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+Require Import ZArith Coq.Program.Wf Coq.Program.Utils.
+
+Parameter data:Set.
+
+Inductive t : Set :=
+ | Leaf : t
+ | Node : t -> data -> t -> Z -> t.
+
+Parameter avl : t -> Prop.
+Parameter bst : t -> Prop.
+Parameter In : data -> t -> Prop.
+Parameter cardinal : t -> nat.
+Definition card2 (s:t*t) := let (s1,s2) := s in cardinal s1 + cardinal s2.
+
+Parameter split : data -> t -> t*(bool*t).
+Parameter join : t -> data -> t -> t.
+Parameter add : data -> t -> t.
+
+Program Fixpoint union
+ (s:t*t)
+ (hb1: bst (fst s))(ha1: avl (fst s))(hb2: bst (snd s))(hb2: avl (snd s))
+ { measure card2 s } :
+ {s' : t | bst s' /\ avl s' /\ forall x, In x s' <-> In x (fst s) \/ In x (snd
+s)} :=
+ match s with
+ | (Leaf,t2) => t2
+ | (t1,Leaf) => t1
+ | (Node l1 v1 r1 h1, Node l2 v2 r2 h2) =>
+ if (Z_ge_lt_dec h1 h2) then
+ if (Z_eq_dec h2 1)
+ then add v2 (fst s)
+ else
+ let (l2', r2') := split v1 (snd s) in
+ join (union (l1,l2') _ _ _ _) v1 (union (r1,snd r2') _ _ _ _)
+ else
+ if (Z_eq_dec h1 1)
+ then add v1 (snd s)
+ else
+ let (l1', r1') := split v2 (fst s) in
+ join (union (l1',l2) _ _ _ _) v2 (union (snd r1',r2) _ _ _ _)
+ end.