lemma fsle_frees_trans:
∀L1,L2,T1,T2. ❪L1,T1❫ ⊆ ❪L2,T2❫ →
∀f2. L2 ⊢ 𝐅+❪T2❫ ≘ f2 →
- â\88\83â\88\83n1,n2,f1. L1 â\8a¢ ð\9d\90\85+â\9dªT1â\9d« â\89\98 f1 & L1 â\89\8bâ\93§*[n1,n2] L2 & ⫱*[n1]f1 â\8a\86 ⫱*[n2]f2.
+ â\88\83â\88\83n1,n2,f1. L1 â\8a¢ ð\9d\90\85+â\9dªT1â\9d« â\89\98 f1 & L1 â\89\8bâ\93§*[n1,n2] L2 & â«°*[n1]f1 â\8a\86 â«°*[n2]f2.
#L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
lemma fsle_frees_conf:
∀L1,L2,T1,T2. ❪L1,T1❫ ⊆ ❪L2,T2❫ →
∀f1. L1 ⊢ 𝐅+❪T1❫ ≘ f1 →
- â\88\83â\88\83n1,n2,f2. L2 â\8a¢ ð\9d\90\85+â\9dªT2â\9d« â\89\98 f2 & L1 â\89\8bâ\93§*[n1,n2] L2 & ⫱*[n1]f1 â\8a\86 ⫱*[n2]f2.
+ â\88\83â\88\83n1,n2,f2. L2 â\8a¢ ð\9d\90\85+â\9dªT2â\9d« â\89\98 f2 & L1 â\89\8bâ\93§*[n1,n2] L2 & â«°*[n1]f1 â\8a\86 â«°*[n2]f2.
#L1 #L2 #T1 #T2 * #n1 #n2 #g1 #g2 #Hg1 #Hg2 #HL #Hn #f1 #Hf1
lapply (frees_mono … Hg1 … Hf1) -Hg1 -Hf1 #Hgf1
lapply (tls_eq_repl n1 … Hgf1) -Hgf1 #Hgf1
#L1 #L2 #HL #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
elim (lveq_inj_void_sn_ge … H1n1 … H1n2) -H1n2 // #H1 #H2 #H3 destruct
-elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
+elim (sor_isfin_ex f1 (â«°f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
<tls_xn in H2n2; #H2n2
/4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
qed.
* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p #I1
elim (lveq_inj_length … H1L) // #H1 #H2 destruct
elim (lveq_inj_length … H2L) // -HL -H2L #H1 #H2 destruct
-elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+elim (sor_isfin_ex f1 (â«°f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
+elim (sor_isfin_ex g1 (â«°g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
/4 width=15 by frees_bind_void, frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
qed.
* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
elim (lveq_inv_pair_pair … H2L) -H2L #H2L #H1 #H2 destruct
elim (lveq_inj … H2L … H1L) -H1L #H1 #H2 destruct
-elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+elim (sor_isfin_ex f1 (â«°f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
+elim (sor_isfin_ex g1 (â«°g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
/4 width=15 by frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
qed.