Section BEQ.
Variables (A: Type) (A_beq: A -> A -> bool).
Hypothesis A_eqb_eq: forall x y, A_beq x y = true <-> x = y.
Definition eqb:
forall {m n} (v1: t A m) (v2: t A n), bool :=
fix fix_beq {m n} v1 v2 :=
match v1, v2 with
|[], [] => true
|_ :: _, [] |[], _ :: _ => false
|h1 :: t1, h2 :: t2 => A_beq h1 h2 && fix_beq t1 t2
end%bool.

Lemma eqb_nat_eq: forall m n (v1: t A m) (v2: t A n)
(Hbeq: eqb v1 v2 = true), m = n.