update in static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / static / feqg_feqg.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/feqg_feqg.ma b/matita/matita/contribs/lambdadelta/static_2/static/feqg_feqg.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/static/reqg_reqg.ma".
+include "static_2/static/feqg.ma".
+
+(* GENERIC EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *********************)
+
+(* Advanced properties ******************************************************)
+
+lemma feqg_sym (S):
+      reflexive … S → symmetric … S →
+      tri_symmetric … (feqg S).
+#S #H1S #H2S #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1
+/3 width=1 by feqg_intro_dx, reqg_sym, teqg_sym/
+qed-.
+
+lemma feqg_dec (S):
+      (∀s1,s2. Decidable … (S s1 s2)) →
+      ∀G1,G2,L1,L2,T1,T2. Decidable (❪G1,L1,T1❫ ≛[S] ❪G2,L2,T2❫).
+#S #HS #G1 #G2 #L1 #L2 #T1 #T2
+elim (eq_genv_dec G1 G2) #HnG destruct
+[ elim (reqg_dec … HS L1 L2 T1) #HnL
+  [ elim (teqg_dec … HS T1 T2) #HnT
+    [ /3 width=1 by feqg_intro_sn, or_introl/ ]
+  ]
+]
+@or_intror #H
+elim (feqg_inv_gen_sn … H) -H #H #HL #HT destruct
+/2 width=1 by/
+qed-. 
+
+(* Main properties **********************************************************)
+
+theorem feqg_trans (S):
+        reflexive … S → symmetric … S → Transitive … S →
+        tri_transitive … (feqg S).
+#S #H1S #H2S #H3S #G1 #G #L1 #L #T1 #T * -G -L -T
+#L #T #HL1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
+/4 width=8 by feqg_intro_sn, reqg_trans, teqg_reqg_div, teqg_trans/
+qed-.
+
+theorem feqg_canc_sn (S):
+        reflexive … S → symmetric … S → Transitive … S →
+        ∀G,G1,L,L1,T,T1. ❪G,L,T❫ ≛[S] ❪G1,L1,T1❫ →
+        ∀G2,L2,T2. ❪G,L,T❫ ≛[S] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≛[S] ❪G2,L2,T2❫.
+/3 width=5 by feqg_trans, feqg_sym/ qed-.
+
+theorem feqg_canc_dx (S):
+        reflexive … S → symmetric … S → Transitive … S →
+        ∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≛[S] ❪G,L,T❫ →
+        ∀G2,L2,T2. ❪G2,L2,T2❫ ≛[S] ❪G,L,T❫ → ❪G1,L1,T1❫ ≛[S] ❪G2,L2,T2❫.
+/3 width=5 by feqg_trans, feqg_sym/ qed-.
+
+(* Main inversion lemmas with generic equivalence on terms ******************)
+
+theorem feqg_tneqg_repl_dx (S):
+        reflexive … S → symmetric … S → Transitive … S →
+        ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≛[S] ❪G2,L2,T2❫ →
+        ∀U1,U2. ❪G1,L1,U1❫ ≛[S] ❪G2,L2,U2❫ →
+        (T2 ≛[S] U2 → ⊥) → (T1 ≛[S] U1 → ⊥).
+#S #H1S #H2S #H3S #G1 #G2 #L1 #L2 #T1 #T2 #HT #U1 #U2 #HU #HnTU2 #HTU1
+elim (feqg_inv_gen_sn … HT) -HT #_ #_ #HT
+elim (feqg_inv_gen_sn … HU) -HU #_ #_ #HU
+/3 width=5 by teqg_repl/
+qed-.