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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-.