proposition: Lemma inject_Q_AR_lt_rev : ∀ q r : Q, ARltT (inject_Q_AR q) (inject_Q_AR r) → q < r.
proof: Proof. intros. destruct (ARtoCR_preserves_ltT (' (' q)%CR) (' (' r)%CR)) as [c _]. pose proof (inject_Q_AR_CR q) as H6. pose proof (inject_Q_AR_CR r) as H7. apply (ARltT_wd _ _ H6 _ _ H7) in X. apply c in X. clear c H6 H7. pose proof (CRAR_id ('q)%CR). pose proof (CRAR_id ('r)%CR). apply (@CRltT_wd _ _ H5 _ _ H6) in X. apply Qlt_from_CRlt, X. Qed.
imports: CoRN.algebra.RSetoid CoRN.metric2.Metric CoRN.metric2.UniformContinuity Coq.Program.Program Coq.setoid_ring.Ring Coq.QArith.Qabs CoRN.stdlib_omissions.Q MathClasses.misc.workaround_tactics CoRN.model.totalorder.QMinMax CoRN.model.totalorder.QposMinMax CoRN.util.Qdlog CoRN.metric2.Complete CoRN.metric2.Prelength CoRN.model.metric2.Qmetric CoRN.metric2.MetricMorphisms CoRN.reals.fast.CRArith CoRN.reals.fast.CRpower MathClasses.implementations.stdlib_binary_naturals MathClasses.orders.minmax MathClasses.implementations.positive_semiring_elements CoRN.reals.faster.ApproximateRationals CoRN.reals.faster.AQmetric