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diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx.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_2A/notation/relations/cosn_5.ma".
+include "basic_2A/computation/lsx.ma".
+
+(* SN EXTENDED STRONGLY CONORMALIZING LOCAL ENVIRONMENTS ********************)
+
+inductive lcosx (h) (g) (G): relation2 ynat lenv ≝
+| lcosx_sort: ∀l. lcosx h g G l (⋆)
+| lcosx_skip: ∀I,L,T. lcosx h g G 0 L → lcosx h g G 0 (L.ⓑ{I}T)
+| lcosx_pair: ∀I,L,T,l. G ⊢ ⬊*[h, g, T, l] L →
+              lcosx h g G l L → lcosx h g G (⫯l) (L.ⓑ{I}T)
+.
+
+interpretation
+   "sn extended strong conormalization (local environment)"
+   'CoSN h g l G L = (lcosx h g G l L).
+
+(* Basic properties *********************************************************)
+
+lemma lcosx_O: ∀h,g,G,L. G ⊢ ~⬊*[h, g, 0] L.
+#h #g #G #L elim L /2 width=1 by lcosx_skip/
+qed.
+
+lemma lcosx_drop_trans_lt: ∀h,g,G,L,l. G ⊢ ~⬊*[h, g, l] L →
+                            ∀I,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → i < l →
+                            G ⊢ ~⬊*[h, g, ⫰(l-i)] K ∧ G ⊢ ⬊*[h, g, V, ⫰(l-i)] K.
+#h #g #G #L #l #H elim H -L -l
+[ #l #J #K #V #i #H elim (drop_inv_atom1 … H) -H #H destruct
+| #I #L #T #_ #_ #J #K #V #i #_ #H elim (ylt_yle_false … H) -H //
+| #I #L #T #l #HT #HL #IHL #J #K #V #i #H #Hil
+  elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK destruct
+  [ >ypred_succ /2 width=1 by conj/
+  | lapply (ylt_pred … Hil ?) -Hil /2 width=1 by ylt_inj/ >ypred_succ #Hil
+    elim (IHL … HLK ?) -IHL -HLK <yminus_inj >yminus_SO2 //
+    <(ypred_succ l) in ⊢ (%→%→?); >yminus_pred /2 width=1 by ylt_inj, conj/
+  ]
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lcosx_inv_succ_aux: ∀h,g,G,L,x. G ⊢ ~⬊*[h, g, x] L → ∀l. x = ⫯l →
+                         L = ⋆ ∨
+                         ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
+                                  G ⊢ ⬊*[h, g, V, l] K.
+#h #g #G #L #l * -L -l /2 width=1 by or_introl/
+[ #I #L #T #_ #x #H elim (ysucc_inv_O_sn … H)
+| #I #L #T #l #HT #HL #x #H <(ysucc_inj … H) -x
+  /3 width=6 by ex3_3_intro, or_intror/
+]
+qed-.
+
+lemma lcosx_inv_succ: ∀h,g,G,L,l. G ⊢ ~⬊*[h, g, ⫯l] L → L = ⋆ ∨
+                      ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
+                               G ⊢ ⬊*[h, g, V, l] K.
+/2 width=3 by lcosx_inv_succ_aux/ qed-.
+
+lemma lcosx_inv_pair: ∀h,g,I,G,L,T,l. G ⊢ ~⬊*[h, g, ⫯l] L.ⓑ{I}T →
+                      G ⊢ ~⬊*[h, g, l] L ∧ G ⊢ ⬊*[h, g, T, l] L.
+#h #g #I #G #L #T #l #H elim (lcosx_inv_succ … H) -H
+[ #H destruct
+| * #Z #Y #X #H destruct /2 width=1 by conj/
+]
+qed-.