Definition beq: forall (A: Type), (A -> A -> bool) -> t A -> t A -> bool := PTree.beq.

(* Specifications: *)
Theorem beq_sound:
forall (A: Type) (eqA: A -> A -> bool) (t1 t2: t A),
beq eqA t1 t2 = true ->
forall (x: elt),
match get x t1, get x t2 with
| None, None => True
| Some y1, Some y2 => eqA y1 y2 = true
| _, _ => False
end.
Theorem beq_sound:
forall (A: Type) (eqA: A -> A -> bool) (t1 t2: t A),
beq eqA t1 t2 = true ->
forall (x: elt),
match get x t1, get x t2 with
| None, None => True
| Some y1, Some y2 => eqA y1 y2 = true
| _, _ => False
end.