Definition relation_mul {A: Type} (R0 R1: relation A) (x y: A) := exists z, R0 x z /\ R1 z y.Fixpoint relation_power {A: Type} (n: nat) (R: relation A) :=
  match n with
  | O => eq
  | S n0 => relation_mul R (relation_power n0 R)
  end.Fixpoint partial_fun_power {A: Type} (n: nat) (f: A -> option A) (x: A) :=
  match n with
  | O => Some x
  | S n0 => match f x with
            | Some fx => partial_fun_power n0 f fx
            | None => None
            end
  end.

Lemma clos_refl_trans_addone: forall (A : Type) (R : relation A) (x y z: A), R x y -> clos_refl_trans A R y z -> clos_refl_trans A R x z.