Section COMBINE.
Variable f: option L.t -> option L.t -> option L.t.
Hypothesis f_none_none: f None None = None.
Definition opt_eq (ox oy: option L.t) : Prop :=
match ox, oy with
| None, None => True
| Some x, Some y => L.eq x y
| _, _ => False
end.
Definition opt_beq (ox oy: option L.t) : bool :=
match ox, oy with
| None, None => true
| Some x, Some y => L.beq x y
| _, _ => false
end.
Definition tree_eq (m1 m2: PTree.t L.t) : Prop :=
forall i, opt_eq (PTree.get i m1) (PTree.get i m2).

Lemma tree_eq_refl: forall m, tree_eq m m.