Commit 2509e6f8 by Robbert Krebbers

Shorten proof of Qp_lower_bound a bit.

parent 39d2538d
 ... ... @@ -551,22 +551,18 @@ 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. Lemma Qp_lower_bound q1 q2 : ∃ q q1' q2', (q1 = q + q1' ∧ q2 = q + q2')%Qp. Proof. revert q1 q2. cut (∀ q1 q2 : Qp, (q1 ≤ q2)%Qc → ∃ q q1' q2', (q1 = q + q1' ∧ q2 = q + q2')%Qp). { intros help q1 q2. destruct (Qc_le_dec q1 q2) as [LE|LE%Qclt_nge%Qclt_le_weak]; [by eauto|]. destruct (help q2 q1) as (q&q1'&q2'&?&?); eauto. } intros q1 q2 Hq. exists (q1 / 2)%Qp, (q1 / 2)%Qp. assert (0 < q2 - q1 / 2)%Qc as Hq2'. { eapply Qclt_le_trans; [|by apply Qcplus_le_mono_r, Hq]. replace (q1 - q1 / 2)%Qc with (q1 * (1 - 1/2))%Qc by ring. replace 0%Qc with (0 * (1-1/2))%Qc by ring. by apply Qcmult_lt_compat_r. } exists (mk_Qp (q2 - q1 / 2%Z) Hq2'). split; [by rewrite Qp_div_2|]. apply Qp_eq; simpl. ring. Qed.
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