(** More or less generic evolution predicates (Definitions 1.2)  *)
    Definition evolve_1 l R S := diagram (TX l) R (Weak TY l) S.Definition evolve_t   R S := evolve_1 (T A) R S.Definition evolve_a   R S := forall a, evolve_1 (L a) R S .Definition evolve     R S := forall l, evolve_1 l R S.(** (Silent) simulation, expansion (Definitions 1.3) *)
    Definition simulation_t R := evolve_t R R.Definition simulation   R := evolve   R R.(** (Relaxed) expansion  (on one side only) *)
    Definition expansion1   R := diagram_r TX R (EWeak TY) R.Definition wexpansion1  R := diagram_r TX R (REWeak TY) R.(** Monotonicity: Definition 1.5 *)
    Record monotonic: Prop := mkmon {
      mon_m:> increasing F;
      mon_t:  forall R S, evolve_t R S -> incl R S -> evolve_t (F R) (F S);
      mon_a:  forall R S, evolve   R S -> incl R S -> evolve_a (F R) (F S)
    }.(** Monotonicity: Definition 1.5 *)
    Record monotonic: Prop := mkmon {
      mon_m:> increasing F;
      mon_t:  forall R S, evolve_t R S -> incl R S -> evolve_t (F R) (F S);
      mon_a:  forall R S, evolve   R S -> incl R S -> evolve_a (F R) (F S)
    }.(** Monotonicity: Definition 1.5 *)
    Record monotonic: Prop := mkmon {
      mon_m:> increasing F;
      mon_t:  forall R S, evolve_t R S -> incl R S -> evolve_t (F R) (F S);
      mon_a:  forall R S, evolve   R S -> incl R S -> evolve_a (F R) (F S)
    }.(** Monotonicity: Definition 1.5 *)
    Record monotonic: Prop := mkmon {
      mon_m:> increasing F;
      mon_t:  forall R S, evolve_t R S -> incl R S -> evolve_t (F R) (F S);
      mon_a:  forall R S, evolve   R S -> incl R S -> evolve_a (F R) (F S)
    }.(** Monotonicity: Definition 1.5 *)
    Record monotonic: Prop := mkmon {
      mon_m:> increasing F;
      mon_t:  forall R S, evolve_t R S -> incl R S -> evolve_t (F R) (F S);
      mon_a:  forall R S, evolve   R S -> incl R S -> evolve_a (F R) (F S)
    }.(** Weak monotonicity: Definition 2.1 *)
    Record wmonotonic: Prop := mkwmon {
      wmon_m:> increasing F;
      wmon_t:  forall R, simulation_t R -> simulation_t (F R);
      wmon_a:  forall R S, simulation_t R -> simulation_t S -> evolve_a R S -> incl R S -> evolve_a (F R) (F S)
    }.(** Weak monotonicity: Definition 2.1 *)
    Record wmonotonic: Prop := mkwmon {
      wmon_m:> increasing F;
      wmon_t:  forall R, simulation_t R -> simulation_t (F R);
      wmon_a:  forall R S, simulation_t R -> simulation_t S -> evolve_a R S -> incl R S -> evolve_a (F R) (F S)
    }.(** Weak monotonicity: Definition 2.1 *)
    Record wmonotonic: Prop := mkwmon {
      wmon_m:> increasing F;
      wmon_t:  forall R, simulation_t R -> simulation_t (F R);
      wmon_a:  forall R S, simulation_t R -> simulation_t S -> evolve_a R S -> incl R S -> evolve_a (F R) (F S)
    }.(** Weak monotonicity: Definition 2.1 *)
    Record wmonotonic: Prop := mkwmon {
      wmon_m:> increasing F;
      wmon_t:  forall R, simulation_t R -> simulation_t (F R);
      wmon_a:  forall R S, simulation_t R -> simulation_t S -> evolve_a R S -> incl R S -> evolve_a (F R) (F S)
    }.(** Weak monotonicity: Definition 2.1 *)
    Record wmonotonic: Prop := mkwmon {
      wmon_m:> increasing F;
      wmon_t:  forall R, simulation_t R -> simulation_t (F R);
      wmon_a:  forall R S, simulation_t R -> simulation_t S -> evolve_a R S -> incl R S -> evolve_a (F R) (F S)
    }.Record controlled: Prop := mkctrl {
      ctrl_t:  forall R, evolve_t R (comp (star B) R) -> simulation_t (comp (star B) R);
      ctrl_a:  forall R S, evolve_t R (comp (star B) R) -> simulation_t S -> 
	                  evolve_a R S -> incl R S -> evolve_a (comp (star B) R) (comp (star B) S)
    }.Record controlled: Prop := mkctrl {
      ctrl_t:  forall R, evolve_t R (comp (star B) R) -> simulation_t (comp (star B) R);
      ctrl_a:  forall R S, evolve_t R (comp (star B) R) -> simulation_t S -> 
	                  evolve_a R S -> incl R S -> evolve_a (comp (star B) R) (comp (star B) S)
    }.Record controlled: Prop := mkctrl {
      ctrl_t:  forall R, evolve_t R (comp (star B) R) -> simulation_t (comp (star B) R);
      ctrl_a:  forall R S, evolve_t R (comp (star B) R) -> simulation_t S -> 
	                  evolve_a R S -> incl R S -> evolve_a (comp (star B) R) (comp (star B) S)
    }.Record controlled: Prop := mkctrl {
      ctrl_t:  forall R, evolve_t R (comp (star B) R) -> simulation_t (comp (star B) R);
      ctrl_a:  forall R S, evolve_t R (comp (star B) R) -> simulation_t S -> 
	                  evolve_a R S -> incl R S -> evolve_a (comp (star B) R) (comp (star B) S)
    }.

Lemma union2_evolve_left: forall l R S S', evolve_1 l R S -> evolve_1 l R (union2 S S').