matita/matita/contribs/lambdadelta/basic_2/syntax/voids_length.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "basic_2/syntax/lenv_length.ma".
  16 include "basic_2/syntax/voids.ma".
  17
  18 (* EXTENSION OF A LOCAL ENVIRONMENT WITH EXCLUSION BINDERS ******************)
  19
  20 (* Properties with length for local environments ****************************)
  21
  22 lemma length_void: ∀L,n. n+|L| = |ⓧ*[n]L|.
  23 #L #n elim n -n //
  24 #n #IH <voids_succ >length_bind <IH -IH //
  25 qed.
  26
  27 lemma length_void_le: ∀L,n. |L| ≤ |ⓧ*[n]L|.
  28 // qed.
  29
  30 (* Main forward properties with length for local environments ***************)
  31
  32 theorem voids_inj_length: ∀n1,n2,L1,L2. ⓧ*[n1]L1 = ⓧ*[n2]L2 →
  33                           |L1| = |L2| → n1 = n2 ∧ L1 = L2.
  34 #n1 elim n1 -n1
  35 [ * /2 width=1 by conj/ #n2 #L1 #L2 | #n1 #IH * [ | #n2 ] #L1 #L2 ]
  36 [ <voids_zero #H destruct
  37   <length_void <commutative_plus #H
  38   elim (plus_xSy_x_false … H)
  39 | <voids_zero #H destruct
  40   <length_void <commutative_plus #H
  41   elim (plus_xSy_x_false … (sym_eq … H))
  42 | <voids_succ <voids_succ #H #HL12
  43   elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *)
  44   #H #_ elim (IH … H HL12) -IH -H -HL12 /2 width=1 by conj/
  45 ]
  46 qed-.