feat(ModelTheory): Semilinear sets in ℕ are closed under intersection and complement

[staroperator]

Separated from #27100.

---

• depends on: feat(Order/WellFoundedSet): add Set.IsPWO.exists_le_minimal, Set.IsPWO.exists_minimal, Set.IsPWO.exists_minimalFor #27082
• depends on: [Merged by Bors] - feat(LinearAlgebra/Matrix): add LinearMap.toMatrix'_mulVec #27086
• depends on: [Merged by Bors] - feat(Algebra/Order/Sub): add map_tsub_of_le #27088
• depends on: [Merged by Bors] - refactor(LinearAlgebra/LinearIndependent): generalize some LinearIndepOn theorems #27096
• depends on: [Merged by Bors] - feat(LinearAlgebra/Finsupp/LinearCombination): add Submodule.mem_span_finset' #27097
• depends on: feat(ModelTheory): define linear and semilinear sets #27342

---

[github-actions]

PR summary 0a739db8b7

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
11 files Mathlib.Algebra.SkewMonoidAlgebra.Basic Mathlib.LinearAlgebra.Basis.Basic Mathlib.LinearAlgebra.Basis.Fin Mathlib.LinearAlgebra.Basis.Flag Mathlib.LinearAlgebra.Basis.Prod Mathlib.LinearAlgebra.Basis.SMul Mathlib.LinearAlgebra.Basis.Submodule Mathlib.LinearAlgebra.FreeModule.Basic Mathlib.LinearAlgebra.LinearIndependent.Basic Mathlib.LinearAlgebra.LinearIndependent.Defs Mathlib.RingTheory.AlgebraTower	1
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic (new file)	978
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Nat (new file)	1210

---

Declarations diff

+ Finite.semilinear
+ IsPWO.exists_le_minimal
+ IsPWO.exists_minimal
+ IsPWO.exists_minimalFor
+ Linear
+ Linear.add
+ Linear.iff_eq_setOf_vadd_mulVec
+ Linear.image
+ Linear.inter
+ Linear.of_homogeneous_equation
+ Linear.preimage
+ Linear.proper_semilinear
+ Linear.semilinear
+ Linear.singleton
+ Linear.span
+ Linear.span_finset
+ Linear.univ
+ Linear.vadd
+ ProperLinear
+ ProperLinear.linear
+ ProperLinear.proper_semilinear
+ ProperSemilinear
+ ProperSemilinear.biUnion_finset
+ ProperSemilinear.empty
+ ProperSemilinear.iUnion_fintype
+ ProperSemilinear.sUnion_finset
+ ProperSemilinear.semilinear
+ ProperSemilinear.union
+ Semilinear
+ Semilinear.add
+ Semilinear.biInter_finset
+ Semilinear.biUnion_finset
+ Semilinear.compl
+ Semilinear.empty
+ Semilinear.finset
+ Semilinear.iInter_fintype
+ Semilinear.iUnion_fintype
+ Semilinear.image
+ Semilinear.inter
+ Semilinear.of_linear_equation
+ Semilinear.preimage
+ Semilinear.proj
+ Semilinear.proj'
+ Semilinear.proper_semilinear
+ Semilinear.reindex
+ Semilinear.sInter_finset
+ Semilinear.sUnion_finset
+ Semilinear.singleton
+ Semilinear.span
+ Semilinear.span_finset
+ Semilinear.union
+ Semilinear.univ
+ Semilinear.vadd
+ compl
+ linearIndepOn_finset_iffₒ
+ linearIndependent_iffₒ
+ not_linearIndepOn_finset_iffₒ
+ not_linearIndependent_iffₒ

You can run this locally as follows

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

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

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

---

Increase in tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type
818	2	erw

Current commit e38912cceb
Reference commit 0a739db8b7

You can run this locally as

./scripts/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).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• feat(Order/WellFoundedSet): add Set.IsPWO.exists_le_minimal, Set.IsPWO.exists_minimal, Set.IsPWO.exists_minimalFor #27082
• [Merged by Bors] - feat(LinearAlgebra/Matrix): add LinearMap.toMatrix'_mulVec #27086
• [Merged by Bors] - feat(Algebra/Order/Sub): add map_tsub_of_le #27088
• [Merged by Bors] - refactor(LinearAlgebra/LinearIndependent): generalize some LinearIndepOn theorems #27096
• [Merged by Bors] - feat(LinearAlgebra/Finsupp/LinearCombination): add Submodule.mem_span_finset' #27097
• feat(ModelTheory): define linear and semilinear sets #27342
  By Dependent Issues (🤖). Happy coding!