[Merged by Bors] - feat(NumberTheory/Padics): the Amice transform

[loefflerd]

Prove the Amice (aka Leopoldt) transform isomorphism, identifying Zp-valued measures on Zp with the power series ring ℤ_[p]⟦X⟧.

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[github-actions]

PR summary a5079844eb

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.NumberTheory.Padics.Measure.AmiceTransform (new file)	2113

---

Declarations diff (regex)

+ AbstractMeasure.ext_mahler
+ amiceTransform
+ amiceTransformEquiv
+ amiceTransformEquiv_apply
+ amiceTransform_invTransform
+ arrowCongrLeft
+ arrowCongrLeft_apply
+ arrowCongrLeft_symm
+ coe_arrowCongrLeft
+ coeff_amiceTransform
+ coeff_amiceTransformEquiv
+ dense_span_mahler
+ ext_mahler
+ injective_amiceTransform
+ invTransform
+ invTransformSummable
+ invTransform_apply
+ invTransformₗ

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 a507984).

• +18 new declarations
• −0 removed declarations

+AbstractMeasure.amiceTransform
+AbstractMeasure.amiceTransform.congr_simp
+AbstractMeasure.amiceTransformEquiv
+AbstractMeasure.amiceTransformEquiv_apply
+AbstractMeasure.amiceTransform_invTransform
+AbstractMeasure.arrowCongrLeft
+AbstractMeasure.arrowCongrLeft_apply
+AbstractMeasure.arrowCongrLeft_symm
+AbstractMeasure.coe_arrowCongrLeft
+AbstractMeasure.coeff_amiceTransform
+AbstractMeasure.coeff_amiceTransformEquiv
+AbstractMeasure.ext_mahler
+AbstractMeasure.injective_amiceTransform
+AbstractMeasure.invTransform
+AbstractMeasure.invTransform_apply
+PadicInt.dense_span_mahler
+PadicInt.ext_mahler
+PadicInt.mahlerEquiv.congr_simp

---

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

Current commit a5079844eb
Reference commit 61eda68641

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!

[faenuccio]

Thanks! I made some comments concerning the files Padics/Measure/Basic, Algebra/Group/ForwardDiff and up to the end of the defs section in AmiceTransform.lean.

[loefflerd]

Many thanks for the very careful and detailed review, Filippo!

[faenuccio]

Thanks again, I've finished reviewing the Amice file and have suggested a bit of golfing. Besides this, I guess the main "point" is whether we want to move things a bit as discussed further up.

[loefflerd]

Thanks for all these great comments! I believe I've now dealt with everything except the query about scalar multiplication with 1 : C(ℤ_[p], R). Let me know how you think we should proceed with that one.

[faenuccio]

bors d+

[mathlib-bors]

✌️ loefflerd can now approve this pull request until 2026-09-09 20:27 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.

[faenuccio]

Thanks for all these great comments! I believe I've now dealt with everything except the query about scalar multiplication with 1 : C(ℤ_[p], R). Let me know how you think we should proceed with that one.

I've replied in a stream which might look "outdated", but I hope you can find the reply nonetheless.

[loefflerd]

bors r+

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

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• ci (staging) / Build
• ci (staging) / Lint style
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