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 |
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/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
/-!
# Gram-Schmidt Orthogonalization and Orthonor... | · simp [h]
· refine mem_span_singleton.2 ⟨‖gramSchmidt 𝕜 f i‖, smul_inv_smul₀ ?_ _⟩
exact mod_cast norm_ne_zero_iff.2 h
theorem span_gramSchmidtNormed_range (f : ι → E) :
span 𝕜 (range (gramSchmidtNormed 𝕜 f)) = span 𝕜 (range (gramSchmidt 𝕜 f)) := by
simpa only [image_univ.symm] using span_gramSchmi... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 285 | 294 |
/-
Copyright (c) 2021 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Adam Topaz, Johan Commelin
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.AlgebraicTopology.MooreComplex
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Catego... | refine Finset.sum_congr rfl fun x _ => ?_
rw [comp_zsmul, zsmul_comp]
congr 1
symm
apply f.naturality
| Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | 315 | 320 |
/-
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.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | Mathlib/Data/Int/Bitwise.lean | 453 | 454 | |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... |
It is equivalent in a natural way to multisets of cardinality 2 (see
| Mathlib/Data/Sym/Sym2.lean | 91 | 92 |
/-
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... |
lemma baseChange_negY :
(W'.baseChange B).toAffine.negY (f x) (f y) = f ((W'.baseChange A).toAffine.negY x y) := by
rw [← RingHom.coe_coe, ← map_negY, map_baseChange]
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 860 | 863 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... |
theorem gcd_eq_right {i j : ℤ} (H : j ∣ i) : gcd i j = natAbs j := by rw [gcd_comm, gcd_eq_left H]
| Mathlib/Data/Int/GCD.lean | 267 | 268 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... | tensorμ Z₁ Z₂ X₁ X₂ ≫ (Z₁ ◁ f₁ ⊗ Z₂ ◁ f₂) := by
convert tensorμ_natural (𝟙 Z₁) (𝟙 Z₂) f₁ f₂ using 1 <;> simp
@[reassoc]
theorem tensor_left_unitality (X₁ X₂ : C) :
(λ_ (X₁ ⊗ X₂)).hom =
((λ_ (𝟙_ C)).inv ▷ (X₁ ⊗ X₂)) ≫
tensorμ (𝟙_ C) (𝟙_ C) X₁ X₂ ≫ ((λ_ X₁).hom ⊗ (λ_ X₂).hom) := by
dsimp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 594 | 608 |
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... | · rw [Sym2.eq_iff, ← and_or_left] at h
exact .inl h.1
· rw [Sym2.eq_iff] at h
obtain (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩) := h <;> simp [hc]
| Mathlib/Data/List/Sym.lean | 63 | 66 |
/-
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... | theorem lt_div_iff_of_neg' (hc : c < 0) : a < b / c ↔ b < c * a := by
rw [mul_comm, lt_div_iff_of_neg hc]
| Mathlib/Algebra/Order/Field/Basic.lean | 359 | 360 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | theorem degree_scaleRoots (p : R[X]) {s : R} : degree (scaleRoots p s) = degree p := by
haveI := Classical.propDecidable
by_cases hp : p = 0
· rw [hp, zero_scaleRoots]
refine le_antisymm (Finset.sup_mono (support_scaleRoots_le p s)) (degree_le_degree ?_)
rw [coeff_scaleRoots_natDegree]
intro h
have := lea... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 67 | 74 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... |
@[simp]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 193 | 194 |
/-
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.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,352 | 1,367 | |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | Mathlib/Data/Part.lean | 785 | 786 | |
/-
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.Algebra.ZMod
import Mathlib.Data.Nat.Multiplicity
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.... | rw [frobeniusFun, coeff_mk]
variable (p) in
/-- `frobeniusFun` is tautologically a polynomial function.
| Mathlib/RingTheory/WittVector/Frobenius.lean | 203 | 207 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | -/
| Mathlib/Tactic/Ring/Basic.lean | 324 | 324 |
/-
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
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,108 | 1,115 | |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | · by_contra! length_lt_one
have : cs.lengthParity (s i) = Multiplicative.ofAdd 0 := by
rw [lengthParity_eq_ofAdd_length, Nat.lt_one_iff.mp length_lt_one, Nat.cast_zero]
have : Multiplicative.ofAdd (0 : ZMod 2) = Multiplicative.ofAdd 1 :=
this.symm.trans (cs.lengthParity_simple i)
contradiction... | Mathlib/GroupTheory/Coxeter/Length.lean | 152 | 159 |
/-
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.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | Continuous (f ⋆[L, μ] g) := by
rw [← convolution_flip]
exact hbf.continuous_convolution_right_of_integrable L.flip hg hf
end CommGroup
| Mathlib/Analysis/Convolution.lean | 703 | 707 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... |
instance isFiniteMeasure_restrict_Icc (x y : ℝ) : IsFiniteMeasure (volume.restrict (Icc x y)) :=
⟨by simp⟩
instance isFiniteMeasure_restrict_Ico (x y : ℝ) : IsFiniteMeasure (volume.restrict (Ico x y)) :=
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 193 | 197 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Connected.Clopen
/-!
# Totally disconnected and totally separated topological spaces
## Main definitions
We define th... |
theorem Continuous.connectedComponentsLift_unique (h : Continuous f) (g : ConnectedComponents α → β)
(hg : g ∘ (↑) = f) : g = h.connectedComponentsLift :=
| Mathlib/Topology/Connected/TotallyDisconnected.lean | 268 | 270 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
have : Continuous eval :=
((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
rw [eq_comm, ← sub_eq_zero]; show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
| Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 422 | 425 |
/-
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.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... | cases n.unpair.2
· rfl
simp only [decode_eq_ofNat, Option.some.injEq]
rw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)]
cases evaln k' _ _
· rfl
simp [k, hg (Nat.pair_lt_pair_right _ lg)]
· have lf := encode_lt_rfind' cf
rw [hg (Nat.pair_lt_pair_righ... | Mathlib/Computability/PartrecCode.lean | 946 | 954 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.TensorProduct.Opposite
import... | theorem toBaseChange_comp_reverseOp (Q : QuadraticForm R V) :
(toBaseChange A Q).op.comp reverseOp =
((Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q)).toAlgHom.comp <|
(Algebra.TensorProduct.map
(AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q))).comp
(toBaseChange A... | Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean | 125 | 138 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
/-!
# Identically distributed random variables
Two random variable... | exact hf.2
@[deprecated (since := "2025-02-21")]
alias memℒp_snd := memLp_snd
theorem memLp_iff [NormedAddCommGroup γ] [BorelSpace γ] {p : ℝ≥0∞} (h : IdentDistrib f g μ ν) :
MemLp f p μ ↔ MemLp g p ν :=
⟨fun hf => h.memLp_snd hf, fun hg => h.symm.memLp_snd hg⟩
@[deprecated (since := "2025-02-21")]
alias memℒ... | Mathlib/Probability/IdentDistrib.lean | 209 | 221 |
/-
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.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | Mathlib/Algebra/Quaternion.lean | 1,347 | 1,348 | |
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Algebra towers for multivariate polynomial
This file proves some basic resu... |
theorem aeval_algebraMap_eq_zero_iff_of_injective {x : σ → A} {p : MvPolynomial σ R}
(h : Function.Injective (algebraMap A B)) :
aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by
| Mathlib/RingTheory/MvPolynomial/Tower.lean | 56 | 59 |
/-
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.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... |
variable (r)
| Mathlib/Combinatorics/SimpleGraph/Density.lean | 66 | 67 |
/-
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.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Adjoint of operators on Hilbert spaces
Given ... | norm_mul_self_le x := le_of_eq <| Eq.symm <| norm_adjoint_comp_self x
theorem isAdjointPair_inner (A : E →L[𝕜] F) :
LinearMap.IsAdjointPair (sesqFormOfInner : E →ₗ[𝕜] E →ₗ⋆[𝕜] 𝕜)
(sesqFormOfInner : F →ₗ[𝕜] F →ₗ⋆[𝕜] 𝕜) A (A†) := by
intro x y
simp only [sesqFormOfInner_apply_apply, adjoint_inner_l... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 223 | 240 |
/-
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.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | simp [normUnit]
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 218 | 219 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... | { Equiv.piCongrRight fun _ => e.toEquiv with
uniformContinuous_toFun := UniformFun.postcomp_uniformContinuous e.uniformContinuous
uniformContinuous_invFun := UniformFun.postcomp_uniformContinuous e.symm.uniformContinuous }
| Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 412 | 415 |
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | Mathlib/Computability/Primrec.lean | 1,651 | 1,653 | |
/-
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.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
/-!
# Adhesive c... | Nonempty (IsColimit (BinaryCofan.mk (c.inr ≫ fE) iY)) ↔ IsPushout f c.inl iY fE := by
constructor
· rintro ⟨h⟩
refine ⟨H, ⟨Limits.PushoutCocone.isColimitAux' _ ?_⟩⟩
intro s
dsimp only [PushoutCocone.inr, PushoutCocone.mk] -- Porting note: Originally `dsimp`
refine ⟨h.desc (BinaryCofan.mk (c.inr ... | Mathlib/CategoryTheory/Adhesive.lean | 114 | 144 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.List.Lemmas
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Li... | theorem map_permutations (f : α → β) (ts : List α) :
map (map f) (permutations ts) = permutations (map f ts) := by
rw [permutations, permutations, map, map_permutationsAux, map]
theorem map_permutations' (f : α → β) (ts : List α) :
| Mathlib/Data/List/Permutation.lean | 223 | 227 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Tactic.FinCases
import Mathlib.Topology.Connected.LocallyConnected
import Mathlib.Topology.Sets.Closeds
/-!
# L... | f.map fun f => f a
/-- If α is finite, this constructs a locally constant function to `α → β` given a
family of locally constant functions with values in β indexed by α. -/
def unflip {X α β : Type*} [Finite α] [TopologicalSpace X] (f : α → LocallyConstant X β) :
| Mathlib/Topology/LocallyConstant/Basic.lean | 364 | 368 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupT... | @[ext]
theorem hom_ext {F₁ F₂ : FreeLieAlgebra R X →ₗ⁅R⁆ L} (h : ∀ x, F₁ (of R x) = F₂ (of R x)) :
| Mathlib/Algebra/Lie/Free.lean | 252 | 253 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... | @[simp]
theorem ShortComplex.mapToComposableArrows_comp {S₁ S₂ S₃ : ShortComplex C} (φ : S₁ ⟶ S₂)
(ψ : S₂ ⟶ S₃) : ShortComplex.mapToComposableArrows (φ ≫ ψ) =
ShortComplex.mapToComposableArrows φ ≫ ShortComplex.mapToComposableArrows ψ := by
aesop_cat
| Mathlib/Algebra/Homology/ExactSequence.lean | 61 | 66 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.MellinTransform
/-!
# Abstract functional equations for Mellin transforms
This file formalises a general version of an argument used to pro... | correction term of the form `A / s + B / (k - s)`. -/
def Λ₀ : ℂ → E := mellin P.f_modif
/-- A meromorphic function which agrees with the Mellin transform of `f - f₀` where defined -/
| Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean | 389 | 392 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and lower... | lemma isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s := by
rw [topology_eq α]
rfl
| Mathlib/Topology/Order/UpperLowerSetTopology.lean | 225 | 227 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | theorem IsRotated.mem_iff (h : l ~r l') {a : α} : a ∈ l ↔ a ∈ l' :=
h.perm.mem_iff
@[simp]
theorem isRotated_nil_iff : l ~r [] ↔ l = [] :=
⟨fun ⟨n, hn⟩ => by simpa using hn, fun h => h ▸ by rfl⟩
| Mathlib/Data/List/Rotate.lean | 404 | 410 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
/-!
# One-dimensional derivatives
This ... | on a neighborhood of `x₀` then its derivative at `x₀` has norm bounded by `C`. -/
theorem HasDerivAt.le_of_lipschitzOn {f : 𝕜 → F} {f' : F} {x₀ : 𝕜} (hf : HasDerivAt f f' x₀)
{s : Set 𝕜} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s) : ‖f'‖ ≤ C := by
| Mathlib/Analysis/Calculus/Deriv/Basic.lean | 800 | 802 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | -- protected lemma Measurable.iSup'' {_ : MeasurableSpace γ} {ι : Sort*} [Countable ι]
-- {f : ι → γ → δ → α} {h : γ → δ} (hf : ∀ i, Measurable ↿(f i)) (hh : Measurable h) :
-- Measurable fun a ↦ (⨆ i, f i a) (h a) := by
-- simp_rw [iSup_apply]
-- exact .iSup fun i ↦ by fun_prop
| Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 773 | 778 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Transfer
/-!
# The Schur-Zassenhaus Theorem
In this file we prove the Schur-Zassenhaus theorem.
## Main results
- `Subgroup.exists_ri... | rwa [inj.eq_iff, inj.eq_iff] at key
/-- Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` -/
private theorem step4 : (Nat.card N).minFac.Prime :=
Nat.minFac_prime (N.one_lt_card_iff_ne_bot.mpr (step0 h1 h3)).ne'
/-- Do not use this lemma: It is made obsolete by `exists_right_comp... | Mathlib/GroupTheory/SchurZassenhaus.lean | 222 | 230 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Multiset coercion to type
This module defines a `CoeSort` instance for multi... | `v ::ₘ m` is equivalent to `Option m` by mapping one `v` to `none` and everything else to `m`.
-/
def consEquiv {v : α} : v ::ₘ m ≃ Option m where
toFun x := if h : x.1 = v ∧ x.2.val = m.count v then none else some ⟨x.1, ⟨x.2, by
| Mathlib/Data/Multiset/Fintype.lean | 247 | 250 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Order.Mono... | rw [monovaryOn_inv_left₀ hf, antivaryOn_inv_right₀ hg]
lemma antivaryOn_inv₀ (hf : ∀ i ∈ s, 0 < f i) (hg : ∀ i ∈ s, 0 < g i) :
AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s := by
rw [antivaryOn_inv_left₀ hf, monovaryOn_inv_right₀ hg]
| Mathlib/Algebra/Order/Monovary.lean | 330 | 333 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | {s : Set β} (hs : MeasurableSet s) (a : α) :
Measurable (fun x ↦ density κ ν a x s) := by
change Measurable ((fun (p : α × γ) ↦ density κ ν p.1 p.2 s) ∘ (fun x ↦ (a, x)))
exact (measurable_density κ ν hs).comp measurable_prodMk_left
lemma density_mono_set (hκν : fst κ ≤ ν) (a : α) (x : γ) {s s' : Set β} (h... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 459 | 464 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 96 | 96 | |
/-
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.Topology.Order.Basic
/-!
# Set neighborhoods of intervals
In this file we prove basic theorems about `𝓝ˢ s`,
where `s` is one of the intervals
`Se... | theorem Ioi_mem_nhdsSet_Ici_iff : Ioi a ∈ 𝓝ˢ (Ici b) ↔ a < b := by
rw [isOpen_Ioi.mem_nhdsSet, Ici_subset_Ioi]
| Mathlib/Topology/Order/NhdsSet.lean | 57 | 58 |
/-
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.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | show s = a :: t ∨ s <:+ t ↔ s <:+ a :: t from
⟨fun o =>
match s, t, o with
| Mathlib/Data/List/Infix.lean | 173 | 175 |
/-
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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
/-!
# The `arctan` function.
Inequalit... | · rw [← lt_div_iff₀ hy, ← inv_eq_one_div] at h
replace h : arctan x < arctan y⁻¹ := tanOrderIso.symm.strictMono h
rwa [arctan_inv_of_pos hy, lt_tsub_iff_right] at h
theorem arctan_add {x y : ℝ} (h : x * y < 1) :
arctan x + arctan y = arctan ((x + y) / (1 - x * y)) := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 218 | 223 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
/-!
# Criteria under which a Dedekind domain is a PID
This file contains some ... | rw [Subtype.ext_iff, val_eq_coe, coe_mul] at hinv
let s := hf.toFinset
haveI := Classical.decEq (Ideal R)
have coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤ := by
simp_rw [s, Finset.mem_erase, hf.mem_toFinset]
rintro M hM M' ⟨hne, hM'⟩
exact Ideal.IsMaximal.coprime_of_ne hM hM' hne.symm
have nle... | Mathlib/RingTheory/DedekindDomain/PID.lean | 109 | 168 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | theorem left_ne_right_of_oangle_eq_pi {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = π) : p₁ ≠ p₃ :=
left_ne_right_of_oangle_ne_zero (h.symm ▸ Real.Angle.pi_ne_zero : ∡ p₁ p₂ p₃ ≠ 0)
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 85 | 86 |
/-
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 | 1,169 | 1,189 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | /-- The ith bit is the ith element of `n.bits`. -/
theorem testBit_eq_inth (n i : ℕ) : n.testBit i = n.bits.getI i := by
induction i generalizing n with
| zero =>
simp only [testBit, zero_eq, shiftRight_zero, one_and_eq_mod_two, mod_two_of_bodd,
bodd_eq_bits_head, List.getI_zero_eq_headI]
cases List.h... | Mathlib/Data/Nat/Bitwise.lean | 175 | 186 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 765 | 771 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,505 | 1,506 | |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.... | · rw [smul_mul_smul_comm, hf, smul_neg, ← Algebra.algebraMap_eq_smul_one, ← sub_eq_add_neg,
← RingHom.map_mul, ← RingHom.map_sub]
· rw [Algebra.smul_def, Algebra.smul_def, Algebra.smul_def, ← Algebra.right_comm _ x₂,
← mul_assoc, ← add_mul, ← RingHom.map_mul, ← RingHom.map_mul, ← RingHom... | Mathlib/Data/Complex/Module.lean | 287 | 291 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | repeat rw [nmul_assoc]
exact nmul_nadd_lt₃' ha hb hc
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 655 | 656 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | exact (-o).cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h
/-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side. -/
theorem sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle x (x + ... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 144 | 150 |
/-
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.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
rw [adjoin_eq_span, Finsupp.mem_span_iff_linearCombination] at hx
rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
use... | Mathlib/RingTheory/Adjoin/Basic.lean | 134 | 141 |
/-
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.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,239 | 2,241 | |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 295 | 295 | |
/-
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.Topology.Sets.Closeds
import Mathlib.Topology.Sets.OpenCover
/-!
# Sober spaces
A quasi-sober space is a topological space where every irreducible closed s... | mem_setOf_eq, and_imp, singleton_subset_iff]
| Mathlib/Topology/Sober.lean | 96 | 97 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
/-!
# Quotients of Polynomial Functors
We assume the following:
* `P`: a polynomial functor
* `W`: its W-type
* `M`: its M... | rfl
constructor
· rw [← yeq, ← abs_repr u, h, ← abs_map]
rfl
intro i
exact (f i).property
rintro ⟨a, f₀, f₁, xeq, yeq, h⟩
use abs ⟨a, fun i => ⟨(f₀ i, f₁ i), h i⟩⟩
constructor
· rw [xeq, ← abs_map]
rfl
rw [yeq, ← abs_map]; rfl
end
/-
Think of trees in the `W` type correspondi... | Mathlib/Data/QPF/Univariate/Basic.lean | 134 | 153 |
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
/-!
# p-groups
This file contains a proof that if `G` is a `p`-group ac... |
/-- If `gcd(p,n) = 1`, then the `n`th power map is a bijection. -/
noncomputable def powEquiv {n : ℕ} (hn : p.Coprime n) : G ≃ G :=
let h : ∀ g : G, (Nat.card (Subgroup.zpowers g)).Coprime n := fun g =>
| Mathlib/GroupTheory/PGroup.lean | 84 | 87 |
/-
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... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 448 | 451 | |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... |
section HasZeroHasOne
variable [Zero α] [One α]
@[simp]
| Mathlib/Data/Matrix/Block.lean | 281 | 286 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | rw [isOfFinOrder_iff_pow_eq_one] at *
rcases h with ⟨m, hm, ha⟩
exact ⟨n * m, mul_pos hn.bot_lt hm, by rwa [pow_mul]⟩
| Mathlib/GroupTheory/OrderOfElement.lean | 107 | 110 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
/-!
# Functions defined piecewise on a fins... | ext i
simp [piecewise]
variable [∀ j, Decidable (j ∈ s)]
| Mathlib/Data/Finset/Piecewise.lean | 35 | 38 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.List.Lemmas
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Li... | induction' ts with a ts ih; · rfl
simp only [permutations'Aux, ih, cons_append, permutationsAux2_snd_cons, append_nil, id_eq,
cons.injEq, true_and]
simp +singlePass only [← permutationsAux2_append]
simp [map_permutationsAux2]
| Mathlib/Data/List/Permutation.lean | 133 | 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.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 967 | 971 | |
/-
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.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,486 | 1,499 | |
/-
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... | | add p q ph qh =>
intro n
simp only [add_mul, smeval_add, ph, qh, Nat.cast_add]
| monomial n a =>
intro n
rw [smeval_monomial, smeval_monomial, nsmul_eq_mul, smul_eq_mul, Nat.cast_mul, Nat.cast_npow]
theorem smeval_at_zero : p.smeval (0 : S) = (p.coeff 0) • (1 : S) := by
induction p using Polyn... | Mathlib/Algebra/Polynomial/Smeval.lean | 192 | 200 |
/-
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.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | · simp [hy, eq_comm]
· rw [o.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero hx hy]
simp [hx, hy]
/-- The angle between two vectors is `θ` if and only if they are nonzero and the second vector
is the first rotated by `θ` and scaled by a positive real, or `θ` and at least one of the
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 314 | 319 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... |
section CommRing
variable [CommRing R] [Module R M]
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 124 | 127 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.Sigma
/-!
# Association Lists
This file defines association lists. An association list is a list where every element consists o... |
@[simp]
theorem insertRec_empty {C : AList β → Sort*} (H0 : C ∅)
| Mathlib/Data/List/AList.lean | 332 | 334 |
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Pointwise.Stabilizer
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTh... | @[to_additive]
lemma isBlock_iff_pairwiseDisjoint_range_smul :
IsBlock G B ↔ (range fun g : G ↦ g • B).PairwiseDisjoint id := pairwiseDisjoint_range_iff.symm
@[to_additive]
lemma isBlock_iff_smul_eq_smul_or_disjoint :
IsBlock G B ↔ ∀ g₁ g₂ : G, g₁ • B = g₂ • B ∨ Disjoint (g₁ • B) (g₂ • B) :=
forall₂_congr fu... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 176 | 183 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Data.DFinsupp.Small
import Mathlib.Data.Fintype.Pi
import Mathlib.LinearA... | p ∈ restrictTotalDegree σ R m ↔ p.totalDegree ≤ m := by
rw [totalDegree, Finset.sup_le_iff]
rfl
| Mathlib/RingTheory/MvPolynomial/Basic.lean | 107 | 110 |
/-
Copyright (c) 2023 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.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | lemma in₁₀_iff' :
(graph t).Adj (in₁ b) (in₀ a) ↔ ∃ x : α × β × γ, x ∈ t ∧ x.2.1 = b ∧ x.1 = a where
mp := by rintro ⟨⟩; exact ⟨_, ‹_›, by simp⟩
mpr := by rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩; constructor; assumption
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 89 | 92 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 550 | 552 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... | @[simp]
theorem tail_val' (B : Fin m.succ → n' → α) (j : n') :
(vecTail fun i => B i j) = fun i => vecTail B i j := rfl
| Mathlib/Data/Matrix/Notation.lean | 168 | 170 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... | def homFromEquiv (α : X ≅ Y) {Z : C} : (X ⟶ Z) ≃ (Y ⟶ Z) where
toFun f := α.inv ≫ f
| Mathlib/CategoryTheory/Iso.lean | 219 | 220 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... | closedBall 1 δ * s = cthickening δ s := by rw [mul_comm, hs.mul_closedBall_one hδ]
| Mathlib/Analysis/Normed/Group/Pointwise.lean | 207 | 207 |
/-
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.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.PSeriesComplex
/-!
# Definition of the Riemann zeta function
## Main definitions:
* `rieman... | real variable. -/
theorem tendsto_sub_mul_tsum_nat_rpow :
Tendsto (fun s : ℝ ↦ (s - 1) * ∑' (n : ℕ), 1 / (n : ℝ) ^ s) (𝓝[>] 1) (𝓝 1) := by
rw [← tendsto_ofReal_iff, ofReal_one]
have : Tendsto (fun s : ℝ ↦ (s : ℂ)) (𝓝[>] 1) (𝓝[{s | 1 < re s}] 1) :=
| Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 213 | 217 |
/-
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.Topology.Order.Basic
/-!
# Set neighborhoods of intervals
In this file we prove basic theorems about `𝓝ˢ s`,
where `s` is one of the intervals
`Se... | theorem nhdsSet_Ico (h : a < b) : 𝓝ˢ (Ico a b) = 𝓝 a ⊔ 𝓟 (Ioo a b) := by
rw [← Ioo_insert_left h, nhdsSet_insert, nhdsSet_Ioo]
| Mathlib/Topology/Order/NhdsSet.lean | 41 | 42 |
/-
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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | theorem fst_sup_snd : Submodule.fst R M M₂ ⊔ Submodule.snd R M M₂ = ⊤ := by
| Mathlib/LinearAlgebra/Prod.lean | 571 | 571 |
/-
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
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 1,023 | 1,032 | |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar... |
theorem memLp_lineDeriv (hf : LipschitzWith C f) (v : E) :
MemLp (fun x ↦ lineDeriv ℝ f x v) ∞ μ :=
memLp_top_of_bound (aestronglyMeasurable_lineDeriv hf.continuous μ)
(C * ‖v‖) (.of_forall fun _x ↦ norm_lineDeriv_le_of_lipschitz ℝ hf)
@[deprecated (since := "2025-02-21")]
alias memℒp_lineDeriv := memLp_lin... | Mathlib/Analysis/Calculus/Rademacher.lean | 63 | 77 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.FunLike.Equiv
import Mathlib.Data.Quot
import Mathlib.Data.Subtype
import Mathlib.Logic.U... | @[simp] theorem equivCongr_refl {α β} :
(Equiv.refl α).equivCongr (Equiv.refl β) = Equiv.refl (α ≃ β) := by ext; rfl
| Mathlib/Logic/Equiv/Defs.lean | 338 | 339 |
/-
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 prod_coe_symm (e : PartialEquiv α β) (e' : PartialEquiv γ δ) :
((e.prod e').symm : β × δ → α × γ) = fun p => (e.symm p.1, e'.symm p.2) :=
rfl
| Mathlib/Logic/Equiv/PartialEquiv.lean | 779 | 781 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... | variable {I}
lemma dual_div_dual :
dual A K J / dual A K I = I / J := by
rw [dual_eq_mul_inv A K J, dual_eq_mul_inv A K I, mul_div_mul_comm, div_self, one_mul]
| Mathlib/RingTheory/DedekindDomain/Different.lean | 343 | 347 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Topology.Bases
import Mathlib.Topology.Inseparable
import Mathlib.Topology.Compactness.Exterior
/-!
# Alexandrov-discrete to... | ⟨fun h ↦ ⟨exterior s, isOpen_exterior, subset_exterior, h⟩,
fun ⟨_U, hU, hsU, hUt⟩ ↦ (exterior_minimal hsU hU).trans hUt⟩
| Mathlib/Topology/AlexandrovDiscrete.lean | 158 | 159 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,460 | 1,464 | |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | IsTheta.const_mul_left hp <| isTheta_refl _ _
lemma growsPolynomially_deriv_rpow_p_mul_one_sub_smoothingFn (p : ℝ) :
GrowsPolynomially fun x => ‖deriv (fun z => z ^ p * (1 - ε z)) x‖ := by
cases eq_or_ne p 0 with
| inl hp => -- p = 0
have h₁ : (fun x => ‖deriv (fun z => z ^ p * (1 - ε z)) x‖)
| Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 892 | 898 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 542 | 547 | |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 1,115 | 1,120 | |
/-
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.GradedMonoid
import Mathlib.Algebra.DirectSum.Basic
/-!
# Additively-graded multiplicative structures on `⨁ i, A i`
This module provides a set of h... | instance semiring : Semiring (⨁ i, A i) :=
{ (inferInstance : NonUnitalNonAssocSemiring _) with
one_mul := one_mul A
mul_one := mul_one A
mul_assoc := mul_assoc A
toNatCast := instNatCast _
natCast_zero := by simp only [NatCast.natCast, GSemiring.natCast_zero, map_zero]
natCast_succ := fun n =... | Mathlib/Algebra/DirectSum/Ring.lean | 248 | 261 |
/-
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.Data.List.Forall2
/-!
# List sections
This file proves some stuff about `List.sections` (definition in `Data.List.Defs`). A section of a
list of lists... | · induction L generalizing f
· cases mem_singleton.1 h
exact Forall₂.nil
simp only [sections, bind_eq_flatMap, mem_flatMap, mem_map] at h
rcases h with ⟨_, _, _, _, rfl⟩
simp only [*, forall₂_cons, true_and]
· induction h with
| nil => simp only [sections, mem_singleton]
| @cons a l f ... | Mathlib/Data/List/Sections.lean | 23 | 34 |
/-
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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | · simp [hz]
· simp_rw [hz, or_false, ← not_lt, not_and_self_iff, or_false]
@[simp]
theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by
by_cases h : arg x = π <;> simp [arg_conj, h]
@[simp]
theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by
by_cases h : arg x ... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 389 | 401 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | refine limsup_le_limsup ?_ ?_ ?_
· exact Eventually.of_forall (fun n ↦ densityProcess_mono_set hκν n a x h)
· exact isCoboundedUnder_le_of_le atTop (fun i ↦ densityProcess_nonneg _ _ _ _ _ _)
· exact isBoundedUnder_of ⟨1, fun n ↦ densityProcess_le_one hκν _ _ _ _⟩
lemma density_nonneg (hκν : fst κ ≤ ν) (a : α)... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 466 | 471 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | rfl
| Mathlib/Order/SymmDiff.lean | 204 | 205 |
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