lemma cnr_inv_zeta: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃+ⓓV.T⦄ → ⊥.
#G #L #V #T #H elim (is_lift_dec T 0 1)
[ * #U #HTU
  lapply (H U ?) -H /2 width=3 by cpr_zeta/ #H destruct
  elim (lift_inv_pair_xy_y … HTU)
| #HT
  elim (cpr_delift G (⋆) V T (⋆. ⓓV) 0) //
  #T2 #T1 #HT2 #HT12 lapply (H (+ⓓV.T2) ?) -H /4 width=1 by tpr_cpr, cpr_bind/ -HT2
  #H destruct /3 width=2 by ex_intro/
]
qed-.

lemma cnr_lref_free: ∀G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄.
#G #L #i #Hi #X #H elim (cpr_inv_lref1 … H) -H // *
#K #V1 #V2 #HLK lapply (drop_fwd_length_lt2 … HLK) -HLK
#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
qed.