feat(RingTheory/Spectrum/Prime/Noetherian): rank of an Artinian ring over a field

[tb65536]

This PR proves that the rank of an Artinian ring over a field is the sum of the ranks of the localizations.

This will be applied in #37130 to prove the ramification-inertia formula (this is where the sum first shows up).

I had to upgrade the Finite instance on the prime spectrum of an Artinian ring to a Fintype instance in order to write down the sum, but this should be fine since this fintype instance can already be inferred from heavier imports: https://live.lean-lang.org/#codez=JYWwDg9gTgLgBAWQIYwBYBtgCMBQOB2E%2BAxhOAK4xJboCmctAHkuHXAN4BKcAXHACoBPMLQBUAXzgBtAMJkQnYPgDmcTgF1pASQDOAQVhLgSfIpVrNfAGJKYw%2BgAoAClFC0AyiOIwo5EGoBKXgBeHDg4JQAzWigtfB0qElocIA

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[github-actions]

PR summary c4037aa6b4

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.RingTheory.Spectrum.Prime.Noetherian	1783	1784	+1 (+0.06%)
Mathlib.RingTheory.OrderOfVanishing.Noetherian	2269	2270	+1 (+0.04%)

Import changes for all files

Files

Import difference

46 files Mathlib.AlgebraicGeometry.AffineSpace Mathlib.AlgebraicGeometry.AffineTransitionLimit Mathlib.AlgebraicGeometry.AlgClosed.Basic Mathlib.AlgebraicGeometry.Artinian Mathlib.AlgebraicGeometry.Fiber Mathlib.AlgebraicGeometry.Geometrically.Basic Mathlib.AlgebraicGeometry.Geometrically.Connected Mathlib.AlgebraicGeometry.Geometrically.Integral Mathlib.AlgebraicGeometry.Geometrically.Irreducible Mathlib.AlgebraicGeometry.Geometrically.Reduced Mathlib.AlgebraicGeometry.IdealSheaf.Functorial Mathlib.AlgebraicGeometry.IdealSheaf.IrreducibleComponent Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation Mathlib.AlgebraicGeometry.Morphisms.FiniteType Mathlib.AlgebraicGeometry.Morphisms.Finite Mathlib.AlgebraicGeometry.Morphisms.FlatDescent Mathlib.AlgebraicGeometry.Morphisms.FlatMono Mathlib.AlgebraicGeometry.Morphisms.Immersion Mathlib.AlgebraicGeometry.Morphisms.Integral Mathlib.AlgebraicGeometry.Morphisms.Proper Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant Mathlib.AlgebraicGeometry.Morphisms.Separated Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen Mathlib.AlgebraicGeometry.Noetherian Mathlib.AlgebraicGeometry.PointsPi Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper Mathlib.AlgebraicGeometry.QuasiAffine Mathlib.AlgebraicGeometry.RationalMap Mathlib.AlgebraicGeometry.Sites.ConstantSheaf Mathlib.AlgebraicGeometry.Sites.Fpqc Mathlib.AlgebraicGeometry.SpreadingOut Mathlib.AlgebraicGeometry.ValuativeCriterion Mathlib.RingTheory.AdicCompletion.Noetherian Mathlib.RingTheory.HopkinsLevitzki Mathlib.RingTheory.Ideal.KrullsHeightTheorem Mathlib.RingTheory.Jacobson.Artinian Mathlib.RingTheory.KrullDimension.Polynomial Mathlib.RingTheory.KrullDimension.Regular Mathlib.RingTheory.OrderOfVanishing.Basic Mathlib.RingTheory.OrderOfVanishing.Noetherian Mathlib.RingTheory.RegularLocalRing.Defs Mathlib.RingTheory.Spectrum.Prime.Jacobson Mathlib.RingTheory.Spectrum.Prime.LTSeries Mathlib.RingTheory.Spectrum.Prime.Noetherian

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Declarations diff

+ exists_not_mem_forall_mem_of_ne
+ finrank_eq_sum_primeSpectrum
+ instance : Fintype (PrimeSpectrum R)
- _root_.IsArtinianRing.exists_not_mem_forall_mem_of_ne
- instance : Finite (PrimeSpectrum R)

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./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).

[tb65536]

These unnamed instances technically had their name changed from the namespace, but I assume that's fine given that the autogenerated names aren't supposed to be used anyway?