(* Properties with r-equivalence for terms **********************************)
lemma nta_conv_cnv (h) (a) (G) (L) (T):
- â\88\80U1. â\9dªG,Lâ\9d« ⊢ T :[h,a] U1 →
- â\88\80U2. â\9dªG,Lâ\9d« â\8a¢ U1 â¬\8c*[h] U2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ U2 ![h,a] â\86\92 â\9dªG,Lâ\9d« ⊢ T :[h,a] U2.
+ â\88\80U1. â\9d¨G,Lâ\9d© ⊢ T :[h,a] U1 →
+ â\88\80U2. â\9d¨G,Lâ\9d© â\8a¢ U1 â¬\8c*[h] U2 â\86\92 â\9d¨G,Lâ\9d© â\8a¢ U2 ![h,a] â\86\92 â\9d¨G,Lâ\9d© ⊢ T :[h,a] U2.
#h #a #G #L #T #U1 #H1 #U2 #HU12 #HU2
elim (cnv_inv_cast … H1) -H1 #X1 #HU1 #HT #HUX1 #HTX1
lapply (cpcs_cprs_conf … HUX1 … HU12) -U1 #H
(* Basic_1: was by definition: ty3_conv *)
(* Basic_2A1: was by definition: nta_conv ntaa_conv *)
lemma nta_conv (h) (a) (G) (L) (T):
- â\88\80U1. â\9dªG,Lâ\9d« ⊢ T :[h,a] U1 →
- â\88\80U2. â\9dªG,Lâ\9d« ⊢ U1 ⬌*[h] U2 →
- â\88\80W2. â\9dªG,Lâ\9d« â\8a¢ U2 :[h,a] W2 â\86\92 â\9dªG,Lâ\9d« ⊢ T :[h,a] U2.
+ â\88\80U1. â\9d¨G,Lâ\9d© ⊢ T :[h,a] U1 →
+ â\88\80U2. â\9d¨G,Lâ\9d© ⊢ U1 ⬌*[h] U2 →
+ â\88\80W2. â\9d¨G,Lâ\9d© â\8a¢ U2 :[h,a] W2 â\86\92 â\9d¨G,Lâ\9d© ⊢ T :[h,a] U2.
#h #a #G #L #T #U1 #H1 #U2 #HU12 #W2 #H2
/3 width=3 by nta_conv_cnv, nta_fwd_cnv_sn/
qed-.
(* Basic_1: was: ty3_gen_sort *)
(* Basic_2A1: was: nta_inv_sort1 *)
lemma nta_inv_sort_sn (h) (a) (G) (L) (X2):
- â\88\80s. â\9dªG,Lâ\9d« ⊢ ⋆s :[h,a] X2 →
- â\88§â\88§ â\9dªG,Lâ\9d« â\8a¢ â\8b\86(⫯[h]s) â¬\8c*[h] X2 & â\9dªG,Lâ\9d« ⊢ X2 ![h,a].
+ â\88\80s. â\9d¨G,Lâ\9d© ⊢ ⋆s :[h,a] X2 →
+ â\88§â\88§ â\9d¨G,Lâ\9d© â\8a¢ â\8b\86(⫯[h]s) â¬\8c*[h] X2 & â\9d¨G,Lâ\9d© ⊢ X2 ![h,a].
#h #a #G #L #X2 #s #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #_ #HX21 #H
lapply (cpms_inv_sort1 … H) -H #H destruct
qed-.
lemma nta_inv_ldec_sn_cnv (h) (a) (G) (K) (V):
- â\88\80X2. â\9dªG,K.â\93\9bVâ\9d« ⊢ #0 :[h,a] X2 →
- â\88\83â\88\83U. â\9dªG,Kâ\9d« â\8a¢ V ![h,a] & â\87§[1] V â\89\98 U & â\9dªG,K.â\93\9bVâ\9d« â\8a¢ U â¬\8c*[h] X2 & â\9dªG,K.â\93\9bVâ\9d« ⊢ X2 ![h,a].
+ â\88\80X2. â\9d¨G,K.â\93\9bVâ\9d© ⊢ #0 :[h,a] X2 →
+ â\88\83â\88\83U. â\9d¨G,Kâ\9d© â\8a¢ V ![h,a] & â\87§[1] V â\89\98 U & â\9d¨G,K.â\93\9bVâ\9d© â\8a¢ U â¬\8c*[h] X2 & â\9d¨G,K.â\93\9bVâ\9d© ⊢ 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