lemma nta_inv_ldef_sn (h) (a) (G) (K) (V):
∀X2. ❪G,K.ⓓV❫ ⊢ #0 :[h,a] X2 →
- ∃∃W,U. ❪G,K❫ ⊢ V :[h,a] W & ⇧*[1] W ≘ U & ❪G,K.ⓓV❫ ⊢ U ⬌*[h] X2 & ❪G,K.ⓓV❫ ⊢ X2 ![h,a].
+ ∃∃W,U. ❪G,K❫ ⊢ V :[h,a] W & ⇧[1] W ≘ U & ❪G,K.ⓓV❫ ⊢ U ⬌*[h] X2 & ❪G,K.ⓓV❫ ⊢ X2 ![h,a].
#h #a #G #Y #X #X2 #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
lemma nta_inv_lref_sn (h) (a) (G) (L):
∀X2,i. ❪G,L❫ ⊢ #↑i :[h,a] X2 →
- ∃∃I,K,T2,U2. ❪G,K❫ ⊢ #i :[h,a] T2 & ⇧*[1] T2 ≘ U2 & ❪G,K.ⓘ[I]❫ ⊢ U2 ⬌*[h] X2 & ❪G,K.ⓘ[I]❫ ⊢ X2 ![h,a] & L = K.ⓘ[I].
+ ∃∃I,K,T2,U2. ❪G,K❫ ⊢ #i :[h,a] T2 & ⇧[1] T2 ≘ U2 & ❪G,K.ⓘ[I]❫ ⊢ U2 ⬌*[h] X2 & ❪G,K.ⓘ[I]❫ ⊢ X2 ![h,a] & L = K.ⓘ[I].
#h #a #G #L #X2 #i #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_lref … H1) -H1 #I #K #Hi #H destruct
lemma nta_inv_lref_sn_drops_cnv (h) (a) (G) (L):
∀X2,i. ❪G,L❫ ⊢ #i :[h,a] X2 →
- ∨∨ ∃∃K,V,W,U. ⇩*[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V :[h,a] W & ⇧*[↑i] W ≘ U & ❪G,L❫ ⊢ U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a]
- | ∃∃K,W,U. ⇩*[i] L ≘ K. ⓛW & ❪G,K❫ ⊢ W ![h,a] & ⇧*[↑i] W ≘ U & ❪G,L❫ ⊢ U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a].
+ ∨∨ ∃∃K,V,W,U. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V :[h,a] W & ⇧[↑i] W ≘ U & ❪G,L❫ ⊢ U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a]
+ | ∃∃K,W,U. ⇩[i] L ≘ K. ⓛW & ❪G,K❫ ⊢ W ![h,a] & ⇧[↑i] W ≘ U & ❪G,L❫ ⊢ U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a].
#h #a #G #L #X2 #i #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_lref_drops … H1) -H1 #I #K #V #HLK #HV