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

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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 "static_2/static/reqg_fqus.ma".
  16 include "static_2/static/feqg.ma".
  17
  18 (* GENERIC EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *********************)
  19
  20 (* Properties with star-iterated structural successor for closures **********)
  21
  22 lemma feqg_fquq_trans (S) (b):
  23       reflexive … S → symmetric … S → Transitive … S →
  24       ∀G1,G,L1,L,T1,T. ❨G1,L1,T1❩ ≛[S] ❨G,L,T❩ →
  25       ∀G2,L2,T2. ❨G,L,T❩ ⬂⸮[b] ❨G2,L2,T2❩ →
  26       ∃∃G,L0,T0. ❨G1,L1,T1❩ ⬂⸮[b] ❨G,L0,T0❩ & ❨G,L0,T0❩ ≛[S] ❨G2,L2,T2❩.
  27 #S #b #H1S #H2S #H3S #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H2
  28 elim(feqg_inv_gen_dx … H1) -H1 // #HG #HL1 #HT1 destruct
  29 elim (reqg_fquq_trans … H2 … HL1) -L // #L #T0 #H2 #HT02 #HL2
  30 elim (teqg_fquq_trans … H2 … HT1) -T // #L0 #T #H2 #HT0 #HL0
  31 lapply (teqg_reqg_conf_sn … HT02 … HL0) -HL0 // #HL0
  32 /4 width=7 by feqg_intro_dx, reqg_trans, teqg_trans, ex2_3_intro/
  33 qed-.
  34
  35 lemma feqg_fqus_trans (S) (b):
  36       reflexive … S → symmetric … S → Transitive … S →
  37       ∀G1,G,L1,L,T1,T. ❨G1,L1,T1❩ ≛[S] ❨G,L,T❩ →
  38       ∀G2,L2,T2. ❨G,L,T❩ ⬂*[b] ❨G2,L2,T2❩ →
  39       ∃∃G,L0,T0. ❨G1,L1,T1❩ ⬂*[b] ❨G,L0,T0❩ & ❨G,L0,T0❩ ≛[S] ❨G2,L2,T2❩.
  40 #S #b #H1S #H2S #H3S #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H2
  41 elim(feqg_inv_gen_dx … H1) -H1 // #HG #HL1 #HT1 destruct
  42 elim (reqg_fqus_trans … H2 … HL1) -L // #L #T0 #H2 #HT02 #HL2
  43 elim (teqg_fqus_trans … H2 … HT1) -T // #L0 #T #H2 #HT0 #HL0
  44 lapply (teqg_reqg_conf_sn … HT02 … HL0) -HL0 // #HL0
  45 /4 width=7 by feqg_intro_dx, reqg_trans, teqg_trans, ex2_3_intro/
  46 qed-.