feat(Analysis): characterizing and congr lemma for toMeromorphicNFAt

[wwylele]

The first two new lemma toMeromorphicNFAt_of_meromorphicOrderAt_ne_zero and MeromorphicAt.exists_analytic_and_toMeromorphicNFAt_eventuallyEq characterize the behavior of toMeromorphicNFAt for meromorphic function, so one no longer needs to unfold the definition of it.

The third lemma toMeromorphicNFAt_congr shows that two functions have equal toMeromorphicNFAt value at a point as long as they equal in the punctured nhds.

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[github-actions]

PR summary 1833ce983e

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff (regex)

+ MeromorphicAt.exists_analytic_and_toMeromorphicNFAt_eventuallyEq
+ toMeromorphicNFAt_congr
+ toMeromorphicNFAt_of_meromorphicOrderAt_ne_zero

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 1833ce9).

• +3 new declarations
• −0 removed declarations

+MeromorphicAt.exists_analytic_and_toMeromorphicNFAt_eventuallyEq
+toMeromorphicNFAt_congr
+toMeromorphicNFAt_of_meromorphicOrderAt_ne_zero

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No changes to strong technical debt.

No changes to weak technical debt.

Current commit 1833ce983e
Reference commit ba1e3bb0bf

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).

[Copilot]

Pull request overview

This PR adds API lemmas in Analysis/Meromorphic/NormalForm to avoid unfolding toMeromorphicNFAt in common proofs, including a characterization for the order-0 case and a congruence lemma under punctured-neighborhood equality.

Changes:

• Added a simp lemma describing toMeromorphicNFAt f x x when meromorphicOrderAt f x ≠ 0.
• Added an existence/characterization lemma producing an analytic representative g with toMeromorphicNFAt f x =ᶠ[𝓝 x] g when meromorphicOrderAt f x = 0.
• Added a [gcongr] lemma toMeromorphicNFAt_congr for equality of values at x under =ᶠ[𝓝[≠] x].

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