∧∧ IH_cnv_cpms_conf_lpr a h G L T
& IH_cnv_cpm_trans_lpr a h G L T.
#a #h #G #L #T #HT
-letin o ≝ (sd_O h)
-lapply (cnv_fwd_fsb … o … HT) -HT #H
+lapply (cnv_fwd_fsb … HT) -HT #H
@(fsb_ind_fpbg … H) -G -L -T #G #L #T #_ #IH
@conj [ letin aux ≝ cnv_cpms_conf_lpr_aux | letin aux ≝ cnv_cpm_trans_lpr_aux ]
-@(aux … o … G L T) // #G0 #L0 #T0 #H
+@(aux … G L T) // #G0 #L0 #T0 #H
elim (IH … H) -IH -H //
qed-.
#a #h #G #L1 #T #HT #L2 #H
@(lprs_ind_dx … H) -L2 /2 width=3 by cnv_lpr_trans/
qed-.
+
+(* 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_cpm_SO … (ⓛ{p}W.U))
+ [|*: /2 width=8 by cnv_cpms_trans/ ] #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-.