Use a different representation of values and prove equivalence of UCC evalutor
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@ -1,22 +1,22 @@
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Require Import Coq.Lists.List.
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Require Import Dawn.
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Require Import DawnV2.
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Require Import Coq.Program.Equality.
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From Ltac2 Require Import Ltac2.
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Inductive step_result :=
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| err
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| middle (e : expr) (s : list expr)
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| final (s : list expr).
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| middle (e : expr) (s : value_stack)
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| final (s : value_stack).
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Fixpoint eval_step (s : list expr) (e : expr) : step_result :=
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Fixpoint eval_step (s : value_stack) (e : expr) : step_result :=
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match e, s with
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| e_int swap, v' :: v :: vs => final (v :: v' :: vs)
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| e_int clone, v :: vs => final (v :: v :: vs)
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| e_int drop, v :: vs => final vs
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| e_int quote, v :: vs => final (e_quote v :: vs)
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| e_int compose, (e_quote v2) :: (e_quote v1) :: vs => final (e_quote (e_comp v1 v2) :: vs)
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| e_int apply, (e_quote v1) :: vs => middle v1 vs
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| e_quote e', vs => final (e_quote e' :: vs)
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| e_int quote, v :: vs => final (v_quote (value_to_expr v) :: vs)
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| e_int compose, (v_quote v2) :: (v_quote v1) :: vs => final (v_quote (e_comp v1 v2) :: vs)
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| e_int apply, (v_quote v1) :: vs => middle v1 vs
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| e_quote e', vs => final (v_quote e' :: vs)
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| e_comp e1 e2, vs =>
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match eval_step vs e1 with
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| final vs' => middle e2 vs'
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@ -26,12 +26,10 @@ Fixpoint eval_step (s : list expr) (e : expr) : step_result :=
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| _, _ => err
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end.
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Definition strip_value_list (vs : value_stack) := @map value expr (@projT1 expr IsValue) vs.
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Theorem eval_step_correct : forall (e : expr) (vs vs' : value_stack), Sem_expr vs e vs' ->
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(eval_step (strip_value_list vs) e = final (strip_value_list vs')) \/
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(eval_step vs e = final vs') \/
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(exists (ei : expr) (vsi : value_stack),
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eval_step (strip_value_list vs) e = middle ei (strip_value_list vsi) /\
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eval_step vs e = middle ei vsi /\
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Sem_expr vsi ei vs').
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Proof.
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intros e vs vs' Hsem.
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@ -39,9 +37,9 @@ Proof.
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induction Hsem.
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- inversion H; (* The expression is just an intrnsic. *)
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(* Dismiss all the straightforward "final" cases,
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of which most intrinsics are *)
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of which most intrinsics are. *)
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try (left; reflexivity).
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(* We are in an intermediate / middle case. *)
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(* Only apply remains; We are in an intermediate / middle case. *)
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right.
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(* The semantics guarantee that the expression in the
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quote evaluates to the final state. *)
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@ -63,26 +61,26 @@ Proof.
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eapply Sem_e_comp. apply Hsem'. apply Hsem2.
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Qed.
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Inductive eval_chain (s : list expr) (e : expr) (s' : list expr) : Prop :=
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| chain_final (P : eval_step s e = final s')
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| chain_middle (ei : expr) (si : list expr)
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(P : eval_step s e = middle ei si) (rest : eval_chain si ei s').
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Inductive eval_chain (vs : value_stack) (e : expr) (vs' : value_stack) : Prop :=
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| chain_final (P : eval_step vs e = final vs')
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| chain_middle (ei : expr) (vsi : value_stack)
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(P : eval_step vs e = middle ei vsi) (rest : eval_chain vsi ei vs').
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Lemma eval_chain_merge : forall (e1 e2 : expr) (s s' s'' : list expr),
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eval_chain s e1 s' -> eval_chain s' e2 s'' -> eval_chain s (e_comp e1 e2) s''.
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Lemma eval_chain_merge : forall (e1 e2 : expr) (vs vs' vs'' : value_stack),
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eval_chain vs e1 vs' -> eval_chain vs' e2 vs'' -> eval_chain vs (e_comp e1 e2) vs''.
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Proof.
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intros e1 e2 s s' s'' ch1 ch2.
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intros e1 e2 vs vs' vs'' ch1 ch2.
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induction ch1;
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eapply chain_middle; simpl; try (rewrite P); auto.
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Qed.
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Lemma eval_chain_split : forall (e1 e2 : expr) (s s'' : list expr),
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eval_chain s (e_comp e1 e2) s'' -> exists s', (eval_chain s e1 s') /\ (eval_chain s' e2 s'').
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Lemma eval_chain_split : forall (e1 e2 : expr) (vs vs'' : value_stack),
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eval_chain vs (e_comp e1 e2) vs'' -> exists vs', (eval_chain vs e1 vs') /\ (eval_chain vs' e2 vs'').
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Proof.
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intros e1 e2 s ss'' ch.
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intros e1 e2 vs vss'' ch.
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ltac1:(dependent induction ch).
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- simpl in P. destruct (eval_step s e1); inversion P.
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- simpl in P. destruct (eval_step s e1) eqn:Hval; try (inversion P).
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- simpl in P. destruct (eval_step vs e1); inversion P.
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- simpl in P. destruct (eval_step vs e1) eqn:Hval; try (inversion P).
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+ injection P as Hinj; subst. specialize (IHch e e2 H0) as [s'0 [ch1 ch2]].
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eexists. split.
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* eapply chain_middle. apply Hval. apply ch1.
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@ -93,9 +91,9 @@ Proof.
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Qed.
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Theorem val_step_sem : forall (e : expr) (vs vs' : value_stack),
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Sem_expr vs e vs' -> eval_chain (strip_value_list vs) e (strip_value_list vs')
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Sem_expr vs e vs' -> eval_chain vs e vs'
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with eval_step_int : forall (i : intrinsic) (vs vs' : value_stack),
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Sem_int vs i vs' -> eval_chain (strip_value_list vs) (e_int i) (strip_value_list vs').
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Sem_int vs i vs' -> eval_chain vs (e_int i) vs'.
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Proof.
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- intros e vs vs' Hsem.
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induction Hsem.
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@ -106,7 +104,7 @@ Proof.
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apply chain_final. auto.
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+ (* In compoition, by induction, we know that the two sub-expressions produce
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proper evaluation chains. Chains can be composed (via eval_chain_merge). *)
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eapply eval_chain_merge with (strip_value_list vs2); auto.
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eapply eval_chain_merge with vs2; auto.
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- intros i vs vs' Hsem.
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(* The evaluation chain depends on the specific intrinsic in use. *)
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inversion Hsem; subst;
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@ -114,15 +112,9 @@ Proof.
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try (apply chain_final; auto; fail).
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(* Only apply is non-final. The first step is popping the quote from the stack,
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and the rest of the steps are given by the evaluation of the code in the quote. *)
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apply chain_middle with e (strip_value_list vs0); auto.
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apply chain_middle with e vs0; auto.
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Qed.
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Theorem eval_step_sem_back : forall (e : expr) (vs vs' : value_stack),
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eval_chain (strip_value_list vs) e (strip_value_list vs') -> Sem_expr vs e vs'.
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Proof.
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(* Thoughts: the issue is with the apparent nondeterminism of evalution. *)
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Admitted.
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Ltac2 Type exn ::= [ | Not_intrinsic ].
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Ltac2 rec destruct_n (n : int) (vs : constr) : unit :=
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@ -146,15 +138,6 @@ Ltac2 int_arity (int : constr) : int :=
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Ltac2 destruct_int_stack (int : constr) (va: constr) := destruct_n (int_arity int) va.
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Ltac2 ensure_valid_value_stack () := Control.enter (fun () =>
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match! goal with
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| [h : eval_step (strip_value_list ?a) (e_int ?b) = ?c |- _] =>
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let h := Control.hyp h in
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destruct_int_stack b a;
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try (inversion $h; fail)
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| [|- _ ] => ()
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end).
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Ltac2 ensure_valid_stack () := Control.enter (fun () =>
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match! goal with
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| [h : eval_step ?a (e_int ?b) = ?c |- _] =>
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@ -164,15 +147,77 @@ Ltac2 ensure_valid_stack () := Control.enter (fun () =>
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| [|- _ ] => ()
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end).
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Theorem test : forall (s s': list expr), eval_step s (e_int swap) = final s' ->
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exists v1 v2 s'', s = v1 :: v2 :: s'' /\ s' = v2 :: v1 :: s''.
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Theorem test : forall (vs vs': value_stack), eval_step vs (e_int swap) = final vs' ->
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exists v1 v2 vs'', vs = v1 :: v2 :: vs'' /\ vs' = v2 :: v1 :: vs''.
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Proof.
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intros s s' Heq.
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ensure_valid_stack ().
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simpl in Heq. injection Heq as Hinj. subst. eauto.
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Qed.
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Theorem eval_step_final_sem : forall (e : expr) (vs vs' : value_stack),
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eval_step vs e = final vs' -> Sem_expr vs e vs'.
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Proof.
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intros e vs vs' Hev. destruct e.
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- destruct i; ensure_valid_stack ();
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(* Get rid of trivial cases that match one-to-one. *)
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simpl in Hev; try (injection Hev as Hinj; subst; solve_basic ()).
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+ (* compose with one quoted value *)
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destruct v. inversion Hev.
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+ (* compose with two quoted values. *)
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destruct v; destruct v0.
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injection Hev as Hinj; subst; solve_basic ().
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+ (* Apply is not final. *) destruct v. inversion Hev.
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- (* Quote is always final, trivially. *)
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simpl in Hev. injection Hev as Hinj; subst. solve_basic ().
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- (* Compose is never final. *)
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simpl in Hev. destruct (eval_step vs e1); inversion Hev.
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Qed.
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Theorem eval_step_middle_sem : forall (e ei: expr) (vs vsi vs' : value_stack),
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eval_step vs e = middle ei vsi ->
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Sem_expr vsi ei vs' ->
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Sem_expr vs e vs'.
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Proof.
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intros e. induction e; intros ei vs vsi vs' Hev Hsem.
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- destruct i; ensure_valid_stack ().
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+ (* compose with one quoted value. *)
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destruct v. inversion Hev.
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+ (* compose with two quoted values. *)
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destruct v; destruct v0. inversion Hev.
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+ (* Apply *)
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destruct v. injection Hev as Hinj; subst.
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solve_basic (). auto.
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- inversion Hev.
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- simpl in Hev.
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destruct (eval_step vs e1) eqn:Hev1.
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+ inversion Hev.
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+ injection Hev as Hinj; subst. inversion Hsem; subst.
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specialize (IHe1 e vs vsi vs2 Hev1 H2).
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eapply Sem_e_comp. apply IHe1. apply H4.
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+ injection Hev as Hinj; subst.
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specialize (eval_step_final_sem e1 vs vsi Hev1) as Hsem1.
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eapply Sem_e_comp. apply Hsem1. apply Hsem.
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Qed.
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Theorem eval_step_sem_back : forall (e : expr) (vs vs' : value_stack),
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eval_chain vs e vs' -> Sem_expr vs e vs'.
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Proof.
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intros e vs vs' ch.
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ltac1:(dependent induction ch).
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- apply eval_step_final_sem. auto.
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- specialize (eval_step_middle_sem e ei vs vsi vs' P IHch). auto.
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Qed.
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Corollary eval_step_no_sem : forall (e : expr) (vs vs' : value_stack),
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~ (Sem_expr vs e vs') -> ~(eval_chain vs e vs').
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Proof.
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intros e vs vs' Hnsem Hch.
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specialize (eval_step_sem_back _ _ _ Hch). auto.
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Qed.
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Require Extraction.
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Require Import ExtrHaskellBasic.
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Extraction Language Haskell.
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Set Extraction KeepSingleton.
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Extraction "UccGen.hs" expr eval_step true false or.
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179
code/dawn/DawnV2.v
Normal file
179
code/dawn/DawnV2.v
Normal file
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@ -0,0 +1,179 @@
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Require Import Coq.Lists.List.
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From Ltac2 Require Import Ltac2.
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Inductive intrinsic :=
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| swap
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| clone
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| drop
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| quote
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| compose
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| apply.
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Inductive expr :=
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| e_int (i : intrinsic)
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| e_quote (e : expr)
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| e_comp (e1 e2 : expr).
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Definition e_compose (e : expr) (es : list expr) := fold_left e_comp es e.
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Inductive value := v_quote (e : expr).
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Definition value_stack := list value.
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Definition value_to_expr (v : value) : expr :=
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match v with
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| v_quote e => e_quote e
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end.
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Inductive Sem_int : value_stack -> intrinsic -> value_stack -> Prop :=
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| Sem_swap : forall (v v' : value) (vs : value_stack), Sem_int (v' :: v :: vs) swap (v :: v' :: vs)
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| Sem_clone : forall (v : value) (vs : value_stack), Sem_int (v :: vs) clone (v :: v :: vs)
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| Sem_drop : forall (v : value) (vs : value_stack), Sem_int (v :: vs) drop vs
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| Sem_quote : forall (v : value) (vs : value_stack), Sem_int (v :: vs) quote ((v_quote (value_to_expr v)) :: vs)
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| 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)
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| Sem_apply : forall (e : expr) (vs vs': value_stack), Sem_expr vs e vs' -> Sem_int (v_quote e :: vs) apply vs'
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with Sem_expr : value_stack -> expr -> value_stack -> Prop :=
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| Sem_e_int : forall (i : intrinsic) (vs vs' : value_stack), Sem_int vs i vs' -> Sem_expr vs (e_int i) vs'
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| Sem_e_quote : forall (e : expr) (vs : value_stack), Sem_expr vs (e_quote e) (v_quote e :: vs)
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| Sem_e_comp : forall (e1 e2 : expr) (vs1 vs2 vs3 : value_stack),
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Sem_expr vs1 e1 vs2 -> Sem_expr vs2 e2 vs3 -> Sem_expr vs1 (e_comp e1 e2) vs3.
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Definition false : expr := e_quote (e_int drop).
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Definition false_v : value := v_quote (e_int drop).
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Definition true : expr := e_quote (e_comp (e_int swap) (e_int drop)).
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Definition true_v : value := v_quote (e_comp (e_int swap) (e_int drop)).
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Theorem false_correct : forall (v v' : value) (vs : value_stack), Sem_expr (v' :: v :: vs) (e_comp false (e_int apply)) (v :: vs).
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Proof.
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intros v v' vs.
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eapply Sem_e_comp.
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- apply Sem_e_quote.
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- apply Sem_e_int. apply Sem_apply. apply Sem_e_int. apply Sem_drop.
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Qed.
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Theorem true_correct : forall (v v' : value) (vs : value_stack), Sem_expr (v' :: v :: vs) (e_comp true (e_int apply)) (v' :: vs).
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Proof.
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intros v v' vs.
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eapply Sem_e_comp.
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- apply Sem_e_quote.
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- apply Sem_e_int. apply Sem_apply. eapply Sem_e_comp.
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* apply Sem_e_int. apply Sem_swap.
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* apply Sem_e_int. apply Sem_drop.
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Qed.
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Definition or : expr := e_comp (e_int clone) (e_int apply).
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Theorem or_false_v : forall (v : value) (vs : value_stack), Sem_expr (false_v :: v :: vs) or (v :: vs).
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Proof with apply Sem_e_int.
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intros v vs.
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eapply Sem_e_comp...
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- apply Sem_clone.
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- apply Sem_apply... apply Sem_drop.
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Qed.
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Theorem or_true : forall (v : value) (vs : value_stack), Sem_expr (true_v :: v :: vs) or (true_v :: vs).
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Proof with apply Sem_e_int.
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intros v vs.
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eapply Sem_e_comp...
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- apply Sem_clone...
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- apply Sem_apply. eapply Sem_e_comp...
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* apply Sem_swap.
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* apply Sem_drop.
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Qed.
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Definition or_false_false := or_false_v false_v.
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Definition or_false_true := or_false_v true_v.
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Definition or_true_false := or_true false_v.
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Definition or_true_true := or_true true_v.
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Fixpoint quote_n (n : nat) :=
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match n with
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| O => e_int quote
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| S n' => e_compose (quote_n n') (e_int swap :: e_int quote :: e_int swap :: e_int compose :: nil)
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end.
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Theorem quote_2_correct : forall (v1 v2 : value) (vs : value_stack),
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Sem_expr (v2 :: v1 :: vs) (quote_n 1) (v_quote (e_comp (value_to_expr v1) (value_to_expr v2)) :: vs).
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Proof with apply Sem_e_int.
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intros v1 v2 vs. simpl.
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repeat (eapply Sem_e_comp)...
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- apply Sem_quote.
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- apply Sem_swap.
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- apply Sem_quote.
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- apply Sem_swap.
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- apply Sem_compose.
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Qed.
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Theorem quote_3_correct : forall (v1 v2 v3 : value) (vs : value_stack),
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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).
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Proof with apply Sem_e_int.
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intros v1 v2 v3 vs. simpl.
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repeat (eapply Sem_e_comp)...
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- apply Sem_quote.
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- apply Sem_swap.
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- apply Sem_quote.
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- apply Sem_swap.
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- apply Sem_compose.
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- apply Sem_swap.
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- apply Sem_quote.
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- apply Sem_swap.
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- apply Sem_compose.
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Qed.
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Ltac2 rec solve_basic () := Control.enter (fun _ =>
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match! goal with
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| [|- Sem_int ?vs1 swap ?vs2] => apply Sem_swap
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| [|- Sem_int ?vs1 clone ?vs2] => apply Sem_clone
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| [|- Sem_int ?vs1 drop ?vs2] => apply Sem_drop
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| [|- Sem_int ?vs1 quote ?vs2] => apply Sem_quote
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| [|- Sem_int ?vs1 compose ?vs2] => apply Sem_compose
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| [|- Sem_int ?vs1 apply ?vs2] => apply Sem_apply
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| [|- Sem_expr ?vs1 (e_comp ?e1 ?e2) ?vs2] => eapply Sem_e_comp; solve_basic ()
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| [|- Sem_expr ?vs1 (e_int ?e) ?vs2] => apply Sem_e_int; solve_basic ()
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| [|- Sem_expr ?vs1 (e_quote ?e) ?vs2] => apply Sem_e_quote
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| [_ : _ |- _] => ()
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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.
|
||||
64
code/dawn/Ucc.hs
Normal file
64
code/dawn/Ucc.hs
Normal file
|
|
@ -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
|
||||
Loading…
Reference in New Issue
Block a user