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diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
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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                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/prednotreducible_5.ma".
-include "basic_2/reduction/cir.ma".
-include "basic_2/reduction/crx.ma".
-
-(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ***************)
-
-definition cix: ∀h. sd h → relation3 genv lenv term ≝
-                λh,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ⊥.
-
-interpretation "irreducibility for context-sensitive extended reduction (term)"
-   'PRedNotReducible h o G L T = (cix h o G L T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cix_inv_sort: ∀h,o,G,L,s,d. deg h o s (d+1) → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃⋆s⦄ → ⊥.
-/3 width=2 by crx_sort/ qed-.
-
-lemma cix_inv_delta: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃#i⦄ → ⊥.
-/3 width=4 by crx_delta/ qed-.
-
-lemma cix_inv_ri2: ∀h,o,I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃②{I}V.T⦄ → ⊥.
-/3 width=1 by crx_ri2/ qed-.
-
-lemma cix_inv_ib2: ∀h,o,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄ →
-                   ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄.
-/4 width=1 by crx_ib2_sn, crx_ib2_dx, conj/ qed-.
-
-lemma cix_inv_bind: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄ →
-                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & ib2 a I.
-#h #o #a * [ elim a -a ]
-#G #L #V #T #H [ elim H -H /3 width=1 by crx_ri2, or_introl/ ]
-elim (cix_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/
-qed-.
-
-lemma cix_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓐV.T⦄ →
-                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & 𝐒⦃T⦄.
-#h #o #G #L #V #T #HVT @and3_intro /3 width=1 by crx_appl_sn, crx_appl_dx/
-generalize in match HVT; -HVT elim T -T //
-* // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crx_beta, crx_theta/
-qed-.
-
-lemma cix_inv_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓕ{I}V.T⦄ →
-                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
-#h #o * #G #L #V #T #H
-[ elim (cix_inv_appl … H) -H /2 width=1 by and4_intro/
-| elim (cix_inv_ri2 … H) -H //
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cix_inv_cir: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄.
-/3 width=1 by crr_crx/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma cix_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃⋆s⦄.
-#h #o #G #L #s #Hk #H elim (crx_inv_sort … H) -L #d #Hkd
-lapply (deg_mono … Hk Hkd) -h -s <plus_n_Sm #H destruct
-qed.
-
-lemma tix_lref: ∀h,o,G,i. ⦃G, ⋆⦄ ⊢ ➡[h, o] 𝐈⦃#i⦄.
-#h #o #G #i #H elim (trx_inv_atom … H) -H #s #d #_ #H destruct
-qed.
-
-lemma cix_gref: ∀h,o,G,L,p. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃§p⦄.
-#h #o #G #L #p #H elim (crx_inv_gref … H)
-qed.
-
-lemma cix_ib2: ∀h,o,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ →
-                               ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄.
-#h #o #a #I #G #L #V #T #HI #HV #HT #H
-elim (crx_inv_ib2 … HI H) -HI -H /2 width=1 by/
-qed.
-
-lemma cix_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ →  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓐV.T⦄.
-#h #o #G #L #V #T #HV #HT #H1 #H2
-elim (crx_inv_appl … H2) -H2 /2 width=1 by/
-qed.