path stringlengths 11 71 | content stringlengths 75 124k |
|---|---|
Algebra\Polynomial\Degree\Lemmas.lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval
/-!
# Theory of degrees of polynomials
Some of the main results include
- `natDeg... |
Algebra\Polynomial\Degree\TrailingDegree.lean | /-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDe... |
Algebra\Polynomial\Module\AEval.lean | /-
Copyright (c) 2022 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Finiteness
/-!
# Action of the polynomial ring on module induced by an algebra element.
Giv... |
Algebra\Polynomial\Module\Basic.lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.AEval
/-!
# Polynomial module
In this file, we define the polynomial module for an `R`-module `M`, i.e. the `R[X]`-module `M[X]`.... |
Algebra\Polynomial\Module\FiniteDimensional.lean | /-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.Algebra.Polynomial.Module.AEval
/-!
# Polynomial modules in finite dimensions
This file is a place to collect resu... |
Algebra\PUnitInstances\Algebra.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.GCDMonoid.Basic
/-!
# Instances on PUnit
This file collects facts about algebraic structures on the one-element type, e.g. that it is a
commutative rin... |
Algebra\PUnitInstances\Module.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.PUnitInstances.Algebra
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Basic
/-!
# Instances on PUnit
This file collects facts ab... |
Algebra\PUnitInstances\Order.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.PUnitInstances.Algebra
import Mathlib.Algebra.Order.AddGroupWithTop
import Mathlib.Order.Heyting.Basic
/-!
# Instances on PUnit
This file collects fact... |
Algebra\Regular\Basic.lean | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Bas... |
Algebra\Regular\Pow.lean | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Regular.Basic
/-!
# Regular elements
## Implement... |
Algebra\Regular\SMul.lean | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.GroupAction.Hom
/-!
# Action of regular elements on a module
We introduce `M`-regular elements, in the... |
Algebra\Ring\AddAut.lean | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.GroupTheory.GroupAction.Group
import Mathlib.Algebra.Module.Defs
/-!
# Multiplication on the left/right as additive automorphisms
In this file we d... |
Algebra\Ring\Aut.lean | /-
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
-/
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Ring.Action.Group
import Mathlib.Algebra.Ring.Equiv
/-!
# Ring automorphisms
... |
Algebra\Ring\Basic.lean | /-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.G... |
Algebra\Ring\BooleanRing.lean | /-
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, Yaël Dillies
-/
import Mathlib.Algebra.PUnitInstances.Algebra
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Ring
import Mathlib.Order.Hom.Lattice
import Mathli... |
Algebra\Ring\Center.lean | /-
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
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
/-!
# Centers of rings
-/
variable {M : Type*}
namespace Set
variable (M)
@[si... |
Algebra\Ring\Centralizer.lean | /-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Algebra.Ring.Defs
/-!
# Centralizers of rings
-/
variable {M : Type*} {S T : Set M}
namespace ... |
Algebra\Ring\CentroidHom.lean | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Module.Hom
import Mathlib.GroupTheory.GroupAction.Ri... |
Algebra\Ring\Commute.lean | /-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... |
Algebra\Ring\CompTypeclasses.lean | /-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Ring.Equiv
/-!
# Propositional typeclasses on several ring homs
This file contains three typeclasses used in the definitio... |
Algebra\Ring\Defs.lean | /-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... |
Algebra\Ring\Equiv.lean | /-
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
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mat... |
Algebra\Ring\Ext.lean | /-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... |
Algebra\Ring\Fin.lean | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Logic.Equiv.Fin
/-!
# Rings and `Fin`
This file collects some basic results involving r... |
Algebra\Ring\Idempotents.lean | /-
Copyright (c) 2022 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Subtype
import Mathlib.Order... |
Algebra\Ring\Identities.lean | /-
Copyright (c) 2020 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Kevin Lacker
-/
import Mathlib.Tactic.Ring
/-!
# Identities
This file contains some "named" commutative ring identities.
-/
variable {R : Type*} [CommRing R] ... |
Algebra\Ring\InjSurj.lean | /-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Opposites
import Mathlib.Algebra.GroupWit... |
Algebra\Ring\Int.lean | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Ring.Parity
/-!
# The integers are a ring
This file contains the commutat... |
Algebra\Ring\Invertible.lean | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Defs
/-!
# Theorems about invertible elements in rings
-/
universe u
variable {α : Type u}
/... |
Algebra\Ring\MinimalAxioms.lean | /-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.MinimalAxioms
/-!
# Minimal Axioms for a Ring
This file defines constr... |
Algebra\Ring\Nat.lean | /-
Copyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Nat
import Math... |
Algebra\Ring\NegOnePow.lean | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Johan Commelin
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.ZMod.IntUnitsPower
/-!
# Integer powers of (-1)
This file defines the map `negOnePow : ℤ → ℤˣ` which sends... |
Algebra\Ring\Opposite.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.GroupWithZero.Opposite
import Mathlib.Algebra.Ring.Hom.Defs
/-!
# Ring structures on the multiplicative opposite
-/
variable {α : Type*}
namespace Mul... |
Algebra\Ring\Parity.lean | /-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Function.Iterate
/-!
# Even and odd elements ... |
Algebra\Ring\Pi.lean | /-
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
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.GroupWithZero.Pi
import Mathlib.Algebra.Ring.CompTypeclasses
import Mathlib.Algebra.Ring.Hom.D... |
Algebra\Ring\Prod.lean | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Chris Hughes, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Int.Cast.Prod
import Mathlib.Algebra.GroupWithZero.Prod
import Mathlib.Algebra.Ring.CompTypeclasse... |
Algebra\Ring\Rat.lean | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Rat.Defs
/-!
# The rational numbers are a commut... |
Algebra\Ring\Regular.lean | /-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.Algebra.Ring.Defs
/-!
# Lemmas about regular... |
Algebra\Ring\Semiconj.lean | /-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Semirings and ... |
Algebra\Ring\SumsOfSquares.lean | /-
Copyright (c) 2024 Florent Schaffhauser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Florent Schaffhauser
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Ring.Defs
/... |
Algebra\Ring\ULift.lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.ULift
import Mathlib.Algebra.Ring.Equiv
/-!
# `ULift` instances for ring
This file defines instances for ring, semiring and related str... |
Algebra\Ring\Units.lean | /-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra... |
Algebra\Ring\WithZero.lean | /-
Copyright (c) 2020 Mario Carneiro, Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johan Commelin
-/
import Mathlib.Algebra.GroupWithZero.WithZero
import Mathlib.Algebra.Ring.Defs
/-!
# Adjoining a zero to a semiring
-/
namespace Wit... |
Algebra\Ring\Action\Basic.lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Ring.Hom.Defs
/-!
# Group action on rings
This file defines the typeclass of monoid acting on semiring... |
Algebra\Ring\Action\Field.lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
/-!
# Group action on fields
-/
variable {M} [Mon... |
Algebra\Ring\Action\Group.lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Group
import Mathlib.Algebra.Ring.Equiv
/-!
# If a group acts multiplicatively on a semiring, e... |
Algebra\Ring\Action\Invariant.lean | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Algebra.Ring.Subring.Basic
/-! # Subrings invariant under an action -/
section Ring
variable (M R : Type*) [Mo... |
Algebra\Ring\Action\Subobjects.lean | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.Algebra.Ring.Action.Basic
/-!
# Instances of `MulSemirin... |
Algebra\Ring\Divisibility\Basic.lean | /-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib... |
Algebra\Ring\Divisibility\Lemmas.lean | /-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.GroupTheory.GroupAction.R... |
Algebra\Ring\Hom\Basic.lean | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Alge... |
Algebra\Ring\Hom\Defs.lean | /-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic... |
Algebra\Ring\Semireal\Defs.lean | /-
Copyright (c) 2024 Florent Schaffhauser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Florent Schaffhauser
-/
import Mathlib.Algebra.Ring.SumsOfSquares
import Mathlib.Algebra.Order.Field.Defs
/-!
# Semireal rings
A semireal ring is a non-trivial commutative rin... |
Algebra\Ring\Subring\Basic.lean | /-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subsemiring.Basic
/-!
# Subrings
Let `R` be a ring. Th... |
Algebra\Ring\Subring\MulOpposite.lean | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.Algebra.Ring.Subsemiring.MulOpposite
import Mathlib.Algebra.Ring.Subring.Basic
/-!
# Subring of opposite rings
For every ring `R`, we construct an equivalence betwe... |
Algebra\Ring\Subring\Order.lean | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Ring.Subring.Basic
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Algebra.Order.Ring.InjSurj
/-!
# Subrings of ordered rings
We study su... |
Algebra\Ring\Subring\Pointwise.lean | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Ring.Subring.Basic
import Mathlib.Algebra.Ring.Subsemiring.Pointwise
import Mathlib.Data.Set.Pointwis... |
Algebra\Ring\Subring\Units.lean | /-
Copyright (c) 2021 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Order.Monoid.Submonoid
import Mathlib.Algebra.Order.Ring.Defs
/-!
# Unit subgroups of a ring
-/
... |
Algebra\Ring\Subsemiring\Basic.lean | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv... |
Algebra\Ring\Subsemiring\MulOpposite.lean | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.Algebra.Group.Submonoid.MulOpposite
import Mathlib.Algebra.Ring.Subsemiring.Basic
/-!
# Subsemiring of opposite semirings
For every semiring `R`, we construct an eq... |
Algebra\Ring\Subsemiring\Order.lean | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Order.Ring.InjSurj
import Mathlib.Algebra.Ring.Subsemiring.Basic
/-!
# `Order`ed instances for `SubsemiringClass` and `Subsemiring`.
-/
namespa... |
Algebra\Ring\Subsemiring\Pointwise.lean | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.Algebra.Ring.Subsemiring.Basic
import Mathlib.Data.Set.Pointwise.Ba... |
Algebra\Squarefree\Basic.lean | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
/-!
# Squarefree elements of monoids
An element of a monoid is squarefr... |
Algebra\Star\Basic.lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Regular.Basic
... |
Algebra\Star\BigOperators.lean | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Star.Basic
/-! # Big-operators lemmas about `star` algebraic operations
These results are kept sep... |
Algebra\Star\Center.lean | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Star.Pointwise
import Mathlib.Algebra.Group.Center
/-! # `Set.center`, `Set.centralizer` and the `star` operat... |
Algebra\Star\CentroidHom.lean | /-
Copyright (c) 2024 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.CentroidHom
import Mathlib.Algebra.Star.StarRingHom
import Mathlib.Algebra.Star.Subsemiring
import Mathlib.Algebra.Star.Basic
/-... |
Algebra\Star\CHSH.lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Real.Sqrt
import Mathlib.Tactic.Polyrith
/-!
# The Clauser-Horne-Shimon... |
Algebra\Star\Free.lean | /-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.FreeAlgebra
/-!
# A *-algebra structure on the free algebra.
Reversing words gives a *-structure on the free mono... |
Algebra\Star\Module.lean | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Frédéric Dupuis
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Module.Equiv.Defs
import Mathlib.Algebra.Module.LinearMa... |
Algebra\Star\NonUnitalSubalgebra.lean | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
/-!
# Non-unital Star Subalgebras
In this... |
Algebra\Star\NonUnitalSubsemiring.lean | /-
Copyright (c) 2024 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Subsemigroup.Basic
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.Algebra.S... |
Algebra\Star\Order.lean | /-
Copyright (c) 2023 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.StarRingHom
import Mathlib.Algebra.Regular.Basic... |
Algebra\Star\Pi.lean | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Ring.Pi
/-!
# `star` on pi types
We put a `Star` structure on pi types that operates elementwise, such that it de... |
Algebra\Star\Pointwise.lean | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.Basic
/-!
# Pointwise star operation on sets
This file defines the ... |
Algebra\Star\Prod.lean | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Action.Prod
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Star.Basic
/-!
# `Star` on product types
We put a `Star` structure on pro... |
Algebra\Star\Rat.lean | /-
Copyright (c) 2024 Yael Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yael Dillies
-/
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.NNRat.Defs
import Mathlib.Data.Rat.Cast.Defs
/-!
# *-ring structure on ℚ and ... |
Algebra\Star\RingQuot.lean | /-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.RingQuot
import Mathlib.Algebra.Star.Basic
/-!
# The *-ring structure on suitable quotients of a *-ring.
-/
namespace RingQuot
universe u
variabl... |
Algebra\Star\SelfAdjoint.lean | /-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
import Mathlib.Algebra.Star.Rat
/-!
# Self-adjoint, s... |
Algebra\Star\StarAlgHom.lean | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Star.Prod
import Mathlib.Alg... |
Algebra\Star\StarRingHom.lean | /-
Copyright (c) 2024 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Star.Basic
/-!
# Morphisms of star rings
This file defines a new type of morphism between (non-unita... |
Algebra\Star\Subalgebra.lean | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Jireh Loreaux
-/
import Mathlib.Algebra.Star.Center
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Star.P... |
Algebra\Star\Subsemiring.lean | /-
Copyright (c) 2024 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Star.NonUnitalSubsemiring
import Mathlib.Algebra.Ring.Subsemiring.Basic
/-!
# Star subrings
A *-subring is a subring of a *-ring whic... |
Algebra\Star\Unitary.lean | /-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Algebra.Spectrum
/-!
# Unitary elem... |
Algebra\Tropical\Basic.lean | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.SMulWithZero
import Mathlib.Order.Hom.Basic
import Mathlib.Algebra.Order.Ring.Nat
import Ma... |
Algebra\Tropical\BigOperators.lean | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLat... |
Algebra\Tropical\Lattice.lean | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
/-!
# Order on tropical algebraic structure
This file defines the o... |
Algebra\Vertex\HVertexOperator.lean | /-
Copyright (c) 2024 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.RingTheory.HahnSeries.Multiplication
/-!
# Vertex operators
In this file we introduce heterogeneous vertex operators using Hahn series. When `R = ℂ`,... |
AlgebraicGeometry\AffineScheme.lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.CategoryTheory.Lim... |
AlgebraicGeometry\FunctionField.lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Properties
/-!
# Function field of integral schemes
We define the function field of an irreducible scheme as the stalk of the generic poi... |
AlgebraicGeometry\GammaSpecAdjunction.lean | /-
Copyright (c) 2021 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
/-!
# Adjunction between `Γ` and ... |
AlgebraicGeometry\Gluing.lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.Cover.Open
/-!
# Gluing Schemes
Given a family of gluing data of schemes, we m... |
AlgebraicGeometry\GluingOneHypercover.lean | /-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne, Joël Riou, Ravi Vakil
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.AlgebraicGeometry.Sites.BigZariski
import Mathlib.CategoryTheory.Sites.OneHypercover
/-!
# The ... |
AlgebraicGeometry\Limits.lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.AlgebraicGeometry.AffineScheme
/-!
# (Co)Limits of Schemes
We construct various limits and colimits in the categ... |
AlgebraicGeometry\Noetherian.lean | /-
Copyright (c) 2024 Geno Racklin Asher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Geno Racklin Asher
-/
import Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.Localization.Sub... |
AlgebraicGeometry\OpenImmersion.lean | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.AlgebraicGeometry.Scheme
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
/-!
# Open immersions... |
AlgebraicGeometry\Properties.lean | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sheaves.SheafCondition.Sites
import Mathlib.Algebra.Categor... |
AlgebraicGeometry\Pullbacks.lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... |
AlgebraicGeometry\Restrict.lean | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Scheme.restrict`: The restriction of a sche... |
AlgebraicGeometry\Scheme.lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
/-!
# The category of schemes
A s... |
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