matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_lreq.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 "ground_2/lib/arith_2a.ma".
  16 include "basic_2A/substitution/drop_lreq.ma".
  17 include "basic_2A/multiple/llpx_sn.ma".
  18
  19 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
  20
  21 (* Properties on equivalence for local environments *************************)
  22
  23 lemma lreq_llpx_sn_trans: ∀R,L2,L,T,l. llpx_sn R l T L2 L →
  24                           ∀L1. L1 ⩬[l, ∞] L2 → llpx_sn R l T L1 L.
  25 #R #L2 #L #T #l #H elim H -L2 -L -T -l
  26 /4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/
  27 [ #I #L2 #L #K2 #K #V2 #V #l #i #Hli #HLK2 #HLK #HK2 #HV2 #_ #L1 #HL12
  28   elim (lreq_drop_trans_be … HL12 … HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
  29   lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
  30 | /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_trans_aux, trans_eq/
  31 | /4 width=1 by llpx_sn_bind, lreq_succ/
  32 ]
  33 qed-.
  34
  35 lemma llpx_sn_lreq_trans: ∀R,L,L1,T,l. llpx_sn R l T L L1 →
  36                           ∀L2. L1 ⩬[l, ∞] L2 → llpx_sn R l T L L2.
  37 #R #L #L1 #T #l #H elim H -L -L1 -T -l
  38 /4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/
  39 [ #I #L #L1 #K #K1 #V #V1 #l #i #Hli #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12
  40   elim (lreq_drop_conf_be … HL12 … HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2
  41   lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
  42 | /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_conf_aux, trans_eq/
  43 | /4 width=1 by llpx_sn_bind, lreq_succ/
  44 ]
  45 qed-.
  46
  47 lemma llpx_sn_lreq_repl: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → ∀K1. K1 ⩬[l, ∞] L1 →
  48                          ∀K2. L2 ⩬[l, ∞] K2 → llpx_sn R l T K1 K2.
  49 /3 width=4 by llpx_sn_lreq_trans, lreq_llpx_sn_trans/ qed-.
  50
  51 lemma llpx_sn_bind_repl_SO: ∀R,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
  52                             ∀J1,J2,W1,W2. llpx_sn R 1 T (L1.ⓑ{J1}W1) (L2.ⓑ{J2}W2).
  53 #R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R … 1 … HT) -HT
  54 /3 width=7 by llpx_sn_lreq_repl, lreq_succ/
  55 qed-.