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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/substitution/llneq_alt.ma".

(* NEGATED LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **************************)

(* Advanced inversion lemmas ************************************************)

lemma llneq_inv_atom1: ∀L1,L2,T,d. L1 ⧣⧣[T, d] L2 → |L1| ≤ d → ⊥.
#L1 #L2 #T #d #H elim H -L1 -L2 -T -d /2 width=1 by/
[ #I1 #I2 #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #_ #_ #_
  >(ldrop_fwd_length … HLK1) -HLK1 normalize
  #H lapply (yle_trans … H … Hdi) -d
  /3 width=4 by yle_inv_inj, le_plus_xySz_x_false/
| #I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #_ #_ #_
  >(ldrop_fwd_length … HLK1) -HLK1 normalize
  #H lapply (yle_trans … H … Hdi) -d
  /3 width=4 by yle_inv_inj, le_plus_xySz_x_false/
| #a #I #L1 <yplus_inj /4 width=1 by yle_succ/
]
qed-.