Section Def'.
Variables X Y Z X' Y' Z' X'' Y'': Type.
Definition transparent (B: relation X) (F: function2 X Y X Y') :=
forall R, incl (F (comp (star B) R)) (comp (star B) (F R)).
(** Chaining functions *)
Definition chaining_l (S: relation2 X Y): function2 Y Z X Z := comp S.
Definition chaining_r (S: relation2 Y Z): function2 X Y X Z := fun R => comp R S.
Definition Chain (F: function2 X Y X' Y') (G: function2 X Y Y' Z') := fun R => comp (F R) (G R).
(** Composition *)
Definition Comp (G: function2 X' Y' X'' Y'') (F: function2 X Y X' Y') := fun R => G (F R).
Variable F: function X Y.
Variable R: relation2 X Y.
(** Simple iteration function *)
Fixpoint Exp(n: nat): relation2 X Y :=
match n with
| O => R
| S n => F (Exp n)
end.
Definition Iter := union Exp.
(** Increasing iteration function *)
(* inverser n et R ? *)
Fixpoint UExp(n: nat): relation2 X Y :=
match n with
| O => R
| S n => union2 (UExp n) (F (UExp n))
end.
Definition UIter := union UExp.

Lemma UExp_UExp: forall R m n, UExp F (UExp F R n) m = UExp F R (m+n).