(* Properties on supclosure *************************************************)
lemma fqu_cpxs_trans (b):
- â\88\80G1,G2,L1,L2,T2,U2. â\9dªG2,L2â\9d« ⊢ T2 ⬈* U2 →
- â\88\80T1. â\9dªG1,L1,T1â\9d« â¬\82[b] â\9dªG2,L2,T2â\9d« →
- â\88\83â\88\83U1. â\9dªG1,L1â\9d« â\8a¢ T1 â¬\88* U1 & â\9dªG1,L1,U1â\9d« â¬\82[b] â\9dªG2,L2,U2â\9d«.
+ â\88\80G1,G2,L1,L2,T2,U2. â\9d¨G2,L2â\9d© ⊢ T2 ⬈* U2 →
+ â\88\80T1. â\9d¨G1,L1,T1â\9d© â¬\82[b] â\9d¨G2,L2,T2â\9d© →
+ â\88\83â\88\83U1. â\9d¨G1,L1â\9d© â\8a¢ T1 â¬\88* U1 & â\9d¨G1,L1,U1â\9d© â¬\82[b] â\9d¨G2,L2,U2â\9d©.
#b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
lemma fquq_cpxs_trans (b):
- â\88\80G1,G2,L1,L2,T2,U2. â\9dªG2,L2â\9d« ⊢ T2 ⬈* U2 →
- â\88\80T1. â\9dªG1,L1,T1â\9d« â¬\82⸮[b] â\9dªG2,L2,T2â\9d« →
- â\88\83â\88\83U1. â\9dªG1,L1â\9d« â\8a¢ T1 â¬\88* U1 & â\9dªG1,L1,U1â\9d« â¬\82⸮[b] â\9dªG2,L2,U2â\9d«.
+ â\88\80G1,G2,L1,L2,T2,U2. â\9d¨G2,L2â\9d© ⊢ T2 ⬈* U2 →
+ â\88\80T1. â\9d¨G1,L1,T1â\9d© â¬\82⸮[b] â\9d¨G2,L2,T2â\9d© →
+ â\88\83â\88\83U1. â\9d¨G1,L1â\9d© â\8a¢ T1 â¬\88* U1 & â\9d¨G1,L1,U1â\9d© â¬\82⸮[b] â\9d¨G2,L2,U2â\9d©.
#b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fquq_cpx_trans … HT1 … HT2) -T
#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
lemma fqup_cpxs_trans (b):
- â\88\80G1,G2,L1,L2,T2,U2. â\9dªG2,L2â\9d« ⊢ T2 ⬈* U2 →
- â\88\80T1. â\9dªG1,L1,T1â\9d« â¬\82+[b] â\9dªG2,L2,T2â\9d« →
- â\88\83â\88\83U1. â\9dªG1,L1â\9d« â\8a¢ T1 â¬\88* U1 & â\9dªG1,L1,U1â\9d« â¬\82+[b] â\9dªG2,L2,U2â\9d«.
+ â\88\80G1,G2,L1,L2,T2,U2. â\9d¨G2,L2â\9d© ⊢ T2 ⬈* U2 →
+ â\88\80T1. â\9d¨G1,L1,T1â\9d© â¬\82+[b] â\9d¨G2,L2,T2â\9d© →
+ â\88\83â\88\83U1. â\9d¨G1,L1â\9d© â\8a¢ T1 â¬\88* U1 & â\9d¨G1,L1,U1â\9d© â¬\82+[b] â\9d¨G2,L2,U2â\9d©.
#b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
lemma fqus_cpxs_trans (b):
- â\88\80G1,G2,L1,L2,T2,U2. â\9dªG2,L2â\9d« ⊢ T2 ⬈* U2 →
- â\88\80T1. â\9dªG1,L1,T1â\9d« â¬\82*[b] â\9dªG2,L2,T2â\9d« →
- â\88\83â\88\83U1. â\9dªG1,L1â\9d« â\8a¢ T1 â¬\88* U1 & â\9dªG1,L1,U1â\9d« â¬\82*[b] â\9dªG2,L2,U2â\9d«.
+ â\88\80G1,G2,L1,L2,T2,U2. â\9d¨G2,L2â\9d© ⊢ T2 ⬈* U2 →
+ â\88\80T1. â\9d¨G1,L1,T1â\9d© â¬\82*[b] â\9d¨G2,L2,T2â\9d© →
+ â\88\83â\88\83U1. â\9d¨G1,L1â\9d© â\8a¢ T1 â¬\88* U1 & â\9d¨G1,L1,U1â\9d© â¬\82*[b] â\9d¨G2,L2,U2â\9d©.
#b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqus_cpx_trans … HT1 … HT2) -T
#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
(* Note: a proof based on fqu_cpx_trans_tneqx might exist *)
(* Basic_2A1: uses: fqu_cpxs_trans_neq *)
lemma fqu_cpxs_trans_tneqg (S) (b):
- â\88\80G1,G2,L1,L2,T1,T2. â\9dªG1,L1,T1â\9d« â¬\82[b] â\9dªG2,L2,T2â\9d« →
- â\88\80U2. â\9dªG2,L2â\9d« ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
- â\88\83â\88\83U1. â\9dªG1,L1â\9d« â\8a¢ T1 â¬\88* U1 & T1 â\89\9b[S] U1 â\86\92 â\8a¥ & â\9dªG1,L1,U1â\9d« â¬\82[b] â\9dªG2,L2,U2â\9d«.
+ â\88\80G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â¬\82[b] â\9d¨G2,L2,T2â\9d© →
+ â\88\80U2. â\9d¨G2,L2â\9d© ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
+ â\88\83â\88\83U1. â\9d¨G1,L1â\9d© â\8a¢ T1 â¬\88* U1 & T1 â\89\9b[S] U1 â\86\92 â\8a¥ & â\9d¨G1,L1,U1â\9d© â¬\82[b] â\9d¨G2,L2,U2â\9d©.
#S #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❨1❩)
#U2 #HVU2 @(ex3_intro … U2)
(* Basic_2A1: uses: fquq_cpxs_trans_neq *)
lemma fquq_cpxs_trans_tneqg (S) (b):
- â\88\80G1,G2,L1,L2,T1,T2. â\9dªG1,L1,T1â\9d« â¬\82⸮[b] â\9dªG2,L2,T2â\9d« →
- â\88\80U2. â\9dªG2,L2â\9d« ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
- â\88\83â\88\83U1. â\9dªG1,L1â\9d« â\8a¢ T1 â¬\88* U1 & T1 â\89\9b[S] U1 â\86\92 â\8a¥ & â\9dªG1,L1,U1â\9d« â¬\82⸮[b] â\9dªG2,L2,U2â\9d«.
+ â\88\80G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â¬\82⸮[b] â\9d¨G2,L2,T2â\9d© →
+ â\88\80U2. â\9d¨G2,L2â\9d© ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
+ â\88\83â\88\83U1. â\9d¨G1,L1â\9d© â\8a¢ T1 â¬\88* U1 & T1 â\89\9b[S] U1 â\86\92 â\8a¥ & â\9d¨G1,L1,U1â\9d© â¬\82⸮[b] â\9d¨G2,L2,U2â\9d©.
#S #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
[ #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tneqg … H12 … HTU2 H) -T2
/3 width=4 by fqu_fquq, ex3_intro/
(* Basic_2A1: uses: fqup_cpxs_trans_neq *)
lemma fqup_cpxs_trans_tneqg (S) (b):
- â\88\80G1,G2,L1,L2,T1,T2. â\9dªG1,L1,T1â\9d« â¬\82+[b] â\9dªG2,L2,T2â\9d« →
- â\88\80U2. â\9dªG2,L2â\9d« ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
- â\88\83â\88\83U1. â\9dªG1,L1â\9d« â\8a¢ T1 â¬\88* U1 & T1 â\89\9b[S] U1 â\86\92 â\8a¥ & â\9dªG1,L1,U1â\9d« â¬\82+[b] â\9dªG2,L2,U2â\9d«.
+ â\88\80G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â¬\82+[b] â\9d¨G2,L2,T2â\9d© →
+ â\88\80U2. â\9d¨G2,L2â\9d© ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
+ â\88\83â\88\83U1. â\9d¨G1,L1â\9d© â\8a¢ T1 â¬\88* U1 & T1 â\89\9b[S] U1 â\86\92 â\8a¥ & â\9d¨G1,L1,U1â\9d© â¬\82+[b] â\9d¨G2,L2,U2â\9d©.
#S #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tneqg … H12 … HTU2 H) -T2
/3 width=4 by fqu_fqup, ex3_intro/
(* Basic_2A1: uses: fqus_cpxs_trans_neq *)
lemma fqus_cpxs_trans_tneqg (S) (b):
- â\88\80G1,G2,L1,L2,T1,T2. â\9dªG1,L1,T1â\9d« â¬\82*[b] â\9dªG2,L2,T2â\9d« →
- â\88\80U2. â\9dªG2,L2â\9d« ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
- â\88\83â\88\83U1. â\9dªG1,L1â\9d« â\8a¢ T1 â¬\88* U1 & T1 â\89\9b[S] U1 â\86\92 â\8a¥ & â\9dªG1,L1,U1â\9d« â¬\82*[b] â\9dªG2,L2,U2â\9d«.
+ â\88\80G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â¬\82*[b] â\9d¨G2,L2,T2â\9d© →
+ â\88\80U2. â\9d¨G2,L2â\9d© ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
+ â\88\83â\88\83U1. â\9d¨G1,L1â\9d© â\8a¢ T1 â¬\88* U1 & T1 â\89\9b[S] U1 â\86\92 â\8a¥ & â\9d¨G1,L1,U1â\9d© â¬\82*[b] â\9d¨G2,L2,U2â\9d©.
#S #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
[ #H12 elim (fqup_cpxs_trans_tneqg … H12 … HTU2 H) -T2
/3 width=4 by fqup_fqus, ex3_intro/