Integers.v

Require Import HoTT.
Require Export HoTT.

Module Export Ints.

  Private Inductive Z : Type :=
  | zero_Z : Z
  | succ : Z -> Z
  | pred : Z -> Z.

  Axiom inv1 : forall n : Z, n = pred(succ n).
  Axiom inv2 : forall n : Z, n = succ(pred n).
  Axiom ZisSet : IsHSet Z.

  Section Z_induction.
    Variable (P : Z -> Type)
             (H : forall n, IsHSet (P n))
             (a : P zero_Z)
             (s : forall (n : Z), P n -> P (succ n))
             (p : forall (n : Z), P n -> P (pred n))
             (i1 : forall (n : Z) (m : P n), (inv1 n) # m = p (succ n) (s (n) m))
             (i2 : forall (n : Z) (m : P n), (inv2 n) # m = s (pred n) (p (n) m)).

    Fixpoint Z_ind
             (x : Z)
             {struct x}
      : P x
      := 
        (match x return _ -> _ -> P x with
         | zero_Z => fun _ => fun _ => a
         | succ n => fun _ => fun _ => s n (Z_ind n)
         | pred n => fun _ => fun _ => p n (Z_ind n)
         end) i1 i2.

    Axiom Z_ind_beta_inv1 : forall (n : Z), apD Z_ind (inv1 n) = i1 n (Z_ind n).

    Axiom Z_ind_beta_inv2 : forall (n : Z), apD Z_ind (inv2 n) = i2 n (Z_ind n).
  End Z_induction.

  Section Z_recursion.
    Variable  (P : Type)
              (H : IsHSet P)
              (a : P)
              (s : P -> P)
              (p : P -> P)
              (i1 : forall (m : P), m = p(s m))
              (i2 : forall (m : P), m = s(p m)).

    Definition Z_rec : Z -> P.
    Proof.
      simple refine (Z_ind _ _ _ _ _ _) ; simpl.
      - apply a.
      - intro ; apply s.
      - intro ; apply p.
      - intros.
        refine (transport_const _ _ @ (i1 _)).
      - intros.
        refine (transport_const _ _ @ (i2 _)).
    Defined.
    
    Definition Z_rec_beta_inv1 (n : Z) : ap Z_rec (inv1 n) = i1 (Z_rec n).
    Proof.
      unfold Z_rec.
      eapply (cancelL (transport_const (inv1 n) _)).
      simple refine ((apD_const _ _)^ @ _).
      apply Z_ind_beta_inv1.
    Defined.

    Definition Z_rec_beta_inv2 (n : Z) : ap Z_rec (inv2 n) = i2 (Z_rec n).
    Proof.
      unfold Z_rec.
      eapply (cancelL (transport_const (inv2 n) _)).
      simple refine ((apD_const _ _)^ @ _).
      apply Z_ind_beta_inv2.
    Defined.

  End Z_recursion.
  
End Ints.

Section ring_Z.
  Fixpoint nat_to_Z (n : nat) : Z := 
    match n with
    | 0 => zero_Z
    | S m => succ (nat_to_Z m)
    end.

  Definition plus : Z -> Z -> Z := fun x => Z_rec Z x succ pred inv1 inv2.

  Lemma plus_0n : forall x, plus zero_Z x = x.
  Proof.
    simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros y Hy.
      apply (ap succ Hy).
    - intros y Hy.
      apply (ap pred Hy).
  Defined.

  Definition plus_n0 x :  plus x zero_Z = x := idpath x.

  Lemma plus_Sn x : forall y, plus (succ x) y = succ(plus x y).
  Proof.
    simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros y Hy.
      apply (ap succ Hy).
    - intros y Hy.
      apply (ap pred Hy @ (inv1 (plus x y))^ @ inv2 (plus x y)).
  Defined.

  Definition plus_nS x y : plus x (succ y) = succ(plus x y) := idpath.

  Lemma plus_Pn x : forall y, plus (pred x) y = pred (plus x y).
  Proof.
    simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros y Hy.
      apply (ap succ Hy @ (inv2 (plus x y))^ @ inv1 (plus x y)).
    - intros y Hy.
      apply (ap pred Hy).
  Defined.

  Definition plus_nP x y : plus x (pred y) = pred(plus x y) := idpath.      

  Lemma plus_comm x : forall y : Z, plus x y = plus y x.
  Proof.
    simple refine (Z_ind _ _ _ _ _ _)
    ; simpl ; try (intros ; apply set_path2).
    - apply (plus_0n x)^.
    - intros n H1.
      apply (ap succ H1 @ (plus_Sn _ _)^).
    - intros n H1.
      apply (ap pred H1 @ (plus_Pn _ _)^).
  Defined.

  Lemma plus_assoc x y : forall z : Z, plus (plus x y) z = plus x (plus y z).
  Proof.
    simple refine (Z_ind _ _ _ _ _ _)
    ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros Sz HSz.
      refine (ap succ HSz).
    - intros Pz HPz.
      apply (ap pred HPz).
  Defined.

  Definition negate : Z -> Z := Z_rec Z zero_Z pred succ inv2 inv1.

  Lemma negate_negate : forall x, negate(negate x) = x.
  Proof.
    simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros Sy HSy.
      apply (ap succ HSy).
    - intros Py HPy.
      apply (ap pred HPy).
  Defined.

  Definition minus x y : Z := plus x (negate y).

  Lemma plus_negatex : forall x, plus x (negate x) = zero_Z.
  Proof.
    simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros Sx HSx.
      refine (ap pred (plus_Sn _ _) @ _).
      refine ((inv1 _)^ @ HSx).
    - intros Px HPx.
      refine (ap succ (plus_Pn _ _) @ _).
      refine ((inv2 _)^ @ HPx).
  Defined.

  Definition plus_xnegate x : plus (negate x) x = zero_Z :=
    plus_comm (negate x) x @ plus_negatex x.

  Lemma plus_negate x : forall y, plus (negate x) (negate y) = negate (plus x y).
  Proof.
    simple refine (Z_ind _ _ _ _ _ _) ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros Sy HSy.
      apply (ap pred HSy).
    - intros Py HPy.
      apply (ap succ HPy).
  Defined.
  
  Definition times (x : Z) : Z -> Z.
  Proof.
    simple refine (Z_rec _ _ _ _ _ _).
    - apply zero_Z.
    - apply (plus x).
    - apply (fun z => minus z x).
    - intros ; unfold minus.
      symmetry.
      refine (ap (fun z => plus z (negate x)) (plus_comm x m) @ _).
      refine (plus_assoc _ _ _ @ _).
      refine (ap (fun z => plus m z) (plus_negatex _) @ _).
      apply plus_n0.
    - intros ; unfold minus.
      symmetry.
      refine (ap (fun z => plus x z) (plus_comm _ _) @ _).
      refine ((plus_assoc _ _ _)^ @ _).
      refine (ap (fun z => plus z m) (plus_negatex _) @ _).
      apply plus_0n.    
  Defined.

  Lemma times_0n : forall x, times zero_Z x = zero_Z.
  Proof.
    simple refine (Z_ind _ _ _ _ _ _) ; try (intros ; apply set_path2) ; simpl.
    - reflexivity.
    - intros Sx HSx.
      apply (plus_0n _ @ HSx).
    - intros Px HPx.
      unfold minus ; simpl ; apply HPx.
  Defined.

  Definition times_n0 n : times n zero_Z = zero_Z := idpath.

  Lemma times_Sn x : forall y, times (succ x) y = plus y (times x y).
  Proof.
    simple refine (Z_ind _ _ _ _ _ _) ; try (intros ; apply set_path2) ; simpl.
    - reflexivity.
    - intros Sy HSy.
      refine (ap (fun z => plus (succ x) z) HSy @ _).
      refine (plus_Sn _ _ @ _).
      refine (_ @ (plus_Sn _ _)^).
      refine (ap succ _).
      refine ((plus_assoc _ _ _)^ @ _).
      refine (_ @ plus_assoc _ _ _).
      refine (ap (fun z => plus z (times x Sy)) (plus_comm _ _)).
    - intros Py HPy ; unfold minus.
      refine (ap (fun z => plus z (negate (succ x))) HPy @ _) ; simpl.
      refine (_ @ (plus_Pn _ _)^).
      refine (ap pred _).
      apply plus_assoc.
  Defined.

  Definition times_nS x y : times x (succ y) = plus x (times x y) := idpath.

  Lemma times_Pn x : forall y, times (pred x) y = minus (times x y) y.
  Proof.
    simple refine (Z_ind _ _ _ _ _ _) ; try (intros ; apply set_path2) ; simpl.
    - reflexivity.
    - intros Sy HSy.
      refine (ap (fun z => plus (pred x) z) HSy @ _) ; unfold minus.
      refine (plus_Pn _ _ @ _) ; simpl.
      refine (ap pred _).
      apply (plus_assoc _ _ _)^.
    - intros Py HPy.
      refine (ap (fun z => minus z (pred x)) HPy @ _) ; unfold minus ; simpl.
      refine (ap succ _).
      refine (plus_assoc _ _ _ @ _).
      refine (_ @ (plus_assoc _ _ _)^).
      refine (ap (fun z => plus (times x Py) z) (plus_comm _ _)).
  Defined.

  Definition times_nP x y : times x (pred y) = minus (times x y) x := idpath.

  Lemma times_comm x : forall y, times x y = times y x.
  Proof.
    simple refine (Z_ind _ _ _ _ _ _)
    ; simpl ; try (intros ; apply set_path2).
    - apply (times_0n x)^.
    - intros Sx HSx.
      apply (ap (fun z => plus x z) HSx @ (times_Sn _ _)^).
    - intros Py HPy.
      apply (ap (fun z => minus z x) HPy @ (times_Pn _ _)^).
  Defined.

  Lemma times_negatex x : forall y, times x (negate y) = negate (times x y).
  Proof.
    simple refine (Z_ind _ _ _ _ _ _)
    ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros Sy HSy.
      unfold minus.
      refine (ap (fun z => plus z (negate x)) HSy @ _).
      refine (plus_negate _ _ @ _).
      apply (ap negate (plus_comm _ _)).
    - intros Py HPy.
      refine (ap (plus x) HPy @ _).
      unfold minus.
      refine (ap (fun z => plus z (negate (times x Py))) (negate_negate _)^ @ _).
      refine (plus_negate _ _ @ _).
      refine (ap negate (plus_comm _ _)).
  Defined.     
  
  Definition times_xnegate x y : times (negate x) y = negate (times x y) :=
    times_comm (negate x) y @ times_negatex y x @ ap negate (times_comm y x).

  Lemma dist_times_plus x y : forall z, times x (plus y z) = plus (times x y) (times x z).
  Proof.
    simple refine (Z_ind _ _ _ _ _ _)
    ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros Sz HSz.
      refine (ap (plus x) HSz @ _).
      refine ((plus_assoc _ _ _)^ @ _).
      refine (_ @ plus_assoc _ _ _).
      refine (ap (fun z => plus z (times x Sz)) (plus_comm _ _)).
    - intros Pz HPz.
      refine (ap (fun z => minus z x) HPz @ _).
      unfold minus ; simpl.
      apply plus_assoc.
  Defined.

  Lemma dist_plus_times x y z : times (plus x y) z = plus (times x z) (times y z).
  Proof.
    refine (times_comm _ _ @ _).
    refine (dist_times_plus _ _ _ @ _).
    refine (ap (plus (times z x)) (times_comm _ _) @ _).
    apply (ap (fun a => plus a (times y z)) (times_comm _ _)).
  Defined.
  
  Lemma times_assoc x y : forall z, times (times x y) z = times x (times y z).
  Proof.
    simple refine (Z_ind _ _ _ _ _ _)
    ; simpl ; try (intros ; apply set_path2).
    - reflexivity.
    - intros Sz HSz.
      refine (ap (plus (times x y)) HSz @ _).
      symmetry ; apply dist_times_plus.
    - intros Pz HPz.
      refine (ap (fun z => minus z (times x y)) HPz @ _).
      unfold minus.
      refine (_ @ (dist_times_plus _ _ _)^).
      refine (ap (plus (times x (times y Pz))) _).
      apply (times_negatex _ _)^.
  Defined.

End ring_Z.

(*
Definition Z_to_S : Z -> S1.
Proof.
  refine (Z_rec _ _ _ _ _ _).
  Unshelve. 

  Focus 1.
  apply base.

  Focus 3.
  intro x.
  apply x.

  Focus 3.
  intro x.
  apply x.

  Focus 1.
  intro m.
  compute.
  reflexivity.

  refine (S1_ind _ _ _).
  Unshelve.

  Focus 2.
  apply loop.

  rewrite @HoTT.Types.Paths.transport_paths_FlFr.
  rewrite ap_idmap.
  rewrite concat_Vp.
  rewrite concat_1p.
  reflexivity.
Defined.

Lemma eq1 : ap Z_to_S (inv1 (pred (succ (pred nul)))) = reflexivity base.
Proof.
  rewrite Z_rec_beta_inv1.
  reflexivity.
Qed.


Lemma eq2 : ap Z_to_S (ap pred (inv2 (succ (pred nul)))) = loop.
Proof.
  rewrite <- ap_compose.
  enough (forall (n m : Z) (p : n = m), ap Z_to_S p = ap (fun n : Z => Z_to_S(pred n)) p).
  Focus 2.
  compute.
  reflexivity.

  rewrite Z_rec_beta_inv2.
  compute.
  reflexivity.
Qed.

Module Export AltInts.

  Private Inductive Z' : Type0 :=
  | positive : nat -> Z'
  | negative : nat -> Z'.

  Axiom path : positive 0 = negative 0.

  Fixpoint Z'_ind
           (P : Z' -> Type)
           (po : forall (x : nat), P (positive x))
           (ne : forall (x : nat), P (negative x))
           (i : path # (po 0) = ne 0)
           (x : Z')
           {struct x}
    : P x
    := 
      (match x return _ -> P x with
       | positive n => fun _ => po n
       | negative n => fun _ => ne n
       end) i.

  Axiom Z'_ind_beta_inv1 : forall
      (P : Z' -> Type)
      (po : forall (x : nat), P (positive x))
      (ne : forall (x : nat), P (negative x))
      (i : path # (po 0) = ne 0)
    , apD (Z'_ind P po ne i) path = i.
End AltInts.

Definition succ_Z' : Z' -> Z'.
Proof.
  refine (Z'_ind _ _ _ _).
  Unshelve.
  Focus 2.
  intro n.
  apply (positive (S n)).

  Focus 2.
  intro n.
  induction n.
  apply (positive 1).

  apply (negative n).

  simpl.
  rewrite transport_const.
  reflexivity.
Defined.

Definition pred_Z' : Z' -> Z'.
Proof.
  refine (Z'_ind _ _ _ _).
  Unshelve.
  Focus 2.
  intro n.
  induction n.
  apply (negative 1).

  apply (positive n).

  Focus 2.
  intro n.
  apply (negative (S n)).

  simpl.
  rewrite transport_const.
  reflexivity.
Defined.


Fixpoint Nat_to_Pos (n : nat) : Pos :=
  match n with
  | 0 => Int.one
  | S k => succ_pos (Nat_to_Pos k)
  end.

Definition Z'_to_Int : Z' -> Int.
Proof.
  refine (Z'_ind _ _ _ _).
  Unshelve.

  Focus 2.
  intro x.
  induction x.
  apply (Int.zero).
  apply (succ_int IHx).

  Focus 2.
  intro x.
  induction x.
  apply (Int.zero).
  apply (pred_int IHx).

  simpl.
  rewrite transport_const.
  reflexivity.
Defined.

Definition Pos_to_Nat : Pos -> nat.
Proof.
  intro x.
  induction x.
  apply 1.
  apply (S IHx).
Defined.


Definition Int_to_Z' (x : Int) : Z'.
Proof.
  induction x.
  apply (negative (Pos_to_Nat p)).
  apply (positive 0).
  apply (positive (Pos_to_Nat p)).
Defined.

Lemma Z'_to_int_pos_homomorphism : 
  forall n : nat, Z'_to_Int (positive (S n)) = succ_int (Z'_to_Int (positive n)).
Proof.
  intro n.
  reflexivity.
Qed.

Lemma Z'_to_int_neg_homomorphism : 
  forall n : nat, Z'_to_Int (negative (S n)) = pred_int (Z'_to_Int (negative n)).
Proof.
  intro n.
  reflexivity.
Qed.

Theorem isoEq1 : forall x : Int, Z'_to_Int(Int_to_Z' x) = x.
Proof.
  intro x.
  induction x.
  induction p.
  reflexivity.

  rewrite Z'_to_int_neg_homomorphism.
  rewrite IHp.
  reflexivity.

  reflexivity.

  induction p.
  reflexivity.

  rewrite Z'_to_int_pos_homomorphism.
  rewrite IHp.
  reflexivity.
Defined.

Lemma Int_to_Z'_succ_homomorphism :
  forall x : Int, Int_to_Z' (succ_int x) = succ_Z' (Int_to_Z' x).
Proof.
  simpl.
  intro x.
  simpl.
  induction x.
  induction p.
  compute.
  apply path.

  reflexivity.

  reflexivity.

  induction p.
  reflexivity.

  reflexivity.
Qed.

Lemma Int_to_Z'_pred_homomorphism :
  forall x : Int, Int_to_Z' (pred_int x) = pred_Z' (Int_to_Z' x).
Proof.
  intro x.
  induction x.
  induction p.
  reflexivity.

  reflexivity.

  reflexivity.

  induction p.
  reflexivity.

  reflexivity.

Qed.

Theorem isoEq2 : forall x : Z', Int_to_Z'(Z'_to_Int x) = x.
Proof.
  refine (Z'_ind _ _ _ _).
  Unshelve.

  Focus 2.
  intro x.
  induction x.
  reflexivity.
  rewrite Z'_to_int_pos_homomorphism.
  rewrite Int_to_Z'_succ_homomorphism.
  rewrite IHx.
  reflexivity.

  Focus 2.
  intro x.
  induction x.
  apply path.
  rewrite Z'_to_int_neg_homomorphism.
  rewrite Int_to_Z'_pred_homomorphism.
  rewrite IHx.
  reflexivity.

  simpl.
  rewrite @HoTT.Types.Paths.transport_paths_FlFr.
  rewrite concat_p1.
  rewrite ap_idmap.

  enough (ap (fun x : Z' => Z'_to_Int x) path = reflexivity Int.zero).
  rewrite ap_compose.
  rewrite X.
  apply concat_1p.

  apply axiomK_hset.
  apply hset_int.
Defined.

Theorem adj : 
  forall x : Z', isoEq1 (Z'_to_Int x) = ap Z'_to_Int (isoEq2 x).
Proof.
  intro x.
  apply hset_int.
Defined.

Definition isomorphism : IsEquiv Z'_to_Int.
Proof.
  apply (BuildIsEquiv Z' Int Z'_to_Int Int_to_Z' isoEq1 isoEq2 adj).
Qed.

Axiom everythingSet : forall T : Type, IsHSet T.

Definition Z_to_Z' : Z -> Z'.
Proof.
  refine (Z_rec _ _ _ _ _ _).
  Unshelve.
  Focus 1.
  apply (positive 0).

  Focus 3.
  apply succ_Z'.

  Focus 3.
  apply pred_Z'.

  Focus 1.
  refine (Z'_ind _ _ _ _).
  Unshelve.
  Focus 2.
  intros.
  reflexivity.

  Focus 2.
  intros.
  induction x.
  Focus 1.
  compute.
  apply path^.

  reflexivity.

  apply everythingSet.

  refine (Z'_ind _ _ _ _).
  Unshelve.
  Focus 2.
  intros.
  induction x.
  Focus 1.
  compute.
  apply path.

  reflexivity.

  Focus 2.
  intros.
  reflexivity.

  apply everythingSet.

Defined.

Definition Z'_to_Z : Z' -> Z.
Proof.
  refine (Z'_ind _ _ _ _).
  Unshelve.
  Focus 2.
  induction 1.
  apply nul.

  apply (succ IHx).

  Focus 2.
  induction 1.
  Focus 1.
  apply nul.

  apply (pred IHx).

  simpl.
  rewrite transport_const.
  reflexivity.

Defined.

Theorem isoZEq1 : forall n : Z', Z_to_Z'(Z'_to_Z n) = n.
Proof.
  refine (Z'_ind _ _ _ _).
  Unshelve.
  Focus 3.
  intros.
  induction x.
  compute.
  apply path.

  transitivity (Z_to_Z' (pred (Z'_to_Z (negative x)))).
  enough (Z'_to_Z (negative x.+1) = pred (Z'_to_Z (negative x))).
  rewrite X.
  reflexivity.

  reflexivity.

  transitivity (pred_Z' (Z_to_Z' (Z'_to_Z (negative x)))).
  Focus 1.
  reflexivity.

  rewrite IHx.
  reflexivity.

  Focus 2.
  intros.
  induction x.
  Focus 1.
  reflexivity.

  transitivity (Z_to_Z' (succ (Z'_to_Z (positive x)))).
  reflexivity.

  transitivity (succ_Z' (Z_to_Z' (Z'_to_Z (positive x)))).
  reflexivity.

  rewrite IHx.
  reflexivity.

  apply everythingSet.
Defined.

Theorem isoZEq2 : forall n : Z, Z'_to_Z(Z_to_Z' n) = n.
Proof.
  refine (Z_ind _ _ _ _ _ _).
  Unshelve.
  Focus 1.
  reflexivity.

  Focus 1.
  intros.
  apply everythingSet.

  Focus 1.
  intros.
  apply everythingSet.

  Focus 1.
  intro n.
  intro X.
  transitivity (Z'_to_Z (succ_Z' (Z_to_Z' n))).
  reflexivity.

  transitivity (succ (Z'_to_Z (Z_to_Z' n))).
  Focus 2.
  rewrite X.
  reflexivity.

  enough (forall m : Z', Z'_to_Z (succ_Z' m) = succ (Z'_to_Z m)).
  rewrite X0.
  reflexivity.

  refine (Z'_ind _ _ _ _).
  Unshelve.
  Focus 2.
  intros.
  reflexivity.

  Focus 2.
  intros.
  induction x.
  Focus 1.
  reflexivity.

  compute.
  rewrite <- inv2.
  reflexivity.

  apply everythingSet.

  intros.
  transitivity (Z'_to_Z (pred_Z' (Z_to_Z' n))).
  reflexivity.

  transitivity (pred (Z'_to_Z (Z_to_Z' n))).
  Focus 2.
  rewrite X.
  reflexivity.

  enough (forall m : Z', Z'_to_Z (pred_Z' m) = pred (Z'_to_Z m)).
  rewrite X0.
  reflexivity.

  refine (Z'_ind _ _ _ _).
  Unshelve.
  Focus 1.
  apply everythingSet.

  Focus 1.
  intro x.
  induction x.
  reflexivity.

  compute.
  rewrite <- inv1.
  reflexivity.

  intro x.
  reflexivity.
Defined.

Theorem adj2 : 
  forall x : Z', isoZEq2 (Z'_to_Z x) = ap Z'_to_Z (isoZEq1 x).
Proof.
  intro x.
  apply everythingSet.
Defined.

Definition isomorphism2 : IsEquiv Z'_to_Z.
Proof.
  apply (BuildIsEquiv Z' Z Z'_to_Z Z_to_Z' isoZEq2 isoZEq1 adj2).
Qed.
*)