(* Properties with generic slicing for local environments *******************)
+(* Note: it should use drops_split_trans_pair2 *)
lemma cpg_lifts: ∀c,h,G. d_liftable2 (cpg h c G).
#c #h #G #K #T generalize in match c; -c
@(fqup_wf_ind_eq … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * *
elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct
lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL0 #HY
- lapply (drops_inv_tls_at … Hf … HY) -HY #HY
+ lapply (drops_tls_at … Hf … HY) -HY #HY
elim (drops_inv_skip2 … HY) -HY #L0 #W #HLK0 #HVW #H destruct
elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2
elim (lifts_total W2 (𝐔❴⫯i2❵)) #U2 #HWU2
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_inv_tls_at … Hf … HLY0) -HLY0 #HLY0
+ lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
elim (drops_inv_skip1 … HLY0) -HLY0 #K0 #V #HLK0 #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