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(*       ___                                                              *)
(*      ||M||                                                             *)
(*      ||A||       A project by Andrea Asperti                           *)
(*      ||T||                                                             *)
(*      ||I||       Developers:                                           *)
(*      ||T||         The HELM team.                                      *)
(*      ||A||         http://helm.cs.unibo.it                             *)
(*      \   /                                                             *)
(*       \ /        This file is distributed under the terms of the       *)
(*        v         GNU General Public License Version 2                  *)
(*                                                                        *)
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include "basic_2/reduction/cnr.ma".

(* Basic inversion lemmas ***************************************************)

lemma cnx_inv_zeta: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃+ⓓV.T⦄ → ⊥.
#h #o #G #L #V #T #H elim (is_lift_dec T 0 1)
[ * #U #HTU
  lapply (H U ?) -H /2 width=3 by cpx_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 /5 width=1 by cpr_cpx, tpr_cpr, cpr_bind/ -HT2
  #H destruct /3 width=2 by ex_intro/
]
qed-.

(* Basic forward lemmas *****************************************************)

lemma cnx_fwd_cnr: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
#h #o #G #L #T #H #U #HTU
@H /2 width=1 by cpr_cpx/ (**) (* auto fails because a δ-expansion gets in the way *)
qed-.

(* Basic properties *********************************************************)

lemma cnx_lref_free: ∀h,o,G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃#i⦄.
#h #o #G #L #i #HL @cnx_lref_free >(drop_fwd_length … HL) -HL //
qed.

axiom cnx_dec: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T1⦄ ∨
               ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 & (T1 = T2 → ⊥).