Section assoc.
Fixpoint assoc {A B : Type}(eq : A -> A -> bool)(x : A)(l : list (A * B)) : option B :=
match l with
| nil => None
| (x', y) :: l' => if eq x x' then Some y else assoc eq x l'
end.
Definition assoc_default {A B : Type}
(eqA : A -> A -> bool) (d : B) (l : list (A * B)) (x : A) : B :=
match assoc eqA x l with
| None => d
| Some b => b
end.

Lemma assoc_app_eq_None (A B:Type) (eq: A -> A -> bool) (x: A) (l1 l2: list (A * B)) :
assoc eq x (l1 ++ l2) = None <-> (assoc eq x l1 = None /\ assoc eq x l2 = None).