Inductive changed2 : Type :=
  | Same
  | Same1
  | Same2
  | CC(m: PTree.t L.t).Inductive changed2 : Type :=
  | Same
  | Same1
  | Same2
  | CC(m: PTree.t L.t).Inductive changed2 : Type :=
  | Same
  | Same1
  | Same2
  | CC(m: PTree.t L.t).Inductive changed2 : Type :=
  | Same
  | Same1
  | Same2
  | CC(m: PTree.t L.t).Inductive changed2 : Type :=
  | Same
  | Same1
  | Same2
  | CC(m: PTree.t L.t).Fixpoint xcombine (m1 m2 : PTree.t L.t) {struct m1} : changed2 :=
    match m1, m2 with
    | PTree.Leaf, PTree.Leaf =>
        Same
    | PTree.Leaf, _ =>
        match combine_r m2 with
        | Unchanged => Same2
        | Changed m => CC m
        end
    | _, PTree.Leaf =>
        match combine_l m1 with
        | Unchanged => Same1
        | Changed m => CC m
        end
    | PTree.Node l1 o1 r1, PTree.Node l2 o2 r2 =>
        let o := f o1 o2 in
        match xcombine l1 l2, xcombine r1 r2 with
        | Same, Same =>
            match opt_beq o o1, opt_beq o o2 with
            | true, true => Same
            | true, false => Same1
            | false, true => Same2
            | false, false => CC(PTree.Node' l1 o r1)
            end
        | Same1, Same | Same, Same1 | Same1, Same1 =>
            if opt_beq o o1 then Same1 else CC(PTree.Node' l1 o r1)
        | Same2, Same | Same, Same2 | Same2, Same2 =>
            if opt_beq o o2 then Same2 else CC(PTree.Node' l2 o r2)
        | Same1, Same2 => CC(PTree.Node' l1 o r2)
        | (Same|Same1), CC r => CC(PTree.Node' l1 o r)
        | Same2, Same1 => CC(PTree.Node' l2 o r1)
        | Same2, CC r => CC(PTree.Node' l2 o r)
        | CC l, (Same|Same1) => CC(PTree.Node' l o r1)
        | CC l, Same2 => CC(PTree.Node' l o r2)
        | CC l, CC r => CC(PTree.Node' l o r)
        end
    end.

(* Specifications: *)
Lemma xcombine_eq: forall m1 m2, match xcombine m1 m2 with | Same => tree_eq m1 (PTree.combine f m1 m2) /\ tree_eq m2 (PTree.combine f m1 m2) | Same1 => tree_eq m1 (PTree.combine f m1 m2) | Same2 => tree_eq m2 (PTree.combine f m1 m2) | CC m => tree_eq m (PTree.combine f m1 m2) end.
Lemma xcombine_eq: forall m1 m2, match xcombine m1 m2 with | Same => tree_eq m1 (PTree.combine f m1 m2) /\ tree_eq m2 (PTree.combine f m1 m2) | Same1 => tree_eq m1 (PTree.combine f m1 m2) | Same2 => tree_eq m2 (PTree.combine f m1 m2) | CC m => tree_eq m (PTree.combine f m1 m2) end.