-(* Advanced inversion lemmas ************************************************)
-
-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
-]
+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/