Interderivability of big and small step

src/Hoas.v

@@ -10,7 +10,7 @@
 (* begin hide *)
 Require Import Arith Eqdep String List.
 
-Require Import Tactics.
+Require Import Axioms DepList Tactics.
 
 Set Implicit Arguments.
 (* end hide *)
@@ -86,7 +86,8 @@ Hint Constructors Val.
 
 Inductive Step : forall t, Exp t -> Exp t -> Prop :=
 | Beta : forall dom ran (B : Exp1 dom ran) (X : Exp dom),
-  App (Abs B) X ==> Subst X B
+  Val X
+  -> App (Abs B) X ==> Subst X B
 | AppCong1 : forall dom ran (F : Exp (dom --> ran)) (X : Exp dom) F',
   F ==> F'
   -> App F X ==> App F' X
@@ -101,7 +102,8 @@ Inductive Step : forall t, Exp t -> Exp t -> Prop :=
   E1 ==> E1'
   -> Plus E1 E2 ==> Plus E1' E2
 | PlusCong2 : forall E1 E2 E2',
-  E2 ==> E2'
+  Val E1
+  -> E2 ==> E2'
   -> Plus E1 E2 ==> Plus E1 E2'
 
   where "E1 ==> E2" := (Step E1 E2).
@@ -124,12 +126,43 @@ Inductive Closed : forall t, Exp t -> Prop :=
 
 Axiom closed : forall t (E : Exp t), Closed E.
 
-Ltac my_crush :=
+Ltac my_crush' :=
   crush;
   repeat (match goal with
             | [ H : _ |- _ ] => generalize (inj_pairT2 _ _ _ _ _ H); clear H
           end; crush).
 
+Ltac my_crush :=
+  my_crush';
+  try (match goal with
+         | [ H : ?F = ?G |- _ ] =>
+           match goal with
+             | [ _ : F (fun _ => unit) = G (fun _ => unit) |- _ ] => fail 1
+             | _ =>
+               let H' := fresh "H'" in
+                 assert (H' : F (fun _ => unit) = G (fun _ => unit)); [ congruence
+                   | discriminate || injection H' ];
+                 clear H'
+           end
+       end; my_crush');
+  repeat match goal with
+           | [ H : ?F = ?G, H2 : ?X (fun _ => unit) = ?Y (fun _ => unit) |- _ ] =>
+             match X with
+               | Y => fail 1
+               | _ =>
+                 assert (X = Y); [ unfold Exp; apply ext_eq; intro var;
+                   let H' := fresh "H'" in
+                     assert (H' : F var = G var); [ congruence
+                       | match type of H' with
+                           | ?X = ?Y =>
+                             let X := eval hnf in X in
+                               let Y := eval hnf in Y in
+                                 change (X = Y) in H'
+                         end; injection H'; clear H'; my_crush' ]
+                   | my_crush'; clear H2 ]
+             end
+         end.
+
 Lemma progress' : forall t (E : Exp t),
   Closed E
   -> Val E \/ exists E', E ==> E'.
@@ -144,3 +177,194 @@ Theorem progress : forall t (E : Exp t),
   intros; apply progress'; apply closed.
 Qed.
 
+
+(** * Big-Step Semantics *)
+
+Reserved Notation "E1 ===> E2" (no associativity, at level 90).
+
+Inductive BigStep : forall t, Exp t -> Exp t -> Prop :=
+| SConst : forall n,
+  Const n ===> Const n
+| SPlus : forall E1 E2 n1 n2,
+  E1 ===> Const n1
+  -> E2 ===> Const n2
+  -> Plus E1 E2 ===> Const (n1 + n2)
+
+| SApp : forall dom ran (E1 : Exp (dom --> ran)) E2 B V2 V,
+  E1 ===> Abs B
+  -> E2 ===> V2
+  -> Subst V2 B ===> V
+  -> App E1 E2 ===> V
+| SAbs : forall dom ran (B : Exp1 dom ran),
+  Abs B ===> Abs B
+
+  where "E1 ===> E2" := (BigStep E1 E2).
+
+Hint Constructors BigStep.
+
+Reserved Notation "E1 ==>* E2" (no associativity, at level 90).
+
+Inductive MultiStep : forall t, Exp t -> Exp t -> Prop :=
+| Done : forall t (E : Exp t), E ==>* E
+| OneStep : forall t (E E' E'' : Exp t),
+  E ==> E'
+  -> E' ==>* E''
+  -> E ==>* E''
+
+  where "E1 ==>* E2" := (MultiStep E1 E2).
+
+Hint Constructors MultiStep.
+
+Theorem MultiStep_trans : forall t (E1 E2 E3 : Exp t),
+  E1 ==>* E2
+  -> E2 ==>* E3
+  -> E1 ==>* E3.
+  induction 1; eauto.
+Qed.
+
+Hint Resolve MultiStep_trans.
+
+Theorem Big_Val : forall t (E V : Exp t),
+  E ===> V
+  -> Val V.
+  induction 1; crush.
+Qed.
+
+Theorem Val_Big : forall t (V : Exp t),
+  Val V
+  -> V ===> V.
+  destruct 1; crush.
+Qed.
+
+Hint Resolve Big_Val Val_Big.
+
+Ltac uiper :=
+  repeat match goal with
+           | [ pf : _ = _ |- _ ] =>
+             generalize pf; subst; intro;
+               rewrite (UIP_refl _ _ pf); simpl;
+                 repeat match goal with
+                          | [ H : forall pf : ?X = ?X, _ |- _ ] =>
+                            generalize (H (refl_equal _)); clear H; intro H
+                        end
+         end.
+
+Lemma Multi_PlusCong1' : forall E2 t (pf : t = Nat) (E1 E1' : Exp t),
+  E1 ==>* E1'
+  -> Plus (match pf with refl_equal => E1 end) E2
+  ==>* Plus (match pf with refl_equal => E1' end) E2.
+  induction 1; crush; uiper; eauto.
+Qed.
+
+Lemma Multi_PlusCong1 : forall E1 E2 E1',
+  E1 ==>* E1'
+  -> Plus E1 E2 ==>* Plus E1' E2.
+  intros; generalize (Multi_PlusCong1' E2 (refl_equal _)); auto.
+Qed.
+
+Lemma Multi_PlusCong2' : forall E1 (_ : Val E1) t (pf : t = Nat) (E2 E2' : Exp t),
+  E2 ==>* E2'
+  -> Plus E1 (match pf with refl_equal => E2 end)
+  ==>* Plus E1 (match pf with refl_equal => E2' end).
+  induction 2; crush; uiper; eauto.
+Qed.
+
+Lemma Multi_PlusCong2 : forall E1 E2 E2',
+  Val E1
+  -> E2 ==>* E2'
+  -> Plus E1 E2 ==>* Plus E1 E2'.
+  intros E1 E2 E2' H; generalize (Multi_PlusCong2' H (refl_equal _)); auto.
+Qed.
+
+Lemma Multi_AppCong1' : forall dom ran E2 t (pf : t = dom --> ran) (E1 E1' : Exp t),
+  E1 ==>* E1'
+  -> App (match pf in _ = t' return Exp t' with refl_equal => E1 end) E2
+  ==>* App (match pf in _ = t' return Exp t' with refl_equal => E1' end) E2.
+  induction 1; crush; uiper; eauto.
+Qed.
+
+Lemma Multi_AppCong1 : forall dom ran (E1 : Exp (dom --> ran)) E2 E1',
+  E1 ==>* E1'
+  -> App E1 E2 ==>* App E1' E2.
+  intros; generalize (Multi_AppCong1' (ran := ran) E2 (refl_equal _)); auto.
+Qed.
+
+Lemma Multi_AppCong2 : forall dom ran (E1 : Exp (dom --> ran)) E2 E2',
+  Val E1
+  -> E2 ==>* E2'
+  -> App E1 E2 ==>* App E1 E2'.
+  induction 2; crush; eauto.
+Qed.
+
+Hint Resolve Multi_PlusCong1 Multi_PlusCong2 Multi_AppCong1 Multi_AppCong2.
+
+Theorem Big_Multi : forall t (E V : Exp t),
+  E ===> V
+  -> E ==>* V.
+  induction 1; crush; eauto 8.
+Qed.
+
+Lemma Big_Val' : forall t (V1 V2 : Exp t),
+  Val V2
+  -> V1 = V2
+  -> V1 ===> V2.
+  crush.
+Qed.
+
+Hint Resolve Big_Val'.
+
+Lemma Multi_Big' : forall t (E E' : Exp t),
+  E ==> E'
+  -> forall E'', E' ===> E''
+    -> E ===> E''.
+  induction 1; crush; eauto;
+    match goal with
+      | [ H : _ ===> _ |- _ ] => inversion H; my_crush; eauto
+    end.
+Qed.
+
+Hint Resolve Multi_Big'.
+
+Theorem Multi_Big : forall t (E V : Exp t),
+  E ==>* V
+  -> Val V
+  -> E ===> V.
+  induction 1; crush; eauto.
+Qed.
+
+
+(** * Constant folding *)
+
+Section cfold.
+  Variable var : type -> Type.
+
+  Fixpoint cfold t (e : exp var t) {struct e} : exp var t :=
+    match e in exp _ t return exp _ t with
+      | Const' n => Const' n
+      | Plus' e1 e2 =>
+        let e1' := cfold e1 in
+        let e2' := cfold e2 in
+          match e1', e2' with
+            | Const' n1, Const' n2 => Const' (n1 + n2)
+            | _, _ => Plus' e1' e2'
+          end
+
+      | Var _ x => Var x
+      | App' _ _ e1 e2 => App' (cfold e1) (cfold e2)
+      | Abs' _ _ e' => Abs' (fun x => cfold (e' x))
+    end.
+End cfold.
+
+Definition Cfold t (E : Exp t) : Exp t := fun _ => cfold (E _).
+
+
+Definition ExpN (G : list type) (t : type) := forall var, hlist var G -> exp var t.
+
+Definition ConstN G (n : nat) : ExpN G Nat :=
+  fun _ _ => Const' n.
+Definition PlusN G (E1 E2 : ExpN G Nat) : ExpN G Nat :=
+  fun _ s => Plus' (E1 _ s) (E2 _ s).
+Definition AppN G dom ran (F : ExpN G (dom --> ran)) (X : ExpN G dom) : ExpN G ran :=
+  fun _ s => App' (F _ s) (X _ s).
+Definition AbsN G dom ran (B : ExpN (dom :: G) ran) : ExpN G (dom --> ran) :=
+  fun _ s => Abs' (fun x => B _ (x ::: s)).
\ No newline at end of file

src/Tactics.v

@@ -48,8 +48,9 @@ Ltac simplHyp invOne :=
   match goal with
     | [ H : ex _ |- _ ] => destruct H
 
-    | [ H : ?F _ = ?F _ |- _ ] => injection H;
+    | [ H : ?F ?X = ?F ?Y |- _ ] => injection H;
       match goal with
+        | [ |- F X = F Y -> _ ] => fail 1
         | [ |- _ = _ -> _ ] => try clear H; intros; try subst
       end
     | [ H : ?F _ _ = ?F _ _ |- _ ] => injection H;