# 「SF-LC」14 ImpCEvalFun

## Logical Foundations - An Evaluation Function For Imp

Posted by Hux on January 14, 2019

## 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.