(* Properties with generic slicing for local environments *******************)
lemma fqup_drops_succ: ∀b,G,K,T,i,L,U. ⇩*[↑i] L ≘ K → ⇧*[↑i] T ≘ U →
- â¦\83G,L,Uâ¦\84 â¬\82+[b] â¦\83G,K,Tâ¦\84.
+ â\9dªG,L,Uâ\9d« â¬\82+[b] â\9dªG,K,Tâ\9d«.
#b #G #K #T #i elim i -i
[ #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
#I #Y #HY #H destruct <(drops_fwd_isid … HY) -K //
/3 width=2 by fqu_fqup, fqu_drop/
| #l #IH #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
#I #Y #HY #H destruct
- elim (lifts_split_trans â\80¦ HTU â\80¦ (ð\9d\90\94â\9d´â\86\91lâ\9dµ) (ð\9d\90\94â\9d´1â\9dµ)) -HTU
+ elim (lifts_split_trans â\80¦ HTU â\80¦ (ð\9d\90\94â\9d¨â\86\91lâ\9d©) (ð\9d\90\94â\9d¨1â\9d©)) -HTU
/4 width=5 by fqup_strap2, fqu_drop/
]
qed.
lemma fqup_drops_strap1: ∀b,G1,G2,L1,K1,K2,T1,T2,U1,i. ⇩*[i] L1 ≘ K1 → ⇧*[i] T1 ≘ U1 →
- â¦\83G1,K1,T1â¦\84 â¬\82[b] â¦\83G2,K2,T2â¦\84 â\86\92 â¦\83G1,L1,U1â¦\84 â¬\82+[b] â¦\83G2,K2,T2â¦\84.
+ â\9dªG1,K1,T1â\9d« â¬\82[b] â\9dªG2,K2,T2â\9d« â\86\92 â\9dªG1,L1,U1â\9d« â¬\82+[b] â\9dªG2,K2,T2â\9d«.
#b #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 *
[ #HLK1 #HTU1 #HT12
>(drops_fwd_isid … HLK1) -L1 //
]
qed-.
-lemma fqup_lref: ∀b,I,G,L,K,V,i. ⇩*[i] L ≘ K.ⓑ{I}V → ⦃G,L,#i⦄ ⬂+[b] ⦃G,K,V⦄.
+lemma fqup_lref: ∀b,I,G,L,K,V,i. ⇩*[i] L ≘ K.ⓑ[I]V → ❪G,L,#i❫ ⬂+[b] ❪G,K,V❫.
/2 width=6 by fqup_drops_strap1/ qed.