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].