feat(GroupTheory/SpecificGroups/Cyclic): add IsCyclic.subgroup_le/eq_subgroup_iff

[xroblot]

Add the following results about subgroup comparison in cyclic groups.

• IsCyclic.subgroup_le_iff_index_dvd: in a cyclic group, H ≤ K ↔ K.index ∣ H.index
• IsCyclic.subgroup_le_subgroup_iff: if K is finite, H ≤ K ↔ Nat.card H ∣ Nat.card K
• IsCyclic.subgroup_eq_subgroup_iff: if H and K are finite, H = K ↔ Nat.card H = Nat.card K
• IsCyclic.infinite_of_ne_bot: a nontrivial subgroup of an infinite cyclic group is infinite
• Subgroup.exists_zpowers_eq_of_zpowers_eq_top: if g generates, every subgroup is of the form zpowers (g ^ i)
• Subgroup.index_eq_card_div: for H finite, H.index = Nat.card G / Nat.card H

🤖 This PR was extracted from the SKW project by Claude.

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• depends on: feat(GroupTheory/SpecificGroups/Cyclic): comparison and index of subgroups generated by powers #42754

[github-actions]

PR summary 4ae95a3c8f

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.GroupTheory.SpecificGroups.Cyclic	1092	1093	+1 (+0.09%)

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Mathlib.Probability.Kernel.Representation 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.Binomial Mathlib.Probability.ProbabilityMassFunction.Constructions Mathlib.Probability.ProbabilityMassFunction.Integrals Mathlib.Probability.Process.Adapted Mathlib.Probability.Process.Filtration Mathlib.Probability.Process.FiniteDimensionalLaws Mathlib.Probability.Process.HittingTime Mathlib.Probability.Process.LocalProperty Mathlib.Probability.Process.PartitionFiltration Mathlib.Probability.Process.Predictable Mathlib.Probability.Process.Stopping Mathlib.Probability.ProductMeasure Mathlib.Probability.StrongLaw Mathlib.RepresentationTheory.AlgebraRepresentation.Basic Mathlib.RepresentationTheory.Character Mathlib.RepresentationTheory.FDRep Mathlib.RepresentationTheory.FinGroupCharZero Mathlib.RepresentationTheory.Homological.GroupCohomology.FiniteCyclic Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90 Mathlib.RepresentationTheory.Homological.GroupCohomology.LongExactSequence Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree Mathlib.RepresentationTheory.Homological.GroupHomology.FiniteCyclic Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality Mathlib.RepresentationTheory.Homological.GroupHomology.LongExactSequence Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree Mathlib.RepresentationTheory.Homological.TateCohomology.Basic Mathlib.RepresentationTheory.Invariants Mathlib.RepresentationTheory.Irreducible Mathlib.RepresentationTheory.Tannaka Mathlib.RingTheory.AdicCompletion.Algebra Mathlib.RingTheory.AdicCompletion.AsTensorProduct Mathlib.RingTheory.AdicCompletion.Completeness Mathlib.RingTheory.AdicCompletion.Exactness Mathlib.RingTheory.AdicCompletion.Functoriality Mathlib.RingTheory.AdicCompletion.LocalRing Mathlib.RingTheory.AdicCompletion.Noetherian Mathlib.RingTheory.AdicCompletion.RingHom Mathlib.RingTheory.AlgebraicIndependent.AlgebraicClosure Mathlib.RingTheory.Complex Mathlib.RingTheory.DedekindDomain.AdicValuation Mathlib.RingTheory.DedekindDomain.Different Mathlib.RingTheory.DedekindDomain.Dvr Mathlib.RingTheory.DedekindDomain.Factorization Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing Mathlib.RingTheory.DedekindDomain.GaussLemma Mathlib.RingTheory.DedekindDomain.Instances Mathlib.RingTheory.DedekindDomain.IntegralClosure Mathlib.RingTheory.DedekindDomain.LinearDisjoint Mathlib.RingTheory.DedekindDomain.PID Mathlib.RingTheory.DedekindDomain.SInteger Mathlib.RingTheory.DedekindDomain.SelmerGroup Mathlib.RingTheory.Depth.Rees Mathlib.RingTheory.DiscreteValuationRing.TFAE Mathlib.RingTheory.Discriminant Mathlib.RingTheory.Etale.Basic Mathlib.RingTheory.Etale.Descent Mathlib.RingTheory.Etale.Field Mathlib.RingTheory.Etale.Finite Mathlib.RingTheory.Etale.Kaehler Mathlib.RingTheory.Etale.Locus Mathlib.RingTheory.Etale.Pi Mathlib.RingTheory.Etale.QuasiFinite Mathlib.RingTheory.Etale.StandardEtale Mathlib.RingTheory.Etale.Weakly Mathlib.RingTheory.Extension.Basic Mathlib.RingTheory.Extension.Cotangent.BaseChange Mathlib.RingTheory.Extension.Cotangent.Basic Mathlib.RingTheory.Extension.Cotangent.Basis Mathlib.RingTheory.Extension.Cotangent.Free Mathlib.RingTheory.Extension.Cotangent.LocalizationAway Mathlib.RingTheory.Extension.ExtendScalars Mathlib.RingTheory.Extension.Generators Mathlib.RingTheory.Extension.Presentation.Basic Mathlib.RingTheory.Extension.Presentation.Core Mathlib.RingTheory.Extension.Presentation.Submersive Mathlib.RingTheory.Flat.LocallyFree Mathlib.RingTheory.Flat.Rank Mathlib.RingTheory.Flat.TorsionFree Mathlib.RingTheory.FractionalIdeal.Norm Mathlib.RingTheory.Frobenius Mathlib.RingTheory.Grassmannian Mathlib.RingTheory.HopkinsLevitzki Mathlib.RingTheory.Ideal.CotangentBaseChange Mathlib.RingTheory.Ideal.Cotangent Mathlib.RingTheory.Ideal.Int Mathlib.RingTheory.Ideal.KrullsHeightTheorem Mathlib.RingTheory.Ideal.Norm.AbsNorm Mathlib.RingTheory.Ideal.Norm.RelNorm Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients Mathlib.RingTheory.Ideal.UFD Mathlib.RingTheory.IntegralClosure.IntegralRestrict Mathlib.RingTheory.IntegralDomain Mathlib.RingTheory.Invariant.Galois Mathlib.RingTheory.Jacobson.Artinian Mathlib.RingTheory.Kaehler.Basic Mathlib.RingTheory.Kaehler.JacobiZariski Mathlib.RingTheory.Kaehler.Polynomial Mathlib.RingTheory.Kaehler.TensorProduct Mathlib.RingTheory.KrullDimension.Polynomial Mathlib.RingTheory.KrullDimension.Regular Mathlib.RingTheory.LaurentSeries Mathlib.RingTheory.LinearDisjoint Mathlib.RingTheory.LittleWedderburn Mathlib.RingTheory.LocalProperties.Semilocal Mathlib.RingTheory.LocalRing.Etale Mathlib.RingTheory.LocalRing.Module Mathlib.RingTheory.LocalRing.ResidueField.Polynomial Mathlib.RingTheory.Localization.AtPrime.Extension Mathlib.RingTheory.Localization.NormTrace Mathlib.RingTheory.MvPowerSeries.Equiv Mathlib.RingTheory.Nilpotent.GeometricallyReduced Mathlib.RingTheory.Norm.Basic Mathlib.RingTheory.Norm.Defs Mathlib.RingTheory.Norm.Transitivity Mathlib.RingTheory.NormTrace Mathlib.RingTheory.NormalClosure Mathlib.RingTheory.Nullstellensatz Mathlib.RingTheory.OrderOfVanishing.Basic Mathlib.RingTheory.OrderOfVanishing.Noetherian Mathlib.RingTheory.Perfection Mathlib.RingTheory.Perfectoid.BDeRham Mathlib.RingTheory.Perfectoid.FontaineTheta Mathlib.RingTheory.Perfectoid.Untilt Mathlib.RingTheory.PicardGroup Mathlib.RingTheory.Polynomial.Cyclotomic.Basic Mathlib.RingTheory.Polynomial.Cyclotomic.Eval Mathlib.RingTheory.Polynomial.Cyclotomic.Expand Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization Mathlib.RingTheory.Polynomial.Cyclotomic.Roots Mathlib.RingTheory.Polynomial.DegreeLT Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral Mathlib.RingTheory.Polynomial.Resultant.Basic Mathlib.RingTheory.Polynomial.Selmer Mathlib.RingTheory.Polynomial.UniversalFactorizationRing Mathlib.RingTheory.QuasiFinite.Basic Mathlib.RingTheory.QuasiFinite.Polynomial Mathlib.RingTheory.QuasiFinite.Weakly Mathlib.RingTheory.RamificationInertia.Basic Mathlib.RingTheory.RamificationInertia.Inertia Mathlib.RingTheory.RamificationInertia.Ramification Mathlib.RingTheory.Regular.Depth Mathlib.RingTheory.Regular.Flat Mathlib.RingTheory.Regular.Free Mathlib.RingTheory.Regular.IsSMulRegular Mathlib.RingTheory.Regular.LinearMap Mathlib.RingTheory.Regular.ProjectiveDimension Mathlib.RingTheory.Regular.RegularSequence Mathlib.RingTheory.RegularLocalRing.Defs Mathlib.RingTheory.RegularLocalRing.Polynomial Mathlib.RingTheory.RingHom.Etale Mathlib.RingTheory.RingHom.LocallyStandardSmooth Mathlib.RingTheory.RingHom.PurelyInseparable Mathlib.RingTheory.RingHom.QuasiFinite Mathlib.RingTheory.RingHom.Smooth Mathlib.RingTheory.RingHom.StandardSmooth Mathlib.RingTheory.RingHom.Unramified Mathlib.RingTheory.RootsOfUnity.AlgebraicallyClosed Mathlib.RingTheory.RootsOfUnity.Basic Mathlib.RingTheory.RootsOfUnity.Complex Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity Mathlib.RingTheory.RootsOfUnity.Lemmas Mathlib.RingTheory.RootsOfUnity.Minpoly Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots Mathlib.RingTheory.SimpleModule.IsAlgClosed Mathlib.RingTheory.Smooth.AdicCompletion Mathlib.RingTheory.Smooth.Basic Mathlib.RingTheory.Smooth.Fiber Mathlib.RingTheory.Smooth.Field Mathlib.RingTheory.Smooth.Flat Mathlib.RingTheory.Smooth.IntegralClosure Mathlib.RingTheory.Smooth.Kaehler Mathlib.RingTheory.Smooth.Local Mathlib.RingTheory.Smooth.Locus Mathlib.RingTheory.Smooth.NoetherianDescent Mathlib.RingTheory.Smooth.Pi Mathlib.RingTheory.Smooth.Quotient Mathlib.RingTheory.Smooth.StandardSmoothCotangent Mathlib.RingTheory.Smooth.StandardSmoothOfFree Mathlib.RingTheory.Smooth.StandardSmooth Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity Mathlib.RingTheory.Spectrum.Prime.Chevalley Mathlib.RingTheory.Spectrum.Prime.FreeLocus Mathlib.RingTheory.Spectrum.Prime.Homeomorph Mathlib.RingTheory.Spectrum.Prime.LTSeries Mathlib.RingTheory.Spectrum.Prime.Polynomial Mathlib.RingTheory.Teichmuller Mathlib.RingTheory.TensorProduct.IsBaseChangeHom Mathlib.RingTheory.TotallySplit Mathlib.RingTheory.Trace.Basic Mathlib.RingTheory.Trace.Defs Mathlib.RingTheory.Trace.Quotient Mathlib.RingTheory.Unramified.Basic Mathlib.RingTheory.Unramified.Dedekind Mathlib.RingTheory.Unramified.Field Mathlib.RingTheory.Unramified.Finite Mathlib.RingTheory.Unramified.LocalRing Mathlib.RingTheory.Unramified.LocalStructure Mathlib.RingTheory.Unramified.Locus Mathlib.RingTheory.Unramified.Pi Mathlib.RingTheory.Valuation.Archimedean Mathlib.RingTheory.Valuation.Discrete.Basic Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing Mathlib.RingTheory.Valuation.Discrete.RankOne Mathlib.RingTheory.Valuation.DiscreteValuativeRel Mathlib.RingTheory.Valuation.FiniteField Mathlib.RingTheory.WittVector.Basic Mathlib.RingTheory.WittVector.Compare Mathlib.RingTheory.WittVector.Complete Mathlib.RingTheory.WittVector.Defs Mathlib.RingTheory.WittVector.DiscreteValuationRing Mathlib.RingTheory.WittVector.Domain Mathlib.RingTheory.WittVector.FrobeniusFractionField Mathlib.RingTheory.WittVector.Frobenius Mathlib.RingTheory.WittVector.Identities Mathlib.RingTheory.WittVector.InitTail Mathlib.RingTheory.WittVector.IsPoly Mathlib.RingTheory.WittVector.Isocrystal Mathlib.RingTheory.WittVector.MulCoeff Mathlib.RingTheory.WittVector.MulP Mathlib.RingTheory.WittVector.StructurePolynomial Mathlib.RingTheory.WittVector.TeichmullerSeries Mathlib.RingTheory.WittVector.Teichmuller Mathlib.RingTheory.WittVector.Truncated Mathlib.RingTheory.WittVector.Verschiebung Mathlib.RingTheory.ZMod.Torsion Mathlib.RingTheory.ZMod.UnitsCyclic Mathlib.RingTheory.ZariskisMainTheorem Mathlib.Tactic.NormNum.LegendreSymbol Mathlib.Tactic Mathlib.Topology.Algebra.Group.Matrix Mathlib.Topology.Algebra.Module.Determinant Mathlib.Topology.Algebra.Module.FiniteDimensionBilinear Mathlib.Topology.Algebra.Module.FiniteDimension Mathlib.Topology.Algebra.Module.TopDualPairing Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete Mathlib.Topology.Algebra.Ring.Compact Mathlib.Topology.Algebra.Valued.LocallyCompact Mathlib.Topology.Algebra.Valued.WithVal Mathlib.Topology.Algebra.Valued.WithZeroMulInt Mathlib.Topology.CWComplex.Abstract.Basic Mathlib.Topology.Category.TopCat.Sphere Mathlib.Topology.Compactification.OnePoint.ProjectiveLine Mathlib.Topology.Compactification.OnePoint.Sphere Mathlib.Topology.MetricSpace.HausdorffDimension

1

---

Declarations diff (regex)

+ AddSubgroup.index_zmultiples_zsmul
+ AddSubgroup.zmultiples_le_zmultiples_iff
+ Int.zmultiples_le_zmultiples_iff
+ IsCyclic.infinite_of_ne_bot
+ IsCyclic.subgroup_eq_subgroup_iff
+ IsCyclic.subgroup_le_iff_index_dvd
+ IsCyclic.subgroup_le_subgroup_iff
+ Subgroup.exists_zpowers_eq_of_zpowers_eq_top
+ Subgroup.index_zpowers_zpow
+ Subgroup.zpowers_eq_zpowers_iff'
+ Subgroup.zpowers_le_zpowers_iff
+ Subgroup.zpowers_le_zpowers_of_dvd
+ Subgroup.zpowers_zpow_sup
+ _root_.Subgroup.map_top
+ dvd_antisymm_iff
+ index_eq_card_div
+ instance [Group G] [IsCyclic G] {H : Subgroup G} [h : Nontrivial H] : Infinite H
+ orderOf_pow_natAbs
+ orderOf_zpow
+ orderOf_zpow'
+ zpowers_mul_le_sup

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 4ae95a3).

• +36 new declarations
• −0 removed declarations

+AddSubgroup.exists_zmultiples_eq_of_zmultiples_eq_top
+AddSubgroup.index_eq_card_sub
+AddSubgroup.index_zmultiples_zsmul
+AddSubgroup.map_top
+AddSubgroup.zmultiples_add_le_sup
+AddSubgroup.zmultiples_eq_zmultiples_iff'
+AddSubgroup.zmultiples_le_zmultiples_iff
+AddSubgroup.zmultiples_le_zmultiples_of_dvd
+AddSubgroup.zmultiples_zsmul_sup
+Int.zmultiples_le_zmultiples_iff
+IsAddCyclic.addSubgroup_eq_addSubgroup_iff
+IsAddCyclic.addSubgroup_le_addSubgroup_iff
+IsAddCyclic.addSubgroup_le_iff_index_dvd
+IsAddCyclic.infinite_of_ne_bot
+IsCyclic.infinite_of_ne_bot
+IsCyclic.subgroup_eq_subgroup_iff
+IsCyclic.subgroup_le_iff_index_dvd
+IsCyclic.subgroup_le_subgroup_iff
+Nat.dvd_antisymm_iff
+Subgroup.exists_zpowers_eq_of_zpowers_eq_top
+Subgroup.index_eq_card_div
+Subgroup.index_zpowers_zpow
+Subgroup.map_top
+Subgroup.zpowers_eq_zpowers_iff'
+Subgroup.zpowers_le_zpowers_iff
+Subgroup.zpowers_le_zpowers_of_dvd
+Subgroup.zpowers_mul_le_sup
+Subgroup.zpowers_zpow_sup
+addOrderOf_nsmul_natAbs
+addOrderOf_zsmul
+addOrderOf_zsmul'
+instInfiniteSubtypeMemAddSubgroupOfIsAddCyclicOfNontrivial
+instInfiniteSubtypeMemSubgroupOfIsCyclicOfNontrivial
+orderOf_pow_natAbs
+orderOf_zpow
+orderOf_zpow'

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 4ae95a3c8f
Reference commit caa4dbde35

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).

[mathlib-merge-conflicts]

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

[mathlib-merge-conflicts]

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

[tb65536]

splice-bot maintainer merge

[mathlib-splicebot]

Split PR created and commented

Split off the changes to Mathlib/Data/Nat/GCD/Basic.lean in #42712 and posted a comment via splice-bot command maintainer.

[xroblot]

@tb65536 Yeah, this PR is getting a bit longer than I expected. I could split it if that helps

[tb65536]

Was mostly just annoyed about the build time to get to the cyclic file, but if you have a nice split, I wouldn't complain :)

[xroblot]

Well, let me see...

[xroblot]

@tb65536 I extracted a first PR #42754

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(GroupTheory/SpecificGroups/Cyclic): comparison and index of subgroups generated by powers #42754
  By Dependent Issues (🤖). Happy coding!