# Library Coq.field.LegacyField_Compl

Require Import List.
Definition assoc_2nd :=
(fix assoc_2nd_rec (A:Type) (B:Set)
(eq_dec:forall e1 e2:B, {e1 = e2} + {e1 <> e2})
(lst:list (prod A B)) {struct lst} :
B -> A -> A :=
fun (key:B) (default:A) =>
match lst with
| nil => default
| (v,e) :: l =>
match eq_dec e key with
| left _ => v
| right _ => assoc_2nd_rec A B eq_dec l key default
end
end).
Definition mem :=
(fix mem (A:Set) (eq_dec:forall e1 e2:A, {e1 = e2} + {e1 <> e2})
(a:A) (l:list A) {struct l} : bool :=
match l with
| nil => false
| a1 :: l1 =>
match eq_dec a a1 with
| left _ => true
| right _ => mem A eq_dec a l1
end
end).