Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | @[reassoc (attr := simp)]
theorem tensor_hom_inv_id {V W X Y Z : C} (f : V ≅ W) (g : X ⟶ Y) (h : Y ⟶ Z) :
(g ⊗ f.hom) ≫ (h ⊗ f.inv) = (g ⊗ 𝟙 V) ≫ (h ⊗ 𝟙 V) := by
| Mathlib/CategoryTheory/Monoidal/Category.lean | 652 | 654 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Birkhoff sums
In this file we define `birkhoffSum f g n x` ... | theorem Function.IsFixedPt.birkhoffSum_eq {f : α → α} {x : α} (h : IsFixedPt f x) (g : α → M)
(n : ℕ) : birkhoffSum f g n x = n • g x := by
simp [birkhoffSum, (h.iterate _).eq]
| Mathlib/Dynamics/BirkhoffSum/Basic.lean | 55 | 57 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | rw [smeval_eq_sum, smeval_eq_sum]
simp only [Polynomial.smul_pow, sum_def, Int.cast_sum, Int.cast_mul, Int.cast_npow]
| Mathlib/Algebra/Polynomial/Smeval.lean | 161 | 162 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
| Mathlib/RingTheory/Coprime/Lemmas.lean | 213 | 216 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Single
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-!
# Homology is an additive functor
When `V` is preadditive, `Homolog... | rw [← F.map_comp, sq n, Functor.map_zero] := by
refine HomologicalComplex.ext rfl ?_
rintro i j (rfl : j + 1 = i)
simp only [CategoryTheory.Functor.mapHomologicalComplex_obj_d, of_d, eqToHom_refl, comp_id,
id_comp]
| Mathlib/Algebra/Homology/Additive.lean | 218 | 223 |
/-
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
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | Mathlib/Data/Set/Image.lean | 1,608 | 1,611 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,673 | 1,679 | |
/-
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
-/
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.Ring.Hom.Defs
i... | @[simp]
| Mathlib/Algebra/Order/Ring/WithTop.lean | 69 | 69 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.PartrecCode
import Mathlib.Data.Set.Subsingleton
/-!
# Computability theory and the halting problem
A universal partial recursive funct... | Mathlib/Computability/Halting.lean | 430 | 435 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b
rcases ne_or_eq x y with (xy | rfl)
· exact ⟨y, us ⟨hab hy, xy.symm⟩⟩
| Mathlib/Topology/Order/DenselyOrdered.lean | 358 | 360 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Adam Topaz
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Endomorphism
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Finit... | `uSwitch.{u, v} : FintypeCat.{u} ⥤ FintypeCat.{v}` by sending
`X : FintypeCat.{u}` to `ULift.{v} (Fin (Fintype.card X))`. -/
noncomputable def uSwitch : FintypeCat.{u} ⥤ FintypeCat.{v} where
| Mathlib/CategoryTheory/FintypeCat.lean | 211 | 213 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | · rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
abs_of_nonneg hr.le, div_eq_inv_mul]
simp_rw [mem_ball_zero_iff, norm_neg]
exact fun _ => id
@[simp]
theorem gauge_closure_zero : gauge (closure (0 : Set E)) = 0 := funext fun x ↦ by
simp only [← singleton_ze... | Mathlib/Analysis/Convex/Gauge.lean | 534 | 556 |
/-
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.Logic.Function.Basic
/-!
# Semiconjugate and commuting maps
We define the following predicates:
* `Function.Semiconj`: `f : α → β` semiconjugates ... | then `fbc ∘ fab` semiconjugates `ga` to `gc`.
See also `Function.Semiconj.comp_left` for a version with reversed arguments. -/
| Mathlib/Logic/Function/Conjugate.lean | 51 | 53 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | change Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I)))
infer_instance
obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure
h_fund (convexBodyLT'_neg_mem K f w₀) (convexBodyLT'_convex K f w₀) h
rw [mem_toAddSubgroup, mem_span_fractionalIdealLattic... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 503 | 513 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.Data.PNat.Prime
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib... | rw [norm_eq_neg_one_pow (K := K) hζ, IsCyclotomicExtension.finrank _ hirr]
norm_cast
· exact hζ.norm_eq_one hn hirr
end CommRing
| Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 314 | 318 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | Mathlib/Data/Nat/Factorization/Basic.lean | 688 | 690 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,237 | 2,239 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ := by
obtain rfl | hμ := eq_or_ne μ 0
| Mathlib/MeasureTheory/Integral/Average.lean | 149 | 150 |
/-
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.Pi.Lemmas
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Topology.Algebra.Monoid
/-!
# Topological group with zero
I... | simpa only [div_eq_mul_inv] using hf.mul tendsto_const_nhds
variable [TopologicalSpace α]
| Mathlib/Topology/Algebra/GroupWithZero.lean | 52 | 54 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | /-- The oriented angle between two vectors is not zero or `π` if and only if those two vectors
are linearly independent. -/
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 405 | 406 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Hom
import Mathlib.LinearAlgebra.Dual.Lemmas
/-!
# Bilinear form
This file defines various properties of ... | Mathlib/LinearAlgebra/BilinearForm/Properties.lean | 484 | 486 | |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... | f = eqToHom h ≫ g ≫ eqToHom h'.symm ↔ HEq f g := by
cases h
cases h'
simp
| Mathlib/CategoryTheory/EqToHom.lean | 65 | 68 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.TensorProduct.Associator
/-!
# The `A`-module structure on `M ⊗[R] N`
Whe... | theorem rightComm_symm_tmul (m : M) (p : P) (q : Q) :
(rightComm R A M P Q).symm ((m ⊗ₜ q) ⊗ₜ p) = (m ⊗ₜ p) ⊗ₜ q :=
rfl
end rightComm
section tensorTensorTensorComm
variable [Module R P] [IsScalarTower R A P]
/-- Heterobasic version of `tensorTensorTensorComm`. -/
def tensorTensorTensorComm :
(M ⊗[R] N) ⊗[... | Mathlib/LinearAlgebra/TensorProduct/Tower.lean | 495 | 510 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
/-!
# Basic definitions about impartial (pre-)games
We will d... | @[simp]
theorem not_equiv_zero_iff : ¬ G ≈ 0 ↔ G ‖ 0 :=
⟨(equiv_or_fuzzy_zero G).resolve_left, Fuzzy.not_equiv⟩
| Mathlib/SetTheory/Game/Impartial.lean | 124 | 127 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | map <| Algebra.ofId F K
lemma map_baseChange [Algebra F K] [IsScalarTower R F K] [Algebra F L] [IsScalarTower R F L]
(f : K →ₐ[F] L) (P : W'⟮F⟯) : map f (baseChange F K P) = baseChange F L P := by
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 936 | 939 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... |
open scoped Classical in
theorem count_coe {J : Ideal R} (hJ : J ≠ 0) :
count K v J = (Associates.mk v.asIdeal).count (Associates.mk J).factors := by
| Mathlib/RingTheory/DedekindDomain/Factorization.lean | 506 | 509 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | [Submodule.IsPrincipal (I : Submodule R₁ K)] :
I * I⁻¹ = 1 ↔ generator (I : Submodule R₁ K) ≠ 0 := by
constructor
| Mathlib/RingTheory/DedekindDomain/Ideal.lean | 183 | 185 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... |
section GroupWithZero
| Mathlib/Algebra/GroupWithZero/Basic.lean | 309 | 310 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... |
variable {v : Set E}
open Module Submodule Set
/-- An orthonormal set in an `InnerProductSpace` is maximal, if and only if the orthogonal
complement of its span is empty. -/
theorem maximal_orthonormal_iff_orthogonalComplement_eq_bot (hv : Orthonormal 𝕜 ((↑) : v → E)) :
(∀ u ⊇ v, Orthonormal 𝕜 ((↑) : u → E) → ... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,312 | 1,328 |
/-
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
-/
import Mathlib.Algebra.Field.ZMod
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.LocalRing.ResidueField.... |
theorem ker_toZMod : RingHom.ker (toZMod : ℤ_[p] →+* ZMod p) = maximalIdeal ℤ_[p] := by
ext x
rw [RingHom.mem_ker]
constructor
· intro h
simpa only [h, ZMod.cast_zero, sub_zero] using toZMod_spec x
· intro h
| Mathlib/NumberTheory/Padics/RingHoms.lean | 273 | 280 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
import Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation... | ## Periodicity and conjugation
-/
/-- The two-variable Jacobi theta function is periodic in `τ` with period 2. -/
lemma jacobiTheta₂_add_right (z τ : ℂ) : jacobiTheta₂ z (τ + 2) = jacobiTheta₂ z τ := by
refine tsum_congr (fun n ↦ ?_)
simp_rw [jacobiTheta₂_term, Complex.exp_add]
| Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 379 | 385 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
/-!
# Polynomia... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | 57 | 57 | |
/-
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
-/
import Mathlib.Algebra.DirectSum.Algebra
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.DirectSum.Internal
... | @GradedMonoid.GMul.mul ι (fun i => 𝒜 i) _ _ _ _ (decompose 𝒜 a i) b :=
Subtype.ext <| coe_decompose_mul_add_of_right_mem 𝒜 b.2
theorem coe_decompose_mul_of_left_mem_zero [AddMonoid ι] [GradedRing 𝒜] {a b : A}
| Mathlib/RingTheory/GradedAlgebra/Basic.lean | 142 | 145 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Data.Int.Interval
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Binom... | universe u
open scoped PowerSeries
| Mathlib/RingTheory/LaurentSeries.lean | 87 | 89 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Ma... | · exact one_dvd _
use (b.primeFactorsList.diff a.succ.succ.primeFactorsList).prod
nth_rw 1 [← Nat.prod_primeFactorsList ha]
| Mathlib/Data/Nat/Factors.lean | 232 | 234 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.FieldTheory.RatFunc.AsPolynomial
/-!
# Laurent expansions of rational functions
## Main declarations
* `R... | Mathlib/FieldTheory/Laurent.lean | 116 | 117 | |
/-
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.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... | inj' _ _ h := Subtype.ext (frobenius_inj R p <| Subtype.ext_iff.1 h)
| Mathlib/FieldTheory/Perfect.lean | 336 | 337 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | exact ⟨integralBasis K i, (canonicalEmbedding.latticeBasis_apply K i).symm⟩
ext1 φ
rw [Pi.zero_apply]
by_cases hφ : ComplexEmbedding.IsReal φ
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 293 | 296 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,484 | 1,489 | |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Computability.Language
import Mathlib.Tactic.AdaptationNote
/-!
# Regular Expressions
This file contains the formal definition for regular expressions and ... | Mathlib/Computability/RegularExpressions.lean | 391 | 394 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... | decidable_of_iff _ wcovBy_iff.symm
| Mathlib/Order/Cover.lean | 435 | 436 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,541 | 1,545 | |
/-
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.Algebra.Module.Torsion
import Mathlib.FieldTheory.Perfect
import Mathlib.LinearAlgebra.AnnihilatingPolynomial
import Mathlib.RingTheory.Artinian.Instances
im... | ← (OrderIso.Iic _ _).complementedLattice_iff, Iic.complementedLattice_iff, Subtype.forall,
Subtype.exists, Subtype.mk_le_mk, Sublattice.mk_inf_mk, Sublattice.mk_sup_mk, Subtype.mk.injEq,
exists_and_left, exists_and_right, invtSubmodule.mk_eq_bot_iff, exists_prop, and_assoc]
rfl
/-- A linear endomorphism ... | Mathlib/LinearAlgebra/Semisimple.lean | 85 | 91 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | C.map_add a b
@[simp]
theorem smul_C {S} [SMulZeroClass S R] (s : S) (r : R) : s • C r = C (s • r) :=
| Mathlib/Algebra/Polynomial/Basic.lean | 474 | 477 |
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Antidiag.Pi
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
im... | multinomial {a, b} (Function.update f b (f b).succ) := by
simp only [binomial_eq_choose, Function.update_apply,
h, Ne, ite_true, ite_false, not_false_eq_true]
| Mathlib/Data/Nat/Choose/Multinomial.lean | 112 | 114 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Order.SuccPred.Archimedean
/-!
# Relations on types with a `SuccOrder`
This file contains properties about relations on types with a `SuccOrder`
... | variable {α : Type*} [PartialOrder α] [PredOrder α] [IsPredArchimedean α]
/-- For `m ≤ n`, `(n, m)` is in the reflexive-transitive closure of `~` if `i ~ pred i`
for all `i` between `n` and `m`. -/
| Mathlib/Order/SuccPred/Relation.lean | 84 | 87 |
/-
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
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 305 | 319 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 811 | 816 | |
/-
Copyright (c) 2023 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.Lemmas
/-!
# `compute_degree` and `monicity`: tactics for explicit polynomials
This file defines two related tactics: `comp... | split_ifs with h
· subst h h_pow_bas
exact coeff_pow_of_natDegree_le ‹_›
· apply coeff_eq_zero_of_natDegree_lt
apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ‹_› ?_)
· exact natDegree_pow_le_of_le m ‹_›
· exact Iff.mp ne_comm h
theorem natDegree_smul_le_of_le {n : ℕ} {a : R} {f : R[X]} (hf : natDegree f... | Mathlib/Tactic/ComputeDegree.lean | 117 | 126 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.B... | isOfFinOrder_iff_pow_eq_one.mpr <| by peel 2 h; exact this g
@[to_additive]
theorem ExponentExists.orderOf_pos (h : ExponentExists G) (g : G) : 0 < orderOf g :=
| Mathlib/GroupTheory/Exponent.lean | 92 | 95 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | @[simp]
theorem irrational_div_natCast_iff : Irrational (x / n) ↔ n ≠ 0 ∧ Irrational x := by
| Mathlib/Data/Real/Irrational.lean | 647 | 648 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 197 | 199 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,152 | 1,163 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... |
theorem symm_image_eq_source_inter_preimage {s : Set β} (h : s ⊆ e.target) :
| Mathlib/Logic/Equiv/PartialEquiv.lean | 418 | 419 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... |
@[simp]
theorem re_inner_self_nonpos {x : E} : re ⟪x, x⟫ ≤ 0 ↔ x = 0 := by
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 295 | 297 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Down-compressions
This file defines down-compression.
Down-compressing `𝒜 : Finset (Fins... | (𝒜 ∪ ℬ).nonMemberSubfamily a = 𝒜.nonMemberSubfamily a ∪ ℬ.nonMemberSubfamily a :=
filter_union _ _ _
theorem memberSubfamily_union (a : α) (𝒜 ℬ : Finset (Finset α)) :
(𝒜 ∪ ℬ).memberSubfamily a = 𝒜.memberSubfamily a ∪ ℬ.memberSubfamily a := by
| Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 74 | 78 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... | lemma mem_Ioc_succ : a ∈ Ioc b (b + 1) ↔ a = b + 1 := by simp
| Mathlib/Order/Interval/Finset/Nat.lean | 138 | 139 |
/-
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.NonUnitalHom
import Mathlib.Data.Set.UnionLift
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.RingTheory.NonUnitalSubring.Basic
... | mul_mem (mem_sup_left hx) (mem_sup_right hy)
| Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | 746 | 747 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... |
@[gcongr]
| Mathlib/Order/Heyting/Basic.lean | 513 | 514 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... |
theorem Eventually.forall_mem {α : Type*} {f : Filter α} {s : Set α} {P : α → Prop}
| Mathlib/Order/Filter/Basic.lean | 676 | 677 |
/-
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.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.... | lemma trace_eq_sum_trace_restrict' (h : IsInternal N) (hN : {i | N i ≠ ⊥}.Finite)
{f : M →ₗ[R] M} (hf : ∀ i, MapsTo f (N i) (N i)) :
trace R M f = ∑ i ∈ hN.toFinset, trace R (N i) (f.restrict (hf i)) := by
let _ : Fintype {i // N i ≠ ⊥} := hN.fintype
let _ : Fintype {i | N i ≠ ⊥} := hN.fintype
rw [← Finse... | Mathlib/Algebra/DirectSum/LinearMap.lean | 73 | 79 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Option.Basic
import Batteries.Tactic.Congr
import Mathlib.Data.Set.Basic
import Mathlib.Tactic.Contrapose
/-!
# Partial Equivalences
In this file, ... | /-- Creates a `PEquiv` that is the identity on `s`, and `none` outside of it. -/
def ofSet (s : Set α) [DecidablePred (· ∈ s)] :
α ≃. α where
toFun a := if a ∈ s then some a else none
invFun a := if a ∈ s then some a else none
inv a b := by
split_ifs with hb ha ha
· simp [eq_comm]
· simp [ne_of_me... | Mathlib/Data/PEquiv.lean | 183 | 193 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | @Fin.addCases _ _ (fun _ => α) a b
@[simp]
theorem append_left (u : Fin m → α) (v : Fin n → α) (i : Fin m) :
append u v (Fin.castAdd n i) = u i :=
addCases_left _
| Mathlib/Data/Fin/Tuple/Basic.lean | 317 | 322 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | @[reassoc, simp]
theorem whiskerLeft_rightUnitor_inv (X Y : C) :
X ◁ (ρ_ Y).inv = (ρ_ (X ⊗ Y)).inv ≫ (α_ X Y (𝟙_ C)).hom :=
| Mathlib/CategoryTheory/Monoidal/Category.lean | 592 | 594 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.Complex.Norm
/-!
# The partial order on the complex numbers
This order is defined by `z ≤ w ↔ z.re ≤ w.re ∧ z.im = w.im`.
This is a natural order o... | @[deprecated (since := "2025-02-16")] alias re_eq_abs := re_eq_norm
@[deprecated (since := "2025-02-16")] alias neg_re_eq_abs := neg_re_eq_norm
@[deprecated (since := "2025-02-16")] alias re_eq_neg_abs := re_eq_neg_norm
| Mathlib/Data/Complex/Order.lean | 113 | 116 |
/-
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.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | on all elements `x ∈ s`. -/
@[to_additive
"Sum of a `s : Multiset α` with `[AddMonoid α]`, given a proof that `+` commutes
on all elements `x ∈ s`."]
| Mathlib/Data/Finset/NoncommProd.lean | 103 | 106 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | exact coeff_add_eq_left_of_lt pn
open scoped Function -- required for scoped `on` notation
theorem degree_sum_eq_of_disjoint (f : S → R[X]) (s : Finset S)
(h : Set.Pairwise { i | i ∈ s ∧ f i ≠ 0 } (Ne on degree ∘ f)) :
degree (s.sum f) = s.sup fun i => degree (f i) := by
| Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 181 | 187 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... | simp [fiberPullbackEquiv]
change ((Types.pullbackIsoPullback _ _).inv ≫ _ ≫
(F ⋙ FintypeCat.incl).map (pullback.fst f g)) _ = _
erw [PreservesPullback.iso_inv_fst, Types.pullbackIsoPullback_inv_fst]
@[simp]
lemma fiberPullbackEquiv_symm_snd_apply {X A B : C} {f : A ⟶ X} {g : B ⟶ X}
(a : F.obj A) (b : F.o... | Mathlib/CategoryTheory/Galois/Basic.lean | 264 | 271 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | · rw [rpow_def_of_pos (abs_pos.2 hx), log_abs]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 158 | 158 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | exact S.adj_sub (c ainA binA (Subtype.coe_ne_coe.mp anb)).2.2
lemma IsClique.sdiff_of_sup_edge {v w : α} {s : Set α} (hc : (G ⊔ edge v w).IsClique s) :
G.IsClique (s \ {v}) := by
intro _ hx _ hy hxy
have := hc hx.1 hy.1 hxy
simp_all [sup_adj, edge_adj]
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 159 | 166 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,806 | 1,812 | |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... |
end Real
end HarmMeanLEGeomMean
section Young
/-! ### Young's inequality -/
namespace Real
/-- **Young's inequality**, a version for nonnegative real numbers. -/
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
| Mathlib/Analysis/MeanInequalities.lean | 394 | 408 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Extreme
import Mathlib.Analysis.Convex.Function
import Mathlib.Topology.Algebra.Module.LinearMap
import Mathlib... | · rintro x ⟨⟨hxA, hxB⟩, ⟨-, hxC⟩⟩
exact ⟨hxA, fun z hz => add_le_add (hxB z hz) (hxC z hz)⟩
rintro x ⟨hxA, hx⟩
refine ⟨⟨hxA, fun y hy => ?_⟩, hxA, fun y hy => ?_⟩
· exact
(add_le_add_iff_right (l₂ x)).1 ((add_le_add (hwB.2 y hy) (hwC.2 x hxA)).trans (hx w hwB.1))
· exact
(add_le_add_iff_left (... | Mathlib/Analysis/Convex/Exposed.lean | 128 | 137 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... | theorem trim_add (m₁ m₂ : OuterMeasure α) : (m₁ + m₂).trim = m₁.trim + m₂.trim :=
ext <| trim_binop (add_apply m₁ m₂)
/-- `trim` respects scalar multiplication. -/
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 426 | 429 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | exact le_csSup G.fintype_cliqueNum_bddAbove (Exists.intro t ⟨tc, rfl⟩)
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 634 | 635 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
-/
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# The Lucas-L... | -- the order of ω divides 2^p
· exact Nat.prime_two
· intro o
have ω_pow := orderOf_dvd_iff_pow_eq_one.1 o
replace ω_pow :=
congr_arg (Units.coeHom (X (q (p' + 2))) : Units (X (q (p' + 2))) → X (q (p' + 2))) ω_pow
simp? at ω_pow says
simp only [Units.coeHom_apply, Units.val_pow_eq_pow_val,... | Mathlib/NumberTheory/LucasLehmer.lean | 433 | 444 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
/-!
# Topological sums and functor... | classical
apply le_nhds_of_cauchy_adhp h
simp only [← mapClusterPt_def, mapClusterPt_iff_frequently, frequently_atTop, ge_iff_le,
| Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 218 | 220 |
/-
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.LinearAlgebra.DirectSum.Finsupp
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
/-!
# Some results on tensor product of s... | rfl
theorem mulMap_op :
mulMap (equivOpposite.symm (MulOpposite.op M)) (equivOpposite.symm (MulOpposite.op N)) =
(MulOpposite.opLinearEquiv R).toLinearMap ∘ₗ mulMap N M ∘ₗ
| Mathlib/LinearAlgebra/TensorProduct/Submodule.lean | 74 | 78 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Patrick Massot
-/
import Mathlib.Topology.Neighborhoods
/-!
# Neighborhoods of a set
In this file we define the filter `𝓝ˢ s` or `nhdsSet s` consisting of all ne... | rw [nhdsSet, ← range_diag, ← range_comp]
rfl
theorem mem_nhdsSet_iff_forall : s ∈ 𝓝ˢ t ↔ ∀ x : X, x ∈ t → s ∈ 𝓝 x := by
| Mathlib/Topology/NhdsSet.lean | 35 | 38 |
/-
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.Ring.Action.Basic
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.RingTheory.Congruence.Defs
/-!
# Congruence relations on rings
This fi... | Mathlib/RingTheory/Congruence/Basic.lean | 471 | 473 | |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | apply sublist_append_left
| cons₂ a s IH => simpa only [sublistsLen_succ_cons] using IH.append ((IHn s).map _)
| Mathlib/Data/List/Sublists.lean | 255 | 257 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | open ChartedSpace
namespace StructureGroupoid
| Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 185 | 188 |
/-
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.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Data.Complex... | /-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreco... | Mathlib/Analysis/Complex/AbsMax.lean | 221 | 226 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Order.CompactlyGenerated.Basic
/-!
# Generators for boolean algebras
In this file, we provide an alternative constructor for boolean algebras.
A set... | let _i := hS.distribLattice_of_sSup_eq_top h
have := hS.complementedLattice_of_sSup_eq_top h
DistribLattice.booleanAlgebraOfComplemented α
lemma sSup_le_sSup_iff_of_atoms (hS : BooleanGenerators S) (X Y : Set α) (hX : X ⊆ S) (hY : Y ⊆ S) :
sSup X ≤ sSup Y ↔ X ⊆ Y := by
| Mathlib/Order/BooleanGenerators.lean | 160 | 165 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,061 | 1,065 | |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | theorem ncard_preimage_of_injective_subset_range {s : Set β} (H : f.Injective)
(hs : s ⊆ Set.range f) :
(f ⁻¹' s).ncard = s.ncard := by
rw [← ncard_image_of_injective _ H, image_preimage_eq_iff.mpr hs]
| Mathlib/Data/Set/Card.lean | 693 | 697 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | alias CompleteLattice.Independent.injective := iSupIndep.injective
theorem iSupIndep_pair {i j : ι} (hij : i ≠ j) (huniv : ∀ k, k = i ∨ k = j) :
iSupIndep t ↔ Disjoint (t i) (t j) := by
constructor
· exact fun h => h.pairwiseDisjoint hij
· rintro h k
obtain rfl | rfl := huniv k
· refine h.mono_right ... | Mathlib/Order/SupIndep.lean | 429 | 438 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
imp... | /-! ### Numerator and denominator -/
| Mathlib/Data/NNRat/Defs.lean | 310 | 311 |
/-
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.Commute.Units
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Logic.Basic
import Mathlib.Data.Set.Basic
/-!
# Cent... |
namespace IsMulCentral
| Mathlib/Algebra/Group/Center.lean | 68 | 69 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.Int.GCD
/-!
# Congruences modulo a natural number
This file def... | Mathlib/Data/Nat/ModEq.lean | 501 | 502 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | ⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
ext
simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
rw [this]
repeat'
| Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 218 | 222 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | (Nat.lt_succ_self _) (dvd_of_mul_right_eq _ this)
| Mathlib/Algebra/Polynomial/Div.lean | 654 | 655 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 1,179 | 1,183 | |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed
import Mathlib.CategoryTheory.Monoidal.Rigid.Basic
import Mathlib.CategoryTheory.Monoidal.Subcategory
im... | Mathlib/Algebra/Category/FGModuleCat/Basic.lean | 276 | 280 | |
/-
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.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | smul_add _ _ _ := mul_add _ _ _
add_smul s t P := by
simp_rw [HSMul.hSMul, SetSemiring.down_add, span_union, sup_mul, add_eq_sup]
mul_smul s t P := by
simp_rw [HSMul.hSMul, SetSemiring.down_mul, ← mul_assoc, span_mul_span]
one_smul P := by
| Mathlib/Algebra/Algebra/Operations.lean | 749 | 754 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 819 | 820 | |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... |
theorem isUnit_nonsing_inv_det_iff {A : Matrix n n α} : IsUnit A⁻¹.det ↔ IsUnit A.det := by
rw [Matrix.det_nonsing_inv, isUnit_ringInverse]
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 426 | 429 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... | the only element that is right-orthogonal to every other element is `0`; i.e.,
for every nonzero `y` in `M₂`, there exists `x` in `M₁` with `B x y ≠ 0`. -/
def SeparatingRight (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) : Prop :=
∀ y : M₂, (∀ x : M₁, B x y = 0) → y = 0
/-- A bilinear map is called non-degenerate if it is left-sep... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 657 | 665 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | Mathlib/Data/PFun.lean | 598 | 603 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,387 | 1,389 |
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