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 |
|---|---|---|---|---|---|
4,851 | lean_file_chunk | 6efd0045ce22485a689d8d190726acebe2df1314db8d3c7afc0c8e36cb07ca5b | 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, Yi Song
-/
module
public import Mathlib.Algebra.Module.FinitePresentation
public import Mathlib.LinearAlgebra.Dual.Lemmas
public import Mathlib.RingTheory.Ideal.AssociatedPr... |
1,919 | lean_file_chunk | f00fc2c47e1ac4d02c689ac461593c7deaf4fd18436571350c247dd6cdf43708 | 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: Jingting Wang, Wanyi He, Nailin Guan
-/
module
public import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
public import Mathlib.RingTheory.QuotSMulTop
/-!
# Categorical constructions... |
7,204 | lean_file_chunk | f5322875f7b77cfefdc9991894bec4c856e7c400dce65e6ed4946fcd003fc2c1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
module
public import Mathlib.Algebra.Module.Torsion.Basic
public import Mathlib.RingTheory.Flat.Basic
public import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
pu... |
23,987 | lean_file_chunk | 865c30e7a2c72f351cb785b439c67a68c330f1b712462723ad43c1b0c645493e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
module
public import Mathlib.RingTheory.Artinian.Module
public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
public import Mathlib.RingTheory.Nakayama
publi... |
8,307 | lean_file_chunk | 502796e7ade3eb5e2410af62966c4a5c812086ed7d56346ffd1216a0372fefa4 | lean_source_file | pkg_mathlib_source_file_chunks | | [], _ => rfl
| r :: rs, h => by
rw [isRegular_iff, ne_comm, ← lt_top_iff_ne_top, Ideal.ofList_cons,
sup_smul, ideal_span_singleton_smul, isWeaklyRegular_cons_iff] at h
refine absurd ?_ (ne_of_lt (lt_of_le_of_lt le_sup_left h.right))
exact Eq.trans (Submodule.map_top _) <| LinearMap.range_eq_top.mp... |
2,278 | lean_file_chunk | 81a1b64d24a758b5408f0e25af19e87c65f4321cebeabe806b061d9f1d5154b9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies, Michał Mrugała. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Michał Mrugała
-/
module
public import Mathlib.RingTheory.HopfAlgebra.Basic
public import Mathlib.RingTheory.Bialgebra.GroupLike
/-!
# Group-like elements... |
2,890 | lean_file_chunk | 2056e85ba66a7a5e5aff7149afc9fb293c90201e75768e3358a441402d1fd048 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Andrew Yang
-/
module
public import Mathlib.RingTheory.HopfAlgebra.Basic
public import Mathlib.RingTheory.Bialgebra.TensorProduct
/-!
# Tensor products of Hopf ... |
4,907 | lean_file_chunk | 712aa4ef9b29428d97177b1cfa19b15c11633b5bca37331477d0704cb194bbbd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey
-/
module
public import Mathlib.RingTheory.Bialgebra.Basic
/-!
# Hopf algebras
In this file we define `HopfAlgebra`, and provide instances for:
* Commutative semirings: `Com... |
2,873 | lean_file_chunk | c0cf7f5ec24ff831bf311a034bde7906f505f544f3b24490a18646bde3776bab | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
module
public import Mathlib.RingTheory.Bialgebra.MonoidAlgebra
public import Mathlib.RingTheory.HopfAlgebra.Basic
/-!
# The Hopf algebra structure on group a... |
12,445 | lean_file_chunk | 346d55a71c91c7f93f577fd86b1ca2628c03c1a4d29bf5e4fe7a45e9f9cf2431 | 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.Ideal.IdempotentFG
public import Mathlib.RingTheory.Unramified.Basic
public import Mathlib.RingTheory.Flat.Stability
/-!
# Various... |
14,540 | lean_file_chunk | 28f795764463300cfdf12e3a18758caecf3b23629664712327f7bd5a7345f007 | 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.LocalRing.Module
public import Mathlib.RingTheory.LocalRing.ResidueField.Ideal
public import Mathlib.RingTheory.Unramified.Field
pu... |
4,567 | lean_file_chunk | 5982fc1e0ee21d06b02f1fe634e7992d0959e7673b8846ef891f2a2db2257d1b | 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.Unramified.Basic
/-!
# Formal-unramification of finite products of rings
## Main result
- `Algebra.FormallyUnramified.pi_iff`: ... |
22,858 | lean_file_chunk | 7d554ce3dbc84f878cdc24ca682d284d9fdc9472369ffcef78a6c1759dbc6a7e | 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.Etale.Locus
public import Mathlib.RingTheory.Etale.StandardEtale
public import Mathlib.RingTheory.LocalRing.ResidueField.Instances
... |
14,541 | lean_file_chunk | 9c23ff983b7178c3f855efbd4f143d2d563f129fe86a08d0717045eeb43f523c | 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.FiniteStability
public import Mathlib.RingTheory.Ideal.Quotient.Nilpotent
public import Mathlib.RingTheory.Kaehler.Basic
public imp... |
5,681 | lean_file_chunk | 2ad66cfa352302b45a8ce3176b2f161f33e2c07b852380e8b269d81cdf09d5e6 | 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.Etale.Kaehler
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
public import Mathlib.RingTheory.Support
import Mathli... |
11,174 | lean_file_chunk | bf23e9c675a806c5782d440244aab43b9bf6596d941c78174098a08497f58fb0 | 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.FieldTheory.PurelyInseparable.Basic
public import Mathlib.RingTheory.Artinian.Ring
public import Mathlib.RingTheory.Unramified.Finite
public i... |
2,266 | lean_file_chunk | d243386cb164c1a131ec15d614a921384579ce2186e33bc65f1a1d9873c823d7 | 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.DedekindDomain.Dvr
public import Mathlib.RingTheory.Finiteness.Quotient
public import Mathlib.RingTheory.Unramified.Field
... |
4,063 | lean_file_chunk | 4ce9b8b8ba5e07470170a48708b429e7ed44fc2c718089413ffb39cc6888dbd8 | 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.IsIntegrallyClosed
public import Mathlib.RingTheory.Ideal.GoingDown
public import Mathlib.RingTheory.IntegralClosure.Algeb... |
17,540 | lean_file_chunk | 78213af1629b15ea1679bbd00e19cbf464c0c5504a5ee1c6266fb591340d7e21 | 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.Localization.Integral
public import Mathlib.RingTheory.Localization.LocalizationLocalization
public import Mathlib.Algebra.Ring.Hom... |
23,995 | lean_file_chunk | ed87398470f772aafb8806180d8d5db61da470b14fff28cfc9bcee0878ca592f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.DedekindDomain.IntegralClosure
public import Mathlib.RingTheory.RingHom.Finite
public import Mathlib.RingTheory.Local... |
2,355 | lean_file_chunk | 4085e0aadcb1c0786384a76936cbc49a6365379d759b66924d96ea1023bcbfc4 | lean_source_file | pkg_mathlib_source_file_chunks | simp only [MulEquiv.toEquiv_eq_coe, EquivLike.coe_coe]
convert (prod_galRestrict_eq_norm A (FractionRing A) (FractionRing B) B x).symm
open Polynomial IsScalarTower in
theorem Algebra.dvd_algebraMap_intNorm_self (x : B) : x ∣ algebraMap A B (intNorm A B x) := by
classical
haveI : FiniteDimensional (FractionRing ... |
9,679 | lean_file_chunk | e6ce6554975da5ad5264fa60390c19246e21bfb8d444cfaa7e1d46472dea17b1 | 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.RingTheory.DedekindDomain.PID
public import Mathlib.FieldTheory.Separable
public import Mathlib.RingTheory.RingHom.Finite
/-!
# Insta... |
23,964 | lean_file_chunk | 77f45b9b9acacb324790041a65cc4da35c78145222260d95c8fbe6220a9b4201 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.NumberTheory.RamificationInertia.Unramified
public import Mathlib.RingTheory.Conductor
public import Mathlib.RingTheory.FractionalIdeal.Extend... |
18,078 | lean_file_chunk | a183a1c78d66245686c06ccf16066c1b6ab363291cc2d7052e66619bcbe651b1 | lean_source_file | pkg_mathlib_source_file_chunks | have : (Subalgebra.toSubmodule (Algebra.adjoin A {x})) =
Submodule.span A (Set.range pb.basis) := by
rw [← span_range_natDegree_eq_adjoin (minpoly.monic hAx) (minpoly.aeval _ _)]
congr; ext y
have : natDegree (minpoly A x) = natDegree (minpoly K x) := by
rw [minpoly.isIntegrallyClosed_eq_field_f... |
3,028 | lean_file_chunk | b3be39243d774f999514039be85b364d032e353966e7bb68cda969f068ee4626 | 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.RingTheory.DedekindDomain.AdicValuation
public import Mathlib.RingTheory.Polynomial.ContentIdeal
public import Mathlib.RingTheory.... |
5,725 | lean_file_chunk | 908a6f766d94c80d859d557c197f66f604ca9949ec80808f0b35339c890332b8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 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
-/
module
public import Mathlib.RingTheory.DedekindDomain.AdicValuation
public import Mathlib.RingTheory.DedekindDomain.Factorization
pu... |
10,012 | lean_file_chunk | ce8bea049de9a6f22cf6c58ae68b9fe8309bb53b11bfe336cc3a296166e8022e | 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.Algebra.Group.Equiv.TypeTags
public import Mathlib.Data.ZMod.QuotientGroup
public import Mathlib.RingTheory.Dedekind... |
12,409 | lean_file_chunk | 402a86cdafba5658d0cbe89b60a3abbf269f67b4cdc0fa38edbd8e0bd2284249 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kenny Lau
-/
module
public import Mathlib.RingTheory.DedekindDomain.Dvr
public import Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
public import Mathlib.RingTheory.Princip... |
7,807 | lean_file_chunk | f0029939a61d8d651fafaab0ade1e3816f86f19e1d2ffc113532fe312ca7556b | 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.Ideal.GoingUp
public import Mathlib.RingTheory.Polynomial.RationalRoot
/-!
# Dedekind rin... |
23,993 | lean_file_chunk | 4627213606aee9ab20ac3cf5550329dd42a622949c7c5e3c5b95d1708087535f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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
-/
module
public import Mathlib.Algebra.Order.Ring.IsNonarchimedean
public import Mathlib.Data.Int.WithZero
public import Mathlib.RingTh... |
12,535 | lean_file_chunk | a3981e589665c761cc6be28ff1cb29c7dc44549d6adf098fa8b66427e8ec8957 | lean_source_file | pkg_mathlib_source_file_chunks | theorem adicValued_apply' (x : WithVal (v.valuation K)) :
v.adicValued.v (WithVal.equiv _ x) = v.valuation K (WithVal.equiv _ x) :=
rfl
variable (K)
/-- The completion of `K` with respect to its `v`-adic valuation. -/
abbrev adicCompletion := (v.valuation K).Completion
theorem valuedAdicCompletion_def {x : v.a... |
23,967 | lean_file_chunk | dc883701c9a3c52fbe2ce6285d15a78d4c874984274218b2449c97ded4144086 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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
-/
module
public import Mathlib.NumberTheory.RamificationInertia.Basic
public import Mathlib.Order.Filter.Cofinite
/-!
# Factorization ... |
17,864 | lean_file_chunk | 7b1c34df3576572a091c21af47cb94880649d46e2fa907ff3d106829122a065d | lean_source_file | pkg_mathlib_source_file_chunks | · rw [negSucc_eq, count_neg_zpow, ← Int.natCast_succ, zpow_natCast, count_pow]
ring
/-- `val_v(v^n) = n` for every `n ∈ ℤ`. -/
theorem count_zpow_self (n : ℤ) :
count K v ((v.asIdeal : FractionalIdeal R⁰ K) ^ n) = n := by
rw [count_zpow, count_self, mul_one]
/-- If `v ≠ w` are two maximal ideals of `R`, the... |
8,671 | lean_file_chunk | ef0f903c943ed514aef66a20dab25c010aadd65b31827062e8cb787755d5ee89 | 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, Yongle Hu
-/
module
public import Mathlib.RingTheory.DiscreteValuationRing.TFAE
public import Mathlib.RingTheory.LocalProperties.Int... |
11,616 | lean_file_chunk | 9d96e886162eed9e52bc8418ee76264c116619590f314e52eb36436cd84c5c5f | 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.LinearAlgebra.BilinearForm.DualLattice
public import Mathlib.LinearAlgebra.FreeModule.PID
public impo... |
4,577 | lean_file_chunk | ea0df0e968a01edd99173fe933c9359c6906f4d9713f28ad76fef233a6fea1c3 | 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.DedekindDomain.AdicValuation
/-!
# `S`-integers and `S`-units of fraction fields of Dedekind domains
Le... |
15,159 | lean_file_chunk | 6e7281cb53d488bb285c9f73a6c40e5241183151b29f4d010125da8565bd47ce | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.FieldTheory.LinearDisjoint
public import Mathlib.RingTheory.DedekindDomain.Different
/-!
# Disjoint extensions with coprime different ide... |
23,954 | lean_file_chunk | 51cf102c3fbede3c20223ac4d989d6bbcfbf501cb414642fdf847d3bd73d1ca8 | 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.LinearAlgebra.TensorProduct.RightExactness
public import... |
408 | lean_file_chunk | cdc6085e786f5077f8d9dfea1b43c06db27491cbe589a944d9d92be7aef4529f | lean_source_file | pkg_mathlib_source_file_chunks | simp only [toFun_eq_rTensor_φ_toFun' _ ha, π]
rw [Function.comp_apply, toFun_eq_rTensor_φ_toFun' _ hb, toFun_eq_rTensor_φ_toFun' _ ha,
comp_toFun', Function.comp_apply]
theorem toFun_comp_apply (S : Type*) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :
(g.comp f).toFun S m = (g.toFun S) (f.toFun S m) := by
... |
23,947 | lean_file_chunk | 333d29e50875fe16ef49738596affb6dc766dd1c064e31f9462e5a0d02cba83e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
module
public import Mathlib.RingTheory.Localization.Away.Basic
public import Mathlib.RingTheory.Ideal.GoingUp
public import Mathlib.RingTheory.Jacobson.Polynomial
public im... |
13,056 | lean_file_chunk | db3a1380cbd210ce521e48e7495f7594c4af3dd5a8dd1fb29c47fe6aa325af81 | lean_source_file | pkg_mathlib_source_file_chunks | exact @isIntegral_isLocalization_polynomial_quotient R _
(Localization M) (Localization M') _ _ P m hmem_P _ _ _ isloc
rw [(map_bot.symm :
(⊥ : Ideal (Localization M')) = Ideal.map (algebraMap (R[X] ⧸ P) (Localization M')) ⊥)]
let bot_maximal := (bot_quotient_isMaximal_iff _).mpr hP
refine bot_maximal.m... |
2,519 | lean_file_chunk | a248540c2fdb1c1882f68a8909375fb2b50d70d068b7c2b5a039cb11550940b0 | 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.RingTheory.Jacobson.Radical
public import Mathlib.RingTheory.SimpleModule.Basic
/-!
# Semiprimary rings
## Main definition
* `IsSemiprimaryRing... |
15,546 | lean_file_chunk | cc79462fba67f079c8430bc789bff4d7717c3b9a8ac09557d48461edd6453605 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Wojciech Nawrocki
-/
module
public import Mathlib.RingTheory.Ideal.Quotient.Operations
public import Mathlib.RingTheory.TwoSidedIdeal.Operations
public import Mathl... |
1,794 | lean_file_chunk | 468f84b992d75fb1a8652a4b8c3e3b0d2a3593748c33be399757e11ea46e821a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
module
public import Mathlib.RingTheory.Jacobson.Ideal
public import Mathlib.RingTheory.Polynomial.Quotient
/-!
# Jacobson radical of polynomial ring
-/
public section
na... |
2,240 | lean_file_chunk | c50883346434ac2841f37d358d8af8c69deeff94356cc7b6c07e1e74869f9170 | 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: Junyan Xu, Andrew Yang
-/
module
public import Mathlib.RingTheory.HopkinsLevitzki
public import Mathlib.RingTheory.Jacobson.Ring
/-!
# Artinian rings over Jacobson rings
## Main result... |
8,189 | lean_file_chunk | befbe99599a7be45449879827c256c44c65c0d5bfd03f8fe18f08f26fa58f0d3 | 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.Quotient.Basic
public import Mathlib.RingTheory.Finiteness.Basic
public import Mathlib.RingTheory.Ideal.Maps
public import Mathlib.R... |
3,302 | lean_file_chunk | 289e13e8e6dd9d979873dd32d66a9bcb441a5e69531809f66b3002d9c7b818d1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies, Michał Mrugała. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Michał Mrugała
-/
module
public import Mathlib.RingTheory.Bialgebra.Basic
public import Mathlib.RingTheory.Coalgebra.GroupLike
/-!
# Group-like elements i... |
6,651 | lean_file_chunk | 9769878cb1881f4eb6aaa797a85cfd0c92603337e15abcdbae25b7d59c1ab566 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Andrew Yang
-/
module
public import Mathlib.RingTheory.Bialgebra.Equiv
public import Mathlib.RingTheory.Coalgebra.TensorProduct
public import Mathlib.RingTheory.... |
8,385 | lean_file_chunk | 11ff3997c66d486500180995c86a903fc41a19ad5d0f7c16ec2731d308a86995 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Kevin Buzzard
-/
module
public import Mathlib.RingTheory.Coalgebra.Basic
public import Mathlib.RingTheory.TensorProduct.Maps
/-!
# Bialgebras
In this file we define `Bialgeb... |
3,999 | lean_file_chunk | 6afb5e5258a8ec306ba1e3b465377b0bc620e758cba140642891fe8bb58d7907 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Yaël Dillies, Michał Mrugała
-/
module
public import Mathlib.RingTheory.Bialgebra.Hom
public import Mathlib.RingTheory.Coalgebra.MonoidAlgebra
/-!
# The bialgeb... |
10,364 | lean_file_chunk | 54ff54eca4929cceb7204f6eb2dbfa39ef3e67ce18166a64c1aa3afd8459fa8a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
module
public import Mathlib.RingTheory.Coalgebra.Equiv
public import Mathlib.RingTheory.Bialgebra.Hom
/-!
# Isomorphisms of `R`-bialgebras
This file defines... |
10,350 | lean_file_chunk | 02be4005181cff5b0002e93eaca4ff11067cc0f28bbd3a4b8978c004501a9025 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov, Amelia Livingston
-/
module
public import Mathlib.RingTheory.Coalgebra.Hom
public import Mathlib.RingTheory.Bialgebra.Basic
/-!
# Homomorphisms of `R`-... |
16,751 | lean_file_chunk | 251b1c7571d2ab05117e571152c144b6ff35e9f4aba405ccfa56031ec8093221 | 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.Valuation.Basic
/-!
# Ring of integers under a given valuation
The elements with valuation less than or equal to 1.
TODO: Define cha... |
921 | lean_file_chunk | 36f75a9d483da2e58405d190d490a90f168d970abe072a5faf2a40d1da7a5aaf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
module
public import Mathlib.RingTheory.Multiplicity
public import Mathlib.RingTheory.Valuation.Basic
/-!
# `multiplicity` of a prime in an integral... |
2,030 | lean_file_chunk | 5152108eac83c98247a76b162f3f3dc71597eee2d8b05a1c836c329b1a9f8e8f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Michail Karatarakis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michail Karatarakis
-/
module
public import Mathlib.RingTheory.LocalRing.ResidueField.Basic
public import Mathlib.RingTheory.Valuation.ValuationSubring
/-!
# Ramification groups... |
19,224 | lean_file_chunk | 5e5a94c402d9c88f21079e85ac908470fad20207866e76cbfe151aa34202f009 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
module
public import Mathlib.RingTheory.Bezout
public import Mathlib.RingTheory.LocalRing.Basic
public import Mathlib.RingTheory.Localization.FractionRing
public import Math... |
2,598 | lean_file_chunk | b403cf4d36cc78aa2c0f6e8a25b621910a03eb2638af04fe17ea39899814d9b1 | 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, Yakov Pechersky
-/
module
public import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
public import Mathlib.RingTheory.Valuation.ValuationRing
/-!
# Integral elements ove... |
2,394 | lean_file_chunk | 723e3a725f642ba7cf94d1938704179039a669e15fcb13bf27bfb3b50b94dd0b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
module
public import Mathlib.RingTheory.Localization.Defs
public import Mathlib.RingTheory.Valuation.Basic
/-!
# Extending valuations to a localization
We show that, give... |
3,639 | lean_file_chunk | c370b74aa2a34847d4aeab178e147fd53ee56b5a0e77075acf8a8c32b3c711fe | 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.GroupTheory.ArchimedeanDensely
public import Mathlib.RingTheory.Valuation.ValuationRing
/-!
# Ring of integers under a given valuatio... |
12,604 | lean_file_chunk | 0141380db2996c8cd45564900272a91cdcaa45cb3babf033d2cc219e1a94133b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang, Yaël Dillies, Javier López-Contreras, Daniel Funck, Junyan Xu.
All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yaël Dillies, Javier López-Contreras, Daniel Funck, Junyan Xu
-/
module
public import Mathlib.RingTheory.Ideal... |
8,464 | lean_file_chunk | 79c77a954d8d829160713d6dbaa25f93c3814d15f541dc134e1c93dbcb0f6f43 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jiedong Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiedong Jiang, Bichang Lei, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
module
public import Mathlib.RingTheory.Valuation.ValuationSubring
public import Mathlib.Algebra.No... |
700 | lean_file_chunk | 390f898c2a525cb729e8e05c6e142ee8bd63468427e945f0f88914fedbb58364 | 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 -- shake: keep-all
public import Mathlib.FieldTheory.Finite.Valuation
deprecated_module (since := "2026-03-31")
public sec... |
23,979 | lean_file_chunk | 13c69c001df620de2254b2b501d8a329d9b8d8a36bf2dc3bfe46c436f84d7ffa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Junyan Xu, Jack McKoen
-/
module
public import Mathlib.RingTheory.Valuation.ValuationRing
public import Mathlib.RingTheory.Localization.AsSubring
public import Mathlib.Algebra... |
9,746 | lean_file_chunk | 3604b70ee8920db74da050367cbe66e3ed6b2e1ba1e0bd9315a4618cbb835513 | lean_source_file | pkg_mathlib_source_file_chunks | theorem principal_units_le_units : A.principalUnitGroup ≤ A.unitGroup := fun a h => by
simpa only [add_sub_cancel] using A.valuation.map_one_add_of_lt h
theorem mem_principalUnitGroup_iff (x : Kˣ) :
x ∈ A.principalUnitGroup ↔ A.valuation ((x : K) - 1) < 1 :=
Iff.rfl
theorem principalUnitGroup_le_principalUnit... |
23,988 | lean_file_chunk | b5d8720cf43f118ff8739a4eec03dc3501541b4d29c0bf4b197f8779efc8c244 | 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: Kevin Buzzard, Johan Commelin, Patrick Massot, Filippo A. E. Nuccio
-/
module
public import Mathlib.Algebra.Order.GroupWithZero.Range
public import Mathlib.Algebra.Order.Hom.Monoid
pu... |
23,939 | lean_file_chunk | 42ede501230f1bdeea5642f6b79dc7475026914ac4aa6c9d44e12e4c640e9d4a | lean_source_file | pkg_mathlib_source_file_chunks | theorem isNontrivial_iff_exists_lt_one {Γ₀ : Type*} [LinearOrderedCommGroupWithZero Γ₀]
(v : Valuation K Γ₀) : v.IsNontrivial ↔ ∃ x ≠ 0, v x < 1 :=
⟨fun h ↦ by simpa using h.exists_lt_one (v := v), fun ⟨x, hx0, hx1⟩ ↦ ⟨x, by simp [hx0, hx1.ne]⟩⟩
lemma IsNontrivial.exists_one_lt {Γ₀ : Type*} [LinearOrderedCommGro... |
1,820 | lean_file_chunk | 82cba97049638199aa78200dfd2f3cb6fd36841291201adb12347ee414668bf0 | lean_source_file | pkg_mathlib_source_file_chunks | { toFun := fun x ↦ .ofAdd <| .toDual <| .toDual <| .ofMul x
invFun := fun x ↦ x.toAdd.ofDual.ofDual.toMul
map_mul' := fun _x _y ↦ rfl
map_le_map_iff' := .rfl }) (AddValuation.ofValuation (R := R) (Γ₀ := (Additive Γ₀)ᵒᵈ))
/-- The `Valuation` associated to a `AddValuation`.
-/
def ofAddValuation : AddV... |
4,003 | lean_file_chunk | 6d1171eb729a39d771736113cb6916a137829f11b804a56cfcaeca5ed820952c | 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: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
module
public import Mathlib.RingTheory.Ideal.Quotient.Operations
public import Mathlib.RingTheory.Valuation.Basic
/-!
# The valuatio... |
10,131 | lean_file_chunk | 02f4d3d15db58b7af29edb3ef27e73058829f6683b4db9b6c5b35648ce6576d3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.Algebra.Order.Group.Units
public import Mathlib.Algebra.Order.GroupWithZero.WithZe... |
2,697 | lean_file_chunk | 8b8fd57f254501965f570bd5d0d3b9c12172b5347747d4eb93d17a23978140b2 | 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.Algebra.GroupWithZero.Range
public import Mathlib.GroupTheory.ArchimedeanDensely
public import Mathlib.RingTheory.Valuation.RankOne
/... |
3,499 | lean_file_chunk | 1b7fb3a2615b137947411faeadff4eff3555f3a3656a632745715633e5f54cc3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.RingTheory.IntegralClosure.IsIntegralClosure.Basic
public im... |
2,355 | lean_file_chunk | 4b9dba74229dee0648cffe0d44cf1cb74ea02c7f081a7df2e7635748f475f3ff | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.FieldTheory.IntermediateField.Adjoin.Basic
public import Mat... |
21,087 | lean_file_chunk | 2d998d6c1206231cfb094b3f171b4bf93ccd7c3e66770a4dc23affad5824da0e | 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
-/
module
public import Mathlib.Algebra.Algebra.Basic
public import Mathlib.Algebra.CharP.Defs
public import Mathlib.Algebra.Polynomial.Degree.Lemmas
public import Mathlib.... |
6,457 | lean_file_chunk | 23469c9fa918f5d174e26a49e590b1fac6248478ca9cb1a740b3378c2d329157 | 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.Polynomial.Basic
public import Mathlib.RingTheory.Polynomial.Content
public import Mathlib.RingTheory.UniqueFactorizationDomain.Basic
p... |
4,945 | lean_file_chunk | b85132f3750b1b4e8c032ca682804446c52be9e95a10ebaac2d948fd4fef26b6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Junqi Liu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junqi Liu, Jinzhao Pan
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
/-!
# shifted Legendre Polynomials
In this file, ... |
8,294 | lean_file_chunk | 2521de7725c76127bc6d6a22311823e2f4b1251e3d16f61ab81b496a6a2ef16e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Uthaiwat, Oliver Nash
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Div
public import Mathlib.Algebra.Polynomial.Identitie... |
8,410 | lean_file_chunk | ca2e18355e8bc0fe78e7565d464ad37a0e67d83621fa4b72a3e677006550cd7b | 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, Andrew Yang, Yuyang Zhao
-/
module
public import Mathlib.Algebra.Polynomial.Degree.Lemmas
public import Mathlib.RingTheory.Po... |
16,091 | lean_file_chunk | a0724d84133496dd632cbd4c7a6babd7583d4fc43c8dc5506b402c2957a2f17d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.FieldTheory.SplittingField.Construction
public import Mathlib.RingTheory.Localization.Integral
public import Mathlib.RingTheory.Integral... |
5,301 | lean_file_chunk | c05aa6f081837dae6a4c86af6b3aac69ed907a6f514847b501e0cc01846d519f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.FieldTheory.Separable
/-!
# Separable degree
This file contains basics about the separab... |
2,692 | lean_file_chunk | da4144e9d97b3bcfb9495682669e0bee1014ed28555319ac9b8e59de968f545c | 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, Yuyang Zhao
-/
module
public import Mathlib.Algebra.Algebra.Tower
public import Mathlib.Algebra.Polynomial.AlgebraMap
/-!
# Algebra towers for polynomial
This file proves some... |
11,266 | lean_file_chunk | cdfad5a0d001260d9ed21ea5b04ebba47366f4a626c615973190289a3af00f25 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
module
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.Algebra.CharP.Invertible
public import Mathlib.Algebra.CharP.Lemmas
public import Ma... |
8,116 | lean_file_chunk | a549dc1ff5ce73ba8da082c1e4811821fd469007ec0b438ec32c540ce2385a17 | 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.RingTheory.Ideal.Quotient.Operations
public import Mathlib.RingTheory.Polynomial.Content
/-!
# The content ideal of a polynomial
... |
3,852 | lean_file_chunk | 6c73aafb5b2a136b31fd799271001a8378b4744d282641b53ecc4fadec3e80e8 | 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, María Inés de Frutos-Fernández
-/
module
public import Mathlib.Algebra.Polynomial.Degree.Defs
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Algebra.Ri... |
3,391 | lean_file_chunk | 2db6e2a1d05ad73cc6849effb33ea5ae614ced60b0530f4ab759ec10832286a3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.Analysis.Complex.Polynomial.UnitTrinomial
public import Mathlib.RingTheory.Polynomial.GaussLemma
public import Mathlib.Tactic.LinearCo... |
23,970 | lean_file_chunk | a1c11be07eb50604354ddab8c41baa50aa09b5393bbfa343672b495052c1c541 | 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.MvPolynomial.CommRing
public import Mathlib.Algebra.MvPolynomial.Equiv
public import Mathlib.Algebra.Polynomial.BigOperators
public import... |
17,499 | lean_file_chunk | 97dfc92449741b333bcbfca2b0af95941e3308d2b1df0e536566e7087360dc1e | lean_source_file | pkg_mathlib_source_file_chunks | instance isPrime_map_C_of_isPrime {P : Ideal R} [IsPrime P] :
IsPrime (map (C : R →+* R[X]) P : Ideal R[X]) :=
(isPrime_map_C_iff_isPrime P).mpr ‹_›
theorem is_fg_degreeLE [IsNoetherianRing R] (I : Ideal R[X]) (n : ℕ) :
Submodule.FG (I.degreeLE n) :=
letI := Classical.decEq R
isNoetherian_submodule_left.... |
16,506 | lean_file_chunk | a19412c2280fd019cf28eefef8fb5acee345832b2241d35f30ccbcdfa07216bd | 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
-/
module
public import Mathlib.Algebra.MvPolynomial.PDeriv
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
public i... |
12,638 | lean_file_chunk | 77bc3d996c7c72f23d031cabe9dbadd5f883396a5784471445809850ae54733d | 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, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
-/
module
public import Mathlib.Algebra.Field.Equiv
public import Mathlib.Algebra.Polynomial.Div
public import Mathlib.Alge... |
11,536 | lean_file_chunk | a90446c8f4d8eefe313828c0710d0e75ac4a61409ad0ed6df477e386570330ea | 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.Data.Multiset.Fintype
public import Mathlib.RingTheory.AdjoinRoot
public import Mathlib.RingTheory.Polynomial.RationalRoot
public import Mathl... |
3,949 | lean_file_chunk | 8043288bb599e1d4b742b7c33669d30514baad365a0efb18c3f01a2b459981a2 | 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.RingDivision
public import Mathlib.RingTheory.Adjoin.Polynomial.Basic
public import Mathlib.RingTheory.Ideal.Maps
/-!
# Ideals... |
18,086 | lean_file_chunk | d500032d205317eb93a062f7b546a3575953c1c5df9d19a41901d87e212571c0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.Algebra.GCDMonoid.Finset
public import Mathlib.Algebra.Polynomial.CancelLeads
public import Mathlib.Algebra.Polynomial.EraseLead
public ... |
8,220 | lean_file_chunk | 4eb988666431841d28760bb209a55a180871599008c4b11a0c3ff85dcb593144 | 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: Anne Baanen, Kenny Lau
-/
module
public import Mathlib.Algebra.Polynomial.Div
public import Mathlib.Algebra.Polynomial.Taylor
public import Mathlib.LinearAlgebra.Determinant
public import ... |
13,217 | lean_file_chunk | be31bba1ec2a38e4cab08e52aa1065d3f886a5744d7d3189bdc3d2446ef7ce1d | 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.Algebra.Order.Ring.IsNonarchimedean
public import Mathlib.RingTheory.PowerSeries.GaussNorm
/-!
# Gauss norm for polynomials
This ... |
5,677 | lean_file_chunk | a7e6c45118481c5d30143bda56d48cc0ff96794dec5a7a20915f2adba8084d4b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Qinchuan Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Qinchuan Zhang
-/
module
public import Mathlib.Tactic.FieldSimp
public import Mathlib.Tactic.LinearCombination
public import Mathlib.RingTheory.Polynomial.Vieta
/-!
# Vieta's Formula... |
4,240 | lean_file_chunk | 06f14e5be75ca5d71ceb63afca6663f620007df8d1d358f85582a584e4f95778 | 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.Algebra.Polynomial.Splits
public import Mathlib.GroupTheory.Perm.ClosureSwap
public import Mathlib.RingTheory.Ideal.Quotient.Operation... |
7,038 | lean_file_chunk | 3577f8fd9ade30cdd0ea1b435ff5ab84d3314afa1a2f635360ee788d04385da7 | 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
-/
module
public import Mathlib.Algebra.Polynomial.Splits
public import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
/-!
# Vieta's Formula
The main result is `Multise... |
2,827 | lean_file_chunk | 50e5212d98e8175ffaff4cc4fee190a8b1bba509e981569e097d8d8130beaf40 | 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.Polynomial.Eval.Irreducible
public import Mathlib.RingTheory.Polynomial.Nilpotent
/-!
# Polynomials over an irreducible ring
This file contai... |
5,127 | lean_file_chunk | bcfad2dc2d66906fcb2689ba54d6fdacfd80e775e488c64c4998d348913c4cdf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
module
public import Mathlib.Algebra.MonoidAlgebra.MapDomain
public import Mathlib.Algebra.Polynomial.Degree.Support
public import Mathlib.Tactic.NoncommRing
/-! # In... |
13,185 | lean_file_chunk | 79c5d6738e9368d934cab47c800bb2f9919d0c553fe8908c36f7e0953b25e079 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Fangming Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fangming Li, Jujian Zhang
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Eval.SMul
public import Mathlib.Algebra.Polynomial.Roots
... |
23,953 | lean_file_chunk | 44fa8643568fe45275b801b042e88f7a7a9bf8e74a3f20bf24da53ec943dc790 | 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, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
pub... |
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