update in static_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_computation / lpxs_reqg.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_reqg.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_reqg.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 "basic_2/rt_transition/lpx_reqg.ma".
+include "basic_2/rt_computation/lpxs_lpx.ma".
+
+(* EXTENDED PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS *************)
+
+(* Properties with generic equivalence on referred entries ******************)
+
+(* Basic_2A1: uses: lleq_lpxs_trans *)
+lemma reqg_lpxs_trans (S) (G) (T:term):
+      reflexive … S → symmetric … S →
+      ∀L2,K2. ❪G,L2❫ ⊢ ⬈* K2 → ∀L1. L1 ≛[S,T] L2 →
+      ∃∃K1. ❪G,L1❫ ⊢ ⬈* K1 & K1 ≛[S,T] K2.
+#S #G #T #H1S #H2S #L2 #K2 #H @(lpxs_ind_sn … H) -L2 /2 width=3 by ex2_intro/
+#L #L2 #HL2 #_ #IH #L1 #HT
+elim (reqg_lpx_trans … HL2 … HT) // -L #L #HL1 #HT
+elim (IH … HT) -L2 #K #HLK #HT
+/3 width=3 by lpxs_step_sn, ex2_intro/
+qed-.
+
+(* Basic_2A1: uses: lpxs_nlleq_inv_step_sn *)
+lemma lpxs_rneqg_inv_step_sn (S) (G) (T:term):
+      reflexive … S → symmetric … S → Transitive … S →
+      (∀s1,s2. Decidable (S s1 s2)) →
+      ∀L1,L2. ❪G,L1❫ ⊢ ⬈* L2 → (L1 ≛[S,T] L2 → ⊥) →
+      ∃∃L,L0. ❪G,L1❫ ⊢ ⬈ L & L1 ≛[S,T] L → ⊥ & ❪G,L❫ ⊢ ⬈* L0 & L0 ≛[S,T] L2.
+#S #G #T #H1S #H2S #H3S #H4S #L1 #L2 #H @(lpxs_ind_sn … H) -L1
+[ #H elim H -H /2 width=1 by reqg_refl/
+| #L1 #L #H1 #H2 #IH2 #H12 elim (reqg_dec S … L1 L T) // #H
+  [ -H1 -H2 elim IH2 -IH2 /3 width=3 by reqg_trans/ -H12
+    #L0 #L3 #H1 #H2 #H3 #H4 lapply (reqg_rneqg_trans … H … H2) -H2 //
+    #H2 elim (reqg_lpx_trans … H1 … H) -L //
+    #L #H1 #H lapply (rneqg_reqg_div … H … H2) -H2 //
+    #H2 elim (reqg_lpxs_trans … H3 … H) -L0
+    /3 width=8 by reqg_trans, ex4_2_intro/
+  | -H12 -IH2 /3 width=6 by reqg_refl, ex4_2_intro/
+  ]
+]
+qed-.