matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_lfeq.ma

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  14
  15 include "basic_2/syntax/ext2_tc.ma".
  16 include "basic_2/relocation/lexs_tc.ma".
  17 include "basic_2/relocation/lex.ma".
  18 include "basic_2/static/lfeq_fqup.ma".
  19 include "basic_2/static/lfxs_lfxs.ma".
  20 include "basic_2/i_static/tc_lfxs_fqup.ma".
  21
  22 (* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
  23
  24 lemma tc_lfxs_inv_lex_lfeq: ∀R. c_reflexive … R →
  25                             (lexs_frees_confluent (cext2 R) cfull) →
  26                             (∀f. 𝐈⦃f⦄ → s_rs_transitive … (cext2 R) (λ_.lexs cfull (cext2 R) f)) →
  27                             ∀L1,L2,T. L1 ⪤**[R, T] L2 →
  28                             ∃∃L. L1 ⪤[LTC … R] L & L ≡[T] L2.
  29 #R #H1R #H2R #H3R #L1 #L2 #T #H
  30 @(tc_lfxs_ind_sn … H1R … H) -H -L2
  31 [ /4 width=3 by lfeq_refl, lex_refl, inj, ex2_intro/
  32 | #L0 #L2 #_ #HL02 * #L * #f0 #Hf0 #HL1 #HL0
  33   lapply (lfeq_lfxs_trans … HL0 … HL02) -L0 * #f1 #Hf1 #HL2
  34   elim (lexs_sdj_split … ceq_ext … HL2 f0 ?) -HL2
  35   [ #L0 #HL0 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ]
  36   lapply (lexs_sdj … HL0 f1 ?) /2 width=1 by sdj_isid_sn/ #H
  37   elim (H2R … Hf1 … H) -H #f2 #Hf2 #Hf21
  38   lapply (sle_lexs_trans … HL02 … Hf21) -f1 // #HL02
  39   lapply (lexs_co ?? cfull (LTC … (cext2 R)) … HL1) -HL1 /2 width=1 by ext2_inv_tc/ #HL1
  40   lapply (lexs_inv_tc_dx … HL1) -HL1 /2 width=1 by ext2_refl/ #HL1
  41   lapply (step ????? HL1 … HL0) -L #HL10
  42   lapply (lexs_tc_dx … H3R … HL10) -HL10 // #HL10
  43   lapply (lexs_co … cfull (cext2 (LTC … R)) … HL10) -HL10 /2 width=1 by ext2_tc/ #HL10
  44   /3 width=5 by ex2_intro/
  45 ]
  46 qed-.