(* Properties with advanced rt_computation for terms ************************)
(* Basic_2A1: was by definition nta_appl ntaa_appl *)
-lemma nta_appl_abst (a) (h) (p) (G) (K):
+lemma nta_appl_abst (a) (h) (p) (G) (K): yinj 0 < a →
∀V,W. ⦃G,K⦄ ⊢ V :[a,h] W →
∀T,U. ⦃G,K.ⓛW⦄ ⊢ T :[a,h] U → ⦃G,K⦄ ⊢ ⓐV.ⓛ{p}W.T :[a,h] ⓐV.ⓛ{p}W.U.
-#a #h #p #G #K #V #W #H1 #T #U #H2
+#a #h #p #G #K #Ha #V #W #H1 #T #U #H2
elim (cnv_inv_cast … H1) -H1 #X1 #HW #HV #HWX1 #HVX1
elim (cnv_inv_cast … H2) -H2 #X2 #HU #HT #HUX2 #HTX2
/4 width=7 by cnv_bind, cnv_appl, cnv_cast, cpms_appl_dx, cpms_bind_dx/
(* Basic_1: was by definition: ty3_appl *)
(* Basic_2A1: was nta_appl_old *)
-lemma nta_appl (a) (h) (p) (G) (L):
+lemma nta_appl (a) (h) (p) (G) (L): yinj 1 < a →
∀V,W. ⦃G,L⦄ ⊢ V :[a,h] W →
∀T,U. ⦃G,L⦄ ⊢ T :[a,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T :[a,h] ⓐV.ⓛ{p}W.U.
-#a #h #p #G #L #V #W #H1 #T #U #H2
+#a #h #p #G #L #Ha #V #W #H1 #T #U #H2
elim (cnv_inv_cast … H1) -H1 #X1 #HW #HV #HWX1 #HVX1
elim (cnv_inv_cast … H2) -H2 #X2 #HU #HT #HUX2 #HTX2
elim (cpms_inv_abst_sn … HUX2) #W0 #U0 #HW0 #HU0 #H destruct
elim (cprs_conf … HWX1 … HW0) -HW0 #X0 #HX10 #HWX0
@(cnv_cast … (ⓐV.ⓛ{p}W0.U0)) (**) (* full auto too slow *)
-[ /3 width=7 by cnv_appl, cpms_bind/
+[ /3 width=7 by cnv_appl, cpms_bind, lt_ylt_trans/
| /4 width=11 by cnv_appl, cpms_cprs_trans, cpms_bind/
| /2 width=1 by cpms_appl_dx/
| /2 width=1 by cpms_appl_dx/