+
+lemma cnv_fwd_cpms_abst_dx_le (h) (a) (G) (L) (W) (p):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∀n1,U1. ⦃G,L⦄ ⊢ T ➡*[n1,h] ⓛ{p}W.U1 → ∀n2. n1 ≤ n2 →
+ ∃∃U2. ⦃G,L⦄ ⊢ T ➡*[n2,h] ⓛ{p}W.U2 & ⦃G,L.ⓛW⦄ ⊢ U1 ➡*[n2-n1,h] U2.
+#h #a #G #L #W #p #T #H
+elim (cnv_fwd_aaa … H) -H #A #HA
+/2 width=2 by cpms_abst_dx_le_aaa/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cnv_appl_ge (h) (a) (n1) (p) (G) (L):
+ ∀n2. n1 ≤ n2 → ad a n2 →
+ ∀V. ⦃G,L⦄ ⊢ V ![h,a] → ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∀X. ⦃G,L⦄ ⊢ V ➡*[1,h] X → ∀W. ⦃G,L⦄ ⊢ W ➡*[h] X →
+ ∀U. ⦃G,L⦄ ⊢ T ➡*[n1,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![h,a].
+#h #a #n1 #p #G #L #n2 #Hn12 #Ha #V #HV #T #HT #X #HVX #W #HW #X #HTX
+elim (cnv_fwd_cpms_abst_dx_le … HT … HTX … Hn12) #U #HTU #_ -n1
+/4 width=11 by cnv_appl, cpms_bind, cpms_cprs_trans/
+qed.