Section defs.
(** * Trees and heap trees *)
(** ** Definition of trees over an ordered set *)
Variable A : Type.
Variable leA : relation A.
Variable eqA : relation A.
Let gtA (x y:A) := ~ leA x y.
Hypothesis leA_dec : forall x y:A, {leA x y} + {leA y x}.
Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Hypothesis leA_refl : forall x y:A, eqA x y -> leA x y.
Hypothesis leA_trans : forall x y z:A, leA x y -> leA y z -> leA x z.
Hypothesis leA_antisym : forall x y:A, leA x y -> leA y x -> eqA x y.
Hint Resolve leA_refl : core.
Hint Immediate eqA_dec leA_dec leA_antisym : core.
Let emptyBag := EmptyBag A.
Let singletonBag := SingletonBag _ eqA_dec.
Inductive Tree :=
| Tree_Leaf : Tree
| Tree_Node : A -> Tree -> Tree -> Tree.
(** [a] is lower than a Tree [T] if [T] is a Leaf or [T] is a Node holding [b>a] *)
Definition leA_Tree (a:A) (t:Tree) :=
match t with
| Tree_Leaf => True
| Tree_Node b T1 T2 => leA a b
end.

Lemma leA_Tree_Node :
forall (a b:A) (G D:Tree), leA a b -> leA_Tree a (Tree_Node b G D).