-
-lemma fpb_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 elim (fpb_fpbu … H1) -H1
-/3 width=5 by fleq_fpbg_trans, fpbg_strap2, fpbu_fpbc/
-qed-.
-
-lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpb_fpbg_trans/
-qed-.
-
-(* Note: this is used in the closure proof *)
-lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpb_trans/
+lemma fpbs_fpb_trans: ∀h,o,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, o] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, o] ⦃G2, L2, U2⦄.
+#h #o #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_fpbg … H) -H
+[ #H12 #G2 #L2 #U2 #H2 elim (fleq_fpb_trans … H12 … H2) -F2 -K2 -T2
+ /3 width=5 by fleq_fpbs, ex2_3_intro/
+| * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
+ @(ex2_3_intro … H4) -H4 /3 width=5 by fpbs_strap1, fpb_fpbq/
+]