+ elim (IH … HgT1 … HL12T … HT12) // -L1 -T1 #gT2 #HgT2 #HgT21
+ lapply (frees_inv_lifts_SO (Ⓣ) … HgT2 … L2 … HUT2) [ /3 width=1 by drops_refl, drops_drop/ ] -V1 -T2
+ /5 width=6 by sor_inv_sle_dx, sle_trans, sle_tl, ex2_intro/
+ ]
+| #I #V1 #T0 #HG #HL #HU #g1 #H1 #L2 #H2 #U2 #H0 destruct
+ elim (frees_inv_flat … H1) -H1 #gV1 #gT0 #HgV1 #HgT0 #Hg1
+ elim (cpx_inv_flat1 … H0) -H0 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
+ lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
+ elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
+ elim (IH … HgT0 … HL12T … HT12) // -IH -HgT0 -HL12T -HT12 #gT2 #HgT2 #HgT21
+ elim (sor_isfin_ex gV2 gT2) /2 width=3 by frees_fwd_isfin/
+ /3 width=10 by frees_flat, monotonic_sle_sor, ex2_intro/
+ | #HU2 #H destruct -HgV1
+ lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
+ elim (IH … HgT0 … HL12T … HU2) // -L1 -T0 -V1
+ /4 width=6 by sor_inv_sle_dx, sle_trans, ex2_intro/
+ | #HU2 #H destruct -HgT0
+ lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ -H2 #HL12V
+ elim (IH … HgV1 … HL12V … HU2) // -L1 -T0 -V1
+ /4 width=6 by sor_inv_sle_sn, sle_trans, ex2_intro/
+ | #p #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #H0 #H1 #H destruct
+ elim (frees_inv_bind … HgT0) -HgT0 #gW1 #gT1 #HgW1 #HgT1 #HgT0
+ lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
+ lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #H2
+ lapply (sle_lexs_trans … H2 gW1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12W
+ lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
+ lapply (lexs_inv_tl … Abst … HL12T … HW12 ?) // -HL12T #HL12T
+ elim (sor_isfin_ex gV1 gW1) /2 width=3 by frees_fwd_isfin/ #g0 #Hg0 #_
+ lapply (sor_trans2 … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
+ lapply (sor_sym … Hg0) -Hg0 #Hg0
+ elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
+ elim (IH … HgW1 … HL12W … HW12) // -HgW1 -HL12W -HW12 #gW2 #HgW2 #HgW21
+ elim (IH … HgT1 … HL12T … HT12) // -IH -HgT1 -HL12T -HT12 #gT2 #HgT2 #HgT21
+ elim (sor_isfin_ex gV2 (⫱gT2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #gVT2 #HgVT2 #_
+ elim (lsubf_frees_trans … HgT2 (⫯gVT2) … (L2.ⓓⓝW2.V2))
+ [2: /4 width=4 by lsubf_refl, lsubf_beta, sor_inv_sle_dx, sor_inv_sle_sn, sle_inv_tl_sn/ ]
+ -HgT2 #gT0 #HgT0 #HgT20
+ elim (sor_isfin_ex gW2 gV2) /2 width=3 by frees_fwd_isfin/ #gV0 #HgV0 #H
+ elim (sor_isfin_ex gV0 (⫱gT0)) /3 width=3 by frees_fwd_isfin, isfin_tl/ -H #g2 #Hg2 #_
+ @(ex2_intro … g2)
+ [ @(frees_bind … Hg2) /2 width=5 by frees_flat/
+ | -L2 @(sor_inv_sle … Hg2) -Hg2
+ [ /3 width=11 by sor_inv_sle_sn_trans, monotonic_sle_sor/
+ | @sle_xn_tl [2:|*: // ] @(sle_trans … HgT20) -HgT20
+ /4 width=8 by monotonic_sle_sor, sor_inv_sle_dx_trans, sle_tl, sle_next/
+ ] (**) (* full auto too slow *)
+ ]
+ | #p #V2 #V #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #H0 #H1 #H destruct
+ elim (frees_inv_bind … HgT0) -HgT0 #gW1 #gT1 #HgW1 #HgT1 #HgT0
+ lapply (sle_lexs_trans … H2 gV1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12V
+ lapply (sle_lexs_trans … H2 gT0 ?) /2 width=2 by sor_inv_sle_dx/ -H2 #H2
+ lapply (sle_lexs_trans … H2 gW1 ?) /2 width=2 by sor_inv_sle_sn/ #HL12W
+ lapply (sle_lexs_trans … H2 (⫱gT1) ?) /2 width=2 by sor_inv_sle_dx/ -H2 #HL12T
+ lapply (lexs_inv_tl … Abbr … HL12T … HW12 ?) // -HL12T #HL12T
+ elim (sor_isfin_ex gV1 gW1) /2 width=3 by frees_fwd_isfin/ #g0 #Hg0 #_
+ lapply (sor_trans2 … Hg1 … HgT0 … Hg0) -Hg1 -HgT0 #Hg1
+ elim (IH … HgV1 … HL12V … HV12) // -HgV1 -HL12V -HV12 #gV2 #HgV2 #HgV21
+ elim (IH … HgW1 … HL12W … HW12) // -HgW1 -HL12W -HW12 #gW2 #HgW2 #HgW21