[Merged by Bors] - refactor(RingTheory/RamificationInertia/Ramification): generalize ramificationIdx_pos

[tb65536]

Positivity of ramification index currently requires Module.Finite, but it is also true for Dedekind domains. This PR extracts a common generalization ramificationIdx_pos_of_mem_minimalPrimes and deduces these two facts as corollaries. This allows for a few more removals of the old ramificationIdx'.

---

• depends on: [Merged by Bors] - chore(RingTheory/QuasiFinite/Basic): extract instance from ramification proof #41378
• depends on: [Merged by Bors] - chore(RingTheory/*): remove domain assumptions by generalizing from torsion free to faithful smul #41379

[github-actions]

PR summary a10433c0de

Import changes exceeding 2%

%	File
+3.87%	Mathlib.NumberTheory.RamificationInertia.Valuation

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.NumberTheory.RamificationInertia.Valuation	2715	2820	+105 (+3.87%)

Import changes for all files

Files	Import difference
Mathlib.NumberTheory.RamificationInertia.Valuation	105

---

Declarations diff (regex)

+ Ideal.mem_minimalPrimes_of_ne_bot
+ ramificationIdx_pos_of_isDedekindDomain
+ ramificationIdx_pos_of_isDedekindDomain'
+ ramificationIdx_pos_of_mem_minimalPrimes

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 a10433c).

• +4 new declarations
• −0 removed declarations

+Ideal.mem_minimalPrimes_of_ne_bot
+Ideal.ramificationIdx_pos_of_isDedekindDomain
+Ideal.ramificationIdx_pos_of_isDedekindDomain'
+Ideal.ramificationIdx_pos_of_mem_minimalPrimes

---

No changes to strong technical debt.
No changes to weak technical debt.

Current commit a10433c0de
Reference commit 9d63ad78de

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).

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - chore(RingTheory/QuasiFinite/Basic): extract instance from ramification proof #41378
• [Merged by Bors] - chore(RingTheory/*): remove domain assumptions by generalizing from torsion free to faithful smul #41379
  By Dependent Issues (🤖). Happy coding!

[mathlib-merge-conflicts]

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

[mathlib-merge-conflicts]

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

[riccardobrasca]

Thanks!

bors d+

[mathlib-bors]

✌️ tb65536 can now approve this pull request until 2026-10-07 07:39 UTC (in 2 weeks). To approve and merge, reply with bors r+. More detailed instructions are available here.

⚠️ This delegation only covers changes within Archive/**, Counterexamples/**, docs/**, DownstreamTest/**, Mathlib/**, MathlibTest/**, Wanted/**, widget/**, Archive.lean, Counterexamples.lean, docs.lean, Mathlib.lean, Wanted.lean; an author commit touching anything else will revoke it. Bors also revokes it if a later push changes too many files for it to check the full list — even if it stays within scope.

[tb65536]

bors r+

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

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