matita/matita/contribs/lambdadelta/static_2/static/rex_lex.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "static_2/relocation/lex.ma".
  16 include "static_2/static/rex_fsle.ma".
  17 include "static_2/static/req.ma".
  18
  19 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
  20
  21 (* Properties with generic extension of a context-sensitive relation ********)
  22
  23 lemma rex_lex (R):
  24       ∀L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤[R,T] L2.
  25 #R #L1 #L2 * #f #Hf #HL12 #T
  26 elim (frees_total L1 T) #g #Hg
  27 /4 width=5 by sex_sdj, sdj_isid_sn, ex2_intro/
  28 qed.
  29
  30 (* Inversion lemmas with generic extension of a context sensitive relation **)
  31
  32 lemma rex_inv_req_lex (R):
  33       c_reflexive … R → f_confluent1_next R ceq →
  34       ∀L1,L2,T. L1 ⪤[R,T] L2 →
  35       ∃∃L. L1 ≡[T] L & L ⪤[R] L2.
  36 #R #H1R #H2R #L1 #L2 #T * #f1 #Hf1 #HL
  37 elim (sex_sdj_split_dx … ceq_ext … HL 𝐢) -HL
  38 [ #L0 #HL10 #HL02
  39   lapply (sex_sdj … HL02 f1 ?) /2 width=1 by sdj_isid_sn/ #H
  40   /3 width=5 by (* 2x *) ex2_intro/
  41 |*: /2 width=1 by ext2_refl, sdj_isid_dx/
  42   #g #I #K #n #HLK #Hg @H2R /width=7 by/ (**) (* no auto with H2R *)
  43 ]
  44 qed-.
  45
  46 (* Forward lemmas with generic extension of a context sensitive relation **)
  47
  48 lemma rex_fwd_lex_req (R):
  49       c_reflexive … R → rex_fsge_compatible R →
  50       ∀L1,L2,T. L1 ⪤[R,T] L2 →
  51       ∃∃L. L1 ⪤[R] L & L ≡[T] L2.
  52 #R #H1R #H2R #L1 #L2 #T * #f1 #Hf1 #HL
  53 elim (sex_sdj_split_sn … ceq_ext … HL 𝐢 ?) -HL
  54 [ #L0 #HL10 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ] -H1R
  55 lapply (sex_sdj … HL10 f1 ?) /2 width=1 by sdj_isid_sn/ #H
  56 elim (frees_sex_conf_fsge … Hf1 … H) // -H2R -H #f0 #Hf0 #Hf01
  57 /4 width=7 by sle_sex_trans, (* 2x *) ex2_intro/
  58 qed-.