-lemma cpg_delift: ∀c,h,I,G,K,V,T1,L,i. ⬇*[i] L ≡ (K.ⓑ{I}V) →
- ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h, 𝟘𝟘] T2 & ⬆*[↑1] T ≡ T2.
-#h #c #I #G #K #V #T1 elim T1 -T1
-[ * #i #L #l /2 width=4 by cpg_atom, lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpg_atom, lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i) /3 width=7 by cpg_delta, ex2_2_intro/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpg_bind, drop_drop, lift_bind, ex2_2_intro/
- | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpg_flat, lift_flat, ex2_2_intro/
+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 * *
+[ #s #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_sort1 … 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/
+| #i1 #HG #HK #HT #c #T2 #H2 #b #f #L #HLK #X1 #H1 destruct
+ elim (cpg_inv_lref1_drops … H2) -H2 *
+ [ #H1 #H2 destruct /2 width=3 by ex2_intro/ ]
+ #cV #K0 #V #V2 #HK0 #HV2 #HVT2 #H destruct
+ 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
+ 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
+ lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2
+ lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ]
+ /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/
+| #l #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_gref1 … 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 #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
+ elim (lifts_inv_bind1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (cpg_inv_bind1 … H2) -H2 *
+ [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
+ | #cT #T2 #HT12 #HXT2 #H1 #H2 #H3 destruct
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] #U2 #HTU2 #HU12
+ lapply (lifts_trans … HXT2 … HTU2 ??) -T2 [3: |*: // ] #HXU2
+ elim (lifts_split_trans … HXU2 f (𝐔❴⫯O❵)) [2: /2 width=1 by after_uni_one_dx/ ]
+ /3 width=5 by cpg_zeta, ex2_intro/
+ ]
+| #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
+ elim (lifts_inv_flat1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (cpg_inv_flat1 … H2) -H2 *
+ [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=5 by cpg_flat, lifts_flat, ex2_intro/
+ | #cT #HT12 #H1 #H2 destruct
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=3 by cpg_eps, ex2_intro/
+ | #cV #HV12 #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 //
+ /3 width=3 by cpg_ee, ex2_intro/
+ | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct
+ elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HY12 … HLK … HYZ1) -HY12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
+ | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct
+ elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
+ elim (IH … HY12 … HLK … HYZ1) -HY12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ elim (lifts_total W2 (𝐔❴1❵)) #W20 #HW20
+ lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
+ lapply (lifts_conf … HV20 … H (↑f) ?) -V2 /2 width=3 by after_uni_one_sn/
+ /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/