chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
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values | text stringlengths 40 24k |
|---|---|---|---|---|---|
9,566 | lean_file_chunk | 7d3c046b65bc42da55b487626df5f9e300e44dd782fd10b6e967946f300b1860 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.EssentialFiniteness
public import Mathlib.RingTheory.LocalRing.ResidueField.Basic
public import Mathlib.RingTheory.Localization.Fra... |
1,052 | lean_file_chunk | ce5d4e247466f16ac488f9e576a0d1f939349efa45ce87d6df79390e7d943b4e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
module
public import Mathlib.RingTheory.Ideal.Quotient.Basic
public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
/-!
# Residue Field ... |
7,156 | lean_file_chunk | 4eb5cbf64cce0538eaa788054a062b1d698de8ae8530011423f3989b873c6fd0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.FieldTheory.RatFunc.AsPolynomial
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
/-!
# Residue field of primes in polynomial al... |
6,104 | lean_file_chunk | ddd19d7ff4cdf3187b3ee44b18223f9beb754f3344aa4d17fc3f2ab1f87531a3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
module
public import Mathlib.Algebra.Group.Units.Hom
public import Mathlib.Data.ZMod.Basic
public import Mathlib.RingTheory.LocalRing.MaximalIdea... |
4,634 | lean_file_chunk | 540b35010f93c6ade4f6bda692c48147460303a1ce9b8418e49768f4354cedf1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
module
public import Mathlib.RingTheory.Jacobson.Ideal
public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Defs
public import Mathlib.RingThe... |
849 | lean_file_chunk | 92132cb1a72072564ee1cfef59159158cbc4865ce88681abaa8dddd92c8d79f9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
module
public import Mathlib.RingTheory.LocalRing.Basic
/-!
# Maximal ideal of local rings
We define the maximal ideal of a local ring as the ... |
23,932 | lean_file_chunk | 8096d3c16c799fd1b1aa1cff7032f343d4a8e3c740c86ab3fecec59760267214 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Kaehler.Polynomial
public import Mathlib.Algebra.Module.FinitePresentation
public import Mathlib.RingTheory.Extension.Presentation.... |
3,082 | lean_file_chunk | 168e977269966d658124142242632dc54279243b92f3d17f5dca3a74df04c07b | lean_source_file | pkg_mathlib_source_file_chunks | abbrev H1Cotangent : Type _ := (Generators.self R S).toExtension.H1Cotangent
/-- The induced map on the first homology of the (naive) cotangent complex of `S` over `R`. -/
noncomputable
def H1Cotangent.map : H1Cotangent R S' →ₗ[S'] H1Cotangent S T :=
Extension.H1Cotangent.map (Generators.defaultHom _ _).toExtensionH... |
11,636 | lean_file_chunk | 5999d6ab41a5c7909560463c3cf00673273967b90980fb60bbfca8f1913956ee | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.RingTheory.Extension.Presentation.Basic
public import Mathlib.RingTheory.Smooth.StandardSmoothCotangent
public import Mathlib.RingTh... |
16,319 | lean_file_chunk | f803c952cb0036dad1120ef9eeec5f1f45a83904f435742d04360c326ec1b63e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.RingTheory.Extension.Cotangent.Basic
public import Mathlib.RingTheory.Smooth.StandardSmoothCotangent
public import Mathlib.RingTheor... |
9,829 | lean_file_chunk | c883a56aa73fdbdf990070731a0f12bab4fe104da2a5ce32a0bf1e404bab80ee | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.RingTheory.Ideal.CotangentBaseChange
public import Mathlib.RingTheory.Extension.Cotangent.Basic
public import Mathlib.Algebra.FiveLe... |
6,066 | lean_file_chunk | e4f0390be83b2a93e57a60eb74626eefd31a3d8d1d2c6a3f5046fc783364eeae | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.LinearAlgebra.Basis.Exact
public import Mathlib.RingTheory.Extension.Cotangent.Basic
public import Mathlib.RingTheory.Extension.Pres... |
23,976 | lean_file_chunk | 0cc6da82dd2df0b6c6744a4281a91215ff825301b9de7860a6d1ddc2d5d10cf0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jung Tao Cheng, Christian Merten, Andrew Yang
-/
module
public import Mathlib.Algebra.MvPolynomial.PDeriv
public import Mathlib.LinearAlgebra.Determinant
public import Mathlib.RingT... |
758 | lean_file_chunk | 3f38ea1f73f848a6997353d81b5baca1f5756ac6caf5831c6057db8515c1bef6 | lean_source_file | pkg_mathlib_source_file_chunks | P.aevalDifferentialEquiv x = P.aevalDifferential x :=
rfl
/-- If `P` is a submersive presentation, the partial derivatives of `P.relation i` by
`P.map j` form a basis of `σ → S`. -/
noncomputable def basisDeriv (P : SubmersivePresentation R S ι σ) : Basis σ S (σ → S) :=
Basis.map (Pi.basisFun S σ) P.aevalDifferent... |
15,635 | lean_file_chunk | 40d384af43dac37e7ef19a8ff13e7a7a58d9bc85387881a870de92c1b2886ab3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.RingTheory.Extension.Presentation.Submersive
/-!
# Presentations on subrings
In this file we establish the API for realising a pre... |
22,588 | lean_file_chunk | 538a42adf00064b630bce0af67107279c0b81a6481f35806752a059a19947f9a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jung Tao Cheng, Christian Merten, Andrew Yang
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.RingTheory.FinitePresentation
public i... |
3,854 | lean_file_chunk | 9e4b65371586b6d076f0e0180e4a61c48f251777c415112b02393f7d3ff6d185 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
public import Mathlib.RingTheory.GradedAlgebra.RingHom
/-!
# Maps on homogeneous ideals
In this file ... |
5,004 | lean_file_chunk | 6f25d9ae27e9429c83667bf855f73b356dca3d83f15229d19c471f5d7753dbe0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Fangming Li
-/
module
public import Mathlib.Algebra.DirectSum.Decomposition
public import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Homogeneous subsemirings of a grade... |
22,132 | lean_file_chunk | b2713242edf9038c1eae6bbfad548a514a3e365fab54a4808537b10b94692941 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.Finsupp.SumProd
public import Mathlib.RingTheory.GradedAlgebra.Basic
public import Mathlib.RingTheory.Ideal.Basic... |
4,619 | lean_file_chunk | 77648e4e5bcb6b1afc02c15fffc7ea07a8e1a384aadba46bcf18df69a67adde5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
module
public import Mathlib.RingTheory.GradedAlgebra.Basic
public import Mathlib.Algebra.GradedMulAction
/-!
# Homogeneous submodules of a graded module
... |
1,237 | lean_file_chunk | 9daf6ef9c935954cb474e1503e1faa07bbbe94670c0cb231f8787a74d0afe661 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
module
public import Mathlib.RingTheory.MvPolynomial.MonomialOrder
public import Mathlib.Data.Finsupp.MonomialOrder.DegLex
/-! # Some lemmas about the... |
16,395 | lean_file_chunk | 0da5e21d6ce29fd1dfcf309003f5c3fec4dacb799d25ee88e4686d8bafad3ed7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
public import Mathlib.RingTheory.MvPolynomial.Tower
public import Mathlib.Data.Finsupp.Notation
public impo... |
12,258 | lean_file_chunk | 7f3d8427289c459c1cb1a70951cba04dd22d1e7a4d0640873868ec470f56fff1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Michael Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Lee
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Basic
public import Mathlib.Algebra.MvPolynomial.CommRing
public import Mathlib.Algebra.MvPolynomial.Rename
public ... |
14,898 | lean_file_chunk | df3b4368084aff1bf5817bfba112aa925c32d7d14419eabc408318e62b714862 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang, Johan Commelin
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Basic
public import Mathlib.Algebra.MvPolynomial.CommRing
public import Mathlib.Combinatorics.E... |
4,593 | lean_file_chunk | c8810a2226bb74838ef5e4f31c74e57cf1476d483a8ffe48cd3bbc09edc7bd35 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.RingTheory.Finiteness.Ideal
public import Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian
public import Mathlib.RingTheory.Noetheria... |
7,249 | lean_file_chunk | 1a399ddba0bbe6d2b34345dfde220b9dee3d51a777b45a67aa6e93c5a260020e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Ideal.IsPrimary
public import Mathlib.RingTheory.Ideal.Over
public import Mathlib.Order.Minimal
/-!
# Minimal primes
We provide ... |
10,327 | lean_file_chunk | b17726d235e38a1b8ecc186d0b0f00981467a9770b95ab203532b9d90107fa54 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Ideal.MinimalPrime.Basic
public import Mathlib.RingTheory.Localization.AtPrime.Basic
/-!
# Minimal primes and localization
We pr... |
1,655 | lean_file_chunk | 6bcb532bcab52963366f9d8422ec410bbdb768aa66e50ac9210599ec1a29e915 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.RingTheory.Ideal.MinimalPrime.Basic
public import Mathlib.RingTheory.Noetherian.Defs
/-!
# Finiteness of minimal primes
We prove f... |
12,022 | lean_file_chunk | c4116525d4cd470c9f8d4619d039889dcad6c4eccd7b0db47af9da553eada284 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Algebra.Exact
public import Mathlib.RingTheory.Finiteness.Ideal
public import Mathlib.RingTheory.Ideal.MinimalPrime.Colon
public import Mathli... |
6,870 | lean_file_chunk | 6adc67e6c12048801ab65254895c189d414260ef0d14bc5992eff97149430e99 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
module
public import Mathlib.Algebra.Module.LocalizedModule.AtPrime
public import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
public import Mathlib.RingTheory.Support
... |
10,968 | lean_file_chunk | 924b17202e0dc0de7458ea181efd007517b4b0e314197d13a861c402c5baa216 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jinzhao Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jinzhao Pan
-/
module
public import Mathlib.Order.RelSeries
public import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
public import Mathlib.RingTheory.Localization.FractionRing
public... |
21,400 | lean_file_chunk | 99bec7354b1516f73c746b50e3c6d2bb6350cc9f87df6a52d6778b6e737fdfda | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
module
public import Mathlib.Algebra.GroupWithZero.Torsion
public import Mathlib.NumberTheory.RamificationInertia.Galois
public import Mathlib.RingTheory.Ded... |
18,244 | lean_file_chunk | 2fd3aab21b3e591a0ebffcab6460491de57894d6cde48f29e4d52a61faffd6f0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
module
public import Mathlib.Algebra.CharP.Quotient
public import Mathlib.LinearAlgebra.FreeModule.Determinant
public import Mathlib.LinearAlgebra.FreeModule... |
3,439 | lean_file_chunk | c7e366b03ee08a88d7d33326d75ba48da120cdbcded792939bdcfce449b71325 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
module
public import Mathlib.RingTheory.Ideal.Quotient.Operations
public import Mathlib.RingTheory.Nilpotent.Lemmas
/-!
# Nilpotent elements in quotient rings
-/
public ... |
8,715 | lean_file_chunk | 7227cf6ffe09fde5a78d4fee8e95191de53650273842f64ce3a95a085ffd276e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jiedong Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan, Jiedong Jiang
-/
module
public import Mathlib.LinearAlgebra.Quotient.Basic
public import Mathlib.RingTheory.Ideal.Quotient.Defs
public import Mathlib.Algebra.Algebra.Opera... |
5,271 | lean_file_chunk | 42aef4fa08976a63bf738d87ae9a2810dc5d0d8fc608ffc71158e8bdd4793dd7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Algebra.Ring.GeomSum
public import Mathlib.Data.Finsupp.Fintype
public import Mathlib.GroupTheory.Index
public import Mathlib.LinearAlgebra.Di... |
7,142 | lean_file_chunk | b5459410c2b63b17a3666540c5901eef25e498a1bdd398610d171994c2775451 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.Data.ZMod.QuotientRing
public import Mathlib.NumberTheory.Height.Northcott
public import Mathlib.RingTheory.DedekindDomain.Basic
public im... |
9,096 | lean_file_chunk | 6a699de1130a519c0874f680e48dbda0eb9fba3c35282dae634a7a7cd736bf36 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
module
public import Mathlib.GroupTheory.QuotientGroup.Finite
public import Mathlib.LinearAlgebra.Quotient.Basic
public import Mathl... |
23,994 | lean_file_chunk | 89cd48aa1804a14afff268b93adf9abe0d38519857d01b838ba5538dc5c6846a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Operations
public import Mathlib.Algebra.Ring.Fin
public import Mathlib.LinearAlgebra.Quotient.Basic
pu... |
22,548 | lean_file_chunk | 396ea869358f005521c518f338bc0b3169445f095d58ead4a6efcdff8d939e86 | lean_source_file | pkg_mathlib_source_file_chunks | @[simp]
theorem quotientMap_mk {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided]
{f : R →+* S} {H : J ≤ I.comap f} {x : R} :
quotientMap I f H (Quotient.mk J x) = Quotient.mk I (f x) :=
Quotient.lift_mk J _ _
@[simp]
theorem quotientMap_algebraMap {J : Ideal A} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSi... |
7,414 | lean_file_chunk | 800c77912d9affd330824d331b08c1dced220802c6d5c6afa743d9e81e4b3796 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
module
public import Mathlib.LinearAlgebra.Quotient.Defs
public import Mathlib.RingTheory.Congruence.Defs
public import Mathlib.Ring... |
2,302 | lean_file_chunk | 1ac1605c7b9d03ab06b5174d7ea870447740ed8b22748939e0127f6b9e6d538f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Pi
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.RingTheory.Ideal.Quotient.Op... |
554 | lean_file_chunk | 71fcc7f0810addec3d8328d39b7c57593e947a6c86b4bb9c93c865a6c78bd5b7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.RingTheory.Ideal.Quotient.Operations
public import Mathlib.RingTheory.Noetherian.Basic
/-!
# Noetherian quotient rings and quotient modules
-... |
2,653 | lean_file_chunk | 19d7947abd8c06966156fc15aab6ea651684271d84f9c0dfaa7fe7d3bc8c31b6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Xavier Généreux, María Inés de Frutos Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, María Inés de Frutos Fernández
-/
module
public import Mathlib.Algebra.Polynomial.Bivariate
public import Mathlib.Algebra.Ring.Defs
... |
2,949 | lean_file_chunk | cf765b7eebf599423659c19c20962737c116d2eaa15c4e35352334dab2e97644 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
/-!
# Polynomials and adjoining roots
## Main results
* `Alge... |
23,937 | lean_file_chunk | 1de8e1d333ed1650a6c5013464f0ac2da725ed8bf4eeb6d1e3d9d9e0eb690a0b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies, Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Andrew Yang
-/
module
public import Mathlib.Algebra.Order.SuccPred.WithBot
public import Mathlib.Algebra.Polynomial.CoeffMem
public import Mathlib.Data.DFinsu... |
19,331 | lean_file_chunk | 5bc06584c797f82d36ae58c6bb2a7e37cb8234c9a55136ef5e213b3e8fab585a | lean_source_file | pkg_mathlib_source_file_chunks | suffices hij : c'.coeffSubmodule R₀ ≤ c.coeffSubmodule R₀ ^ (c.val j).degree.succ by
by_cases hi' : c.val i = 1
· gcongr
· exact c.one_le_coeffSubmodule
· refine Submodule.span_le.mpr (Set.union_subset ?_ ?_)
· exact Set.subset_union_left.trans Submodule.subset_span
... |
4,369 | lean_file_chunk | 508c601efe8e2805bb9621d5b469d770daea2d7e9ea9dde0b2a07d240a36d0f5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.FiniteStability
public import Mathlib.RingTheory.Ideal.GoingDown
public import Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexit... |
4,397 | lean_file_chunk | c305c9998e238bd3b6891989c6e3939fe266cf32a19991b22e6435138e175adb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten, Junyan Xu
-/
module
public import Mathlib.FieldTheory.PurelyInseparable.Basic
public import Mathlib.RingTheory.Flat.Basic
public import Mathlib.RingTheory.Spectrum... |
2,052 | lean_file_chunk | 9a6edf2f2b4f4d27fe26b8b682c63a41d92b223a567c9f147a003c2b3522b348 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Spectrum.Prime.Topology
public import Mathlib.RingTheory.Support
/-!
# Subsets of prime spectra related to modules
## Main resul... |
5,185 | lean_file_chunk | e5c60c301ad15780aea249c244a2cbd3b3aa0ae15c5ccf5c3913b3bf77cd581b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yongle Hu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yongle Hu
-/
module
public import Mathlib.RingTheory.Ideal.KrullsHeightTheorem
/-!
# Lemmas about `LTSeries` in the prime spectrum
## Main results
* `PrimeSpectrum.exist_ltSeries_mem_on... |
3,730 | lean_file_chunk | 5c975646f463add3d61f5cc020f0be6118bc57c83a4c54b5680de1ae67145880 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Spectrum.Prime.Topology
public import Mathlib.RingTheory.SurjectiveOnStalks
/-!
# Lemmas regarding the prime spectrum of tensor p... |
23,953 | lean_file_chunk | f838f97f2ea6b79032c66dc628bbee4e089b79e4d56950ca0116970101b5a82d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
module
public import Mathlib.Algebra.Order.Ring.Idempotent
public import Mathlib.Order.Heyting.Hom
public import Mathlib.RingTheory.Finiteness.Ideal
public import Ma... |
23,915 | lean_file_chunk | f329b7902975d39acc7e2c0db9af713d565cbdb99ae9dd791fa6f170d2d2f96c | lean_source_file | pkg_mathlib_source_file_chunks | theorem basicOpen_zero : basicOpen (0 : R) = ⊥ :=
TopologicalSpace.Opens.ext <| by simp
theorem basicOpen_le_basicOpen_iff (f g : R) :
basicOpen f ≤ basicOpen g ↔ f ∈ (Ideal.span ({g} : Set R)).radical := by
rw [← SetLike.coe_subset_coe, basicOpen_eq_zeroLocus_compl, basicOpen_eq_zeroLocus_compl,
Set.compl... |
18,155 | lean_file_chunk | f087f703210970bb9ae08f887c407143b565f1e45b357125ccaad20c0be11fee | lean_source_file | pkg_mathlib_source_file_chunks | obtain ⟨n, hn⟩ := Ideal.exists_pow_le_of_le_radical_of_fg this (Submodule.FG.mul hI hJ)
have hnz : n ≠ 0 := by rintro rfl; simp at hn
rw [mul_pow, Ideal.zero_eq_bot] at hn
have : I ^ n ⊔ J ^ n = ⊤ := by
rw [eq_top_iff, ← Ideal.span_pow_eq_top (I ∪ J : Set R) _ n, Ideal.span_le, Set.image_union,
Set.unio... |
19,465 | lean_file_chunk | 35fcdf3e97541a2bb9aee813f90fad8f402b3ab2405e9deedd1159f8fee5f816 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Filippo A. E. Nuccio, Andrew Yang
-/
module
public import Mathlib.RingTheory.Ideal.MinimalPrime.Basic
public import Mathlib.RingTheory.Nilpotent.Lemmas
public import M... |
14,091 | lean_file_chunk | 3325a80a173c0bb6073a221ca56926a0a2f69df32355431578712ba68af2f5fd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Filippo A. E. Nuccio, Andrew Yang
-/
module
public import Mathlib.RingTheory.Spectrum.Prime.Basic
public import Mathlib.RingTheory.LocalRing.ResidueField.Ideal
public ... |
2,166 | lean_file_chunk | 7c329b9f67013b22895d782590cbde6c1fd5fbf35f2de0a52260d767e25e77ef | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Filippo A. E. Nuccio, Andrew Yang
-/
module
public import Mathlib.RingTheory.Ideal.Prime
/-!
# Prime spectrum of a commutative (semi)ring as a type
The prime spectru... |
18,376 | lean_file_chunk | 47f858d492b88e1fffcd966456e9a733c74516a8b4f2fcb33540fcbe2066b7de | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Flat.Stability
public import Mathlib.RingTheory.LocalProperties.Projective
public import Mathlib.RingTheory.LocalRing.Module
public... |
7,369 | lean_file_chunk | 539c930f2e424a3a5272abae9145657ca9f466c59556b53867601c318706e8d9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies, Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Andrew Yang
-/
module
public import Mathlib.Order.SuccPred.WithBot
public import Mathlib.RingTheory.Spectrum.Prime.Topology
/-!
# Constructible sets in the p... |
10,686 | lean_file_chunk | fec2f5137c70bc572a0b012343f1a1fa1f8567de06a5b2e33aca849696481cd2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.LinearAlgebra.Charpoly.BaseChange
public import Mathlib.LinearAlgebra.Eigenspace.Zero
public import Mathlib.RingTheory.AdjoinRoot
public impor... |
4,485 | lean_file_chunk | 1b6c0c32f4e090211644cab05dfe50a74a2c60352f925f368baced70dc633d73 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Jacobson.Ring
public import Mathlib.RingTheory.Spectrum.Prime.Noetherian
public import Mathlib.Topology.JacobsonSpace
/-!
# The pr... |
2,602 | lean_file_chunk | b27c4d3a321ed2439594581a59fd854814537aff5cb1f2048560a37f8ff56623 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Andrew Yang
-/
module
public import Mathlib.RingTheory.Artinian.Module
public import Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian
public import Mathlib... |
2,886 | lean_file_chunk | 339a0fbe7ea32f5bf238d447be8c325977214a83c63b587428f6b1089367ead7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
module
public import Mathlib.RingTheory.Polynomial.Basic
public import Mathlib.RingTheory.Spectrum.Prime.Topology
/-!
The morphism `Spec R[x] --> Spec R` induced by t... |
1,672 | lean_file_chunk | 6aa31b4730e16d66baf947267f3382cd1792523869948d550235d70043c0ce0b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
module
public import Mathlib.RingTheory.Spectrum.Maximal.Basic
public import Mathlib.RingTheory.Spectrum.Prime.Topology
/-!
# The Zariski topolo... |
1,741 | lean_file_chunk | 35caf65935d38b63c8f232f1f0c109f8d6afb74e745bc82d23ccca0f90f44b17 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
module
public import Mathlib.RingTheory.Ideal.Operations
public import Mathlib.RingTheory.Spectrum.Maximal.Defs
public import Mathlib.RingTheory.... |
995 | lean_file_chunk | 9338dfcd018eda6921c66af306c9da5dbc5e07e7e6d5ef69dea4c22bde1d9689 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
module
public import Mathlib.RingTheory.Ideal.Maximal
/-!
# Maximal spectrum of a commutative (semi)ring
The maximal spectrum of a commutative ... |
11,825 | lean_file_chunk | 3bf61eb8c647634952abd709465d26a95da0094934582e3d549927c898a4b743 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
module
public import Mathlib.RingTheory.Localization.AsSubring
public import Mathlib.RingTheory.Spectrum.Maximal.Basic
public import Mathlib.Ring... |
23,989 | lean_file_chunk | 1dbe801bd68b9776103aa5e4db465166431063e6c0e3867df0001cb9a64cd199 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.Algebra.Polynomial.Roots
public import Mathlib.RingTheory.FiniteType
public import Mathlib.RingTheory.IntegralClosure.Algebra.Basic
public import ... |
5,030 | lean_file_chunk | b47fbddc9a6a440029d5c416cda23cb451a0aa502ea47dc98b5bdee3adbf5325 | lean_source_file | pkg_mathlib_source_file_chunks | theorem IsIntegral.tower_bot_of_field {R A B : Type*} [CommRing R] [Field A]
[Ring B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]
{x : A} (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
h.tower_bot (algebraMap A B).injective
theorem RingHom.isIntegralElem.of_comp ... |
1,058 | lean_file_chunk | 39b8ce9827ad8e4f36edafa427209d545901e0af9ae0042657ff0099c074e376 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs
/-!
# Integral closure as a characteristic predicate
## Main definitions
Let `R` be a `CommRing` and... |
10,546 | lean_file_chunk | 972c9303ee7d3d1cc93ded3d4b513296e44e71e43047de6b007864849e8b84d6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs
public import Mathlib.Algebra.Polynomial.Expand
public import Mathlib.RingTheory.Adjoin.Polynomial.Basi... |
1,741 | lean_file_chunk | 962115f5ee87ca450d0b5298f2c2c09e76d1fa88bcbb178e7142f9792e6822af | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.Algebra.Polynomial.Degree.Defs
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Tactic.Algebraize
/-!
# Integral closure ... |
5,744 | lean_file_chunk | 2572a56ba78ad3c82c662ba45102c23b461acde9322ed25d717c5a482fae69a3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.FieldTheory.Minpoly.Basic
public import Mathlib.RingTheory.IntegralClosure.Algebra.Basic
public import Mathlib.RingTheory.Localization.Fractio... |
11,059 | lean_file_chunk | c78352e1e5caf1d9c05f1c5f5a3b881e540ec2fd805d5c9199e0f03c119baf9d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
public import Mathlib.RingTheory.IntegralClosure.Algebra.Defs
public import Mathlib.RingTheory.IntegralClo... |
4,265 | lean_file_chunk | 5f8274a31076dcaf641976ae0fef63967cc6a2126fe86913f8e78b35a41d5767 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.IntegralClosure.Algebra.Basic
/-!
# Integrality over ideals
## Main results
- `Polynomial.exists_monic_aeval_eq_zero_forall_mem_... |
1,079 | lean_file_chunk | 6a2c22ba7ed6b1eb14089ce7231735a09dfce9800f52d984186e4cc78a0560e6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs
/-!
# Integral algebras
## Main definitions
Let `R` be a `CommRing` and let `A` be an R-algebra.
* ... |
23,989 | lean_file_chunk | 740598aa40ee50f3d9f63a19b679fa28ff4acea81e08b9f50cba53a0ebb65c2b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
module
public import Mathlib.Algebra.Polynomial.FieldDivision
public import Mathlib.Algebra.Squarefree.Basic
public import Mathlib... |
23,929 | lean_file_chunk | 0e5edef3a0c3bcc3411c1d6662f67d9855ec4ab5c6ac50dd50dd56c139b14255 | lean_source_file | pkg_mathlib_source_file_chunks | all_goals rw [← MaximalSpectrum.iInf_localization_eq_bot, Algebra.mem_iInf]
· exact fun hx ⟨v, hv⟩ => hx ((equivMaximalSpectrum hR).symm ⟨v, hv⟩)
· exact fun hx ⟨v, hv, hbot⟩ => hx ⟨v, hv.isMaximal hbot⟩
section RingEquiv
variable {R} {S : Type*} [CommRing S]
/-- A surjective ring homomorphism `f : R →+* S` indu... |
7,402 | lean_file_chunk | d3cf6d90774aa3d5727d46999ddbf0cd1e458e4adecc5457f297da2c1b1e4d6a | lean_source_file | pkg_mathlib_source_file_chunks | /-- The bijection `normalizedFactorsEquivSpanNormalizedFactors` between the set of prime
factors of `r` and the set of prime factors of the ideal `⟨r⟩` preserves multiplicities. See
`count_normalizedFactorsSpan_eq_count` for the version stated in terms of multisets `count`. -/
theorem emultiplicity_normalizedFactorsEqu... |
20,139 | lean_file_chunk | b4aa59cf74cafec4f542bcd15a0c3b85fc57436fc052dc336b5a9890f8bfa50c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
public import Mathlib.RingTheory.DedekindDomain.Basic
public imp... |
23,342 | lean_file_chunk | 0dd7cee61e014b291179ff88422b837abda3c4e926655c86c00dba95bc39bb42 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 María Inés de Frutos-Fernández, Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
module
public import Mathlib.Algebra.GroupWithZero.Range
public import Mathlib.Algebra.Or... |
5,873 | lean_file_chunk | 19ee95b9a9ee99157f9e675f0946b037479584feb66899429cc33f6dede17ac5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
module
public import Mathlib.RingTheory.Valuation.Discrete.Basic
public import Mathlib.RingTheory.Valuation.Ran... |
3,095 | lean_file_chunk | dd6b304ac2ba6c879653b0a6367248a36341ddc6598e46d9543527f1c4c4a781 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
module
public import Mathlib.RingTheory.Valuation.ValuativeRel.Basic
/-!
# Trivial Valuative Relations
Trivial valuative relations relate all non-zero elements ... |
23,987 | lean_file_chunk | d42f4a0f0b4bc5b4d7828781849de7b214aef1b41c00ede0b0c177dacf5f7947 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Liu, Adam Topaz
-/
module
public import Mathlib.RingTheory.Valuation.Basic
public import Mathlib.Data.NNReal.Defs
public import Mathlib.Topology.Defs.Filter
public import Mathlib.Or... |
23,984 | lean_file_chunk | 26758a780474e398381445d946016f23c65011981c37b9960a82178bf8d008f2 | lean_source_file | pkg_mathlib_source_file_chunks | apply ValueGroupWithZero.sound
· simpa [mul_comm] using h₂
· simpa [mul_comm] using h₁
@[simp]
theorem ValueGroupWithZero.inv_mk (x : R) (y : posSubmonoid R) (hx : ¬x ≤ᵥ 0) :
(ValueGroupWithZero.mk x y)⁻¹ = ValueGroupWithZero.mk (y : R) ⟨x, hx⟩ := dif_neg hx
/-- The value group-with-zero is a linearly... |
4,838 | lean_file_chunk | f26fb4237381a725e9c94a486c49c49d9b36d99c79564d9c803737ead43ecb01 | lean_source_file | pkg_mathlib_source_file_chunks | embedding_embed_valuation_eq γ
/-- The map `orderMonoidIso v` is strictly monotone. -/
lemma orderMonoidIso_strictMono [v.Compatible] : StrictMono (orderMonoidIso v) :=
embed_strictMono v
/-- The map `embedding ∘ orderMonoidIso (valuation R))` is identity. -/
lemma leftInverse_embedding_orderMonoidIso : Function.Le... |
9,691 | lean_file_chunk | 87e82afdcff87b70b1c70ff1db894ad5671dbb6fb224f1f348ff64f36fa1b679 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.Algebra.Algebra.ZMod
public import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
/-!
# Cyclotomic polynomials and `expand`.
We gath... |
14,836 | lean_file_chunk | e451e6b1de85ef605a0264895177c5ae3c07f73d087223a608bb9c764a8eadd0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
module
public import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
public import Mathlib.Tactic.ByContra
public import Mathlib.Topology.Algebra.Polynomial
public im... |
23,969 | lean_file_chunk | 5003c6bc9c97653cf6d6709831185e6c4808dcdb50d9f4976c87903f1bbddc65 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.Algebra.Polynomial.Splits
public import Mathlib.FieldTheory.RatFunc.AsPolynomial
public import Mathlib.NumberTheory.ArithmeticFunction... |
7,467 | lean_file_chunk | 2caad9c646a586722b02be48db3cfd150ef1e0ca45a6fe30bdef1baa5ec25fd2 | lean_source_file | pkg_mathlib_source_file_chunks | have := eq_cyclotomic_iff (R := R) (P := ∑ i ∈ range p, (X ^ p ^ m) ^ i)
(pow_pos hp.pos (m + 1))
rw [eq_comm] at this
rw [this, Nat.prod_properDivisors_prime_pow hp]
induction n with
| zero => haveI := Fact.mk hp; simp [cyclotomic_prime]
| succ n_n n_ih =>
rw [← (eq_cyclotomic_iff (pow_pos hp.p... |
8,634 | lean_file_chunk | b8154ab82fb75d620cc6dbc19c139014d9937fa93e0646ff55d665d67d79028c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.FieldTheory.Finite.GaloisField
public import Mathlib.RingTheory.SimpleModule.Basic
public import Mathlib.RingTheory.Polynomial.Cycloto... |
11,640 | lean_file_chunk | 6fc64d3fc38a72fbad5c0431a7701cb67204acc72883d9692d03083c5fa70d3c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
public import Mathlib.RingTheory.RootsOfUnity.Minpoly
/-!
# Roots of cyclotomic polynomials.
... |
2,579 | lean_file_chunk | 5b1ee5c565fdf3b5f596e52dbe1624f3e89c327e703c09423ed62a9b330249bb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle, Jake Levinson
-/
module
public import Mathlib.RingTheory.Polynomial.Hermite.Basic
public import Mathlib.Analysis.Calculus.Deriv.Add
public import Mathlib.Analysis.Calculus.D... |
7,009 | lean_file_chunk | a1d80a13f11b254c0c9a69dde27053544744dd3b42f0a480c171c68f48417fbd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
module
public import Mathlib.Algebra.Polynomial.Derivative
public import Mathlib.Data.Nat.Factorial.DoubleFactorial
/-!
# Hermite polynomials
This file defines `Polynomi... |
8,995 | lean_file_chunk | b80d208da2a11cc4b00e61a0e74b203226b1732d40e76c059f3460495c43e4b1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Antoine Chambert-Loir
-/
module
public import Mathlib.RingTheory.Ideal.Maps
public import Mathlib.RingTheory.Polynomial.Content
public import Mathlib.RingTheory.I... |
8,980 | lean_file_chunk | ecef3791e3bf6f908da1fb0418d3dccc27fa39c23866c985a837e308619d8c9a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.RingTheory.Ideal.BigOperators
public import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
public import Mathlib.RingTheory.Polyno... |
19,051 | lean_file_chunk | da4494f559d3daee871f4159abb065bf9547a3d2917d3fbb7db026b5c9909bc0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.Data.Nat.Choose.Dvd
public import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
public import Mathlib.RingTheory.Norm.Transitivi... |
3,425 | lean_file_chunk | 301076fa1eb5155d0b20e2f98a9cbff7301d8dd937982af2d445a59d9aea77a6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
module
public import Mathlib.RingTheory.Polynomial.Eisenstein.Basic
public import Mathlib.RingTheory.PowerSeries.Order
/-!
# Distinguished polynomial
In this file we def... |
23,970 | lean_file_chunk | e272321dcd559c4ede1473f4f83b190dd6dc88b264a207006125721f9da1391e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen, Andrew Yang
-/
module
public import Mathlib.Algebra.Polynomial.Derivative
public import Mathlib.Algebra.Polynomial.Div
public import Mathlib.FieldTheory.SplittingFi... |
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