include "basic_2/reduction/cpx_lift.ma".
include "basic_2/reduction/cnx.ma".
-(* CONTEXT-SENSITIVE EXTENDED NORMAL TERMS **********************************)
-
-(* Advanced properties ******************************************************)
-
-lemma cnx_lref_atom: ∀h,g,G,L,i. ⇩[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃#i⦄.
-#h #g #G #L #i #HL #X #H
-elim (cpx_inv_lref1 … H) -H // *
-#I #K #V1 #V2 #HLK #_ #_
-lapply (ldrop_mono … HL … HLK) -L #H destruct
-qed.
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************)
(* Relocation properties ****************************************************)
-lemma cnx_lift: ∀h,g,G,L0,L,T,T0,s,d,e. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⇩[s, d, e] L0 ≡ L →
- ⇧[d, e] T ≡ T0 → ⦃G, L0⦄ ⊢ 𝐍[h, g]⦃T0⦄.
+lemma cnx_lift: ∀h,g,G,L0,L,T,T0,s,d,e. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⇩[s, d, e] L0 ≡ L →
+ ⇧[d, e] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡[h, g] 𝐍⦃T0⦄.
#h #g #G #L0 #L #T #T0 #s #d #e #HLT #HL0 #HT0 #X #H
elim (cpx_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
<(HLT … HT1) in HT0; -L #HT0
>(lift_mono … HT10 … HT0) -T1 -X //
qed.
-lemma cnx_inv_lift: ∀h,g,G,L0,L,T,T0,s,d,e. ⦃G, L0⦄ ⊢ 𝐍[h, g]⦃T0⦄ → ⇩[s, d, e] L0 ≡ L →
- ⇧[d, e] T ≡ T0 → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄.
+lemma cnx_inv_lift: ∀h,g,G,L0,L,T,T0,s,d,e. ⦃G, L0⦄ ⊢ ➡[h, g] 𝐍⦃T0⦄ → ⇩[s, d, e] L0 ≡ L →
+ ⇧[d, e] T ≡ T0 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
#h #g #G #L0 #L #T #T0 #s #d #e #HLT0 #HL0 #HT0 #X #H
elim (lift_total X d e) #X0 #HX0
lapply (cpx_lift … H … HL0 … HT0 … HX0) -L #HTX0