feat(Topology/EMetricSpace/BoundedVariation): more BoundedVariationOn and eVariationOn API

[teorth]

Various API for BoundedVariationOn and eVariationOn, including a formula for the eVariationOn for finite sets such as pairs, or the variation of the identity function, or more generally a monotone function.

---

Note: to obtain the variation formula for finite sets, Data.Finset.Sort was added to the imports.

[github-actions]

PR summary e74f9ee260

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Topology.EMetricSpace.BoundedVariation	1339	1341	+2 (+0.15%)

Import changes for all files

Files	Import difference
4 files Mathlib.Analysis.ConstantSpeed Mathlib.MeasureTheory.VectorMeasure.BoundedVariation Mathlib.Topology.EMetricSpace.BoundedVariation Mathlib.Topology.EMetricSpace.VariationOnFromTo	2

---

Declarations diff (regex)

+ BoundedVariationOn.id_Icc
+ MonotoneOn.eVariationOn_eq
+ _root_.BoundedVariationOn.of_finite
+ _root_.BoundedVariationOn.of_finset
+ eVariationOn_id
+ eVariationOn_id_Icc
+ image_range_of_monotone
+ pair
+ union'

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit e74f9ee).

• +8 new declarations
• −0 removed declarations

+BoundedVariationOn.id_Icc
+BoundedVariationOn.of_finite
+MonotoneOn.eVariationOn_eq
+eVariationOn.image_range_of_monotone
+eVariationOn.pair
+eVariationOn.union'
+eVariationOn_id
+eVariationOn_id_Icc

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit e74f9ee260
Reference commit 69b866d83a

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/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).

[github-actions]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions.
Thank you!

[CoolRmal]

Most comments are about the coding style and relatively minor. The only major concern I have is MonotoneOn.eVariationOn_eq and theorems dependent on it.

[CoolRmal]

awaiting-author

[teorth]

I have now folded the proof of MonotoneOn.eVariation_le into that of MonotoneOn.eVariation_eq and made MonotoneOn.eVariation_le a deprecated corollary. Also, I have removed BoundedVariationOn.of_singleton (and made BoundedVariationOn.of_finset private) as both are superseded by BoundedVariationOn.of_finite.

-awaiting-author

[CoolRmal]

I think this LGTM now, I'll put it on the maintainer queue after you address the comments.

awaiting-author

[teorth]

-awaiting-author

[CoolRmal]

maintainer merge

[github-actions]

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