matita/matita/contribs/lambda_delta/basic_2/reducibility/tpr_tps.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/reducibility/ltpr_ldrop.ma".
  16
  17 (* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
  18
  19 (* Properties on parallel substitution for terms ****************************)
  20
  21 (* Basic_1: was: pr0_subst1_fwd *)
  22 lemma ltpr_tpr_conf: ∀L1,T,U1,d,e. L1 ⊢ T ▶ [d, e] U1 → ∀L2. L1 ➡ L2 →
  23                      ∃∃U2. U1 ➡ U2 & L2 ⊢ T ▶ [d, e] U2.
  24 #L1 #T #U1 #d #e #H elim H -L1 -T -U1 -d -e
  25 [ /2 width=3/
  26 | #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
  27   elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
  28   elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
  29   elim (lift_total V2 0 (i+1)) #W2 #HVW2
  30   lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=6/
  31 | #L1 #a #I #V #W1 #T #U1 #d #e #_ #_ #IHV #IHT #L2 #HL12
  32   elim (IHV … HL12) -IHV #W2 #HW12
  33   elim (IHT (L2.ⓑ{I}W2) ?) -IHT /2 width=1/ -L1 /3 width=5/
  34 | #L1 #I #V #W1 #T #U1 #d #e #_ #_ #IHV #IHT #L2 #HL12
  35   elim (IHV … HL12) -IHV
  36   elim (IHT … HL12) -IHT -HL12 /3 width=5/
  37 ]
  38 qed.
  39
  40 (* Basic_1: was: pr0_subst1_back *)
  41 lemma ltpr_tps_trans: ∀L2,T,U2,d,e. L2 ⊢ T ▶ [d, e] U2 → ∀L1. L1 ➡ L2 →
  42                       ∃∃U1. U1 ➡ U2 & L1 ⊢ T ▶ [d, e] U1.
  43 #L2 #T #U2 #d #e #H elim H -L2 -T -U2 -d -e
  44 [ /2 width=3/
  45 | #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12
  46   elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
  47   elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2
  48   elim (lift_total V1 0 (i+1)) #W1 #HVW1
  49   lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=6/
  50 | #L2 #a #I #V #W2 #T #U2 #d #e #_ #_ #IHV #IHT #L1 #HL12
  51   elim (IHV … HL12) -IHV #W1 #HW12
  52   elim (IHT (L1.ⓑ{I}W1) ?) -IHT /2 width=1/ -L2 /3 width=5/
  53 | #L2 #I #V #W2 #T #U2 #d #e #_ #_ #IHV #IHT #L1 #HL12
  54   elim (IHV … HL12) -IHV
  55   elim (IHT … HL12) -IHT -HL12 /3 width=5/
  56 ]
  57 qed.