- cprs and cnx on the way

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tdeq.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/syntax/tdeq_tdeq.ma".
+include "basic_2/rt_computation/cpxs.ma".
+include "basic_2/rt_transition/cpx_lfdeq.ma".
+include "basic_2/static/lfdeq_fqup.ma".
+include "basic_2/rt_transition/lfpx_fqup.ma".
+
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+
+axiom tdeq_dec: ∀h,o,T1,T2. Decidable (tdeq h o T1 T2).
+
+axiom tdeq_canc_sn: ∀h,o. left_cancellable … (tdeq h o).
+
+lemma tdeq_cpx_trans: ∀h,o,U1,T1. U1 ≡[h, o] T1 → ∀G,L,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → 
+                      ∃∃U2.  ⦃G, L⦄ ⊢ U1 ⬈[h] U2 & U2 ≡[h, o] T2.
+#h #o #U1 #T1 #HUT1 #G #L #T2 #HT12
+elim (cpx_tdeq_conf_lexs … o … HT12 … U1 … L … L) /3 width=3 by tdeq_sym, ex2_intro/
+qed-.
+
+lemma tdeq_cpxs_trans: ∀h,o,U1,T1. U1 ≡[h, o] T1 → ∀G,L,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → 
+                       ∃∃U2.  ⦃G, L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≡[h, o] T2.
+#h #o #U1 #T1 #HUT1 #G #L #T2 #HT12 @(cpxs_ind … HT12) -T2 /2 width=3 by ex2_intro/
+#T #T2 #_ #HT2 * #U #HU1 #HUT elim (tdeq_cpx_trans … HUT … HT2) -T -T1
+/3 width=3 by ex2_intro, cpxs_strap1/
+qed-.
+
+(* Note: this requires tdeq to be symmetric *)
+lemma cpxs_tdneq_inv_step_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≡[h, o] T2 → ⊥) →
+                              ∃∃T,T0. ⦃G, L⦄ ⊢ T1 ⬈[h] T & T1 ≡[h, o] T → ⊥ & ⦃G, L⦄ ⊢ T ⬈*[h] T0 & T0 ≡[h, o] T2.
+#h #o #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
+[ #H elim H -H //
+| #T1 #T #H1 #H2 #IH #Hn12 elim (tdeq_dec h o T1 T) #H destruct
+  [ -H1 -H2 elim IH -IH /3 width=3 by tdeq_trans/ -Hn12
+    #X #X2 #HTX #HnTX #HX2 #HXT2 elim (tdeq_cpx_trans … H … HTX) -HTX
+    #X1 #HTX1 #HX1 elim (tdeq_cpxs_trans … HX1 … HX2) -HX2
+    /5 width=8 by tdeq_canc_sn, tdeq_trans, ex4_2_intro/ (* Note: 2 tdeq_trans *)
+  | -IH -Hn12 /3 width=6 by ex4_2_intro/
+  ]
+]
+qed-.
+
+(* Basic_2A1: removed theorems 1: cpxs_neq_inv_step_sn *)