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(* 02110-1301 USA                                                     *)

Theorem bout_ind :
forall Pr : proc -> proc -> PP -> PP -> type -> Prop,
(forall (p q : PP) (x : VV) (t : type) (P P' : proc),
fresh q P ->
q <> p ->
sem (subs_var_proc P (pname q) x) (aout p q) P' -> Pr (res x t P) P' p q t) ->
(forall (p q : PP) (x y : VV) (t t' : type) (P P' : proc),
(forall r : PP,
sem (subs_var_proc P (pname r) x) (about p q t)
(subs_var_proc P' (pname r) y)) ->
(forall r : PP,
Pr (subs_var_proc P (pname r) x) (subs_var_proc P' (pname r) y) p q t) ->
Pr (res x t' P) (res y t' P') p q t) ->
(forall (P P' : proc) (p q : PP) (t : type),
sem (par (ban P) P) (about p q t) P' ->
Pr (par (ban P) P) P' p q t -> Pr (ban P) P' p q t) ->
(forall (P P' Q : proc) (p q : PP) (t : type),
fresh q Q ->
sem P (about p q t) P' -> Pr P P' p q t -> Pr (par P Q) (par P' Q) p q t) ->
(forall (P P' Q : proc) (p q : PP) (t : type),
fresh q Q ->
sem P (about p q t) P' -> Pr P P' p q t -> Pr (par Q P) (par Q P') p q t) ->
(forall (P P' Q : proc) (p q : PP) (t : type),
sem P (about p q t) P' -> Pr P P' p q t -> Pr (sum P Q) P' p q t) ->
(forall (P P' Q : proc) (p q : PP) (t : type),
sem P (about p q t) P' -> Pr P P' p q t -> Pr (sum Q P) P' p q t) ->
(forall (P P' : proc) (p q r : PP) (t : type),
sem P (about p q t) P' ->
Pr P P' p q t -> Pr (mat (pname r) (pname r) P) P' p q t) ->
forall (P P' : proc) (p q : PP) (t : type),
sem P (about p q t) P' -> Pr P P' p q t.