update in static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / syntax / teqg_teqg.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/teqg_teqg.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/teqg_teqg.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "static_2/syntax/teqg.ma".
+
+(* GENERIC EQUIVALENCE ON TERMS *********************************************)
+
+(* Main properties **********************************************************)
+
+theorem teqg_trans (S):
+        Transitive … S → Transitive … (teqg S).
+#S #HS #T1 #T #H elim H -T1 -T //
+[ #s1 #s #Hs1 #X #H
+  elim (teqg_inv_sort1 … H) -H /3 width=3 by teqg_sort/
+| #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
+  elim (teqg_inv_pair1 … H) -H /3 width=1 by teqg_pair/
+]
+qed-.
+
+theorem teqg_canc_sn (S):
+        symmetric … S → Transitive … S →
+        left_cancellable … (teqg S).
+/3 width=3 by teqg_trans, teqg_sym/ qed-.
+
+theorem teqg_canc_dx (S):
+        symmetric … S → Transitive … S →
+        right_cancellable … (teqg S).
+/3 width=3 by teqg_trans, teqg_sym/ qed-.
+
+theorem teqg_repl (S):
+        symmetric … S → Transitive … S →
+        replace_2 … (teqg S) (teqg S) (teqg S).
+/3 width=3 by teqg_canc_sn, teqg_trans/ qed-.
+
+(* Negated main properies ***************************************************)
+
+theorem teqg_tneqg_trans (S):
+        symmetric … S → Transitive … S →
+        ∀T1,T. T1 ≛[S] T → ∀T2. (T ≛[S] T2 → ⊥) → T1 ≛[S] T2 → ⊥.
+/3 width=3 by teqg_canc_sn/ qed-.
+
+theorem tneqg_teqg_canc_dx (S):
+        Transitive … S →
+        ∀T1,T. (T1 ≛[S] T → ⊥) → ∀T2. T2 ≛[S] T → T1 ≛[S] T2 → ⊥.
+/3 width=3 by teqg_trans/ qed-.