(* Properties with degree-based equivalence for closures ********************)
(* Basic_2A1: uses: fleq_fpb_trans *)
-lemma feqx_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ❪F1,K1,T1❫ ≛ ❪F2,K2,T2❫ →
- ∀G2,L2,U2. ❪F2,K2,T2❫ ≻[h] ❪G2,L2,U2❫ →
- ∃∃G1,L1,U1. ❪F1,K1,T1❫ ≻[h] ❪G1,L1,U1❫ & ❪G1,L1,U1❫ ≛ ❪G2,L2,U2❫.
-#h #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
+lemma feqx_fpb_trans:
+ ∀F1,F2,K1,K2,T1,T2. ❪F1,K1,T1❫ ≛ ❪F2,K2,T2❫ →
+ ∀G2,L2,U2. ❪F2,K2,T2❫ ≻ ❪G2,L2,U2❫ →
+ ∃∃G1,L1,U1. ❪F1,K1,T1❫ ≻ ❪G1,L1,U1❫ & ❪G1,L1,U1❫ ≛ ❪G2,L2,U2❫.
+#F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
#K2 #T2 #HK12 #HT12 #G2 #L2 #U2 #H12
elim (teqx_fpb_trans … HT12 … H12) -T2 #K0 #T0 #H #HT0 #HK0
elim (reqx_fpb_trans … HK12 … H) -K2 #L0 #U0 #H #HUT0 #HLK0
(* Inversion lemmas with degree-based equivalence for closures **************)
(* Basic_2A1: uses: fpb_inv_fleq *)
-lemma fpb_inv_feqx: ∀h,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ →
- ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫ → ⊥.
-#h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+lemma fpb_inv_feqx:
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≻ ❪G2,L2,T2❫ →
+ ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫ → ⊥.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #H elim (feqx_inv_gen_sn … H) -H
/3 width=11 by reqx_fwd_length, fqu_inv_teqx/
| #T2 #_ #HnT #H elim (feqx_inv_gen_sn … H) -H /2 width=1 by/