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(* Regarding rationals *)
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Definition frac (a b : Z) := ((IZR a) / (IZR b))%R.Definition is_rat (r : R) :=
  exists pq : Z * Z, let (p,q) := pq in ~(q = 0) /\ r = (frac p q).Definition is_ratp (c : R * R) := let (x,y) := c in (is_rat x) /\ (is_rat y).

Lemma frac_rat : forall a b : Z, b <> 0 -> (frac a b >= 0)%R -> (frac a b <= 1)%R -> a >= 0 /\ b > 0 /\ a <= b \/ a <= 0 /\ b < 0 /\ b <= a.