### Add lemma about the existence of a lower bound of two fractions.

parent 62a55d05
 ... ... @@ -550,3 +550,23 @@ Lemma Qp_div_2 x : (x / 2 + x / 2 = x)%Qp. Proof. change 2%positive with (2 * 1)%positive. by rewrite Qp_div_S, Qp_div_1. Qed. Lemma Qp_lower_bound q1 q2: ∃ q q1' q2', (q1 = q + q1' ∧ q2 = q + q2')%Qp. Proof. assert (Hdiff : ∀ a b:Qp, (a ≤ b)%Qc → ∃ c, (b - a / 2)%Qp = Some c ∧ (a/2 + c)%Qp = b). { intros a b Hab. unfold Qp_minus. destruct decide as [|[]]. - eexists. split. done. apply Qp_eq. simpl. ring. - eapply Qclt_le_trans; [|by apply Qcplus_le_mono_r, Hab]. change (0 < a - a/2)%Qc. replace (a - a / 2)%Qc with (a * (1 - 1/2))%Qc by ring. replace 0%Qc with (0 * (1-1/2))%Qc by ring. by apply Qcmult_lt_compat_r. } destruct (Qc_le_dec q1 q2) as [LE|LE%Qclt_nge%Qclt_le_weak]. - destruct (Hdiff _ _ LE) as [q2' [EQ <-]]. exists (q1 / 2)%Qp, (q1 / 2)%Qp, q2'. split; apply Qp_eq. by rewrite Qp_div_2. ring. - destruct (Hdiff _ _ LE) as [q1' [EQ <-]]. exists (q2 / 2)%Qp, q1', (q2 / 2)%Qp. split; apply Qp_eq. ring. by rewrite Qp_div_2. Qed.
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