Module VarDFunctor (Map : VWS
with Definition E.t := string
with Definition E.eq := @eq string).
Definition key_eq_dec := string_dec.
Definition value_eq_dec := string_dec.
Definition val_eq_dec : forall v v' : option value, {v = v'} + {v <> v'}.
decide equality.
auto using value_eq_dec.
Defined.
Definition trace_state_correct (trace : list (input * output)) (st : data) (st' : data) :=
forall k,
fst (interpret k (rev (inputs_with_key trace k)) (Map.find k st)) = Map.find k st'.
Ltac vard_unfold :=
unfold runHandler,
getk,
setk,
delk,
resp in *; monad_unfold.

Lemma inputs_with_key_plus_key :
forall l k i o,
input_key i = k ->
inputs_with_key (l ++ [(i, o)]) k =
(inputs_with_key l k) ++ [i].