(* Advanced properties ******************************************************)
(* Basic_2A1: was: aaa_lref *)
-lemma aaa_lref_drops: â\88\80I,G,K,V,B,i,L. â¬\87*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ⁝ B → ⦃G,L⦄ ⊢ #i ⁝ B.
+lemma aaa_lref_drops: â\88\80I,G,K,V,B,i,L. â\87©*[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/
(* Basic_2A1: was: aaa_inv_lref *)
lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G,L⦄ ⊢ #i ⁝ A →
- â\88\83â\88\83I,K,V. â¬\87*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A.
+ â\88\83â\88\83I,K,V. â\87©*[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
]
qed-.
+lemma aaa_pair_inv_lref (G) (L) (i):
+ ∀A. ⦃G,L⦄ ⊢ #i ⁝ A → ∀I,K,V. ⇩*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ⁝ A.
+#G #L #i #A #H #I #K #V #HLK
+elim (aaa_inv_lref_drops … H) -H #J #Y #X #HLY #HX
+lapply (drops_mono … HLY … HLK) -L -i #H destruct //
+qed-.
+
(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: includes: aaa_lift *)
(* Note: it should use drops_split_trans_pair2 *)
-lemma aaa_lifts: â\88\80G,L1,T1,A. â¦\83G,L1â¦\84 â\8a¢ T1 â\81\9d A â\86\92 â\88\80b,f,L2. â¬\87*[b,f] L2 ≘ L1 →
- â\88\80T2. â¬\86*[f] T1 ≘ T2 → ⦃G,L2⦄ ⊢ T2 ⁝ A.
+lemma aaa_lifts: â\88\80G,L1,T1,A. â¦\83G,L1â¦\84 â\8a¢ T1 â\81\9d A â\86\92 â\88\80b,f,L2. â\87©*[b,f] L2 ≘ L1 →
+ â\88\80T2. â\87§*[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
(* Inversion lemmas with generic slicing for local environments *************)
(* Basic_2A1: includes: aaa_inv_lift *)
-lemma aaa_inv_lifts: â\88\80G,L2,T2,A. â¦\83G,L2â¦\84 â\8a¢ T2 â\81\9d A â\86\92 â\88\80b,f,L1. â¬\87*[b,f] L2 ≘ L1 →
- â\88\80T1. â¬\86*[f] T1 ≘ T2 → ⦃G,L1⦄ ⊢ T1 ⁝ A.
+lemma aaa_inv_lifts: â\88\80G,L2,T2,A. â¦\83G,L2â¦\84 â\8a¢ T2 â\81\9d A â\86\92 â\88\80b,f,L1. â\87©*[b,f] L2 ≘ L1 →
+ â\88\80T1. â\87§*[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