[Merged by Bors] - feat(Geometry/Euclidean/SignedDist): signedDist between two points

[JovanGerb]

original PR: #24245

This PR defines signedDist, the signed distance between two points.

It also redefines signedInfDist so that it uses signedDist.

The motivation is to use this in IMO2020Q6. Additionally this definition will be useful for properly reasoning about the power of a point

#mathlib4 > Signed distance between points

some comments:

• should signedDistLinear be private?
• I'm not too happy about the hypothesis in signedDist_congr (h : ∃ r > (0 : ℝ), r • v = w). This relationship between v and w is a symmetric one that should have some API around it, similar to SameRay. It could also be spelled as (ℝ≥0 ∙ v) = ℝ≥0 ∙ w

---

• depends on: [Merged by Bors] - feat(Topology/Algebra/ContinuousAffineMap): define id #27259
• depends on : [Merged by Bors] - feat(MetricSpace): tag dist_nonneg with @[simp] #27285
  

[github-actions]

PR summary c2c4e1f99f

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Geometry.Euclidean.SignedDist	2042	2040	-2 (-0.10%)

Import changes for all files

Files	Import difference
Mathlib.Geometry.Euclidean.Incenter Mathlib.Geometry.Euclidean.SignedDist	-2

---

Declarations diff

+ abs_signedDist_eq_dist_iff_vsub_mem_span
+ abs_signedDist_le_dist
+ signedDist
+ signedDist_anticomm
+ signedDist_apply
+ signedDist_apply_apply
+ signedDist_apply_linear
+ signedDist_apply_linear_apply
+ signedDist_congr
+ signedDist_eq_dist_iff_vsub_mem_span
+ signedDist_eq_zero_of_orthogonal
+ signedDist_le_dist
+ signedDist_left_congr
+ signedDist_linear_apply
+ signedDist_linear_apply_apply
+ signedDist_neg
+ signedDist_right_congr
+ signedDist_self
+ signedDist_smul
+ signedDist_triangle
+ signedDist_triangle_left
+ signedDist_triangle_right
+ signedDist_vadd_left
+ signedDist_vadd_left_swap
+ signedDist_vadd_right
+ signedDist_vadd_right_swap
+ signedDist_vadd_vadd
+ signedDist_vsub_self
+ signedDist_vsub_self_rev
+ signedDist_zero
+ signedInfDist_def
+ signedInfDist_eq_signedDist_of_mem
+ signedInfDist_eq_signedDist_orthogonalProjection
+ signedInfDist_singleton
- signedInfDist_apply
- signedInfDist_linear
- signedInfDist_linear_apply

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - feat(Topology/Algebra/ContinuousAffineMap): define id #27259
  By Dependent Issues (🤖). Happy coding!

[leanprover-community-bot-assistant]

This pull request has conflicts, please merge master and resolve them.

[leanprover-community-bot-assistant]

This pull request has conflicts, please merge master and resolve them.

[jsm28]

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by jsm28.

[jcommelin]

Thanks 🎉

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• CI Success