-lemma aaa_inv_lift: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,c,l,k. ⬇[c, l, k] L2 ≡ L1 →
- ∀T1. ⬆[l, k] T1 ≡ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
-#G #L2 #T2 #A #H elim H -G -L2 -T2 -A
-[ #G #L2 #s #L1 #c #l #k #_ #T1 #H
- >(lift_inv_sort2 … H) -H //
-| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #c #l #k #HL21 #T1 #H
- elim (lift_inv_lref2 … H) -H * #Hil #H destruct
- [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=9 by aaa_lref/
- | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /3 width=9 by aaa_lref/
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: includes: aaa_lift *)
+lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀L2,c,f. ⬇*[c, f] L2 ≡ L1 →
+ ∀T2. ⬆*[f] T1 ≡ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+@fqup_wf_ind_eq #G0 #L0 #T0 #IH #G #L1 * *
+[ #s #HG #HL #HT #A #H #L2 #c #f #HL21 #X #HX -IH -c
+ lapply (aaa_inv_sort … H) -H #H destruct
+ >(lifts_inv_sort1 … HX) -HX //
+| #i1 #HG #HL #HT #A #H #L2 #c #f #HL21 #X #HX
+ elim (aaa_inv_lref_gen … H) -H #J #K1 #V1 #HLK1 #HA
+ elim (lifts_inv_lref1 … HX) -HX #i2 #Hf #H destruct
+ lapply (drops_trans … HL21 … HLK1 ??) -HL21 [1,2: // ] #H
+ elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK2 #HY
+ lapply (drops_inv_tls_at … Hf … HY) -HY #HY -Hf
+ elim (drops_inv_skip2 … HY) -HY #K2 #V2 #HK21 #HV12 #H destruct
+ /4 width=12 by aaa_lref_gen, fqup_lref, drops_inv_gen/
+| #l #HG #HL #HT #A #H #L2 #c #f #HL21 #X #HX -IH -c -f
+ elim (aaa_inv_gref … H)
+| #p * #V1 #T1 #HG #HL #HT #A #H #L2 #c #f #HL21 #X #HX
+ [ elim (aaa_inv_abbr … H) -H #B #HB #HA
+ elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=9 by aaa_abbr, drops_skip/
+ | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
+ elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=8 by aaa_abst, drops_skip/
+ ]
+| * #V1 #T1 #HG #HL #HT #A #H #L2 #c #f #HL21 #X #HX
+ [ elim (aaa_inv_appl … H) -H #B #HB #HA
+ elim (lifts_inv_flat1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /3 width=10 by aaa_appl/
+ | elim (aaa_inv_cast … H) -H #H1A #H2A
+ elim (lifts_inv_flat1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /3 width=8 by aaa_cast/
+ ]
+]
+qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: includes: aaa_inv_lift *)
+lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,c,f. ⬇*[c, f] L2 ≡ L1 →
+ ∀T1. ⬆*[f] T1 ≡ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
+@fqup_wf_ind_eq #G0 #L0 #T0 #IH #G #L2 * *
+[ #s #HG #HL #HT #A #H #L1 #c #f #HL21 #X #HX -IH -c
+ lapply (aaa_inv_sort … H) -H #H destruct
+ >(lifts_inv_sort2 … HX) -HX //
+| #i2 #HG #HL #HT #A #H #L1 #c #f #HL21 #X #HX
+ elim (aaa_inv_lref_gen … H) -H #J #K2 #V2 #HLK2 #HA
+ elim (lifts_inv_lref2 … HX) -HX #i1 #Hf #H destruct
+ lapply (drops_split_div … HL21 (𝐔❴i1❵) ???) -HL21 [4: * |*: // ] #Y #HLK1 #HY
+ lapply (drops_conf … HLK2 … HY ??) -HY [1,2: /2 width=6 by after_uni_dx/ ] #HY
+ lapply (drops_inv_tls_at … Hf … HY) -HY #HY -Hf
+ elim (drops_inv_skip1 … HY) -HY #K1 #V1 #HK21 #HV12 #H destruct
+ /4 width=12 by aaa_lref_gen, fqup_lref, drops_inv_F/
+| #l #HG #HL #HT #A #H #L1 #c #f #HL21 #X #HX -IH -c -f
+ elim (aaa_inv_gref … H)
+| #p * #V2 #T2 #HG #HL #HT #A #H #L1 #c #f #HL21 #X #HX
+ [ elim (aaa_inv_abbr … H) -H #B #HB #HA
+ elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /4 width=9 by aaa_abbr, drops_skip/
+ | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
+ elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /4 width=8 by aaa_abst, drops_skip/
+ ]
+| * #V2 #T2 #HG #HL #HT #A #H #L1 #c #f #HL21 #X #HX
+ [ elim (aaa_inv_appl … H) -H #B #HB #HA
+ elim (lifts_inv_flat2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /3 width=10 by aaa_appl/
+ | elim (aaa_inv_cast … H) -H #H1A #H2A
+ elim (lifts_inv_flat2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /3 width=8 by aaa_cast/