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

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   3 (*      ||M||                                                             *)
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  13 (**************************************************************************)
  14
  15 include "ground/notation/functions/one_0.ma".
  16 include "ground/notation/functions/uparrow_1.ma".
  17 include "ground/lib/relations.ma".
  18
  19 (* POSITIVE INTEGERS ********************************************************)
  20
  21 inductive pnat: Type[0] ≝
  22 | punit: pnat
  23 | psucc: pnat → pnat
  24 .
  25
  26 interpretation
  27   "unit (positive integers)"
  28   'One = (punit).
  29
  30 interpretation
  31   "successor (positive integers)"
  32   'UpArrow p = (psucc p).
  33
  34 (* Basic inversions *********************************************************)
  35
  36 (* Note: destruct *)
  37 lemma eq_inv_psucc_bi: injective … psucc.
  38 #p #q #H destruct //
  39 qed.
  40
  41 (* Basic constructions ******************************************************)
  42
  43 lemma eq_pnat_dec (p1,p2:pnat): Decidable (p1 = p2).
  44 #p1 elim p1 -p1 [| #p1 #IH ] * [2,4: #p2 ]
  45 [1,4: @or_intror #H destruct
  46 | elim (IH p2) -IH #H destruct
  47   /4 width=1 by eq_inv_psucc_bi, or_intror, or_introl/
  48 | /2 width=1 by or_introl/
  49 ]
  50 qed-.