(* Properties with generic equivalence for closures *************************)
-(**) (* to update *)
-lemma feqg_cpx_trans (S):
+lemma feqg_cpx_trans_cpx (S):
reflexive … S → symmetric … S →
∀G1,G2,L1,L2,T1,T. ❪G1,L1,T1❫ ≛[S] ❪G2,L2,T❫ →
- ∀T2. ❪G2,L2❫ ⊢ T ⬈ T2 →
- ∃∃T0. ❪G1,L1❫ ⊢ T1 ⬈ T0 & ❪G1,L1,T0❫ ≛[S] ❪G2,L2,T2❫.
+ ∀T2. ❪G2,L2❫ ⊢ T ⬈ T2 → ❪G1,L1❫ ⊢ T1 ⬈ T2.
#S #H1S #H2S #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
elim (feqg_inv_gen_dx … H) -H // #H #HL12 #HT1 destruct
-lapply (reqg_cpx_trans … HL12 … HT2) // #H
+@(cpx_teqg_repl_reqg … HT2)
+/2 width=7 by reqg_sym, teqg_sym, teqg_refl/
+qed-.
+
+lemma feqg_cpx_trans_feqg (S):
+ reflexive … S → symmetric … S →
+ ∀G1,G2,L1,L2,T1,T. ❪G1,L1,T1❫ ≛[S] ❪G2,L2,T❫ →
+ ∀T2. ❪G2,L2❫ ⊢ T ⬈ T2 → ❪G1,L1,T2❫ ≛[S] ❪G2,L2,T2❫.
+#S #H1S #H2S #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
+elim (feqg_inv_gen_dx … H) -H // #H #HL12 #_ destruct
lapply (cpx_reqg_conf_dx … HT2 … HL12) -HT2 -HL12 // #HL12
-lapply (teqg_cpx_trans … HT1 … H) -T // #HT12
-/4 width=4 by feqg_intro_sn, teqg_refl, ex2_intro/
+/3 width=1 by feqg_intro_sn, teqg_refl/
qed-.