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

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  14
  15 include "basic_2/syntax/lenv.ma".
  16
  17 (* LENGTH OF A LOCAL ENVIRONMENT ********************************************)
  18
  19 rec definition length L ≝ match L with
  20 [ LAtom     ⇒ 0
  21 | LBind L _ ⇒ ↑(length L)
  22 ].
  23
  24 interpretation "length (local environment)" 'card L = (length L).
  25
  26 (* Basic properties *********************************************************)
  27
  28 lemma length_atom: |⋆| = 0.
  29 // qed.
  30
  31 (* Basic_2A1: uses: length_pair *)
  32 lemma length_bind: ∀I,L. |L.ⓘ{I}| = ↑|L|.
  33 // qed.
  34
  35 (* Basic inversion lemmas ***************************************************)
  36
  37 lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆.
  38 * // #L #I >length_bind
  39 #H destruct
  40 qed-.
  41
  42 lemma length_inv_zero_sn: ∀L. 0 = |L| → L = ⋆.
  43 /2 width=1 by length_inv_zero_dx/ qed-.
  44
  45 (* Basic_2A1: was: length_inv_pos_dx *)
  46 lemma length_inv_succ_dx: ∀n,L. |L| = ↑n →
  47                           ∃∃I,K. |K| = n & L = K. ⓘ{I}.
  48 #n *
  49 [ >length_atom #H destruct
  50 | #L #I >length_bind /3 width=4 by ex2_2_intro, injective_S/
  51 ]
  52 qed-.
  53
  54 (* Basic_2A1: was: length_inv_pos_sn *)
  55 lemma length_inv_succ_sn: ∀n,L. ↑n = |L| →
  56                           ∃∃I,K. n = |K| & L = K. ⓘ{I}.
  57 #n #L #H lapply (sym_eq ??? H) -H
  58 #H elim (length_inv_succ_dx … H) -H /2 width=4 by ex2_2_intro/
  59 qed-.
  60
  61 (* Basic_2A1: removed theorems 1: length_inj *)