feat (Topology/Compactness/CompactSystem): closed and compact square cylinders form a compact system

[pfaffelh]

feat (Topology/Compactness/CompactSystem): Closed and compact square cylinders form a
compact system

A square cylinder is a set of the form s.pi t in a product space. A closed compact square cylinder has IsClosed (t i) and IsCompact (t i) for all i ∈ s. This set system is a compact system (as introduced in PR #36013).

Streamline the definition of square cylinders in MeasureTheory/Constructions/Cylinders on the way.

Co-authors: Rémy Degenne remy.degenne@inria.fr

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• depends on: [Merged by Bors] - feat(Topology/Compactness/CompactSystem): introduce compact Systems #36013

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[github-actions]

PR summary d9804c6aab

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.MeasureTheory.PiSystem	599	602	+3 (+0.50%)
Mathlib.MeasureTheory.Constructions.Cylinders	738	741	+3 (+0.41%)

Import changes for all files

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Mathlib.Probability.Distributions.Fernique Mathlib.Probability.Distributions.Gamma Mathlib.Probability.Distributions.Gaussian.Basic Mathlib.Probability.Distributions.Gaussian.CharFun Mathlib.Probability.Distributions.Gaussian.Fernique Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Def Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Def Mathlib.Probability.Distributions.Gaussian.Real Mathlib.Probability.Distributions.Geometric Mathlib.Probability.Distributions.Pareto Mathlib.Probability.Distributions.Poisson Mathlib.Probability.Distributions.SetBernoulli Mathlib.Probability.Distributions.Uniform Mathlib.Probability.HasLawExists Mathlib.Probability.HasLaw Mathlib.Probability.IdentDistribIndep Mathlib.Probability.IdentDistrib Mathlib.Probability.Independence.Basic 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Mathlib.Probability.Kernel.Irreducible Mathlib.Probability.Kernel.MeasurableIntegral Mathlib.Probability.Kernel.MeasurableLIntegral Mathlib.Probability.Kernel.Posterior Mathlib.Probability.Kernel.Proper Mathlib.Probability.Kernel.RadonNikodym Mathlib.Probability.Kernel.SetIntegral Mathlib.Probability.Kernel.WithDensity Mathlib.Probability.Martingale.Basic Mathlib.Probability.Martingale.BorelCantelli Mathlib.Probability.Martingale.Centering Mathlib.Probability.Martingale.Convergence Mathlib.Probability.Martingale.OptionalSampling Mathlib.Probability.Martingale.OptionalStopping Mathlib.Probability.Martingale.Upcrossing Mathlib.Probability.Moments.Basic Mathlib.Probability.Moments.ComplexMGF Mathlib.Probability.Moments.CovarianceBilinDual Mathlib.Probability.Moments.CovarianceBilin Mathlib.Probability.Moments.Covariance Mathlib.Probability.Moments.IntegrableExpMul Mathlib.Probability.Moments.MGFAnalytic Mathlib.Probability.Moments.SubGaussian Mathlib.Probability.Moments.Tilted Mathlib.Probability.Moments.Variance Mathlib.Probability.Notation Mathlib.Probability.ProbabilityMassFunction.Basic Mathlib.Probability.ProbabilityMassFunction.Binomial Mathlib.Probability.ProbabilityMassFunction.Constructions Mathlib.Probability.ProbabilityMassFunction.Integrals Mathlib.Probability.ProbabilityMassFunction.Monad Mathlib.Probability.Process.Adapted Mathlib.Probability.Process.Filtration Mathlib.Probability.Process.FiniteDimensionalLaws Mathlib.Probability.Process.HittingTime Mathlib.Probability.Process.Kolmogorov Mathlib.Probability.Process.PartitionFiltration Mathlib.Probability.Process.Predictable Mathlib.Probability.Process.Stopping Mathlib.Probability.ProductMeasure Mathlib.Probability.StrongLaw Mathlib.Probability.UniformOn Mathlib.RingTheory.Polynomial.Selmer Mathlib.Tactic.NormNum.LegendreSymbol Mathlib.Tactic Mathlib.Topology.Baire.BaireMeasurable Mathlib.Topology.EMetricSpace.PairReduction Mathlib.Topology.MetricSpace.HausdorffDimension	1
4 files Mathlib.MeasureTheory.Constructions.Cylinders Mathlib.MeasureTheory.MeasurableSpace.Pi Mathlib.MeasureTheory.MeasurableSpace.Prod Mathlib.MeasureTheory.PiSystem	3
Mathlib.Topology.Compactness.CompactSystem (new file)	783

---

Declarations diff

+ IsCompactSystem
+ IsCompactSystem.compactClosedOrUnivPi
+ IsCompactSystem.compactClosedPi
+ IsPiSystem.dissipate_mem
+ accumulate_eq_biInter_lt
+ biInter_univ_pi_empty_iff
+ compactClosedSquareCylinders
+ directed_accumulate
+ directed_dissipate
+ dissipate_eq_biInter_lt
+ exists_accumulate_eq_univ_iff_of_directed
+ exists_dissipate_eq_empty_iff_of_directed
+ exists_subset_accumulate_of_directed
+ exists_subset_dissipate_of_directed
+ iInter_pi_empty_iff
+ iInter_univ_pi_empty_iff
+ iff_nonempty_iInter
+ insert_empty
+ insert_univ
+ isCompactSystem.compactClosedSquareCylinders
+ isCompactSystem_iff_nonempty_iInter_of_directed
+ isCompactSystem_iff_nonempty_iInter_of_lt
+ isCompactSystem_iff_of_directed
+ isCompactSystem_insert_empty_iff
+ isCompactSystem_insert_univ_iff
+ isCompactSystem_insert_univ_isCompact_isClosed
+ isCompactSystem_isCompact
+ isCompactSystem_isCompact_isClosed
+ mono
+ nonempty_iInter
+ of_IsEmpty
+ of_nonempty_iInter
+ pi
+ pi_image_eq_of_subset
+ squareCylinders_eq_iUnion_image'
+ squareCylinders_subset_of_or_univ

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.

---

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).

[github-actions]

🚨 PR Title Needs Formatting

Please update the title to match our commit style conventions.

Errors from script:

error: the PR title should be of the form
  abbrev: main title
or
  abbrev(scope): main title

Details on the required title format

The title should fit the following format:

<kind>(<optional-scope>): <subject>

<kind> is:

• feat (feature)
• fix (bug fix)
• doc (documentation)
• style (formatting, missing semicolons, ...)
• refactor
• test (when adding missing tests)
• chore (maintain)
• perf (performance improvement, optimization, ...)
• ci (changes to continuous integration, repo automation, ...)

<optional-scope> is a name of module or a directory which contains changed modules.
This is not necessary to include, but may be useful if the <subject> is insufficient.
The Mathlib directory prefix is always omitted.
For instance, it could be

• Data/Nat/Basic
• Algebra/Group/Defs
• Topology/Constructions

<subject> has the following constraints:

• do not capitalize the first letter
• no dot(.) at the end
• use imperative, present tense: "change" not "changed" nor "changes"

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - feat(Topology/Compactness/CompactSystem): introduce compact Systems #36013
  By Dependent Issues (🤖). Happy coding!

[mathlib-merge-conflicts]

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