lambdadelta

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / dynamic / lsubsv_cpds.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpds.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpds.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/dynamic/snv_cpcs.ma".
+include "basic_2/dynamic/lsubsv_ssta.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Properties for the preservation results **********************************)
+
+fact lsubsv_sstas_aux: ∀h,g,L0,T0.
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
+                       ∀L2,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L2, T⦄ → ⦃h, L2⦄ ⊢ T ¡[g] →
+                       ∀L1. h ⊢ L1 ¡⊑[g] L2 → ∀U2. ⦃h, L2⦄ ⊢ T •*[g] U2 →
+                       ∃∃U1. ⦃h, L1⦄ ⊢ T •*[g] U1 & L1 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 #T #HLT0 #HT #L1 #HL12 #U2 #H @(sstas_ind … H) -U2 [ /2 width=3/ ]
+#U2 #W #l #HTU2 #HU2W * #U1 #HTU1 #HU12
+lapply (IH1 … HT … HL12) // #H
+lapply (snv_sstas_aux … IH2 … HTU1) // /3 width=4 by ygt_yprs_trans, lsubsv_yprs/ -H #HU1
+lapply (snv_sstas_aux … IH2 … HTU2) // #H
+lapply (IH1 … H … HL12) [ /3 width=4 by ygt_yprs_trans, sstas_yprs/ ] -H #HU2
+elim (snv_fwd_ssta … HU1) #l1 #W1 #HUW1
+elim (lsubsv_ssta_trans … HU2W … HL12) -HU2W #W2 #HUW2 #HW2
+elim (ssta_cpcs_lpr_aux … IH4 IH3 … HU1 HU2 … HUW1 … HUW2 … HU12 L1) -HU1 -HU2 -HUW2 -HU12 //
+[2,3: /4 width=4 by ygt_yprs_trans, sstas_yprs, lsubsv_yprs/ ] -L2 -L0 -T0 -U2 #H #HW12 destruct
+lapply (cpcs_trans … HW12 … HW2) -W2 /3 width=4/
+qed-.
+
+fact lsubsv_cpds_aux: ∀h,g,L0,T0.
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
+                      ∀L2,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L2, T1⦄ → ⦃h, L2⦄ ⊢ T1 ¡[g] →
+                      ∀L1. h ⊢ L1 ¡⊑[g] L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 →
+                      ∃∃T. ⦃h, L1⦄ ⊢ T1 •*➡*[g] T & L1 ⊢ T2 ➡* T.
+#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 #T1 #HLT0 #HT1 #L1 #HL12 #T2 * #T #HT1T #HTT2
+lapply (lsubsv_cprs_trans … HL12 … HTT2) -HTT2 #HTT2
+elim (lsubsv_sstas_aux … IH4 IH3 IH2 IH1 … HLT0 … HL12 … HT1T) // -L2 -L0 -T0 #T0 #HT10 #HT0
+lapply (cpcs_cprs_strap1 … HT0 … HTT2) -T #HT02
+elim (cpcs_inv_cprs … HT02) -HT02 /3 width=3/
+qed-.