matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_tc.etc

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  14
  15 include "basic_2/substitution/lpx_sn.ma".
  16
  17 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
  18
  19 (* Properties on transitive_closure *****************************************)
  20
  21 lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
  22                             ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
  23                             TC … (lpx_sn R) L1 L2.
  24 #R #HR #L1 #L2 #H elim H -L1 -L2
  25 /2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
  26 qed-.
  27
  28 (* Inversion lemmas on transitive closure ***********************************)
  29
  30 lemma TC_lpx_sn_ind: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
  31                      ∀S:relation lenv.
  32                      S (⋆) (⋆) → (
  33                         ∀I,K1,K2,V1,V2.
  34                         TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
  35                         S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
  36                      ) →
  37                      ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
  38 #R #HR #S #IH1 #IH2 #L2 elim L2 -L2
  39 [ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
  40 | #L2 #I #V2 #IHL2 #X #H
  41   elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
  42   #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
  43 ]
  44 qed-.
  45
  46 lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
  47                                 ∀L1,L2. TC … (lpx_sn R) L1 L2 →
  48                                 lpx_sn (LTC … R) L1 L2.
  49 /3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.