「SF-LC」14 ImpCEvalFun

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Step-Indexed Evaluator

…Copied from 12-imp.md:

Chapter ImpCEvalFun provide some workarounds to make functional evalution works:

  1. step-indexed evaluator, i.e. limit the recursion depth. (think about Depth-Limited Search).
  2. return option to tell if it’s a normal or abnormal termination.
  3. use LETOPT...IN... to reduce the “optional unwrapping” (basicaly Monadic binding >>=!)
    this approach of let-binding became so popular in ML family.
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Notation "'LETOPT' x <== e1 'IN' e2"
   := (match e1 with
         | Some x ⇒ e2
         | None ⇒ None
       end)
   (right associativity, at level 60).

Open Scope imp_scope.
Fixpoint ceval_step (st : state) (c : com) (i : nat)
                    : option state :=
  match i with
  | O ⇒ None       (* depth-limit hit! *)
  | S i' ⇒
    match c with
      | SKIP ⇒
          Some st
      | l ::= a1 ⇒
          Some (l !-> aeval st a1 ; st)
      | c1 ;; c2 ⇒
          LETOPT st' <== ceval_step st c1 i' IN    (* option bind *)
          ceval_step st' c2 i'
      | TEST b THEN c1 ELSE c2 FI ⇒
          if (beval st b)
            then ceval_step st c1 i'
            else ceval_step st c2 i'
      | WHILE b1 DO c1 END ⇒
          if (beval st b1)
          then LETOPT st' <== ceval_step st c1 i' IN
               ceval_step st' c i'
          else Some st
    end
  end.
Close Scope imp_scope.

Relational vs. Step-Indexed Evaluation

Prove ceval_step is equiv to ceval

->

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Theorem ceval_step__ceval: forall c st st',
      (exists i, ceval_step st c i = Some st') ->
      st =[ c ]=> st'.

The critical part of proof:

  • destruct for the i.
  • induction i, generalize on all st st' c.
    1. i = 0 case contradiction
    2. i = S i' case;
      destruct c.
      • destruct (ceval_step ...) for the option
        1. None case contradiction
        2. Some case, use induction hypothesis…

<-

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Theorem ceval__ceval_step: forall c st st',
      st =[ c ]=> st' ->
      exists i, ceval_step st c i = Some st'.
Proof.
  intros c st st' Hce.
  induction Hce.