proposition: Lemma AQarctan_small_Qprf : -1 < AQtoQ num / AQtoQ den < 1.
proof: Proof. split. - apply Qlt_shift_div_l. pose proof (rings.preserves_0 (f:=cast AQ Q)). rewrite <- H5. apply (strictly_order_preserving (cast AQ Q)), dpos. setoid_replace (-1 * AQtoQ den) with (-AQtoQ den) by reflexivity. rewrite <- rings.preserves_negate. apply (strictly_order_preserving (cast AQ Q)), Pnd. - apply Qlt_shift_div_r. pose proof (rings.preserves_0 (f:=cast AQ Q)). rewrite <- H5. apply (strictly_order_preserving (cast AQ Q)), dpos. rewrite Qmult_1_l. apply (strictly_order_preserving (cast AQ Q)), Pnd. Qed.
imports: MathClasses.interfaces.abstract_algebra MathClasses.theory.nat_pow MathClasses.theory.int_pow CoRN.algebra.RSetoid CoRN.stdlib_omissions.Q CoRN.metric2.Metric CoRN.metric2.UniformContinuity CoRN.reals.fast.CRarctan_small CoRN.reals.fast.CRarctan CoRN.reals.fast.CRstreams CoRN.reals.fast.CRAlternatingSum CoRN.reals.faster.ARAlternatingSum CoRN.reals.faster.ARsin CoRN.reals.faster.ARArith