-(* 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/
+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/