Module Type TREE.
Parameter elt: Type.
Parameter elt_eq: forall (a b: elt), {a = b} + {a <> b}.
Parameter t: Type -> Type.
Parameter empty: forall (A: Type), t A.
Parameter get: forall (A: Type), elt -> t A -> option A.
Parameter set: forall (A: Type), elt -> A -> t A -> t A.
Parameter remove: forall (A: Type), elt -> t A -> t A.
(** Extensional equality between trees. *)
Parameter beq: forall (A: Type), (A -> A -> bool) -> t A -> t A -> bool.
(** Applying a function to all data of a tree. *)
Parameter map:
forall (A B: Type), (elt -> A -> B) -> t A -> t B.
(** Same as [map], but the function does not receive the [elt] argument. *)
Parameter map1:
forall (A B: Type), (A -> B) -> t A -> t B.
(** Applying a function pairwise to all data of two trees. *)
Parameter combine:
forall (A B C: Type), (option A -> option B -> option C) -> t A -> t B -> t C.
(** Enumerating the bindings of a tree. *)
Parameter elements:
forall (A: Type), t A -> list (elt * A).

Axiom elements_remove:
forall (A: Type) i v (m: t A),
get i m = Some v ->
exists l1 l2, elements m = l1 ++ (i,v) :: l2 /\ elements (remove i m) = l1 ++ l2.