Use a different representation of values and prove equivalence of UCC evalutor

code/dawn/DawnEval.v

@@ -1,22 +1,22 @@
 Require Import Coq.Lists.List.
-Require Import Dawn.
+Require Import DawnV2.
 Require Import Coq.Program.Equality.
 From Ltac2 Require Import Ltac2.
 
 Inductive step_result :=
   | err
-  | middle (e : expr) (s : list expr)
+  | middle (e : expr) (s : value_stack)
-  | final (s : list expr).
+  | final (s : value_stack).
 
-Fixpoint eval_step (s : list expr) (e : expr) : step_result :=
+Fixpoint eval_step (s : value_stack) (e : expr) : step_result :=
   match e, s with
   | e_int swap, v' :: v :: vs => final (v :: v' :: vs)
   | e_int clone, v :: vs => final (v :: v :: vs)
   | e_int drop, v :: vs => final vs
-  | e_int quote, v :: vs => final (e_quote v :: vs)
+  | e_int quote, v :: vs => final (v_quote (value_to_expr v) :: vs)
-  | e_int compose, (e_quote v2) :: (e_quote v1) :: vs => final (e_quote (e_comp v1 v2) :: vs)
+  | e_int compose, (v_quote v2) :: (v_quote v1) :: vs => final (v_quote (e_comp v1 v2) :: vs)
-  | e_int apply, (e_quote v1) :: vs => middle v1 vs
+  | e_int apply, (v_quote v1) :: vs => middle v1 vs
-  | e_quote e', vs => final (e_quote e' :: vs)
+  | e_quote e', vs => final (v_quote e' :: vs)
   | e_comp e1 e2, vs =>
       match eval_step vs e1 with
       | final vs' => middle e2 vs'
@@ -26,12 +26,10 @@ Fixpoint eval_step (s : list expr) (e : expr) : step_result :=
   | _, _ => err
   end.
 
-Definition strip_value_list (vs : value_stack) := @map value expr (@projT1 expr IsValue) vs.
-
 Theorem eval_step_correct : forall (e : expr) (vs vs' : value_stack), Sem_expr vs e vs' ->
-  (eval_step (strip_value_list vs) e = final (strip_value_list vs')) \/
+  (eval_step vs e = final vs') \/
   (exists (ei : expr) (vsi : value_stack),
-    eval_step (strip_value_list vs) e = middle ei (strip_value_list vsi) /\
+    eval_step vs e = middle ei vsi /\
     Sem_expr vsi ei vs').
 Proof.
   intros e vs vs' Hsem.
@@ -39,9 +37,9 @@ Proof.
   induction Hsem.
   - inversion H; (* The expression is just an intrnsic. *)
     (* Dismiss all the straightforward "final" cases,
-       of which most intrinsics are *)
+       of which most intrinsics are. *)
     try (left; reflexivity).
-    (* We are in an intermediate / middle case. *)
+    (* Only apply remains; We are in an intermediate / middle case. *)
     right.
     (* The semantics guarantee that the expression in the
        quote evaluates to the final state. *)
@@ -63,26 +61,26 @@ Proof.
       eapply Sem_e_comp. apply Hsem'. apply Hsem2.
 Qed.
 
-Inductive eval_chain (s : list expr) (e : expr) (s' : list expr) : Prop :=
+Inductive eval_chain (vs : value_stack) (e : expr) (vs' : value_stack) : Prop :=
-  | chain_final (P : eval_step s e = final s')
+  | chain_final (P : eval_step vs e = final vs')
-  | chain_middle (ei : expr) (si : list expr)
+  | chain_middle (ei : expr) (vsi : value_stack)
-      (P : eval_step s e = middle ei si) (rest : eval_chain si ei s').
+      (P : eval_step vs e = middle ei vsi) (rest : eval_chain vsi ei vs').
 
-Lemma eval_chain_merge : forall (e1 e2 : expr) (s s' s'' : list expr),
+Lemma eval_chain_merge : forall (e1 e2 : expr) (vs vs' vs'' : value_stack),
-  eval_chain s e1 s' -> eval_chain s' e2 s'' -> eval_chain s (e_comp e1 e2) s''.
+  eval_chain vs e1 vs' -> eval_chain vs' e2 vs'' -> eval_chain vs (e_comp e1 e2) vs''.
 Proof.
-  intros e1 e2 s s' s'' ch1 ch2.
+  intros e1 e2 vs vs' vs'' ch1 ch2.
   induction ch1;
   eapply chain_middle; simpl; try (rewrite P); auto.
 Qed.
 
-Lemma eval_chain_split : forall (e1 e2 : expr) (s s'' : list expr),
+Lemma eval_chain_split : forall (e1 e2 : expr) (vs vs'' : value_stack),
-  eval_chain s (e_comp e1 e2) s'' -> exists s', (eval_chain s e1 s') /\ (eval_chain s' e2 s'').
+  eval_chain vs (e_comp e1 e2) vs'' -> exists vs', (eval_chain vs e1 vs') /\ (eval_chain vs' e2 vs'').
 Proof.
-  intros e1 e2 s ss'' ch.
+  intros e1 e2 vs vss'' ch.
   ltac1:(dependent induction ch).
-  - simpl in P. destruct (eval_step s e1); inversion P.
+  - simpl in P. destruct (eval_step vs e1); inversion P.
-  - simpl in P. destruct (eval_step s e1) eqn:Hval; try (inversion P).
+  - simpl in P. destruct (eval_step vs e1) eqn:Hval; try (inversion P).
     + injection P as Hinj; subst. specialize (IHch e e2 H0) as [s'0 [ch1 ch2]].
       eexists. split.
       * eapply chain_middle. apply Hval. apply ch1.
@@ -93,9 +91,9 @@ Proof.
 Qed.
 
 Theorem val_step_sem : forall (e : expr) (vs vs' : value_stack),
-  Sem_expr vs e vs' -> eval_chain (strip_value_list vs) e (strip_value_list vs')
+  Sem_expr vs e vs' -> eval_chain vs e vs'
 with eval_step_int : forall (i : intrinsic) (vs vs' : value_stack),
-  Sem_int vs i vs' -> eval_chain (strip_value_list vs) (e_int i) (strip_value_list vs').
+  Sem_int vs i vs' -> eval_chain vs (e_int i) vs'.
 Proof.
   - intros e vs vs' Hsem.
     induction Hsem.
@@ -106,7 +104,7 @@ Proof.
       apply chain_final. auto.
     + (* In compoition, by induction, we know that the two sub-expressions produce
          proper evaluation chains. Chains can be composed (via eval_chain_merge). *)
-      eapply eval_chain_merge with (strip_value_list vs2); auto.
+      eapply eval_chain_merge with vs2; auto.
   - intros i vs vs' Hsem.
     (* The evaluation chain depends on the specific intrinsic in use. *)
     inversion Hsem; subst;
@@ -114,15 +112,9 @@ Proof.
     try (apply chain_final; auto; fail).
     (* Only apply is non-final. The first step is popping the quote from the stack,
        and the rest of the steps are given by the evaluation of the code in the quote. *)
-    apply chain_middle with e (strip_value_list vs0); auto.
+    apply chain_middle with e vs0; auto.
 Qed.
 
-Theorem eval_step_sem_back : forall (e : expr) (vs vs' : value_stack),
-  eval_chain (strip_value_list vs) e (strip_value_list vs') -> Sem_expr vs e vs'.
-Proof.
-  (* Thoughts: the issue is with the apparent nondeterminism of evalution. *)
-Admitted.
-
 Ltac2 Type exn ::= [ | Not_intrinsic ].
 
 Ltac2 rec destruct_n (n : int) (vs : constr) : unit :=
@@ -146,15 +138,6 @@ Ltac2 int_arity (int : constr) : int :=
 
 Ltac2 destruct_int_stack (int : constr) (va: constr) := destruct_n (int_arity int) va.
 
-Ltac2 ensure_valid_value_stack () := Control.enter (fun () =>
-  match! goal with
-  | [h : eval_step (strip_value_list ?a) (e_int ?b) = ?c |- _] =>
-      let h := Control.hyp h in
-      destruct_int_stack b a;
-      try (inversion $h; fail)
-  | [|- _ ] => ()
-  end).
-
 Ltac2 ensure_valid_stack () := Control.enter (fun () =>
   match! goal with
   | [h : eval_step ?a (e_int ?b) = ?c |- _] =>
@@ -164,15 +147,77 @@ Ltac2 ensure_valid_stack () := Control.enter (fun () =>
   | [|- _ ] => ()
   end).
 
-Theorem test : forall (s s': list expr), eval_step s (e_int swap) = final s' ->
+Theorem test : forall (vs vs': value_stack), eval_step vs (e_int swap) = final vs' ->
-  exists v1 v2 s'', s = v1 :: v2 :: s'' /\ s' = v2 :: v1 :: s''.
+  exists v1 v2 vs'', vs = v1 :: v2 :: vs'' /\ vs' = v2 :: v1 :: vs''.
 Proof.
   intros s s' Heq.
   ensure_valid_stack ().
   simpl in Heq. injection Heq as Hinj. subst. eauto.
 Qed.
 
+Theorem eval_step_final_sem : forall (e : expr) (vs vs' : value_stack),
+  eval_step vs e = final vs' -> Sem_expr vs e vs'.
+Proof.
+  intros e vs vs' Hev. destruct e.
+  - destruct i; ensure_valid_stack ();
+    (* Get rid of trivial cases that match one-to-one. *)
+    simpl in Hev; try (injection Hev as Hinj; subst; solve_basic ()).
+    + (* compose with one quoted value *)
+      destruct v. inversion Hev.
+    + (* compose with two quoted values. *)
+      destruct v; destruct v0.
+      injection Hev as Hinj; subst; solve_basic ().
+    + (* Apply is not final. *) destruct v. inversion Hev.
+  - (* Quote is always final, trivially. *)
+    simpl in Hev. injection Hev as Hinj; subst. solve_basic ().
+  - (* Compose is never final. *)
+    simpl in Hev. destruct (eval_step vs e1); inversion Hev.
+Qed.
+
+Theorem eval_step_middle_sem : forall (e ei: expr) (vs vsi vs' : value_stack),
+  eval_step vs e = middle ei vsi ->
+  Sem_expr vsi ei vs' ->
+  Sem_expr vs e vs'.
+Proof.
+  intros e. induction e; intros ei vs vsi vs' Hev Hsem.
+  - destruct i; ensure_valid_stack ().
+    + (* compose with one quoted value. *)
+      destruct v. inversion Hev.
+    + (* compose with two quoted values. *)
+      destruct v; destruct v0. inversion Hev.
+    + (* Apply *)
+      destruct v. injection Hev as Hinj; subst.
+      solve_basic (). auto.
+  - inversion Hev.
+  - simpl in Hev.
+    destruct (eval_step vs e1) eqn:Hev1.
+    + inversion Hev.
+    + injection Hev as Hinj; subst. inversion Hsem; subst.
+      specialize (IHe1 e vs vsi vs2 Hev1 H2).
+      eapply Sem_e_comp. apply IHe1. apply H4.
+    + injection Hev as Hinj; subst.
+      specialize (eval_step_final_sem e1 vs vsi Hev1) as Hsem1.
+      eapply Sem_e_comp. apply Hsem1. apply Hsem.
+Qed.
+
+Theorem eval_step_sem_back : forall (e : expr) (vs vs' : value_stack),
+  eval_chain vs e vs' -> Sem_expr vs e vs'.
+Proof.
+  intros e vs vs' ch.
+  ltac1:(dependent induction ch).
+  - apply eval_step_final_sem. auto.
+  - specialize (eval_step_middle_sem e ei vs vsi vs' P IHch). auto.
+Qed.
+
+Corollary eval_step_no_sem : forall (e : expr) (vs vs' : value_stack),
+  ~ (Sem_expr vs e vs') -> ~(eval_chain vs e vs').
+Proof.
+  intros e vs vs' Hnsem Hch.
+  specialize (eval_step_sem_back _ _ _ Hch). auto.
+Qed.
+
 Require Extraction.
 Require Import ExtrHaskellBasic.
 Extraction Language Haskell.
+Set Extraction KeepSingleton.
 Extraction "UccGen.hs" expr eval_step true false or.

code/dawn/DawnV2.v

@@ -0,0 +1,179 @@
+Require Import Coq.Lists.List.
+From Ltac2 Require Import Ltac2.
+
+Inductive intrinsic :=
+  | swap
+  | clone
+  | drop
+  | quote
+  | compose
+  | apply.
+
+Inductive expr :=
+  | e_int (i : intrinsic)
+  | e_quote (e : expr)
+  | e_comp (e1 e2 : expr).
+
+Definition e_compose (e : expr) (es : list expr) := fold_left e_comp es e.
+
+Inductive value := v_quote (e : expr).
+Definition value_stack := list value.
+
+Definition value_to_expr (v : value) : expr :=
+  match v with
+  | v_quote e => e_quote e
+  end.
+
+Inductive Sem_int : value_stack -> intrinsic -> value_stack -> Prop :=
+  | Sem_swap : forall (v v' : value) (vs : value_stack), Sem_int (v' :: v :: vs) swap (v :: v' :: vs)
+  | Sem_clone : forall (v : value) (vs : value_stack), Sem_int (v :: vs) clone (v :: v :: vs)
+  | Sem_drop : forall (v : value) (vs : value_stack), Sem_int (v :: vs) drop vs
+  | Sem_quote : forall (v : value) (vs : value_stack), Sem_int (v :: vs) quote ((v_quote (value_to_expr v)) :: vs)
+  | Sem_compose : forall (e e' : expr) (vs : value_stack), Sem_int (v_quote e' :: v_quote e :: vs) compose (v_quote (e_comp e e') :: vs)
+  | Sem_apply : forall (e : expr) (vs vs': value_stack), Sem_expr vs e vs' -> Sem_int (v_quote e :: vs) apply vs'
+
+with Sem_expr : value_stack -> expr -> value_stack -> Prop :=
+  | Sem_e_int : forall (i : intrinsic) (vs vs' : value_stack), Sem_int vs i vs' -> Sem_expr vs (e_int i) vs'
+  | Sem_e_quote : forall (e : expr) (vs : value_stack), Sem_expr vs (e_quote e) (v_quote e :: vs)
+  | Sem_e_comp : forall (e1 e2 : expr) (vs1 vs2 vs3 : value_stack),
+      Sem_expr vs1 e1 vs2 -> Sem_expr vs2 e2 vs3 -> Sem_expr vs1 (e_comp e1 e2) vs3.
+
+Definition false : expr := e_quote (e_int drop).
+Definition false_v : value := v_quote (e_int drop).
+
+Definition true : expr := e_quote (e_comp (e_int swap) (e_int drop)).
+Definition true_v : value := v_quote (e_comp (e_int swap) (e_int drop)).
+
+Theorem false_correct : forall (v v' : value) (vs : value_stack), Sem_expr (v' :: v :: vs) (e_comp false (e_int apply)) (v :: vs).
+Proof.
+  intros v v' vs.
+  eapply Sem_e_comp.
+  - apply Sem_e_quote.
+  - apply Sem_e_int. apply Sem_apply. apply Sem_e_int. apply Sem_drop.
+Qed.
+
+Theorem true_correct : forall (v v' : value) (vs : value_stack), Sem_expr (v' :: v :: vs) (e_comp true (e_int apply)) (v' :: vs).
+Proof.
+  intros v v' vs.
+  eapply Sem_e_comp.
+  - apply Sem_e_quote.
+  - apply Sem_e_int. apply Sem_apply. eapply Sem_e_comp.
+    * apply Sem_e_int. apply Sem_swap.
+    * apply Sem_e_int. apply Sem_drop.
+Qed.
+
+Definition or : expr := e_comp (e_int clone) (e_int apply).
+
+Theorem or_false_v : forall (v : value) (vs : value_stack), Sem_expr (false_v :: v :: vs) or (v :: vs).
+Proof with apply Sem_e_int.
+  intros v vs.
+  eapply Sem_e_comp...
+  - apply Sem_clone.
+  - apply Sem_apply... apply Sem_drop.
+Qed.
+
+Theorem or_true : forall (v : value) (vs : value_stack), Sem_expr (true_v :: v :: vs) or (true_v :: vs).
+Proof with apply Sem_e_int.
+  intros v vs.
+  eapply Sem_e_comp...
+  - apply Sem_clone...
+  - apply Sem_apply. eapply Sem_e_comp...
+    * apply Sem_swap.
+    * apply Sem_drop.
+Qed.
+
+Definition or_false_false := or_false_v false_v.
+Definition or_false_true := or_false_v true_v.
+Definition or_true_false := or_true false_v.
+Definition or_true_true := or_true true_v.
+
+Fixpoint quote_n (n : nat) :=
+  match n with
+  | O => e_int quote
+  | S n' => e_compose (quote_n n') (e_int swap :: e_int quote :: e_int swap :: e_int compose :: nil)
+  end.
+
+Theorem quote_2_correct : forall (v1 v2 : value) (vs : value_stack),
+    Sem_expr (v2 :: v1 :: vs) (quote_n 1) (v_quote (e_comp (value_to_expr v1) (value_to_expr v2)) :: vs).
+Proof with apply Sem_e_int.
+  intros v1 v2 vs. simpl.
+  repeat (eapply Sem_e_comp)...
+  - apply Sem_quote.
+  - apply Sem_swap.
+  - apply Sem_quote.
+  - apply Sem_swap.
+  - apply Sem_compose.
+Qed.
+
+Theorem quote_3_correct : forall (v1 v2 v3 : value) (vs : value_stack),
+  Sem_expr (v3 :: v2 :: v1 :: vs) (quote_n 2) (v_quote (e_comp (value_to_expr v1) (e_comp (value_to_expr v2) (value_to_expr v3))) :: vs).
+Proof with apply Sem_e_int.
+  intros v1 v2 v3 vs. simpl.
+  repeat (eapply Sem_e_comp)...
+  - apply Sem_quote.
+  - apply Sem_swap.
+  - apply Sem_quote.
+  - apply Sem_swap.
+  - apply Sem_compose.
+  - apply Sem_swap.
+  - apply Sem_quote.
+  - apply Sem_swap.
+  - apply Sem_compose.
+Qed.
+
+Ltac2 rec solve_basic () := Control.enter (fun _ =>
+  match! goal with
+  | [|- Sem_int ?vs1 swap ?vs2] => apply Sem_swap
+  | [|- Sem_int ?vs1 clone ?vs2] => apply Sem_clone
+  | [|- Sem_int ?vs1 drop ?vs2] => apply Sem_drop
+  | [|- Sem_int ?vs1 quote ?vs2] => apply Sem_quote
+  | [|- Sem_int ?vs1 compose ?vs2] => apply Sem_compose
+  | [|- Sem_int ?vs1 apply ?vs2] => apply Sem_apply
+  | [|- Sem_expr ?vs1 (e_comp ?e1 ?e2) ?vs2] => eapply Sem_e_comp; solve_basic ()
+  | [|- Sem_expr ?vs1 (e_int ?e) ?vs2] => apply Sem_e_int; solve_basic ()
+  | [|- Sem_expr ?vs1 (e_quote ?e) ?vs2] => apply Sem_e_quote
+  | [_ : _ |- _] => ()
+  end).
+
+Theorem quote_2_correct' : forall (v1 v2 : value) (vs : value_stack),
+    Sem_expr (v2 :: v1 :: vs) (quote_n 1) (v_quote (e_comp (value_to_expr v1) (value_to_expr v2)) :: vs).
+Proof. intros. simpl. solve_basic (). Qed.
+
+Theorem quote_3_correct' : forall (v1 v2 v3 : value) (vs : value_stack),
+  Sem_expr (v3 :: v2 :: v1 :: vs) (quote_n 2) (v_quote (e_comp (value_to_expr v1) (e_comp (value_to_expr v2) (value_to_expr v3))) :: vs).
+Proof. intros. simpl. solve_basic (). Qed.
+
+Definition rotate_n (n : nat) := e_compose (quote_n n) (e_int swap :: e_int quote :: e_int compose :: e_int apply :: nil).
+
+Lemma eval_value : forall (v : value) (vs : value_stack),
+  Sem_expr vs (value_to_expr v) (v :: vs).
+Proof.
+  intros v vs.
+  destruct v. 
+  simpl. apply Sem_e_quote.
+Qed.
+
+Theorem rotate_3_correct : forall (v1 v2 v3 : value) (vs : value_stack),
+  Sem_expr (v3 :: v2 :: v1 :: vs) (rotate_n 1) (v1 :: v3 :: v2 :: vs).
+Proof.
+  intros. unfold rotate_n. simpl. solve_basic ().
+  repeat (eapply Sem_e_comp); apply eval_value.
+Qed.
+
+Theorem rotate_4_correct : forall (v1 v2 v3 v4 : value) (vs : value_stack),
+  Sem_expr (v4 :: v3 :: v2 :: v1 :: vs) (rotate_n 2) (v1 :: v4 :: v3 :: v2 :: vs).
+Proof.
+  intros. unfold rotate_n. simpl. solve_basic ().
+  repeat (eapply Sem_e_comp); apply eval_value.
+Qed.
+
+Theorem e_comp_assoc : forall (e1 e2 e3 : expr) (vs vs' : value_stack),
+  Sem_expr vs (e_comp e1 (e_comp e2 e3)) vs' <-> Sem_expr vs (e_comp (e_comp e1 e2) e3) vs'.
+Proof.
+  intros e1 e2 e3 vs vs'.
+  split; intros Heval.
+  - inversion Heval; subst. inversion H4; subst.
+    eapply Sem_e_comp. eapply Sem_e_comp. apply H2. apply H3. apply H6.
+  - inversion Heval; subst. inversion H2; subst.
+    eapply Sem_e_comp. apply H3. eapply Sem_e_comp. apply H6. apply H4.
+Qed.

code/dawn/Ucc.hs

@@ -0,0 +1,64 @@
+module Ucc where
+import UccGen
+import Text.Parsec
+import Data.Functor.Identity
+import Control.Applicative hiding ((<|>))
+import System.IO
+
+instance Show Intrinsic where
+    show Swap = "swap"
+    show Clone = "clone"
+    show Drop = "drop"
+    show Quote = "quote"
+    show Compose = "compose"
+    show Apply = "apply"
+
+instance Show Expr where
+    show (E_int i) = show i
+    show (E_quote e) = "[" ++ show e ++ "]"
+    show (E_comp e1 e2) = show e1 ++ " " ++ show e2
+
+instance Show Value where
+    show (V_quote e) = show (E_quote e)
+
+type Parser a = ParsecT String () Identity a
+
+intrinsic :: Parser Intrinsic
+intrinsic = (<* spaces) $ foldl1 (<|>) $ map (\(s, i) -> try (string s >> return i))
+    [ ("swap", Swap)
+    , ("clone", Clone)
+    , ("drop", Drop)
+    , ("quote", Quote)
+    , ("compose", Compose)
+    , ("apply", Apply)
+    ]
+
+expression :: Parser Expr
+expression = foldl1 E_comp <$> many1 single
+    where
+        single
+            = (E_int <$> intrinsic)
+            <|> (fmap E_quote $ char '[' *> spaces *> expression <* char ']' <* spaces)
+
+parseExpression :: String -> Either ParseError Expr
+parseExpression = runParser expression () "<inline>"
+
+eval :: [Value] -> Expr -> Maybe [Value]
+eval s e =
+    case eval_step s e of
+        Err -> Nothing
+        Final s' -> Just s'
+        Middle e' s' -> eval s' e'
+
+main :: IO ()
+main = do
+    putStr "> "
+    hFlush stdout
+    str <- getLine
+    case parseExpression str of
+        Right e ->
+            case eval [] e of
+                Just st -> putStrLn $ show st
+                _ -> putStrLn "Evaluation error"
+        _ -> putStrLn "Parse error"
+    main