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 | LPair 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 lemma length_pair: ∀I,L,V. |L.ⓑ{I}V| = ⫯|L|.
  32 // qed.
  33
  34 (* Basic inversion lemmas ***************************************************)
  35
  36 lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆.
  37 * // #L #I #V >length_pair
  38 #H destruct
  39 qed-.
  40
  41 lemma length_inv_zero_sn: ∀L. 0 = |L| → L = ⋆.
  42 /2 width=1 by length_inv_zero_dx/ qed-.
  43
  44 (* Basic_2A1: was: length_inv_pos_dx *)
  45 lemma length_inv_succ_dx: ∀n,L. |L| = ⫯n →
  46                           ∃∃I,K,V. |K| = n & L = K. ⓑ{I}V.
  47 #n * [ >length_atom #H destruct ]
  48 #L #I #V >length_pair /3 width=5 by ex2_3_intro, injective_S/
  49 qed-.
  50
  51 (* Basic_2A1: was: length_inv_pos_sn *)
  52 lemma length_inv_succ_sn: ∀n,L. ⫯n = |L| →
  53                           ∃∃I,K,V. n = |K| & L = K. ⓑ{I}V.
  54 #l #L #H lapply (sym_eq ??? H) -H
  55 #H elim (length_inv_succ_dx … H) -H /2 width=5 by ex2_3_intro/
  56 qed-.
  57
  58 (* Basic_2A1: removed theorems 1: length_inj *)