(* *)
(**************************************************************************)
-include "basic_2/relocation/drops.ma".
+include "basic_2/relocation/drops_drops.ma".
+include "basic_2/s_computation/fqup_weight.ma".
+include "basic_2/s_computation/fqup_drops.ma".
include "basic_2/static/aaa.ma".
(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-(* Properties with generic slicing for local environments *******************)
+(* Advanced properties ******************************************************)
(* Basic_2A1: was: aaa_lref *)
-lemma aaa_lref_gen: ∀I,G,K,V,B,i,L. ⬇*[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
+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_pair2_isuni_next … H) -H // *
- #Z #Y #X #HY #H destruct /3 width=1 by aaa_lref/
+| #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.
-(* 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.
-#G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #G #L1 #s #L2 #c #f #_ #T2 #H
- >(lifts_inv_sort1 … H) -H //
-| #I #G #L1 #V1 #B #_ #IHB #L2 #c #f #HL21 #T2 #H
- elim (lifts_inv_lref1 … H) -H #i2 #Hi #H destruct
- @aaa_lref_gen [5: @IHB ]
-| #I #G #L1 #V1 #B #i1 #_ #IHB #L2 #c #f #HL21 #T2 #H
- elim (lifts_inv_lref1 … H) -H #i2 #Hi #H destruct
- lapply (at_inv_nxx … H)
-
-
-
-
-| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #c #l #k #HL21 #T2 #H
- elim (lift_inv_lref1 … H) -H * #Hil #H destruct
- [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=9 by aaa_lref/
- | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
- /3 width=9 by aaa_lref, drop_inv_gen/
- ]
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #c #l #k #HL21 #X #H
- elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=5 by aaa_abbr, drop_skip/
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #c #l #k #HL21 #X #H
- elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=5 by aaa_abst, drop_skip/
-| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #c #l #k #HL21 #X #H
- elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=5 by aaa_appl/
-| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #c #l #k #HL21 #X #H
- elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=5 by aaa_cast/
-]
-qed.
-
-(* Inversion lemmas with generic slicing for local environments *************)
+(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: was: aaa_inv_lref *)
-lemma aaa_inv_lref_gen: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
- ∃∃I,K,V. ⬇*[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+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 #V #H #HA destruct elim (IH … HA) -IH -HA /3 width=5 by drops_drop, ex2_3_intro/
+ #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-.
-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/
+(* 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/
]
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #c #l #k #HL21 #X #H
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /4 width=5 by aaa_abbr, drop_skip/
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #c #l #k #HL21 #X #H
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /4 width=5 by aaa_abst, drop_skip/
-| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #c #l #k #HL21 #X #H
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /3 width=5 by aaa_appl/
-| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #c #l #k #HL21 #X #H
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /3 width=5 by aaa_cast/
]
qed-.
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