adamc@170: (* Copyright (c) 2008, Adam Chlipala
adamc@170:  * 
adamc@170:  * This work is licensed under a
adamc@170:  * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@170:  * Unported License.
adamc@170:  * The license text is available at:
adamc@170:  *   http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@170:  *)
adamc@170: 
adamc@170: (* begin hide *)
adamc@170: Require Import String List.
adamc@170: 
adamc@170: Require Import Axioms Tactics.
adamc@170: 
adamc@170: Set Implicit Arguments.
adamc@170: (* end hide *)
adamc@170: 
adamc@170: 
adamc@170: (** %\chapter{Type-Theoretic Interpreters}% *)
adamc@170: 
adamc@170: (** TODO: Prose for this chapter *)
adamc@170: 
adamc@170: 
adamc@170: (** * Simply-Typed Lambda Calculus *)
adamc@170: 
adamc@170: Module STLC.
adamc@170:   Inductive type : Type :=
adamc@170:   | Nat : type
adamc@170:   | Arrow : type -> type -> type.
adamc@170: 
adamc@170:   Infix "-->" := Arrow (right associativity, at level 60).
adamc@170: 
adamc@170:   Section vars.
adamc@170:     Variable var : type -> Type.
adamc@170: 
adamc@170:     Inductive exp : type -> Type :=
adamc@170:     | Var : forall t,
adamc@170:       var t
adamc@170:       -> exp t
adamc@170: 
adamc@170:     | Const : nat -> exp Nat
adamc@170:     | Plus : exp Nat -> exp Nat -> exp Nat
adamc@170: 
adamc@170:     | App : forall t1 t2,
adamc@170:       exp (t1 --> t2)
adamc@170:       -> exp t1
adamc@170:       -> exp t2
adamc@170:     | Abs : forall t1 t2,
adamc@170:       (var t1 -> exp t2)
adamc@170:       -> exp (t1 --> t2).
adamc@170:   End vars.
adamc@170: 
adamc@170:   Definition Exp t := forall var, exp var t.
adamc@170: 
adamc@170:   Implicit Arguments Var [var t].
adamc@170:   Implicit Arguments Const [var].
adamc@170:   Implicit Arguments Plus [var].
adamc@170:   Implicit Arguments App [var t1 t2].
adamc@170:   Implicit Arguments Abs [var t1 t2].
adamc@170: 
adamc@170:   Notation "# v" := (Var v) (at level 70).
adamc@170: 
adamc@170:   Notation "^ n" := (Const n) (at level 70).
adamc@171:   Infix "+^" := Plus (left associativity, at level 79).
adamc@170: 
adamc@170:   Infix "@" := App (left associativity, at level 77).
adamc@170:   Notation "\ x , e" := (Abs (fun x => e)) (at level 78).
adamc@170:   Notation "\ ! , e" := (Abs (fun _ => e)) (at level 78).
adamc@170: 
adamc@170:   Fixpoint typeDenote (t : type) : Set :=
adamc@170:     match t with
adamc@170:       | Nat => nat
adamc@170:       | t1 --> t2 => typeDenote t1 -> typeDenote t2
adamc@170:     end.
adamc@170: 
adamc@170:   Fixpoint expDenote t (e : exp typeDenote t) {struct e} : typeDenote t :=
adamc@170:     match e in (exp _ t) return (typeDenote t) with
adamc@170:       | Var _ v => v
adamc@170:         
adamc@170:       | Const n => n
adamc@170:       | Plus e1 e2 => expDenote e1 + expDenote e2
adamc@170:         
adamc@170:       | App _ _ e1 e2 => (expDenote e1) (expDenote e2)
adamc@170:       | Abs _ _ e' => fun x => expDenote (e' x)
adamc@170:     end.
adamc@170: 
adamc@170:   Definition ExpDenote t (e : Exp t) := expDenote (e _).
adamc@170: 
adamc@170:   Section cfold.
adamc@170:     Variable var : type -> Type.
adamc@170: 
adamc@170:     Fixpoint cfold t (e : exp var t) {struct e} : exp var t :=
adamc@170:       match e in exp _ t return exp _ t with
adamc@170:         | Var _ v => #v
adamc@170: 
adamc@170:         | Const n => ^n
adamc@170:         | Plus e1 e2 =>
adamc@170:           let e1' := cfold e1 in
adamc@170:           let e2' := cfold e2 in
adamc@170:           match e1', e2' with
adamc@170:             | Const n1, Const n2 => ^(n1 + n2)
adamc@171:             | _, _ => e1' +^ e2'
adamc@170:           end
adamc@170: 
adamc@170:         | App _ _ e1 e2 => cfold e1 @ cfold e2
adamc@170:         | Abs _ _ e' => Abs (fun x => cfold (e' x))
adamc@170:       end.
adamc@170:   End cfold.
adamc@170: 
adamc@170:   Definition Cfold t (E : Exp t) : Exp t := fun _ => cfold (E _).
adamc@170: 
adamc@170:   Lemma cfold_correct : forall t (e : exp _ t),
adamc@170:     expDenote (cfold e) = expDenote e.
adamc@170:     induction e; crush; try (ext_eq; crush);
adamc@170:       repeat (match goal with
adamc@170:                 | [ |- context[cfold ?E] ] => dep_destruct (cfold E)
adamc@170:               end; crush).
adamc@170:   Qed.
adamc@170: 
adamc@170:   Theorem Cfold_correct : forall t (E : Exp t),
adamc@170:     ExpDenote (Cfold E) = ExpDenote E.
adamc@170:     unfold ExpDenote, Cfold; intros; apply cfold_correct.
adamc@170:   Qed.
adamc@170: End STLC.
adamc@171: 
adamc@171: 
adamc@171: (** * Adding Products and Sums *)
adamc@171: 
adamc@171: Module PSLC.
adamc@171:   Inductive type : Type :=
adamc@171:   | Nat : type
adamc@171:   | Arrow : type -> type -> type
adamc@171:   | Prod : type -> type -> type
adamc@171:   | Sum : type -> type -> type.
adamc@171: 
adamc@171:   Infix "-->" := Arrow (right associativity, at level 62).
adamc@171:   Infix "**" := Prod (right associativity, at level 61).
adamc@171:   Infix "++" := Sum (right associativity, at level 60).
adamc@171: 
adamc@171:   Section vars.
adamc@171:     Variable var : type -> Type.
adamc@171: 
adamc@171:     Inductive exp : type -> Type :=
adamc@171:     | Var : forall t,
adamc@171:       var t
adamc@171:       -> exp t
adamc@171: 
adamc@171:     | Const : nat -> exp Nat
adamc@171:     | Plus : exp Nat -> exp Nat -> exp Nat
adamc@171: 
adamc@171:     | App : forall t1 t2,
adamc@171:       exp (t1 --> t2)
adamc@171:       -> exp t1
adamc@171:       -> exp t2
adamc@171:     | Abs : forall t1 t2,
adamc@171:       (var t1 -> exp t2)
adamc@171:       -> exp (t1 --> t2)
adamc@171: 
adamc@171:     | Pair : forall t1 t2,
adamc@171:       exp t1
adamc@171:       -> exp t2
adamc@171:       -> exp (t1 ** t2)
adamc@171:     | Fst : forall t1 t2,
adamc@171:       exp (t1 ** t2)
adamc@171:       -> exp t1
adamc@171:     | Snd : forall t1 t2,
adamc@171:       exp (t1 ** t2)
adamc@171:       -> exp t2
adamc@171: 
adamc@171:     | Inl : forall t1 t2,
adamc@171:       exp t1
adamc@171:       -> exp (t1 ++ t2)
adamc@171:     | Inr : forall t1 t2,
adamc@171:       exp t2
adamc@171:       -> exp (t1 ++ t2)
adamc@171:     | SumCase : forall t1 t2 t,
adamc@171:       exp (t1 ++ t2)
adamc@171:       -> (var t1 -> exp t)
adamc@171:       -> (var t2 -> exp t)
adamc@171:       -> exp t.
adamc@171:   End vars.
adamc@171: 
adamc@171:   Definition Exp t := forall var, exp var t.
adamc@171: 
adamc@171:   Implicit Arguments Var [var t].
adamc@171:   Implicit Arguments Const [var].
adamc@171:   Implicit Arguments Abs [var t1 t2].
adamc@171:   Implicit Arguments Inl [var t1 t2].
adamc@171:   Implicit Arguments Inr [var t1 t2].
adamc@171: 
adamc@171:   Notation "# v" := (Var v) (at level 70).
adamc@171: 
adamc@171:   Notation "^ n" := (Const n) (at level 70).
adamc@171:   Infix "+^" := Plus (left associativity, at level 79).
adamc@171: 
adamc@171:   Infix "@" := App (left associativity, at level 77).
adamc@171:   Notation "\ x , e" := (Abs (fun x => e)) (at level 78).
adamc@171:   Notation "\ ! , e" := (Abs (fun _ => e)) (at level 78).
adamc@171: 
adamc@171:   Notation "[ e1 , e2 ]" := (Pair e1 e2).
adamc@171:   Notation "#1 e" := (Fst e) (at level 75).
adamc@171:   Notation "#2 e" := (Snd e) (at level 75).
adamc@171: 
adamc@171:   Notation "'case' e 'of' x => e1 | y => e2" := (SumCase e (fun x => e1) (fun y => e2))
adamc@171:     (at level 79).
adamc@171: 
adamc@171:   Fixpoint typeDenote (t : type) : Set :=
adamc@171:     match t with
adamc@171:       | Nat => nat
adamc@171:       | t1 --> t2 => typeDenote t1 -> typeDenote t2
adamc@171:       | t1 ** t2 => typeDenote t1 * typeDenote t2
adamc@171:       | t1 ++ t2 => typeDenote t1 + typeDenote t2
adamc@171:     end%type.
adamc@171: 
adamc@171:   Fixpoint expDenote t (e : exp typeDenote t) {struct e} : typeDenote t :=
adamc@171:     match e in (exp _ t) return (typeDenote t) with
adamc@171:       | Var _ v => v
adamc@171:         
adamc@171:       | Const n => n
adamc@171:       | Plus e1 e2 => expDenote e1 + expDenote e2
adamc@171:         
adamc@171:       | App _ _ e1 e2 => (expDenote e1) (expDenote e2)
adamc@171:       | Abs _ _ e' => fun x => expDenote (e' x)
adamc@171: 
adamc@171:       | Pair _ _ e1 e2 => (expDenote e1, expDenote e2)
adamc@171:       | Fst _ _ e' => fst (expDenote e')
adamc@171:       | Snd _ _ e' => snd (expDenote e')
adamc@171: 
adamc@171:       | Inl _ _ e' => inl _ (expDenote e')
adamc@171:       | Inr _ _ e' => inr _ (expDenote e')
adamc@171:       | SumCase _ _ _ e' e1 e2 =>
adamc@171:         match expDenote e' with
adamc@171:           | inl v => expDenote (e1 v)
adamc@171:           | inr v => expDenote (e2 v)
adamc@171:         end
adamc@171:     end.
adamc@171: 
adamc@171:   Definition ExpDenote t (e : Exp t) := expDenote (e _).
adamc@171: 
adamc@171:   Section cfold.
adamc@171:     Variable var : type -> Type.
adamc@171: 
adamc@171:     Fixpoint cfold t (e : exp var t) {struct e} : exp var t :=
adamc@171:       match e in exp _ t return exp _ t with
adamc@171:         | Var _ v => #v
adamc@171: 
adamc@171:         | Const n => ^n
adamc@171:         | Plus e1 e2 =>
adamc@171:           let e1' := cfold e1 in
adamc@171:           let e2' := cfold e2 in
adamc@171:           match e1', e2' with
adamc@171:             | Const n1, Const n2 => ^(n1 + n2)
adamc@171:             | _, _ => e1' +^ e2'
adamc@171:           end
adamc@171: 
adamc@171:         | App _ _ e1 e2 => cfold e1 @ cfold e2
adamc@171:         | Abs _ _ e' => Abs (fun x => cfold (e' x))
adamc@171: 
adamc@171:         | Pair _ _ e1 e2 => [cfold e1, cfold e2]
adamc@171:         | Fst _ _ e' => #1 (cfold e')
adamc@171:         | Snd _ _ e' => #2 (cfold e')
adamc@171: 
adamc@171:         | Inl _ _ e' => Inl (cfold e')
adamc@171:         | Inr _ _ e' => Inr (cfold e')
adamc@171:         | SumCase _ _ _ e' e1 e2 =>
adamc@171:           case cfold e' of
adamc@171:             x => cfold (e1 x)
adamc@171:           | y => cfold (e2 y)
adamc@171:       end.
adamc@171:   End cfold.
adamc@171: 
adamc@171:   Definition Cfold t (E : Exp t) : Exp t := fun _ => cfold (E _).
adamc@171: 
adamc@171:   Lemma cfold_correct : forall t (e : exp _ t),
adamc@171:     expDenote (cfold e) = expDenote e.
adamc@171:     induction e; crush; try (ext_eq; crush);
adamc@171:       repeat (match goal with
adamc@171:                 | [ |- context[cfold ?E] ] => dep_destruct (cfold E)
adamc@171:                 | [ |- match ?E with inl _ => _ | inr _ => _ end = _ ] => destruct E
adamc@171:               end; crush).
adamc@171:   Qed.
adamc@171: 
adamc@171:   Theorem Cfold_correct : forall t (E : Exp t),
adamc@171:     ExpDenote (Cfold E) = ExpDenote E.
adamc@171:     unfold ExpDenote, Cfold; intros; apply cfold_correct.
adamc@171:   Qed.
adamc@171: End PSLC.