matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_cpss.ma

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   3 (*      ||M||                                                             *)
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   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "basic_2/grammar/lpx_sn_lpx_sn.ma".
  16 include "basic_2/substitution/lpss_ldrop.ma".
  17
  18 (* SN PARALLEL SUBSTITUTION FOR LOCAL ENVIRONMENTS **************************)
  19
  20 (* Main properties on context-sensitive parallel substitution for terms *****)
  21
  22 theorem cpss_trans_lpss: lpx_sn_transitive cpss cpss.
  23 #L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * [|*]
  24 [ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct
  25   elim (cpss_inv_atom1 … H1) -H1
  26   [ #H destruct
  27     elim (cpss_inv_atom1 … HT2) -HT2
  28     [ #H destruct //
  29     | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
  30       elim (lpss_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
  31       elim (lpss_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
  32       lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) /3 width=9/
  33     ]
  34   | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
  35     elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
  36     elim (lpss_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
  37     lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
  38     elim (cpss_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
  39     lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) /3 width=9/
  40   ]
  41 | #a #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
  42   elim (cpss_inv_bind1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
  43   elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
  44 | #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
  45   elim (cpss_inv_flat1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
  46   elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
  47 ]
  48 qed-.
  49
  50 (* Basic_1: was only: subst1_trans *)
  51 theorem cpss_trans: ∀L. Transitive … (cpss L).
  52 /2 width=5 by cpss_trans_lpss/ qed-.
  53
  54 (* Properties on context-sensitive parallel substitution for terms **********)
  55
  56 (* Basic_1: was only: subst1_subst1 *)
  57 lemma lpss_cpss_trans: ∀L1,L2. L1 ⊢ ▶* L2 →
  58                        ∀T1,T2. L2 ⊢ T1 ▶* T2 → L1 ⊢ T1 ▶* T2.
  59 /2 width=5 by cpss_trans_lpss/ qed-.