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

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  14
  15 include "ground/notation/functions/zero_0.ma".
  16 include "ground/arith/pnat.ma".
  17
  18 (* NON-NEGATIVE INTEGERS ****************************************************)
  19
  20 (*** nat *)
  21 inductive nat: Type[0] ≝
  22 | nzero: nat
  23 | ninj : pnat → nat
  24 .
  25
  26 coercion ninj.
  27
  28 interpretation "zero (non-negative integers" 'Zero = nzero.
  29
  30 (* Basic inversions *********************************************************)
  31
  32 (* Note: destruct *)
  33 lemma eq_inv_ninj_bi: injective … ninj.
  34 #p #q #H destruct //
  35 qed-.
  36
  37 (* Basic constructions ******************************************************)
  38
  39 lemma eq_nat_dec (n1,n2:nat): Decidable (n1 = n2).
  40 * [| #p1 ] * [2,4: #p2 ]
  41 [1,4: @or_intror #H destruct
  42 | elim (eq_pnat_dec p1 p2)
  43   /4 width=1 by eq_inv_ninj_bi, or_intror, or_introl/
  44 | /2 width=1 by or_introl/
  45 ]
  46 qed-.