+
+(* Forward lemmas with t_bound rt_computation for terms *********************)
+
+lemma cnv_fwd_cpms_total (a) (h) (n) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
+#a #h #n #G #L #T #H
+elim (cnv_fwd_aaa … H) -H #A #HA
+/2 width=2 by cpms_total_aaa/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cnv_inv_appl_pred (a) (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![yinj a,h] →
+ ∃∃p,W0,U0. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+ ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[↓a,h] ⓛ{p}W0.U0.
+#a #h #G #L #V #T #H
+elim (cnv_inv_appl … H) -H #n #p #W #U #Ha #HV #HT #HVW #HTU
+lapply (ylt_inv_inj … Ha) -Ha #Ha
+elim (cnv_fwd_aaa … HT) #A #HA
+elim (cpms_total_aaa h … (a-↑n) … (ⓛ{p}W.U))
+[|*: /2 width=8 by cpms_aaa_conf/ ] -HA #X #HU0
+elim (cpms_inv_abst_sn … HU0) #W0 #U0 #HW0 #_ #H destruct
+lapply (cpms_trans … HVW … HW0) -HVW -HW0 #HVW0
+lapply (cpms_trans … HTU … HU0) -HTU -HU0
+>(arith_m2 … Ha) -Ha #HTU0
+/2 width=5 by ex4_3_intro/
+qed-.