proposition: Lemma AQarctan_pos_prf2 {a : AQ} : ¬a ≤ 1 ≪ (-1) → 0 < a.
proof: Proof. intros. apply orders.lt_le_trans with (1 ≪ (-1)). apply (strictly_order_reflecting (cast AQ Q)). rewrite aq_shift_opp_1, rings.preserves_0, rings.preserves_1. split; easy. now apply orders.le_flip. Qed.
imports: CoRN.algebra.RSetoid CoRN.metric2.Metric CoRN.metric2.UniformContinuity CoRN.metric2.MetricMorphisms CoRN.reals.fast.CRarctan_small CoRN.reals.fast.CRarctan CoRN.reals.fast.CRpi_fast CoRN.reals.faster.ARpi CoRN.reals.faster.ARarctan_small CoRN.reals.faster.ARArith