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_in_Some_simple {A B:Type} beq k (l: list (A*B)):
(∀ a b:A, beq a b = true ↔ a = b) →
k ∈ map fst l ↔ (∃ v, assoc beq k l = Some v).