--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/drops_drops.ma".
+include "static_2/s_computation/fqup_weight.ma".
+include "static_2/s_computation/fqup_drops.ma".
+include "static_2/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: was: aaa_lref *)
+lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
+#I #G #K #V #B #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H //
+ #H destruct /2 width=1 by aaa_zero/
+| #i #IH #L <uni_succ #H #HB lapply (drops_inv_bind2_isuni_next … H) -H // *
+ #Z #Y #HY #H destruct /3 width=1 by aaa_lref/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_2A1: was: aaa_inv_lref *)
+lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
+ ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+#G #A #i elim i -i
+[ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/
+| #i #IH #L #H elim (aaa_inv_lref … H) -H
+ #I #K #H #HA destruct elim (IH … HA) -IH -HA /3 width=5 by drops_drop, ex2_3_intro/
+]
+qed-.
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: includes: aaa_lift *)
+(* Note: it should use drops_split_trans_pair2 *)
+lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b, 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 #b #f #L2 #HL21 #X #HX -b -IH
+ lapply (aaa_inv_sort … H) -H #H destruct
+ >(lifts_inv_sort1 … HX) -HX //
+| #i1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
+ elim (aaa_inv_lref_drops … 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_tls_at … Hf … HY) -HY #HY -Hf
+ elim (drops_inv_skip2 … HY) -HY #Z #K2 #HK21 #HZ #H destruct
+ elim (liftsb_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct
+ /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_gen/
+| #l #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -f -IH
+ elim (aaa_inv_gref … H)
+| #p * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
+ [ elim (aaa_inv_abbr … H) -H #B #HB #HA
+ elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
+ | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
+ elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /5 width=8 by aaa_abst, drops_skip, ext2_pair/
+ ]
+| * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #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 → ∀b,f,L1. ⬇*[b, 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 #b #f #L1 #HL21 #X #HX -b -IH
+ lapply (aaa_inv_sort … H) -H #H destruct
+ >(lifts_inv_sort2 … HX) -HX //
+| #i2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
+ elim (aaa_inv_lref_drops … 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_tls_at … Hf … HY) -HY #HY -Hf
+ elim (drops_inv_skip1 … HY) -HY #Z #K1 #HK21 #HZ #H destruct
+ elim (liftsb_inv_pair_dx … HZ) -HZ #V1 #HV12 #H destruct
+ /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_F/
+| #l #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -IH -b -f
+ elim (aaa_inv_gref … H)
+| #p * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
+ [ elim (aaa_inv_abbr … H) -H #B #HB #HA
+ elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
+ | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
+ elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /5 width=8 by aaa_abst, drops_skip, ext2_pair/
+ ]
+| * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #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/
+ ]
+]
+qed-.
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