main part of lfpxs_cpxs ...

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfpxs_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_lfpx.ma".
+include "basic_2/rt_computation/lfpxs_fqup.ma".
+
+(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
+
+(* Properties with uncounted context-sensitive rt-computation for terms *****)
+
+lemma lfpxs_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
+                  ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
+#h #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=3 by lfpxs_strap1, lfpx_pair/
+qed.
+
+(* Basic_2A1: was just: lpxs_cpx_trans *)
+lemma lfpxs_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (lfpxs h G).
+/3 width=5 by s_r_trans_LTC2, lfpx_cpxs_trans/ qed-.
+
+(* Note: lfpxs_cpx_conf does not hold, thus we cannot invoke s_r_trans_LTC1 *)
+(* Basic_2A1: was just: lpxs_cpxs_trans *)
+lemma lfpxs_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (lfpxs h G).
+#h #G @s_r_to_s_rs_trans @s_r_trans_LTC2
+@s_rs_trans_TC1 /2 width=3 by lfpx_cpxs_trans/ (**) (* full auto too slow *)
+qed-.
+
+(* Advanced properties on uncounted rt-computation for terms ****************)
+
+lemma cpxs_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
+                  ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
+                  ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+/4 width=3 by lfpxs_cpxs_trans, lfpxs_pair, cpxs_bind/ qed.
+
+(* Advanced inversion lemmas on uncounted rt-computation for terms **********)
+
+lemma cpxs_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 &
+                               U2 = ⓛ{p}V2.T2.
+#h #p #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
+#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
+elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
+lapply (lfpxs_cpx_trans … HT02 (L.ⓛV1) ?)
+/3 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro/
+qed-.
+
+lemma cpxs_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 → (
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 &
+                               U2 = ⓓ{p}V2.T2
+                      ) ∨
+                      ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≡ T2 & p = true.
+#h #p #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
+#U0 #U2 #_ #HU02 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+  elim (cpx_inv_abbr1 … HU02) -HU02 *
+  [ #V2 #T2 #HV02 #HT02 #H destruct
+    lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?)
+    /4 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
+  | #T2 #HT02 #HUT2
+    lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
+    /4 width=3 by lfpxs_pair, cpxs_trans, ex3_intro, or_intror/
+  ]
+| #U1 #HTU1 #HU01
+  elim (cpx_lifts … HU02 (Ⓣ) … (L.ⓓV1) … HU01)
+  /4 width=3 by cpxs_strap1, drops_refl, drops_drop, ex3_intro, or_intror/
+]
+qed-.