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 |
|---|---|---|---|---|---|
2,983 | lean_file_chunk | 1d4b2965e151eeff495614373581765dc15cdc750a6e4f19d632f1de1af8419f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
module
public import Mathlib.Algebra.Order.SuccPred.PartialSups
public import Mathlib.Data.Nat.SuccPred
public import Mathlib.Order.Disjointed
/-!
# `Disjointed` fo... |
6,540 | lean_file_chunk | 37146e22d602fd3b1a0c13a2419b22d3b18cb3d56aeca7107efe117859faea1f | 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.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Order.Monoid.Defs
/-!
# Ordered scalar multiplication and vector addition
This ... |
7,598 | lean_file_chunk | 83e23a36152e4d2cf239263bf63a48a39536b96fbc7ec8ab4c67e4e5a005f5b2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
module
public import Mathlib.Algebra.Order.Floor.Ring
/-!
# Rounding
This file defines the `round` function, which uses the `floor` or `ceil` fun... |
22,069 | lean_file_chunk | 3a762ac2941c69447541f671f0f7718f77935a6e14f23d81b306f19accb655d9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Algebra.Order.Module.Defs
public import Mathlib.Algebra.Order.Module.Synonym
public import Mathlib.... |
23,960 | lean_file_chunk | 9d7fd6f67fc181a0103cf6f9247e6faed90fcfb673a8de352304c547c18de905 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Algebra.Group.ModEq
public import Mathlib.Algebra.Order.Archimedean.Basic
public import Mathlib.Algebra.Ring.Periodic
public import Mathlib.... |
23,984 | lean_file_chunk | 263b3df409e759091f3017b4797a085f7ccdff7a66c8aac70f74d12d02a0d4e4 | lean_source_file | pkg_mathlib_source_file_chunks | theorem modEq_iff_toIcoMod_ne_toIocMod : a ≡ b [PMOD p] ↔ toIcoMod hp a b ≠ toIocMod hp a b :=
(tfae_modEq hp a b).out 0 2
theorem modEq_iff_toIcoMod_add_period_eq_toIocMod :
a ≡ b [PMOD p] ↔ toIcoMod hp a b + p = toIocMod hp a b :=
(tfae_modEq hp a b).out 0 3
theorem not_modEq_iff_toIcoMod_eq_toIocMod : ¬a ≡... |
6,446 | lean_file_chunk | dfb71a8f9bf4d5d81999647dcf40fbaf1cad49f1240f305054de3128c39f1950 | lean_source_file | pkg_mathlib_source_file_chunks | simpa using toIcoMod_sub_zsmul hp a b m
@[simp]
theorem toIcoMod_sub_natCast_mul (a b : R) (m : ℕ) :
toIcoMod hp a (b - m * p) = toIcoMod hp a b :=
mod_cast toIcoMod_sub_intCast_mul hp a b m
@[simp]
theorem toIcoMod_sub_ofNat_mul (a b : R) (m : ℕ) [m.AtLeastTwo] :
toIcoMod hp a (b - ofNat(m) * p) = toIcoMod... |
1,567 | lean_file_chunk | 23d39b80f5ccb36a16c3f60fe34be38b4e61f792a2da9e3cb47a04f62f26627f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Algebra.Order.Ring.Defs
public import Mathlib.Algebra.Ring.Invertible
public import Mathlib.Data.Nat.Cast.Order.Ring
/-!
# Lemmas abo... |
5,571 | lean_file_chunk | 32078bc517eefb3919a44cdb815bd98464d6325c006a0291f7e17d49026adf5c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
module
public import Mathlib.Algebra.Notation.Lemmas
public import Mathlib.Algebra.Order.Monoid.Canonical.Defs
public import Mathlib.Algebra.Order.Ring.Def... |
8,077 | lean_file_chunk | 89e0f5c7568fcceb91fa2a517582c503f6ce7d217ae5caa72f0381928b9f34af | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
public import Mathlib.Algebra.Group.Pointwise.Set.Lattice
public import Mathlib.Algebra.Order.Group... |
23,962 | lean_file_chunk | 32a5b3dadbfdf369eadb1449c0d10cd37515973a9cae233210e6bef337f40766 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Mantas Bakšys. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys
-/
module
public import Mathlib.Algebra.Order.Module.Defs
public import Mathlib.Algebra.Order.Module.Synonym
public import Mathlib.Algebra.Order.Monoid.OrderDual
public i... |
4,252 | lean_file_chunk | ec55a4f5b0fd075a7888c0a8251125a77e95415f0b7740506ec134b312c17113 | lean_source_file | pkg_mathlib_source_file_chunks | together. Stated by permuting the entries of `g`. -/
theorem Monovary.sum_mul_comp_perm_eq_sum_mul_iff (hfg : Monovary f g) :
∑ i, f i * g (σ i) = ∑ i, f i * g i ↔ Monovary f (g ∘ σ) :=
hfg.sum_smul_comp_perm_eq_sum_smul_iff
/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise multiplication... |
2,239 | lean_file_chunk | dcf18c6ec5a4e7fc67c36b4fedba2880600fb500c42272fc0e79c18633b9e030 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Algebra.Notation.Pi.Defs
public import Mathlib.Algebra.Notation.Prod
public import Math... |
7,951 | lean_file_chunk | 7f3660311e6d732c1590475d8aaffe5acf43cd45ea04ff9f7a7a6b98c2a58ba8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Pieter Cuijpers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pieter Cuijpers
-/
module
public import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
public import Mathlib.Order.CompleteLattice.Basic
public import Mathlib.Tactic.Variable
/-!
# Th... |
11,033 | lean_file_chunk | 9b3f251a8ec3ee4eb4eccc2873e8d73da5b26109874d418c3815972ec64732a2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
module
public import Mathlib.Algebra.CharZero.Defs
public import Mathlib.Algebra.Group.Hom.Defs
public import Mathli... |
7,396 | lean_file_chunk | 2a26b3d165075a6bd4dad43cf97e61491aca16db022105a8fb5f46038a87b3e0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Mantas Bakšys, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys, Yaël Dillies
-/
module
public import Mathlib.Algebra.Order.Monovary
public import Mathlib.Algebra.Order.Rearrangement
public import Mathlib.GroupTheory.Per... |
4,874 | lean_file_chunk | 86770bb2bdb5a088ed02f5d4acee8eab9df37335bb3163bf79d55c956c1d766f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Kim Morrison
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.Order.Module.Defs
public import Mathlib.Tactic.Positivity.Core
/-!
# Order... |
13,244 | lean_file_chunk | b48d20573e1ff3be786f38832965749c773b620ad34edf37741b42496c007b19 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Siddhartha Prasad, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Siddhartha Prasad, Yaël Dillies
-/
module
public import Mathlib.Algebra.Order.Monoid.Canonical.Defs
public import Mathlib.Algebra.Ring.InjSurj
public import Mathlib.A... |
788 | lean_file_chunk | 13f76d9547a2088cda97ada8de4474a92da2c601ea633c4df78f4f78bec5ef6f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.GroupTheory.Finiteness
/-!
# Affine monoids
This file defines affine monoids as finitely generated cancellative torsion-free commutative
m... |
1,108 | lean_file_chunk | 9a9dc904c710eb2dd80c89308afb1b26adf174e11222897ce92c908608795f5f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Group.UniqueProds.Basic
public import Mathlib.GroupTheory.Finiteness
import Mathlib.Algebra.AffineMonoid.Embedding
import Mathlib.A... |
1,592 | lean_file_chunk | 1fcc37ff6bd5a4549df20953d0c5fee4dd7803d41aa27d20c61b3fb57c632c7c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies, Patrick Luo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Patrick Luo
-/
module
public import Mathlib.GroupTheory.Finiteness
public import Mathlib.GroupTheory.FreeAbelianGroup
public import Mathlib.GroupTheory.Monoid... |
6,432 | lean_file_chunk | 52ea1fb93e560ef729dce6fff17fd4859e7be4c681d73a180b896389f64eadca | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies, Patrick Luo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Patrick Luo
-/
module
public import Mathlib.Algebra.Group.Irreducible.Defs
public import Mathlib.GroupTheory.Finiteness
/-!
# An affine monoid with no non-t... |
2,687 | lean_file_chunk | f47dad36e1806f0009959d2d1624ab06d15c66f7b3b7fad6cca84a0a2cbec497 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yunzhou Xie, Jujian Zhang
-/
module
public import Mathlib.Algebra.Azumaya.Defs
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Matrix algebra is an Azumaya algebra ove... |
3,061 | lean_file_chunk | 1535180ec877823fd8395b4181fa8f9c4722b2b5e738da26e69d7248f34aef33 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yunzhou Xie, Jujian Zhang
-/
module
public import Mathlib.Algebra.Azumaya.Defs
public import Mathlib.Algebra.Central.End
public import Mathlib.Algebra.Central.TensorProduct
public import... |
2,379 | lean_file_chunk | 9523c9e894f80f086bff6192b83a84d4b5e04fe628b83c3cc2611f5e5f36bfed | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yunzhou Xie, Jujian Zhang
-/
module
public import Mathlib.Algebra.Module.Projective
public import Mathlib.RingTheory.Finiteness.Defs
public import Mathlib.RingTheory.TensorProduct.Basic
... |
2,590 | lean_file_chunk | 72dd96cfa029f71803ab7785d56b57cbe24a182de0e9bb2d1bf6244b206febb5 | 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.FreeAlgebra
public import Mathlib.SetTheory.Cardinal.Free
/-!
# Cardinality of free algebras
This file contains some results about the cardina... |
11,603 | lean_file_chunk | 58a3f767c58923bb64351bd70b66394de1bc3e4c4466274888507f048681fb1c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
module
public import Mathlib.Algebra.DirectSum.Module
public import Mathlib.Algebra.Module.Submodule.Basic
/-!
# Decompositions of additive monoids, groups,... |
23,581 | lean_file_chunk | 91d897308f551489906e04cc80e078a71b8d127a5c24f3f66369178048455b91 | 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
-/
module
public import Mathlib.Algebra.DirectSum.Basic
public import Mathlib.LinearAlgebra.DFinsupp
public import Mathlib.LinearAlgebra.Basis.Defs
/-!
# Direct sum of modules
... |
23,058 | lean_file_chunk | 9bb1cde7eb9357993d3195a946496fc92f46a7e18b20a1ba0593ad8b905826dd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
module
public import Mathlib.Algebra.GradedMonoid
public import Mathlib.Algebra.DirectSum.Basic
public import Mathlib.Algebra.Ring.Associator
/-!
# Additively-graded mult... |
21,273 | lean_file_chunk | 88c81425059beff0216e37eba9d8c7ded3fdf924afd2c7b626c034816500abb1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li
-/
module
public import Mathlib.Algebra.Algebra.Operations
public import Mathlib.Algebra.Algebra.Subalgebra.Basic
public import Math... |
16,791 | lean_file_chunk | 3c7b944c43310f5b4a150d6193986c5c1599e28de6677ec31660b4a30d259b35 | 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.Group.Submonoid.Operations
public import Mathlib.Data.DFinsupp.Sigma
public import Mathlib.Data.DFinsupp.Submonoid
/-!
# Direct sum
This... |
1,822 | lean_file_chunk | e0db33be66642c91a43d9857f5ee8c4327c51f846724b3d647ae9e8f326e74e9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Algebra.DirectSum.Module
public import Mathlib.Data.Finsupp.ToDFinsupp
/-!
# Results on direct sums and finitely supported functions.
... |
8,152 | lean_file_chunk | 1f8c3ec9bbefca5303c7ad88c78b9644aa394ed811a9504b15281018df76b037 | 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: Oliver Nash
-/
module
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.LinearAlgebra.FreeModule.PID
public import Mathlib.LinearAlgebra.Eigenspace.Basic
... |
6,505 | lean_file_chunk | 963149d4d5d6f4c2695ab9f4d4ae2a77d0ce8fac1883e7a133ff6b7625bf0de3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.DirectSum.Module
public import Mathlib.Algebra.DirectSum.Ring
/-! # Additively-graded alge... |
2,191 | lean_file_chunk | b181637c4bb71251d4afe3e680f38235dfb1532eb4f2b014e8669985357cd9a6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yunzhou Xie, Jujian Zhang, Andrew Yang
-/
module
public import Mathlib.RingTheory.Idempotents
public import Mathlib.Algebra.DirectSum.Decomposition
/-!
# Decomposition of the identity o... |
1,024 | lean_file_chunk | 1ac05ee0e56ddeebd468ecc9035a82ca57e11876d6e54dd16df421a16d8180b7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.DirectSum.Basic
public import Mathlib.Algebra.Group.AddChar
/-!
# Direct sum of additive characters
This file defines the direct s... |
9,244 | lean_file_chunk | 08e28889da1f5ee11660735141c2cc4e42f1d4578affcd47cb41dc507f3881a1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies, Michał Mrugała, Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Michał Mrugała, Andrew Yang
-/
module
public import Mathlib.Algebra.Category.CommAlgCat.Monoidal
public import Mathlib.CategoryTheory.Monoidal... |
3,979 | lean_file_chunk | 820c636de9cabb4649b246933d82e4c2ab2b17cb63959a221c61caa99a21f40b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Category.Ring.Basic
public import Mathlib.Algebra.Ring.BooleanRing
public import Mathlib.Order.Category.BoolAlg
/-!
# The category ... |
2,579 | lean_file_chunk | d5c3bfe74ef7a04c33581f9670b11cce9b287160e4c7cfe4958117c746ff5c11 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Category.MonCat.Basic
public import Mathlib.Algebra.GroupWithZero.WithZero
public import Mathlib.CategoryTheory.Category.Bipointed
... |
11,023 | lean_file_chunk | 1d03e7fd0c5736cec407a4f43949101601dd18511ef3bae82267ba5a0e98bc90 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
module
public import Mathlib.Algebra.QuadraticAlgebra.Defs
public import Mathlib.Algebra.Star.Unitary
/-! # Quadratic algebras : involution and norm.
... |
17,057 | lean_file_chunk | a5c00c79a5c132d166e8ffc4b798c6cba9beb869d9210393272ddcb65b1d6fa9 | 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: Yunzhou Xie, Kenny Lau, Jiayang Hong
-/
module
public import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quadra... |
1,249 | lean_file_chunk | 2b849b59975f59ef04a0efc4efd25d5ae697f059e5510738b2825ded9be52437 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Snir Broshi. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Snir Broshi
-/
module
public import Mathlib.Algebra.QuadraticAlgebra.Basic
public import Mathlib.LinearAlgebra.Determinant
/-!
# Quadratic Algebra
We prove that the expression for the ... |
11,292 | lean_file_chunk | 30a5b851149fde4653d5f5a3b052d5680576cc55d1801bd09f9f33ca674bc6a8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Torsion
import Mathlib.Tactic.TermCongr
import Mathlib.Tactic.Use
/-!
# Equality modulo... |
3,645 | lean_file_chunk | e3e2fd8b15fed872894abcad7ff5a810ecad52f3223b96fdf2eaec0037eade04 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
module
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Algebra.Group.Commute.Defs
public import Mathlib.Algebra.Group.Hom.Defs
public import ... |
10,419 | lean_file_chunk | c58ae55c5aa13455f87a4da248a6aaa0d9fa35e401646788beebc61021ab40eb | 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
-/
module
public import Mathlib.Algebra.Group.Commute.Defs
public import Mathlib.Algebra.Group.InjSurj
public import Mathlib.Algebra.Group.Torsion
public import Mathlib.Algebra.O... |
8,956 | lean_file_chunk | 26417b2f0b1752e5e84118cc87f6b89487870d9c386a67ac7f3eb609ed759d99 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
module
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Algebra.Group.Hom.Defs
public import Mathlib.Algebra.Group.Pi.Basic
public import Mathlib.Algebr... |
1,695 | lean_file_chunk | 27dd0b5f5971f1b6bf902a23aa2ab65b2b78087c45b74b6cd3b17fa7549105ea | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Algebra.Notation.Support
/-!
# Support of a function
In this file we prove basic propertie... |
6,982 | lean_file_chunk | f5c869448c30dddf92c7186b96b2392b04d41f2d4aff4a48aca3f5a81e9c1a57 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Algebra.Group.Equiv.Defs
public import Mathlib.Algebra.Group.InjSurj
public import Mathlib.Data.Fintype.Basic
/-!
# Transfer algebraic ... |
1,324 | lean_file_chunk | 93dc832b07f947a42a81272cd0dbc82ce8ac1c3063c979a16c105300fed12832 | 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.Group.Defs
/-!
# `PUnit` is a commutative group
This file collects facts about algebraic structures on the one-element type, e.g. that i... |
11,015 | lean_file_chunk | 2478d5b938d1a51c55773040a1727d1662e600a528d05c4fd598f0e6abc8f4b7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, David Loeffler
-/
module
public import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Vertical line test for group homs
This file proves the vertical line test for monoid homo... |
1,263 | lean_file_chunk | 3fc0ec147e0db74eaa140b6ac010e166263c402d9cfa4940ee24f57ff5d27881 | 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.Algebra.Group.Conj
public import Mathlib.Data.Fintype.Units
/-!
# Conjugacy of elements of finite groups
-/
@[expose] public section
... |
22,637 | lean_file_chunk | 321c770f4a6317948ec45a643c86aa925eb31f58a45df8f7fd4625d1a1b21168 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Group.Equiv.Defs
public import Mathlib.Algebra.Group.Hom.Basic
public import Mathlib.Algebra.Grou... |
1,183 | lean_file_chunk | 5d87ce224ab454b9ced284db938ef21692222c54129214da85c92390cd36c20d | 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.Algebra.Group.Defs
public import Mathlib.Data.Bracket
/-!
# The bracket on a group given by commutator.
## Notation
After `open sco... |
3,271 | lean_file_chunk | 5d8cd40c96d88281a7949e8e5621bb1ae5def0e37781de41fff324a9f72bddfe | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Algebra.Notation.Prod
/-!
# Typeclasses for power-associative structures
In this file we define ... |
11,033 | lean_file_chunk | b897ae7baf16270769eece4f87760577e5484e8a1132c5639a6f2073cfadedc7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
module
public import Mathlib.Algebra.Group.Invertible.Basic
public import Mathlib.Algebra.Notation.Prod
public import Mathlib.Data.Set.Basic
public import M... |
23,998 | lean_file_chunk | f482418785552a9c7c2f1bd1720521d9f269ca748d325b1dc1e4c45da0cf3edf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
module
public import Aesop
public import Mathlib.Algebra.Group.Defs
public import Mathlib.Data.Int.Init
public import M... |
13,161 | lean_file_chunk | 46bc042b6b377e52f3750e90bdb572f4641ad6e2b3e9fb9e9daa398a631eb1f0 | lean_source_file | pkg_mathlib_source_file_chunks | fun x ↦ mul_div_cancel_right x c
@[to_additive]
theorem leftInverse_mul_left_div (c : G) : Function.LeftInverse (fun x ↦ x * c) fun x ↦ x / c :=
fun x ↦ div_mul_cancel x c
@[to_additive]
theorem leftInverse_mul_right_inv_mul (c : G) :
Function.LeftInverse (fun x ↦ c * x) fun x ↦ c⁻¹ * x :=
fun x ↦ mul_inv_can... |
4,460 | lean_file_chunk | 4ebe901d1018d8c45aa1db873a09c839ccb61fe691434c89cca862239a4d596f | 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.Group.Equiv.Defs
public import Mathlib.Algebra.Group.InjSurj
/-!
# `ULift` instances for groups and monoids
This file defines inst... |
3,440 | lean_file_chunk | 88281ba545aa8d1ac91d3cec99790f7813cd84c199620844beb720c63b78aa41 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.BigOperators.Pi
public import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
public import Mathlib.Algebra.Group.Pi.Basic
public i... |
4,288 | lean_file_chunk | cc1ca36889340cc5c7ed57e8b1bf445ec7340b5e8b2a48ce272bd1039c01396d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Dexin Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dexin Zhang
-/
module
public import Mathlib.GroupTheory.GroupAction.SubMulAction.Closure
public import Mathlib.Algebra.Group.Pointwise.Set.Basic
/-!
# Semigroup ideals
This file define... |
23,989 | lean_file_chunk | 20bc6c658f2eb3528e4d75435f7146917c05917eabc51cb79cbb2b4207da4fb4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Group.Equiv.TypeTags
public import Mathlib.Algebra.Group.Pi.Basic
public import Mathlib.Algebra.... |
3,928 | lean_file_chunk | ad0af897669e90dba8fb1ea7b30e72c0ae7c23fd153542f9298566ea77574d5b | lean_source_file | pkg_mathlib_source_file_chunks | namespace AddAut
variable (A) [Add A]
/-- The group operation on additive automorphisms is defined by `g h => AddEquiv.trans h g`.
This means that multiplication agrees with composition, `(g*h)(x) = g (h x)`.
-/
instance : Group (AddAut A) where
mul g h := AddEquiv.trans h g
one := AddEquiv.refl _
inv := AddEqu... |
9,635 | lean_file_chunk | 516bece40ba2f9d34c88314d09ef2dd39a515237e1ad14553d361507ea5396df | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Chris Hughes, Michael Howes
-/
module
public import Mathlib.Algebra.Group.End
public import Mathlib.Algebra.Group.Semiconj.Units
/-!
# Conjugacy of group elements
Se... |
6,468 | lean_file_chunk | b7e0243c21d42a7797bc950fcdb1f7386c1bc737bd76c194f56545bfe2eae893 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yury Kudryashov
-/
module
public import Mathlib.Algebra.Group.Hom.Defs
/-!
# Extensionality lemmas for monoid and group structures
In this file we prove extens... |
23,960 | lean_file_chunk | 1e62583891148d18bd5dd3e1d01f0e116faf0be5d19bb77c285d7af61e88252c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
module
public import Mathlib.Algebra.Group.Defs
public import Mathlib.Logic.Function.Basic
public import Mathlib.Tactic.Spread
/-!
# Lifting algebraic data classes ... |
420 | lean_file_chunk | 618b817098166f8a3a14432535b05f4a64e662391eba52797711b45db9c5af36 | lean_source_file | pkg_mathlib_source_file_chunks | protected abbrev commGroup [CommGroup M₁] (f : M₁ → M₂) (hf : Surjective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f x⁻¹ = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n) : CommGroup M₂ :=
{ hf.... |
23,995 | lean_file_chunk | fbc9a2f6007211ca5da69d3e01810f5b147bf55bcb88b0df2522cc856f7a39f7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
module
public import Batteries.Logic
public import Batteries.Util.LibraryNote
public import Mathlib.Algebra.Notation.De... |
23,916 | lean_file_chunk | b9c2182d14a10f3ca8595efd3ab71a4a79def0b880314db391906dc062cbf0f8 | lean_source_file | pkg_mathlib_source_file_chunks | | k + 1 =>
rw [npowRec, npowRec'_succ k.succ_ne_zero, ← mul_assoc]
congr
simp [ih]
@[to_additive]
theorem npowBinRec.go_spec {M : Type*} [Semigroup M] [One M] (k : ℕ) (m n : M) :
npowBinRec.go (k + 1) m n = m * npowRec' (k + 1) n := by
unfold go
generalize hk : k + 1 = k'
replace hk : k' ≠ ... |
12,044 | lean_file_chunk | bd37d5c6eb06b819bd7648d200d3993ae513af71bc6ee531a87566528603847e | lean_source_file | pkg_mathlib_source_file_chunks | /-- Despite the asymmetry of `inv_eq_of_mul`, the symmetric version is true thanks to the
involutivity of inversion. -/
protected inv_eq_of_mul (a b : G) : a * b = 1 → a⁻¹ = b
section DivisionMonoid
variable [DivisionMonoid G] {a b : G}
@[to_additive (attr := simp) neg_add_rev]
theorem mul_inv_rev (a b : G) : (a... |
3,776 | lean_file_chunk | 7e9a576af408e9f620cd524198edce88c13401fb748fe031a4ea6eb3d3bfa286 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 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.Group.Action.TransferInstance
public import Mathlib.Logic.Small.Defs
/-!
# Transfer group structures from `α` to `Shrink α`
-/
@[e... |
18,766 | lean_file_chunk | fb563137bbbfd538ef4c24f1ea3eeea91cc946c082d68b3c2c1466cd04707ee8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
module
public import Mathlib.Algebra.Group.Equiv.Defs
public import Mathlib.Algebra.Group.Pi.Lemmas
public import Mathlib.Data.Finset.Max
public import... |
6,247 | lean_file_chunk | f16ff6fbc44f082baebe0d60e192a9760246e217ae9ec282ba0b4a035eda34c1 | 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.Group.Equiv.Basic
public import Mathlib.Algebra.Group.Equiv.Opposite
public import Mathlib.Algebra.Group.TypeTags.Basic
public imp... |
3,880 | lean_file_chunk | 6e361454b1bfed86aa1556ad8b7957e9db1825ac63c0d64ccf989f6ef2b72b5e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies, Patrick Luo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Patrick Luo
-/
module
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Tactic.MkIffOfInductiveProp
/-!
# Torsion-free monoids and groups
This... |
1,440 | lean_file_chunk | 1bc692dc3f6535dd6cff0587bf02fcd3adb802591e384c80529123fc128e5976 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
module
public import Mathlib.Logic.Embedding.Basic
public import Mathlib.Algebra.Group.Defs
/-!
# The embedding of a cancellative semigroup into itself by multiplicat... |
5,249 | lean_file_chunk | 11a72aea9eeba26e8ecd1c15e05c86d1c58f909f19574dc7b9006ab510d65d81 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
public import Mathlib.Algebra.Group.Action.Pi
public import Mathlib.Algebra.Module.Defs
/-!
# E... |
3,075 | lean_file_chunk | cc6a1762c28ee3f7f0f7230824b1a289bc1dd2b6776521b42ff771defd50dad5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.Algebra.Group.Defs
/-!
# Minimal Axioms for a Group
This file defines constructors to define a group structure on a Type, while proving on... |
4,220 | lean_file_chunk | 2b29d305c95009a70a27cb69e94f976485c1f07a378dc4d4b7910cb32b332f6a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
module
public import Mathlib.Algebra.Group.Pi.Basic
public import Mathlib.Algebra.Group.Prod
public import Mathlib.Algebra.Ring.Int.Defs
public import Mathlib.Data.N... |
12,135 | lean_file_chunk | bae59ae995f4d8d61499f794f66fb82411ca3e304b6edf77765360240a003425 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Giulio Caflisch, David Loeffler, Yu Shao, Weijie Jiang, BeiBei Xiong
-/
module
public import Mathlib.Algebra.BigOperators.Pi
public import Mathlib.Algebra.Group.AddChar
public import ... |
19,247 | lean_file_chunk | 588ac3f66e74041804684a9066c3c4104e2805d81361579759f178b3d33df256 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
module
public import Mathlib.Algebra.BigOperators.Pi
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Algebra.Group.Subgroup.Ker
public imp... |
10,411 | lean_file_chunk | 82374625ac4bf06012499a26b30b663fbe6b50cb65b5323f66778ddded819e6a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Yaël Dillies, Andrew Yang
-/
module
public import Mathlib.Algebra.BigOperators.Field
public import Mathlib.Algebra.MvPolynomial.Equiv
public import Mathlib.Algebra.MvPol... |
5,569 | lean_file_chunk | d9adf0796050b2acb3cfdd0e87af4756702987ce2f7f91dbc7c9ad5d25c3d798 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Yury Kudryashov
-/
module
public import Mathlib.Algebra.MvPolynomial.Derivation
public import Mathlib.Algebra.MvPolynomial.Equiv
/-!
# Partial derivatives of polynomia... |
6,096 | lean_file_chunk | 902b73ef0fdb4d1537df7b8fbfa06cc667ee58476a539ac0c953a199edab4749 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Algebra.MvPolynomial.Supported
public import Mathlib.RingTheory.Derivation.Basic
/-!
# Derivations of multivariate polynomials
In th... |
3,964 | lean_file_chunk | fcb324e6d29d605205a7d3ce4f333ebd29b9869c4fc354636d6dd856793c72cf | 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.MvPolynomial.Homogeneous
public import Mathlib.RingTheory.Polynomial.Nilpotent
/-!
# Nilpotents and units in multivariate polynomi... |
3,439 | lean_file_chunk | 6caa97c358aef24a531cbbd6a252e7ad4b6a9587742b3fd489b5ca256f2bcfde | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
module
public import Mathlib.Algebra.MvPolynomial.Rename
/-!
# `comap` operation on `MvPolynomial`
This file defines the `comap` function on `MvPolynomial`.
`MvPo... |
10,912 | lean_file_chunk | c2a8b7b71c6dc5de8d159e43e8cede70de60a7874482b373c5ce27dc0cec9133 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
module
public import Mathlib.Data.Finsupp.Lex
public import Mathlib.Algebra.MvPolynomial.Degrees
/-!
# Variables of polynomials
Thi... |
7,166 | lean_file_chunk | 017a263d6440d6c3042ff38843d6a8e30c8fe61b65cb2810c10543c1a9dd1382 | 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, Robert Y. Lewis
-/
module
public import Mathlib.Algebra.MvPolynomial.Monad
public import Mathlib.Algebra.MvPolynomial.Nilpotent
public import Mathlib.Algebra.Order.Rin... |
15,127 | lean_file_chunk | c4eb1c9521e1fa0908075774b26aebf020f0213c164d44b31d1cfccb13a5fc6b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
module
public import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Renaming variables of polynomials
This file establishes the `rename` o... |
23,980 | lean_file_chunk | ce9ffa5ae36ddc6e03e3e4657206befb903b6d8909359d116c02c37dc8fd2523 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
module
public import Mathlib.Algebra.MvPolynomial.Basic
/-!
# Multivariate polynomials
This file defines functions for evaluating m... |
7,434 | lean_file_chunk | e0ec153da4dfc57b7e875fb9093764ce93709e3d5135246229da89d658033daf | lean_source_file | pkg_mathlib_source_file_chunks | theorem eval₂Hom_zero'_apply (f : R →+* S₂) (p : MvPolynomial σ R) :
eval₂Hom f (fun _ => 0 : σ → S₂) p = f (constantCoeff p) :=
eval₂Hom_zero_apply f p
@[simp]
theorem eval₂_zero_apply (f : R →+* S₂) (p : MvPolynomial σ R) :
eval₂ f (0 : σ → S₂) p = f (constantCoeff p) :=
eval₂Hom_zero_apply _ _
@[simp]
... |
23,989 | lean_file_chunk | 29d8e58f9a195f2963988150ac122ccb4dcf9e885fe903bd19e64db16601298a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
module
public import Mathlib.Algebra.MonoidAlgebra.Degree
public import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynom... |
1,324 | lean_file_chunk | f33fd736dfefdbd819c9f05295eb167e130844d31bde87a5b8a242671fc4405d | lean_source_file | pkg_mathlib_source_file_chunks | rw [le_antisymm_iff, Submodule.mul_le]
refine ⟨fun x hx ↦ x.as_sum ▸ sum_mem fun i hi ↦ ?_,
fun x hx y hy ↦ degrees_mul_le.trans (add_le_add hx hy)⟩
replace hi : i.toMultiset ≤ s + t := (Finset.le_sup hi).trans hx
let a := (i.toMultiset - t).toFinsupp
let b := (i.toMultiset ⊓ t).toFinsupp
have : a + b = i... |
1,077 | lean_file_chunk | 6abaa539b7bcfbbaefdaf9bee492e5f1e4ca0641c079d906664c5a81e80ba090 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
module
public import Mathlib.Algebra.MvPolynomial.Basic
public import Mathlib.RingTheory.AlgebraTower
/-!
# Invertible polynomials... |
14,161 | lean_file_chunk | 652ef7ce8007cb1e7aac6006030903dca306a41a0526d3595216715fbf4b8657 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
module
public import Mathlib.Algebra.MvPolynomial.Rename
public import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Monad operati... |
5,679 | lean_file_chunk | 4df6ef0e68a10aad628e4351bb022d62585c8070a791feacd459cccb3ffc6278 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
module
public import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Multivariate polynomials over a ring
Many results about polynomia... |
6,179 | lean_file_chunk | c1cdafdcdec006d02e9c0900f7ae7152b5dc808146fe9a1170a095a934af5ce0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Bolton Bailey
-/
module
public import Mathlib.Algebra.MvPolynomial.Variables
public import Mathlib.Algebra.MvPolynomial.Equiv
public import Mathlib.RingTheory.Mv... |
23,994 | lean_file_chunk | 1436b5288ebc75dd805be69576302eae713d51227a0528611784a603ab93ed63 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Lattice
public import Mathlib.Algebra.Algebra.Tower
public import Mathlib.Alg... |
14,606 | lean_file_chunk | 1fec3f0658ca1dbb590300834a15aacc372900c43a37469e03fa385e978720b5 | lean_source_file | pkg_mathlib_source_file_chunks | @[simp]
theorem coeff_mul_X (m) (s : σ) (p : MvPolynomial σ R) :
coeff (m + Finsupp.single s 1) (p * X s) = coeff m p :=
(coeff_mul_monomial _ _ _ _).trans (mul_one _)
@[simp]
theorem coeff_X_mul (m) (s : σ) (p : MvPolynomial σ R) :
coeff (Finsupp.single s 1 + m) (X s * p) = coeff m p :=
(coeff_monomial_mu... |
2,809 | lean_file_chunk | 5879313847b611610a4307be9e7350ea8ef24408278a1e3ab24615932405a554 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
module
public import Mathlib.Algebra.MvPolynomial.Eval
/-!
## Counit morphisms for multivariate polynomials
One may consider the ring of multivariate polynomials `... |
13,339 | lean_file_chunk | 02ef2a41e956268b89ef98d9ffae4a4019f043bcd2be82b543112ff8f810f6fa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
module
public import Mathlib.Algebra.MonoidAlgebra.Division
public import Mathlib.Algebra.MvPolynomial.CommRing
public import Mathlib.Data.Finsupp.Weight
/-!
# Division o... |
3,969 | lean_file_chunk | a0e9ea0af14b9b79a52b82c740200503d687b79496feacdf4f34abdb0f4221bd | 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.Variables
/-!
# Polynomials supported by a set of variables
This file contains the definition and lemmas about `MvPol... |
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