lemma lsubr_cpxs_trans: ∀h,o,G. lsub_trans … (cpxs h o G) lsubr.
/3 width=5 by lsubr_cpx_trans, LTC_lsub_trans/
qed-.

lemma cprs_cpxs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈* T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2.
#h #o #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /3 width=3 by cpxs_strap1,
cpr_cpx/
qed.

lemma cpxs_inv_cnx1: ∀h,o,G,L,T,U. ⦃G, L⦄ ⊢ T ⬈*[h] U → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → T = U.
#h #o #G #L #T #U #H @(cpxs_ind_dx … H) -T //
#T0 #T #H1T0 #_ #IHT #H2T0
lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
qed-.

lemma cpxs_neq_inv_step_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 = T2 → ⊥) →
                            ∃∃T. ⦃G, L⦄ ⊢ T1 ⬈[h] T & T1 = T → ⊥ & ⦃G, L⦄ ⊢ T ⬈*[h] T2.
#h #o #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
[ #H elim H -H //
| #T1 #T #H1 #H2 #IH2 #H12 elim (eq_term_dec T1 T) #H destruct
  [ -H1 -H2 /3 width=1 by/
  | -IH2 /3 width=4 by ex3_intro/ (**) (* auto fails without clear *)
  ]
]
qed-.