(* BEGIN NEW MATERIAL *)
(*D1*)
Parameter tree_height_zero : forall t, {tree_height t = 0} + {tree_height t <> 0}.
Instance tree_heightable : heightable t :=
Heightable tree_height tree_height_zero.
(*D2*)
Parameter tree_round_left : nat -> t -> option t.
Instance  roundableL_tree : roundableLeft t :=
RoundableLeft tree_round_left.
(*D3*)
Parameter tree_round_right : nat -> t -> option t.
Instance  roundableR_tree : roundableRight t :=
RoundableRight tree_round_right.
(*D4*)
Parameter tree_avg : nat -> t -> t -> option t.
Instance avgable_tree : avgable t :=
Avgable tree_avg.
(*D5*)
Parameter countBLeafCT : nat -> t -> nat.
(*D6*)
Parameter share_metric : nat -> t -> nat.
(*D7*)
Parameter tree_decompose : t -> (t * t).
Instance decompose_tree : decomposible t :=
Decomposible tree_decompose.
(*D8*)
Parameter recompose : (t * t) -> t.
(*D9*)
Parameter power : nat -> nat -> nat.
(*D10*)
Parameter add : t -> t -> option t.
(*D11*)
Parameter sub : t -> t -> option t.

(*L4*)
Axiom decompose_height : forall n t1 t2 t3,
height t1 = S n ->
decompose t1 = (t2, t3) ->
(height t2 <= n)%nat /\ (height t3 <= n)%nat.