-| #l #HG #HL #HU #U2 #H0 #L2 #_ #g1 #H1 #g2 #H2 -IH -G0 -L0 -U0
- lapply (cpx_inv_gref1 … H0) -H0 #H destruct
- /3 width=3 by frees_inv_gref, sle_isid_sn/
-
-| #j #HG #HL #HU #U2 #H1 #L2 #HL12 #i #H2 elim (cpx_inv_lref1 … H1) -H1
- [ #H destruct elim (frees_inv_lref … H2) -H2 //
- * #I #K2 #W2 #Hj #Hji #HLK2 #HW2
- elim (lpx_drop_trans_O1 … HL12 … HLK2) -HL12 #Y #HLK1 #H
- elim (lpx_inv_pair2 … H) -H #K1 #W1 #HK12 #HW12 #H destruct
- /4 width=11 by frees_lref_be, fqup_lref/
- | * #I #K1 #W1 #W0 #HLK1 #HW10 #HW0U2
- lapply (drop_fwd_drop2 … HLK1) #H0
- elim (lpx_drop_conf … H0 … HL12) -H0 -HL12 #K2 #HK12 #HLK2
- elim (ylt_split i (j+1)) >yplus_SO2 #Hji
- [ -IH elim (frees_inv_lift_be … H2 … HLK2 … HW0U2) /2 width=1 by ylt_fwd_succ2/
- | lapply (frees_inv_lift_ge … H2 … HLK2 … HW0U2 ?) -L2 -U2 // destruct
- /4 width=11 by frees_lref_be, fqup_lref, yle_succ1_inj/
- ]
- ]
-| -IH #p #HG #HL #HU #U2 #H1 >(cpx_inv_gref1 … H1) -H1 destruct
- #L2 #_ #i #H2 elim (frees_inv_gref … H2)
-| #a #I #W1 #U1 #HG #HL #HU #X #HX #L2 #HL12 #i #Hi destruct
- elim (cpx_inv_bind1 … HX) -HX *
- [ #W2 #U2 #HW12 #HU12 #H destruct
- elim (frees_inv_bind_O … Hi) -Hi
- /4 width=7 by frees_bind_dx_O, frees_bind_sn, lpx_pair/
- | #U2 #HU12 #HXU2 #H1 #H2 destruct
- lapply (frees_lift_ge … Hi (L2.ⓓW1) (Ⓕ) … HXU2 ?)
- /4 width=7 by frees_bind_dx_O, lpx_pair, drop_drop/
- ]
-| #I #W1 #X1 #HG #HL #HU #X2 #HX2 #L2 #HL12 #i #Hi destruct
- elim (cpx_inv_flat1 … HX2) -HX2 *
- [ #W2 #U2 #HW12 #HU12 #H destruct
- elim (frees_inv_flat … Hi) -Hi /3 width=7 by frees_flat_dx, frees_flat_sn/
- | #HU12 #H destruct /3 width=7 by frees_flat_dx/
- | #HW12 #H destruct /3 width=7 by frees_flat_sn/
- | #b #W2 #V1 #V2 #U1 #U2 #HW12 #HV12 #HU12 #H1 #H2 #H3 destruct
- elim (frees_inv_bind … Hi) -Hi #Hi
- [ elim (frees_inv_flat … Hi) -Hi
- /4 width=7 by frees_flat_dx, frees_flat_sn, frees_bind_sn/
- | lapply (lreq_frees_trans … Hi (L2.ⓛV2) ?) /2 width=1 by lreq_succ/ -Hi #HU2
- lapply (frees_weak … HU2 0 ?) -HU2
- /5 width=7 by frees_bind_dx_O, frees_flat_dx, lpx_pair/
+axiom monotonic_sle_sor: ∀f1,g1. f1 ⊆ g1 → ∀f2,g2. f2 ⊆ g2 →
+ ∀f. f1 ⋓ f2 ≡ f → ∀g. g1 ⋓ g2 ≡ g → f ⊆ g.
+
+axiom sle_tl: ∀f1,f2. f1 ⊆ f2 → ⫱f1 ⊆ ⫱f2.
+
+axiom frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f →
+ ∀K. ⬇*[b, 𝐔❴1❵] L ≡ K → ∀T. ⬆*[1] T ≡ U →
+ K ⊢ 𝐅*⦃T⦄ ≡ ⫱f.
+
+(* Basic_2A1: was: lpx_cpx_frees_trans *)
+lemma cpx_frees_trans_lexs: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
+ ∀L2. L1 ⦻*[cpx h G, cfull, f1] L2 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
+ ∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
+#h #G #L1 #T1 @(fqup_wf_ind_eq … G L1 T1) -G -L1 -T1
+#G0 #L0 #U0 #IH #G #L1 * *
+[ -IH #s #HG #HL #HU #g1 #H1 #L2 #_ #U2 #H0 destruct
+ lapply (frees_inv_sort … H1) -H1 #Hg1
+ elim (cpx_inv_sort1 … H0) -H0 #H destruct
+ /3 width=3 by frees_sort_gen, sle_refl, ex2_intro/
+| #i #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
+ elim (frees_inv_lref_drops … H1) -H1 *
+ [ -IH #HL1 #Hg1
+ elim (cpx_inv_lref1_drops … H0) -H0
+ [ #H destruct lapply (pippo … HL1 … H2) -HL1 -H2
+ /3 width=3 by frees_lref_atom, sle_refl, ex2_intro/
+ | * -H2 -Hg1 #I #K1 #V1 #V2 #HLK1
+ lapply (drops_TF … HLK1) -HLK1 #HLK1
+ lapply (drops_mono … HLK1 … HL1) -L1 #H destruct