(* Properties on supclosure *************************************************)
-lemma fqu_cpx_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83 ⦃G2, L2, T2⦄ →
+lemma fqu_cpx_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90 ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\83 ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\90 ⦃G2, L2, U2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
[ #I #G #L #V2 #U2 #HVU2
]
qed-.
-lemma fqu_ssta_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83 ⦃G2, L2, T2⦄ →
+lemma fqu_ssta_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90 ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 •[h, g] U2 →
∀l. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] l+1 →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\83 ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\90 ⦃G2, L2, U2⦄.
/3 width=5 by fqu_cpx_trans, ssta_cpx/ qed-.
-lemma fquq_cpx_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83⸮ ⦃G2, L2, T2⦄ →
+lemma fquq_cpx_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90⸮ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\83⸮ ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\90⸮ ⦃G2, L2, U2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
[ #HT12 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fquq_ssta_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83⸮ ⦃G2, L2, T2⦄ →
+lemma fquq_ssta_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90⸮ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 •[h, g] U2 →
∀l. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] l+1 →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\83⸮ ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\90⸮ ⦃G2, L2, U2⦄.
/3 width=5 by fquq_cpx_trans, ssta_cpx/ qed-.
-lemma fqup_cpx_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83+ ⦃G2, L2, T2⦄ →
+lemma fqup_cpx_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90+ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\83+ ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\90+ ⦃G2, L2, U2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
/3 width=3 by fqu_fqup, ex2_intro/
]
qed-.
-lemma fqus_cpx_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83* ⦃G2, L2, T2⦄ →
+lemma fqus_cpx_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90* ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\83* ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & â¦\83G1, L1, U1â¦\84 â\8a\90* ⦃G2, L2, U2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
[ #HT12 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fqu_cpx_trans_neq: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83 ⦃G2, L2, T2⦄ →
+lemma fqu_cpx_trans_neq: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90 ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & T1 = U1 â\86\92 â\8a¥ & â¦\83G1, L1, U1â¦\84 â\8a\83 ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & T1 = U1 â\86\92 â\8a¥ & â¦\83G1, L1, U1â¦\84 â\8a\90 ⦃G2, L2, U2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
#U2 #HVU2 @(ex3_intro … U2)
]
qed-.
-lemma fquq_cpx_trans_neq: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83⸮ ⦃G2, L2, T2⦄ →
+lemma fquq_cpx_trans_neq: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90⸮ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & T1 = U1 â\86\92 â\8a¥ & â¦\83G1, L1, U1â¦\84 â\8a\83⸮ ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & T1 = U1 â\86\92 â\8a¥ & â¦\83G1, L1, U1â¦\84 â\8a\90⸮ ⦃G2, L2, U2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
[ #H12 elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fquq, ex3_intro/
]
qed-.
-lemma fqup_cpx_trans_neq: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83+ ⦃G2, L2, T2⦄ →
+lemma fqup_cpx_trans_neq: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90+ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & T1 = U1 â\86\92 â\8a¥ & â¦\83G1, L1, U1â¦\84 â\8a\83+ ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & T1 = U1 â\86\92 â\8a¥ & â¦\83G1, L1, U1â¦\84 â\8a\90+ ⦃G2, L2, U2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fqup, ex3_intro/
]
qed-.
-lemma fqus_cpx_trans_neq: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83* ⦃G2, L2, T2⦄ →
+lemma fqus_cpx_trans_neq: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90* ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & T1 = U1 â\86\92 â\8a¥ & â¦\83G1, L1, U1â¦\84 â\8a\83* ⦃G2, L2, U2⦄.
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] U1 & T1 = U1 â\86\92 â\8a¥ & â¦\83G1, L1, U1â¦\84 â\8a\90* ⦃G2, L2, U2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
[ #H12 elim (fqup_cpx_trans_neq … H12 … HTU2 H) -T2
/3 width=4 by fqup_fqus, ex3_intro/