matita/matita/contribs/lambdadelta/ground/etc/relocation/trace_snot.etc

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   2 (*       ___                                                              *)
   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 "ground_2/notation/functions/complement_1.ma".
  16 include "ground_2/relocation/trace.ma".
  17
  18 (* RELOCATION TRACE *********************************************************)
  19
  20 let rec snot (t:trace) on t ≝ match t with
  21 [ nil      ⇒ ◊
  22 | cons b t ⇒ (¬ b) @ snot t
  23 ].
  24
  25 interpretation
  26    "complement (trace)"
  27    'Complement t = (snot t).
  28
  29 (* Basic properties *********************************************************)
  30
  31 lemma snot_empty: ∁ (◊) = ◊.
  32 // qed.
  33
  34 lemma snot_inh: ∀t,b. ∁ (b@t) = (¬ b) @ ∁ t.
  35 // qed.
  36
  37 lemma snot_true: ∀t. ∁ (Ⓣ @ t) = Ⓕ @ ∁ t.
  38 // qed.
  39
  40 lemma snot_false: ∀t. ∁ (Ⓕ @ t) = Ⓣ @ ∁ t.
  41 // qed.
  42
  43 lemma snot_length: ∀t. |∁ t| = |t|.
  44 #t elim t -t normalize //
  45 qed.
  46
  47 lemma snot_colength: ∀t. ∥∁ t∥ = |t| - ∥t∥.
  48 #t elim t -t //
  49 * /2 width=1 by minus_Sn_m/
  50 qed.