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fairness.v
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(****************************************************************************)
(* *)
(* *)
(* Solange Coupet-Grimal *)
(* *)
(* *)
(* Laboratoire d'Informatique Fondamentale de Marseille *)
(* CMI-Technopole de Chateau-Gombert *)
(* 39, Rue F. Joliot Curie *)
(* 13453 MARSEILLE Cedex 13 *)
(* [email protected] *)
(* *)
(* *)
(* Coq V7.0 *)
(* Juillet 2002 *)
(* *)
(****************************************************************************)
(* fairness.v *)
(****************************************************************************)
Require Export ltl congruence.
Section fairness.
Variables (state label : Set) (transition : label -> relation state)
(fair : label -> Prop).
Notation Stream := (stream state) (only parsing).
Notation Stream_formula := (stream_formula state) (only parsing).
Notation Fair_step := (fair_step transition fair) (only parsing).
Notation Fairstr := (fairstr transition fair) (only parsing).
Notation Strong_fairstr := (strong_fairstr transition fair) (only parsing).
(************************* Fairness comparison ********************************)
Lemma strong_fairstr_implies_fairstr :
forall str : stream state,
strong_fairstr transition fair str -> fairstr transition fair str.
unfold fairstr in |- *; unfold strong_fairstr in |- *;
unfold infinitely_often in |- *.
intros str H_fairstr.
apply
implies_always
with
(P := eventually
(fun str' : stream state =>
fair_step transition fair (head_str str')
(head_str (tl_str str')))); auto.
clear H_fairstr.
generalize str; clear str; unfold implies in |- *; cofix strong_fairstr_implies_fairstr.
intros str; case str; clear str.
constructor; auto.
clear strong_fairstr_implies_fairstr.
intro H.
apply
implies_eventually
with
(P := fun str' : stream state =>
fair_step transition fair (head_str str') (head_str (tl_str str')));
try assumption.
clear H.
apply lift_implies_stream; auto.
Qed.
End fairness.