blog-static/code/dawn/DawnEval.v

Require Import Coq.Lists.List.
Require Import Dawn.
Require Import Coq.Program.Equality.

Inductive step_result :=
  | err
  | middle (e : expr) (s : list expr)
  | final (s : list expr).

Fixpoint eval_step (s : list expr) (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 compose, (e_quote v2) :: (e_quote v1) :: vs => final (e_quote (e_comp v1 v2) :: vs)
  | e_int apply, (e_quote v1) :: vs => middle v1 vs
  | e_quote e', vs => final (e_quote e' :: vs)
  | e_comp e1 e2, vs =>
      match eval_step vs e1 with
      | final vs' => middle e2 vs'
      | middle e1' vs' => middle (e_comp e1' e2) vs'
      | err => err
      end
  | _, _ => 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')) \/
  (exists (ei : expr) (vsi : value_stack),
    eval_step (strip_value_list vs) e = middle ei (strip_value_list vsi) /\
    Sem_expr vsi ei vs').
Proof.
  intros e vs vs' Hsem.
  (* Proceed by induction on the semantics. *)
  induction Hsem.
  - inversion H; (* The expression is just an intrnsic. *)
    (* Dismiss all the straightforward "final" cases,
       of which most intrinsics are *)
    try (left; reflexivity).
    (* We are in an intermediate / middle case. *)
    right.
    (* The semantics guarantee that the expression in the
       quote evaluates to the final state. *)
    exists e, vs0. auto.
  - (* The expression is a quote. This is yet another final case. *)
    left; reflexivity.
  - (* The composition is never a final step, since we have to evaluate both
       branches to "finish up". *)
    destruct IHHsem1; right.
    + (* If the left branch finihed, only the right branch needs to be evaluted. *)
      simpl. rewrite H. exists e2, vs2. auto.
    + (* Otherwise, the left branch has an intermediate evaluation, guaranteed
         by induction to be consitent. *)
      destruct H as [ei [vsi [Heval Hsem']]].
      (* We compose the remaining part of the left branch with the right branch. *)
      exists (e_comp ei e2), vsi. simpl.
      (* The evaluation is trivially to a "middle" state. *)
      rewrite Heval. split. auto.
      eapply Sem_e_comp. apply Hsem'. apply Hsem2.
Qed.

Inductive eval_chain (s : list expr) (e : expr) (s' : list expr) : Prop :=
  | chain_final (P : eval_step s e = final s')
  | chain_middle (ei : expr) (si : list expr)
      (P : eval_step s e = middle ei si) (rest : eval_chain si ei s').

Lemma eval_chain_merge : forall (e1 e2 : expr) (s s' s'' : list expr),
  eval_chain s e1 s' -> eval_chain s' e2 s'' -> eval_chain s (e_comp e1 e2) s''.
Proof.
  intros e1 e2 s s' s'' 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),
  eval_chain s (e_comp e1 e2) s'' -> exists s', (eval_chain s e1 s') /\ (eval_chain s' e2 s'').
Proof.
  intros e1 e2 s ss'' ch.
  dependent induction ch.
  - simpl in P. destruct (eval_step s e1); inversion P.
  - simpl in P. destruct (eval_step s 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.
      * apply ch2.
    + subst. eexists. split.
      * eapply chain_final. apply Hval.
      * apply ch.
Qed.

Theorem eval_step_preservation : forall (e : expr) (vs : value_stack) (s' : list expr),
  ((exists e', eval_step (strip_value_list vs) e = middle e' s') ->
    exists vs', (strip_value_list vs') = s') /\
  ((eval_step (strip_value_list vs) e = final s') ->
    exists vs', (strip_value_list vs') = s').
Proof.
  intros e vs. induction e; split.
  - intros [e' Hst].
    destruct i; try (destruct vs as [|v1 [|v2 vs]]; inversion Hst; fail).
    * destruct vs as [|v1 [|v2 vs]]; inversion Hst;
      destruct (projT1 v1); inversion H0.
      destruct (projT1 v2); inversion H1.
    * destruct vs as [|v1 vs]; inversion Hst.
      destruct (projT1 v1); inversion H0.
      eauto.
  - intros Hst.
    destruct i; try (destruct vs as [|v1 [|v2 vs]]; inversion Hst; fail).
    * destruct vs as [|v1 [|v2 vs]]; inversion Hst. exists (v2 :: v1 :: vs). reflexivity.
    * destruct vs as [|v1 vs]; inversion Hst. exists (v1 :: v1 :: vs). reflexivity.
    * destruct vs as [|v1 vs]; inversion Hst. exists vs. reflexivity.
    * destruct vs as [|v1 vs]; inversion Hst. exists (v_quote (projT1 v1) :: vs). reflexivity.
    * destruct vs as [|v1 [|v2 vs]]; inversion Hst.
      + destruct (projT1 v1); inversion H0.
      + destruct (projT1 v1); destruct (projT1 v2); inversion H0.
        exists (v_quote (e_comp e0 e) :: vs). reflexivity.
    * destruct vs as [|v1 vs]; inversion Hst;
      destruct (projT1 v1); inversion H0.
  - intros [e' Hst]. inversion Hst.
  - intros Hst. inversion Hst. exists (v_quote e :: vs). reflexivity.
  - intros [e' Hst]. inversion Hst.
    destruct (eval_step (strip_value_list vs) e1) eqn:Hst1.
    * inversion H0.
    * specialize (IHe1 s') as [IHe1 _]. apply IHe1. exists e.
      injection H0 as Hinj; subst; auto.
    * specialize (IHe1 s') as [_ IHe1]. apply IHe1. 
      injection H0 as Hinj; subst; auto.
  - intros Hst.
    inversion Hst. destruct (eval_step (strip_value_list vs) e1); inversion H0.
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')
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').
Proof.
  - intros e vs vs' Hsem.
    induction Hsem.
    + (* This is an intrinsic, which is handled by the second
         theorem, eval_step_int. This lemma is used here. *)
      auto.
    + (* A quote doesn't have a next step, and so is final. *)
      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.
  - intros i vs vs' Hsem.
    (* The evaluation chain depends on the specific intrinsic in use. *)
    inversion Hsem; subst;
    (* Most intrinsics produce a final value, and the evaluation chain is trivial. *)
    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.
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'
with eval_step_int_back : forall (i : intrinsic) (vs vs' : value_stack),
  eval_chain (strip_value_list vs) (e_int i) (strip_value_list vs') -> Sem_int vs i vs'.
Proof.
  (* Thoughts: prove a "step value stack preservation" property. A step suggested by
     eval_step will ensure the stack contains values if it contained values initially.

     Done! eval_step_preservation.

     From there, by induction, a chain preserves value stacks. *)
Admitted.


Theorem eval_step_sem_not : forall (e : expr) (vs : value_stack),
  ~ (exists vs', Sem_expr vs e vs') -> ~(exists vs', eval_chain (strip_value_list vs) e (strip_value_list vs'))
with eval_step_int_not : forall (i : intrinsic) (vs : value_stack),
  ~ (exists vs', Sem_int vs i vs') -> ~(exists vs', eval_chain (strip_value_list vs) (e_int i) (strip_value_list vs')).
Proof.
(*
  - intros e vs Hnsem [vs' Hev].
    destruct e.
    + specialize (eval_step_sem_not _ _ Hnsem). apply eval_step_sem_not.
      eexists. apply Hev.
    + apply Hnsem. eexists. apply Sem_e_quote.
    + inversion Hev.
      * simpl in P. destruct (eval_step (strip_value_list vs) e1); inversion P.
      * specialize (eval_chain_split e1 e2 (strip_value_list vs) (strip_value_list vs') Hev) as [s' [ch1 ch2]].
        assert (Hnboth : ~ (exists vsi, Sem_expr vs e1 vsi /\ Sem_expr vsi e2 vs')).
        { intros [vsi [H1 H2]]. apply Hnsem. exists vs'. eapply Sem_e_comp. apply H1. apply H2. }

        assert (Hncomp : ~ (exists vsi, Sem_expr vs e1 vsi) \/ exists vsi, Sem_expr vs e1 vsi /\ ~(Sem_expr vsi e2 vs')).
        { 
  - intros i vs Hnint [vs' Hev]. destruct i.
    + destruct vs as [|v [|v' vs]]; inversion Hev; simpl in P; inversion P.
      apply Hnint. eexists. apply Sem_swap.
    + destruct vs as [|v vs]; inversion Hev; simpl in P; inversion P.
      apply Hnint. eexists. apply Sem_clone.
    + destruct vs as [|v vs]; inversion Hev; simpl in P; inversion P.
      apply Hnint. eexists. apply Sem_drop.
    + destruct vs as [|v vs]; inversion Hev; simpl in P; inversion P.
      apply Hnint. eexists. apply Sem_quote.
    + destruct vs as [|v [|v' vs]];
      try (destruct v; destruct x);
      try (destruct v'; destruct x0);
      simpl in Hev; inversion Hev; simpl in P; inversion P; subst.
      destruct i; destruct i0.
      apply Hnint. eexists. apply Sem_compose.
    + destruct vs as [|v vs].
      * simpl in Hev; inversion Hev; simpl in P; inversion P.
      * (* Monkey at keyboard mode engaged here. *)
        destruct v eqn:Hv. destruct i eqn:Hi.
        simpl in Hev; inversion Hev; simpl in P; inversion P.
        injection P as Heq. subst.
        assert (Hna : ~(exists vs', Sem_expr vs ei vs')).
        { intros [vs'0 Hsem]. apply Hnint. eexists. apply Sem_apply. apply Hsem. }
        specialize (eval_step_sem_not _ _ Hna). apply eval_step_sem_not. eexists.
        apply rest.
 *)
Admitted.

Require Extraction.
Require Import ExtrHaskellBasic.
Extraction Language Haskell.
Extraction "UccGen.hs" eval_step true false or.