(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Axiom
typest_writing :
forall t t' : type,
typest t t' -> capst (getcap t') Write -> typest (getobj t') (getobj t).
Axiom
typest_reading :
forall t t' : type,
typest t t' -> capst (getcap t') Read -> typest (getobj t) (getobj t').
Axiom
typest_tail :
forall t t' : type, typest t t' -> capst (getcap t) (getcap t').

(* Specifications: *)
Theorem subtyping_extension :
forall (G D : env) (p : PP) (t1 t2 : type),
envst D G -> typest t1 t2 -> envst (addenv D p t1) (addenv G p t2).
Theorem subtyping :
forall (G : env) (P : proc),
typing G P -> forall D : env, envst D G -> typing D P.
Theorem weaken_env :
forall (G : env) (P : proc) (q : PP) (t : type),
typest t (G q) -> typing G P -> typing (addenv G q t) P.
Theorem est_refl : forall G : env, envst G G.
Theorem type_with_other_subs :
forall (G : env) (P : proc) (p q : PP) (x : VV) (t : type),
fresh p P ->
fresh q P ->
typing (addenv G p t) (subs_var_proc P (pname p) x) ->
typing (addenv G q t) (subs_var_proc P (pname q) x).
Theorem subs_typing :
forall (G : env) (P : proc),
typing G P ->
forall p q : PP,
typest (G p) (G q) -> typing G (subs_par_proc P (pname p) q).