Commit 2509e6f8 authored by Robbert Krebbers's avatar 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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