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fixes #1231 (space ball lemma) #1247

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Merged
CohenCyril merged 1 commit into from
Jun 24, 2024
Merged

fixes #1231 (space ball lemma) #1247

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8 changes: 8 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Original file line number Diff line number Diff line change
Expand Up @@ -4,8 +4,16 @@

### Added

- in `topology.v`:
+ lemma `ball_subspace_ball`

### Changed

- in `topology.v`:
+ lemmas `subspace_pm_ball_center`, `subspace_pm_ball_sym`,
`subspace_pm_ball_triangle`, `subspace_pm_entourage` turned
into local `Let`'s

### Renamed

- in `constructive_ereal.v`:
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13 changes: 9 additions & 4 deletions theories/topology.v
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Expand Up @@ -6403,25 +6403,26 @@ End SubspaceUniform.

Section SubspacePseudoMetric.
Context {R : numDomainType} {X : pseudoMetricType R} (A : set X).
Implicit Type e : R.

Definition subspace_ball (x : subspace A) (r : R) :=
if x \in A then A `&` ball (x : X) r else [set x].

Lemma subspace_pm_ball_center x (e : R) : 0 < e -> subspace_ball x e x.
Let subspace_pm_ball_center x e : 0 < e -> subspace_ball x e x.
Proof.
rewrite /subspace_ball; case: ifP => //= /asboolP ? ?.
by split=> //; exact: ballxx.
Qed.

Lemma subspace_pm_ball_sym x y (e : R) :
Let subspace_pm_ball_sym x y e :
subspace_ball x e y -> subspace_ball y e x.
Proof.
rewrite /subspace_ball; case: ifP => //= /asboolP ?.
by move=> [] Ay /ball_sym yBx; case: ifP => /asboolP.
by move=> ->; case: ifP => /asboolP.
Qed.

Lemma subspace_pm_ball_triangle x y z e1 e2 :
Let subspace_pm_ball_triangle x y z e1 e2 :
subspace_ball x e1 y -> subspace_ball y e2 z -> subspace_ball x (e1 + e2) z.
Proof.
rewrite /subspace_ball; (repeat case: ifP => /asboolP).
Expand All @@ -6431,7 +6432,7 @@ rewrite /subspace_ball; (repeat case: ifP => /asboolP).
- by move=> _ _ -> ->.
Qed.

Lemma subspace_pm_entourageE :
Let subspace_pm_entourageE :
@entourage (subspace A) = entourage_ subspace_ball.
Proof.
rewrite eqEsubset; split; rewrite /subspace_ball.
Expand All @@ -6452,6 +6453,10 @@ HB.instance Definition _ :=
subspace_pm_ball_center subspace_pm_ball_sym subspace_pm_ball_triangle
subspace_pm_entourageE.

Lemma ball_subspace_ball (x : subspace A) r (y : subspace A) :
ball x r y = subspace_ball x r y.
Proof. by []. Qed.

End SubspacePseudoMetric.

Section SubspaceWeak.
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