feat(RingTheory/MvPowerSeries): multivariable power series ring is a noetherian ring when the index is finite

[BryceT233]

This is a split of #36507 which includes isEmptyEquiv, optionEquivLeft and finSuccEquiv. We use finSuccEquiv to show multivariable power series ring over a noetherian ring is a noetherian ring when the index is finite.

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[github-actions]

PR summary 9250306a2a

Import changes exceeding 2%

%	File
+3.80%	Mathlib.RingTheory.MvPowerSeries.Equiv

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.RingTheory.MvPowerSeries.Equiv	1788	1856	+68 (+3.80%)

Import changes for all files

Files	Import difference
Mathlib.RingTheory.AdicCompletion.LocalRing	33
Mathlib.RingTheory.AdicCompletion.Completeness	51
Mathlib.RingTheory.MvPowerSeries.Equiv	68

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Declarations diff

+ coeff_coeff_finSuccEquiv
+ coeff_coeff_optionEquivLeft
+ coeff_coeff_optionFunLeft
+ coeff_optionInvFunLeft
+ embDomain_finSuccEquiv_cons
+ finSuccEquiv
+ finSuccEquiv_C
+ finSuccEquiv_X_succ
+ finSuccEquiv_X_zero
+ finSuccEquiv_comp_C
+ image_optionElim_product_antidiagonal
+ isEmptyEquiv
+ isNoetherianRing
+ isNoetherianRing_fin
+ optionEquivLeft
+ optionEquivLeft_C
+ optionEquivLeft_X_none
+ optionEquivLeft_X_some
+ optionEquivLeft_monomial
+ optionFunLeft
+ optionFunLeft_monomial
+ optionFunLeft_mul
+ optionInvFunLeft

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.

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No changes to strong technical debt.

No changes to weak technical debt.

Current commit 9250306a2a
Reference commit 97f12126be

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).