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Test_Convert.v
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Test_Convert.v
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Require Import Coq.Lists.List.
Require Import Coq.Strings.String.
Require Import StaticSemA.
Require Import IdModTypeA.
Require Import ModNat1A.
Require Import AbbrevA.
Require Import Eqdep FunctionalExtensionality Coq.Program.Tactics.
Import ListNotations.
Open Scope string_scope.
Module Convert2.
Module LM := Abbrev ModNat1.
Export LM.
Definition Id := LM.Id.
Definition IdEqDec := LM.IdEqDec.
Definition IdEq := LM.IdEq.
Definition W := LM.W.
Definition Loc_PI := LM.Loc_PI.
Definition BInit := LM.BInit.
Definition WP := LM.WP.
(************************************************************************)
(* syntax from Pip *)
Definition state : Type := nat.
Inductive result (A : Type) : Type :=
| val : A -> result A
| undef : nat -> state-> result A.
Definition LLI (A :Type) : Type := state -> result (A * state).
Arguments val [ A ].
Arguments undef [ A ].
Definition ret {A : Type} (a : A) : LLI A :=
fun s => val (a , s) .
Definition bind {A B : Type} (m : LLI A)(f : A -> LLI B) : LLI B :=
fun s => match m s with
| val (a, s') => f a s'
| undef a s' => undef a s'
end.
Definition put (s : state) : LLI unit :=
fun _ => val (tt, s).
Definition get : LLI state :=
fun s => val (s, s).
Definition undefined {A : Type} (code : nat ): LLI A :=
fun s => undef code s.
Definition run {A : Type} (m : LLI A) (s : state) : option A :=
match m s with
|undef _ _=> None
| val (a, _) => Some a
end.
Notation "'perform' x ':=' m 'in' e" := (bind m (fun x => e))
(at level 60, x ident, m at level 200, e at level 60,
format "'[v' '[' 'perform' x ':=' m 'in' ']' '/' '[' e ']' ']'")
: state_scope.
Notation "m1 ;; m2" := (bind m1 (fun _ => m2))
(at level 60, right associativity) : state_scope.
Definition page := nat.
Definition next (x : LLI nat) := bind x (fun x => ret (S x)).
Open Scope state_scope.
(**********************************************************************)
(* shallow representation of recursive function using 'iterate' *)
Fixpoint iterate {A B : Type} (f0 : A -> LLI B)
(f1: (A -> LLI B) -> (A -> LLI B))
(n: nat) (a: A) : LLI B :=
match n with
| 0 => f0 a
| S n' => f1 (iterate f0 f1 n') a
end.
(* example 1 *)
Fixpoint fact (n: nat) : nat :=
match n with 0 => 1 | (S m) => (S m) * fact m end.
Definition fact0 := fun n:nat => 1.
Definition fact1 := fun (f: nat -> nat) (n: nat) =>
match n with 0 => 1 | (S m) => (S m) * f m end.
Definition factM0 := fun n:nat => ret 1.
Definition factM1 := fun (f: nat -> LLI nat) (n: nat) =>
match n with 0 => ret 1 | (S m) =>
bind (f m) (fun x => ret (x * (S m))) end.
Definition factM (n: nat) := iterate factM0 factM1 n n.
(* example 2 *)
Fixpoint lengthN (ls: list nat) : nat :=
match ls with [] => 0 | h::tl => 1 + lengthN tl end.
Definition lengthN0 := fun ls: list nat => 0.
Definition lengthN1 := fun (f: list nat -> nat) (ls: list nat) =>
match ls with [] => 0 | h::tl => 1 + f tl end.
Definition lengthNM0 := fun ls: list nat => ret 0.
Definition lengthNM1 := fun (f: list nat -> LLI nat) (ls: list nat) =>
match ls with [] => ret 0 | h::tl =>
bind (f tl) (fun x => ret (x + 1)) end.
Definition lengthNM (n: nat) (ls: list nat) :=
iterate lengthNM0 lengthNM1 n ls.
(* example 3 *)
Fixpoint dummyRec2 timeout (arg : nat) : LLI nat :=
match timeout with
| 0 => ret arg
| S timeout1 =>
bind (ret (S arg)) (dummyRec2 timeout1)
end.
Definition dummyRecI0 := fun n:nat => ret n.
Definition dummyRecI1 := fun (f: nat -> LLI nat) (n: nat) =>
bind (ret (S n)) f.
Definition dummyRecI timeout (arg : nat) : LLI nat :=
iterate dummyRecI0 dummyRecI1 timeout arg.
Lemma dummyEq (timeout arg : nat) :
dummyRec2 timeout arg = dummyRecI timeout arg.
revert arg.
induction timeout.
auto.
intros.
simpl.
simpl.
unfold bind.
simpl.
eapply functional_extensionality_dep.
rewrite IHtimeout.
intros.
reflexivity.
Qed.
(************************************************************************)
(* Pip function definition *)
(* original definition *)
Fixpoint dummyRec timeout (arg : nat) : LLI nat :=
match timeout with
| 0 => ret arg
| S timeout1 =>
perform n := next (ret arg) in
dummyRec timeout1 n
end.
(* desugared definition *)
Fixpoint dummyRec1 timeout (arg : nat) : LLI nat :=
match timeout with
| 0 => ret arg
| S timeout1 =>
bind (ret (S arg)) (dummyRec1 timeout1)
end.
(**************************************************************************)
(* translation *)
Definition dummyRecD0 (x: Id) : Exp := VLift (Var x).
Definition dummyRecD1 (x: Id) (y: Id) (n: nat) : Exp :=
BindS x (Val (cst nat (S n))) (Apply (FVar y) (PS [VLift (Var x)])).
Definition dummyRecD (arg timeout: nat) : Fun := FC
nil
[("arg", Nat)]
(dummyRecD0 "arg")
(dummyRecD1 "arg" "dummyRec" arg)
"dummyRec"
timeout.
(* alternatives *)
(* 1 *)
Definition dummyRecD0x (n: nat) : Exp := Val (cst nat n).
Definition dummyRecD1x (y: Id) (n: nat) : Exp :=
Apply (FVar y) (PS [(Val (cst nat (S n)))]).
Definition dummyRecDx (arg timeout: nat) : Fun := FC
nil
[("arg", Nat)]
(dummyRecD0x arg)
(dummyRecD1x "dummyRec" arg)
"dummyRec"
timeout.
Definition dummyRecDxx (arg: nat) : Fun := FC
nil
[("arg", Nat)]
(dummyRecD0x arg)
(dummyRecD1x "dummyRec" arg)
"dummyRec"
arg.
(* 2 *)
Definition dummyRecD0v (n: ValueI nat) : Exp := Val (existT ValueI nat n).
Definition dummyRecD1v (y: Id) (n: ValueI nat) : Exp :=
match n with Cst _ v =>
Apply (FVar y) (PS [(Val (cst nat (S v)))]) end.
Definition dummyRecDv (arg: ValueI nat) (timeout: nat) : Fun := FC
nil
[("arg", Nat)]
(dummyRecD0v arg)
(dummyRecD1v "dummyRec" arg)
"dummyRec"
timeout.
Definition dummyRecDvv (arg: ValueI nat) : Fun := FC
nil
[("arg", Nat)]
(dummyRecD0v arg)
(dummyRecD1v "dummyRec" arg)
"dummyRec"
(ValueI2T nat arg).
(************************************************************************)
(* Essayons de convertir la fonction suivante, de type : page -> LLI page *)
(* Definition getPd partition := *)
(* perform idxPD := getPDidx in *)
(* perform idx := MALInternal.Index.succ idxPD in *)
(* readPhysical partition idx. *)
(* sample manual translation from Pip *)
Definition PartitionType := vtyp nat.
Definition IdxType := vtyp nat.
(*** to be implemented using Modify *)
Variable IndexSuccImpl : QValue -> Exp.
Variable ReadPhysicalImpl : QValue -> QValue -> Exp.
(***)
Definition IndexSucc : Fun := FC
nil
[("i_arg", IdxType)]
(ReadPhysicalImpl (Var "p_arg")
(Var "i_arg"))
(Val (cst nat 0))
("ReadPhysical")
0.
Definition ReadPhysical : Fun := FC
nil
[("p_arg", PartitionType);("i_arg", IdxType)]
(ReadPhysicalImpl (Var "p_arg")
(Var "i_arg"))
(Val (cst nat 0))
"ReadPhysical"
0.
Definition PerformIdxInReadPhysical : Exp :=
BindS "idxPD"
(VLift (Var "getPDix"))
(BindS "idx"
(Apply (FVar "IndexSucc")
(PS [(VLift (Var "idxPD"))]))
(Apply (QF ReadPhysical)
(PS [(VLift (Var "partition"));
(VLift (Var "idx"))]))).
Definition getPd : Fun := FC
[("ReadPhysical", ReadPhysical); ("IndexSucc", IndexSucc)]
[("partition", PartitionType)]
PerformIdxInReadPhysical
(Val (cst nat 0))
"getPd"
0.
(**********************************************************************)
(* simpler example - typing a recursive function *)
Definition dRecD0 (x: Id): Exp := Return LL (Var x).
Definition dRecD1 (x rec: Id): Exp :=
Apply (FVar rec) (PS [Return LL (Var x)]).
Definition dRecD (timeout: nat) : Fun := FC nil
[("arg", Nat)]
(dRecD0 "arg")
(dRecD1 "arg" "dRecD")
"dRecD"
timeout.
Definition dRecFT : FTyp := FT [("arg", Nat)] Nat.
Definition dRecPT : PTyp := PT [Nat].
Definition dApp : Exp :=
Apply (QF (dRecD 2)) (PS [Val (cst nat 2)]).
Definition expTypingTest (e: Exp) (t: VTyp): Type :=
ExpTyping emptyE emptyE emptyE e t.
Definition expTypingTestA (ftenv: funTC) (tenv: valTC) (fenv: funEnv)
(e: Exp) (t: VTyp): Type :=
ExpTyping ftenv tenv fenv e t.
Lemma expTypingTestDAppA (ftenv: funTC) (tenv: valTC) (fenv: funEnv)
(k1: MatchEnvsT FunTyping fenv ftenv) :
expTypingTestA ftenv tenv fenv dApp Nat.
unfold expTypingTestA.
unfold dApp.
econstructor.
(* pt = env2ptyp fps *)
instantiate (2:= dRecPT).
instantiate (1:= [("arg",Nat)]).
auto.
apply k1.
(* QFunTyping 2 *)
constructor.
econstructor.
econstructor.
(* dRecD1 1 *)
econstructor.
(* pt = env2ptyp fps *)
instantiate (2:= dRecPT).
instantiate (1:= [("arg",Nat)]).
auto.
constructor.
(* FunTyping 1 *)
econstructor.
econstructor.
(* dRecD1 0 *)
econstructor.
(* pt = env2ptyp fps *)
instantiate (2:= dRecPT).
instantiate (1:= [("arg",Nat)]).
auto.
econstructor.
(* FunTyping 0 *)
econstructor.
econstructor.
repeat constructor.
constructor.
(* QFunTyping 0 *)
econstructor.
econstructor.
instantiate (1:=dRecD 0).
econstructor.
repeat econstructor.
repeat econstructor.
instantiate (1:=emptyE).
econstructor.
instantiate (1:=emptyE).
econstructor.
repeat econstructor.
(* update 0 *)
simpl.
unfold updateE.
unfold dRecD.
auto.
auto.
(* PrmsTyping *)
econstructor.
constructor.
constructor.
repeat constructor.
constructor.
(* FunTyping 0 *)
econstructor.
econstructor.
repeat constructor.
constructor.
(* QFunTyping 1 *)
econstructor.
econstructor 1 with (ls1:=nil) (ls3:=nil) (ls2:=nil) (ls4:=nil).
instantiate (1:=dRecD 1).
econstructor.
econstructor.
econstructor.
instantiate (1:=[("arg", Nat)]).
reflexivity.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
simpl.
reflexivity.
econstructor.
econstructor.
instantiate (1:=dRecD 0).
econstructor 1 with (ls1:=nil) (ls3:=nil) (ls2:=nil) (ls4:=nil).
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
econstructor.
simpl.
reflexivity.
econstructor.
econstructor.
simpl.
reflexivity.
econstructor.
simpl.
reflexivity.
(* PrmsTyping *)
econstructor.
constructor.
constructor.
repeat constructor.
constructor.
(* FunTyping 0 *)
econstructor.
econstructor.
repeat constructor.
constructor.
econstructor.
simpl.
reflexivity.
(* PrmsTyping *)
econstructor.
constructor.
constructor.
repeat constructor.
(* FunTyping 1 *)
econstructor.
econstructor.
(* dRecD1 *)
econstructor.
(* pt = env2ptyp fps *)
instantiate (2:= dRecPT).
instantiate (1:= [("arg",Nat)]).
auto.
repeat constructor.
econstructor.
econstructor.
repeat constructor.
econstructor.
econstructor.
instantiate (1:=dRecD 0).
econstructor.
econstructor.
repeat constructor.
instantiate (1:=nil).
econstructor.
instantiate (1:=nil).
econstructor.
constructor.
repeat constructor.
repeat constructor.
(* PrmsTyping *)
econstructor.
constructor.
constructor.
repeat constructor.
constructor.
(* FunTyping 0 *)
econstructor.
econstructor.
repeat constructor.
constructor.
repeat constructor.
Defined.
(*
Definition succIndexInternal (idx: index) : Exp :=
BindS i (extract1 idx) (BindS P (extract2 idx))
(IfThenElse (Lt_dec i) (apply DSome
(Apply IndexSucc (Var i))) DNone) .
*)
(*
Definition plusR_X (n:nat) := QF (FC emptyE [("i",Nat)]
(VLift (Var "i"))
(SuccR_X (Apply (FVar "plusR") (PS [VLift (Var "i")])))
"plusR" n).
Definition plusX := FC emptyE [("i",Nat),("j",Nat)]
(Val 0)
(IfThenElse (EEqual 0 "j")
(VLift (Var "i"))
(Apply (FVar "plusR") (PS [VLift (Var "i"), PredR "j"]))).
*)
End Convert2.
(*
(* il faudrait bien commencer par définir l’ensemble des identifiants possibles ?
est-ce que tricher avec un type comme string serait faisable ? *)
Inductive Id : Type := partition | idxPD | idx.
(* Mais je n’arrive pas à créer de valeur de type TPipStatic20.Id, qui
est le type attendu, est-ce normal ? *)
(* comment se fait qu’About Fun n’indique pas de dépendance à un type Id ? *)
Definition corps : Exp.
Admitted.
Definition getPdFun : Fun := FC nil nil corps corps partition 0.
(* FC : pas d’autre choix de toute façon *)
(* [] : pas d’environnements, si ? que pourraient-ils contenir ? *)
(* [] : des constantes, comme par exemple la valeur de N ? *)
(* corps *)
(* corps : ici, il n’est pas nécessaire de faire une différence entre les cas 0 et 1, si ? *)
(* partition : c’est bien l’argument ? *)
(* 0. *)
*)