matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpys0/lpys_ldrop.etc

   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/relocation/lpx_sn_ldrop.ma".
  16 include "basic_2/substitution/cpys_lift.ma".
  17 include "basic_2/substitution/lpys.ma".
  18
  19 (* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
  20
  21 (* Properies on local environment slicing ***********************************)
  22
  23 lemma lpys_ldrop_conf: ∀G. dropable_sn (lpys G).
  24 /3 width=6 by lpx_sn_deliftable_dropable, cpys_inv_lift1/ qed-.
  25
  26 lemma ldrop_lpys_trans: ∀G. dedropable_sn (lpys G).
  27 /3 width=10 by lpx_sn_liftable_dedropable, cpys_lift/ qed-.
  28
  29 lemma lpys_ldrop_trans_O1: ∀G. dropable_dx (lpys G).
  30 /2 width=3 by lpx_sn_dropable/ qed-.
  31
  32 (* Properties on supclosure *************************************************)
  33
  34 lemma fqu_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
  35                       ∀K2. ⦃G2, L2⦄ ⊢ ▶* K2 →
  36                       ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶* K1 & ⦃G1, L1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
  37 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
  38 /3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpys_pair, ex3_2_intro/
  39 [ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpys_inv_pair1 … H) -H
  40   #K2 #W2 #HLK2 #HVW2 #H destruct
  41   /3 width=5 by fqu_fquq, cpys_pair_sn, fqu_bind_dx, ex3_2_intro/
  42 | #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #K2 #HK12
  43   elim (ldrop_lpys_trans … HLK1 … HK12) -HK12
  44   /3 width=7 by fqu_drop, ex3_2_intro/
  45 ]
  46 qed-.
  47
  48 lemma fquq_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
  49                        ∀K2. ⦃G2, L2⦄ ⊢ ▶* K2 →
  50                        ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶* K1 & ⦃G1, L1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
  51 #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
  52 [ #HT12 elim (fqu_lpys_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
  53 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
  54 ]
  55 qed-.
  56
  57 lemma lpys_fqu_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
  58                       ∀K1. ⦃G1, K1⦄ ⊢ ▶* L1 →
  59                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶* L2.
  60 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
  61 /3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpys_pair, ex3_2_intro/
  62 [ #I #G1 #L1 #V1 #X #H elim (lpys_inv_pair2 … H) -H
  63   #K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1)
  64   /4 width=7 by cpys_delta, fqu_drop, ldrop_drop, ex3_2_intro/
  65 | #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #L0 #HL01
  66   elim (lpys_ldrop_trans_O1 … HL01 … HLK1) -L1
  67   /3 width=5 by fqu_drop, ex3_2_intro/
  68 ]
  69 qed-.
  70
  71 lemma lpys_fquq_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
  72                        ∀K1. ⦃G1, K1⦄ ⊢ ▶* L1 →
  73                        ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶* L2.
  74 #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
  75 [ #HT12 elim (lpys_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
  76 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
  77 ]
  78 qed-.