feat(Algebra/Ring): add sum_range_id_mul_geometric_add

[matt-w-horn]

Over a CommRing: ∑ k < m, k * (x ^ k * (1 - x)) + m * x ^ m = ∑ k ∈ Icc 1 m, x ^ k. Summation by parts for the geometric sum; it computes the mean of a censored geometric distribution, which #42381 adds.

Used in Overload, a library on retry and overload dynamics, where it's stated over ℝ with a side condition that turns out to be unnecessary: https://github.com/matt-w-horn/overload/blob/4ecbebf/Overload/Retry/Amplification.lean

Claude Fable wrote the Lean from a spec I gave it. I reviewed and tested it myself, including adversarial tests and the suites in the library above, and did the same for its dependencies.

Assisted-by: Claude Fable 5

[github-actions]

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[github-actions]

PR summary cc5213c2be

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ sum_range_id_mul_geometric_add

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 cc5213c).

• +1 new declarations
• −0 removed declarations

+sum_range_id_mul_geometric_add

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit cc5213c2be
Reference commit 375d54da29

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).

[matt-w-horn]

LLM-generated

[matt-w-horn]

easy