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).
Hint Resolve opt_beq_correct opt_eq_refl opt_eq_sym
tree_eq_refl tree_eq_sym
tree_eq_node tree_eq_node' tree_eq_node'' : combine.
Inductive changed: Type := Unchanged | Changed (m: PTree.t L.t).
Fixpoint combine_l (m : PTree.t L.t) {struct m} : changed :=
match m with
| PTree.Leaf =>
Unchanged
| PTree.Node l o r =>
let o' := f o None in
match combine_l l, combine_l r with
| Unchanged, Unchanged => if opt_beq o' o then Unchanged else Changed (PTree.Node' l o' r)
| Unchanged, Changed r' => Changed (PTree.Node' l o' r')
| Changed l', Unchanged => Changed (PTree.Node' l' o' r)
| Changed l', Changed r' => Changed (PTree.Node' l' o' r')
end
end.

Lemma combine_l_eq:
forall m,
tree_eq (match combine_l m with Unchanged => m | Changed m' => m' end)
(PTree.xcombine_l f m).