(* Properties with context-sensitive free variables *************************)
-axiom lsubf_frees_trans: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 → ∀f,L1. ⦃L1, f⦄ ⫃𝐅* ⦃L2, f2⦄ →
+lemma lsubf_frees_trans: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 → ∀f,L1. ⦃L1, f⦄ ⫃𝐅* ⦃L2, f2⦄ →
∃∃f1. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 & f1 ⊆ f.
-(*
#f2 #L2 #T #H elim H -f2 -L2 -T
[ #f2 #I #Hf2 #f #L1 #H elim (lsubf_inv_atom2 … H) -H
#H #_ destruct /3 width=3 by frees_atom, sle_isid_sn, ex2_intro/
[ #K1 #H elim (sle_inv_nx … H ??) -H [ <tl_next_rew |*: // ]
#g2 #_ #H1 #H12 #H2 destruct elim (IH … H12) -K2
/3 width=7 by frees_zero, sle_next, ex2_intro/
- | #g #K1 #V #Hg <tl_next_rew #Hf lapply (sor_sym … Hf) -Hf
- #Hf #H elim (sle_inv_nx … H ??) -H [|*: // ]
+ | #g #K1 #V #Hg #Hf #H elim (sle_inv_nx … H ??) -H [ <tl_next_rew |*: // ]
#g2 #_ #H1 #H12 #H2 #H3 destruct elim (IH … H12) -K2
#f1 #Hf1 elim (sor_isfin_ex … f1 g ??)
- /5 width=10 by frees_fwd_isfin, frees_flat, frees_zero, monotonic_sle_sor, sor_inv_sle_dx, sor_sym, sor_sle_sn, sle_next, ex2_intro/
+ /4 width=7 by frees_fwd_isfin, frees_flat, frees_zero, sor_inv_sle, sle_next, ex2_intro/
]
| #f2 #I #K2 #W #i #_ #IH #f #L1 #H elim (lsubf_inv_pair2 … H) -H *
[ #K1 #_ #H12 #H | #g #K1 #V #Hg #Hf #_ #H12 #H1 #H2 ]
destruct elim (IH … H12) -K2
/3 width=3 by frees_gref, sle_inv_tl_dx, ex2_intro/
| #f2V #f2T #f2 #p #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f #L1 #H12
+ elim (IHV f L1) -IHV [2: /3 width=4 by lsubf_sle_div, sor_inv_sle_sn/ ]
+ elim (IHT (⫯f) (L1.ⓑ{I}V)) -IHT [2: /4 width=4 by lsubf_sle_div, lsubf_pair_nn, sor_inv_sle_dx, sor_inv_tl_dx/ ]
+ -f2V -f2T -f2 -L2 #f1T #HT #Hf1T #f1V #HV #Hf1V elim (sor_isfin_ex … f1V (⫱f1T) ??)
+ /4 width=9 by frees_fwd_isfin, frees_bind, sor_inv_sle, sle_xn_tl, isfin_tl, ex2_intro/
| #f2V #f2T #f2 #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f #L1 #H12
-*)
+ elim (IHV f L1) -IHV [2: /3 width=4 by lsubf_sle_div, sor_inv_sle_sn/ ]
+ elim (IHT f L1) -IHT [2: /3 width=4 by lsubf_sle_div, sor_inv_sle_dx/ ]
+ -f2V -f2T -f2 -L2 #f1T #HT #Hf1T #f1V #HV #Hf1V elim (sor_isfin_ex … f1V f1T ??)
+ /3 width=7 by frees_fwd_isfin, frees_flat, sor_inv_sle, ex2_intro/
+]
+qed-.