+lemma cpg_lifts_bi: ∀Rt. reflexive … Rt →
+ ∀c,h,G. d_liftable2_bi … lifts (cpg Rt h c G).
+/3 width=12 by cpg_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+lemma cpg_inv_lifts_sn: ∀Rt. reflexive … Rt →
+ ∀c,h,G. d_deliftable2_sn … lifts (cpg Rt h c G).
+#Rt #HRt #c #h #G #L #U generalize in match c; -c
+@(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * *
+[ #s #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_sort2 … H1) -H1 #H destruct
+ elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
+ /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
+| #i2 #HG #HL #HU #c #U2 #H2 #b #f #K #HLK #X1 #H1 destruct
+ elim (cpg_inv_lref1_drops … H2) -H2 *
+ [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
+ #cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
+ elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
+ lapply (drops_split_div … HLK (𝐔❴i1❵) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
+ lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
+ lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
+ elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct
+ elim (liftsb_inv_pair_dx … HZ) -HZ #V #HVW #H destruct
+ elim (IH … HW2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -L -L0 -W #V2 #HVW2 #HV2
+ lapply (lifts_trans … HVW2 … HWU2 ??) -W2 [3,6: |*: // ] #HVU2
+ elim (lifts_split_trans … HVU2 ? f) -HVU2 [1,4: |*: /2 width=4 by after_uni_succ_sn/ ]
+ /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_F, ex2_intro/
+| #l #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_gref2 … H1) -H1 #H destruct
+ elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
+ /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
+| #p #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
+ elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (cpg_inv_bind1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
+ /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
+ | #cU #U2 #HU21 #HUX2 #H1 #H2 #H3 destruct
+ elim (lifts_div4_one … HTU1 … HU21) -HTU1 #T2 #HT21 #HTU2
+ elim (IH … HUX2 … HLK … HTU2) [| /3 width=1 by fqup_zeta/ ] -L -W1 -U1 -U2
+ /3 width=5 by cpg_zeta, ex2_intro/
+ ]
+| * #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
+ elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
+ [ elim (cpg_inv_appl1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
+ | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
+ /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
+ | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12
+ elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
+ lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H
+ elim (lifts_split_trans … H ? (⫯f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ]
+ /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
+ ]
+ | elim (cpg_inv_cast1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #HcWU #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
+ | #cU #HU12 #H destruct
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=3 by cpg_eps, ex2_intro/
+ | #cW #HW12 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 //
+ /3 width=3 by cpg_ee, ex2_intro/
+ ]
+ ]