advances on cpxs and cnx (cnxa removed) ,,,

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.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_computation/cpxs_tdeq.ma".
+include "basic_2/rt_computation/cpxs_cpxs.ma".
+include "basic_2/rt_computation/csx_csx.ma".
+
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+
+(* Properties with uncounted context-sensitive rt-computation for terms *****)
+
+(* Basic_1: was just: sn3_intro *)
+lemma csx_intro_cpxs: ∀h,o,G,L,T1.
+                      (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≡[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
+                      ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
+/4 width=1 by cpx_cpxs, csx_intro/ qed-.
+
+(* Basic_1: was just: sn3_pr3_trans *)
+lemma csx_cpxs_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+                      ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
+#h #o #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
+/2 width=3 by csx_cpx_trans/
+qed-.
+
+(* Eliminators with uncounted context-sensitive rt-computation for terms ****)
+
+lemma csx_ind_cpxs_tdeq: ∀h,o,G,L. ∀R:predicate term.
+                         (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+                               (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≡[h, o] T2 → ⊥) → R T2) → R T1
+                         ) →
+                         ∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+                         ∀T0. ⦃G, L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≡[h, o] T2 → R T2.
+#h #o #G #L #R #IH #T1 #H @(csx_ind … H) -T1
+#T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02
+@IH -IH /3 width=3 by csx_cpxs_trans, csx_tdeq_trans/ -HT1 #V2 #HTV2 #HnTV2
+lapply (tdeq_tdneq_trans … HT02 … HnTV2) -HnTV2 #H
+elim (tdeq_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
+lapply (tndeq_tdeq_canc_dx … H … HV02) -H #HnTV0
+elim (tdeq_dec h o T1 T0) #H
+[ lapply (tdeq_tdneq_trans … H … HnTV0) -H -HnTV0 #Hn10
+  lapply (cpxs_trans … HT10 … HTV0) -T0 #H10
+  elim (cpxs_tdneq_inv_step_sn … H10 …  Hn10) -H10 -Hn10
+  /3 width=8 by tdeq_trans/
+| elim (cpxs_tdneq_inv_step_sn … HT10 … H) -HT10 -H #T #V #HT1 #HnT1 #HTV #HVT0
+  elim (tdeq_cpxs_trans … HVT0 … HTV0) -T0
+  /3 width=8 by cpxs_trans, tdeq_trans/
+]
+qed-.
+
+lemma csx_ind_alt: ∀h,o,G,L. ∀R:predicate term.
+                   (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+                         (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≡[h, o] T2 → ⊥) → R T2) → R T1
+                   ) →
+                   ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → R T.
+#h #o #G #L #R #IH #T #HT
+@(csx_ind_cpxs_tdeq … IH … HT) -IH -HT // (**) (* full auto fails *)
+qed-.