matita/matita/contribs/lambdadelta/basic_2/etc_new/lenv/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_2/ynat/ynat_lt.ma".
  16 include "basic_2/grammar/lenv.ma".
  17
  18 (* LENGTH OF A LOCAL ENVIRONMENT ********************************************)
  19
  20 let rec length L ≝ match L with
  21 [ LAtom       ⇒ 0
  22 | LPair 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 lemma length_pair: ∀I,L,V. |L.ⓑ{I}V| = ⫯|L|.
  33 // qed.
  34
  35 lemma length_inj: ∀L. |L| < ∞.
  36 #L elim L -L /2 width=1 by ylt_succ_Y/
  37 qed.
  38
  39 (* Basic inversion lemmas ***************************************************)
  40
  41 lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆.
  42 * // #L #I #V >length_pair
  43 #H elim (ysucc_inv_O_dx … H)
  44 qed-.
  45
  46 lemma length_inv_zero_sn: ∀L. yinj 0 = |L| → L = ⋆.
  47 /2 width=1 by length_inv_zero_dx/ qed-.
  48
  49 lemma length_inv_pos_dx: ∀l,L. |L| = ⫯l →
  50                          ∃∃I,K,V. |K| = l & L = K. ⓑ{I}V.
  51 #l * /3 width=5 by ysucc_inj, ex2_3_intro/
  52 >length_atom #H elim (ysucc_inv_O_sn … H)
  53 qed-.
  54
  55 lemma length_inv_pos_sn: ∀l,L. ⫯l = |L| →
  56                          ∃∃I,K,V. l = |K| & L = K. ⓑ{I}V.
  57 #l #L #H lapply (sym_eq ??? H) -H
  58 #H elim (length_inv_pos_dx … H) -H /2 width=5 by ex2_3_intro/
  59 qed-.