter_aux nil.

    -  Definition filter (f : elt → bool) (s : t) : t := rev_filter f s.
    +  Definition filter (f : elt → bool) (s : t) : t := rev_filter f s.

    -  Definition remove x s :=
    -    rev_filter (fun y ⇒ match X.compare x y with Eq ⇒ false | _ ⇒ true end) s.
    +  Definition remove x s :=
    +    rev_filter (fun y ⇒ match X.compare x y with Eq ⇒ false | _ ⇒ true end) s.

    -  Definition union (s1 s2 : t) : t :=
    -    List.rev_append s2 s1.
    +  Definition union (s1 s2 : t) : t :=
    +    List.rev_append s2 s1.

    -  Definition inter (s1 s2 : t) : t :=
    -    rev_filter (fun y ⇒ mem y s2) s1.
    +  Definition inter (s1 s2 : t) : t :=
    +    rev_filter (fun y ⇒ mem y s2) s1.

    -  Definition elements (x : t) : list elt := x.
    +  Definition elements (x : t) : list elt := x.

    -  Definition elements_dist (x : t) : list elt :=
    -    RDFS.remove_dups_by_sortingA x.
    +  Definition elements_dist (x : t) : list elt :=
    +    RDFS.remove_dups_by_sortingA x.

    -  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    -    fold_left (flip f) (elements s) i.
    +  Definition fold {B : Type} (f : elt → B → B) (s : t) (i : B) : B :=
    +    fold_left (flip f) (elements s) i.

    -  Definition diff (s s' : t) : t := fold remove s' s.
    +  Definition diff (s s' : t) : t := fold remove s' s.

    -  Definition subset (s s' : t) : bool :=
    -    List.forallb (fun x ⇒ mem x s') s.
    +  Definition subset (s s' : t) : bool :=
    +    List.forallb (fun x ⇒ mem x s') s.

    -  Definition equal (s s' : t) : bool := andb (subset s s') (subset s' s).
    +  Definition equal (s s' : t) : bool := andb (subset s s') (subset s' s).

    -  Fixpoint for_all (f : elt → bool) (s : t) : bool :=
    -    match s with
    -    | nil ⇒ true
    -    | x :: l ⇒ if f x then for_all f l else false
    +  Fixpoint for_all (f : elt → bool) (s : t) : bool :=
    +    match s with
    +    | nil ⇒ true
    +    | x :: l ⇒ if f x then for_all f l else false
        end.

    -  Fixpoint exists_ (f : elt → bool) (s : t) : bool :=
    -    match s with
    -    | nil ⇒ false
    -    | x :: l ⇒ if f x then true else exists_ f l
    +  Fixpoint exists_ (f : elt → bool) (s : t) : bool :=
    +    match s with
    +    | nil ⇒ false
    +    | x :: l ⇒ if f x then true else exists_ f l
        end.

    -  Fixpoint partition_aux (a1 a2 : t) (f : elt → bool) (s : t) : t × t :=
    -    match s with
    -    | nil ⇒ (a1, a2)
    -    | x :: l ⇒
    -        if f x then partition_aux (x :: a1) a2 f l else
    -                    partition_aux a1 (x :: a2) f l
    +  Fixpoint partition_aux (a1 a2 : t) (f : elt → bool) (s : t) : t × t :=
    +    match s with
    +    | nil ⇒ (a1, a2)
    +    | x :: l ⇒
    +        if f x then partition_aux (x :: a1) a2 f l else
    +                    partition_aux a1 (x :: a2) f l
        end.

    -  Definition partition := partition_aux nil nil.
    +  Definition partition := partition_aux nil nil.

    -  Definition cardinal (s : t) : nat := length (elements_dist s).
    +  Definition cardinal (s : t) : nat := length (elements_dist s).

    -  Definition choose (s : t) : option elt :=
    -     match s with
    -      | nil ⇒ None
    -      | x::_ ⇒ Some x
    +  Definition choose (s : t) : option elt :=
    +     match s with
    +      | nil ⇒ None
    +      | x::_ ⇒ Some x
         end.

    @@ -526,7 +525,7 @@

    Library MSetsExtra.MSetListWithDups

    -

    Main Module

    +

    Main Module

    @@ -540,38 +539,38 @@

    Library MSetsExtra.MSetListWithDups

    -Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
    +Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
      Include Ops E.
    -  Import MX.
    +  Import MX.

    -

    Proofs of set operation specifications.

    +

    Proofs of set operation specifications.

    Logical predicates
    -  Definition In x (s : t) := SetoidList.InA E.eq x s.
    +  Definition In x (s : t) := SetoidList.InA E.eq x s.

    -  Instance In_compat : Proper (E.eq==>eq==>iff) In.
    +  #[local] Instance In_compat : Proper (E.eq==>eq==>iff) In.
      Proof. repeat red. intros. rewrite H H0. auto. Qed.

    -  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    -  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    -  Definition Empty s := ∀ a : elt, ¬ In a s.
    -  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    -  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.
    +  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    +  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    +  Definition Empty s := ∀ a : elt, ¬ In a s.
    +  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    +  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.

    -  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
    -  Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).
    +  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
    +  Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).

    -  Definition eq : t → t → Prop := Equal.
    -  Lemma eq_equiv : Equivalence eq.
    +  Definition eq : t → t → Prop := Equal.
    +  Lemma eq_equiv : Equivalence eq.
      Proof.
        constructor. {
          done.
    @@ -593,52 +592,52 @@

    Library MSetsExtra.MSetListWithDups


    -  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    +  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).

    -  Lemma mem_spec : ∀ s x, mem x s = true ↔ In x s.
    +  Lemma mem_spec : ∀ s x, mem x s = true ↔ In x s.
      Proof.
        induction s as [| y s' IH]. {
          move ⇒ x.
    -      rewrite /= /In InA_nil.
    +      rewrite /= /In InA_nil.
          split ⇒ //.
        } {
          move ⇒ x.
    -      rewrite /= /In InA_cons.
    +      rewrite /= /In InA_cons.
          move : (MX.compare_eq_iff x y).
          case (E.compare x y). {
            tauto.
          } {
    -        rewrite IH; intuition; inversion H.
    +        rewrite IH; intuition; inversion H1.
          } {
    -        rewrite IH; intuition; inversion H.
    +        rewrite IH; intuition; inversion H1.
          }
        }
     Qed.

    -  Lemma subset_spec : ∀ s s', subset s s' = true ↔ s[<=]s'.
    +  Lemma subset_spec : ∀ s s', subset s s' = true ↔ s[<=]s'.
      Proof.
        intros s s'.
    -    rewrite /subset forallb_forall /Subset /In.
    +    rewrite /subset forallb_forall /Subset /In.
        split. {
    -      move ⇒ H z /InA_alt [] x [H_z_eq] H_in.
    +      move ⇒ H z /InA_alt [] x [H_z_eq] H_in.
          move : (H _ H_in).
          rewrite mem_spec.
          setoid_replace z with x ⇒ //.
        } {
          move ⇒ H z H_in.
          rewrite mem_spec.
    -      apply H, In_InA ⇒ //.
    +      apply H, In_InA ⇒ //.
          apply E.eq_equiv.
        }
      Qed.

    -  Lemma equal_spec : ∀ s s', equal s s' = true ↔ s[=]s'.
    +  Lemma equal_spec : ∀ s s', equal s s' = true ↔ s[=]s'.
      Proof.
        intros s s'.
    -    rewrite /Equal /equal Bool.andb_true_iff !subset_spec /Subset.
    +    rewrite /Equal /equal Bool.andb_true_iff !subset_spec /Subset.
        split. {
          move ⇒ [H1 H2] a.
          split.
    @@ -651,70 +650,65 @@

    Library MSetsExtra.MSetListWithDups

      Qed.

    -  Lemma eq_dec : ∀ x y : t, {eq x y} + {¬ eq x y}.
    +  Lemma eq_dec : ∀ x y : t, {eq x y} + {¬ eq x y}.
      Proof.
        intros x y.
    -    change ({Equal x y}+{¬Equal x y}).
    +    change ({Equal x y}+{¬Equal x y}).
        destruct (equal x y) eqn:H; [left|right];
         rewrite <- equal_spec; congruence.
      Qed.

    -  Lemma empty_spec : Empty empty.
    -  Proof. rewrite /Empty /empty /In. move ⇒ a /InA_nil //. Qed.
    +  Lemma empty_spec : Empty empty.
    +  Proof. rewrite /Empty /empty /In. move ⇒ a /InA_nil //. Qed.

    -  Lemma is_empty_spec : ∀ s, is_empty s = true ↔ Empty s.
    +  Lemma is_empty_spec : ∀ s, is_empty s = true ↔ Empty s.
      Proof.
        rewrite /is_empty /Empty /In.
        case; split ⇒ //. {
          move ⇒ _ a.
    -      rewrite InA_nil //.
    +      rewrite InA_nil //.
        } {
          move ⇒ H; contradiction (H a).
    -      apply InA_cons_hd.
    -      apply Equivalence_Reflexive.
    +      apply InA_cons_hd.
    +      apply Equivalence_Reflexive.
        }
      Qed.

    -  Lemma add_spec : ∀ s x y, In y (add x s) ↔ E.eq y x ∨ In y s.
    +  Lemma add_spec : ∀ s x y, In y (add x s) ↔ E.eq y x ∨ In y s.
      Proof.
        intros s x y.
    -    rewrite /add /In InA_cons //.
    +    rewrite /add /In InA_cons //.
      Qed.

    -  Lemma singleton_spec : ∀ x y, In y (singleton x) ↔ E.eq y x.
    +  Lemma singleton_spec : ∀ x y, In y (singleton x) ↔ E.eq y x.
      Proof.
        intros x y.
    -    rewrite /singleton /In InA_cons.
    +    rewrite /singleton /In InA_cons.
        split. {
    -      move ⇒ [] // /InA_nil //.
    +      move ⇒ [] // /InA_nil //.
        } {
          by left.
        }
      Qed.

    -  Hint Resolve (@Equivalence_Reflexive _ _ E.eq_equiv).
    -  Hint Immediate (@Equivalence_Symmetric _ _ E.eq_equiv).
    -  Hint Resolve (@Equivalence_Transitive _ _ E.eq_equiv).
    - -
    -  Lemma rev_filter_aux_spec : ∀ s acc x f, compatb f →
    -    (In x (rev_filter_aux acc f s) ↔ (In x s ∧ f x = true) ∨ (In x acc)).
    +  Lemma rev_filter_aux_spec : ∀ s acc x f, compatb f →
    +    (In x (rev_filter_aux acc f s) ↔ (In x s ∧ f x = true) ∨ (In x acc)).
      Proof.
        intros s acc x f H_compat.
        move : x acc.
        induction s as [| y s' IH]. {
          intros x acc.
    -      rewrite /rev_filter_aux /In InA_nil.
    +      rewrite /rev_filter_aux /In InA_nil.
          tauto.
        } {
          intros x acc.
          rewrite /= IH /In.
    -      case_eq (f y) ⇒ H_fy; rewrite !InA_cons; intuition. {
    +      case_eq (f y) ⇒ H_fy; rewrite !InA_cons; intuition. {
            left.
            split; first by left.
            setoid_replace x with y ⇒ //.
    @@ -727,30 +721,30 @@

    Library MSetsExtra.MSetListWithDups

      Qed.

    -  Lemma filter_spec : ∀ s x f, compatb f →
    -    (In x (filter f s) ↔ In x s ∧ f x = true).
    +  Lemma filter_spec : ∀ s x f, compatb f →
    +    (In x (filter f s) ↔ In x s ∧ f x = true).
      Proof.
        intros s x f H_compat.
    -    rewrite /filter /rev_filter rev_filter_aux_spec /In InA_nil.
    +    rewrite /filter /rev_filter rev_filter_aux_spec /In InA_nil.
        tauto.
      Qed.

    -  Lemma remove_spec : ∀ s x y, In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    +  Lemma remove_spec : ∀ s x y, In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
      Proof.
        intros s x y.
        rewrite /remove /rev_filter.
    -    have H_compat : compatb ((fun y0 : elt ⇒
    -         match E.compare x y0 with
    -         | Eq ⇒ false
    -         | _ ⇒ true
    +    have H_compat : compatb ((fun y0 : elt ⇒
    +         match E.compare x y0 with
    +         | Eq ⇒ false
    +         | _ ⇒ true
             end)). {
          repeat red; intros.
          setoid_replace x0 with y0 ⇒ //.
        }
    -    rewrite rev_filter_aux_spec /In InA_nil.
    -    have → : (E.eq y x ↔ E.eq x y). {
    -      split; move ⇒ ?; by apply Equivalence_Symmetric.
    +    rewrite rev_filter_aux_spec /In InA_nil.
    +    have → : (E.eq y x ↔ E.eq x y). {
    +      split; move ⇒ ?; by apply Equivalence_Symmetric.
        }
        rewrite -compare_eq_iff.
        case (E.compare x y). {
    @@ -765,74 +759,74 @@

    Library MSetsExtra.MSetListWithDups

      Qed.

    -  Lemma union_spec : ∀ s s' x, In x (union s s') ↔ In x s ∨ In x s'.
    +  Lemma union_spec : ∀ s s' x, In x (union s s') ↔ In x s ∨ In x s'.
      Proof.
        intros s s' x.
    -    rewrite /union /In rev_append_rev InA_app_iff InA_rev; tauto.
    +    rewrite /union /In rev_append_rev InA_app_iff InA_rev; tauto.
      Qed.

    -  Lemma inter_spec : ∀ s s' x, In x (inter s s') ↔ In x s ∧ In x s'.
    +  Lemma inter_spec : ∀ s s' x, In x (inter s s') ↔ In x s ∧ In x s'.
      Proof.
        intros s s' x.
    -    have H_compat : compatb (fun y : elt ⇒ mem y s'). {
    +    have H_compat : compatb (fun y : elt ⇒ mem y s'). {
          repeat red; intros.
    -      suff : ( mem x0 s' = true ↔ mem y s' = true). {
    +      suff : ( mem x0 s' = true ↔ mem y s' = true). {
            case (mem y s'), (mem x0 s'); intuition.
          }
          rewrite !mem_spec /In.
          setoid_replace x0 with y ⇒ //.
        }
    -    rewrite /inter rev_filter_aux_spec mem_spec /In InA_nil.
    +    rewrite /inter rev_filter_aux_spec mem_spec /In InA_nil.
        tauto.
      Qed.

    -  Lemma fold_spec : ∀ s (A : Type) (i : A) (f : elt → A → A),
    -    fold f s i = fold_left (flip f) (elements s) i.
    +  Lemma fold_spec : ∀ s (A : Type) (i : A) (f : elt → A → A),
    +    fold f s i = fold_left (flip f) (elements s) i.
      Proof. done. Qed.

    -  Lemma elements_spec1 : ∀ s x, InA E.eq x (elements s) ↔ In x s.
    +  Lemma elements_spec1 : ∀ s x, InA E.eq x (elements s) ↔ In x s.
      Proof.
        intros s x.
        rewrite /elements /In //.
      Qed.

    -  Lemma diff_spec : ∀ s s' x, In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +  Lemma diff_spec : ∀ s s' x, In x (diff s s') ↔ In x s ∧ ¬In x s'.
      Proof.
        intros s s' x.
        rewrite /diff fold_spec -(elements_spec1 s').
        move : s.
        induction (elements s') as [| y ys IH] ⇒ s. {
    -      rewrite InA_nil /=; tauto.
    +      rewrite InA_nil /=; tauto.
        } {
    -      rewrite /= IH InA_cons /flip remove_spec.
    +      rewrite /= IH InA_cons /flip remove_spec.
          tauto.
        }
      Qed.

    -  Lemma cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    +  Lemma cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
      Proof. rewrite /cardinal //. Qed.

    -  Lemma for_all_spec : ∀ s f, compatb f →
    -    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +  Lemma for_all_spec : ∀ s f, compatb f →
    +    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
      Proof.
        intros s f H_compat.
        rewrite /For_all.
        induction s as [| x xs IH]. {
          rewrite /= /In.
          split ⇒ //.
    -      move ⇒ _ x /InA_nil //.
    +      move ⇒ _ x /InA_nil //.
        } {
          rewrite /=.
          case_eq (f x) ⇒ H_fx. {
            rewrite IH.
            split. {
    -          move ⇒ H x' /InA_cons []. {
    +          move ⇒ H x' /InA_cons []. {
                move ⇒ → //.
              } {
                apply H.
    @@ -840,50 +834,50 @@

    Library MSetsExtra.MSetListWithDups

            } {
              move ⇒ H x' H_in.
              apply H.
    -          apply InA_cons.
    +          apply InA_cons.
              by right.
            }
          } {
            split ⇒ //.
            move ⇒ H.
    -        suff : f x = true. {
    +        suff : f x = true. {
              rewrite H_fx //.
            }
            apply H.
    -        apply InA_cons_hd.
    -        done.
    +        apply InA_cons_hd.
    +        apply (@Equivalence_Reflexive _ _ E.eq_equiv).
          }
        }
      Qed.

    -  Lemma exists_spec : ∀ s f, compatb f →
    -    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +  Lemma exists_spec : ∀ s f, compatb f →
    +    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
      Proof.
        intros s f H_compat.
        rewrite /Exists.
        induction s as [| x xs IH]. {
          rewrite /= /In.
          split ⇒ //.
    -      move ⇒ [x] [] /InA_nil //.
    +      move ⇒ [x] [] /InA_nil //.
        } {
          rewrite /=.
          case_eq (f x) ⇒ H_fx. {
            split ⇒ // _.
            âˆƒ x.
            split ⇒ //.
    -        apply InA_cons_hd.
    -        done.
    +        apply InA_cons_hd.
    +        apply (@Equivalence_Reflexive _ _ E.eq_equiv).
          } {
            rewrite IH.
            split. {
              move ⇒ [x'] [H_in] H_fx'.
              âˆƒ x'.
              split ⇒ //.
    -          apply InA_cons.
    +          apply InA_cons.
              by right.
            } {
    -          move ⇒ [x'] [] /InA_cons []. {
    +          move ⇒ [x'] [] /InA_cons []. {
                move ⇒ →.
                rewrite H_fx //.
              } {
    @@ -895,8 +889,8 @@

    Library MSetsExtra.MSetListWithDups

      Qed.

    -  Lemma partition_aux_spec : ∀ a1 a2 s f,
    -    (partition_aux a1 a2 f s = (rev_filter_aux a1 f s, rev_filter_aux a2 (fun x ⇒ negb (f x)) s)).
    +  Lemma partition_aux_spec : ∀ a1 a2 s f,
    +    (partition_aux a1 a2 f s = (rev_filter_aux a1 f s, rev_filter_aux a2 (fun x ⇒ negb (f x)) s)).
      Proof.
        move ⇒ a1 a2 s f.
        move : a1 a2.
    @@ -912,40 +906,40 @@

    Library MSetsExtra.MSetListWithDups

      Qed.

    -  Lemma partition_spec1 : ∀ s f, compatb f →
    -    fst (partition f s) [=] filter f s.
    +  Lemma partition_spec1 : ∀ s f, compatb f →
    +    fst (partition f s) [=] filter f s.
      Proof.
        move ⇒ s f _.
    -    rewrite /partition partition_aux_spec /fst /filter /rev_filter //.
    +    rewrite /partition partition_aux_spec /fst /filter /rev_filter //.
      Qed.

    -  Lemma partition_spec2 : ∀ s f, compatb f →
    -    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    +  Lemma partition_spec2 : ∀ s f, compatb f →
    +    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
      Proof.
        move ⇒ s f _.
    -    rewrite /partition partition_aux_spec /snd /filter /rev_filter //.
    +    rewrite /partition partition_aux_spec /snd /filter /rev_filter //.
      Qed.

    -  Lemma choose_spec1 : ∀ s x, choose s = Some x → In x s.
    +  Lemma choose_spec1 : ∀ s x, choose s = Some x → In x s.
      Proof.
       move ⇒ [] // y s' x [->].
       rewrite /In.
    -   apply InA_cons_hd.
    -   apply Equivalence_Reflexive.
    +   apply InA_cons_hd.
    +   apply Equivalence_Reflexive.
      Qed.

    -  Lemma choose_spec2 : ∀ s, choose s = None → Empty s.
    -  Proof. move ⇒ [] // _ a. rewrite /In InA_nil //. Qed.
    +  Lemma choose_spec2 : ∀ s, choose s = None → Empty s.
    +  Proof. move ⇒ [] // _ a. rewrite /In InA_nil //. Qed.

    -  Lemma elements_dist_spec_full :
    -    âˆ€ s,
    -      Sorted E.lt (elements_dist s) ∧
    -      NoDupA E.eq (elements_dist s) ∧
    -      (∀ x, InA E.eq x (elements_dist s) ↔ InA E.eq x (elements s)).
    +  Lemma elements_dist_spec_full :
    +    âˆ€ s,
    +      Sorted E.lt (elements_dist s) ∧
    +      NoDupA E.eq (elements_dist s) ∧
    +      (∀ x, InA E.eq x (elements_dist s) ↔ InA E.eq x (elements s)).
      Proof.
        move ⇒ s.
        rewrite /elements_dist /elements.
    @@ -955,12 +949,12 @@

    Library MSetsExtra.MSetListWithDups

      Qed.

    -  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    -                                          InA E.eq x (elements s).
    +  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    +                                          InA E.eq x (elements s).
      Proof. intros; apply elements_dist_spec_full. Qed.

    -  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    +  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
      Proof. intros; apply elements_dist_spec_full. Qed.

    diff --git a/html/MSetsExtra.MSetWithDups.html b/html/MSetsExtra.MSetWithDups.html index 9ba7eed..85666d1 100644 --- a/html/MSetsExtra.MSetWithDups.html +++ b/html/MSetsExtra.MSetWithDups.html @@ -24,7 +24,7 @@

    Library MSetsExtra.MSetWithDups

    -

    Signature for weak sets which may contain duplicates

    +

    Signature for weak sets which may contain duplicates

    @@ -66,14 +66,14 @@

    Library MSetsExtra.MSetWithDups


    -Require Import Coq.MSets.MSetInterface.
    +Require Import Coq.MSets.MSetInterface.
    Require Import mathcomp.ssreflect.ssreflect.

    -

    WSetsOnWithDups

    +

    WSetsOnWithDups

    @@ -91,68 +91,68 @@

    Library MSetsExtra.MSetWithDups

    -Module Type WSetsOnWithDups (E : DecidableType).
    -  Include WOps E.
    +Module Type WSetsOnWithDups (E : DecidableType).
    +  Include WOps E.

    -  Parameter In : elt → t → Prop.
    -  Declare Instance In_compat : Proper (E.eq==>eq==>iff) In.
    +  Parameter In : elt → t → Prop.
    +  #[local] Declare Instance In_compat : Proper (E.eq==>eq==>iff) In.

    -  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    -  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    -  Definition Empty s := ∀ a : elt, ¬ In a s.
    -  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    -  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.
    +  Definition Equal s s' := ∀ a : elt, In a s ↔ In a s'.
    +  Definition Subset s s' := ∀ a : elt, In a s → In a s'.
    +  Definition Empty s := ∀ a : elt, ¬ In a s.
    +  Definition For_all (P : elt → Prop) s := ∀ x, In x s → P x.
    +  Definition Exists (P : elt → Prop) s := ∃ x, In x s ∧ P x.

    -  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
    -  Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).
    +  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
    +  Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).

    -  Definition eq : t → t → Prop := Equal.
    -  Include IsEq.
    +  Definition eq : t → t → Prop := Equal.
    +  Include IsEq.
    eq is obviously an equivalence, for subtyping only
    -  Include HasEqDec.
    - -
    -  Section Spec.
    -  Variable s s': t.
    -  Variable x y : elt.
    -  Variable f : elt → bool.
    -  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    - -
    -  Parameter mem_spec : mem x s = true ↔ In x s.
    -  Parameter equal_spec : equal s s' = true ↔ s[=]s'.
    -  Parameter subset_spec : subset s s' = true ↔ s[<=]s'.
    -  Parameter empty_spec : Empty empty.
    -  Parameter is_empty_spec : is_empty s = true ↔ Empty s.
    -  Parameter add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    -  Parameter remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    -  Parameter singleton_spec : In y (singleton x) ↔ E.eq y x.
    -  Parameter union_spec : In x (union s s') ↔ In x s ∨ In x s'.
    -  Parameter inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
    -  Parameter diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
    -  Parameter fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    -    fold f s i = fold_left (flip f) (elements s) i.
    -  Parameter filter_spec : compatb f →
    -    (In x (filter f s) ↔ In x s ∧ f x = true).
    -  Parameter for_all_spec : compatb f →
    -    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    -  Parameter exists_spec : compatb f →
    -    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    -  Parameter partition_spec1 : compatb f →
    -    fst (partition f s) [=] filter f s.
    -  Parameter partition_spec2 : compatb f →
    -    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    -  Parameter elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    -  Parameter choose_spec1 : choose s = Some x → In x s.
    -  Parameter choose_spec2 : choose s = None → Empty s.
    +  Include HasEqDec.
    + +
    +  Section Spec.
    +  Variable s s': t.
    +  Variable x y : elt.
    +  Variable f : elt → bool.
    +  Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).
    + +
    +  Parameter mem_spec : mem x s = true ↔ In x s.
    +  Parameter equal_spec : equal s s' = true ↔ s[=]s'.
    +  Parameter subset_spec : subset s s' = true ↔ s[<=]s'.
    +  Parameter empty_spec : Empty empty.
    +  Parameter is_empty_spec : is_empty s = true ↔ Empty s.
    +  Parameter add_spec : In y (add x s) ↔ E.eq y x ∨ In y s.
    +  Parameter remove_spec : In y (remove x s) ↔ In y s ∧ ¬E.eq y x.
    +  Parameter singleton_spec : In y (singleton x) ↔ E.eq y x.
    +  Parameter union_spec : In x (union s s') ↔ In x s ∨ In x s'.
    +  Parameter inter_spec : In x (inter s s') ↔ In x s ∧ In x s'.
    +  Parameter diff_spec : In x (diff s s') ↔ In x s ∧ ¬In x s'.
    +  Parameter fold_spec : ∀ (A : Type) (i : A) (f : elt → A → A),
    +    fold f s i = fold_left (flip f) (elements s) i.
    +  Parameter filter_spec : compatb f →
    +    (In x (filter f s) ↔ In x s ∧ f x = true).
    +  Parameter for_all_spec : compatb f →
    +    (for_all f s = true ↔ For_all (fun x ⇒ f x = true) s).
    +  Parameter exists_spec : compatb f →
    +    (exists_ f s = true ↔ Exists (fun x ⇒ f x = true) s).
    +  Parameter partition_spec1 : compatb f →
    +    fst (partition f s) [=] filter f s.
    +  Parameter partition_spec2 : compatb f →
    +    snd (partition f s) [=] filter (fun x ⇒ negb (f x)) s.
    +  Parameter elements_spec1 : InA E.eq x (elements s) ↔ In x s.
    +  Parameter choose_spec1 : choose s = Some x → In x s.
    +  Parameter choose_spec2 : choose s = None → Empty s.

      End Spec.
    @@ -164,7 +164,7 @@

    Library MSetsExtra.MSetWithDups

    -

    WSetsOnWithDupsExtra

    +

    WSetsOnWithDupsExtra

    @@ -174,7 +174,7 @@

    Library MSetsExtra.MSetWithDups

    the original behavior of elements.
    -Module Type WSetsOnWithDupsExtra (E : DecidableType).
    +Module Type WSetsOnWithDupsExtra (E : DecidableType).
      Include WSetsOnWithDups E.

    @@ -184,14 +184,14 @@

    Library MSetsExtra.MSetWithDups

    An operation for getting an elements list without duplicates
    -  Parameter elements_dist : t → list elt.
    +  Parameter elements_dist : t → list elt.

    -  Parameter elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    -                                              InA E.eq x (elements s).
    +  Parameter elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    +                                              InA E.eq x (elements s).

    -  Parameter elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    +  Parameter elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).

    @@ -200,14 +200,14 @@

    Library MSetsExtra.MSetWithDups

    Cardinality can then be specified with respect to elements_dist.
    -  Parameter cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    +  Parameter cardinal_spec : ∀ s, cardinal s = length (elements_dist s).
    End WSetsOnWithDupsExtra.

    -

    WSetOn to WSetsOnWithDupsExtra

    +

    WSetOn to WSetsOnWithDupsExtra

    @@ -219,22 +219,22 @@

    Library MSetsExtra.MSetWithDups


    -Module WSetsOn_TO_WSetsOnWithDupsExtra (E : DecidableType) (W : WSetsOn E) <:
    +Module WSetsOn_TO_WSetsOnWithDupsExtra (E : DecidableType) (W : WSetsOn E) <:
      WSetsOnWithDupsExtra E.

      Include W.

    -  Definition elements_dist := W.elements.
    +  Definition elements_dist := W.elements.

    -  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    -                                          InA E.eq x (elements s).
    +  Lemma elements_dist_spec1 : ∀ x s, InA E.eq x (elements_dist s) ↔
    +                                          InA E.eq x (elements s).
      Proof. done. Qed.

    -  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
    +  Lemma elements_dist_spec2w : ∀ s, NoDupA E.eq (elements_dist s).
      Proof. apply elements_spec2w. Qed.

    diff --git a/html/coqdoc.css b/html/coqdoc.css index dbc930f..48096e5 100644 --- a/html/coqdoc.css +++ b/html/coqdoc.css @@ -230,6 +230,10 @@ tr.infrulemiddle hr { color: rgb(40%,0%,40%); } +.id[title="binder"] { + color: rgb(40%,0%,40%); +} + .id[type="definition"] { color: rgb(0%,40%,0%); } @@ -327,3 +331,8 @@ ul.doclist { margin-top: 0em; margin-bottom: 0em; } + +.code :target { + border: 2px solid #D4D4D4; + background-color: #e5eecc; +} diff --git a/html/index.html b/html/index.html index eb94472..f5b604f 100644 --- a/html/index.html +++ b/html/index.html @@ -20,34 +20,34 @@ Global Index A -B -C +B +C D E F G H -I +I J K -L +L M N O -P +P Q R S -T +T U V W -X -Y +X +Y Z _ other -(596 entries) +(2110 entries) Notation Index @@ -82,6 +82,38 @@ (5 entries) +Binder Index +A +B +C +D +E +F +G +H +I +J +K +L +M +N +O +P +Q +R +S +T +U +V +W +X +Y +Z +_ +other +(1515 entries) + + Module Index A B @@ -255,7 +287,7 @@ J K L -M +M N O P @@ -271,7 +303,7 @@ Z _ other -(11 entries) +(10 entries) Projection Index @@ -338,7 +370,7 @@ (2 entries) -Instance Index +Section Index A B C @@ -351,26 +383,26 @@ J K L -M +M N O P Q -R +R S T U V -W +W X Y Z _ other -(17 entries) +(3 entries) -Section Index +Instance Index A B C @@ -383,23 +415,23 @@ J K L -M +M N O P Q -R +R S T U V -W +W X Y Z _ other -(3 entries) +(17 entries) Abbreviation Index @@ -500,9 +532,286 @@

    Global Index

    -

    A

    +

    A

    +ab:347 [binder, in MSetsExtra.MSetIntervals]
    +ab:363 [binder, in MSetsExtra.MSetIntervals]
    +acc_f:906 [binder, in MSetsExtra.MSetIntervals]
    +acc_t:905 [binder, in MSetsExtra.MSetIntervals]
    +acc_f:899 [binder, in MSetsExtra.MSetIntervals]
    +acc_t:897 [binder, in MSetsExtra.MSetIntervals]
    +acc_f:200 [binder, in MSetsExtra.MSetIntervals]
    +acc_t:199 [binder, in MSetsExtra.MSetIntervals]
    +acc_f:196 [binder, in MSetsExtra.MSetIntervals]
    +acc_t:195 [binder, in MSetsExtra.MSetIntervals]
    +acc':348 [binder, in MSetsExtra.MSetIntervals]
    +acc':364 [binder, in MSetsExtra.MSetIntervals]
    +acc:106 [binder, in MSetsExtra.MSetIntervals]
    +acc:119 [binder, in MSetsExtra.MSetListWithDups]
    +acc:125 [binder, in MSetsExtra.MSetIntervals]
    +acc:141 [binder, in MSetsExtra.MSetIntervals]
    +acc:147 [binder, in MSetsExtra.MSetIntervals]
    +acc:15 [binder, in MSetsExtra.MSetListWithDups]
    +acc:151 [binder, in MSetsExtra.MSetIntervals]
    +acc:168 [binder, in MSetsExtra.MSetIntervals]
    +acc:172 [binder, in MSetsExtra.MSetIntervals]
    +acc:181 [binder, in MSetsExtra.MSetIntervals]
    +acc:218 [binder, in MSetsExtra.MSetIntervals]
    +acc:221 [binder, in MSetsExtra.MSetIntervals]
    +acc:23 [binder, in MSetsExtra.MSetIntervals]
    +acc:250 [binder, in MSetsExtra.MSetFoldWithAbort]
    +acc:311 [binder, in MSetsExtra.MSetIntervals]
    +acc:317 [binder, in MSetsExtra.MSetIntervals]
    +acc:322 [binder, in MSetsExtra.MSetIntervals]
    +acc:326 [binder, in MSetsExtra.MSetIntervals]
    +acc:332 [binder, in MSetsExtra.MSetIntervals]
    +acc:338 [binder, in MSetsExtra.MSetIntervals]
    +acc:344 [binder, in MSetsExtra.MSetIntervals]
    +acc:35 [binder, in MSetsExtra.MSetIntervals]
    +acc:352 [binder, in MSetsExtra.MSetIntervals]
    +acc:356 [binder, in MSetsExtra.MSetIntervals]
    +acc:359 [binder, in MSetsExtra.MSetIntervals]
    +acc:366 [binder, in MSetsExtra.MSetFoldWithAbort]
    +acc:370 [binder, in MSetsExtra.MSetFoldWithAbort]
    +acc:372 [binder, in MSetsExtra.MSetIntervals]
    +acc:378 [binder, in MSetsExtra.MSetFoldWithAbort]
    +acc:389 [binder, in MSetsExtra.MSetFoldWithAbort]
    +acc:393 [binder, in MSetsExtra.MSetFoldWithAbort]
    +acc:401 [binder, in MSetsExtra.MSetFoldWithAbort]
    +acc:414 [binder, in MSetsExtra.MSetIntervals]
    +acc:43 [binder, in MSetsExtra.MSetListWithDups]
    +acc:558 [binder, in MSetsExtra.MSetIntervals]
    +acc:56 [binder, in MSetsExtra.MSetIntervals]
    +acc:561 [binder, in MSetsExtra.MSetIntervals]
    +acc:584 [binder, in MSetsExtra.MSetIntervals]
    +acc:588 [binder, in MSetsExtra.MSetIntervals]
    +acc:600 [binder, in MSetsExtra.MSetIntervals]
    +acc:606 [binder, in MSetsExtra.MSetIntervals]
    +acc:633 [binder, in MSetsExtra.MSetIntervals]
    +acc:637 [binder, in MSetsExtra.MSetIntervals]
    +acc:649 [binder, in MSetsExtra.MSetIntervals]
    +acc:65 [binder, in MSetsExtra.MSetIntervals]
    +acc:657 [binder, in MSetsExtra.MSetIntervals]
    +acc:684 [binder, in MSetsExtra.MSetIntervals]
    +acc:70 [binder, in MSetsExtra.MSetIntervals]
    +acc:832 [binder, in MSetsExtra.MSetIntervals]
    +acc:837 [binder, in MSetsExtra.MSetIntervals]
    +acc:840 [binder, in MSetsExtra.MSetIntervals]
    +acc:852 [binder, in MSetsExtra.MSetIntervals]
    +acc:857 [binder, in MSetsExtra.MSetIntervals]
    +acc:864 [binder, in MSetsExtra.MSetIntervals]
    +acc:869 [binder, in MSetsExtra.MSetIntervals]
    +acc:9 [binder, in MSetsExtra.MSetListWithDups]
    add_add_sub_eq [lemma, in MSetsExtra.MSetIntervals]
    -

    E

    +a1:152 [binder, in MSetsExtra.MSetListWithDups]
    +a1:3 [binder, in MSetsExtra.MSetListWithDups]
    +a1:79 [binder, in MSetsExtra.MSetListWithDups]
    +a2:153 [binder, in MSetsExtra.MSetListWithDups]
    +a2:4 [binder, in MSetsExtra.MSetListWithDups]
    +a2:80 [binder, in MSetsExtra.MSetListWithDups]
    +a:1002 [binder, in MSetsExtra.MSetIntervals]
    +a:1003 [binder, in MSetsExtra.MSetIntervals]
    +A:102 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:11 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:12 [binder, in MSetsExtra.MSetIntervals]
    +A:126 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:130 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:137 [binder, in MSetsExtra.MSetListWithDups]
    +A:140 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:147 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:158 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:175 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:190 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:20 [binder, in MSetsExtra.MSetIntervals]
    +A:201 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:218 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:27 [binder, in MSetsExtra.MSetIntervals]
    +a:275 [binder, in MSetsExtra.MSetIntervals]
    +a:278 [binder, in MSetsExtra.MSetIntervals]
    +a:280 [binder, in MSetsExtra.MSetIntervals]
    +A:3 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:30 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:302 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:308 [binder, in MSetsExtra.MSetIntervals]
    +A:31 [binder, in MSetsExtra.MSetIntervals]
    +A:315 [binder, in MSetsExtra.MSetIntervals]
    +A:316 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:32 [binder, in MSetsExtra.MSetWithDups]
    +A:320 [binder, in MSetsExtra.MSetIntervals]
    +A:324 [binder, in MSetsExtra.MSetIntervals]
    +A:329 [binder, in MSetsExtra.MSetIntervals]
    +A:336 [binder, in MSetsExtra.MSetIntervals]
    +A:341 [binder, in MSetsExtra.MSetIntervals]
    +A:343 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:349 [binder, in MSetsExtra.MSetIntervals]
    +A:35 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:351 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:354 [binder, in MSetsExtra.MSetIntervals]
    +A:357 [binder, in MSetsExtra.MSetIntervals]
    +A:359 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:36 [binder, in MSetsExtra.MSetIntervals]
    +A:369 [binder, in MSetsExtra.MSetIntervals]
    +A:37 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:371 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:382 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:39 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:394 [binder, in MSetsExtra.MSetFoldWithAbort]
    +a:4 [binder, in MSetsExtra.MSetWithDups]
    +A:54 [binder, in MSetsExtra.MSetIntervals]
    +A:58 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:63 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:63 [binder, in MSetsExtra.MSetIntervals]
    +A:65 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:67 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:68 [binder, in MSetsExtra.MSetIntervals]
    +A:7 [binder, in MSetsExtra.MSetFoldWithAbort]
    +a:7 [binder, in MSetsExtra.MSetWithDups]
    +A:817 [binder, in MSetsExtra.MSetIntervals]
    +A:9 [binder, in MSetsExtra.MSetFoldWithAbort]
    +a:9 [binder, in MSetsExtra.MSetWithDups]
    +A:91 [binder, in MSetsExtra.MSetFoldWithAbort]
    +a:92 [binder, in MSetsExtra.MSetListWithDups]
    +a:934 [binder, in MSetsExtra.MSetIntervals]
    +a:937 [binder, in MSetsExtra.MSetIntervals]
    +a:939 [binder, in MSetsExtra.MSetIntervals]
    +a:95 [binder, in MSetsExtra.MSetListWithDups]
    +A:97 [binder, in MSetsExtra.MSetFoldWithAbort]
    +a:97 [binder, in MSetsExtra.MSetListWithDups]
    +A:970 [binder, in MSetsExtra.MSetIntervals]
    +a:985 [binder, in MSetsExtra.MSetIntervals]
    +a:986 [binder, in MSetsExtra.MSetIntervals]
    +a:987 [binder, in MSetsExtra.MSetIntervals]
    +a:988 [binder, in MSetsExtra.MSetIntervals]
    +A:99 [binder, in MSetsExtra.MSetFoldWithAbort]
    +A:994 [binder, in MSetsExtra.MSetIntervals]
    +a:997 [binder, in MSetsExtra.MSetIntervals]
    +

    B

    +base:309 [binder, in MSetsExtra.MSetFoldWithAbort]
    +base:350 [binder, in MSetsExtra.MSetFoldWithAbort]
    +base:358 [binder, in MSetsExtra.MSetFoldWithAbort]
    +base:366 [binder, in MSetsExtra.MSetIntervals]
    +bs:333 [binder, in MSetsExtra.MSetIntervals]
    +bs:337 [binder, in MSetsExtra.MSetIntervals]
    +B:100 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:103 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:127 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:13 [binder, in MSetsExtra.MSetIntervals]
    +B:159 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:21 [binder, in MSetsExtra.MSetIntervals]
    +B:212 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:227 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:228 [binder, in MSetsExtra.MSetIntervals]
    +b:232 [binder, in MSetsExtra.MSetIntervals]
    +b:236 [binder, in MSetsExtra.MSetIntervals]
    +B:237 [binder, in MSetsExtra.MSetFoldWithAbort]
    +b:240 [binder, in MSetsExtra.MSetIntervals]
    +B:253 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:268 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:28 [binder, in MSetsExtra.MSetIntervals]
    +B:287 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:290 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:294 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:297 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:303 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:317 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:32 [binder, in MSetsExtra.MSetIntervals]
    +B:330 [binder, in MSetsExtra.MSetIntervals]
    +B:342 [binder, in MSetsExtra.MSetIntervals]
    +B:344 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:352 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:360 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:37 [binder, in MSetsExtra.MSetIntervals]
    +B:372 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:383 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:395 [binder, in MSetsExtra.MSetFoldWithAbort]
    +B:60 [binder, in MSetsExtra.MSetListWithDups]
    +B:92 [binder, in MSetsExtra.MSetFoldWithAbort]
    +

    C

    +cur:178 [binder, in MSetsExtra.MSetIntervals]
    +cur:182 [binder, in MSetsExtra.MSetIntervals]
    +cur:858 [binder, in MSetsExtra.MSetIntervals]
    +cx:306 [binder, in MSetsExtra.MSetIntervals]
    +c_f:900 [binder, in MSetsExtra.MSetIntervals]
    +c_t:898 [binder, in MSetsExtra.MSetIntervals]
    +c1:298 [binder, in MSetsExtra.MSetIntervals]
    +c1:399 [binder, in MSetsExtra.MSetIntervals]
    +c1:403 [binder, in MSetsExtra.MSetIntervals]
    +c1:485 [binder, in MSetsExtra.MSetIntervals]
    +c1:510 [binder, in MSetsExtra.MSetIntervals]
    +c1:513 [binder, in MSetsExtra.MSetIntervals]
    +c1:517 [binder, in MSetsExtra.MSetIntervals]
    +c1:523 [binder, in MSetsExtra.MSetIntervals]
    +c1:706 [binder, in MSetsExtra.MSetIntervals]
    +c1:714 [binder, in MSetsExtra.MSetIntervals]
    +c1:759 [binder, in MSetsExtra.MSetIntervals]
    +c1:98 [binder, in MSetsExtra.MSetIntervals]
    +c2:100 [binder, in MSetsExtra.MSetIntervals]
    +c2:146 [binder, in MSetsExtra.MSetIntervals]
    +c2:167 [binder, in MSetsExtra.MSetIntervals]
    +c2:297 [binder, in MSetsExtra.MSetIntervals]
    +c2:401 [binder, in MSetsExtra.MSetIntervals]
    +c2:405 [binder, in MSetsExtra.MSetIntervals]
    +c2:410 [binder, in MSetsExtra.MSetIntervals]
    +c2:467 [binder, in MSetsExtra.MSetIntervals]
    +c2:487 [binder, in MSetsExtra.MSetIntervals]
    +c2:511 [binder, in MSetsExtra.MSetIntervals]
    +c2:515 [binder, in MSetsExtra.MSetIntervals]
    +c2:519 [binder, in MSetsExtra.MSetIntervals]
    +c2:525 [binder, in MSetsExtra.MSetIntervals]
    +c2:582 [binder, in MSetsExtra.MSetIntervals]
    +c2:586 [binder, in MSetsExtra.MSetIntervals]
    +c2:631 [binder, in MSetsExtra.MSetIntervals]
    +c2:635 [binder, in MSetsExtra.MSetIntervals]
    +c2:709 [binder, in MSetsExtra.MSetIntervals]
    +c2:717 [binder, in MSetsExtra.MSetIntervals]
    +c2:760 [binder, in MSetsExtra.MSetIntervals]
    +c:102 [binder, in MSetsExtra.MSetIntervals]
    +c:123 [binder, in MSetsExtra.MSetIntervals]
    +c:191 [binder, in MSetsExtra.MSetIntervals]
    +c:198 [binder, in MSetsExtra.MSetIntervals]
    +c:214 [binder, in MSetsExtra.MSetIntervals]
    +c:220 [binder, in MSetsExtra.MSetIntervals]
    +c:242 [binder, in MSetsExtra.MSetIntervals]
    +c:288 [binder, in MSetsExtra.MSetIntervals]
    +c:292 [binder, in MSetsExtra.MSetIntervals]
    +c:295 [binder, in MSetsExtra.MSetIntervals]
    +c:300 [binder, in MSetsExtra.MSetIntervals]
    +c:301 [binder, in MSetsExtra.MSetIntervals]
    +c:310 [binder, in MSetsExtra.MSetIntervals]
    +c:328 [binder, in MSetsExtra.MSetIntervals]
    +c:351 [binder, in MSetsExtra.MSetIntervals]
    +c:361 [binder, in MSetsExtra.MSetIntervals]
    +c:374 [binder, in MSetsExtra.MSetIntervals]
    +c:377 [binder, in MSetsExtra.MSetIntervals]
    +c:384 [binder, in MSetsExtra.MSetIntervals]
    +c:393 [binder, in MSetsExtra.MSetIntervals]
    +c:396 [binder, in MSetsExtra.MSetIntervals]
    +c:420 [binder, in MSetsExtra.MSetIntervals]
    +c:429 [binder, in MSetsExtra.MSetIntervals]
    +c:432 [binder, in MSetsExtra.MSetIntervals]
    +c:434 [binder, in MSetsExtra.MSetIntervals]
    +c:489 [binder, in MSetsExtra.MSetIntervals]
    +c:492 [binder, in MSetsExtra.MSetIntervals]
    +c:496 [binder, in MSetsExtra.MSetIntervals]
    +c:500 [binder, in MSetsExtra.MSetIntervals]
    +c:503 [binder, in MSetsExtra.MSetIntervals]
    +c:507 [binder, in MSetsExtra.MSetIntervals]
    +c:530 [binder, in MSetsExtra.MSetIntervals]
    +c:58 [binder, in MSetsExtra.MSetIntervals]
    +c:67 [binder, in MSetsExtra.MSetIntervals]
    +c:7 [binder, in MSetsExtra.MSetIntervals]
    +c:732 [binder, in MSetsExtra.MSetIntervals]
    +c:767 [binder, in MSetsExtra.MSetIntervals]
    +c:821 [binder, in MSetsExtra.MSetIntervals]
    +c:833 [binder, in MSetsExtra.MSetIntervals]
    +c:838 [binder, in MSetsExtra.MSetIntervals]
    +c:841 [binder, in MSetsExtra.MSetIntervals]
    +c:842 [binder, in MSetsExtra.MSetIntervals]
    +c:846 [binder, in MSetsExtra.MSetIntervals]
    +c:851 [binder, in MSetsExtra.MSetIntervals]
    +c:855 [binder, in MSetsExtra.MSetIntervals]
    +c:862 [binder, in MSetsExtra.MSetIntervals]
    +c:895 [binder, in MSetsExtra.MSetIntervals]
    +

    E

    ElementEncode [module, in MSetsExtra.MSetIntervals]
    ElementEncodeN [module, in MSetsExtra.MSetIntervals]
    ElementEncodeNat [module, in MSetsExtra.MSetIntervals]
    @@ -531,16 +840,288 @@

    Global Index

    ElementEncode.encode [axiom, in MSetsExtra.MSetIntervals]
    ElementEncode.encode_lt [axiom, in MSetsExtra.MSetIntervals]
    ElementEncode.encode_eq [axiom, in MSetsExtra.MSetIntervals]
    -

    F

    +elt:1 [binder, in MSetsExtra.MSetFoldWithAbort]
    +elt:28 [binder, in MSetsExtra.MSetFoldWithAbort]
    +elt:31 [binder, in MSetsExtra.MSetFoldWithAbort]
    +elt:4 [binder, in MSetsExtra.MSetFoldWithAbort]
    +elt:56 [binder, in MSetsExtra.MSetFoldWithAbort]
    +elt:59 [binder, in MSetsExtra.MSetFoldWithAbort]
    +elt:89 [binder, in MSetsExtra.MSetFoldWithAbort]
    +elt:93 [binder, in MSetsExtra.MSetFoldWithAbort]
    +eqA:14 [binder, in MSetsExtra.MSetIntervals]
    +eqB:15 [binder, in MSetsExtra.MSetIntervals]
    +e_pre:261 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e_pre:244 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e_pre:233 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e0:251 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e0:341 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e0:342 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:1007 [binder, in MSetsExtra.MSetIntervals]
    +e1:1009 [binder, in MSetsExtra.MSetIntervals]
    +e1:1014 [binder, in MSetsExtra.MSetIntervals]
    +e1:1016 [binder, in MSetsExtra.MSetIntervals]
    +e1:1021 [binder, in MSetsExtra.MSetIntervals]
    +e1:1023 [binder, in MSetsExtra.MSetIntervals]
    +e1:118 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:122 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:156 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:169 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:172 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:184 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:187 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:198 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:209 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:223 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:225 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:24 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:332 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:336 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:44 [binder, in MSetsExtra.MSetIntervals]
    +e1:47 [binder, in MSetsExtra.MSetIntervals]
    +e1:52 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:81 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e1:85 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:1008 [binder, in MSetsExtra.MSetIntervals]
    +e2:1010 [binder, in MSetsExtra.MSetIntervals]
    +e2:1015 [binder, in MSetsExtra.MSetIntervals]
    +e2:1017 [binder, in MSetsExtra.MSetIntervals]
    +e2:1022 [binder, in MSetsExtra.MSetIntervals]
    +e2:1024 [binder, in MSetsExtra.MSetIntervals]
    +e2:121 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:125 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:157 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:171 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:174 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:186 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:189 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:200 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:211 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:224 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:226 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:27 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:335 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:339 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:45 [binder, in MSetsExtra.MSetIntervals]
    +e2:48 [binder, in MSetsExtra.MSetIntervals]
    +e2:55 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:84 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e2:88 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:1006 [binder, in MSetsExtra.MSetIntervals]
    +e:1013 [binder, in MSetsExtra.MSetIntervals]
    +e:1020 [binder, in MSetsExtra.MSetIntervals]
    +e:115 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:116 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:134 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:136 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:138 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:143 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:145 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:150 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:152 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:154 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:167 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:182 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:196 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:207 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:21 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:22 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:222 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:230 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:232 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:235 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:241 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:243 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:246 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:248 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:256 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:256 [binder, in MSetsExtra.MSetIntervals]
    +e:260 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:264 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:272 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:273 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:274 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:301 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:329 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:330 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:340 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:379 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:380 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:402 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:403 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:405 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:42 [binder, in MSetsExtra.MSetIntervals]
    +e:49 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:50 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:772 [binder, in MSetsExtra.MSetIntervals]
    +e:78 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:79 [binder, in MSetsExtra.MSetFoldWithAbort]
    +e:998 [binder, in MSetsExtra.MSetIntervals]
    +

    F

    foldWithAbortGtLtSpecPred [definition, in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortGtLtType [definition, in MSetsExtra.MSetFoldWithAbort]
    +foldWithAbortGtLt:66 [binder, in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortGtSpecPred [definition, in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortGtType [definition, in MSetsExtra.MSetFoldWithAbort]
    +foldWithAbortGt:38 [binder, in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortPrecomputeSpecPred [definition, in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortPrecomputeType [definition, in MSetsExtra.MSetFoldWithAbort]
    +foldWithAbortPrecompute:101 [binder, in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortSpecPred [definition, in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortType [definition, in MSetsExtra.MSetFoldWithAbort]
    -

    H

    +foldWithAbort:10 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fold:36 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fold:64 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fold:8 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fold:98 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fwasa:221 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fwasa:228 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fwasa:238 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fwasa:254 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fwasa:269 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fwasa:288 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fwasa:291 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fwasa:295 [binder, in MSetsExtra.MSetFoldWithAbort]
    +fwasa:298 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:399 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:397 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_pre:396 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:387 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:385 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_pre:384 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:376 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:374 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_pre:373 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:364 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:362 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_pre:361 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:356 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:354 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_pre:353 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:348 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:346 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_pre:345 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:324 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:323 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_pre:320 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:307 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:305 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_pre:304 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_abort:205 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:194 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:180 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:179 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:165 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:164 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_pre:161 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_abort:149 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:142 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:133 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:131 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:110 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:109 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_precompute:106 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:73 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_lt:72 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_gt:44 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f_abort:16 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f':108 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f':15 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f':163 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f':178 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f':193 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f':204 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f':322 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f':43 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f':71 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:107 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:121 [binder, in MSetsExtra.MSetListWithDups]
    +f:124 [binder, in MSetsExtra.MSetListWithDups]
    +f:132 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:139 [binder, in MSetsExtra.MSetListWithDups]
    +f:14 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:141 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:147 [binder, in MSetsExtra.MSetListWithDups]
    +f:148 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:150 [binder, in MSetsExtra.MSetListWithDups]
    +f:155 [binder, in MSetsExtra.MSetListWithDups]
    +f:158 [binder, in MSetsExtra.MSetListWithDups]
    +f:16 [binder, in MSetsExtra.MSetIntervals]
    +f:160 [binder, in MSetsExtra.MSetListWithDups]
    +f:162 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:177 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:184 [binder, in MSetsExtra.MSetIntervals]
    +f:188 [binder, in MSetsExtra.MSetIntervals]
    +f:192 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:194 [binder, in MSetsExtra.MSetIntervals]
    +f:201 [binder, in MSetsExtra.MSetIntervals]
    +f:203 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:205 [binder, in MSetsExtra.MSetIntervals]
    +f:207 [binder, in MSetsExtra.MSetIntervals]
    +f:211 [binder, in MSetsExtra.MSetIntervals]
    +f:217 [binder, in MSetsExtra.MSetIntervals]
    +f:22 [binder, in MSetsExtra.MSetIntervals]
    +f:222 [binder, in MSetsExtra.MSetIntervals]
    +f:226 [binder, in MSetsExtra.MSetIntervals]
    +f:229 [binder, in MSetsExtra.MSetIntervals]
    +f:234 [binder, in MSetsExtra.MSetIntervals]
    +f:238 [binder, in MSetsExtra.MSetIntervals]
    +f:29 [binder, in MSetsExtra.MSetIntervals]
    +f:306 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:309 [binder, in MSetsExtra.MSetIntervals]
    +f:316 [binder, in MSetsExtra.MSetIntervals]
    +f:321 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:321 [binder, in MSetsExtra.MSetIntervals]
    +f:325 [binder, in MSetsExtra.MSetIntervals]
    +f:33 [binder, in MSetsExtra.MSetIntervals]
    +f:331 [binder, in MSetsExtra.MSetIntervals]
    +f:34 [binder, in MSetsExtra.MSetWithDups]
    +f:343 [binder, in MSetsExtra.MSetIntervals]
    +f:347 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:350 [binder, in MSetsExtra.MSetIntervals]
    +f:355 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:355 [binder, in MSetsExtra.MSetIntervals]
    +f:358 [binder, in MSetsExtra.MSetIntervals]
    +f:363 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:370 [binder, in MSetsExtra.MSetIntervals]
    +f:375 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:38 [binder, in MSetsExtra.MSetIntervals]
    +f:386 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:398 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:42 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:44 [binder, in MSetsExtra.MSetListWithDups]
    +f:48 [binder, in MSetsExtra.MSetListWithDups]
    +f:55 [binder, in MSetsExtra.MSetIntervals]
    +f:61 [binder, in MSetsExtra.MSetListWithDups]
    +f:64 [binder, in MSetsExtra.MSetIntervals]
    +f:69 [binder, in MSetsExtra.MSetIntervals]
    +f:70 [binder, in MSetsExtra.MSetFoldWithAbort]
    +f:71 [binder, in MSetsExtra.MSetListWithDups]
    +f:75 [binder, in MSetsExtra.MSetListWithDups]
    +f:81 [binder, in MSetsExtra.MSetListWithDups]
    +f:819 [binder, in MSetsExtra.MSetIntervals]
    +f:824 [binder, in MSetsExtra.MSetIntervals]
    +f:828 [binder, in MSetsExtra.MSetIntervals]
    +f:854 [binder, in MSetsExtra.MSetIntervals]
    +f:861 [binder, in MSetsExtra.MSetIntervals]
    +f:867 [binder, in MSetsExtra.MSetIntervals]
    +f:876 [binder, in MSetsExtra.MSetIntervals]
    +f:882 [binder, in MSetsExtra.MSetIntervals]
    +f:889 [binder, in MSetsExtra.MSetIntervals]
    +f:894 [binder, in MSetsExtra.MSetIntervals]
    +f:903 [binder, in MSetsExtra.MSetIntervals]
    +f:910 [binder, in MSetsExtra.MSetIntervals]
    +f:913 [binder, in MSetsExtra.MSetIntervals]
    +f:917 [binder, in MSetsExtra.MSetIntervals]
    +f:920 [binder, in MSetsExtra.MSetIntervals]
    +f:923 [binder, in MSetsExtra.MSetIntervals]
    +f:925 [binder, in MSetsExtra.MSetIntervals]
    +f:971 [binder, in MSetsExtra.MSetIntervals]
    +f:974 [binder, in MSetsExtra.MSetIntervals]
    +f:976 [binder, in MSetsExtra.MSetIntervals]
    +f:978 [binder, in MSetsExtra.MSetIntervals]
    +f:980 [binder, in MSetsExtra.MSetIntervals]
    +f:996 [binder, in MSetsExtra.MSetIntervals]
    +

    H

    HasFoldWithAbort [module, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps [module, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.choose_with_abort_spec [lemma, in MSetsExtra.MSetFoldWithAbort]
    @@ -569,7 +1150,125 @@

    Global Index

    HasFoldWithAbortOps.fwasa_f_pre [projection, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbort.foldWithAbortPrecompute [axiom, in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbort.foldWithAbortPrecomputeSpec [axiom, in MSetsExtra.MSetFoldWithAbort]
    -

    M

    +Heq:62 [binder, in MSetsExtra.MSetIntervals]
    +Hs':580 [binder, in MSetsExtra.MSetIntervals]
    +Hs':629 [binder, in MSetsExtra.MSetIntervals]
    +Hs':681 [binder, in MSetsExtra.MSetIntervals]
    +Hs':728 [binder, in MSetsExtra.MSetIntervals]
    +Hs':743 [binder, in MSetsExtra.MSetIntervals]
    +Hs':748 [binder, in MSetsExtra.MSetIntervals]
    +Hs:268 [binder, in MSetsExtra.MSetIntervals]
    +Hs:480 [binder, in MSetsExtra.MSetIntervals]
    +Hs:483 [binder, in MSetsExtra.MSetIntervals]
    +Hs:544 [binder, in MSetsExtra.MSetIntervals]
    +Hs:547 [binder, in MSetsExtra.MSetIntervals]
    +Hs:579 [binder, in MSetsExtra.MSetIntervals]
    +Hs:628 [binder, in MSetsExtra.MSetIntervals]
    +Hs:680 [binder, in MSetsExtra.MSetIntervals]
    +Hs:690 [binder, in MSetsExtra.MSetIntervals]
    +Hs:697 [binder, in MSetsExtra.MSetIntervals]
    +Hs:727 [binder, in MSetsExtra.MSetIntervals]
    +Hs:731 [binder, in MSetsExtra.MSetIntervals]
    +Hs:735 [binder, in MSetsExtra.MSetIntervals]
    +Hs:737 [binder, in MSetsExtra.MSetIntervals]
    +Hs:742 [binder, in MSetsExtra.MSetIntervals]
    +Hs:747 [binder, in MSetsExtra.MSetIntervals]
    +Hs:780 [binder, in MSetsExtra.MSetIntervals]
    +Hs:785 [binder, in MSetsExtra.MSetIntervals]
    +Hs:789 [binder, in MSetsExtra.MSetIntervals]
    +Hs:793 [binder, in MSetsExtra.MSetIntervals]
    +Hs:810 [binder, in MSetsExtra.MSetIntervals]
    +Hs:814 [binder, in MSetsExtra.MSetIntervals]
    +Hs:825 [binder, in MSetsExtra.MSetIntervals]
    +Hs:829 [binder, in MSetsExtra.MSetIntervals]
    +H0:575 [binder, in MSetsExtra.MSetIntervals]
    +H0:624 [binder, in MSetsExtra.MSetIntervals]
    +H0:676 [binder, in MSetsExtra.MSetIntervals]
    +H1:313 [binder, in MSetsExtra.MSetIntervals]
    +H2:314 [binder, in MSetsExtra.MSetIntervals]
    +H:540 [binder, in MSetsExtra.MSetIntervals]
    +H:574 [binder, in MSetsExtra.MSetIntervals]
    +H:59 [binder, in MSetsExtra.MSetIntervals]
    +H:623 [binder, in MSetsExtra.MSetIntervals]
    +H:675 [binder, in MSetsExtra.MSetIntervals]
    +H:693 [binder, in MSetsExtra.MSetIntervals]
    +H:883 [binder, in MSetsExtra.MSetIntervals]
    +H:918 [binder, in MSetsExtra.MSetIntervals]
    +H:921 [binder, in MSetsExtra.MSetIntervals]
    +

    I

    +In:34 [binder, in MSetsExtra.MSetFoldWithAbort]
    +In:6 [binder, in MSetsExtra.MSetFoldWithAbort]
    +In:62 [binder, in MSetsExtra.MSetFoldWithAbort]
    +In:96 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i':105 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i':13 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i':319 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i':41 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i':69 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i1:86 [binder, in MSetsExtra.MSetIntervals]
    +i2:87 [binder, in MSetsExtra.MSetIntervals]
    +i:104 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i:12 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i:138 [binder, in MSetsExtra.MSetListWithDups]
    +i:160 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i:176 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i:191 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i:202 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i:231 [binder, in MSetsExtra.MSetIntervals]
    +i:318 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i:33 [binder, in MSetsExtra.MSetWithDups]
    +i:40 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i:407 [binder, in MSetsExtra.MSetIntervals]
    +i:63 [binder, in MSetsExtra.MSetListWithDups]
    +i:68 [binder, in MSetsExtra.MSetFoldWithAbort]
    +i:818 [binder, in MSetsExtra.MSetIntervals]
    +i:91 [binder, in MSetsExtra.MSetIntervals]
    +i:995 [binder, in MSetsExtra.MSetIntervals]
    +

    L

    +ls:290 [binder, in MSetsExtra.MSetIntervals]
    +lt:33 [binder, in MSetsExtra.MSetFoldWithAbort]
    +lt:61 [binder, in MSetsExtra.MSetFoldWithAbort]
    +lt:95 [binder, in MSetsExtra.MSetFoldWithAbort]
    +l':24 [binder, in MSetsExtra.MSetListWithDups]
    +l':30 [binder, in MSetsExtra.MSetListWithDups]
    +l1:345 [binder, in MSetsExtra.MSetIntervals]
    +l1:425 [binder, in MSetsExtra.MSetIntervals]
    +l1:707 [binder, in MSetsExtra.MSetIntervals]
    +l1:715 [binder, in MSetsExtra.MSetIntervals]
    +l2:346 [binder, in MSetsExtra.MSetIntervals]
    +l2:426 [binder, in MSetsExtra.MSetIntervals]
    +l2:710 [binder, in MSetsExtra.MSetIntervals]
    +l2:718 [binder, in MSetsExtra.MSetIntervals]
    +l:10 [binder, in MSetsExtra.MSetListWithDups]
    +l:103 [binder, in MSetsExtra.MSetIntervals]
    +l:115 [binder, in MSetsExtra.MSetIntervals]
    +l:119 [binder, in MSetsExtra.MSetIntervals]
    +l:134 [binder, in MSetsExtra.MSetIntervals]
    +l:14 [binder, in MSetsExtra.MSetListWithDups]
    +l:16 [binder, in MSetsExtra.MSetListWithDups]
    +l:17 [binder, in MSetsExtra.MSetIntervals]
    +l:23 [binder, in MSetsExtra.MSetListWithDups]
    +l:24 [binder, in MSetsExtra.MSetIntervals]
    +l:254 [binder, in MSetsExtra.MSetIntervals]
    +l:258 [binder, in MSetsExtra.MSetIntervals]
    +l:260 [binder, in MSetsExtra.MSetIntervals]
    +l:28 [binder, in MSetsExtra.MSetListWithDups]
    +l:29 [binder, in MSetsExtra.MSetListWithDups]
    +l:30 [binder, in MSetsExtra.MSetIntervals]
    +l:339 [binder, in MSetsExtra.MSetIntervals]
    +l:34 [binder, in MSetsExtra.MSetListWithDups]
    +l:34 [binder, in MSetsExtra.MSetIntervals]
    +l:367 [binder, in MSetsExtra.MSetIntervals]
    +l:39 [binder, in MSetsExtra.MSetIntervals]
    +l:427 [binder, in MSetsExtra.MSetIntervals]
    +l:50 [binder, in MSetsExtra.MSetIntervals]
    +l:551 [binder, in MSetsExtra.MSetIntervals]
    +l:554 [binder, in MSetsExtra.MSetIntervals]
    +l:698 [binder, in MSetsExtra.MSetIntervals]
    +l:701 [binder, in MSetsExtra.MSetIntervals]
    +l:75 [binder, in MSetsExtra.MSetIntervals]
    +l:849 [binder, in MSetsExtra.MSetIntervals]
    +

    M

    Make [module, in MSetsExtra.MSetListWithDups]
    MakeAVLSetsWithFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    MakeAVLSetsWithFoldA.foldWithAbortPrecompute [definition, in MSetsExtra.MSetFoldWithAbort]
    @@ -627,7 +1326,7 @@

    Global Index

    Make.Subset [definition, in MSetsExtra.MSetListWithDups]
    Make.subset_spec [lemma, in MSetsExtra.MSetListWithDups]
    Make.union_spec [lemma, in MSetsExtra.MSetListWithDups]
    -_ [<=] _ [notation, in MSetsExtra.MSetListWithDups]
    +_ [<=] _ [notation, in MSetsExtra.MSetListWithDups]
    _ [=] _ [notation, in MSetsExtra.MSetListWithDups]
    MSetFoldWithAbort [library]
    MSetIntervals [module, in MSetsExtra.MSetIntervals]
    @@ -693,7 +1392,6 @@

    Global Index

    MSetIntervals.min_elt_spec2 [lemma, in MSetsExtra.MSetIntervals]
    MSetIntervals.min_elt_spec1 [lemma, in MSetsExtra.MSetIntervals]
    MSetIntervals.min_elt [definition, in MSetsExtra.MSetIntervals]
    -MSetIntervals.Mkt [constructor, in MSetsExtra.MSetIntervals]
    MSetIntervals.partition [definition, in MSetsExtra.MSetIntervals]
    MSetIntervals.partition_spec2 [lemma, in MSetsExtra.MSetIntervals]
    MSetIntervals.partition_spec1 [lemma, in MSetsExtra.MSetIntervals]
    @@ -718,10 +1416,16 @@

    Global Index

    MSetIntervals.union_spec [lemma, in MSetsExtra.MSetIntervals]
    MSetListWithDups [library]
    MSetWithDups [library]
    -

    N

    +

    N

    NoDupA_map [lemma, in MSetsExtra.MSetIntervals]
    NOP [module, in MSetsExtra.MSetIntervals]
    -

    O

    +n:1011 [binder, in MSetsExtra.MSetIntervals]
    +n:1018 [binder, in MSetsExtra.MSetIntervals]
    +n:2 [binder, in MSetsExtra.MSetIntervals]
    +n:289 [binder, in MSetsExtra.MSetIntervals]
    +n:4 [binder, in MSetsExtra.MSetIntervals]
    +n:52 [binder, in MSetsExtra.MSetIntervals]
    +

    O

    Ops [module, in MSetsExtra.MSetListWithDups]
    Ops [module, in MSetsExtra.MSetIntervals]
    Ops.acc_pred [lemma, in MSetsExtra.MSetIntervals]
    @@ -824,7 +1528,37 @@

    Global Index

    Ops.union [definition, in MSetsExtra.MSetListWithDups]
    Ops.union [definition, in MSetsExtra.MSetIntervals]
    Ops.union_aux [definition, in MSetsExtra.MSetIntervals]
    -

    R

    +

    P

    +pre_x:392 [binder, in MSetsExtra.MSetFoldWithAbort]
    +pre_x:369 [binder, in MSetsExtra.MSetFoldWithAbort]
    +P:10 [binder, in MSetsExtra.MSetWithDups]
    +P:101 [binder, in MSetsExtra.MSetListWithDups]
    +P:112 [binder, in MSetsExtra.MSetFoldWithAbort]
    +P:13 [binder, in MSetsExtra.MSetWithDups]
    +P:18 [binder, in MSetsExtra.MSetFoldWithAbort]
    +P:220 [binder, in MSetsExtra.MSetFoldWithAbort]
    +p:231 [binder, in MSetsExtra.MSetFoldWithAbort]
    +p:236 [binder, in MSetsExtra.MSetFoldWithAbort]
    +P:239 [binder, in MSetsExtra.MSetFoldWithAbort]
    +p:242 [binder, in MSetsExtra.MSetFoldWithAbort]
    +p:247 [binder, in MSetsExtra.MSetFoldWithAbort]
    +p:257 [binder, in MSetsExtra.MSetFoldWithAbort]
    +p:265 [binder, in MSetsExtra.MSetFoldWithAbort]
    +P:270 [binder, in MSetsExtra.MSetFoldWithAbort]
    +P:281 [binder, in MSetsExtra.MSetIntervals]
    +P:284 [binder, in MSetsExtra.MSetIntervals]
    +P:292 [binder, in MSetsExtra.MSetFoldWithAbort]
    +P:300 [binder, in MSetsExtra.MSetFoldWithAbort]
    +p:319 [binder, in MSetsExtra.MSetIntervals]
    +P:326 [binder, in MSetsExtra.MSetFoldWithAbort]
    +P:46 [binder, in MSetsExtra.MSetFoldWithAbort]
    +p:53 [binder, in MSetsExtra.MSetIntervals]
    +P:75 [binder, in MSetsExtra.MSetFoldWithAbort]
    +P:940 [binder, in MSetsExtra.MSetIntervals]
    +P:943 [binder, in MSetsExtra.MSetIntervals]
    +P:98 [binder, in MSetsExtra.MSetListWithDups]
    +p:982 [binder, in MSetsExtra.MSetIntervals]
    +

    R

    Raw [module, in MSetsExtra.MSetIntervals]
    Raw.addZ_invariant [lemma, in MSetsExtra.MSetIntervals]
    Raw.addZ_InZ [lemma, in MSetsExtra.MSetIntervals]
    @@ -1042,15 +1776,448 @@

    Global Index

    RemoveDupsFromSorted.XTotalLeBool.leb [definition, in MSetsExtra.MSetListWithDups]
    RemoveDupsFromSorted.XTotalLeBool.leb_total [lemma, in MSetsExtra.MSetListWithDups]
    RemoveDupsFromSorted.XTotalLeBool.t [definition, in MSetsExtra.MSetListWithDups]
    -_ <=? _ [notation, in MSetsExtra.MSetListWithDups]
    +_ <=? _ [notation, in MSetsExtra.MSetListWithDups]
    rev_map_alt_def [lemma, in MSetsExtra.MSetIntervals]
    rev_map_aux_alt_def [lemma, in MSetsExtra.MSetIntervals]
    rev_map [definition, in MSetsExtra.MSetIntervals]
    rev_map_aux [definition, in MSetsExtra.MSetIntervals]
    -

    S

    +r:859 [binder, in MSetsExtra.MSetIntervals]
    +

    S

    SetsWithFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    SetsWithFoldA.E [module, in MSetsExtra.MSetFoldWithAbort]
    -

    W

    +st':114 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':120 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':124 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':20 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':26 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':328 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':334 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':338 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':48 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':54 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':77 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':83 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st':87 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st0:313 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st1:314 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st2:315 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:113 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:117 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:119 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:123 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:135 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:137 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:139 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:144 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:146 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:151 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:153 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:155 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:168 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:170 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:173 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:183 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:185 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:185 [binder, in MSetsExtra.MSetIntervals]
    +st:188 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:189 [binder, in MSetsExtra.MSetIntervals]
    +st:19 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:192 [binder, in MSetsExtra.MSetIntervals]
    +st:197 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:199 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:208 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:208 [binder, in MSetsExtra.MSetIntervals]
    +st:210 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:212 [binder, in MSetsExtra.MSetIntervals]
    +st:215 [binder, in MSetsExtra.MSetIntervals]
    +st:23 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:25 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:258 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:262 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:266 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:275 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:327 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:331 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:333 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:337 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:381 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:404 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:47 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:51 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:53 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:76 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:80 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:82 [binder, in MSetsExtra.MSetFoldWithAbort]
    +st:86 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s'':749 [binder, in MSetsExtra.MSetIntervals]
    +s':107 [binder, in MSetsExtra.MSetListWithDups]
    +s':109 [binder, in MSetsExtra.MSetListWithDups]
    +s':130 [binder, in MSetsExtra.MSetListWithDups]
    +s':133 [binder, in MSetsExtra.MSetListWithDups]
    +s':143 [binder, in MSetsExtra.MSetListWithDups]
    +s':162 [binder, in MSetsExtra.MSetIntervals]
    +s':274 [binder, in MSetsExtra.MSetIntervals]
    +s':277 [binder, in MSetsExtra.MSetIntervals]
    +s':3 [binder, in MSetsExtra.MSetWithDups]
    +s':430 [binder, in MSetsExtra.MSetIntervals]
    +s':435 [binder, in MSetsExtra.MSetIntervals]
    +s':577 [binder, in MSetsExtra.MSetIntervals]
    +s':6 [binder, in MSetsExtra.MSetWithDups]
    +s':626 [binder, in MSetsExtra.MSetIntervals]
    +s':65 [binder, in MSetsExtra.MSetListWithDups]
    +s':67 [binder, in MSetsExtra.MSetListWithDups]
    +s':678 [binder, in MSetsExtra.MSetIntervals]
    +s':70 [binder, in MSetsExtra.MSetListWithDups]
    +s':726 [binder, in MSetsExtra.MSetIntervals]
    +s':741 [binder, in MSetsExtra.MSetIntervals]
    +s':746 [binder, in MSetsExtra.MSetIntervals]
    +s':796 [binder, in MSetsExtra.MSetIntervals]
    +s':91 [binder, in MSetsExtra.MSetListWithDups]
    +s':933 [binder, in MSetsExtra.MSetIntervals]
    +s':936 [binder, in MSetsExtra.MSetIntervals]
    +s':94 [binder, in MSetsExtra.MSetListWithDups]
    +s':954 [binder, in MSetsExtra.MSetIntervals]
    +s':956 [binder, in MSetsExtra.MSetIntervals]
    +s':958 [binder, in MSetsExtra.MSetIntervals]
    +s':960 [binder, in MSetsExtra.MSetIntervals]
    +s':962 [binder, in MSetsExtra.MSetIntervals]
    +s':984 [binder, in MSetsExtra.MSetIntervals]
    +s0:245 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s0:249 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s0:252 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s1:138 [binder, in MSetsExtra.MSetIntervals]
    +s1:143 [binder, in MSetsExtra.MSetIntervals]
    +s1:152 [binder, in MSetsExtra.MSetIntervals]
    +s1:155 [binder, in MSetsExtra.MSetIntervals]
    +s1:157 [binder, in MSetsExtra.MSetIntervals]
    +s1:173 [binder, in MSetsExtra.MSetIntervals]
    +s1:176 [binder, in MSetsExtra.MSetIntervals]
    +s1:379 [binder, in MSetsExtra.MSetIntervals]
    +s1:386 [binder, in MSetsExtra.MSetIntervals]
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    +s1:465 [binder, in MSetsExtra.MSetIntervals]
    +s1:497 [binder, in MSetsExtra.MSetIntervals]
    +s1:53 [binder, in MSetsExtra.MSetListWithDups]
    +s1:55 [binder, in MSetsExtra.MSetListWithDups]
    +s1:556 [binder, in MSetsExtra.MSetIntervals]
    +s1:559 [binder, in MSetsExtra.MSetIntervals]
    +s1:562 [binder, in MSetsExtra.MSetIntervals]
    +s1:564 [binder, in MSetsExtra.MSetIntervals]
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    +s1:572 [binder, in MSetsExtra.MSetIntervals]
    +s1:605 [binder, in MSetsExtra.MSetIntervals]
    +s1:616 [binder, in MSetsExtra.MSetIntervals]
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    +s1:621 [binder, in MSetsExtra.MSetIntervals]
    +s1:656 [binder, in MSetsExtra.MSetIntervals]
    +s1:668 [binder, in MSetsExtra.MSetIntervals]
    +s1:671 [binder, in MSetsExtra.MSetIntervals]
    +s1:673 [binder, in MSetsExtra.MSetIntervals]
    +s1:703 [binder, in MSetsExtra.MSetIntervals]
    +s1:720 [binder, in MSetsExtra.MSetIntervals]
    +s1:738 [binder, in MSetsExtra.MSetIntervals]
    +s1:751 [binder, in MSetsExtra.MSetIntervals]
    +s1:753 [binder, in MSetsExtra.MSetIntervals]
    +s1:761 [binder, in MSetsExtra.MSetIntervals]
    +s1:763 [binder, in MSetsExtra.MSetIntervals]
    +s1:765 [binder, in MSetsExtra.MSetIntervals]
    +s1:94 [binder, in MSetsExtra.MSetIntervals]
    +s1:968 [binder, in MSetsExtra.MSetIntervals]
    +s2:140 [binder, in MSetsExtra.MSetIntervals]
    +s2:144 [binder, in MSetsExtra.MSetIntervals]
    +s2:153 [binder, in MSetsExtra.MSetIntervals]
    +s2:156 [binder, in MSetsExtra.MSetIntervals]
    +s2:159 [binder, in MSetsExtra.MSetIntervals]
    +s2:174 [binder, in MSetsExtra.MSetIntervals]
    +s2:177 [binder, in MSetsExtra.MSetIntervals]
    +s2:380 [binder, in MSetsExtra.MSetIntervals]
    +s2:387 [binder, in MSetsExtra.MSetIntervals]
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    +s2:454 [binder, in MSetsExtra.MSetIntervals]
    +s2:458 [binder, in MSetsExtra.MSetIntervals]
    +s2:462 [binder, in MSetsExtra.MSetIntervals]
    +s2:498 [binder, in MSetsExtra.MSetIntervals]
    +s2:54 [binder, in MSetsExtra.MSetListWithDups]
    +s2:557 [binder, in MSetsExtra.MSetIntervals]
    +s2:56 [binder, in MSetsExtra.MSetListWithDups]
    +s2:560 [binder, in MSetsExtra.MSetIntervals]
    +s2:563 [binder, in MSetsExtra.MSetIntervals]
    +s2:565 [binder, in MSetsExtra.MSetIntervals]
    +s2:567 [binder, in MSetsExtra.MSetIntervals]
    +s2:570 [binder, in MSetsExtra.MSetIntervals]
    +s2:571 [binder, in MSetsExtra.MSetIntervals]
    +s2:573 [binder, in MSetsExtra.MSetIntervals]
    +s2:604 [binder, in MSetsExtra.MSetIntervals]
    +s2:617 [binder, in MSetsExtra.MSetIntervals]
    +s2:620 [binder, in MSetsExtra.MSetIntervals]
    +s2:622 [binder, in MSetsExtra.MSetIntervals]
    +s2:655 [binder, in MSetsExtra.MSetIntervals]
    +s2:669 [binder, in MSetsExtra.MSetIntervals]
    +s2:672 [binder, in MSetsExtra.MSetIntervals]
    +s2:674 [binder, in MSetsExtra.MSetIntervals]
    +s2:704 [binder, in MSetsExtra.MSetIntervals]
    +s2:721 [binder, in MSetsExtra.MSetIntervals]
    +s2:723 [binder, in MSetsExtra.MSetIntervals]
    +s2:739 [binder, in MSetsExtra.MSetIntervals]
    +s2:752 [binder, in MSetsExtra.MSetIntervals]
    +s2:754 [binder, in MSetsExtra.MSetIntervals]
    +s2:762 [binder, in MSetsExtra.MSetIntervals]
    +s2:764 [binder, in MSetsExtra.MSetIntervals]
    +s2:766 [binder, in MSetsExtra.MSetIntervals]
    +s2:95 [binder, in MSetsExtra.MSetIntervals]
    +s2:969 [binder, in MSetsExtra.MSetIntervals]
    +s:102 [binder, in MSetsExtra.MSetListWithDups]
    +s:104 [binder, in MSetsExtra.MSetListWithDups]
    +s:106 [binder, in MSetsExtra.MSetListWithDups]
    +s:108 [binder, in MSetsExtra.MSetListWithDups]
    +s:108 [binder, in MSetsExtra.MSetIntervals]
    +s:11 [binder, in MSetsExtra.MSetWithDups]
    +s:111 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:112 [binder, in MSetsExtra.MSetListWithDups]
    +s:112 [binder, in MSetsExtra.MSetIntervals]
    +s:113 [binder, in MSetsExtra.MSetListWithDups]
    +s:114 [binder, in MSetsExtra.MSetIntervals]
    +s:116 [binder, in MSetsExtra.MSetIntervals]
    +s:117 [binder, in MSetsExtra.MSetIntervals]
    +s:118 [binder, in MSetsExtra.MSetListWithDups]
    +s:122 [binder, in MSetsExtra.MSetListWithDups]
    +s:124 [binder, in MSetsExtra.MSetIntervals]
    +s:125 [binder, in MSetsExtra.MSetListWithDups]
    +s:127 [binder, in MSetsExtra.MSetIntervals]
    +s:129 [binder, in MSetsExtra.MSetListWithDups]
    +s:131 [binder, in MSetsExtra.MSetIntervals]
    +s:132 [binder, in MSetsExtra.MSetListWithDups]
    +s:133 [binder, in MSetsExtra.MSetIntervals]
    +s:135 [binder, in MSetsExtra.MSetIntervals]
    +s:136 [binder, in MSetsExtra.MSetListWithDups]
    +s:136 [binder, in MSetsExtra.MSetIntervals]
    +s:14 [binder, in MSetsExtra.MSetWithDups]
    +s:140 [binder, in MSetsExtra.MSetListWithDups]
    +s:142 [binder, in MSetsExtra.MSetListWithDups]
    +s:145 [binder, in MSetsExtra.MSetListWithDups]
    +s:146 [binder, in MSetsExtra.MSetListWithDups]
    +s:148 [binder, in MSetsExtra.MSetIntervals]
    +s:149 [binder, in MSetsExtra.MSetListWithDups]
    +s:154 [binder, in MSetsExtra.MSetListWithDups]
    +s:157 [binder, in MSetsExtra.MSetListWithDups]
    +s:159 [binder, in MSetsExtra.MSetListWithDups]
    +s:161 [binder, in MSetsExtra.MSetIntervals]
    +s:162 [binder, in MSetsExtra.MSetListWithDups]
    +s:164 [binder, in MSetsExtra.MSetListWithDups]
    +s:165 [binder, in MSetsExtra.MSetListWithDups]
    +s:166 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:168 [binder, in MSetsExtra.MSetListWithDups]
    +s:169 [binder, in MSetsExtra.MSetListWithDups]
    +s:169 [binder, in MSetsExtra.MSetIntervals]
    +s:17 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:181 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:195 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:2 [binder, in MSetsExtra.MSetWithDups]
    +s:202 [binder, in MSetsExtra.MSetIntervals]
    +s:206 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:219 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:223 [binder, in MSetsExtra.MSetIntervals]
    +s:229 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:230 [binder, in MSetsExtra.MSetIntervals]
    +s:234 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:235 [binder, in MSetsExtra.MSetIntervals]
    +s:239 [binder, in MSetsExtra.MSetIntervals]
    +s:240 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:243 [binder, in MSetsExtra.MSetIntervals]
    +s:246 [binder, in MSetsExtra.MSetIntervals]
    +s:247 [binder, in MSetsExtra.MSetIntervals]
    +s:249 [binder, in MSetsExtra.MSetIntervals]
    +s:250 [binder, in MSetsExtra.MSetIntervals]
    +s:253 [binder, in MSetsExtra.MSetIntervals]
    +s:255 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:263 [binder, in MSetsExtra.MSetIntervals]
    +s:264 [binder, in MSetsExtra.MSetIntervals]
    +s:267 [binder, in MSetsExtra.MSetIntervals]
    +s:270 [binder, in MSetsExtra.MSetIntervals]
    +s:271 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:272 [binder, in MSetsExtra.MSetIntervals]
    +s:273 [binder, in MSetsExtra.MSetIntervals]
    +s:276 [binder, in MSetsExtra.MSetIntervals]
    +s:279 [binder, in MSetsExtra.MSetIntervals]
    +s:282 [binder, in MSetsExtra.MSetIntervals]
    +s:285 [binder, in MSetsExtra.MSetIntervals]
    +s:289 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:293 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:296 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:299 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:325 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:362 [binder, in MSetsExtra.MSetIntervals]
    +s:365 [binder, in MSetsExtra.MSetIntervals]
    +s:37 [binder, in MSetsExtra.MSetListWithDups]
    +s:371 [binder, in MSetsExtra.MSetIntervals]
    +s:375 [binder, in MSetsExtra.MSetIntervals]
    +s:378 [binder, in MSetsExtra.MSetIntervals]
    +s:385 [binder, in MSetsExtra.MSetIntervals]
    +s:389 [binder, in MSetsExtra.MSetIntervals]
    +s:395 [binder, in MSetsExtra.MSetIntervals]
    +s:41 [binder, in MSetsExtra.MSetListWithDups]
    +s:413 [binder, in MSetsExtra.MSetIntervals]
    +s:416 [binder, in MSetsExtra.MSetIntervals]
    +s:421 [binder, in MSetsExtra.MSetIntervals]
    +s:424 [binder, in MSetsExtra.MSetIntervals]
    +s:436 [binder, in MSetsExtra.MSetIntervals]
    +s:440 [binder, in MSetsExtra.MSetIntervals]
    +s:441 [binder, in MSetsExtra.MSetIntervals]
    +s:444 [binder, in MSetsExtra.MSetIntervals]
    +s:447 [binder, in MSetsExtra.MSetIntervals]
    +s:45 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:45 [binder, in MSetsExtra.MSetListWithDups]
    +s:470 [binder, in MSetsExtra.MSetIntervals]
    +s:472 [binder, in MSetsExtra.MSetIntervals]
    +s:474 [binder, in MSetsExtra.MSetIntervals]
    +s:477 [binder, in MSetsExtra.MSetIntervals]
    +s:478 [binder, in MSetsExtra.MSetIntervals]
    +s:481 [binder, in MSetsExtra.MSetIntervals]
    +s:49 [binder, in MSetsExtra.MSetWithDups]
    +s:49 [binder, in MSetsExtra.MSetListWithDups]
    +s:490 [binder, in MSetsExtra.MSetIntervals]
    +s:494 [binder, in MSetsExtra.MSetIntervals]
    +s:5 [binder, in MSetsExtra.MSetWithDups]
    +s:501 [binder, in MSetsExtra.MSetIntervals]
    +s:504 [binder, in MSetsExtra.MSetIntervals]
    +s:508 [binder, in MSetsExtra.MSetIntervals]
    +s:51 [binder, in MSetsExtra.MSetWithDups]
    +s:51 [binder, in MSetsExtra.MSetListWithDups]
    +s:521 [binder, in MSetsExtra.MSetIntervals]
    +s:528 [binder, in MSetsExtra.MSetIntervals]
    +s:53 [binder, in MSetsExtra.MSetWithDups]
    +s:531 [binder, in MSetsExtra.MSetIntervals]
    +s:535 [binder, in MSetsExtra.MSetIntervals]
    +s:538 [binder, in MSetsExtra.MSetIntervals]
    +s:541 [binder, in MSetsExtra.MSetIntervals]
    +s:546 [binder, in MSetsExtra.MSetIntervals]
    +s:552 [binder, in MSetsExtra.MSetIntervals]
    +s:555 [binder, in MSetsExtra.MSetIntervals]
    +s:56 [binder, in MSetsExtra.MSetWithDups]
    +s:57 [binder, in MSetsExtra.MSetWithDups]
    +s:576 [binder, in MSetsExtra.MSetIntervals]
    +s:583 [binder, in MSetsExtra.MSetIntervals]
    +s:587 [binder, in MSetsExtra.MSetIntervals]
    +s:62 [binder, in MSetsExtra.MSetListWithDups]
    +s:625 [binder, in MSetsExtra.MSetIntervals]
    +s:632 [binder, in MSetsExtra.MSetIntervals]
    +s:636 [binder, in MSetsExtra.MSetIntervals]
    +s:64 [binder, in MSetsExtra.MSetListWithDups]
    +s:66 [binder, in MSetsExtra.MSetListWithDups]
    +s:677 [binder, in MSetsExtra.MSetIntervals]
    +s:683 [binder, in MSetsExtra.MSetIntervals]
    +s:685 [binder, in MSetsExtra.MSetIntervals]
    +s:687 [binder, in MSetsExtra.MSetIntervals]
    +s:69 [binder, in MSetsExtra.MSetListWithDups]
    +s:691 [binder, in MSetsExtra.MSetIntervals]
    +s:694 [binder, in MSetsExtra.MSetIntervals]
    +s:699 [binder, in MSetsExtra.MSetIntervals]
    +s:702 [binder, in MSetsExtra.MSetIntervals]
    +s:71 [binder, in MSetsExtra.MSetIntervals]
    +s:72 [binder, in MSetsExtra.MSetListWithDups]
    +s:725 [binder, in MSetsExtra.MSetIntervals]
    +s:729 [binder, in MSetsExtra.MSetIntervals]
    +s:734 [binder, in MSetsExtra.MSetIntervals]
    +s:736 [binder, in MSetsExtra.MSetIntervals]
    +s:74 [binder, in MSetsExtra.MSetFoldWithAbort]
    +s:74 [binder, in MSetsExtra.MSetIntervals]
    +s:740 [binder, in MSetsExtra.MSetIntervals]
    +s:745 [binder, in MSetsExtra.MSetIntervals]
    +s:755 [binder, in MSetsExtra.MSetIntervals]
    +s:756 [binder, in MSetsExtra.MSetIntervals]
    +s:76 [binder, in MSetsExtra.MSetListWithDups]
    +s:769 [binder, in MSetsExtra.MSetIntervals]
    +s:77 [binder, in MSetsExtra.MSetIntervals]
    +s:770 [binder, in MSetsExtra.MSetIntervals]
    +s:775 [binder, in MSetsExtra.MSetIntervals]
    +s:777 [binder, in MSetsExtra.MSetIntervals]
    +s:781 [binder, in MSetsExtra.MSetIntervals]
    +s:783 [binder, in MSetsExtra.MSetIntervals]
    +s:786 [binder, in MSetsExtra.MSetIntervals]
    +s:79 [binder, in MSetsExtra.MSetIntervals]
    +s:790 [binder, in MSetsExtra.MSetIntervals]
    +s:791 [binder, in MSetsExtra.MSetIntervals]
    +s:794 [binder, in MSetsExtra.MSetIntervals]
    +s:795 [binder, in MSetsExtra.MSetIntervals]
    +s:799 [binder, in MSetsExtra.MSetIntervals]
    +s:8 [binder, in MSetsExtra.MSetWithDups]
    +s:801 [binder, in MSetsExtra.MSetIntervals]
    +s:805 [binder, in MSetsExtra.MSetIntervals]
    +s:806 [binder, in MSetsExtra.MSetIntervals]
    +s:808 [binder, in MSetsExtra.MSetIntervals]
    +s:811 [binder, in MSetsExtra.MSetIntervals]
    +s:815 [binder, in MSetsExtra.MSetIntervals]
    +s:816 [binder, in MSetsExtra.MSetIntervals]
    +s:82 [binder, in MSetsExtra.MSetListWithDups]
    +s:820 [binder, in MSetsExtra.MSetIntervals]
    +s:822 [binder, in MSetsExtra.MSetIntervals]
    +s:823 [binder, in MSetsExtra.MSetIntervals]
    +s:827 [binder, in MSetsExtra.MSetIntervals]
    +s:83 [binder, in MSetsExtra.MSetIntervals]
    +s:85 [binder, in MSetsExtra.MSetListWithDups]
    +s:86 [binder, in MSetsExtra.MSetListWithDups]
    +s:868 [binder, in MSetsExtra.MSetIntervals]
    +s:877 [binder, in MSetsExtra.MSetIntervals]
    +s:881 [binder, in MSetsExtra.MSetIntervals]
    +s:887 [binder, in MSetsExtra.MSetIntervals]
    +s:89 [binder, in MSetsExtra.MSetListWithDups]
    +s:90 [binder, in MSetsExtra.MSetListWithDups]
    +s:904 [binder, in MSetsExtra.MSetIntervals]
    +s:911 [binder, in MSetsExtra.MSetIntervals]
    +s:914 [binder, in MSetsExtra.MSetIntervals]
    +s:916 [binder, in MSetsExtra.MSetIntervals]
    +s:919 [binder, in MSetsExtra.MSetIntervals]
    +s:922 [binder, in MSetsExtra.MSetIntervals]
    +s:924 [binder, in MSetsExtra.MSetIntervals]
    +s:93 [binder, in MSetsExtra.MSetListWithDups]
    +s:931 [binder, in MSetsExtra.MSetIntervals]
    +s:932 [binder, in MSetsExtra.MSetIntervals]
    +s:935 [binder, in MSetsExtra.MSetIntervals]
    +s:938 [binder, in MSetsExtra.MSetIntervals]
    +s:941 [binder, in MSetsExtra.MSetIntervals]
    +s:944 [binder, in MSetsExtra.MSetIntervals]
    +s:947 [binder, in MSetsExtra.MSetIntervals]
    +s:949 [binder, in MSetsExtra.MSetIntervals]
    +s:951 [binder, in MSetsExtra.MSetIntervals]
    +s:953 [binder, in MSetsExtra.MSetIntervals]
    +s:955 [binder, in MSetsExtra.MSetIntervals]
    +s:957 [binder, in MSetsExtra.MSetIntervals]
    +s:959 [binder, in MSetsExtra.MSetIntervals]
    +s:96 [binder, in MSetsExtra.MSetListWithDups]
    +s:961 [binder, in MSetsExtra.MSetIntervals]
    +s:963 [binder, in MSetsExtra.MSetIntervals]
    +s:964 [binder, in MSetsExtra.MSetIntervals]
    +s:965 [binder, in MSetsExtra.MSetIntervals]
    +s:966 [binder, in MSetsExtra.MSetIntervals]
    +s:967 [binder, in MSetsExtra.MSetIntervals]
    +s:972 [binder, in MSetsExtra.MSetIntervals]
    +s:973 [binder, in MSetsExtra.MSetIntervals]
    +s:975 [binder, in MSetsExtra.MSetIntervals]
    +s:977 [binder, in MSetsExtra.MSetIntervals]
    +s:979 [binder, in MSetsExtra.MSetIntervals]
    +s:981 [binder, in MSetsExtra.MSetIntervals]
    +s:983 [binder, in MSetsExtra.MSetIntervals]
    +s:99 [binder, in MSetsExtra.MSetListWithDups]
    +

    T

    +t:2 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:29 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:308 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:32 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:349 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:357 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:365 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:377 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:388 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:400 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:5 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:57 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:60 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:90 [binder, in MSetsExtra.MSetFoldWithAbort]
    +t:94 [binder, in MSetsExtra.MSetFoldWithAbort]
    +

    W

    WSetsOnWithDups [module, in MSetsExtra.MSetWithDups]
    WSetsOnWithDupsExtra [module, in MSetsExtra.MSetWithDups]
    WSetsOnWithDupsExtra.cardinal_spec [axiom, in MSetsExtra.MSetWithDups]
    @@ -1092,7 +2259,7 @@

    Global Index

    WSetsOnWithDups.Subset [definition, in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.subset_spec [axiom, in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.union_spec [axiom, in MSetsExtra.MSetWithDups]
    -_ [<=] _ [notation, in MSetsExtra.MSetWithDups]
    +_ [<=] _ [notation, in MSetsExtra.MSetWithDups]
    _ [=] _ [notation, in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w [lemma, in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1 [lemma, in MSetsExtra.MSetWithDups]
    @@ -1102,24 +2269,1944 @@

    Global Index

    WSetsWithDupsFoldA.E [module, in MSetsExtra.MSetFoldWithAbort]
    WSetsWithFoldA [module, in MSetsExtra.MSetFoldWithAbort]
    WSetsWithFoldA.E [module, in MSetsExtra.MSetFoldWithAbort]
    -

    Z

    +

    X

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    +xs:20 [binder, in MSetsExtra.MSetListWithDups]
    +xs:277 [binder, in MSetsExtra.MSetFoldWithAbort]
    +xx:804 [binder, in MSetsExtra.MSetIntervals]
    +x_pre:312 [binder, in MSetsExtra.MSetFoldWithAbort]
    +x':21 [binder, in MSetsExtra.MSetListWithDups]
    +x':798 [binder, in MSetsExtra.MSetIntervals]
    +x0:278 [binder, in MSetsExtra.MSetFoldWithAbort]
    +x0:281 [binder, in MSetsExtra.MSetFoldWithAbort]
    +x0:284 [binder, in MSetsExtra.MSetFoldWithAbort]
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    +x1:459 [binder, in MSetsExtra.MSetIntervals]
    +x1:463 [binder, in MSetsExtra.MSetIntervals]
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    +x1:522 [binder, in MSetsExtra.MSetIntervals]
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    +x:111 [binder, in MSetsExtra.MSetIntervals]
    +x:113 [binder, in MSetsExtra.MSetIntervals]
    +x:114 [binder, in MSetsExtra.MSetListWithDups]
    +x:116 [binder, in MSetsExtra.MSetListWithDups]
    +x:118 [binder, in MSetsExtra.MSetIntervals]
    +x:12 [binder, in MSetsExtra.MSetWithDups]
    +x:120 [binder, in MSetsExtra.MSetListWithDups]
    +x:120 [binder, in MSetsExtra.MSetIntervals]
    +x:121 [binder, in MSetsExtra.MSetIntervals]
    +x:122 [binder, in MSetsExtra.MSetIntervals]
    +x:123 [binder, in MSetsExtra.MSetListWithDups]
    +x:126 [binder, in MSetsExtra.MSetListWithDups]
    +x:126 [binder, in MSetsExtra.MSetIntervals]
    +x:130 [binder, in MSetsExtra.MSetIntervals]
    +x:131 [binder, in MSetsExtra.MSetListWithDups]
    +x:132 [binder, in MSetsExtra.MSetIntervals]
    +x:134 [binder, in MSetsExtra.MSetListWithDups]
    +x:137 [binder, in MSetsExtra.MSetIntervals]
    +x:141 [binder, in MSetsExtra.MSetListWithDups]
    +x:144 [binder, in MSetsExtra.MSetListWithDups]
    +x:148 [binder, in MSetsExtra.MSetListWithDups]
    +x:15 [binder, in MSetsExtra.MSetWithDups]
    +x:151 [binder, in MSetsExtra.MSetListWithDups]
    +x:156 [binder, in MSetsExtra.MSetListWithDups]
    +x:161 [binder, in MSetsExtra.MSetListWithDups]
    +x:163 [binder, in MSetsExtra.MSetListWithDups]
    +x:166 [binder, in MSetsExtra.MSetListWithDups]
    +x:167 [binder, in MSetsExtra.MSetListWithDups]
    +x:179 [binder, in MSetsExtra.MSetIntervals]
    +x:186 [binder, in MSetsExtra.MSetIntervals]
    +x:19 [binder, in MSetsExtra.MSetListWithDups]
    +x:190 [binder, in MSetsExtra.MSetIntervals]
    +x:193 [binder, in MSetsExtra.MSetIntervals]
    +x:197 [binder, in MSetsExtra.MSetIntervals]
    +x:209 [binder, in MSetsExtra.MSetIntervals]
    +x:213 [binder, in MSetsExtra.MSetIntervals]
    +x:216 [binder, in MSetsExtra.MSetIntervals]
    +x:219 [binder, in MSetsExtra.MSetIntervals]
    +x:25 [binder, in MSetsExtra.MSetListWithDups]
    +x:255 [binder, in MSetsExtra.MSetIntervals]
    +x:257 [binder, in MSetsExtra.MSetIntervals]
    +x:269 [binder, in MSetsExtra.MSetIntervals]
    +x:271 [binder, in MSetsExtra.MSetIntervals]
    +x:283 [binder, in MSetsExtra.MSetIntervals]
    +x:286 [binder, in MSetsExtra.MSetIntervals]
    +x:287 [binder, in MSetsExtra.MSetIntervals]
    +x:291 [binder, in MSetsExtra.MSetIntervals]
    +x:293 [binder, in MSetsExtra.MSetIntervals]
    +x:294 [binder, in MSetsExtra.MSetIntervals]
    +x:296 [binder, in MSetsExtra.MSetIntervals]
    +x:299 [binder, in MSetsExtra.MSetIntervals]
    +x:303 [binder, in MSetsExtra.MSetIntervals]
    +x:305 [binder, in MSetsExtra.MSetIntervals]
    +x:307 [binder, in MSetsExtra.MSetIntervals]
    +x:31 [binder, in MSetsExtra.MSetListWithDups]
    +x:312 [binder, in MSetsExtra.MSetIntervals]
    +x:318 [binder, in MSetsExtra.MSetIntervals]
    +x:32 [binder, in MSetsExtra.MSetListWithDups]
    +x:323 [binder, in MSetsExtra.MSetIntervals]
    +x:327 [binder, in MSetsExtra.MSetIntervals]
    +x:33 [binder, in MSetsExtra.MSetListWithDups]
    +x:340 [binder, in MSetsExtra.MSetIntervals]
    +x:353 [binder, in MSetsExtra.MSetIntervals]
    +x:36 [binder, in MSetsExtra.MSetListWithDups]
    +x:368 [binder, in MSetsExtra.MSetIntervals]
    +x:37 [binder, in MSetsExtra.MSetWithDups]
    +x:373 [binder, in MSetsExtra.MSetIntervals]
    +x:376 [binder, in MSetsExtra.MSetIntervals]
    +x:383 [binder, in MSetsExtra.MSetIntervals]
    +x:39 [binder, in MSetsExtra.MSetWithDups]
    +x:392 [binder, in MSetsExtra.MSetIntervals]
    +x:397 [binder, in MSetsExtra.MSetIntervals]
    +x:40 [binder, in MSetsExtra.MSetListWithDups]
    +x:412 [binder, in MSetsExtra.MSetIntervals]
    +x:415 [binder, in MSetsExtra.MSetIntervals]
    +x:419 [binder, in MSetsExtra.MSetIntervals]
    +x:42 [binder, in MSetsExtra.MSetWithDups]
    +x:42 [binder, in MSetsExtra.MSetListWithDups]
    +x:422 [binder, in MSetsExtra.MSetIntervals]
    +x:431 [binder, in MSetsExtra.MSetIntervals]
    +x:438 [binder, in MSetsExtra.MSetIntervals]
    +x:439 [binder, in MSetsExtra.MSetIntervals]
    +x:443 [binder, in MSetsExtra.MSetIntervals]
    +x:446 [binder, in MSetsExtra.MSetIntervals]
    +x:449 [binder, in MSetsExtra.MSetIntervals]
    +x:468 [binder, in MSetsExtra.MSetIntervals]
    +x:469 [binder, in MSetsExtra.MSetIntervals]
    +x:471 [binder, in MSetsExtra.MSetIntervals]
    +x:473 [binder, in MSetsExtra.MSetIntervals]
    +x:476 [binder, in MSetsExtra.MSetIntervals]
    +x:479 [binder, in MSetsExtra.MSetIntervals]
    +x:48 [binder, in MSetsExtra.MSetWithDups]
    +x:482 [binder, in MSetsExtra.MSetIntervals]
    +x:488 [binder, in MSetsExtra.MSetIntervals]
    +x:491 [binder, in MSetsExtra.MSetIntervals]
    +x:495 [binder, in MSetsExtra.MSetIntervals]
    +x:499 [binder, in MSetsExtra.MSetIntervals]
    +x:5 [binder, in MSetsExtra.MSetListWithDups]
    +x:50 [binder, in MSetsExtra.MSetListWithDups]
    +x:502 [binder, in MSetsExtra.MSetIntervals]
    +x:506 [binder, in MSetsExtra.MSetIntervals]
    +x:509 [binder, in MSetsExtra.MSetIntervals]
    +x:529 [binder, in MSetsExtra.MSetIntervals]
    +x:532 [binder, in MSetsExtra.MSetIntervals]
    +x:536 [binder, in MSetsExtra.MSetIntervals]
    +x:539 [binder, in MSetsExtra.MSetIntervals]
    +x:542 [binder, in MSetsExtra.MSetIntervals]
    +x:545 [binder, in MSetsExtra.MSetIntervals]
    +x:548 [binder, in MSetsExtra.MSetIntervals]
    +x:549 [binder, in MSetsExtra.MSetIntervals]
    +x:55 [binder, in MSetsExtra.MSetWithDups]
    +x:553 [binder, in MSetsExtra.MSetIntervals]
    +x:57 [binder, in MSetsExtra.MSetIntervals]
    +x:578 [binder, in MSetsExtra.MSetIntervals]
    +x:58 [binder, in MSetsExtra.MSetListWithDups]
    +x:59 [binder, in MSetsExtra.MSetListWithDups]
    +x:627 [binder, in MSetsExtra.MSetIntervals]
    +x:640 [binder, in MSetsExtra.MSetIntervals]
    +x:642 [binder, in MSetsExtra.MSetIntervals]
    +x:646 [binder, in MSetsExtra.MSetIntervals]
    +x:647 [binder, in MSetsExtra.MSetIntervals]
    +x:648 [binder, in MSetsExtra.MSetIntervals]
    +x:650 [binder, in MSetsExtra.MSetIntervals]
    +x:653 [binder, in MSetsExtra.MSetIntervals]
    +x:66 [binder, in MSetsExtra.MSetIntervals]
    +x:663 [binder, in MSetsExtra.MSetIntervals]
    +x:679 [binder, in MSetsExtra.MSetIntervals]
    +x:68 [binder, in MSetsExtra.MSetListWithDups]
    +x:682 [binder, in MSetsExtra.MSetIntervals]
    +x:686 [binder, in MSetsExtra.MSetIntervals]
    +x:688 [binder, in MSetsExtra.MSetIntervals]
    +x:692 [binder, in MSetsExtra.MSetIntervals]
    +x:695 [binder, in MSetsExtra.MSetIntervals]
    +x:7 [binder, in MSetsExtra.MSetListWithDups]
    +x:700 [binder, in MSetsExtra.MSetIntervals]
    +x:730 [binder, in MSetsExtra.MSetIntervals]
    +x:733 [binder, in MSetsExtra.MSetIntervals]
    +x:744 [binder, in MSetsExtra.MSetIntervals]
    +x:76 [binder, in MSetsExtra.MSetIntervals]
    +x:768 [binder, in MSetsExtra.MSetIntervals]
    +x:771 [binder, in MSetsExtra.MSetIntervals]
    +x:776 [binder, in MSetsExtra.MSetIntervals]
    +x:778 [binder, in MSetsExtra.MSetIntervals]
    +x:78 [binder, in MSetsExtra.MSetIntervals]
    +x:782 [binder, in MSetsExtra.MSetIntervals]
    +x:784 [binder, in MSetsExtra.MSetIntervals]
    +x:787 [binder, in MSetsExtra.MSetIntervals]
    +x:792 [binder, in MSetsExtra.MSetIntervals]
    +x:797 [binder, in MSetsExtra.MSetIntervals]
    +x:8 [binder, in MSetsExtra.MSetIntervals]
    +x:800 [binder, in MSetsExtra.MSetIntervals]
    +x:802 [binder, in MSetsExtra.MSetIntervals]
    +x:807 [binder, in MSetsExtra.MSetIntervals]
    +x:809 [binder, in MSetsExtra.MSetIntervals]
    +x:812 [binder, in MSetsExtra.MSetIntervals]
    +x:82 [binder, in MSetsExtra.MSetIntervals]
    +x:826 [binder, in MSetsExtra.MSetIntervals]
    +x:830 [binder, in MSetsExtra.MSetIntervals]
    +x:831 [binder, in MSetsExtra.MSetIntervals]
    +x:836 [binder, in MSetsExtra.MSetIntervals]
    +x:839 [binder, in MSetsExtra.MSetIntervals]
    +x:847 [binder, in MSetsExtra.MSetIntervals]
    +x:856 [binder, in MSetsExtra.MSetIntervals]
    +x:863 [binder, in MSetsExtra.MSetIntervals]
    +x:88 [binder, in MSetsExtra.MSetListWithDups]
    +x:888 [binder, in MSetsExtra.MSetIntervals]
    +x:890 [binder, in MSetsExtra.MSetIntervals]
    +x:901 [binder, in MSetsExtra.MSetIntervals]
    +x:902 [binder, in MSetsExtra.MSetIntervals]
    +x:907 [binder, in MSetsExtra.MSetIntervals]
    +x:908 [binder, in MSetsExtra.MSetIntervals]
    +x:909 [binder, in MSetsExtra.MSetIntervals]
    +x:912 [binder, in MSetsExtra.MSetIntervals]
    +x:915 [binder, in MSetsExtra.MSetIntervals]
    +x:926 [binder, in MSetsExtra.MSetIntervals]
    +x:930 [binder, in MSetsExtra.MSetIntervals]
    +x:942 [binder, in MSetsExtra.MSetIntervals]
    +x:945 [binder, in MSetsExtra.MSetIntervals]
    +x:946 [binder, in MSetsExtra.MSetIntervals]
    +x:948 [binder, in MSetsExtra.MSetIntervals]
    +x:950 [binder, in MSetsExtra.MSetIntervals]
    +x:952 [binder, in MSetsExtra.MSetIntervals]
    +x:999 [binder, in MSetsExtra.MSetIntervals]
    +

    Y

    +yc1:89 [binder, in MSetsExtra.MSetIntervals]
    +yc2:88 [binder, in MSetsExtra.MSetIntervals]
    +yc2:93 [binder, in MSetsExtra.MSetIntervals]
    +yc:844 [binder, in MSetsExtra.MSetIntervals]
    +ys':27 [binder, in MSetsExtra.MSetListWithDups]
    +y':26 [binder, in MSetsExtra.MSetListWithDups]
    +y':774 [binder, in MSetsExtra.MSetIntervals]
    +y':835 [binder, in MSetsExtra.MSetIntervals]
    +y':860 [binder, in MSetsExtra.MSetIntervals]
    +y':865 [binder, in MSetsExtra.MSetIntervals]
    +y':866 [binder, in MSetsExtra.MSetIntervals]
    +y':872 [binder, in MSetsExtra.MSetIntervals]
    +y':873 [binder, in MSetsExtra.MSetIntervals]
    +y':878 [binder, in MSetsExtra.MSetIntervals]
    +y0:128 [binder, in MSetsExtra.MSetListWithDups]
    +y1:398 [binder, in MSetsExtra.MSetIntervals]
    +y1:402 [binder, in MSetsExtra.MSetIntervals]
    +y1:408 [binder, in MSetsExtra.MSetIntervals]
    +y1:5 [binder, in MSetsExtra.MSetIntervals]
    +y1:512 [binder, in MSetsExtra.MSetIntervals]
    +y1:516 [binder, in MSetsExtra.MSetIntervals]
    +y1:705 [binder, in MSetsExtra.MSetIntervals]
    +y1:713 [binder, in MSetsExtra.MSetIntervals]
    +y1:757 [binder, in MSetsExtra.MSetIntervals]
    +y1:90 [binder, in MSetsExtra.MSetIntervals]
    +y2:145 [binder, in MSetsExtra.MSetIntervals]
    +y2:166 [binder, in MSetsExtra.MSetIntervals]
    +y2:400 [binder, in MSetsExtra.MSetIntervals]
    +y2:404 [binder, in MSetsExtra.MSetIntervals]
    +y2:409 [binder, in MSetsExtra.MSetIntervals]
    +y2:466 [binder, in MSetsExtra.MSetIntervals]
    +y2:514 [binder, in MSetsExtra.MSetIntervals]
    +y2:518 [binder, in MSetsExtra.MSetIntervals]
    +y2:581 [binder, in MSetsExtra.MSetIntervals]
    +y2:585 [binder, in MSetsExtra.MSetIntervals]
    +y2:6 [binder, in MSetsExtra.MSetIntervals]
    +y2:630 [binder, in MSetsExtra.MSetIntervals]
    +y2:634 [binder, in MSetsExtra.MSetIntervals]
    +y2:708 [binder, in MSetsExtra.MSetIntervals]
    +y2:716 [binder, in MSetsExtra.MSetIntervals]
    +y2:758 [binder, in MSetsExtra.MSetIntervals]
    +y:11 [binder, in MSetsExtra.MSetIntervals]
    +y:111 [binder, in MSetsExtra.MSetListWithDups]
    +y:115 [binder, in MSetsExtra.MSetListWithDups]
    +y:117 [binder, in MSetsExtra.MSetListWithDups]
    +y:127 [binder, in MSetsExtra.MSetListWithDups]
    +y:135 [binder, in MSetsExtra.MSetListWithDups]
    +y:2 [binder, in MSetsExtra.MSetListWithDups]
    +y:279 [binder, in MSetsExtra.MSetFoldWithAbort]
    +y:282 [binder, in MSetsExtra.MSetFoldWithAbort]
    +y:285 [binder, in MSetsExtra.MSetFoldWithAbort]
    +y:302 [binder, in MSetsExtra.MSetIntervals]
    +y:304 [binder, in MSetsExtra.MSetIntervals]
    +y:360 [binder, in MSetsExtra.MSetIntervals]
    +y:381 [binder, in MSetsExtra.MSetIntervals]
    +y:382 [binder, in MSetsExtra.MSetIntervals]
    +y:388 [binder, in MSetsExtra.MSetIntervals]
    +y:390 [binder, in MSetsExtra.MSetIntervals]
    +y:391 [binder, in MSetsExtra.MSetIntervals]
    +y:394 [binder, in MSetsExtra.MSetIntervals]
    +y:406 [binder, in MSetsExtra.MSetIntervals]
    +y:423 [binder, in MSetsExtra.MSetIntervals]
    +y:428 [binder, in MSetsExtra.MSetIntervals]
    +y:433 [binder, in MSetsExtra.MSetIntervals]
    +y:437 [binder, in MSetsExtra.MSetIntervals]
    +y:442 [binder, in MSetsExtra.MSetIntervals]
    +y:445 [binder, in MSetsExtra.MSetIntervals]
    +y:448 [binder, in MSetsExtra.MSetIntervals]
    +y:452 [binder, in MSetsExtra.MSetIntervals]
    +y:493 [binder, in MSetsExtra.MSetIntervals]
    +y:505 [binder, in MSetsExtra.MSetIntervals]
    +y:52 [binder, in MSetsExtra.MSetListWithDups]
    +y:526 [binder, in MSetsExtra.MSetIntervals]
    +y:527 [binder, in MSetsExtra.MSetIntervals]
    +y:533 [binder, in MSetsExtra.MSetIntervals]
    +y:543 [binder, in MSetsExtra.MSetIntervals]
    +y:550 [binder, in MSetsExtra.MSetIntervals]
    +y:566 [binder, in MSetsExtra.MSetIntervals]
    +y:568 [binder, in MSetsExtra.MSetIntervals]
    +y:57 [binder, in MSetsExtra.MSetListWithDups]
    +y:591 [binder, in MSetsExtra.MSetIntervals]
    +y:592 [binder, in MSetsExtra.MSetIntervals]
    +y:593 [binder, in MSetsExtra.MSetIntervals]
    +y:594 [binder, in MSetsExtra.MSetIntervals]
    +y:595 [binder, in MSetsExtra.MSetIntervals]
    +y:596 [binder, in MSetsExtra.MSetIntervals]
    +y:597 [binder, in MSetsExtra.MSetIntervals]
    +y:598 [binder, in MSetsExtra.MSetIntervals]
    +y:599 [binder, in MSetsExtra.MSetIntervals]
    +y:6 [binder, in MSetsExtra.MSetListWithDups]
    +y:601 [binder, in MSetsExtra.MSetIntervals]
    +y:609 [binder, in MSetsExtra.MSetIntervals]
    +y:618 [binder, in MSetsExtra.MSetIntervals]
    +y:641 [binder, in MSetsExtra.MSetIntervals]
    +y:643 [binder, in MSetsExtra.MSetIntervals]
    +y:644 [binder, in MSetsExtra.MSetIntervals]
    +y:645 [binder, in MSetsExtra.MSetIntervals]
    +y:654 [binder, in MSetsExtra.MSetIntervals]
    +y:660 [binder, in MSetsExtra.MSetIntervals]
    +y:670 [binder, in MSetsExtra.MSetIntervals]
    +y:689 [binder, in MSetsExtra.MSetIntervals]
    +y:696 [binder, in MSetsExtra.MSetIntervals]
    +y:711 [binder, in MSetsExtra.MSetIntervals]
    +y:712 [binder, in MSetsExtra.MSetIntervals]
    +y:719 [binder, in MSetsExtra.MSetIntervals]
    +y:722 [binder, in MSetsExtra.MSetIntervals]
    +y:724 [binder, in MSetsExtra.MSetIntervals]
    +y:773 [binder, in MSetsExtra.MSetIntervals]
    +y:779 [binder, in MSetsExtra.MSetIntervals]
    +y:788 [binder, in MSetsExtra.MSetIntervals]
    +y:8 [binder, in MSetsExtra.MSetListWithDups]
    +y:803 [binder, in MSetsExtra.MSetIntervals]
    +y:813 [binder, in MSetsExtra.MSetIntervals]
    +y:845 [binder, in MSetsExtra.MSetIntervals]
    +y:848 [binder, in MSetsExtra.MSetIntervals]
    +y:853 [binder, in MSetsExtra.MSetIntervals]
    +y:896 [binder, in MSetsExtra.MSetIntervals]
    +y:9 [binder, in MSetsExtra.MSetIntervals]
    +

    Z

    Z_to_N_minus_neq_0 [lemma, in MSetsExtra.MSetIntervals]
    Z_lt_le_add_r [lemma, in MSetsExtra.MSetIntervals]
    Z_lt_add_r [lemma, in MSetsExtra.MSetIntervals]
    Z_le_add_r [lemma, in MSetsExtra.MSetIntervals]
    +z:1 [binder, in MSetsExtra.MSetIntervals]
    +z:1004 [binder, in MSetsExtra.MSetIntervals]
    +z:1005 [binder, in MSetsExtra.MSetIntervals]
    +z:1012 [binder, in MSetsExtra.MSetIntervals]
    +z:1019 [binder, in MSetsExtra.MSetIntervals]
    +z:206 [binder, in MSetsExtra.MSetIntervals]
    +z:227 [binder, in MSetsExtra.MSetIntervals]
    +z:233 [binder, in MSetsExtra.MSetIntervals]
    +z:237 [binder, in MSetsExtra.MSetIntervals]
    +z:241 [binder, in MSetsExtra.MSetIntervals]
    +z:3 [binder, in MSetsExtra.MSetIntervals]
    +z:411 [binder, in MSetsExtra.MSetIntervals]
    +z:418 [binder, in MSetsExtra.MSetIntervals]
    +z:475 [binder, in MSetsExtra.MSetIntervals]
    +z:520 [binder, in MSetsExtra.MSetIntervals]
    +z:534 [binder, in MSetsExtra.MSetIntervals]
    +z:537 [binder, in MSetsExtra.MSetIntervals]
    +z:750 [binder, in MSetsExtra.MSetIntervals]
    +z:884 [binder, in MSetsExtra.MSetIntervals]
    +z:885 [binder, in MSetsExtra.MSetIntervals]
    +z:886 [binder, in MSetsExtra.MSetIntervals]
    +z:891 [binder, in MSetsExtra.MSetIntervals]
    +z:892 [binder, in MSetsExtra.MSetIntervals]
    +z:893 [binder, in MSetsExtra.MSetIntervals]


    Notation Index

    -

    M

    -_ [<=] _ [in MSetsExtra.MSetListWithDups]
    +

    M

    +_ [<=] _ [in MSetsExtra.MSetListWithDups]
    _ [=] _ [in MSetsExtra.MSetListWithDups]
    -

    R

    -_ <=? _ [in MSetsExtra.MSetListWithDups]
    -

    W

    -_ [<=] _ [in MSetsExtra.MSetWithDups]
    +

    R

    +_ <=? _ [in MSetsExtra.MSetListWithDups]
    +

    W

    +_ [<=] _ [in MSetsExtra.MSetWithDups]
    _ [=] _ [in MSetsExtra.MSetWithDups]


    +

    Binder Index

    +

    A

    +ab:347 [in MSetsExtra.MSetIntervals]
    +ab:363 [in MSetsExtra.MSetIntervals]
    +acc_f:906 [in MSetsExtra.MSetIntervals]
    +acc_t:905 [in MSetsExtra.MSetIntervals]
    +acc_f:899 [in MSetsExtra.MSetIntervals]
    +acc_t:897 [in MSetsExtra.MSetIntervals]
    +acc_f:200 [in MSetsExtra.MSetIntervals]
    +acc_t:199 [in MSetsExtra.MSetIntervals]
    +acc_f:196 [in MSetsExtra.MSetIntervals]
    +acc_t:195 [in MSetsExtra.MSetIntervals]
    +acc':348 [in MSetsExtra.MSetIntervals]
    +acc':364 [in MSetsExtra.MSetIntervals]
    +acc:106 [in MSetsExtra.MSetIntervals]
    +acc:119 [in MSetsExtra.MSetListWithDups]
    +acc:125 [in MSetsExtra.MSetIntervals]
    +acc:141 [in MSetsExtra.MSetIntervals]
    +acc:147 [in MSetsExtra.MSetIntervals]
    +acc:15 [in MSetsExtra.MSetListWithDups]
    +acc:151 [in MSetsExtra.MSetIntervals]
    +acc:168 [in MSetsExtra.MSetIntervals]
    +acc:172 [in MSetsExtra.MSetIntervals]
    +acc:181 [in MSetsExtra.MSetIntervals]
    +acc:218 [in MSetsExtra.MSetIntervals]
    +acc:221 [in MSetsExtra.MSetIntervals]
    +acc:23 [in MSetsExtra.MSetIntervals]
    +acc:250 [in MSetsExtra.MSetFoldWithAbort]
    +acc:311 [in MSetsExtra.MSetIntervals]
    +acc:317 [in MSetsExtra.MSetIntervals]
    +acc:322 [in MSetsExtra.MSetIntervals]
    +acc:326 [in MSetsExtra.MSetIntervals]
    +acc:332 [in MSetsExtra.MSetIntervals]
    +acc:338 [in MSetsExtra.MSetIntervals]
    +acc:344 [in MSetsExtra.MSetIntervals]
    +acc:35 [in MSetsExtra.MSetIntervals]
    +acc:352 [in MSetsExtra.MSetIntervals]
    +acc:356 [in MSetsExtra.MSetIntervals]
    +acc:359 [in MSetsExtra.MSetIntervals]
    +acc:366 [in MSetsExtra.MSetFoldWithAbort]
    +acc:370 [in MSetsExtra.MSetFoldWithAbort]
    +acc:372 [in MSetsExtra.MSetIntervals]
    +acc:378 [in MSetsExtra.MSetFoldWithAbort]
    +acc:389 [in MSetsExtra.MSetFoldWithAbort]
    +acc:393 [in MSetsExtra.MSetFoldWithAbort]
    +acc:401 [in MSetsExtra.MSetFoldWithAbort]
    +acc:414 [in MSetsExtra.MSetIntervals]
    +acc:43 [in MSetsExtra.MSetListWithDups]
    +acc:558 [in MSetsExtra.MSetIntervals]
    +acc:56 [in MSetsExtra.MSetIntervals]
    +acc:561 [in MSetsExtra.MSetIntervals]
    +acc:584 [in MSetsExtra.MSetIntervals]
    +acc:588 [in MSetsExtra.MSetIntervals]
    +acc:600 [in MSetsExtra.MSetIntervals]
    +acc:606 [in MSetsExtra.MSetIntervals]
    +acc:633 [in MSetsExtra.MSetIntervals]
    +acc:637 [in MSetsExtra.MSetIntervals]
    +acc:649 [in MSetsExtra.MSetIntervals]
    +acc:65 [in MSetsExtra.MSetIntervals]
    +acc:657 [in MSetsExtra.MSetIntervals]
    +acc:684 [in MSetsExtra.MSetIntervals]
    +acc:70 [in MSetsExtra.MSetIntervals]
    +acc:832 [in MSetsExtra.MSetIntervals]
    +acc:837 [in MSetsExtra.MSetIntervals]
    +acc:840 [in MSetsExtra.MSetIntervals]
    +acc:852 [in MSetsExtra.MSetIntervals]
    +acc:857 [in MSetsExtra.MSetIntervals]
    +acc:864 [in MSetsExtra.MSetIntervals]
    +acc:869 [in MSetsExtra.MSetIntervals]
    +acc:9 [in MSetsExtra.MSetListWithDups]
    +a1:152 [in MSetsExtra.MSetListWithDups]
    +a1:3 [in MSetsExtra.MSetListWithDups]
    +a1:79 [in MSetsExtra.MSetListWithDups]
    +a2:153 [in MSetsExtra.MSetListWithDups]
    +a2:4 [in MSetsExtra.MSetListWithDups]
    +a2:80 [in MSetsExtra.MSetListWithDups]
    +a:1002 [in MSetsExtra.MSetIntervals]
    +a:1003 [in MSetsExtra.MSetIntervals]
    +A:102 [in MSetsExtra.MSetFoldWithAbort]
    +A:11 [in MSetsExtra.MSetFoldWithAbort]
    +A:12 [in MSetsExtra.MSetIntervals]
    +A:126 [in MSetsExtra.MSetFoldWithAbort]
    +A:130 [in MSetsExtra.MSetFoldWithAbort]
    +A:137 [in MSetsExtra.MSetListWithDups]
    +A:140 [in MSetsExtra.MSetFoldWithAbort]
    +A:147 [in MSetsExtra.MSetFoldWithAbort]
    +A:158 [in MSetsExtra.MSetFoldWithAbort]
    +A:175 [in MSetsExtra.MSetFoldWithAbort]
    +A:190 [in MSetsExtra.MSetFoldWithAbort]
    +A:20 [in MSetsExtra.MSetIntervals]
    +A:201 [in MSetsExtra.MSetFoldWithAbort]
    +A:218 [in MSetsExtra.MSetFoldWithAbort]
    +A:27 [in MSetsExtra.MSetIntervals]
    +a:275 [in MSetsExtra.MSetIntervals]
    +a:278 [in MSetsExtra.MSetIntervals]
    +a:280 [in MSetsExtra.MSetIntervals]
    +A:3 [in MSetsExtra.MSetFoldWithAbort]
    +A:30 [in MSetsExtra.MSetFoldWithAbort]
    +A:302 [in MSetsExtra.MSetFoldWithAbort]
    +A:308 [in MSetsExtra.MSetIntervals]
    +A:31 [in MSetsExtra.MSetIntervals]
    +A:315 [in MSetsExtra.MSetIntervals]
    +A:316 [in MSetsExtra.MSetFoldWithAbort]
    +A:32 [in MSetsExtra.MSetWithDups]
    +A:320 [in MSetsExtra.MSetIntervals]
    +A:324 [in MSetsExtra.MSetIntervals]
    +A:329 [in MSetsExtra.MSetIntervals]
    +A:336 [in MSetsExtra.MSetIntervals]
    +A:341 [in MSetsExtra.MSetIntervals]
    +A:343 [in MSetsExtra.MSetFoldWithAbort]
    +A:349 [in MSetsExtra.MSetIntervals]
    +A:35 [in MSetsExtra.MSetFoldWithAbort]
    +A:351 [in MSetsExtra.MSetFoldWithAbort]
    +A:354 [in MSetsExtra.MSetIntervals]
    +A:357 [in MSetsExtra.MSetIntervals]
    +A:359 [in MSetsExtra.MSetFoldWithAbort]
    +A:36 [in MSetsExtra.MSetIntervals]
    +A:369 [in MSetsExtra.MSetIntervals]
    +A:37 [in MSetsExtra.MSetFoldWithAbort]
    +A:371 [in MSetsExtra.MSetFoldWithAbort]
    +A:382 [in MSetsExtra.MSetFoldWithAbort]
    +A:39 [in MSetsExtra.MSetFoldWithAbort]
    +A:394 [in MSetsExtra.MSetFoldWithAbort]
    +a:4 [in MSetsExtra.MSetWithDups]
    +A:54 [in MSetsExtra.MSetIntervals]
    +A:58 [in MSetsExtra.MSetFoldWithAbort]
    +A:63 [in MSetsExtra.MSetFoldWithAbort]
    +A:63 [in MSetsExtra.MSetIntervals]
    +A:65 [in MSetsExtra.MSetFoldWithAbort]
    +A:67 [in MSetsExtra.MSetFoldWithAbort]
    +A:68 [in MSetsExtra.MSetIntervals]
    +A:7 [in MSetsExtra.MSetFoldWithAbort]
    +a:7 [in MSetsExtra.MSetWithDups]
    +A:817 [in MSetsExtra.MSetIntervals]
    +A:9 [in MSetsExtra.MSetFoldWithAbort]
    +a:9 [in MSetsExtra.MSetWithDups]
    +A:91 [in MSetsExtra.MSetFoldWithAbort]
    +a:92 [in MSetsExtra.MSetListWithDups]
    +a:934 [in MSetsExtra.MSetIntervals]
    +a:937 [in MSetsExtra.MSetIntervals]
    +a:939 [in MSetsExtra.MSetIntervals]
    +a:95 [in MSetsExtra.MSetListWithDups]
    +A:97 [in MSetsExtra.MSetFoldWithAbort]
    +a:97 [in MSetsExtra.MSetListWithDups]
    +A:970 [in MSetsExtra.MSetIntervals]
    +a:985 [in MSetsExtra.MSetIntervals]
    +a:986 [in MSetsExtra.MSetIntervals]
    +a:987 [in MSetsExtra.MSetIntervals]
    +a:988 [in MSetsExtra.MSetIntervals]
    +A:99 [in MSetsExtra.MSetFoldWithAbort]
    +A:994 [in MSetsExtra.MSetIntervals]
    +a:997 [in MSetsExtra.MSetIntervals]
    +

    B

    +base:309 [in MSetsExtra.MSetFoldWithAbort]
    +base:350 [in MSetsExtra.MSetFoldWithAbort]
    +base:358 [in MSetsExtra.MSetFoldWithAbort]
    +base:366 [in MSetsExtra.MSetIntervals]
    +bs:333 [in MSetsExtra.MSetIntervals]
    +bs:337 [in MSetsExtra.MSetIntervals]
    +B:100 [in MSetsExtra.MSetFoldWithAbort]
    +B:103 [in MSetsExtra.MSetFoldWithAbort]
    +B:127 [in MSetsExtra.MSetFoldWithAbort]
    +B:13 [in MSetsExtra.MSetIntervals]
    +B:159 [in MSetsExtra.MSetFoldWithAbort]
    +B:21 [in MSetsExtra.MSetIntervals]
    +B:212 [in MSetsExtra.MSetFoldWithAbort]
    +B:227 [in MSetsExtra.MSetFoldWithAbort]
    +B:228 [in MSetsExtra.MSetIntervals]
    +b:232 [in MSetsExtra.MSetIntervals]
    +b:236 [in MSetsExtra.MSetIntervals]
    +B:237 [in MSetsExtra.MSetFoldWithAbort]
    +b:240 [in MSetsExtra.MSetIntervals]
    +B:253 [in MSetsExtra.MSetFoldWithAbort]
    +B:268 [in MSetsExtra.MSetFoldWithAbort]
    +B:28 [in MSetsExtra.MSetIntervals]
    +B:287 [in MSetsExtra.MSetFoldWithAbort]
    +B:290 [in MSetsExtra.MSetFoldWithAbort]
    +B:294 [in MSetsExtra.MSetFoldWithAbort]
    +B:297 [in MSetsExtra.MSetFoldWithAbort]
    +B:303 [in MSetsExtra.MSetFoldWithAbort]
    +B:317 [in MSetsExtra.MSetFoldWithAbort]
    +B:32 [in MSetsExtra.MSetIntervals]
    +B:330 [in MSetsExtra.MSetIntervals]
    +B:342 [in MSetsExtra.MSetIntervals]
    +B:344 [in MSetsExtra.MSetFoldWithAbort]
    +B:352 [in MSetsExtra.MSetFoldWithAbort]
    +B:360 [in MSetsExtra.MSetFoldWithAbort]
    +B:37 [in MSetsExtra.MSetIntervals]
    +B:372 [in MSetsExtra.MSetFoldWithAbort]
    +B:383 [in MSetsExtra.MSetFoldWithAbort]
    +B:395 [in MSetsExtra.MSetFoldWithAbort]
    +B:60 [in MSetsExtra.MSetListWithDups]
    +B:92 [in MSetsExtra.MSetFoldWithAbort]
    +

    C

    +cur:178 [in MSetsExtra.MSetIntervals]
    +cur:182 [in MSetsExtra.MSetIntervals]
    +cur:858 [in MSetsExtra.MSetIntervals]
    +cx:306 [in MSetsExtra.MSetIntervals]
    +c_f:900 [in MSetsExtra.MSetIntervals]
    +c_t:898 [in MSetsExtra.MSetIntervals]
    +c1:298 [in MSetsExtra.MSetIntervals]
    +c1:399 [in MSetsExtra.MSetIntervals]
    +c1:403 [in MSetsExtra.MSetIntervals]
    +c1:485 [in MSetsExtra.MSetIntervals]
    +c1:510 [in MSetsExtra.MSetIntervals]
    +c1:513 [in MSetsExtra.MSetIntervals]
    +c1:517 [in MSetsExtra.MSetIntervals]
    +c1:523 [in MSetsExtra.MSetIntervals]
    +c1:706 [in MSetsExtra.MSetIntervals]
    +c1:714 [in MSetsExtra.MSetIntervals]
    +c1:759 [in MSetsExtra.MSetIntervals]
    +c1:98 [in MSetsExtra.MSetIntervals]
    +c2:100 [in MSetsExtra.MSetIntervals]
    +c2:146 [in MSetsExtra.MSetIntervals]
    +c2:167 [in MSetsExtra.MSetIntervals]
    +c2:297 [in MSetsExtra.MSetIntervals]
    +c2:401 [in MSetsExtra.MSetIntervals]
    +c2:405 [in MSetsExtra.MSetIntervals]
    +c2:410 [in MSetsExtra.MSetIntervals]
    +c2:467 [in MSetsExtra.MSetIntervals]
    +c2:487 [in MSetsExtra.MSetIntervals]
    +c2:511 [in MSetsExtra.MSetIntervals]
    +c2:515 [in MSetsExtra.MSetIntervals]
    +c2:519 [in MSetsExtra.MSetIntervals]
    +c2:525 [in MSetsExtra.MSetIntervals]
    +c2:582 [in MSetsExtra.MSetIntervals]
    +c2:586 [in MSetsExtra.MSetIntervals]
    +c2:631 [in MSetsExtra.MSetIntervals]
    +c2:635 [in MSetsExtra.MSetIntervals]
    +c2:709 [in MSetsExtra.MSetIntervals]
    +c2:717 [in MSetsExtra.MSetIntervals]
    +c2:760 [in MSetsExtra.MSetIntervals]
    +c:102 [in MSetsExtra.MSetIntervals]
    +c:123 [in MSetsExtra.MSetIntervals]
    +c:191 [in MSetsExtra.MSetIntervals]
    +c:198 [in MSetsExtra.MSetIntervals]
    +c:214 [in MSetsExtra.MSetIntervals]
    +c:220 [in MSetsExtra.MSetIntervals]
    +c:242 [in MSetsExtra.MSetIntervals]
    +c:288 [in MSetsExtra.MSetIntervals]
    +c:292 [in MSetsExtra.MSetIntervals]
    +c:295 [in MSetsExtra.MSetIntervals]
    +c:300 [in MSetsExtra.MSetIntervals]
    +c:301 [in MSetsExtra.MSetIntervals]
    +c:310 [in MSetsExtra.MSetIntervals]
    +c:328 [in MSetsExtra.MSetIntervals]
    +c:351 [in MSetsExtra.MSetIntervals]
    +c:361 [in MSetsExtra.MSetIntervals]
    +c:374 [in MSetsExtra.MSetIntervals]
    +c:377 [in MSetsExtra.MSetIntervals]
    +c:384 [in MSetsExtra.MSetIntervals]
    +c:393 [in MSetsExtra.MSetIntervals]
    +c:396 [in MSetsExtra.MSetIntervals]
    +c:420 [in MSetsExtra.MSetIntervals]
    +c:429 [in MSetsExtra.MSetIntervals]
    +c:432 [in MSetsExtra.MSetIntervals]
    +c:434 [in MSetsExtra.MSetIntervals]
    +c:489 [in MSetsExtra.MSetIntervals]
    +c:492 [in MSetsExtra.MSetIntervals]
    +c:496 [in MSetsExtra.MSetIntervals]
    +c:500 [in MSetsExtra.MSetIntervals]
    +c:503 [in MSetsExtra.MSetIntervals]
    +c:507 [in MSetsExtra.MSetIntervals]
    +c:530 [in MSetsExtra.MSetIntervals]
    +c:58 [in MSetsExtra.MSetIntervals]
    +c:67 [in MSetsExtra.MSetIntervals]
    +c:7 [in MSetsExtra.MSetIntervals]
    +c:732 [in MSetsExtra.MSetIntervals]
    +c:767 [in MSetsExtra.MSetIntervals]
    +c:821 [in MSetsExtra.MSetIntervals]
    +c:833 [in MSetsExtra.MSetIntervals]
    +c:838 [in MSetsExtra.MSetIntervals]
    +c:841 [in MSetsExtra.MSetIntervals]
    +c:842 [in MSetsExtra.MSetIntervals]
    +c:846 [in MSetsExtra.MSetIntervals]
    +c:851 [in MSetsExtra.MSetIntervals]
    +c:855 [in MSetsExtra.MSetIntervals]
    +c:862 [in MSetsExtra.MSetIntervals]
    +c:895 [in MSetsExtra.MSetIntervals]
    +

    E

    +elt:1 [in MSetsExtra.MSetFoldWithAbort]
    +elt:28 [in MSetsExtra.MSetFoldWithAbort]
    +elt:31 [in MSetsExtra.MSetFoldWithAbort]
    +elt:4 [in MSetsExtra.MSetFoldWithAbort]
    +elt:56 [in MSetsExtra.MSetFoldWithAbort]
    +elt:59 [in MSetsExtra.MSetFoldWithAbort]
    +elt:89 [in MSetsExtra.MSetFoldWithAbort]
    +elt:93 [in MSetsExtra.MSetFoldWithAbort]
    +eqA:14 [in MSetsExtra.MSetIntervals]
    +eqB:15 [in MSetsExtra.MSetIntervals]
    +e_pre:261 [in MSetsExtra.MSetFoldWithAbort]
    +e_pre:244 [in MSetsExtra.MSetFoldWithAbort]
    +e_pre:233 [in MSetsExtra.MSetFoldWithAbort]
    +e0:251 [in MSetsExtra.MSetFoldWithAbort]
    +e0:341 [in MSetsExtra.MSetFoldWithAbort]
    +e0:342 [in MSetsExtra.MSetFoldWithAbort]
    +e1:1007 [in MSetsExtra.MSetIntervals]
    +e1:1009 [in MSetsExtra.MSetIntervals]
    +e1:1014 [in MSetsExtra.MSetIntervals]
    +e1:1016 [in MSetsExtra.MSetIntervals]
    +e1:1021 [in MSetsExtra.MSetIntervals]
    +e1:1023 [in MSetsExtra.MSetIntervals]
    +e1:118 [in MSetsExtra.MSetFoldWithAbort]
    +e1:122 [in MSetsExtra.MSetFoldWithAbort]
    +e1:156 [in MSetsExtra.MSetFoldWithAbort]
    +e1:169 [in MSetsExtra.MSetFoldWithAbort]
    +e1:172 [in MSetsExtra.MSetFoldWithAbort]
    +e1:184 [in MSetsExtra.MSetFoldWithAbort]
    +e1:187 [in MSetsExtra.MSetFoldWithAbort]
    +e1:198 [in MSetsExtra.MSetFoldWithAbort]
    +e1:209 [in MSetsExtra.MSetFoldWithAbort]
    +e1:223 [in MSetsExtra.MSetFoldWithAbort]
    +e1:225 [in MSetsExtra.MSetFoldWithAbort]
    +e1:24 [in MSetsExtra.MSetFoldWithAbort]
    +e1:332 [in MSetsExtra.MSetFoldWithAbort]
    +e1:336 [in MSetsExtra.MSetFoldWithAbort]
    +e1:44 [in MSetsExtra.MSetIntervals]
    +e1:47 [in MSetsExtra.MSetIntervals]
    +e1:52 [in MSetsExtra.MSetFoldWithAbort]
    +e1:81 [in MSetsExtra.MSetFoldWithAbort]
    +e1:85 [in MSetsExtra.MSetFoldWithAbort]
    +e2:1008 [in MSetsExtra.MSetIntervals]
    +e2:1010 [in MSetsExtra.MSetIntervals]
    +e2:1015 [in MSetsExtra.MSetIntervals]
    +e2:1017 [in MSetsExtra.MSetIntervals]
    +e2:1022 [in MSetsExtra.MSetIntervals]
    +e2:1024 [in MSetsExtra.MSetIntervals]
    +e2:121 [in MSetsExtra.MSetFoldWithAbort]
    +e2:125 [in MSetsExtra.MSetFoldWithAbort]
    +e2:157 [in MSetsExtra.MSetFoldWithAbort]
    +e2:171 [in MSetsExtra.MSetFoldWithAbort]
    +e2:174 [in MSetsExtra.MSetFoldWithAbort]
    +e2:186 [in MSetsExtra.MSetFoldWithAbort]
    +e2:189 [in MSetsExtra.MSetFoldWithAbort]
    +e2:200 [in MSetsExtra.MSetFoldWithAbort]
    +e2:211 [in MSetsExtra.MSetFoldWithAbort]
    +e2:224 [in MSetsExtra.MSetFoldWithAbort]
    +e2:226 [in MSetsExtra.MSetFoldWithAbort]
    +e2:27 [in MSetsExtra.MSetFoldWithAbort]
    +e2:335 [in MSetsExtra.MSetFoldWithAbort]
    +e2:339 [in MSetsExtra.MSetFoldWithAbort]
    +e2:45 [in MSetsExtra.MSetIntervals]
    +e2:48 [in MSetsExtra.MSetIntervals]
    +e2:55 [in MSetsExtra.MSetFoldWithAbort]
    +e2:84 [in MSetsExtra.MSetFoldWithAbort]
    +e2:88 [in MSetsExtra.MSetFoldWithAbort]
    +e:1006 [in MSetsExtra.MSetIntervals]
    +e:1013 [in MSetsExtra.MSetIntervals]
    +e:1020 [in MSetsExtra.MSetIntervals]
    +e:115 [in MSetsExtra.MSetFoldWithAbort]
    +e:116 [in MSetsExtra.MSetFoldWithAbort]
    +e:134 [in MSetsExtra.MSetFoldWithAbort]
    +e:136 [in MSetsExtra.MSetFoldWithAbort]
    +e:138 [in MSetsExtra.MSetFoldWithAbort]
    +e:143 [in MSetsExtra.MSetFoldWithAbort]
    +e:145 [in MSetsExtra.MSetFoldWithAbort]
    +e:150 [in MSetsExtra.MSetFoldWithAbort]
    +e:152 [in MSetsExtra.MSetFoldWithAbort]
    +e:154 [in MSetsExtra.MSetFoldWithAbort]
    +e:167 [in MSetsExtra.MSetFoldWithAbort]
    +e:182 [in MSetsExtra.MSetFoldWithAbort]
    +e:196 [in MSetsExtra.MSetFoldWithAbort]
    +e:207 [in MSetsExtra.MSetFoldWithAbort]
    +e:21 [in MSetsExtra.MSetFoldWithAbort]
    +e:22 [in MSetsExtra.MSetFoldWithAbort]
    +e:222 [in MSetsExtra.MSetFoldWithAbort]
    +e:230 [in MSetsExtra.MSetFoldWithAbort]
    +e:232 [in MSetsExtra.MSetFoldWithAbort]
    +e:235 [in MSetsExtra.MSetFoldWithAbort]
    +e:241 [in MSetsExtra.MSetFoldWithAbort]
    +e:243 [in MSetsExtra.MSetFoldWithAbort]
    +e:246 [in MSetsExtra.MSetFoldWithAbort]
    +e:248 [in MSetsExtra.MSetFoldWithAbort]
    +e:256 [in MSetsExtra.MSetFoldWithAbort]
    +e:256 [in MSetsExtra.MSetIntervals]
    +e:260 [in MSetsExtra.MSetFoldWithAbort]
    +e:264 [in MSetsExtra.MSetFoldWithAbort]
    +e:272 [in MSetsExtra.MSetFoldWithAbort]
    +e:273 [in MSetsExtra.MSetFoldWithAbort]
    +e:274 [in MSetsExtra.MSetFoldWithAbort]
    +e:301 [in MSetsExtra.MSetFoldWithAbort]
    +e:329 [in MSetsExtra.MSetFoldWithAbort]
    +e:330 [in MSetsExtra.MSetFoldWithAbort]
    +e:340 [in MSetsExtra.MSetFoldWithAbort]
    +e:379 [in MSetsExtra.MSetFoldWithAbort]
    +e:380 [in MSetsExtra.MSetFoldWithAbort]
    +e:402 [in MSetsExtra.MSetFoldWithAbort]
    +e:403 [in MSetsExtra.MSetFoldWithAbort]
    +e:405 [in MSetsExtra.MSetFoldWithAbort]
    +e:42 [in MSetsExtra.MSetIntervals]
    +e:49 [in MSetsExtra.MSetFoldWithAbort]
    +e:50 [in MSetsExtra.MSetFoldWithAbort]
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    +e:78 [in MSetsExtra.MSetFoldWithAbort]
    +e:79 [in MSetsExtra.MSetFoldWithAbort]
    +e:998 [in MSetsExtra.MSetIntervals]
    +

    F

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    +foldWithAbortPrecompute:101 [in MSetsExtra.MSetFoldWithAbort]
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    +fold:36 [in MSetsExtra.MSetFoldWithAbort]
    +fold:64 [in MSetsExtra.MSetFoldWithAbort]
    +fold:8 [in MSetsExtra.MSetFoldWithAbort]
    +fold:98 [in MSetsExtra.MSetFoldWithAbort]
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    +fwasa:228 [in MSetsExtra.MSetFoldWithAbort]
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    +fwasa:254 [in MSetsExtra.MSetFoldWithAbort]
    +fwasa:269 [in MSetsExtra.MSetFoldWithAbort]
    +fwasa:288 [in MSetsExtra.MSetFoldWithAbort]
    +fwasa:291 [in MSetsExtra.MSetFoldWithAbort]
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    +f:980 [in MSetsExtra.MSetIntervals]
    +f:996 [in MSetsExtra.MSetIntervals]
    +

    H

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    +Hs':629 [in MSetsExtra.MSetIntervals]
    +Hs':681 [in MSetsExtra.MSetIntervals]
    +Hs':728 [in MSetsExtra.MSetIntervals]
    +Hs':743 [in MSetsExtra.MSetIntervals]
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    +Hs:697 [in MSetsExtra.MSetIntervals]
    +Hs:727 [in MSetsExtra.MSetIntervals]
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    +Hs:742 [in MSetsExtra.MSetIntervals]
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    +Hs:785 [in MSetsExtra.MSetIntervals]
    +Hs:789 [in MSetsExtra.MSetIntervals]
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    +Hs:814 [in MSetsExtra.MSetIntervals]
    +Hs:825 [in MSetsExtra.MSetIntervals]
    +Hs:829 [in MSetsExtra.MSetIntervals]
    +H0:575 [in MSetsExtra.MSetIntervals]
    +H0:624 [in MSetsExtra.MSetIntervals]
    +H0:676 [in MSetsExtra.MSetIntervals]
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    +H2:314 [in MSetsExtra.MSetIntervals]
    +H:540 [in MSetsExtra.MSetIntervals]
    +H:574 [in MSetsExtra.MSetIntervals]
    +H:59 [in MSetsExtra.MSetIntervals]
    +H:623 [in MSetsExtra.MSetIntervals]
    +H:675 [in MSetsExtra.MSetIntervals]
    +H:693 [in MSetsExtra.MSetIntervals]
    +H:883 [in MSetsExtra.MSetIntervals]
    +H:918 [in MSetsExtra.MSetIntervals]
    +H:921 [in MSetsExtra.MSetIntervals]
    +

    I

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    +In:6 [in MSetsExtra.MSetFoldWithAbort]
    +In:62 [in MSetsExtra.MSetFoldWithAbort]
    +In:96 [in MSetsExtra.MSetFoldWithAbort]
    +i':105 [in MSetsExtra.MSetFoldWithAbort]
    +i':13 [in MSetsExtra.MSetFoldWithAbort]
    +i':319 [in MSetsExtra.MSetFoldWithAbort]
    +i':41 [in MSetsExtra.MSetFoldWithAbort]
    +i':69 [in MSetsExtra.MSetFoldWithAbort]
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    +i2:87 [in MSetsExtra.MSetIntervals]
    +i:104 [in MSetsExtra.MSetFoldWithAbort]
    +i:12 [in MSetsExtra.MSetFoldWithAbort]
    +i:138 [in MSetsExtra.MSetListWithDups]
    +i:160 [in MSetsExtra.MSetFoldWithAbort]
    +i:176 [in MSetsExtra.MSetFoldWithAbort]
    +i:191 [in MSetsExtra.MSetFoldWithAbort]
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    +i:231 [in MSetsExtra.MSetIntervals]
    +i:318 [in MSetsExtra.MSetFoldWithAbort]
    +i:33 [in MSetsExtra.MSetWithDups]
    +i:40 [in MSetsExtra.MSetFoldWithAbort]
    +i:407 [in MSetsExtra.MSetIntervals]
    +i:63 [in MSetsExtra.MSetListWithDups]
    +i:68 [in MSetsExtra.MSetFoldWithAbort]
    +i:818 [in MSetsExtra.MSetIntervals]
    +i:91 [in MSetsExtra.MSetIntervals]
    +i:995 [in MSetsExtra.MSetIntervals]
    +

    L

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    +lt:33 [in MSetsExtra.MSetFoldWithAbort]
    +lt:61 [in MSetsExtra.MSetFoldWithAbort]
    +lt:95 [in MSetsExtra.MSetFoldWithAbort]
    +l':24 [in MSetsExtra.MSetListWithDups]
    +l':30 [in MSetsExtra.MSetListWithDups]
    +l1:345 [in MSetsExtra.MSetIntervals]
    +l1:425 [in MSetsExtra.MSetIntervals]
    +l1:707 [in MSetsExtra.MSetIntervals]
    +l1:715 [in MSetsExtra.MSetIntervals]
    +l2:346 [in MSetsExtra.MSetIntervals]
    +l2:426 [in MSetsExtra.MSetIntervals]
    +l2:710 [in MSetsExtra.MSetIntervals]
    +l2:718 [in MSetsExtra.MSetIntervals]
    +l:10 [in MSetsExtra.MSetListWithDups]
    +l:103 [in MSetsExtra.MSetIntervals]
    +l:115 [in MSetsExtra.MSetIntervals]
    +l:119 [in MSetsExtra.MSetIntervals]
    +l:134 [in MSetsExtra.MSetIntervals]
    +l:14 [in MSetsExtra.MSetListWithDups]
    +l:16 [in MSetsExtra.MSetListWithDups]
    +l:17 [in MSetsExtra.MSetIntervals]
    +l:23 [in MSetsExtra.MSetListWithDups]
    +l:24 [in MSetsExtra.MSetIntervals]
    +l:254 [in MSetsExtra.MSetIntervals]
    +l:258 [in MSetsExtra.MSetIntervals]
    +l:260 [in MSetsExtra.MSetIntervals]
    +l:28 [in MSetsExtra.MSetListWithDups]
    +l:29 [in MSetsExtra.MSetListWithDups]
    +l:30 [in MSetsExtra.MSetIntervals]
    +l:339 [in MSetsExtra.MSetIntervals]
    +l:34 [in MSetsExtra.MSetListWithDups]
    +l:34 [in MSetsExtra.MSetIntervals]
    +l:367 [in MSetsExtra.MSetIntervals]
    +l:39 [in MSetsExtra.MSetIntervals]
    +l:427 [in MSetsExtra.MSetIntervals]
    +l:50 [in MSetsExtra.MSetIntervals]
    +l:551 [in MSetsExtra.MSetIntervals]
    +l:554 [in MSetsExtra.MSetIntervals]
    +l:698 [in MSetsExtra.MSetIntervals]
    +l:701 [in MSetsExtra.MSetIntervals]
    +l:75 [in MSetsExtra.MSetIntervals]
    +l:849 [in MSetsExtra.MSetIntervals]
    +

    N

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    +n:1018 [in MSetsExtra.MSetIntervals]
    +n:2 [in MSetsExtra.MSetIntervals]
    +n:289 [in MSetsExtra.MSetIntervals]
    +n:4 [in MSetsExtra.MSetIntervals]
    +n:52 [in MSetsExtra.MSetIntervals]
    +

    P

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    +pre_x:369 [in MSetsExtra.MSetFoldWithAbort]
    +P:10 [in MSetsExtra.MSetWithDups]
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    +P:112 [in MSetsExtra.MSetFoldWithAbort]
    +P:13 [in MSetsExtra.MSetWithDups]
    +P:18 [in MSetsExtra.MSetFoldWithAbort]
    +P:220 [in MSetsExtra.MSetFoldWithAbort]
    +p:231 [in MSetsExtra.MSetFoldWithAbort]
    +p:236 [in MSetsExtra.MSetFoldWithAbort]
    +P:239 [in MSetsExtra.MSetFoldWithAbort]
    +p:242 [in MSetsExtra.MSetFoldWithAbort]
    +p:247 [in MSetsExtra.MSetFoldWithAbort]
    +p:257 [in MSetsExtra.MSetFoldWithAbort]
    +p:265 [in MSetsExtra.MSetFoldWithAbort]
    +P:270 [in MSetsExtra.MSetFoldWithAbort]
    +P:281 [in MSetsExtra.MSetIntervals]
    +P:284 [in MSetsExtra.MSetIntervals]
    +P:292 [in MSetsExtra.MSetFoldWithAbort]
    +P:300 [in MSetsExtra.MSetFoldWithAbort]
    +p:319 [in MSetsExtra.MSetIntervals]
    +P:326 [in MSetsExtra.MSetFoldWithAbort]
    +P:46 [in MSetsExtra.MSetFoldWithAbort]
    +p:53 [in MSetsExtra.MSetIntervals]
    +P:75 [in MSetsExtra.MSetFoldWithAbort]
    +P:940 [in MSetsExtra.MSetIntervals]
    +P:943 [in MSetsExtra.MSetIntervals]
    +P:98 [in MSetsExtra.MSetListWithDups]
    +p:982 [in MSetsExtra.MSetIntervals]
    +

    R

    +r:859 [in MSetsExtra.MSetIntervals]
    +

    S

    +st':114 [in MSetsExtra.MSetFoldWithAbort]
    +st':120 [in MSetsExtra.MSetFoldWithAbort]
    +st':124 [in MSetsExtra.MSetFoldWithAbort]
    +st':20 [in MSetsExtra.MSetFoldWithAbort]
    +st':26 [in MSetsExtra.MSetFoldWithAbort]
    +st':328 [in MSetsExtra.MSetFoldWithAbort]
    +st':334 [in MSetsExtra.MSetFoldWithAbort]
    +st':338 [in MSetsExtra.MSetFoldWithAbort]
    +st':48 [in MSetsExtra.MSetFoldWithAbort]
    +st':54 [in MSetsExtra.MSetFoldWithAbort]
    +st':77 [in MSetsExtra.MSetFoldWithAbort]
    +st':83 [in MSetsExtra.MSetFoldWithAbort]
    +st':87 [in MSetsExtra.MSetFoldWithAbort]
    +st0:313 [in MSetsExtra.MSetFoldWithAbort]
    +st1:314 [in MSetsExtra.MSetFoldWithAbort]
    +st2:315 [in MSetsExtra.MSetFoldWithAbort]
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    +st:119 [in MSetsExtra.MSetFoldWithAbort]
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    +s':130 [in MSetsExtra.MSetListWithDups]
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    +s':274 [in MSetsExtra.MSetIntervals]
    +s':277 [in MSetsExtra.MSetIntervals]
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    +s':435 [in MSetsExtra.MSetIntervals]
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    +s':67 [in MSetsExtra.MSetListWithDups]
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    +s':741 [in MSetsExtra.MSetIntervals]
    +s':746 [in MSetsExtra.MSetIntervals]
    +s':796 [in MSetsExtra.MSetIntervals]
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    +s':933 [in MSetsExtra.MSetIntervals]
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    +x:161 [in MSetsExtra.MSetListWithDups]
    +x:163 [in MSetsExtra.MSetListWithDups]
    +x:166 [in MSetsExtra.MSetListWithDups]
    +x:167 [in MSetsExtra.MSetListWithDups]
    +x:179 [in MSetsExtra.MSetIntervals]
    +x:186 [in MSetsExtra.MSetIntervals]
    +x:19 [in MSetsExtra.MSetListWithDups]
    +x:190 [in MSetsExtra.MSetIntervals]
    +x:193 [in MSetsExtra.MSetIntervals]
    +x:197 [in MSetsExtra.MSetIntervals]
    +x:209 [in MSetsExtra.MSetIntervals]
    +x:213 [in MSetsExtra.MSetIntervals]
    +x:216 [in MSetsExtra.MSetIntervals]
    +x:219 [in MSetsExtra.MSetIntervals]
    +x:25 [in MSetsExtra.MSetListWithDups]
    +x:255 [in MSetsExtra.MSetIntervals]
    +x:257 [in MSetsExtra.MSetIntervals]
    +x:269 [in MSetsExtra.MSetIntervals]
    +x:271 [in MSetsExtra.MSetIntervals]
    +x:283 [in MSetsExtra.MSetIntervals]
    +x:286 [in MSetsExtra.MSetIntervals]
    +x:287 [in MSetsExtra.MSetIntervals]
    +x:291 [in MSetsExtra.MSetIntervals]
    +x:293 [in MSetsExtra.MSetIntervals]
    +x:294 [in MSetsExtra.MSetIntervals]
    +x:296 [in MSetsExtra.MSetIntervals]
    +x:299 [in MSetsExtra.MSetIntervals]
    +x:303 [in MSetsExtra.MSetIntervals]
    +x:305 [in MSetsExtra.MSetIntervals]
    +x:307 [in MSetsExtra.MSetIntervals]
    +x:31 [in MSetsExtra.MSetListWithDups]
    +x:312 [in MSetsExtra.MSetIntervals]
    +x:318 [in MSetsExtra.MSetIntervals]
    +x:32 [in MSetsExtra.MSetListWithDups]
    +x:323 [in MSetsExtra.MSetIntervals]
    +x:327 [in MSetsExtra.MSetIntervals]
    +x:33 [in MSetsExtra.MSetListWithDups]
    +x:340 [in MSetsExtra.MSetIntervals]
    +x:353 [in MSetsExtra.MSetIntervals]
    +x:36 [in MSetsExtra.MSetListWithDups]
    +x:368 [in MSetsExtra.MSetIntervals]
    +x:37 [in MSetsExtra.MSetWithDups]
    +x:373 [in MSetsExtra.MSetIntervals]
    +x:376 [in MSetsExtra.MSetIntervals]
    +x:383 [in MSetsExtra.MSetIntervals]
    +x:39 [in MSetsExtra.MSetWithDups]
    +x:392 [in MSetsExtra.MSetIntervals]
    +x:397 [in MSetsExtra.MSetIntervals]
    +x:40 [in MSetsExtra.MSetListWithDups]
    +x:412 [in MSetsExtra.MSetIntervals]
    +x:415 [in MSetsExtra.MSetIntervals]
    +x:419 [in MSetsExtra.MSetIntervals]
    +x:42 [in MSetsExtra.MSetWithDups]
    +x:42 [in MSetsExtra.MSetListWithDups]
    +x:422 [in MSetsExtra.MSetIntervals]
    +x:431 [in MSetsExtra.MSetIntervals]
    +x:438 [in MSetsExtra.MSetIntervals]
    +x:439 [in MSetsExtra.MSetIntervals]
    +x:443 [in MSetsExtra.MSetIntervals]
    +x:446 [in MSetsExtra.MSetIntervals]
    +x:449 [in MSetsExtra.MSetIntervals]
    +x:468 [in MSetsExtra.MSetIntervals]
    +x:469 [in MSetsExtra.MSetIntervals]
    +x:471 [in MSetsExtra.MSetIntervals]
    +x:473 [in MSetsExtra.MSetIntervals]
    +x:476 [in MSetsExtra.MSetIntervals]
    +x:479 [in MSetsExtra.MSetIntervals]
    +x:48 [in MSetsExtra.MSetWithDups]
    +x:482 [in MSetsExtra.MSetIntervals]
    +x:488 [in MSetsExtra.MSetIntervals]
    +x:491 [in MSetsExtra.MSetIntervals]
    +x:495 [in MSetsExtra.MSetIntervals]
    +x:499 [in MSetsExtra.MSetIntervals]
    +x:5 [in MSetsExtra.MSetListWithDups]
    +x:50 [in MSetsExtra.MSetListWithDups]
    +x:502 [in MSetsExtra.MSetIntervals]
    +x:506 [in MSetsExtra.MSetIntervals]
    +x:509 [in MSetsExtra.MSetIntervals]
    +x:529 [in MSetsExtra.MSetIntervals]
    +x:532 [in MSetsExtra.MSetIntervals]
    +x:536 [in MSetsExtra.MSetIntervals]
    +x:539 [in MSetsExtra.MSetIntervals]
    +x:542 [in MSetsExtra.MSetIntervals]
    +x:545 [in MSetsExtra.MSetIntervals]
    +x:548 [in MSetsExtra.MSetIntervals]
    +x:549 [in MSetsExtra.MSetIntervals]
    +x:55 [in MSetsExtra.MSetWithDups]
    +x:553 [in MSetsExtra.MSetIntervals]
    +x:57 [in MSetsExtra.MSetIntervals]
    +x:578 [in MSetsExtra.MSetIntervals]
    +x:58 [in MSetsExtra.MSetListWithDups]
    +x:59 [in MSetsExtra.MSetListWithDups]
    +x:627 [in MSetsExtra.MSetIntervals]
    +x:640 [in MSetsExtra.MSetIntervals]
    +x:642 [in MSetsExtra.MSetIntervals]
    +x:646 [in MSetsExtra.MSetIntervals]
    +x:647 [in MSetsExtra.MSetIntervals]
    +x:648 [in MSetsExtra.MSetIntervals]
    +x:650 [in MSetsExtra.MSetIntervals]
    +x:653 [in MSetsExtra.MSetIntervals]
    +x:66 [in MSetsExtra.MSetIntervals]
    +x:663 [in MSetsExtra.MSetIntervals]
    +x:679 [in MSetsExtra.MSetIntervals]
    +x:68 [in MSetsExtra.MSetListWithDups]
    +x:682 [in MSetsExtra.MSetIntervals]
    +x:686 [in MSetsExtra.MSetIntervals]
    +x:688 [in MSetsExtra.MSetIntervals]
    +x:692 [in MSetsExtra.MSetIntervals]
    +x:695 [in MSetsExtra.MSetIntervals]
    +x:7 [in MSetsExtra.MSetListWithDups]
    +x:700 [in MSetsExtra.MSetIntervals]
    +x:730 [in MSetsExtra.MSetIntervals]
    +x:733 [in MSetsExtra.MSetIntervals]
    +x:744 [in MSetsExtra.MSetIntervals]
    +x:76 [in MSetsExtra.MSetIntervals]
    +x:768 [in MSetsExtra.MSetIntervals]
    +x:771 [in MSetsExtra.MSetIntervals]
    +x:776 [in MSetsExtra.MSetIntervals]
    +x:778 [in MSetsExtra.MSetIntervals]
    +x:78 [in MSetsExtra.MSetIntervals]
    +x:782 [in MSetsExtra.MSetIntervals]
    +x:784 [in MSetsExtra.MSetIntervals]
    +x:787 [in MSetsExtra.MSetIntervals]
    +x:792 [in MSetsExtra.MSetIntervals]
    +x:797 [in MSetsExtra.MSetIntervals]
    +x:8 [in MSetsExtra.MSetIntervals]
    +x:800 [in MSetsExtra.MSetIntervals]
    +x:802 [in MSetsExtra.MSetIntervals]
    +x:807 [in MSetsExtra.MSetIntervals]
    +x:809 [in MSetsExtra.MSetIntervals]
    +x:812 [in MSetsExtra.MSetIntervals]
    +x:82 [in MSetsExtra.MSetIntervals]
    +x:826 [in MSetsExtra.MSetIntervals]
    +x:830 [in MSetsExtra.MSetIntervals]
    +x:831 [in MSetsExtra.MSetIntervals]
    +x:836 [in MSetsExtra.MSetIntervals]
    +x:839 [in MSetsExtra.MSetIntervals]
    +x:847 [in MSetsExtra.MSetIntervals]
    +x:856 [in MSetsExtra.MSetIntervals]
    +x:863 [in MSetsExtra.MSetIntervals]
    +x:88 [in MSetsExtra.MSetListWithDups]
    +x:888 [in MSetsExtra.MSetIntervals]
    +x:890 [in MSetsExtra.MSetIntervals]
    +x:901 [in MSetsExtra.MSetIntervals]
    +x:902 [in MSetsExtra.MSetIntervals]
    +x:907 [in MSetsExtra.MSetIntervals]
    +x:908 [in MSetsExtra.MSetIntervals]
    +x:909 [in MSetsExtra.MSetIntervals]
    +x:912 [in MSetsExtra.MSetIntervals]
    +x:915 [in MSetsExtra.MSetIntervals]
    +x:926 [in MSetsExtra.MSetIntervals]
    +x:930 [in MSetsExtra.MSetIntervals]
    +x:942 [in MSetsExtra.MSetIntervals]
    +x:945 [in MSetsExtra.MSetIntervals]
    +x:946 [in MSetsExtra.MSetIntervals]
    +x:948 [in MSetsExtra.MSetIntervals]
    +x:950 [in MSetsExtra.MSetIntervals]
    +x:952 [in MSetsExtra.MSetIntervals]
    +x:999 [in MSetsExtra.MSetIntervals]
    +

    Y

    +yc1:89 [in MSetsExtra.MSetIntervals]
    +yc2:88 [in MSetsExtra.MSetIntervals]
    +yc2:93 [in MSetsExtra.MSetIntervals]
    +yc:844 [in MSetsExtra.MSetIntervals]
    +ys':27 [in MSetsExtra.MSetListWithDups]
    +y':26 [in MSetsExtra.MSetListWithDups]
    +y':774 [in MSetsExtra.MSetIntervals]
    +y':835 [in MSetsExtra.MSetIntervals]
    +y':860 [in MSetsExtra.MSetIntervals]
    +y':865 [in MSetsExtra.MSetIntervals]
    +y':866 [in MSetsExtra.MSetIntervals]
    +y':872 [in MSetsExtra.MSetIntervals]
    +y':873 [in MSetsExtra.MSetIntervals]
    +y':878 [in MSetsExtra.MSetIntervals]
    +y0:128 [in MSetsExtra.MSetListWithDups]
    +y1:398 [in MSetsExtra.MSetIntervals]
    +y1:402 [in MSetsExtra.MSetIntervals]
    +y1:408 [in MSetsExtra.MSetIntervals]
    +y1:5 [in MSetsExtra.MSetIntervals]
    +y1:512 [in MSetsExtra.MSetIntervals]
    +y1:516 [in MSetsExtra.MSetIntervals]
    +y1:705 [in MSetsExtra.MSetIntervals]
    +y1:713 [in MSetsExtra.MSetIntervals]
    +y1:757 [in MSetsExtra.MSetIntervals]
    +y1:90 [in MSetsExtra.MSetIntervals]
    +y2:145 [in MSetsExtra.MSetIntervals]
    +y2:166 [in MSetsExtra.MSetIntervals]
    +y2:400 [in MSetsExtra.MSetIntervals]
    +y2:404 [in MSetsExtra.MSetIntervals]
    +y2:409 [in MSetsExtra.MSetIntervals]
    +y2:466 [in MSetsExtra.MSetIntervals]
    +y2:514 [in MSetsExtra.MSetIntervals]
    +y2:518 [in MSetsExtra.MSetIntervals]
    +y2:581 [in MSetsExtra.MSetIntervals]
    +y2:585 [in MSetsExtra.MSetIntervals]
    +y2:6 [in MSetsExtra.MSetIntervals]
    +y2:630 [in MSetsExtra.MSetIntervals]
    +y2:634 [in MSetsExtra.MSetIntervals]
    +y2:708 [in MSetsExtra.MSetIntervals]
    +y2:716 [in MSetsExtra.MSetIntervals]
    +y2:758 [in MSetsExtra.MSetIntervals]
    +y:11 [in MSetsExtra.MSetIntervals]
    +y:111 [in MSetsExtra.MSetListWithDups]
    +y:115 [in MSetsExtra.MSetListWithDups]
    +y:117 [in MSetsExtra.MSetListWithDups]
    +y:127 [in MSetsExtra.MSetListWithDups]
    +y:135 [in MSetsExtra.MSetListWithDups]
    +y:2 [in MSetsExtra.MSetListWithDups]
    +y:279 [in MSetsExtra.MSetFoldWithAbort]
    +y:282 [in MSetsExtra.MSetFoldWithAbort]
    +y:285 [in MSetsExtra.MSetFoldWithAbort]
    +y:302 [in MSetsExtra.MSetIntervals]
    +y:304 [in MSetsExtra.MSetIntervals]
    +y:360 [in MSetsExtra.MSetIntervals]
    +y:381 [in MSetsExtra.MSetIntervals]
    +y:382 [in MSetsExtra.MSetIntervals]
    +y:388 [in MSetsExtra.MSetIntervals]
    +y:390 [in MSetsExtra.MSetIntervals]
    +y:391 [in MSetsExtra.MSetIntervals]
    +y:394 [in MSetsExtra.MSetIntervals]
    +y:406 [in MSetsExtra.MSetIntervals]
    +y:423 [in MSetsExtra.MSetIntervals]
    +y:428 [in MSetsExtra.MSetIntervals]
    +y:433 [in MSetsExtra.MSetIntervals]
    +y:437 [in MSetsExtra.MSetIntervals]
    +y:442 [in MSetsExtra.MSetIntervals]
    +y:445 [in MSetsExtra.MSetIntervals]
    +y:448 [in MSetsExtra.MSetIntervals]
    +y:452 [in MSetsExtra.MSetIntervals]
    +y:493 [in MSetsExtra.MSetIntervals]
    +y:505 [in MSetsExtra.MSetIntervals]
    +y:52 [in MSetsExtra.MSetListWithDups]
    +y:526 [in MSetsExtra.MSetIntervals]
    +y:527 [in MSetsExtra.MSetIntervals]
    +y:533 [in MSetsExtra.MSetIntervals]
    +y:543 [in MSetsExtra.MSetIntervals]
    +y:550 [in MSetsExtra.MSetIntervals]
    +y:566 [in MSetsExtra.MSetIntervals]
    +y:568 [in MSetsExtra.MSetIntervals]
    +y:57 [in MSetsExtra.MSetListWithDups]
    +y:591 [in MSetsExtra.MSetIntervals]
    +y:592 [in MSetsExtra.MSetIntervals]
    +y:593 [in MSetsExtra.MSetIntervals]
    +y:594 [in MSetsExtra.MSetIntervals]
    +y:595 [in MSetsExtra.MSetIntervals]
    +y:596 [in MSetsExtra.MSetIntervals]
    +y:597 [in MSetsExtra.MSetIntervals]
    +y:598 [in MSetsExtra.MSetIntervals]
    +y:599 [in MSetsExtra.MSetIntervals]
    +y:6 [in MSetsExtra.MSetListWithDups]
    +y:601 [in MSetsExtra.MSetIntervals]
    +y:609 [in MSetsExtra.MSetIntervals]
    +y:618 [in MSetsExtra.MSetIntervals]
    +y:641 [in MSetsExtra.MSetIntervals]
    +y:643 [in MSetsExtra.MSetIntervals]
    +y:644 [in MSetsExtra.MSetIntervals]
    +y:645 [in MSetsExtra.MSetIntervals]
    +y:654 [in MSetsExtra.MSetIntervals]
    +y:660 [in MSetsExtra.MSetIntervals]
    +y:670 [in MSetsExtra.MSetIntervals]
    +y:689 [in MSetsExtra.MSetIntervals]
    +y:696 [in MSetsExtra.MSetIntervals]
    +y:711 [in MSetsExtra.MSetIntervals]
    +y:712 [in MSetsExtra.MSetIntervals]
    +y:719 [in MSetsExtra.MSetIntervals]
    +y:722 [in MSetsExtra.MSetIntervals]
    +y:724 [in MSetsExtra.MSetIntervals]
    +y:773 [in MSetsExtra.MSetIntervals]
    +y:779 [in MSetsExtra.MSetIntervals]
    +y:788 [in MSetsExtra.MSetIntervals]
    +y:8 [in MSetsExtra.MSetListWithDups]
    +y:803 [in MSetsExtra.MSetIntervals]
    +y:813 [in MSetsExtra.MSetIntervals]
    +y:845 [in MSetsExtra.MSetIntervals]
    +y:848 [in MSetsExtra.MSetIntervals]
    +y:853 [in MSetsExtra.MSetIntervals]
    +y:896 [in MSetsExtra.MSetIntervals]
    +y:9 [in MSetsExtra.MSetIntervals]
    +

    Z

    +z:1 [in MSetsExtra.MSetIntervals]
    +z:1004 [in MSetsExtra.MSetIntervals]
    +z:1005 [in MSetsExtra.MSetIntervals]
    +z:1012 [in MSetsExtra.MSetIntervals]
    +z:1019 [in MSetsExtra.MSetIntervals]
    +z:206 [in MSetsExtra.MSetIntervals]
    +z:227 [in MSetsExtra.MSetIntervals]
    +z:233 [in MSetsExtra.MSetIntervals]
    +z:237 [in MSetsExtra.MSetIntervals]
    +z:241 [in MSetsExtra.MSetIntervals]
    +z:3 [in MSetsExtra.MSetIntervals]
    +z:411 [in MSetsExtra.MSetIntervals]
    +z:418 [in MSetsExtra.MSetIntervals]
    +z:475 [in MSetsExtra.MSetIntervals]
    +z:520 [in MSetsExtra.MSetIntervals]
    +z:534 [in MSetsExtra.MSetIntervals]
    +z:537 [in MSetsExtra.MSetIntervals]
    +z:750 [in MSetsExtra.MSetIntervals]
    +z:884 [in MSetsExtra.MSetIntervals]
    +z:885 [in MSetsExtra.MSetIntervals]
    +z:886 [in MSetsExtra.MSetIntervals]
    +z:891 [in MSetsExtra.MSetIntervals]
    +z:892 [in MSetsExtra.MSetIntervals]
    +z:893 [in MSetsExtra.MSetIntervals]
    +

    Module Index

    -

    E

    +

    E

    ElementEncode [in MSetsExtra.MSetIntervals]
    ElementEncodeN [in MSetsExtra.MSetIntervals]
    ElementEncodeNat [in MSetsExtra.MSetIntervals]
    @@ -1128,10 +4215,10 @@

    Module Index

    ElementEncodeZ [in MSetsExtra.MSetIntervals]
    ElementEncodeZ.E [in MSetsExtra.MSetIntervals]
    ElementEncode.E [in MSetsExtra.MSetIntervals]
    -

    H

    +

    H

    HasFoldWithAbort [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps [in MSetsExtra.MSetFoldWithAbort]
    -

    M

    +

    M

    Make [in MSetsExtra.MSetListWithDups]
    MakeAVLSetsWithFoldA [in MSetsExtra.MSetFoldWithAbort]
    MakeGenTreeFoldA [in MSetsExtra.MSetFoldWithAbort]
    @@ -1146,23 +4233,23 @@

    Module Index

    MSetIntervalsZ [in MSetsExtra.MSetIntervals]
    MSetIntervals.E [in MSetsExtra.MSetIntervals]
    MSetIntervals.Raw [in MSetsExtra.MSetIntervals]
    -

    N

    +

    N

    NOP [in MSetsExtra.MSetIntervals]
    -

    O

    +

    O

    Ops [in MSetsExtra.MSetListWithDups]
    Ops [in MSetsExtra.MSetIntervals]
    Ops.MX [in MSetsExtra.MSetListWithDups]
    Ops.RDFS [in MSetsExtra.MSetListWithDups]
    -

    R

    +

    R

    Raw [in MSetsExtra.MSetIntervals]
    RemoveDupsFromSorted [in MSetsExtra.MSetListWithDups]
    RemoveDupsFromSorted.MX [in MSetsExtra.MSetListWithDups]
    RemoveDupsFromSorted.XSort [in MSetsExtra.MSetListWithDups]
    RemoveDupsFromSorted.XTotalLeBool [in MSetsExtra.MSetListWithDups]
    -

    S

    +

    S

    SetsWithFoldA [in MSetsExtra.MSetFoldWithAbort]
    SetsWithFoldA.E [in MSetsExtra.MSetFoldWithAbort]
    -

    W

    +

    W

    WSetsOnWithDups [in MSetsExtra.MSetWithDups]
    WSetsOnWithDupsExtra [in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra [in MSetsExtra.MSetWithDups]
    @@ -1172,13 +4259,13 @@

    Module Index

    WSetsWithFoldA.E [in MSetsExtra.MSetFoldWithAbort]


    Variable Index

    -

    M

    +

    M

    MSetIntervals.Spec.f [in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.s [in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.s' [in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.x [in MSetsExtra.MSetIntervals]
    MSetIntervals.Spec.y [in MSetsExtra.MSetIntervals]
    -

    W

    +

    W

    WSetsOnWithDups.Spec.f [in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Spec.s [in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Spec.s' [in MSetsExtra.MSetWithDups]
    @@ -1186,16 +4273,16 @@

    Variable Index

    WSetsOnWithDups.Spec.y [in MSetsExtra.MSetWithDups]


    Library Index

    -

    M

    +

    M

    MSetFoldWithAbort
    MSetIntervals
    MSetListWithDups
    MSetWithDups


    Lemma Index

    -

    A

    +

    A

    add_add_sub_eq [in MSetsExtra.MSetIntervals]
    -

    E

    +

    E

    ElementEncodeNat.decode_encode_ok [in MSetsExtra.MSetIntervals]
    ElementEncodeNat.encode_lt [in MSetsExtra.MSetIntervals]
    ElementEncodeNat.encode_eq [in MSetsExtra.MSetIntervals]
    @@ -1205,7 +4292,7 @@

    Lemma Index

    ElementEncodeZ.decode_encode_ok [in MSetsExtra.MSetIntervals]
    ElementEncodeZ.encode_lt [in MSetsExtra.MSetIntervals]
    ElementEncodeZ.encode_eq [in MSetsExtra.MSetIntervals]
    -

    H

    +

    H

    HasFoldWithAbortOps.choose_with_abort_spec [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.exists_with_abort_spec [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.filter_with_abort_spec [in MSetsExtra.MSetFoldWithAbort]
    @@ -1217,7 +4304,7 @@

    Lemma Index

    HasFoldWithAbortOps.foldWithAbortSpec [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.foldWithAbortSpec_Equal [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.forall_with_abort_spec [in MSetsExtra.MSetFoldWithAbort]
    -

    M

    +

    M

    MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetFoldWithAbort]
    MakeGenTreeFoldA.foldWithAbort_RawSpec [in MSetsExtra.MSetFoldWithAbort]
    MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetFoldWithAbort]
    @@ -1282,12 +4369,12 @@

    Lemma Index

    MSetIntervals.singleton_spec [in MSetsExtra.MSetIntervals]
    MSetIntervals.subset_spec [in MSetsExtra.MSetIntervals]
    MSetIntervals.union_spec [in MSetsExtra.MSetIntervals]
    -

    N

    +

    N

    NoDupA_map [in MSetsExtra.MSetIntervals]
    -

    O

    +

    O

    Ops.acc_pred [in MSetsExtra.MSetIntervals]
    Ops.singleton_alt_def [in MSetsExtra.MSetIntervals]
    -

    R

    +

    R

    Raw.addZ_invariant [in MSetsExtra.MSetIntervals]
    Raw.addZ_InZ [in MSetsExtra.MSetIntervals]
    Raw.addZ_alt_def [in MSetsExtra.MSetIntervals]
    @@ -1451,26 +4538,26 @@

    Lemma Index

    RemoveDupsFromSorted.XTotalLeBool.leb_total [in MSetsExtra.MSetListWithDups]
    rev_map_alt_def [in MSetsExtra.MSetIntervals]
    rev_map_aux_alt_def [in MSetsExtra.MSetIntervals]
    -

    W

    +

    W

    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w [in MSetsExtra.MSetWithDups]
    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1 [in MSetsExtra.MSetWithDups]
    -

    Z

    +

    Z

    Z_to_N_minus_neq_0 [in MSetsExtra.MSetIntervals]
    Z_lt_le_add_r [in MSetsExtra.MSetIntervals]
    Z_lt_add_r [in MSetsExtra.MSetIntervals]
    Z_le_add_r [in MSetsExtra.MSetIntervals]


    Axiom Index

    -

    E

    +

    E

    ElementEncode.decode [in MSetsExtra.MSetIntervals]
    ElementEncode.decode_encode_ok [in MSetsExtra.MSetIntervals]
    ElementEncode.encode [in MSetsExtra.MSetIntervals]
    ElementEncode.encode_lt [in MSetsExtra.MSetIntervals]
    ElementEncode.encode_eq [in MSetsExtra.MSetIntervals]
    -

    H

    +

    H

    HasFoldWithAbort.foldWithAbortPrecompute [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbort.foldWithAbortPrecomputeSpec [in MSetsExtra.MSetFoldWithAbort]
    -

    W

    +

    W

    WSetsOnWithDupsExtra.cardinal_spec [in MSetsExtra.MSetWithDups]
    WSetsOnWithDupsExtra.elements_dist_spec2w [in MSetsExtra.MSetWithDups]
    WSetsOnWithDupsExtra.elements_dist_spec1 [in MSetsExtra.MSetWithDups]
    @@ -1498,9 +4585,7 @@

    Axiom Index

    WSetsOnWithDups.union_spec [in MSetsExtra.MSetWithDups]


    Constructor Index

    -

    M

    -MSetIntervals.Mkt [in MSetsExtra.MSetIntervals]
    -

    O

    +

    O

    Ops.ICR_after_touch [in MSetsExtra.MSetIntervals]
    Ops.ICR_after [in MSetsExtra.MSetIntervals]
    Ops.ICR_subsume_2 [in MSetsExtra.MSetIntervals]
    @@ -1510,35 +4595,43 @@

    Constructor Index

    Ops.ICR_overlap_before [in MSetsExtra.MSetIntervals]
    Ops.ICR_before_touch [in MSetsExtra.MSetIntervals]
    Ops.ICR_before [in MSetsExtra.MSetIntervals]
    -

    R

    +

    R

    Raw.ok [in MSetsExtra.MSetIntervals]


    Projection Index

    -

    H

    +

    H

    HasFoldWithAbortOps.fwasa_P' [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.fwasa_f_gt [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.fwasa_f_lt [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.fwasa_f_pre [in MSetsExtra.MSetFoldWithAbort]
    -

    M

    +

    M

    MSetIntervals.is_ok [in MSetsExtra.MSetIntervals]
    MSetIntervals.this [in MSetsExtra.MSetIntervals]
    -

    R

    +

    R

    Raw.ok [in MSetsExtra.MSetIntervals]


    Inductive Index

    -

    O

    +

    O

    Ops.interval_compare_result [in MSetsExtra.MSetIntervals]
    -

    R

    +

    R

    Raw.Ok [in MSetsExtra.MSetIntervals]


    +

    Section Index

    +

    M

    +MSetIntervals.Spec [in MSetsExtra.MSetIntervals]
    +

    R

    +Raw.ForNotations [in MSetsExtra.MSetIntervals]
    +

    W

    +WSetsOnWithDups.Spec [in MSetsExtra.MSetWithDups]
    +

    Instance Index

    -

    M

    +

    M

    Make.In_compat [in MSetsExtra.MSetListWithDups]
    MSetIntervals.eq_equiv [in MSetsExtra.MSetIntervals]
    MSetIntervals.In_compat [in MSetsExtra.MSetIntervals]
    MSetIntervals.lt_compat [in MSetsExtra.MSetIntervals]
    MSetIntervals.lt_strorder [in MSetsExtra.MSetIntervals]
    -

    R

    +

    R

    Raw.add_ok [in MSetsExtra.MSetIntervals]
    Raw.diff_ok [in MSetsExtra.MSetIntervals]
    Raw.empty_ok [in MSetsExtra.MSetIntervals]
    @@ -1550,33 +4643,25 @@

    Instance Index

    Raw.remove_ok [in MSetsExtra.MSetIntervals]
    Raw.singleton_ok [in MSetsExtra.MSetIntervals]
    Raw.union_ok [in MSetsExtra.MSetIntervals]
    -

    W

    +

    W

    WSetsOnWithDups.In_compat [in MSetsExtra.MSetWithDups]


    -

    Section Index

    -

    M

    -MSetIntervals.Spec [in MSetsExtra.MSetIntervals]
    -

    R

    -Raw.ForNotations [in MSetsExtra.MSetIntervals]
    -

    W

    -WSetsOnWithDups.Spec [in MSetsExtra.MSetWithDups]
    -

    Abbreviation Index

    -

    M

    +

    M

    Make.compatb [in MSetsExtra.MSetListWithDups]
    MSetIntervals.compatb [in MSetsExtra.MSetIntervals]
    -

    W

    +

    W

    WSetsOnWithDups.compatb [in MSetsExtra.MSetWithDups]


    Definition Index

    -

    E

    +

    E

    ElementEncodeNat.decode [in MSetsExtra.MSetIntervals]
    ElementEncodeNat.encode [in MSetsExtra.MSetIntervals]
    ElementEncodeN.decode [in MSetsExtra.MSetIntervals]
    ElementEncodeN.encode [in MSetsExtra.MSetIntervals]
    ElementEncodeZ.decode [in MSetsExtra.MSetIntervals]
    ElementEncodeZ.encode [in MSetsExtra.MSetIntervals]
    -

    F

    +

    F

    foldWithAbortGtLtSpecPred [in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortGtLtType [in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortGtSpecPred [in MSetsExtra.MSetFoldWithAbort]
    @@ -1585,7 +4670,7 @@

    Definition Index

    foldWithAbortPrecomputeType [in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortSpecPred [in MSetsExtra.MSetFoldWithAbort]
    foldWithAbortType [in MSetsExtra.MSetFoldWithAbort]
    -

    H

    +

    H

    HasFoldWithAbortOps.choose_with_abort [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.exists_with_abort [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.filter_with_abort [in MSetsExtra.MSetFoldWithAbort]
    @@ -1594,7 +4679,7 @@

    Definition Index

    HasFoldWithAbortOps.foldWithAbortGtLt [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.foldWithAbortSpecArgsForPred [in MSetsExtra.MSetFoldWithAbort]
    HasFoldWithAbortOps.forall_with_abort [in MSetsExtra.MSetFoldWithAbort]
    -

    M

    +

    M

    MakeAVLSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetFoldWithAbort]
    MakeGenTreeFoldA.foldWithAbort_Raw [in MSetsExtra.MSetFoldWithAbort]
    MakeListSetsWithFoldA.foldWithAbortPrecompute [in MSetsExtra.MSetFoldWithAbort]
    @@ -1642,7 +4727,7 @@

    Definition Index

    MSetIntervals.Subset [in MSetsExtra.MSetIntervals]
    MSetIntervals.t [in MSetsExtra.MSetIntervals]
    MSetIntervals.union [in MSetsExtra.MSetIntervals]
    -

    O

    +

    O

    Ops.add [in MSetsExtra.MSetListWithDups]
    Ops.add [in MSetsExtra.MSetIntervals]
    Ops.addZ [in MSetsExtra.MSetIntervals]
    @@ -1729,7 +4814,7 @@

    Definition Index

    Ops.union [in MSetsExtra.MSetListWithDups]
    Ops.union [in MSetsExtra.MSetIntervals]
    Ops.union_aux [in MSetsExtra.MSetIntervals]
    -

    R

    +

    R

    Raw.choose_spec2 [in MSetsExtra.MSetIntervals]
    Raw.choose_spec1 [in MSetsExtra.MSetIntervals]
    Raw.elementsZ_single [in MSetsExtra.MSetIntervals]
    @@ -1767,7 +4852,7 @@

    Definition Index

    RemoveDupsFromSorted.XTotalLeBool.t [in MSetsExtra.MSetListWithDups]
    rev_map [in MSetsExtra.MSetIntervals]
    rev_map_aux [in MSetsExtra.MSetIntervals]
    -

    W

    +

    W

    WSetsOnWithDups.Empty [in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.eq [in MSetsExtra.MSetWithDups]
    WSetsOnWithDups.Equal [in MSetsExtra.MSetWithDups]
    @@ -1777,44 +4862,44 @@

    Definition Index

    WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist [in MSetsExtra.MSetWithDups]


    Record Index

    -

    H

    +

    H

    HasFoldWithAbortOps.FoldWithAbortSpecArg [in MSetsExtra.MSetFoldWithAbort]
    -

    M

    +

    M

    MSetIntervals.t_ [in MSetsExtra.MSetIntervals]
    -

    R

    +

    R

    Raw.Ok [in MSetsExtra.MSetIntervals]


    - - + + - + - + - + - + - - + + - + @@ -1849,6 +4934,38 @@

    Record Index

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + @@ -2022,7 +5139,7 @@

    Record Index

    - + @@ -2038,7 +5155,7 @@

    Record Index

    - + @@ -2105,7 +5222,7 @@

    Record Index

    - + @@ -2118,26 +5235,26 @@

    Record Index

    - + - + - + - + - + @@ -2150,23 +5267,23 @@

    Record Index

    - + - + - + - + diff --git a/html/toc.html b/html/toc.html index cb185ba..ca7807c 100644 --- a/html/toc.html +++ b/html/toc.html @@ -17,7 +17,7 @@
    Global Index ABCBC D E F G HII J KLL M N OPP Q R STT U V WXYXY Z _ other(596 entries)(2110 entries)
    Notation Index (5 entries)
    Binder IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(1515 entries)
    Module Index A B J K LMM N O P Z _ other(11 entries)(10 entries)
    Projection Index (2 entries)
    Instance IndexSection Index A B C J K LMM N O P QRR S T U VWW X Y Z _ other(17 entries)(3 entries)
    Section IndexInstance Index A B C J K LMM N O P QRR S T U VWW X Y Z _ other(3 entries)(17 entries)
    Abbreviation Index