Infix "==" := eqRat (at level 70) : rat_scope.

(* Specifications: *)
Theorem le_Rat_dec : forall r1 r2, {r1 <= r2} + {~r1 <= r2}.
Theorem eqRat_refl : forall r, eqRat r r.
Theorem eqRat_symm : forall r1 r2, eqRat r1 r2 -> eqRat r2 r1.
Theorem eqRat_trans : forall r1 r2 r3, eqRat r1 r2 -> eqRat r2 r3 -> eqRat r1 r3.
Theorem leRat_refl : forall r, leRat r r.
Lemma mult_le_compat_r_iff_h : forall n2 n3 n1, n1 > O -> (n2 * n1 <= n3 * n1)%nat -> (n2 <= n3)%nat.
Lemma mult_le_compat_r_iff : forall n1 n2 n3, n1 > O -> (n2 * n1 <= n3 * n1)%nat -> (n2 <= n3)%nat.
Theorem leRat_trans : forall r1 r2 r3, leRat r1 r2 -> leRat r2 r3 -> leRat r1 r3.
Theorem eqRat_impl_leRat : forall r1 r2, eqRat r1 r2 -> leRat r1 r2.
Theorem leRat_impl_eqRat : forall r1 r2, leRat r1 r2 -> leRat r2 r1 -> eqRat r1 r2.