lemma fpbs_cpx_tneqg_trans (S):
reflexive … S → symmetric … S → Transitive … S →
(∀s1,s2. Decidable (S s1 s2)) →
- â\88\80G1,G2,L1,L2,T1,T2. â\9dªG1,L1,T1â\9d« â\89¥ 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\89¥ â\9dªG2,L2,U2â\9d«.
+ â\88\80G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â\89¥ 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\89¥ â\9d¨G2,L2,U2â\9d©.
#S #H1S #H2S #H3S #H4S #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2
elim (fpbs_inv_star S … H) -H // #G0 #L0 #L3 #T0 #T3 #HT10 #H10 #HL03 #H32
lapply (feqg_cpx_trans_cpx … H32 … HTU2) // #HTU32