(* Properties based on preservation *****************************************)
lemma cnv_cpms_nta (h) (a) (G) (L):
- ∀T. ❪G,L❫ ⊢ T ![h,a] → ∀U.❪G,L❫ ⊢ T ➡*[1,h] U → ❪G,L❫ ⊢ T :[h,a] U.
+ ∀T. ❪G,L❫ ⊢ T ![h,a] → ∀U.❪G,L❫ ⊢ T ➡*[h,1] U → ❪G,L❫ ⊢ T :[h,a] U.
/3 width=4 by cnv_cast, cnv_cpms_trans/ qed.
lemma cnv_nta_sn (h) (a) (G) (L):
(* Basic_1: uses: ty3_sred_wcpr0_pr0 *)
lemma nta_cpr_conf_lpr (h) (a) (G):
- ∀L1,T1,U. ❪G,L1❫ ⊢ T1 :[h,a] U → ∀T2. ❪G,L1❫ ⊢ T1 ➡[h] T2 →
- ∀L2. ❪G,L1❫ ⊢ ➡[h] L2 → ❪G,L2❫ ⊢ T2 :[h,a] U.
+ ∀L1,T1,U. ❪G,L1❫ ⊢ T1 :[h,a] U → ∀T2. ❪G,L1❫ ⊢ T1 ➡[h,0] T2 →
+ ∀L2. ❪G,L1❫ ⊢ ➡[h,0] L2 → ❪G,L2❫ ⊢ T2 :[h,a] U.
#h #a #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
/3 width=6 by cnv_cpm_trans_lpr, cpm_cast/
qed-.
(* Basic_1: uses: ty3_sred_pr2 ty3_sred_pr0 *)
lemma nta_cpr_conf (h) (a) (G) (L):
∀T1,U. ❪G,L❫ ⊢ T1 :[h,a] U →
- ∀T2. ❪G,L❫ ⊢ T1 ➡[h] T2 → ❪G,L❫ ⊢ T2 :[h,a] U.
+ ∀T2. ❪G,L❫ ⊢ T1 ➡[h,0] T2 → ❪G,L❫ ⊢ T2 :[h,a] U.
#h #a #G #L #T1 #U #H #T2 #HT12
/3 width=6 by cnv_cpm_trans, cpm_cast/
qed-.
(* Basic_1: uses: ty3_sred_pr3 ty3_sred_pr1 *)
lemma nta_cprs_conf (h) (a) (G) (L):
∀T1,U. ❪G,L❫ ⊢ T1 :[h,a] U →
- ∀T2. ❪G,L❫ ⊢ T1 ➡*[h] T2 → ❪G,L❫ ⊢ T2 :[h,a] U.
+ ∀T2. ❪G,L❫ ⊢ T1 ➡*[h,0] T2 → ❪G,L❫ ⊢ T2 :[h,a] U.
#h #a #G #L #T1 #U #H #T2 #HT12
/3 width=6 by cnv_cpms_trans, cpms_cast/
qed-.
(* Basic_1: uses: ty3_cred_pr2 *)
lemma nta_lpr_conf (h) (a) (G):
∀L1,T,U. ❪G,L1❫ ⊢ T :[h,a] U →
- ∀L2. ❪G,L1❫ ⊢ ➡[h] L2 → ❪G,L2❫ ⊢ T :[h,a] U.
+ ∀L2. ❪G,L1❫ ⊢ ➡[h,0] L2 → ❪G,L2❫ ⊢ T :[h,a] U.
#h #a #G #L1 #T #U #HTU #L2 #HL12
/2 width=3 by cnv_lpr_trans/
qed-.
(* Basic_1: uses: ty3_cred_pr3 *)
lemma nta_lprs_conf (h) (a) (G):
∀L1,T,U. ❪G,L1❫ ⊢ T :[h,a] U →
- ∀L2. ❪G,L1❫ ⊢ ➡*[h] L2 → ❪G,L2❫ ⊢ T :[h,a] U.
+ ∀L2. ❪G,L1❫ ⊢ ➡*[h,0] L2 → ❪G,L2❫ ⊢ T :[h,a] U.
#h #a #G #L1 #T #U #HTU #L2 #HL12
/2 width=3 by cnv_lprs_trans/
qed-.
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
qed-.
(* Basic_1: uses: ty3_gen_lift *)
-(* Note: "❪G, L❫ ⊢ U2 ⬌*[h] X2" can be "❪G, L❫ ⊢ X2 ➡*[h] U2" *)
+(* Note: "❪G, L❫ ⊢ U2 ⬌*[h] X2" can be "❪G, L❫ ⊢ X2 ➡*[h,0] U2" *)
lemma nta_inv_lifts_sn (h) (a) (G):
∀L,T2,X2. ❪G,L❫ ⊢ T2 :[h,a] X2 →
∀b,f,K. ⇩*[b,f] L ≘ K → ∀T1. ⇧*[f] T1 ≘ T2 →