Inductive Forall2' (A B : Type) (R : A -> B -> Type) : list A -> list B -> Type :=
Forall2'_nil : Forall2' R [] []
| Forall2'_cons : forall (x : A) (y : B) (l : list A) (l' : list B),
R x y -> Forall2' R l l' -> Forall2' R (x :: l) (y :: l').

Lemma evaluates_sound t v c:
(forall (t0 : term) (v0 : value), assoc_cache t0 c = Some v0
-> evaluates t0 v0 /\ max_arity_term (term_from_value variable function v0) <= max_arity) ->
evaluates t v ->
max_arity_term t <= max_arity ->
exists p, wf p /\ proj_right p = v /\ proj_left p = t /\ cache_left p = c /\
max_arity_term (term_from_value variable function v) <= max_arity /\
(forall t0 v0, assoc_cache t0 (cache_right p) = Some v0
-> evaluates t0 v0 /\ max_arity_term (term_from_value variable function v0) <= max_arity).