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 |
|---|---|---|---|---|---|
23,048 | lean_file_chunk | 79589c65452abe1455496e4d03fd92ae937b0a62d65171e1406ed19a291e1ca6 | lean_source_file | pkg_mathlib_source_file_chunks | /-- `S n` is the `n`th rescaled Chebyshev polynomial of the second kind (also known as a
Vieta–Fibonacci polynomial), given by $S_n(2x) = U_n(x)$. See
`Polynomial.Chebyshev.S_comp_two_mul_X`. -/
noncomputable def S : ℤ → R[X]
| 0 => 1
| 1 => X
| (n : ℕ) + 2 => X * S (n + 1) - S n
| -((n : ℕ) + 1) => X * S (-n) ... |
14,678 | lean_file_chunk | 6b3b9c292e44742ef0ac524bb88c34d90c1018105d20f121492078abf415fff4 | 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, Devon Tuma
-/
module
public import Mathlib.Algebra.Polynomial.Splits
/-!
# Scaling the roots of a polynomial
This file defines `scaleRoots p s` for a polynomial `p` in one... |
23,944 | lean_file_chunk | 4ba547c743e9522b5d90b85cc227617a64c7b1cb85428bd76eec78540bd8c319 | 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.Extension.Presentation.Submersive
public import Mathlib.RingTheory.FiniteStability
public import Mathlib.RingTheory.Flat.Faithfully... |
13,118 | lean_file_chunk | 5c4309a94291dce40fe7138c374a603ccfb8424f0f5d24d8408115a2def2c3cd | lean_source_file | pkg_mathlib_source_file_chunks | refine AlgHom.restrictScalars_injective R (Algebra.TensorProduct.ext ?_ ?_)
· ext; simp [MvPolynomial.universalFactorizationMapLiftEquiv, MvPolynomial.mapEquivMonic,
UniversalFactorizationRing.factor₁, coeff_freeMonic]; rfl
· ext; simp [MvPolynomial.universalFactorizationMapLiftEquiv, MvPolynomial.mapEq... |
5,735 | lean_file_chunk | c99a9b3975612cb12f8a0a8a8778a2913a910efeee066238b98016fe08dbad2e | 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
-/
module
public import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
public import Mathlib.RingTheory.Localization.NumDen
public import Mathlib.RingTheory.Polynomial.S... |
5,236 | lean_file_chunk | b2d1457e422df4508976c16b1fe7bbbc843ce8bdea12e2d0a88474bd1068617d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jineon Baek and Seewoo Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jineon Baek, Seewoo Lee
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
public import Mathlib.LinearAlgeb... |
2,628 | lean_file_chunk | 315a208257003700deb6cef4dd04c3437f4fb46b767bde0bf71c6596c450f8c8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jineon Baek, Seewoo Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jineon Baek, Seewoo Lee
-/
module
public import Mathlib.Algebra.Polynomial.FieldDivision
public import Mathlib.RingTheory.Polynomial.Wronskian
public import Mathlib.RingTheor... |
3,907 | lean_file_chunk | eeac220edcd5db6c838a5b4cb188f124ef5fc3604635f2c10e4a0d38373ec0a8 | 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.FieldTheory.IntermediateField.Adjoin.Basic
public import Mathlib.FieldTheory.MvRatFunc.Rank
public import Mathlib.RingTheory.Algebraic.Cardi... |
5,603 | lean_file_chunk | d166e22745527ed185b971510f194ce5886df90dea0bc55eaafcc822ab0ac275 | 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.FieldTheory.AlgebraicClosure
public import Mathlib.RingTheory.Algebraic.Integral
public import Mathlib.RingTheory.AlgebraicIndependent.Transcenden... |
22,153 | lean_file_chunk | 199f952e816b6bf05863832f629672c3473ec4491e8d61d61ae5576953e133e7 | 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.Algebra.Subalgebra.Tower
public import Mathlib.Algebra.MvPolynomial.Equiv
public import Mathlib.Algebra.MvPolynomial.Monad
public im... |
7,878 | lean_file_chunk | 125ca6786c6638681038a319678c2d614031ccb562004d191d39242c9797a6d7 | 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.MvPolynomial.CommRing
/-!
# Algebraic Independence
This file defines algebraic independence of a family of elements of an `R` alge... |
13,263 | lean_file_chunk | 0ed5445c7cca83048ddd0c91c78f93295317e5f8748effbfc72b647cacc17ef1 | 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.Data.Fin.Tuple.Reflection
public import Mathlib.RingTheory.Algebraic.MvPolynomial
public import Mathlib.RingTheory.AlgebraicIndependent.Basi... |
23,929 | lean_file_chunk | d5771881c6d65111de804eb25de275c746835f7d5bc108499168abf98d89061e | 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.Combinatorics.Matroid.IndepAxioms
public import Mathlib.Combinatorics.Matroid.Rank.Cardinal
public import Mathlib.FieldTheory.IntermediateFi... |
4,698 | lean_file_chunk | 4409c45a6965dc8bc1d233a065c7073ef7b993fb72a4ecb542fbb0f98665b892 | lean_source_file | pkg_mathlib_source_file_chunks | rw [← matroid_cRank_eq, ← Matroid.rankFinite_iff_cRank_lt_aleph0] at fin
exact .of_subtype_range hx.injective <| matroid_indep_iff.mpr hx.to_subtype_range
|>.isBase_of_cRank_le <| lift_le.mp <| (matroid_cRank_eq R A ▸ le).trans_eq
(mk_range_eq_of_injective hx.injective).symm
theorem isTranscendenceBasis_of... |
2,754 | lean_file_chunk | 27b0afdbd0470c5abd8f8735280161642ca54a298ff2613bf07ee1b35b655fa1 | 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.FieldTheory.IntermediateField.Adjoin.Algebra
public import Mathlib.RingTheory.AlgebraicIndependent.Defs
/-!
# Algebraic Independence
This ... |
519 | lean_file_chunk | 80af43608fedc154971a1b5e5aa4e9f3fa6bb5c6518fedbba4dd4b6308aeb4ce | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 euprunin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: euprunin
-/
module
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.RingTheory.NonUnitalSubring.Defs
public import Mathlib.RingTheory.TwoSidedIdeal.Basic
/-!
# Additional inst... |
8,635 | lean_file_chunk | b94113ffd87824de0d2ab545a871ee87850971e0f8f9d6e50d53c410b75e1a36 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
module
public import Mathlib.Tactic.Abel
public import Mathlib.Algebra.Ring.Opposite
public import Mathlib.GroupTheory.GroupAction.SubMulAction
public import Mathlib.Rin... |
15,563 | lean_file_chunk | e20b01f05da507d8c72dff85622505824f3ded08f7626bc38f5ea94db347b724 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Jireh Loreaux, Yunzhou Xie
-/
module
public import Mathlib.Algebra.Group.Subgroup.Map
public import Mathlib.Algebra.Module.Opposite
public import Mathlib.Algebra.Module.Su... |
2,015 | lean_file_chunk | 520e4f3fa9573c54a52c8b9bb5aa396738bdc686c201e847e9b4d941f37698ef | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
module
public import Mathlib.RingTheory.Congruence.BigOperators
public import Mathlib.RingTheory.TwoSidedIdeal.Basic
/-!
# Interactions between `∑, ∏` and two-sided ide... |
1,870 | lean_file_chunk | e46b65c201bb29ef68b6dda9f9da72e86ef3118f39db38e7d021dab31ce407ff | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jujian. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Jujian Zhang
-/
module
public import Mathlib.RingTheory.TwoSidedIdeal.Basic
public import Mathlib.RingTheory.TwoSidedIdeal.Lattice
/-!
# Kernel of a ring homomorphism as a tw... |
5,228 | lean_file_chunk | e4dd678ae1fab838502b142ea49b98916feb5b8781f7904e2963d3d64d4e11fc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
module
public import Mathlib.RingTheory.TwoSidedIdeal.Basic
/-!
# The complete lattice structure on two-sided ideals
-/
@[expose] public section
namespace TwoSidedIde... |
23,957 | lean_file_chunk | 51d69855f484e82d7046377c00c390a7c24078bafe659e824d795866c770faba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Algebra.Group.Submonoid.Membership
public import Mathlib.Algebra.Group.Subsemigroup.Membership
public import Mathlib.Algebra.Group.Subsemi... |
6,539 | lean_file_chunk | f575ac2bb23787afff3502a7574efb1e5044033716dfc37f7ba6ea954475fd73 | lean_source_file | pkg_mathlib_source_file_chunks | /-- Product of non-unital subsemirings is isomorphic to their product as semigroups. -/
def prodEquiv (s : NonUnitalSubsemiring R) (t : NonUnitalSubsemiring S) : s.prod t ≃+* s × t :=
{ Equiv.Set.prod (s : Set R) (t : Set S) with
map_mul' := fun _ _ => rfl
map_add' := fun _ _ => rfl }
theorem mem_iSup_of_dir... |
12,652 | lean_file_chunk | 76888fa960f369179a5b45e93bd4cadb01766ecb85d62e88ac1b08f0e1c46119 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Algebra.Ring.Hom.Defs
public import Mathlib.Algebra.Ring.InjSurj
public import Mathlib.Algebra.Group.Submonoid.Defs
public import Mathlib.... |
23,977 | lean_file_chunk | 07716fcd3d20ae2089f90abaab553bf9b4f83241973ae7597386d6f4e69b6c51 | 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.... |
1,579 | lean_file_chunk | d2904f86f301676ab9159c5b6bb5cc29065fcf7028aea27fb6daaec8650b6b10 | lean_source_file | pkg_mathlib_source_file_chunks | /-- Rescale a multivariate power series, as a `MonoidHom` in the scaling parameters. -/
noncomputable def rescaleMonoidHom :
(σ → R) →* MvPowerSeries σ R →+* MvPowerSeries σ R where
toFun := rescale
map_one' := rescale_one
map_mul' a b := by ext; simp [mul_comm, rescale_rescale]
end CommSemiring
section Com... |
10,100 | lean_file_chunk | 05c09812abf0f20bfbe88add3d8662e3c58bae5905d50993af3c9567200e88d4 | 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.Group.Units.Basic
public import Mathlib.RingTheory.MvPowerSeries.Basic
public import Mathlib.RingTheory.MvPowerSeries... |
10,367 | lean_file_chunk | 208e659e6c2e5cf5030b1fdb2e2622b4007e09e957f74eb74993b26e235f38e5 | 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.MvPowerSeries.Substitution
public import Mathlib.Algebra.CharP.Frobenius
public import Mathlib.Algebra.MvPolynomial.Expand
public i... |
16,651 | lean_file_chunk | 2725ca25d93d404f945ccc3532f47d879cb35e363e34ecff64fea5aedc36babd | 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.Basic
public import Mathlib.RingT... |
11,601 | lean_file_chunk | 33ab6d2ba551e0f1ce8cfc677e7284c06cf307efc9a4a386940c6ee2c2dab1eb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Bingyu Xia. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Bingyu Xia
-/
module
public import Mathlib.Order.Filter.TendstoCofinite
public import Mathlib.RingTheory.MvPowerSeries.Basic
public import Mathlib.Algebra.MvPolynomial.Re... |
23,959 | lean_file_chunk | 998662912662569f6cfbd1ddd76b1b8f51142f439094ccd1818f36e5ded14005 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir, María Inés de Frutos Fernandez. 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.Data.ENat.Basic
public import Mathlib.Data.Finsupp.Weight
... |
5,259 | lean_file_chunk | 24aa6142136450a87bf00f4d35884377709da1d01347c811d369b23ab5aee998 | lean_source_file | pkg_mathlib_source_file_chunks | theorem coeff_weightedHomogeneousComponent (p : ℕ) (d : σ →₀ ℕ) (f : MvPowerSeries σ R) :
coeff d (weightedHomogeneousComponent w p f) =
if weight w d = p then coeff d f else 0 :=
rfl
variable {w} in
theorem weightedHomogeneousComponent_of_lt_weightedOrder_eq_zero
{f : MvPowerSeries σ R} {p : ℕ} (hf : ... |
14,714 | lean_file_chunk | 2cc194f34807c0da3a6a6291068f3f6d4e1518470a4ea6c5921c97fe173ef25d | 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.Algebra.MvPolynomial.CommRing
public import Mathlib.RingTh... |
12,650 | lean_file_chunk | 26de9702ae97d718fbbce6d24b23db9b2398cd67041277710eae542c6eac3cdf | 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.MvPowerSeries.Basic
public import Mathlib.Data.Finsupp.Interval
public import Mathlib.Algebra.MvPolynomial.Eval
pu... |
6,834 | lean_file_chunk | a12049e8f358e83416e9ebc33165a85809a8b9cbc3d909ab912b131369750a1f | 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.Data.Finsupp.WellFounded
public import Mathlib.RingTheory.MvPowerSeries.LexOrder
public import Mathlib.RingTheory.MvPowerS... |
23,994 | lean_file_chunk | c5eb78a3d4a3d869e12a27dfcbd70e209f85980751d0475d773afa7328844182 | 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.Order.Antidiag.Finsupp
public import Mathlib.Data.Finsupp.Weight
public import Mathlib.LinearAlgebra.Pi
public import... |
10,803 | lean_file_chunk | 474c423d02d0a50a32d229f56167db6e4b90e602a0b6d3a5db5ce5431a1473b4 | lean_source_file | pkg_mathlib_source_file_chunks | induction s using Finset.cons_induction with
| empty => simp
| cons a s ha h => simp [h, monomial_mul_monomial]
/-- The `d`th coefficient of a power of a multivariate power series
is the sum, indexed by `finsuppAntidiag (Finset.range n) d`, of products of coefficients -/
theorem coeff_pow [DecidableEq σ] (f : MvPo... |
5,230 | lean_file_chunk | 17016923b19cf648316a1c9115cfca3bedab92a79438a4473814c426c2685bc6 | 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.Ring.Basic
public import Mathlib.RingTheory.MvPowerSeries.Basic
/-!
# Gauss norm for multivariate power series
This file... |
7,019 | lean_file_chunk | d63cf2cafb648cf53a57b2e6d7f61e297e1e639fb8d73581122880682eb498c3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Bingyu Xia. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bingyu Xia
-/
module
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.RingTheory.AdicCompletion.Algebra
public import Mathlib.RingTheory.MvPolynomial.Ideal
public imp... |
7,373 | lean_file_chunk | 2bfac6862073de57b3c4a2e203e0ad83735288f7556f9afa85c1839d3a7c9949 | 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.Data.Finsupp.Interval
public import Mathlib.RingTheory.Ide... |
6,198 | lean_file_chunk | 08760bca86d0b31c72287cf325f15377c341408d3018f2e3a3688e2970824c09 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
module
public import Mathlib.RingTheory.MvPowerSeries.Basic
public import Mathlib.Data.Finsupp.WellFounded
/-! LexOrder of multivariate power series
... |
4,397 | lean_file_chunk | 089ed257f7f2b79bf44ca1e5050a5e487255102c429386901984e59514081b77 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yongle Hu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yongle Hu
-/
module
public import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
public import Mathlib.RingTheory.LocalProperties.Basic
public import Mathlib.RingTheory.Spectrum.Maxim... |
23,946 | lean_file_chunk | fdf7b7be4ff1463bed1f6faf2a2953aee035d5274cc131ebf5eaa13a29c7a4f3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.Localization.AtPrime.Basic
public import Mathlib.RingTheory.Localization.BaseChange
public import Mathlib.RingTheory.Localization.L... |
3,157 | lean_file_chunk | 034922f5c2d88cac3af19b3fedbffea2976bd30a087e7e69b7a2ae1600c01345 | lean_source_file | pkg_mathlib_source_file_chunks | rw [map, span, Submodule.localized'_eq_span, Algebra.coe_linearMap]
theorem Ideal.localized₀_eq_restrictScalars_map (I : Ideal R) :
Submodule.localized₀ p (Algebra.linearMap R S) I = (I.map (algebraMap R S)).restrictScalars R :=
congr(Submodule.restrictScalars R $(localized'_eq_map S p I))
theorem Algebra.ideal... |
9,114 | lean_file_chunk | 4bd4233a6adf63449d8ee5b49809d0b1952f03af4003c7c3684c99e4ee71ca17 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
module
public import Mathlib.Algebra.Category.Grp.Zero
public import Mathlib.Algebra.Category.ModuleCat.EnoughInjectives
public import Mathlib.Algebra.Category.ModuleCat.L... |
6,037 | lean_file_chunk | b5440bbb0406d62dc9a99fae062f7a2c245972e19ba9d8023f12510d8e8b5567 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
module
public import Mathlib.Algebra.Module.FinitePresentation
public import Mathlib.Algebra.Module.Injective
public import Mathlib.LinearAlgebra.BilinearMap
public import... |
8,716 | lean_file_chunk | 856abbe8378c09f4d6319d7f0d37d26753ba6ae423d858e7d73f6aaee4468506 | 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, David Swinarski
-/
module
public import Mathlib.Algebra.Module.FinitePresentation
public import Mathlib.Algebra.Module.Projective
public import Mathlib.LinearAlgebra.Dimensi... |
14,805 | lean_file_chunk | 75fbc24151591323297d9d19c1d250c096cda279e943b998640f8b7521ea9691 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sihan Su. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sihan Su, Yongle Hu, Yi Song
-/
module
public import Mathlib.Algebra.Exact
public import Mathlib.RingTheory.LocalProperties.Submodule
public import Mathlib.RingTheory.Localization.Algebra
p... |
9,167 | lean_file_chunk | f491660079a35081fa49c74b5945f391fbb083af5cdac27255a2c2fea1d33828 | 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, David Swinarski
-/
module
public import Mathlib.Algebra.Module.LocalizedModule.Submodule
public import Mathlib.RingTheory.Localization.AtPrime.Basic
public import Mathlib.Ri... |
8,503 | lean_file_chunk | 58430548aeb162b995ab684fed34a9237f60faa8dace9c4e369dbcad8b960928 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
module
public import Mathlib.Algebra.Category.ModuleCat.Localization
public import Mathlib.Algebra.Category.ModuleCat.Projective
public import Mathlib.CategoryTheory.Abeli... |
5,509 | lean_file_chunk | e837f93a79dcb152641c8c9d578dadf109af4ee588746a934b31aee3dadea646 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yiming Fu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yiming Fu
-/
module
public import Mathlib.RingTheory.DedekindDomain.PID
public import Mathlib.RingTheory.KrullDimension.PID
/-!
# Local properties for semilocal rings
This file proves so... |
1,954 | lean_file_chunk | dd20a731ec3d53c1c111e50b1f250ae731f5c70fd969dcabc9ac79d23a6abc1c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.LocalProperties.Basic
public import Mathlib.RingTheory.Nilpotent.Defs
/-!
# `IsReduced` is a local property
In this file, we prov... |
6,102 | lean_file_chunk | 9f5b30ad91b1d447e131a22982efccb34806a7b9a78f8a082d6fdc5dc64805cb | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.RingTheory.Localization.FractionRing
public import Mathlib.RingTheory.Localization... |
23,960 | lean_file_chunk | 9c4dad2dd40987fc30f795f6144a2331d47fffd74633f5a737d0c3af7a619a66 | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.RingTheory.Algebraic.Integral
public import Mathlib.RingTheory.Localization.Algebr... |
3,740 | lean_file_chunk | 82637d3d27f316e4a168b1ff7505218beb67f34abf883ca29124ad8c368638c5 | lean_source_file | pkg_mathlib_source_file_chunks | have := FractionRing.isScalarTower_liftAlgebra R K
rw [IsFractionRing.isAlgebraic_iff R (FractionRing R) K, isAlgebraic_iff_isIntegral]
obtain ⟨a : S, b, ha, rfl⟩ := div_surjective S x
obtain ⟨f, hf₁, hf₂⟩ := h b
rw [div_eq_mul_inv]
refine .mul ?_ (.inv ?_) <;> exact isAlgebraic_iff_isIntegral.mp <|... |
15,328 | lean_file_chunk | c3cc118ad5d086f6b0de19def999145f281f4bb912a3394347514e7f0d7612ed | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Anne Baanen
-/
module
public import Mathlib.Algebra.Module.LocalizedModule.IsLocalization
public import Mathlib.LinearAlgebra.Basis.Basic
public import Mathlib.RingTheory.Loca... |
1,076 | lean_file_chunk | d8e8d70fcd02301a3ff8a3779990ce5e4ef0f6218a9142e9500440df0dc38083 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Ruben Van de Velde. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ruben Van de Velde
-/
module
public import Mathlib.RingTheory.Int.Basic
public import Mathlib.RingTheory.Localization.NumDen
/-!
# Ring-theoretic fractions in `ℚ`
-/
public sect... |
5,478 | lean_file_chunk | ae2b5d2d5f3df641977959b2bd1a63e295070d9ecb3c121d5d54d57f524f4422 | 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
-/
module
public import Mathlib.RingTheory.Localization.LocalizationLocalization
public import Mathlib.LinearAlgebra.FreeModule.Basic
public import Mathlib.Algebra.A... |
5,466 | lean_file_chunk | b7870b4161c444b6a88696b7c2fa6c88c7a2efed43c239284e7e776eccc3b4be | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.Algebra.Group.Pointwise.Set.Scalar
public import Mathlib.Algebra.Ring.Subsemiring.... |
5,060 | lean_file_chunk | a95984d0bb18638c65e0e72ca9d198b1df113759ec3e0f1bb55160cf46ce9c4a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Madison Crim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Madison Crim
-/
module
public import Mathlib.Algebra.Algebra.Pi
public import Mathlib.Algebra.BigOperators.Pi
public import Mathlib.Algebra.Divisibility.Prod
public import Mathlib.Algeb... |
4,257 | lean_file_chunk | e80d30d4368b10baeda30e1a4240fd21590f740ade950e718b2cb66c593f9eab | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.GroupTheory.Submonoid.Inverses
public import Mathlib.RingTheory.FiniteType
public ... |
23,996 | lean_file_chunk | 3e00754ec3ed3e45baeeedb1164cf097f18adf0681db7b5ab809d87bf96e5cb6 | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.Algebra.Algebra.Tower
public import Mathlib.Algebra.Field.IsField
public import Ma... |
2,510 | lean_file_chunk | 4be58fd4d34c5f4d6d25816189078df6400764a660646e512e6234afc3474dde | lean_source_file | pkg_mathlib_source_file_chunks | variable (M) in
/--
Another version of `IsLocalization.map_injective_of_injective` that is more general for the choice
of the localization submonoid but requires it does not contain zero.
-/
theorem map_injective_of_injective' {f : R →+* S} {Rₘ : Type*} [CommRing Rₘ] [Algebra R Rₘ]
[IsLocalization M Rₘ] (Sₘ : Type*... |
3,180 | lean_file_chunk | 6f1635fdbb607c1cdc069a3755ac924aff5703287859fe6145000111685e36f0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
module
public import Mathlib.RingTheory.Localization.FractionRing
public import Mathlib.GroupTheory.MonoidLocalization.Cardinality
public import Mathlib.RingTheory.O... |
18,475 | lean_file_chunk | 757b0cec9bed073cc13d3bec0cf11aaa2943c80257995846a7cd3da297e23440 | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.GroupTheory.MonoidLocalization.Away
public import Mathlib.RingTheory.Ideal.IsPrima... |
5,481 | lean_file_chunk | 3ff9d64140a967a73b7122317980ce6c847788305fc1b2867d956745bfc38601 | 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
-/
module
public import Mathlib.RingTheory.Localization.Module
public import Mathlib.RingTheory.Norm.Basic
public import Mathlib.RingTheory.Discriminant
/-!
# Field/algebra... |
23,990 | lean_file_chunk | 133cd8c75fb74681012e1f72cd4a7db008929771afb722811aac13334d538a22 | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Defs
public import Mathlib.Algebra.Regular.Basic... |
14,142 | lean_file_chunk | 776836b586e29dd1929a45d47693988cc7a4fbba4562610068b6b0946970a367 | lean_source_file | pkg_mathlib_source_file_chunks | (toLocalizationMap M S).lift_injective_iff hg
variable (M) in
include M in
lemma injective_iff_map_algebraMap_eq {T} [CommSemiring T] (f : S →+* T) :
Function.Injective f ↔ ∀ x y,
algebraMap R S x = algebraMap R S y ↔ f (algebraMap R S x) = f (algebraMap R S y) := by
rw [← IsLocalization.lift_of_comp (M :=... |
12,853 | lean_file_chunk | 759330f0c0da97c2e941b033ef99460b92f2d034c9bb3049e8c9e0a207a731f2 | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.RingTheory.Localization.AtPrime.Basic
public import Mathlib.RingTheory.Localizatio... |
7,656 | lean_file_chunk | b19a6991b1ef76bc27869cf21337946cb95a96cf94fc7ec7b6850bc70fba8351 | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.RingTheory.Localization.FractionRing
public import Mathlib.RingTheory.Localization... |
8,807 | lean_file_chunk | 96117dbddd426c012d0565f5cdc8cf03844889dc4b19319d5a6efb9bb1ba7d9a | 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.Module.LocalizedModule.IsLocalization
public import Mathlib.RingTheory.Ideal.Maps
public import Mathlib.RingTheory.Localizat... |
15,931 | lean_file_chunk | 8bad392f894181b5b1f747fd5761fa762a2776a44b98cc97096b3b114160cc0e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
module
public import Mathlib.LinearAlgebra.DirectSum.Finsupp
public import Mathlib.RingTheory.IsTensorProduct
public import Mathlib.RingTheory.Localization.... |
4,478 | lean_file_chunk | ad5f56d6828893d52fc0df8d53c47c8728c195657aed0df6e060f4b4aea9fe93 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Algebra.Module.FinitePresentation
public import Mathlib.RingTheory.Localization.Finiteness
public import Mathlib.LinearAlgebra.FreeModule.Stro... |
13,160 | lean_file_chunk | 7cff58cf19e6bbca143b7f424e7dd5d47213216574ffb00452e398f2d030b265 | 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.Module.LocalizedModule.Int
public import Mathlib.RingTheory.Localization.Algebra
public import Mathlib.RingTheory.Localizati... |
23,995 | lean_file_chunk | ab068c407982081066ed1ec6ee5086ec6aeb9230d019ad7404c9a6dbf78afccf | 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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
module
public import Mathlib.Algebra.Field.Equiv
public import Mathlib.Algebra.Field.Subfield.Basic
public impo... |
3,480 | lean_file_chunk | 0fcd38701410549740bbc6954be5b06c56e506805db207791fe699e4e0738a61 | lean_source_file | pkg_mathlib_source_file_chunks | /-- The fraction ring of a commutative ring `R` as a quotient type.
We instantiate this definition as generally as possible, and assume that the
commutative ring `R` is an integral domain only when this is needed for proving.
In this generality, this construction is also known as the *total fraction ring* of `R`.
-/
... |
23,973 | lean_file_chunk | 62eab0bd4338e7b508d86f2d6f39785234865067cdc7224877e567e3a30712d5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.Ring.Action.Rat
public import Mathlib.RingTheory.HahnSeries.Multiplication
public import Mathlib.Data.Rat.Cast.Lemmas
/-!
# Sum... |
11,288 | lean_file_chunk | 0bca74b24d68eed6ba0e69b577b332165919f13638cea2ab07140ff6d0d9490f | lean_source_file | pkg_mathlib_source_file_chunks | · rw [dif_pos hb] at h
exact Set.mem_iUnion.2 ⟨Classical.choose hb, h⟩
· simp [-Set.mem_range, dif_neg hb] at h
finite_co_support' g :=
((s.finite_co_support g).image f).subset
(by
intro b h
by_cases hb : b ∈ Set.range f
· simp only [Ne, Set.mem_setOf_eq, dif_pos hb] at h
... |
1,984 | lean_file_chunk | e06a78fae2b5cfee5a388977ba4455f4c77bc83f8b2cef2da2eb7d8043313c26 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.RingTheory.HahnSeries.Multiplication
public import Mathlib.RingTheory.Valuation.Basic
/-!
# Valuations on Hahn Series rings
If `Γ` is a... |
4,322 | lean_file_chunk | 8c6f2243a559202cea47cbadc91515882e17669aba8b03a70a383c9552808048 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Weiyi Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Weiyi Wang
-/
module
public import Mathlib.Algebra.Order.Module.HahnEmbedding
public import Mathlib.Algebra.Module.LinearMap.Rat
public import Mathlib.Algebra.Field.Rat
public import Math... |
18,209 | lean_file_chunk | ab4bdd799983e6860dc2cbc5e8dda7c3b149cd72d9852c51ec36c1ab34ec1cc9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Weiyi Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Weiyi Wang
-/
module
public import Mathlib.Algebra.Order.Archimedean.Class
public import Mathlib.Algebra.Order.Ring.Synonym
public import Mathlib.Order.Hom.Lex
public import Mathlib.Order... |
8,089 | lean_file_chunk | 462e30aec0e6e23c11fe8caa43ebddb0380c403b680906b6e6aecfb210cb6176 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
module
public import Mathlib.RingTheory.HahnSeries.Multiplication
public import Mathlib.RingTheory.PowerSeries.Basic
public import Mathlib.RingTheory.MvPowerSeries.N... |
18,173 | lean_file_chunk | c2bd84c16656a4f35d76ece331b8d642623219d439f45fc981b06b055e4672c2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.BigOperators.Group.Finset.Basic
public import Mathlib.Algebra.Group.Pi.Lemmas
public import Mathlib.Algebra.Group.Support
public... |
23,945 | lean_file_chunk | 3fe4c54550983d54082fc3a94d2ab66eb7fb86422351cfc0f3b30b64a1565754 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Lattice
public import Mathlib.Algebra.GroupWithZero.Regular
public import Mathlib.Algebra.Mod... |
17,929 | lean_file_chunk | 192e67fd25a26002ac269b45d537f5994de104005a4713357e6f0374968c8937 | lean_source_file | pkg_mathlib_source_file_chunks | theorem order_single_mul_of_isRegular {g : Γ} {r : R} (hr : IsRegular r)
{x : R⟦Γ⟧} (hx : x ≠ 0) : (((single g) r) * x).order = g + x.order := by
obtain _ | _ := subsingleton_or_nontrivial R
· exact (hx <| Subsingleton.eq_zero x).elim
have hrx : ((single g) r).leadingCoeff * x.leadingCoeff ≠ 0 := by
rwa [... |
22,324 | lean_file_chunk | d18ec8ef63e8d449b64e3e4a2657859fde2a634d6acf19876c777e345a15f488 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.Notation.Support
public import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
public import Mathlib.Data.Finsupp.Defs
public imp... |
8,978 | lean_file_chunk | 19a093e5f69a5b5d8bfb92a80854f8f80b23c6b757e6401adee90c7d8499509e | 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.HahnSeries.Summable
public import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Evaluation of power series in Hahn Series
We d... |
7,383 | lean_file_chunk | d4c746b99b68c8ed22f2adb8a34cd1ceba13b163919d781113aeff7cfcaeac7a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
module
public import Mathlib.Algebra.Field.Subfield.Defs
public import Mathlib.RingTheory.HahnSeries.Summable
public import Mathlib.SetTheory... |
4,878 | lean_file_chunk | c8838b2d557e4ebdaf566ddbe4e39b87ce3365dad34bfe71433aef5415e16617 | 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.HahnSeries.HEval
public import Mathlib.RingTheory.PowerSeries.Binomial
/-!
# Binomial expansions of powers of Hahn Series
We... |
23,959 | lean_file_chunk | 5768129745745a1db594971c3d21e47c14d3e4787cb70a762883a0c07a8b3539 | 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.Algebra.Module.FinitePresentation
public import Mathlib.RingTheory.Artinian.Ring
public import Mathlib.RingTheory.FiniteStability
public impor... |
5,065 | lean_file_chunk | 5a3137463a6512d9f4369cc8d804e82bfe666151582b3fde39d2cc796ff9c449 | lean_source_file | pkg_mathlib_source_file_chunks | let f : Localization.Away e →ₐ[S] Localization.AtPrime p.asIdeal :=
IsLocalization.liftAlgHom (M := .powers e) (f := Algebra.ofId _ _) <| by
simp only [Subtype.forall]
refine Submonoid.powers_le (P := (IsUnit.submonoid _).comap _).mpr ?_
simpa [IsUnit.mem_submonoid_iff] using IsLocalization.map_un... |
14,404 | lean_file_chunk | db1160534274ae16bca6fca6ebca35f9e3bd6ec8d6e0ce6b3c50f4ce11079aa7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 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.QuasiFinite.Basic
/-!
# Weakly Quasi-finite primes
The definition `Algebra.QuasiFiniteAt` is equivalent to the usual definition ... |
4,373 | lean_file_chunk | 7f8fab3338d1929753a876defa1acba64af80c0fbe3aee18563e6ac1570f5a9e | 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.ResidueField.Polynomial
public import Mathlib.RingTheory.QuasiFinite.Weakly
/-! # Quasi-finite primes in polynomial alge... |
2,635 | lean_file_chunk | c72fd91f0b0e71f8c2ee386e20a245880ea7aaf7cd9237dd71dac6261fe46ab0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.RingTheory.RingHom.FaithfullyFlat
public import Mathlib.RingTheory.RingHom.Injective
public import Mathlib.RingTheory.RingHom.Surjec... |
23,928 | lean_file_chunk | 363cad2017655ff719a848f378f1bae1b7a857555115e9afec42f62f311c7a51 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Judith Ludwig, Florent Schaffhauser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Florent Schaffhauser, Yunzhou Xie, Jujian Zhang
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Quotient
public import Mathlib.RingTheo... |
4,516 | lean_file_chunk | eea788f1f280d0632458402df1c2ae9992704e7c23e56e82cedf93e489268b65 | lean_source_file | pkg_mathlib_source_file_chunks | /-- If `S` is a faithfully flat `R`-algebra, then any faithfully flat `S`-Module is faithfully flat
as an `R`-module. -/
theorem trans : FaithfullyFlat R M := by
rw [iff_zero_iff_lTensor_zero]
refine ⟨Module.Flat.trans R S M, @fun N _ _ N' _ _ f => ⟨fun aux => ?_, fun eq => eq ▸ by simp⟩⟩
rw [zero_iff_lTensor_zer... |
6,194 | lean_file_chunk | 7a3593f91f5829a6325dc56ec010d211fb0c89872cc9cbdfcdad3a7e66e53e8c | 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.RingTheory.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.Ideal.Over
public import... |
8,149 | lean_file_chunk | 90d711bab9f59bc6252c8c8db33705219f8c9782adeab1459c1814065e92f6ab | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jingting Wang, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jingting Wang, Junyan Xu, Andrew Yang
-/
module
public import Mathlib.RingTheory.Spectrum.Prime.RingHom
public import Mathlib.RingTheory.Spectrum.Prime.TensorProduct
public ... |
3,841 | lean_file_chunk | 08c27086fae139548abe8de8da51d395af1bbad820a0db255bbb67cc93900fde | 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.ResidueField.Ideal
public import Mathlib.FieldTheory.Separable
public import Mathlib.RingTheory.IntegralClosure.IsIntegra... |
7,445 | lean_file_chunk | 832ab7f9789a63104218b1749f09e7765b1ea7ca4398230ddf3f3225296a4695 | 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.Ring.Action.End
public import Mathlib.RingTheory.Finiteness.Cardinality
public import Mathlib.RingTheory.Loc... |
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