Section Controlled.
Variables A X: Type.
Variable TX: reduction_t A X.
Variables B B': relation X.
Hypothesis HB: bicontrolled TX B.
Hypothesis HB': bicontrolled TX B'.
Let F  := chaining_r (X:=X) (bisim TX TX).
Let G  := Comp (star (X:=X)) (Union2 (identity (X:=X) (Y:=X)) (constant (bisim TX TX))).
Let F_mon: monotonic TX TX F.
Let FG: contains F G.
Let BG: contains (chaining_l (star B)) G.
Let B'G: contains (chaining_l (star B')) G.
Variable R: relation X.
Hypothesis HRt: evolve_t TX TX R (comp (star B) (F R)).
Hypothesis HRa: evolve_a TX TX R (G R).
Hypothesis HRt': evolve_t TX TX (trans R) (comp (star B') (F (trans R))).
Hypothesis HRa': evolve_a TX TX (trans R) (G (trans R)).

(** Theorem 3.6 *)
Theorem upto_ctrl: incl R (bisim TX TX).
End Controlled.