[Merged by Bors] - feat(Data/List/SignVariations): add List.signVariations

[tomaz1502]

This PR adds List.signVariations, which counts the number of variations in a list, ignoring all zeroes, as well as its basic API. This counting is the mechanism of both Descartes' Rule of Signs and of Sturm-Tarski theorem, which will be added in upcoming PRs.

This PR also adapts RuleOfSigns.lean to use the new API, simplifying some of its proofs.

The contents of this PR were manually written. Claude Code (Fable 5.1) was used to review the code.

[github-actions]

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[github-actions]

PR summary 1769435c55

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Algebra.Polynomial.RuleOfSigns	1
Mathlib.Data.List.SignVariations (new file)	800

---

Declarations diff (regex)

+ signVariations_congr
+ signVariations_cons_cons_of_ne_zero
+ signVariations_cons_replicate_zero
+ signVariations_cons_replicate_zero_append
+ signVariations_cons_zero_cons
+ signVariations_filter_ne_zero
+ signVariations_map
+ signVariations_map_of_sign_eq_neg
+ signVariations_map_sign
+ signVariations_nil
+ signVariations_singleton
+ signVariations_zero_cons
++- signVariations

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 1769435).

• +13 new declarations
• −0 removed declarations

+List.signVariations
+List.signVariations_congr
+List.signVariations_cons_cons_of_ne_zero
+List.signVariations_cons_replicate_zero
+List.signVariations_cons_replicate_zero_append
+List.signVariations_cons_zero_cons
+List.signVariations_filter_ne_zero
+List.signVariations_map
+List.signVariations_map_of_sign_eq_neg
+List.signVariations_map_sign
+List.signVariations_nil
+List.signVariations_singleton
+List.signVariations_zero_cons

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
exposed public sections	5067	1

Current commit 1769435c55
Reference commit 64401b008c

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.py 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!

[kim-em]

I'm posting some review comments, identified with the assistance of Claude and Codex. I'm not putting separate ":robot:" attributions on each comment.

[kim-em]

• The docstring of signVariations_X_sub_C_mul_eraseLead_le (L242, untouched by this PR) says the two counts are "the same", but the lemma only proves ≤, and equality is false: P = X^2 + X + 10, η = 2 gives 3 vs 1. Worth fixing while you're here.
• PR description: "This counting mechanism of both ..." is missing a verb ("This is the counting mechanism behind both ...").

[kim-em]

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• ci (staging) / Lint style
• ci (staging) / Build
• ci (staging) / Post-Build Step
• ci (staging) / Post-CI job
• ci (staging) / Test and lint