-lemma cnv_inv_appl_SO (a) (h) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
- ∃∃n,p,W0,U0. a = Ⓣ → n = 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
-* #h #G #L #V #T #H
-elim (cnv_inv_appl … H) -H [ * [| #n ] | #n ] #p #W #U #Ha #HV #HT #HVW #HTU
-[ elim (cnv_fwd_aaa … HT) #A #HA
- elim (aaa_cpm_SO h … (ⓛ{p}W.U))
- [|*: /2 width=8 by cpms_aaa_conf/ ] #X #HU0
- elim (cpm_inv_abst1 … HU0) #W0 #U0 #HW0 #_ #H0 destruct
- lapply (cpms_step_dx … HVW … HW0) -HVW -HW0 #HVW0
- lapply (cpms_step_dx … HTU … HU0) -HTU -HU0 #HTU0
- /2 width=7 by ex5_4_intro/
-| lapply (Ha ?) -Ha [ // ] #Ha
- lapply (le_n_O_to_eq n ?) [ /3 width=1 by le_S_S_to_le/ ] -Ha #H destruct
- /2 width=7 by ex5_4_intro/
-| @(ex5_4_intro … HV HT HVW HTU) #H destruct
-]
-qed-.
-
-lemma cnv_inv_appl_true (h) (G) (L):
- ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h] →
- ∃∃p,W0,U0. ⦃G,L⦄ ⊢ V ![h] & ⦃G,L⦄ ⊢ T ![h] &
- ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[1,h] ⓛ{p}W0.U0.
-#h #G #L #V #T #H
-elim (cnv_inv_appl_SO … H) -H #n #p #W #U #Hn
->Hn -n [| // ] #HV #HT #HVW #HTU
+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