(* Properties with extended structural successor for closures ***************)
-lemma fqu_tdeq_conf: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
+lemma fqu_tdeq_conf: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,T1⦄ →
∀U2. U1 ≛ U2 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & L2 ≛[T1] L & T1 ≛ T2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐[b] ⦃G2,L,T2⦄ & L2 ≛[T1] L & T1 ≛ T2.
#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: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
+lemma tdeq_fqu_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,T1⦄ →
∀U2. U2 ≛ U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
#b #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21
elim (fqu_tdeq_conf … H12 U2) -H12
/3 width=5 by rdeq_sym, tdeq_sym, ex3_2_intro/
qed-.
(* Basic_2A1: uses: lleq_fqu_trans *)
-lemma rdeq_fqu_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐[b] ⦃G2, K2, U⦄ →
+lemma rdeq_fqu_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐[b] ⦃G2,K2,U⦄ →
∀L1. L1 ≛[T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
#b #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
[ #I #G #L2 #V2 #L1 #H elim (rdeq_inv_zero_pair_dx … H) -H
#K1 #V1 #HV1 #HV12 #H destruct
(* Properties with optional structural successor for closures ***************)
-lemma tdeq_fquq_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, T1⦄ →
+lemma tdeq_fquq_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,T1⦄ →
∀U2. U2 ≛ U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐⸮[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐⸮[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
#b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -H
[ #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
/3 width=5 by fqu_fquq, ex3_2_intro/
qed-.
(* Basic_2A1: was just: lleq_fquq_trans *)
-lemma rdeq_fquq_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮[b] ⦃G2, K2, U⦄ →
+lemma rdeq_fquq_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐⸮[b] ⦃G2,K2,U⦄ →
∀L1. L1 ≛[T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐⸮[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐⸮[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
#b #G1 #G2 #L2 #K2 #T #U #H elim H -H
[ #H #L1 #HL12 elim (rdeq_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/
(* Properties with plus-iterated structural successor for closures **********)
(* Basic_2A1: was just: lleq_fqup_trans *)
-lemma rdeq_fqup_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, U⦄ →
+lemma rdeq_fqup_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐+[b] ⦃G2,K2,U⦄ →
∀L1. L1 ≛[T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐+[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐+[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
#b #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
[ #G2 #K2 #U #HTU #L1 #HL12 elim (rdeq_fqu_trans … HTU … HL12) -L2
/3 width=5 by fqu_fqup, ex3_2_intro/
]
qed-.
-lemma tdeq_fqup_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, T1⦄ →
+lemma tdeq_fqup_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,T1⦄ →
∀U2. U2 ≛ U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐+[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐+[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
#b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1
[ #G1 #L1 #U1 #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
/3 width=5 by fqu_fqup, ex3_2_intro/
(* Properties with star-iterated structural successor for closures **********)
-lemma tdeq_fqus_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, T1⦄ →
+lemma tdeq_fqus_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,T1⦄ →
∀U2. U2 ≛ U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐*[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐*[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
#b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H
[ #H elim (tdeq_fqup_trans … H … HU21) -U1 /3 width=5 by fqup_fqus, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
qed-.
(* Basic_2A1: was just: lleq_fqus_trans *)
-lemma rdeq_fqus_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐*[b] ⦃G2, K2, U⦄ →
+lemma rdeq_fqus_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐*[b] ⦃G2,K2,U⦄ →
∀L1. L1 ≛[T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐*[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐*[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
#b #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H
[ #H elim (rdeq_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/