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add lt_nbhs{r,l} variants #1881

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6 changes: 6 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Original file line number Diff line number Diff line change
Expand Up @@ -70,6 +70,9 @@
- in `derive.v`:
+ lemmas `compact_EVT_max`, `compact_EVT_min`, `EVT_max_rV`, `EVT_min_rV`

- in `num_topology.v`:
+ lemmas `lt_le_nbhsr`, `lt_le_nbhsl`

### Changed

- in `constructive_ereal.v`: fixed the infamous `%E` scope bug.
Expand Down Expand Up @@ -251,6 +254,9 @@
+ lemmas `integral_itv_bndo_bndc`, `integral_itv_obnd_cbnd`,
`integral_itv_bndoo`

- in `num_topology.v`:
+ lemma `lt_nbhsl`

### Deprecated

- in `lebesgue_integral_nonneg.v`:
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10 changes: 9 additions & 1 deletion theories/topology_theory/num_topology.v
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Expand Up @@ -374,14 +374,18 @@ exists x => // z /=; rewrite sub0r normrN.
by apply: le_lt_trans; rewrite ler_norm.
Qed.

Lemma lt_nbhsl {R : realType} (x a : R) : x < a ->
Lemma lt_nbhsl {R : realFieldType} (x a : R) : x < a ->
\forall y \near nbhs x, y < a.
Proof.
move=> xb; exists ((a - x) / 2) => /=; first by rewrite divr_gt0// subr_gt0.
move=> r/=; rewrite ltr_pdivlMr// ltrBrDr; apply: le_lt_trans.
by rewrite -lerBlDr -normrN opprB (le_trans (ler_norm _))// ler_peMr// ler1n.
Qed.

Lemma lt_le_nbhsl {R : realFieldType} (t x : R) :
x < t -> \forall y \near nbhs x, y <= t.
Proof. by move/lt_nbhsl; apply: filterS => y /ltW. Qed.

Lemma Nlt_nbhsl {R : realType} (x a : R) :
- x < a -> \forall y \near nbhs x, - y < a.
Proof.
Expand Down Expand Up @@ -426,6 +430,10 @@ have [uz|uz] := leP u z.
by rewrite ltr0_norm ?subr_lt0// opprB addrAC -lerBlDr opprK lerD// ?ltW.
Qed.

Lemma lt_le_nbhsr {R : realFieldType} (t x : R) :
t < x -> \forall y \near nbhs x, t <= y.
Proof. by move/lt_nbhsr; apply: filterS => y /ltW. Qed.

Global Instance Proper_dnbhs_regular_numFieldType (R : numFieldType) (x : R^o) :
ProperFilter x^'.
Proof.
Expand Down