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(** *    Fractional part                                 *)
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Definition Int_part (r:R) : Z := (up r - 1)%Z.(**********)
Definition frac_part (r:R) : R := r - IZR (Int_part r).

Lemma Int_part_spec : forall r z, r - 1 < IZR z <= r -> z = Int_part r.