Section COMBINE.
Variables A B C: Type.
Variable f: option A -> option B -> option C.
Hypothesis f_none_none: f None None = None.
Fixpoint xcombine_l (m : t A) {struct m} : t C :=
match m with
| Leaf => Leaf
| Node l o r => Node' (xcombine_l l) (f o None) (xcombine_l r)
end.
Fixpoint xcombine_r (m : t B) {struct m} : t C :=
match m with
| Leaf => Leaf
| Node l o r => Node' (xcombine_r l) (f None o) (xcombine_r r)
end.
Fixpoint combine (m1: t A) (m2: t B) {struct m1} : t C :=
match m1 with
| Leaf => xcombine_r m2
| Node l1 o1 r1 =>
match m2 with
| Leaf => xcombine_l m1
| Node l2 o2 r2 => Node' (combine l1 l2) (f o1 o2) (combine r1 r2)
end
end.
End COMBINE.

Theorem combine_commut:
forall (A B: Type) (f g: option A -> option A -> option B),
(forall (i j: option A), f i j = g j i) ->
forall (m1 m2: t A),
combine f m1 m2 = combine g m2 m1.