(* Properties with r-equivalence for terms **********************************)
-lemma nta_conv_cnv (a) (h) (G) (L) (T):
- â\88\80U1. â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U1 →
- â\88\80U2. â¦\83G,Lâ¦\84 â\8a¢ U1 â¬\8c*[h] U2 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ U2 ![a,h] â\86\92 â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #HU12 #HU2
+lemma nta_conv_cnv (h) (a) (G) (L) (T):
+ â\88\80U1. â\9d¨G,Lâ\9d© â\8a¢ 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© â\8a¢ 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
elim (cpcs_inv_cprs … H) -H #X2 #HX12 #HU12
(* Basic_1: was by definition: ty3_conv *)
(* Basic_2A1: was by definition: nta_conv ntaa_conv *)
-lemma nta_conv (a) (h) (G) (L) (T):
- â\88\80U1. â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U1 →
- â\88\80U2. â¦\83G,Lâ¦\84 ⊢ U1 ⬌*[h] U2 →
- â\88\80W2. â¦\83G,Lâ¦\84 â\8a¢ U2 :[a,h] W2 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ T :[a,h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #HU12 #W2 #H2
+lemma nta_conv (h) (a) (G) (L) (T):
+ â\88\80U1. â\9d¨G,Lâ\9d© â\8a¢ 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© â\8a¢ 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 (a) (h) (G) (L) (X2):
- ∀s. ⦃G,L⦄ ⊢ ⋆s :[a,h] X2 → ⦃G,L⦄ ⊢ ⋆(next h s) ⬌*[h] X2.
-#a #h #G #L #X2 #s #H
+lemma nta_inv_sort_sn (h) (a) (G) (L) (X2):
+ ∀s. ❨G,L❩ ⊢ ⋆s :[h,a] X2 →
+ ∧∧ ❨G,L❩ ⊢ ⋆(⫯[h]s) ⬌*[h] X2 & ❨G,L❩ ⊢ 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
-/2 width=1 by cpcs_cprs_sn/
+/3 width=1 by cpcs_cprs_sn, conj/
+qed-.
+
+lemma nta_inv_ldec_sn_cnv (h) (a) (G) (K) (V):
+ ∀X2. ❨G,K.ⓛV❩ ⊢ #0 :[h,a] X2 →
+ ∃∃U. ❨G,K❩ ⊢ V ![h,a] & ⇧[1] V ≘ 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
+elim (cpms_inv_ell_sn … H2) -H2 *
+[ #_ #H destruct
+| #m #W #HVW #HWX1 #H destruct
+ elim (lifts_total V (𝐔❨1❩)) #U #HVU
+ lapply (cpms_lifts_bi … HVW (Ⓣ) … (K.ⓛV) … HVU … HWX1) -W
+ [ /3 width=1 by drops_refl, drops_drop/ ] #HUX1
+ /3 width=5 by cprs_div, ex4_intro/
+]
qed-.