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(* 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.             *)
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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
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(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(* correction de l'invariant de compatibilité par rapport à une signature *)

(* Specifications: *)
Lemma pl_produit_correct_wrt_sign_invar :
forall (p0 p1 : prec_list) (n : nat),
pl_tl_length p0 n -> pl_tl_length p1 n -> pl_tl_length (pl_produit p0 p1) n.
Lemma st_produit_l_correct_wrt_sign_invar_with_offset :
forall (s0 : state) (a : ad) (p : prec_list) (sigma : signature)
(pa : pre_ad),
state_correct_wrt_sign_with_offset s0 sigma pa ->
state_correct_wrt_sign_with_offset (M1 prec_list a p) sigma pa ->
state_correct_wrt_sign_with_offset (s_produit_l a p s0) sigma pa.
Lemma st_produit_r_correct_wrt_sign_invar_with_offset :
forall (s0 : state) (a : ad) (p : prec_list) (sigma : signature)
(pa : pre_ad),
state_correct_wrt_sign_with_offset s0 sigma pa ->
state_correct_wrt_sign_with_offset (M1 prec_list a p) sigma pa ->
state_correct_wrt_sign_with_offset (s_produit_r a p s0) sigma pa.
Lemma st_produit_correct_wrt_sign_invar_with_offset :
forall (s0 s1 : state) (sigma : signature) (pa : pre_ad),
state_correct_wrt_sign_with_offset s0 sigma pa ->
state_correct_wrt_sign_with_offset s1 sigma pa ->
state_correct_wrt_sign_with_offset (s_produit s0 s1) sigma pa.
Lemma st_produit_correct_wrt_sign_invar :
forall (s0 s1 : state) (sigma : signature),
state_correct_wrt_sign s0 sigma ->
state_correct_wrt_sign s1 sigma ->
state_correct_wrt_sign (s_produit s0 s1) sigma.
Lemma positive_sum :
forall p : positive,
p = 1%positive \/
(exists q : positive, p = xO q) \/ (exists q : positive, p = xI q).
Lemma preDTA_produit_l_correct_wrt_sign_invar :
forall (d : preDTA) (a : ad) (s : state) (sigma : signature),
predta_correct_wrt_sign d sigma ->
predta_correct_wrt_sign (M1 state a s) sigma ->
predta_correct_wrt_sign (preDTA_produit_l a s d) sigma.
Lemma preDTA_produit_r_correct_wrt_sign_invar :
forall (d : preDTA) (a : ad) (s : state) (sigma : signature),
predta_correct_wrt_sign d sigma ->
predta_correct_wrt_sign (M1 state a s) sigma ->
predta_correct_wrt_sign (preDTA_produit_r a s d) sigma.
Lemma preDTA_produit_correct_wrt_sign_invar :
forall (d0 d1 : preDTA) (sigma : signature),
predta_correct_wrt_sign d0 sigma ->
predta_correct_wrt_sign d1 sigma ->
predta_correct_wrt_sign (preDTA_produit d0 d1) sigma.
Lemma inter_correct_wrt_sign_invar :
forall (d0 d1 : DTA) (sigma : signature),
dta_correct_wrt_sign d0 sigma ->
dta_correct_wrt_sign d1 sigma -> dta_correct_wrt_sign (inter d0 d1) sigma.