include "static_2/relocation/lifts_basic.ma".
include "basic_2/rt_transition/cpx_drops.ma".
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* EXTENDED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS **************)
(* Properties with basic relocation *****************************************)
-lemma cpx_subst (h) (G) (L) (U1) (i):
- ∀I,K,V. ⇩*[i] L ≘ K.ⓑ{I}V →
- ∃∃U2,T2. ⦃G,L⦄ ⊢ U1 ⬈[h] U2 & ⇧[i,1] T2 ≘ U2.
-#h #G #L #U1 @(fqup_wf_ind_eq (Ⓣ) … G L U1) -G -L -U1
+lemma cpx_subst (G) (L) (U1) (i):
+ ∀I,K,V. ⇩[i] L ≘ K.ⓑ[I]V →
+ ∃∃U2,T2. ❪G,L❫ ⊢ U1 ⬈ U2 & ⇧[i,1] T2 ≘ U2.
+#G #L #U1 @(fqup_wf_ind_eq (Ⓣ) … G L U1) -G -L -U1
#G0 #L0 #U0 #IH #G #L * *
[ #s #HG #HL #HT #i #I #K #V #_ destruct -IH
/2 width=4 by lifts_sort, ex2_2_intro/
| #j #HG #HL #HT #i #I #K #V #HLK destruct -IH
elim (lt_or_eq_or_gt i j) #Hij
[ /3 width=4 by lifts_lref_ge_minus, cpx_refl, ex2_2_intro/
- | elim (lifts_total V (ð\9d\90\94â\9d´â\86\91iâ\9dµ)) #U2 #HU2
- elim (lifts_split_trans â\80¦ HU2 (ð\9d\90\94â\9d´iâ\9dµ) (ð\9d\90\81â\9d´i,1â\9dµ)) [2: @(after_basic_rc i 0) ]
+ | elim (lifts_total V (ð\9d\90\94â\9d¨â\86\91iâ\9d©)) #U2 #HU2
+ elim (lifts_split_trans â\80¦ HU2 (ð\9d\90\94â\9d¨iâ\9d©) (ð\9d\90\81â\9d¨i,1â\9d©)) [2: @(after_basic_rc i 0) ]
/3 width=7 by cpx_delta_drops, ex2_2_intro/
| /3 width=4 by lifts_lref_lt, cpx_refl, ex2_2_intro/
]
/2 width=4 by lifts_gref, ex2_2_intro/
| #p #J #W1 #U1 #HG #HL #HT #i #I #K #V #HLK destruct
elim (IH G L W1 … HLK) [| // ] #W2 #V2 #HW12 #HVW2
- elim (IH G (L.ⓑ{J}W1) U1 … (↑i)) [|*: /3 width=4 by drops_drop/ ] #U2 #T2 #HU12 #HTU2
+ elim (IH G (L.ⓑ[J]W1) U1 … (↑i)) [|*: /3 width=4 by drops_drop/ ] #U2 #T2 #HU12 #HTU2
/3 width=9 by cpx_bind, lifts_bind, ex2_2_intro/
| #J #W1 #U1 #HG #HL #HT #i #I #K #V #HLK destruct
elim (IH G L W1 … HLK) [| // ] #W2 #V2 #HW12 #HVW2