(** * Composition of simulation, expansion1, wexpansion1 relations *)
Section Composition.
Variables X Y Z: Type.
Variable TX: reduction_t A X.
Variable TY: reduction_t A Y.
Variable TZ: reduction_t A Z.
Variable R: relation2 X Y.
Variable S: relation2 Y Z.

(* Specifications: *)
Lemma simulation_comp: simulation TX TY R -> simulation TY TZ S -> simulation TX TZ (comp R S).
Lemma expansion1_comp: expansion1 TX TY R -> expansion1 TY TZ S -> expansion1 TX TZ (comp R S).
Let wexpansion1_t: wexpansion1 TY TZ S ->
forall x x' y, star (TY (T _)) x x' -> S x y -> exists2 y', star (TZ (T _)) y y' & S x' y'.
Lemma wexpansion1_comp: wexpansion1 TX TY R -> wexpansion1 TY TZ S -> wexpansion1 TX TZ (comp R S).
End Composition.