(* Properties with supclosure ***********************************************)
lemma fqu_tdeq_conf: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
- â\88\80U2. U1 â\89¡[h, o] U2 →
- â\88\83â\88\83L,T2. â¦\83G1, L1, U2â¦\84 â\8a\90[b] â¦\83G2, L, T2â¦\84 & L2 â\89¡[h, o, T1] L & T1 â\89¡[h, o] T2.
+ â\88\80U2. U1 â\89\9b[h, o] U2 →
+ â\88\83â\88\83L,T2. â¦\83G1, L1, U2â¦\84 â\8a\90[b] â¦\83G2, L, T2â¦\84 & L2 â\89\9b[h, o, T1] L & T1 â\89\9b[h, o] T2.
#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -G1 -G2 -L1 -L2 -U1 -T1
[ #I #G #L #W #X #H >(tdeq_inv_lref1 … H) -X
/2 width=5 by fqu_lref_O, ex3_2_intro/
qed-.
lemma tdeq_fqu_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
- â\88\80U2. U2 â\89¡[h, o] U1 →
- â\88\83â\88\83L,T2. â¦\83G1, L1, U2â¦\84 â\8a\90[b] â¦\83G2, L, T2â¦\84 & T2 â\89¡[h, o] T1 & L â\89¡[h, o, T1] L2.
+ â\88\80U2. U2 â\89\9b[h, o] U1 →
+ â\88\83â\88\83L,T2. â¦\83G1, L1, U2â¦\84 â\8a\90[b] â¦\83G2, L, T2â¦\84 & T2 â\89\9b[h, o] T1 & L â\89\9b[h, o, T1] L2.
#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21
elim (fqu_tdeq_conf … o … H12 U2) -H12
/3 width=5 by lfdeq_sym, tdeq_sym, ex3_2_intro/
(* Basic_2A1: was just: lleq_fqu_trans *)
lemma lfdeq_fqu_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐[b] ⦃G2, K2, U⦄ →
- â\88\80L1. L1 â\89¡[h, o, T] L2 →
- â\88\83â\88\83K1,U0. â¦\83G1, L1, Tâ¦\84 â\8a\90[b] â¦\83G2, K1, U0â¦\84 & U0 â\89¡[h, o] U & K1 â\89¡[h, o, U] K2.
+ â\88\80L1. L1 â\89\9b[h, o, T] L2 →
+ â\88\83â\88\83K1,U0. â¦\83G1, L1, Tâ¦\84 â\8a\90[b] â¦\83G2, K1, U0â¦\84 & U0 â\89\9b[h, o] U & K1 â\89\9b[h, o, U] K2.
#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
[ #I #G #L2 #V2 #L1 #H elim (lfdeq_inv_zero_pair_dx … H) -H
#K1 #V1 #HV1 #HV12 #H destruct
(* Basic_2A1: was just: lleq_fquq_trans *)
lemma lfdeq_fquq_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮[b] ⦃G2, K2, U⦄ →
- â\88\80L1. L1 â\89¡[h, o, T] L2 →
- â\88\83â\88\83K1,U0. â¦\83G1, L1, Tâ¦\84 â\8a\90⸮[b] â¦\83G2, K1, U0â¦\84 & U0 â\89¡[h, o] U & K1 â\89¡[h, o, U] K2.
+ â\88\80L1. L1 â\89\9b[h, o, T] L2 →
+ â\88\83â\88\83K1,U0. â¦\83G1, L1, Tâ¦\84 â\8a\90⸮[b] â¦\83G2, K1, U0â¦\84 & U0 â\89\9b[h, o] U & K1 â\89\9b[h, o, U] K2.
#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -H
[ #H #L1 #HL12 elim (lfdeq_fqu_trans … H … HL12) -L2 /3 width=5 by fqu_fquq, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
(* Basic_2A1: was just: lleq_fqup_trans *)
lemma lfdeq_fqup_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, U⦄ →
- â\88\80L1. L1 â\89¡[h, o, T] L2 →
- â\88\83â\88\83K1,U0. â¦\83G1, L1, Tâ¦\84 â\8a\90+[b] â¦\83G2, K1, U0â¦\84 & U0 â\89¡[h, o] U & K1 â\89¡[h, o, U] K2.
+ â\88\80L1. L1 â\89\9b[h, o, T] L2 →
+ â\88\83â\88\83K1,U0. â¦\83G1, L1, Tâ¦\84 â\8a\90+[b] â¦\83G2, K1, U0â¦\84 & U0 â\89\9b[h, o] U & K1 â\89\9b[h, o, U] K2.
#h #o #b #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
[ #G2 #K2 #U #HTU #L1 #HL12 elim (lfdeq_fqu_trans … HTU … HL12) -L2
/3 width=5 by fqu_fqup, ex3_2_intro/
(* Basic_2A1: was just: lleq_fqus_trans *)
lemma lfdeq_fqus_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐*[b] ⦃G2, K2, U⦄ →
- â\88\80L1. L1 â\89¡[h, o, T] L2 →
- â\88\83â\88\83K1,U0. â¦\83G1, L1, Tâ¦\84 â\8a\90*[b] â¦\83G2, K1, U0â¦\84 & U0 â\89¡[h, o] U & K1 â\89¡[h, o, U] K2.
+ â\88\80L1. L1 â\89\9b[h, o, T] L2 →
+ â\88\83â\88\83K1,U0. â¦\83G1, L1, Tâ¦\84 â\8a\90*[b] â¦\83G2, K1, U0â¦\84 & U0 â\89\9b[h, o] U & K1 â\89\9b[h, o, U] K2.
#h #o #b #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H
[ #H elim (lfdeq_fqup_trans … H … HL12) -L2 /3 width=5 by fqup_fqus, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/