matita/matita/contribs/lambdadelta/static_2/syntax/lenv_length.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/arith/nat_succ.ma".
  16 include "static_2/syntax/lenv.ma".
  17
  18 (* LENGTH OF A LOCAL ENVIRONMENT ********************************************)
  19
  20 rec definition length L ≝ match L with
  21 [ LAtom     ⇒ 𝟎
  22 | LBind L _ ⇒ ↑(length L)
  23 ].
  24
  25 interpretation "length (local environment)" 'card L = (length L).
  26
  27 (* Basic properties *********************************************************)
  28
  29 lemma length_atom: |⋆| = 𝟎.
  30 // qed.
  31
  32 (* Basic_2A1: uses: length_pair *)
  33 lemma length_bind: ∀I,L. |L.ⓘ[I]| = ↑|L|.
  34 // qed.
  35
  36 (* Basic inversion lemmas ***************************************************)
  37
  38 lemma length_inv_zero_dx: ∀L. |L| = 𝟎 → L = ⋆.
  39 * // #L #I
  40 >length_bind #H
  41 elim (eq_inv_nsucc_zero … H)
  42 qed-.
  43
  44 lemma length_inv_zero_sn: ∀L. 𝟎 = |L| → L = ⋆.
  45 /2 width=1 by length_inv_zero_dx/ qed-.
  46
  47 (* Basic_2A1: was: length_inv_pos_dx *)
  48 lemma length_inv_succ_dx: ∀n,L. |L| = ↑n →
  49                           ∃∃I,K. |K| = n & L = K. ⓘ[I].
  50 #n *
  51 [ >length_atom #H
  52   elim (eq_inv_zero_nsucc … H)
  53 | #L #I >length_bind
  54   /3 width=4 by ex2_2_intro, eq_inv_nsucc_bi/
  55 ]
  56 qed-.
  57
  58 (* Basic_2A1: was: length_inv_pos_sn *)
  59 lemma length_inv_succ_sn: ∀n,L. ↑n = |L| →
  60                           ∃∃I,K. n = |K| & L = K. ⓘ[I].
  61 #n #L #H lapply (sym_eq ??? H) -H
  62 #H elim (length_inv_succ_dx … H) -H /2 width=4 by ex2_2_intro/
  63 qed-.
  64
  65 (* Basic_2A1: removed theorems 1: length_inj *)