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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_frac_part_spec : forall r z f, 0 <= f < 1 -> r = (IZR z) + f -> z = Int_part r /\ f = frac_part r.