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(* 02110-1301 USA                                                     *)

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