matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr_2.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/dynamic/snv_ltpr_1.ma".
  16 include "basic_2/dynamic/snv_lift.ma".
  17
  18 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
  19
  20 (* Properties on context-free parallel reduction for local environments *****)
  21
  22 fact snv_ltpr_tpr_aux: ∀h,g,n. (
  23                           ∀L1,T1. ♯{L1, T1} < n →
  24                           ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
  25                           ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
  26                           ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
  27                        ) → (
  28                           ∀L1,T1. ♯{L1, T1} < n → ⦃h, L1⦄ ⊩ T1 :[g] →
  29                           ∀L2. L1 ➡ L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
  30                        ) →
  31                        ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
  32                        ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
  33 #h #g #n #IH2 #IH1 #L1 * * [||||*]
  34 [ #k #Hn #H1 #L2 #_ #X #H2 destruct -IH2 -IH1 -L1
  35   >(tpr_inv_atom1 … H2) -X //
  36 | #i #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
  37   elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
  38   >(tpr_inv_atom1 … H2) -X
  39   elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
  40   elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
  41   lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
  42   lapply (IH1 … HV1 … HK12 V2 ?) -IH1 -HV1 -HK12 // -L1 -K1 /4 width=1/ -HV12 /2 width=5/
  43 | #p #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
  44   elim (snv_inv_gref … H1)
  45 | #a #I #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
  46   elim (snv_inv_bind … H1) -H1 #HV1 #HT1
  47   elim (tpr_inv_bind1 … H2) -H2 *
  48   [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
  49     lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
  50     lapply (cpr_intro (L2.ⓑ{I}V2) … T2 0 1 HT10 ?) -HT10 /2 width=1/ -HT02 #HT12
  51     lapply (IH1 … HV1 … HL12 V2 ?) -HV1 // /4 width=1/ #HV2
  52     lapply (IH1 … HT1 (L2.ⓑ{I}V2) … T2 ?) -IH1 -HT1 /3 width=1/
  53   | #T2 #HT12 #HXT2 #H1 #H2 destruct
  54     lapply (IH1 … HT1 (L2.ⓓV1) … T2 ?) -IH1 -HT1 // /4 width=1/ -HT12 -HL12 #HT2
  55     lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
  56   ]
  57 | #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct
  58   elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
  59   elim (tpr_inv_appl1 … H2) -H2 *
  60   [ #V2 #T2 #HV12 #HT12 #H destruct
  61     lapply (IH1 … HV1 … HL12 V2 ?) // /4 width=1/ #HV2
  62     lapply (IH1 … HT1 … HL12 T2 ?) // /4 width=1/ #HT2
  63     lapply (IH1 … HT1 … HTU1) -IH1 // #H
  64     elim (snv_inv_bind … H) -H #HW1 #HU1
  65     elim (IH2 … HVW1 … HL12 … HV12 HV1) -IH2 -HVW1 -HV12 -HV1 // #W2 #HVW2 #HW12
  66     lapply (fpcs_canc_sn L1 L1 … W10 W1 … HW12) -HW12 /3 width=1/ -W10 #HW12
  67     @(snv_appl … HV2 HT2 HVW2)