feat(Algebra/Order/Chebyshev): the Abel inequality

[teorth]

Adds Finset.sum_mul_le_sum_mul_of_sum_range_le, a form of Abel's inequality: if
the partial sums of f are dominated by those of c up to n, and g is nonnegative
and antitone, then the g-weighted sums satisfy ∑ f i * g i ≤ ∑ c i * g i. Proved by
summation by parts (Finset.sum_range_by_parts), over any ordered commutative ring. Other common variants of Abel's inequality are also deduced as a corollary.

---

I chose to place this inequality in Mathlib.Algebra.Order.Chebyshev,although technically the Chebyshev inequality and Abel inequality have no logical dependency and one could place them instead in a separate file such as, `Mathlib.Algebra.Order.Abel'. Happy to do this instead of the reviewers request it; but the two inequalities are thematically related even if they are logically independent.

This PR is stacked upon a unrelated previous PR to the same file, but if reviewers prefer to move these inequalities to a separate file then the dependency disappears. Both PRs will be used in a forthcoming PR on the prime number theorem.

• depends on: feat(Algebra/Order/Chebyshev): the Cauchy–Schwarz inequality for multisets #42873

AI was used to generate initial proofs of these lemmas, which were then heavily rewritten and golfed by the author.

[github-actions]

PR summary 28af7f245d

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Algebra.Order.Chebyshev	1039	1058	+19 (+1.83%)

Import changes for all files

Files	Import difference
6 files Mathlib.Combinatorics.Additive.Corner.Roth Mathlib.Combinatorics.SimpleGraph.Regularity.Bound Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk Mathlib.Combinatorics.SimpleGraph.Regularity.Increment Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma Mathlib.Combinatorics.SimpleGraph.Triangle.Removal	1
Mathlib.Algebra.Order.Chebyshev	19

---

Declarations diff (regex)

+ Multiset.sq_sum_le_card_mul_sum_sq
+ abs_sum_mul_le_mul_of_abs_sum_range_le
+ mul_le_sum_mul_of_le_sum_range
+ prod_map_eq_prod_toEnumFinset
+ sum_mul_le_mul_of_sum_range_le
+ sum_mul_le_sum_mul_of_sum_range_le

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 28af7f2).

• +7 new declarations
• −0 removed declarations

+Finset.abs_sum_mul_le_mul_of_abs_sum_range_le
+Finset.mul_le_sum_mul_of_le_sum_range
+Finset.sum_mul_le_mul_of_sum_range_le
+Finset.sum_mul_le_sum_mul_of_sum_range_le
+Multiset.prod_map_eq_prod_toEnumFinset
+Multiset.sq_sum_le_card_mul_sum_sq
+Multiset.sum_map_eq_sum_toEnumFinset

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 28af7f245d
Reference commit 1ce330756f

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).

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(Algebra/Order/Chebyshev): the Cauchy–Schwarz inequality for multisets #42873
  By Dependent Issues (🤖). Happy coding!