chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
3,548 | lean_file_chunk | 6517dc2a29737315ba3319140f9666750c9fe9d9aa60aa4a29292927a80e64f2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
module
public import Mathlib.LinearAlgebra.Isomorphisms
public import Mathlib.RingTheory.Ideal.Operations
public import Mathlib.RingTheory.Ideal.Quotient.Defs
/-!... |
5,513 | lean_file_chunk | 7bafeb1c304c1261f3a3080ea6f754e7a702addcc36f8f131717096d70a6dc7e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 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.TensorProduct.Quotient
public import Mathlib.RingTheory.Flat.Tensor
public import Mathlib.RingTheory.Ideal.IdempotentF... |
4,443 | lean_file_chunk | ab50eacf94078d492d435f7d50c8e1c787b42fb2047190105c13c36913eddf0d | 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.Module.Submodule.Defs
public import Mathlib.Tactic.Abel
/-!
# Ideals over a ring
This file defines `Ideal... |
12,309 | lean_file_chunk | cce52dcf64bc7fb9e95ac98cfa69a9e515982a3702e7e8c53272c589fc0b2c2c | 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.GroupWithZero.Associated
public import Mathlib.Algebra.Ring.Idempotent
public import Mathlib.LinearAlgebra.S... |
708 | lean_file_chunk | d22feeddaf3974ab2c0c44bb513395fd2e1f927ca046a4c7cdee10ed22657344 | 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.Module.Submodule.Basic
public import Mathlib.RingTheory.Ideal.Defs
/-!
# Big operators for ideals
This co... |
4,755 | lean_file_chunk | 39bed901592bb51bb440cc31e318417f7ebaaf859d7590da580429dd53964e1d | 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.GroupWithZero.NonZeroDivisors
public import Mathlib.RingTheory.Ideal.Lattice
/-!
# Prime ideals
This file... |
17,416 | lean_file_chunk | 11e8ea7d61fe31b9565ca4ee8a22e596d6f3200a9b162e9ddaf684e815784916 | 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.Module.Torsion.Basic
public import Mathlib.Algebra.Ring.Idempotent
public import Mathlib.LinearAlgebra.Dimension.Finite
public import ... |
2,505 | lean_file_chunk | 67262a44b8a4c64857bbfea37bbf85318b984b264f83bba95267c171ac92672e | 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: Anne Baanen
-/
module
public import Mathlib.Algebra.Algebra.Bilinear
public import Mathlib.LinearAlgebra.Basis.Defs
public import Mathlib.LinearAlgebra.Basis.Submodule
public import Mathli... |
3,142 | lean_file_chunk | 82c75268b947ef1f478f3e40327b763724c00954d281a47f4729ef5100756330 | 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.Module.Submodule.Lattice
public import Mathlib.RingTheory.Ideal.Defs
public import Mathlib.Tactic.NormNum.In... |
22,005 | lean_file_chunk | aabc6a0244c09961286fd0115c313b525b754ab2f77f5c9986881cb4d6073d1b | 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: Wanyi He, Jiedong Jiang, Jingting Wang, Andrew Yang, Shouxin Zhang
-/
module
public import Mathlib.Algebra.Module.SpanRank
public import Mathlib.RingTheory.Ideal.MinimalPrime.Localizatio... |
9,710 | lean_file_chunk | 30c43802bf3cb6141aa4834cb9571705299e5ace2426e5654914e11a5799917e | 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.Prime
public import Mathlib.RingTheory.Ideal.Span
/-!
# Ideals over a ring
This file contains an... |
7,806 | lean_file_chunk | 9fd03c3c7838d93892af98a0c0e0b7697c92be18633e58b8e141189f8c1405e3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.RingTheory.PrincipalIdealDomain
/-!
# Principal Ideals
This file deals with the set of principal ideals of a `CommRing R`.
## Main defi... |
1,532 | lean_file_chunk | 178f1e130abbdbdca0f842401ae6c47dc67f17a360e31cebbba0e69bb6aac77e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.Algebra.Divisibility.Prod
public import Mathlib.Algebra.Polynomial.FieldDivision
public import Mathlib.LinearAlgebra.InvariantBasisNumber
public i... |
23,989 | lean_file_chunk | bdf70c6f985559c1bbd8262182049cdfb8815277b0b0356cff8feb4628409f2d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.Algebra.Group.Units.Opposite
public import Mathlib.Algebra.Regular.Opposite
public import Mathlib.Data.SetLike.Fintype
public import Mathlib... |
5,697 | lean_file_chunk | 17b83fe8152d9a24488ec4e666b63c74c136f2c45ae9e2b69836fb3bb9896251 | lean_source_file | pkg_mathlib_source_file_chunks | Ideal.Quotient.maximal_of_isField _ (isField_of_isDomain _)
lemma isPrime_iff_isMaximal (p : Ideal R) : p.IsPrime ↔ p.IsMaximal :=
⟨fun _ ↦ isMaximal_of_isPrime p, fun h ↦ h.isPrime⟩
theorem mem_minimalPrimes {I p : Ideal R} [hp : p.IsPrime] (hIp : I ≤ p) : p ∈ I.minimalPrimes :=
⟨⟨hp, hIp⟩, fun q ⟨_, _⟩ hqp ↦ ((... |
3,562 | lean_file_chunk | d52436b1039bdd445273191a7adc3de64e06ead7fe276bc2a7fc3aac3005c179 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Junyan Xu, Jujian Zhang
-/
module
public import Mathlib.Algebra.Field.Equiv
public import Mathlib.RingTheory.Artinian.Module
public import Mathlib.RingTheory.Localization.... |
2,588 | lean_file_chunk | 0cba84bd335235dd88d1aec35e6f392ebb41c3967c00e59791035cf761ba877b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.Algebra.Module.Submodule.Defs
/-!
# Artinian rings and modules
A module satisfying these equivalent conditions is said to be an *Artinian*... |
2,372 | lean_file_chunk | 777d91fc2be8448d12b6c9e2c7d7a400582f6e7bf4f56d5b36236b055e00beff | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michal Staromiejski. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michal Staromiejski
-/
module
public import Mathlib.RingTheory.Artinian.Module
public import Mathlib.RingTheory.IntegralClosure.Algebra.Defs
public import Mathlib.RingTheory.Inte... |
8,734 | lean_file_chunk | 23d18e0988b70ab10c8ebf8d286a464d4f8408b72bd08ce9cf401f5e89ed3032 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
module
public import Mathlib.LinearAlgebra.Isomorphisms
public import Mathlib.RingTheory.Finiteness.Basic
public import Mathlib.RingTheory.Finiteness.Bil... |
15,793 | lean_file_chunk | 1cf3108b8a98f3636c9781acf93e7eb8c590ab856c333c2c009e17cceabb694a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Antoine Chambert-Loir and María-Inés de Frutos Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María-Inés de Frutos Fernández
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.DirectLimit
public import M... |
4,947 | lean_file_chunk | e8723cb23c34e2f271ba33240c14cfa3fa093b1a78fb9f425cab1012da98255c | 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: Antoine Chambert-Loir
-/
module
public import Mathlib.RingTheory.TensorProduct.IsBaseChangePi
public import Mathlib.LinearAlgebra.FreeModule.Basic
public import Mathlib.LinearA... |
23,958 | lean_file_chunk | 92f2a30af9f7fa61a949552e236e549b8388145ce4f9ab0969a16ea3a466470d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
module
public import Mathlib.Algebra.Algebra.RestrictScalars
public import Mathlib.Algebra.Algebra.Subalgebra.Lattice
public import Mathlib.Algebra.Modul... |
9,490 | lean_file_chunk | 56c1d645bdb0ddb6b47a0f9dcb6745fd74101cf2668c187a46126b30fa5e770d | lean_source_file | pkg_mathlib_source_file_chunks | (tensorTensorTensorComm R R R R A B C D).toLinearEquiv =
_root_.TensorProduct.tensorTensorTensorComm R A B C D := rfl
lemma map_bijective {f : A →ₐ[R] B} {g : C →ₐ[R] D}
(hf : Function.Bijective f) (hg : Function.Bijective g) :
Function.Bijective (map f g) :=
_root_.TensorProduct.map_bijective hf hg
l... |
7,889 | lean_file_chunk | 19fc75dbde5b8b3bd1d3b610bed6e802bdc58f760a5c878dcd4167dd2a06a84f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yong-Gyu Choi. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yong-Gyu Choi
-/
module
public import Mathlib.Algebra.Category.Ring.Constructions
public import Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra
/-!
# Exactness properties of the differ... |
3,825 | lean_file_chunk | 07f936612f7c5d0310fadf3a97137f42a1f8cdbb39b6b4636ab6d489ef021091 | 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: Christian Merten
-/
module
public import Mathlib.Algebra.Algebra.Pi
public import Mathlib.LinearAlgebra.TensorProduct.Pi
public import Mathlib.LinearAlgebra.TensorProduct.Prod
publi... |
23,953 | lean_file_chunk | c9c2985e813d7b99ac518fa492051539a5fd79049df2b7b5d02f5af4bfda4d2c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
module
public import Mathlib.Algebra.Algebra.Operations
public import Mathlib.Algebra.Star.TensorProduct
public import Mathlib.LinearAlgebra.TensorProduc... |
5,396 | lean_file_chunk | 45c51383d2dbf82fe1ce17309247d66eb45bdac4308692a0dbbb820823fc30c7 | lean_source_file | pkg_mathlib_source_file_chunks | theorem smul_def (a : A) (b : B) (m : M) : a ⊗ₜ[R] b • m = a • b • m :=
rfl
section Lemmas
theorem linearMap_comp_mul' :
Algebra.linearMap R (A ⊗[R] B) ∘ₗ LinearMap.mul' R R =
map (Algebra.linearMap R A) (Algebra.linearMap R B) := by
ext
simp only [AlgebraTensorModule.curry_apply, curry_apply, LinearM... |
6,286 | lean_file_chunk | 4f7fdbca94fa33cd37922d356df1dc8fc7959c55387b877505ab81995fd451b6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten, Yi Song, Sihan Su. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten, Yi Song, Sihan Su
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Quotient
public import Mathlib.LinearAlgebra.TensorProduct.RightEx... |
4,755 | lean_file_chunk | c453339de90ad0b6f83eb8643d329006d62a56fe54925f609ba6aec265d2f06f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michał Mrugała. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michał Mrugała
-/
module
public import Mathlib.Algebra.MonoidAlgebra.Basic
public import Mathlib.RingTheory.IsTensorProduct
/-!
# Monoid algebras commute with base change
In this fi... |
11,663 | lean_file_chunk | aad2935e2b8db42e509c534a2966097a09118528f3f3b889550e8a5f1180c1d3 | 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.LinearAlgebra.DirectSum.Finsupp
public import Mathlib.Algebra.MvPolynomial.Eval
public import Mathlib.RingTheory.TensorPro... |
1,727 | lean_file_chunk | ae0071dbf5613072abb7c04a55502b9f299a5ab07c266adc39acf728af2a481b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.LinearAlgebra.Basis.VectorSpace
public import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.Localization.FractionRing
pu... |
5,246 | lean_file_chunk | e073626306cf6803ca5f9c2bf7aba0614eaa8f51f02b6b7189ccfb2c5942fe50 | 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, Antoine Chambert-Loir
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Pi
public import Mathlib.LinearAlgebra.TensorProduct.Prod
public import Mathlib.R... |
4,497 | lean_file_chunk | 1c8fe8e184caec03f90d67a618897f67b448a094a7fb3d83d26117a329e54f71 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
module
public import Mathlib.LinearAlgebra.DirectSum.Finsupp
public import Mathlib.LinearAlgebra.Finsupp.Pi
public import Mathlib.LinearAlgebra.FreeModul... |
8,858 | lean_file_chunk | 15cc70cc0f66e1f3fc7f1fba8fdfdc855860012efb340f483bb5ffa51a04ac00 | 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: Antoine Chambert-Loir
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Pi
public import Mathlib.LinearAlgebra.TensorProduct.Prod
public import Mathlib.RingTheory.Loc... |
10,993 | lean_file_chunk | f3a2aa80e47cd01cf197428c5e2942bbb00f62d29bc90ba251557652b1bdea3b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Janos Wolosz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Janos Wolosz
-/
module
public import Mathlib.Algebra.Algebra.Basic
public import Mathlib.Algebra.Algebra.Bilinear
public import Mathlib.Algebra.BigOperators.GroupWithZero.Action
public ... |
4,782 | lean_file_chunk | 0d2e924b2037191080d22eba93e4aa652e8644559bd74062fcd918339b5379a6 | 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.LinearAlgebra.Matrix.ToLin
public import Mathlib.LinearAlgebra.Quotient.Basic
public import Mathlib.RingTheory.Ideal.Maps
public import Mathli... |
3,022 | lean_file_chunk | b3acd0669746b76c2c412aa03531bc233bb06d35aef906461c3d417c4f2d77dc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Dion Leijnse. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dion Leijnse
-/
module
public import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
public import Mathlib.RingTheory.Flat.Basic
/-!
# Geometrically reduced algebras
In this file we ... |
8,300 | lean_file_chunk | 2945b30042282dde608047a6ca0b4851f959b4180adbf1d2ebebe2590c41432c | 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.Algebra.BigOperators.Finprod
public import Mathlib.Algebra.FiniteSupport.Defs
public import Mathlib.Algebra.GroupWithZero.Action.Defs
public i... |
5,491 | lean_file_chunk | aedbc9810ced10dd62fbd5456fe2f254945fb7f4491df2e73763dccbaa86af4b | 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.Algebra.GroupWithZero.Hom
public import Mathlib.Algebra.GroupWithZero.Units.Basic
public import Mathlib.Algebra.Ring.Defs
public import Mathli... |
1,021 | lean_file_chunk | 771fdea9c396c31d414a45e38784a62cdfec7a740bbf33068b4c47e46b1262f2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Finiteness.Ideal
public import Mathlib.RingTheory.Nilpotent.Lemmas
public import Mathlib.RingTheory.Noetherian.De... |
17,448 | lean_file_chunk | 3792389c59b1a9facfb5b9e049368cbc542e28e882ec57f527d01580bad8a426 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard, María Inés de Frutos-Fernández
-/
module
public import Mathlib.Algebra.Order.SuccPred.PartialSups
public import Mathlib.LinearAlgebra.Fin... |
1,214 | lean_file_chunk | 0f05d7351612b3c54ba12c30916c4ef9e9f4d5a1948fa9b5d40a6f9a9964f5fa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
module
public import Mathlib.Order.Filter.EventuallyConst
public import Mathlib.RingTheory.Noetherian.Defs
/-!
# Noetherian modules an... |
8,290 | lean_file_chunk | cbabde755cbaa3f526378aa06e581264e172e0ff57314063582ff576777e50d7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
module
public import Mathlib.Order.Filter.AtTopBot.Basic
public import Mathlib.RingTheory.Finiteness.Basic
/-!
# Noetherian rings and ... |
670 | lean_file_chunk | 88b8c60e21c167f7127228a7a15bda06e5bdb56c4c29f6cb65d7eb4f82f98378 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.RingTheory.Noetherian.Defs
public import Mathlib.RingTheory.UniqueFactorizationDomain.Ideal
/-!
# Noetherian domains have unique factori... |
4,094 | lean_file_chunk | fa9136d9277e5a0dc61d26c674381489e9eef7905a2c4054136f7045b0c1b121 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
module
public import Mathlib.Algebra.Module.Submodule.IterateMapComap
public import Mathlib.Order.PartialSups
public import Mathlib.Rin... |
3,221 | lean_file_chunk | 28e9df8811878b1800232b4ae128ec5fbb774f2ed957de55b9a538677adedf1a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Anthony Fernandes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anthony Fernandes, Marc Robin
-/
module
public import Mathlib.RingTheory.Ideal.Oka
public import Mathlib.RingTheory.Noetherian.Defs
public import Mathlib.RingTheory.Ideal.BigOperat... |
12,946 | lean_file_chunk | dca8ef4434dfe57805f7b1c20f2583a895055aa678b2c14d7ce26e5bd0deea81 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
module
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Data.Fintype.Basic
public import Mathlib.Data.Int.GCD
public import ... |
19,572 | lean_file_chunk | a594257132d7ffdfa02b2e46e9b25860196fa32183d5e502dcdb85689bfc815f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
module
public import Mathlib.Algebra.Group.Action.Units
public import Mathlib.Algebra.Group.Nat.Units
public import Mathlib.Algebra.GroupWithZero.Associ... |
4,205 | lean_file_chunk | d187fce89bafaa5aa28330f742c4244aa6f6d3b9a4836865600d78ac02eb9d26 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
module
public import Mathlib.LinearAlgebra.DFinsupp
public import Mathlib.RingTheory.Ideal.BigOperators
public import Mathlib.RingTheory.Ideal.Operat... |
11,520 | lean_file_chunk | c63c6704a961f45b1234b49d7c57734b3468a55d911354f9072b4aaeeef7635c | 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.DedekindDomain.Basic
public import Mathlib.RingTheory.DiscreteValuationRing.Basic
public import Mathlib.RingTheory.Finiteness.Ideal... |
23,987 | lean_file_chunk | b7bd5525249de64ee367fc626b8ff1863d150f2941e34368afc218f9682cc57e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
module
public import Mathlib.RingTheory.AdicCompletion.Basic
public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
public import Mathlib.RingTheory.LocalRing.R... |
1,512 | lean_file_chunk | 8d7a4c91f1409521cbcf4ed777503689b67c5593dcb2044107d2ea09c069aa5b | lean_source_file | pkg_mathlib_source_file_chunks | letI : IsDomain O := hv.hom_inj.isDomain
intro _ _ h
rw [← hv.coe_span_singleton_eq_setOf_le_v_algebraMap, ← h.maximalIdeal_eq]
lemma maximalIdeal_pow_eq_setOf_le_v_algebraMap_pow :
letI : IsDomain O := hv.hom_inj.isDomain
∀ [IsDiscreteValuationRing O] {ϖ : O} (_h : Irreducible ϖ) (n : ℕ),
((IsLocalRin... |
7,899 | lean_file_chunk | 1e6c8aadc21d867a63b90df4e5c1f2dc1f8ecc218bef235e91c577a3ed985f3e | 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.RingTheory.FractionalIdeal.Operations
/-!
# Inverse operator for fractional ideals
This file define... |
7,505 | lean_file_chunk | 4f642250722a78b818465dc01a2d8a80f370f7d4ca30401998d8f5dd6eb7f4f0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.RingTheory.FractionalIdeal.Basic
public import Mathlib.RingTheory.Ideal.Norm.AbsNorm
public import Mathlib.RingTheory.Localization.NormTra... |
23,993 | lean_file_chunk | 1d0bc1ebae0d4dd032d80dfd36f48a9fdf38cf1cc471375aff5938d56b1af68b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
module
public import Mathlib.RingTheory.Localization.Integer
public import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This f... |
871 | lean_file_chunk | 18ebe941a20b424b7ce0a48af9f70b905d5731b3319e1d695e92266f8a952a15 | lean_source_file | pkg_mathlib_source_file_chunks | theorem coeIdeal_finprod [IsLocalization S P] {α : Sort*} {f : α → Ideal R}
(hS : S ≤ nonZeroDivisors R) :
((∏ᶠ a : α, f a : Ideal R) : FractionalIdeal S P) = ∏ᶠ a : α, (f a : FractionalIdeal S P) :=
MonoidHom.map_finprod_of_injective (coeIdealHom S P).toMonoidHom (coeIdeal_injective' hS) f
end Order
sectio... |
23,973 | lean_file_chunk | 4759d04dc42873b6b2f4bcbaa8a51a486a725efa8fcfc2dddf3a697349ce6a1e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
module
public import Mathlib.Algebra.EuclideanDomain.Basic
public import Mathlib.RingTheory.FractionalIdeal.Basic
public import Mathlib.RingTheory.In... |
11,936 | lean_file_chunk | 1c6ceb67a6dbfca92b7ea0b28332c60265c260cd104a2f77354e19d83b05e516 | lean_source_file | pkg_mathlib_source_file_chunks | | zero => rw [pow_zero, pow_zero, spanSingleton_one]
| succ n hn => rw [pow_succ, hn, spanSingleton_mul_spanSingleton, pow_succ]
@[simp]
theorem coeIdeal_span_singleton (x : R) :
(↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) := by
ext y
refine (mem_coeIdeal S).trans... |
11,931 | lean_file_chunk | 34d854b884e1f58c851cb7171bd0e1b94aa006b19771c300cf7d762e466f3c2e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 James Sundstrom. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: James Sundstrom, Xavier Roblot
-/
module
public import Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
/-!
# Extension of fractional ideals
This file defines the extension of a frac... |
1,602 | lean_file_chunk | a7b26a6d0518639b0cb454461e32f7e77aa2ad655fe80db9d7111d67a3d2d9e6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Weijie Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Weijie Jiang
-/
module
public import Mathlib.RingTheory.PowerSeries.Basic
public import Mathlib.Combinatorics.Enumerative.Catalan
/-!
# Catalan Power Series
We introduce the Catalan g... |
6,286 | lean_file_chunk | 3db1b290c9ade2c915f450d95e05ce1525dc9b7f6d1d63df7e9878d16b1ce21c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Ralf Stephan
-/
module
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.Data.Nat.Cast.Field
public import Mathlib.RingTheory.PowerSeries.Derivative
/-!
# Exp... |
22,413 | lean_file_chunk | 7669eec91c31734945b2e707735905abcd8ddbc75b843ad35b0cb69485228d5f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Antoine Chambert-Loir, María Inés de Frutos Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos Fernández
-/
module
public import Mathlib.RingTheory.MvPowerSeries.Substitution
public import Mathli... |
12,659 | lean_file_chunk | 0d98fa7f474bc33ed88c7a6324842af0fcd96709e17b4517a1f8126755549a46 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
module
public import Mathlib.Algebra.Polynomial.FieldDivision
public import Mathlib.RingTheory.Discr... |
4,254 | lean_file_chunk | 731ac30353aba4d68702fec0cdf34b00909448c90df1e40bfbb5aa4329a87367 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Wenrong Zou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wenrong Zou
-/
module
public import Mathlib.RingTheory.PowerSeries.Substitution
public import Mathlib.RingTheory.MvPowerSeries.Expand
/-!
## Expand power series
Given a power series `φ... |
12,548 | lean_file_chunk | 807adad93baa1e48e99d9ee80e1fda287d1a68712555a60c355106837ba5eeec | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
module
public import Mathlib.RingTheory.MvPowerSeries.PiTopology
public import Mathlib.... |
6,853 | lean_file_chunk | 4d8a89cdb279162ff1a8490b75316b1c2b846bfdc05749851b90ac5d80d7917f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Definition of well-known power series
In this file we d... |
17,472 | lean_file_chunk | 876eedd4b9783562657633fd9967c8ffe72a6ff7ff14c6ea5f83437dbcadac0b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
module
public import Mathlib.Algebra.CharP.Defs
public import Mathlib.RingTheory.Multiplicity
public import Mathlib.RingTheory.PowerSeries.Basic
public im... |
9,381 | lean_file_chunk | c46e3916f5c1e463d65121581903c888c443e1fcabf3cd08a972ef78172eef30 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
module
public import Mathlib.RingTheory.MvPowerSeries.Evaluation
public import Mathlib.... |
8,135 | lean_file_chunk | 9f4f72cc8ca5989b24a57a1ab224b675a653579dca71849af59628d8acdfd45b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
module
public import Mathlib.Algebra.Polynomial.Coeff
public import Mathlib.Algebra.Polynomial.Degree.Lemmas
public import Mathlib.RingTheory.PowerSeries.... |
5,846 | lean_file_chunk | b688c4416a50534291270c9083d334dc33718c529e15bbd01bd4761e71ec94a7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 William Coram. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: William Coram
-/
module
public import Mathlib.Analysis.Normed.Group.Ultra
public import Mathlib.Analysis.RCLike.Basic
public import Mathlib.RingTheory.PowerSeries.Basic
public import M... |
4,037 | lean_file_chunk | a759466edf1f431674f6ad2e9100b83166b6e0ecd9788b69335a8a1d0fb64575 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Weijie Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Weijie Jiang
-/
module
public import Mathlib.Combinatorics.Enumerative.Schroder
public import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Schröder Numbers Power Series
This file define... |
2,098 | lean_file_chunk | f9400065218efdb47f17ad043165ca150cdcc0faba585f5e86e7efcb6ea33a1d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
module
public import Mathlib.RingTheory.PowerSeries.Order
public import Mathlib.RingTheory.Ideal.Maps
/-!
# Power series over rings with no zero divisors... |
23,981 | lean_file_chunk | 04a311b9d9857666872f1790efc369fcb80f69e1e82181df35b1a52de1e5ccca | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau, Ralf Stephan
-/
module
public import Mathlib.Algebra.CharP.Defs
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial... |
6,066 | lean_file_chunk | 20462cbad2fcd2ae79c90733914c0d22bb17179fd353c1f4844662cf4e5993b5 | lean_source_file | pkg_mathlib_source_file_chunks | mapAlgHom φ f = f.map φ :=
MvPowerSeries.mapAlgHom_apply φ f
end Algebra
end PowerSeries
namespace Polynomial
open Finsupp Polynomial
section Semiring
variable {R : Type*} [Semiring R] (φ ψ : R[X])
/-- The natural inclusion from polynomials into formal power series. -/
@[coe]
def toPowerSeries : R[X] → PowerSer... |
23,992 | lean_file_chunk | aab15674adcaf9c273ce5a7c24af080eedf2fc1b81ef02d17b7a885904053a81 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.RingTheory.LocalRing.ResidueField.Basic
public import Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished
public import Mathlib.RingTheory.PowerSerie... |
17,028 | lean_file_chunk | 40adbf8abec4532082bb4ca56836bdad8f3043bbceef6f69e758fe13f3e1a717 | lean_source_file | pkg_mathlib_source_file_chunks | split_ifs with hg
· exact degree_trunc_lt _ _
· nontriviality A
rw [Polynomial.degree_zero]
exact WithBot.bot_lt_coe _
section
variable {g} (hg : g.map (IsLocalRing.residue A) ≠ 0)
include hg
theorem isWeierstrassDivision_weierstrassDiv_weierstrassMod
[IsAdicComplete (IsLocalRing.maximalIdeal A) A] :... |
8,162 | lean_file_chunk | c67507b19f7f8123543caa7c4ea5bfa3dbba756b7f0343c159e14860dd9afe29 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill, Ralf Stephan
-/
module
public import Mathlib.Algebra.Polynomial.Derivation
public import Mathlib.RingTheory.Derivation.Basic
public import Mathlib.RingTheory.PowerSe... |
3,108 | lean_file_chunk | a7e8651395df9a9d3afcf3992c653e2b9bc17e94f33201570c90d0447a10bb74 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.RingTheory.Ideal.Operations
public import Mathlib.RingTheory.Ideal.BigOperators
public import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Some results ... |
10,012 | lean_file_chunk | b919236bc51e9762f0dcec39fa7932d8fe38a54499115df328f99f6253563a26 | 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, Anthony Fernandes, Marc Robin
-/
module
public import Mathlib.Algebra.Module.SpanRank
public import Mathlib.RingTheory.Noetherian.OfPrime
public import Mathlib.RingT... |
4,710 | lean_file_chunk | a3161ea1f35431a89332e1fe25680e77ef9179a8c086d150a1e46736743359d0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Fabrizio Barroero. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fabrizio Barroero
-/
module
public import Mathlib.Data.Real.Archimedean
public import Mathlib.RingTheory.PowerSeries.Order
public import Mathlib.RingTheory.MvPowerSeries.GaussNorm
... |
3,296 | lean_file_chunk | 590732514b5f1bf4bdae22620687581c5d628ed84607966fe2a6919b0e8fa9a0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Ralf Stephan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ralf Stephan
-/
module
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.RingTheory.PowerSeries.Derivative
public import Mathlib.RingTheory.PowerSeries.Exp
public import M... |
3,262 | lean_file_chunk | daa4c27fff837a30970b552ff2a4d73d7fdaab85b6b0840c3236ee0f8c2c9f45 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
module
public import Mathlib.RingTheory.Binomial
public import Mathlib.RingTheory.PowerSeries.WellKnown
public import Mathlib.Tactic.SuppressCompilation
/-!
# Binom... |
5,436 | lean_file_chunk | 3453e44f38773517100a6e12096e8de90267d3173796aa82aef041de49826723 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Blake Farman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Blake Farman
-/
module
public import Mathlib.RingTheory.IdealFilter.Basic
public import Mathlib.Topology.Algebra.LinearTopology
public import Mathlib.Topology.Algebra.FilterBasis
/-!
#... |
7,960 | lean_file_chunk | a68ee4455379d30fdad01c463017355d62088b88bb2324d556d1b42e30fe1c77 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Blake Farman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Blake Farman
-/
module
public import Mathlib.Order.PFilter
public import Mathlib.RingTheory.Ideal.Colon
/-!
# Ideal Filters
An **ideal filter** is a filter in the lattice of ideals of... |
5,846 | lean_file_chunk | 4f7953d53e0524f3b0ec6250e506bb9187ef6d9a28863452c0593583ee984e7a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Wenrong Zou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wenrong Zou
-/
module
public import Mathlib.RingTheory.PowerSeries.Substitution
public import Mathlib.Tactic.Ring.NamePowerVars
/-! # Formal group laws over commutative ring
Let `R` be... |
7,235 | lean_file_chunk | 3d87ec37e34c2a28055bc222dfedd85753d0daffa4a086dccbde92d3bd088fd3 | 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.EuclideanDomain.Int
public import Mathlib.Algebra.MvPolynomial.Eval
public import Mathlib.RingTheory.Adjoin.Basic
public import Mathlib.Ri... |
2,413 | lean_file_chunk | f10f47e208c46e1bde014c15ac743022b83b5c66c0dfdfe5d9fea4721f772d35 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Xavier Généreux. 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.RingTheory.Adjoin.Polynomial.Basic
public import Mathlib.RingTheory.Polynomial.Tower
/-!
# Adjoin one ... |
6,120 | lean_file_chunk | 296e44a2bbed535235d6385401e8027c9ee8520d896674ea36f4baa8208d7720 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 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.Adjoin.FG
/-!
# Adjoining elements and being finitely generated in an algebra tower
## Main results
* `Algebra.fg_trans'`: if `S` is... |
9,285 | lean_file_chunk | de2ee367179874d258040f83bd386222bd6e0423b2d90f1f71a65196db3448e0 | 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.Algebra.Subalgebra.Prod
public import Mathlib.Algebra.Algebra.Subalgebra.Tower
public import Mathlib.LinearAlgebra.Basis.Basic
public impo... |
7,854 | lean_file_chunk | cc8e4499cb5cddb47677b59f9aaf8aed00dacee8ef37b05dad22b0678377dd64 | 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.Adjoin.Basic
public import Mathlib.RingTheory.PowerBasis
public import Mathlib.LinearAlgebra.Matrix.Basis
/-!
# Power basis for `R... |
1,182 | lean_file_chunk | b555e94f40308844e7553db121e0f27efebe5b60817e9773faa18b83dc4a25d4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Dion Leijnse. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dion Leijnse
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Finiteness
public import Mathlib.RingTheory.TensorProduct.Maps
public import Mathlib.RingTheory.Adjoin.FG
/-!
#... |
7,619 | lean_file_chunk | 5ca844d383079a04a7bad1f54b6be4f4da64a45970dd6ce86fee447e45641476 | 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
-/
module
public import Mathlib.Algebra.Polynomial.Splits
public import Mathlib.RingTheory.Adjoin.Basic
public import Mathlib.RingTheory.AdjoinRoot
/-!
# Adjoining element... |
126 | lean_file_chunk | b29967ea08deaad240b28bccdbe36add5e74ada19dda95280fcad879946541ab | lean_source_file | pkg_mathlib_source_file_chunks | module -- shake: keep-all
public import Mathlib.RingTheory.Adjoin.Polynomial.Basic
deprecated_module (since := "2026-03-01") |
2,580 | lean_file_chunk | a666c09daa3bde7275c26ae9412a47ce020df1197ed18204e5ce84a99fbdd812 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.Algebra.Group.Pointwise.Set.Card
public import Mathlib.LinearAlgebra.Dimension.Constructions
public import Mathlib.RingTheory.Adjoin.Basic
/-!
# Some r... |
7,071 | lean_file_chunk | e450bc0244f3ebfd6d2e81bc5e20ed8ddd7e53977e3a5196c2ca239b3f16d747 | 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.KrullDimension.Zero
public import Mathlib.RingTheory.LocalProperties.Reduced
/-!
# Strongly transcendental elements
In this file,... |
23,989 | lean_file_chunk | dd51ac2c263c0a70f151187fa78cbffecedfd7291572857e15407216f44a7537 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
module
public import Mathlib.LinearAlgebra.Dimension.Localization
public import Mathlib.RingTheory.Algebraic.Basic
public import Mathlib.RingTheory.IntegralClosure.I... |
3,428 | lean_file_chunk | 61daf3a287241dd304b2088022d0a2b7836d1ab5ae85c0d65e8364d4bae8dff0 | lean_source_file | pkg_mathlib_source_file_chunks | theorem Algebra.IsPushout.isAlgebraic' [IsPushout R R' S S'] : Algebra.IsAlgebraic R' S' :=
(equiv R R' S S').isAlgebraic
theorem Algebra.IsPushout.isAlgebraic [h : IsPushout R S R' S'] : Algebra.IsAlgebraic R' S' :=
have := h.symm; (equiv R R' S S').isAlgebraic
end Pushout
instance [NoZeroDivisors R] : Algebra.... |
2,661 | lean_file_chunk | 44ba0dbe61804d41c528f0116be23e5e4081eb611cb3d9fcbc5897963d83813b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 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.Polynomial.AlgebraMap
/-!
# Algebraic functions
This file defines algebraic functions as the image of the `algebraMap R[X] (R ... |
23,932 | lean_file_chunk | 1ab88a68a06e72b7a0a737e304eca1e0a226ec21b16c7c529a53e7010cc5396a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 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.Polynomial.Expand
public import Mathlib.RingTheory.Adjoin.Polynomial.Basic
public import Mathlib.RingTheory.Algebraic.Defs
publi... |
5,754 | lean_file_chunk | c927b6cb6d31710d9d0acfbd830cf9c26b5e79971e77842f6f5987fc9a313eb7 | lean_source_file | pkg_mathlib_source_file_chunks | /-- Given a transcendental element `s : S` over `R`, the `R`-algebra equivalence
between `R[X]` and `R[s]` given by sending `X` to `s`. -/
noncomputable def algEquivOfTranscendental (s : S) (h : Transcendental R s) :
R[X] ≃ₐ[R] R[s] :=
AlgEquiv.ofBijective (aeval ⟨s, self_mem_adjoin_singleton R s⟩) <| by
refi... |
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