matita/matita/contribs/lambdadelta/static_2/i_static/rexs_lex.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "static_2/relocation/lex_tc.ma".
  16 include "static_2/static/req_fqup.ma".
  17 include "static_2/static/req_fsle.ma".
  18 include "static_2/i_static/rexs_fqup.ma".
  19
  20 (* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
  21
  22 (* Properties with generic extension of a context sensitive relation ********)
  23
  24 lemma rexs_lex: ∀R. c_reflexive … R →
  25                 ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤*[R, T] L2.
  26 #R #HR #L1 #L2 #T *
  27 /5 width=7 by rexs_tc, sex_inv_tc_dx, sex_co, ext2_inv_tc, ext2_refl/
  28 qed.
  29
  30 lemma rexs_lex_req: ∀R. c_reflexive … R →
  31                     ∀L1,L. L1 ⪤[CTC … R] L → ∀L2,T. L ≡[T] L2 →
  32                     L1 ⪤*[R, T] L2.
  33 /3 width=3 by rexs_lex, rexs_step_dx, req_fwd_rex/ qed.
  34
  35 (* Inversion lemmas with generic extension of a context sensitive relation **)
  36
  37 (* Note: s_rs_transitive_lex_inv_isid could be invoked in the last auto but makes it too slow *)
  38 lemma rexs_inv_lex_req: ∀R. c_reflexive … R →
  39                         rex_fsge_compatible R →
  40                         s_rs_transitive … R (λ_.lex R) →
  41                         req_transitive R →
  42                         ∀L1,L2,T. L1 ⪤*[R, T] L2 →
  43                         ∃∃L. L1 ⪤[CTC … R] L & L ≡[T] L2.
  44 #R #H1R #H2R #H3R #H4R #L1 #L2 #T #H
  45 lapply (s_rs_transitive_lex_inv_isid … H3R) -H3R #H3R
  46 @(rexs_ind_sn … H1R … H) -H -L2
  47 [ /4 width=3 by req_refl, lex_refl, inj, ex2_intro/
  48 | #L0 #L2 #_ #HL02 * #L * #f0 #Hf0 #HL1 #HL0
  49   lapply (req_rex_trans … HL0 … HL02) -L0 // * #f1 #Hf1 #HL2
  50   elim (sex_sdj_split … ceq_ext … HL2 f0 ?) -HL2
  51   [ #L0 #HL0 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ]
  52   lapply (sex_sdj … HL0 f1 ?) /2 width=1 by sdj_isid_sn/ #H
  53   elim (frees_sex_conf … Hf1 … H) // -H2R -H #f2 #Hf2 #Hf21
  54   lapply (sle_sex_trans … HL02 … Hf21) -f1 // #HL02
  55   lapply (sex_co ?? cfull (CTC … (cext2 R)) … HL1) -HL1 /2 width=1 by ext2_inv_tc/ #HL1
  56   /8 width=11 by sex_inv_tc_dx, sex_tc_dx, sex_co, ext2_tc, ext2_refl, step, ex2_intro/ (**) (* full auto too slow *)
  57 ]
  58 qed-.