Ltac cgen H := generalize H; clear H.Definition reduction := A -> relation X.Definition incl_r: relation reduction := fun R1 R2 => forall a, incl (R1 a) (R2 a).Definition eeq_r: relation reduction := fun R1 R2 => forall a, eeq (R1 a) (R2 a).Definition diagram_r(RX: reduction A X) R (RY: reduction A Y) S := forall a, diagram (RX a) R (RY a) S.(** A `label' is either the silent action, or a visible one *)  
    Inductive Lbl: Type := T | L(a: A).(** A `label' is either the silent action, or a visible one *)  
    Inductive Lbl: Type := T | L(a: A).(** A `label' is either the silent action, or a visible one *)  
    Inductive Lbl: Type := T | L(a: A).Definition reduction_t := reduction Lbl.(** Weak transition relation *)  
    Definition Weak: reduction_t X := fun l => 
      match l with 
	| T => star (Red T)
	| L a => comp (star (Red T)) (comp (Red (L a)) (star (Red T)))
      end.(** Transition relation for expansion *)
    Definition EWeak: reduction_t X := fun l => 
      match l with 
	| T => union2 (eq (A:=X)) (Red T)
	| L a => Red (L a)
      end.(** Transition relation for relaxed expansion *)
    Definition REWeak: reduction_t X := fun l => 
      match l with 
	| T => union2 (eq (A:=X)) (Red T)
	| L a => comp (Red (L a)) (star (Red T))
      end.

(* Specifications: *)
Lemma weak_refl: forall x, Weak T x x.
Lemma tau_weak: forall y l x z, Red T x y -> Weak l y z -> Weak l x z.
Lemma weak_tau: forall y l x z, Red l x y -> Weak T y z -> Weak l x z.
Lemma red_weak: forall l x y, Red l x y -> Weak l x y.
Lemma taus_weak: forall y l x z, Weak T x y -> Weak l y z -> Weak l x z.
Lemma weak_taus: forall y l x z, Weak l x y -> Weak T y z -> Weak l x z.
Theorem Weak_ind: forall P: Lbl -> X -> X -> Prop, (forall x, P T x x) -> (forall y l x z , Red T x y -> Weak l y z -> P l y z -> P l x z) -> (forall y a x z, Red (L a) x y -> Weak T y z -> P T y z -> P (L a) x z) -> forall l x x', Weak l x x' -> P l x x'.