(* 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.

(*This one looks like L4 Axiom decompose_share_height_decrease : forall sh sh' sh'' n, decompose sh = (sh',sh'') -> height sh = S n -> (height sh' <= n /\ height sh'' <= n)%nat. *)
(*L11*)
Axiom decompose_height_le: forall n s s1 s2,
decompose s = (s1,s2) ->
(height s <= S n)%nat ->
(height s1 <= n)%nat /\ (height s2 <= n)%nat.