matita/matita/contribs/lambdadelta/ground/arith/pnat_pred.ma

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  14
  15 include "ground/notation/functions/downarrow_1.ma".
  16 include "ground/arith/pnat_split.ma".
  17
  18 (* PREDECESSOR FOR POSITIVE INTEGERS ****************************************)
  19
  20 definition ppred (p): pnat ≝ psplit … (𝟏) (λp.p) p.
  21
  22 interpretation
  23   "predecessor (positive integers)"
  24   'DownArrow p = (ppred p).
  25
  26 (* Basic constructions ******************************************************)
  27
  28 lemma ppred_unit: 𝟏 = ↓𝟏.
  29 // qed.
  30
  31 lemma ppred_succ (p): p = ↓↑p.
  32 // qed.
  33
  34 (* Basic inversions *********************************************************)
  35
  36 lemma ppred_inv_refl (p): p = ↓p → 𝟏 = p.
  37 #p elim p -p //
  38 #p #IH #H /2 width=1 by/
  39 qed-.
  40
  41 lemma pnat_split_unit_pos (p): ∨∨ 𝟏 = p | p = ↑↓p.
  42 #p elim p -p
  43 /2 width=1 by or_introl, or_intror/
  44 qed-.