blog-static/code/dawn/DawnEval.v

Require Import Coq.Lists.List.
Require Import Dawn.
Require Import Coq.Program.Equality.
From Ltac2 Require Import Ltac2.

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.
  ltac1:(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 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'.
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 :=
  if Int.le n 0 then () else
    let v := Fresh.in_goal @v in
    let vs' := Fresh.in_goal @vs in
    destruct $vs as [|$v $vs']; Control.enter (fun () =>
      try (destruct_n (Int.sub n 1) (Control.hyp vs'))
    ).

Ltac2 int_arity (int : constr) : int :=
  match! int with
  | swap => 2
  | clone => 1
  | drop => 1
  | quote => 1
  | compose => 2
  | apply => 1
  | _ => Control.throw Not_intrinsic
  end.

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 |- _] =>
      let h := Control.hyp h in
      destruct_int_stack b a;
      try (inversion $h; fail)
  | [|- _ ] => ()
  end).

Theorem test : forall (s s': list expr), eval_step s (e_int swap) = final s' ->
  exists v1 v2 s'', s = v1 :: v2 :: s'' /\ s' = v2 :: v1 :: s''.
Proof.
  intros s s' Heq.
  ensure_valid_stack ().
  simpl in Heq. injection Heq as Hinj. subst. eauto.
Qed.

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