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) 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.Topology.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | Mathlib/Topology/Order.lean | 902 | 903 | |
/-
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.Data.Countable.Defs
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Data... | exact fun p hp hpm _ ↦ ⟨hp, hpm.trans hmn, hn⟩
lemma mem_primeFactors_iff_mem_primeFactorsList : p ∈ n.primeFactors ↔ p ∈ n.primeFactorsList := by
| Mathlib/Data/Nat/PrimeFin.lean | 43 | 45 |
/-
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.MeasureTheory.MeasurableSpace.EventuallyMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.M... |
end MeasurableSingletonClass
| Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 319 | 320 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
imp... | Mathlib/Algebra/Group/Hom/Defs.lean | 1,174 | 1,177 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | lemma ofArrows_le_ofObjects
{I : Type*} (Y : I → C) {X : C} (f : ∀ i, Y i ⟶ X) :
Sieve.ofArrows Y f ≤ Sieve.ofObjects Y X := by
intro W g hg
rw [mem_ofArrows_iff] at hg
obtain ⟨i, a, rfl⟩ := hg
exact ⟨i, ⟨a⟩⟩
| Mathlib/CategoryTheory/Sites/Sieves.lean | 504 | 510 |
/-
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... | obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure
h_fund (fun _ ↦ convexBodySum_neg_mem K B) (convexBodySum_convex K B)
(convexBodySum_compact K B) h
rw [mem_toAddSubgroup, mem_span_fractionalIdealLatticeBasis] at hx
obtain ⟨a, ha, rfl⟩ := hx
refine ⟨a, ha... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 598 | 629 |
/-
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.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
/-!
# Definition... | @[simp] lemma nilpotencyClass_zero [Nontrivial R] :
nilpotencyClass (0 : R) = 1 :=
nilpotencyClass_eq_succ_iff.mpr <| by constructor <;> simp
@[simp] lemma pos_nilpotencyClass_iff [Nontrivial R] :
| Mathlib/RingTheory/Nilpotent/Defs.lean | 137 | 141 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "covering pres... | theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisymm
· apply le_sInf; intro J hJ
intro X S hS
induction hS with
| of X S hS => apply hJ; assumption
| top => apply J.top_mem
| transitive X R S _ _ H1 H2 => exact J.tr... | Mathlib/CategoryTheory/Sites/Coverage.lean | 275 | 286 |
/-
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 | 1,439 | 1,448 | |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 837 | 838 | |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | ∑ i ∈ range n, x ^ i = (x ^ n - 1) / (x - 1) := by
have : x - 1 ≠ 0 := by simp_all [sub_eq_iff_eq_add]
rw [← geom_sum_mul, mul_div_cancel_right₀ _ this]
lemma geom_sum_of_one_lt {x : K} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K]
[CanonicallyOrderedAdd K] [Sub K] [OrderedSub K]
(h : 1 < x) (n... | Mathlib/Algebra/GeomSum.lean | 283 | 300 |
/-
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.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | · haveI := h₁.fst
haveI := h₂.fst
apply NFBelow.oadd
· infer_instance
· rwa [repr_add]
· rw [repr_add, add_lt_add_iff_left]
exact h₂.lt
instance mul_nf : ∀ (o₁ o₂) [NF o₁] [NF o₂], NF (o₁ * o₂)
| 0, o, _, h₂ => by cases o <;> exact NF.zero
| oadd _ _ _, _, ⟨⟨_, hb₁⟩⟩, ⟨⟨_,... | Mathlib/SetTheory/Ordinal/Notation.lean | 532 | 565 |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.DSlope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathl... | eqOn_of_preconnected_of_frequently_eq hf hg isPreconnected_univ (mem_univ z₀) hfg (mem_univ x)
section Mul
/-!
### Vanishing of products of analytic functions
-/
| Mathlib/Analysis/Analytic/IsolatedZeros.lean | 264 | 269 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | rw [← neg_eq_zero, ← inner_neg_right] at h
rw [or_comm, ← neg_ne_zero, or_comm] at h0
rw [sub_eq_add_neg, sin_angle_add_of_inner_eq_zero h h0, norm_neg]
/-- The tangent of an angle in a right-angled triangle as a ratio of sides, version subtracting
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 257 | 261 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... | Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 292 | 303 | |
/-
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.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
/-!
# Phragmen-Lindelöf principle
In this file we prove several ve... | * `‖f z‖` is bounded from above by `A * exp(B * ‖z‖ ^ c)` on the open second quadrant
for some `A`, `B`, and `c < 2`;
* `f` is equal to zero on the boundary of the second quadrant.
Then `f` is equal to zero on the closed second quadrant. -/
theorem eq_zero_on_quadrant_II (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0))
... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 461 | 477 |
/-
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.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | (measurableSet_singleton 0)
/-- The set of points `a : α` such that `κ a ⟂ₘ η a` is measurable. -/
@[measurability]
lemma measurableSet_mutuallySingular (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
MeasurableSet {a | κ a ⟂ₘ η a} := by
simp_rw [← withDensity_rnDeriv_eq_zero_iff_mutuallySingular,... | Mathlib/Probability/Kernel/RadonNikodym.lean | 468 | 478 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | | _, 0 => by simp
| Mathlib/Data/Nat/Choose/Basic.lean | 378 | 378 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... | · have := hs.fintype
have := fintypeImage s f
simp_rw [Nat.card_eq_fintype_card, Set.card_image_of_inj_on hf]
| Mathlib/SetTheory/Cardinal/Finite.lean | 144 | 146 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... | linarith [mul_le_mul_of_nonneg_right this <| (mul_nonneg_iff_of_pos_left two_pos).2 C0]
rw [← toReal_ofReal (le_of_lt ε₂0)]
refine toReal_mono ofReal_ne_top (le_trans ?_ (le_of_lt hU))
trans μ' (⋃ J ∈ B', J)
· simp only [μ', μ.restrict_eq_self <| (un _ hB').trans I.coe_subset_Icc]
exact le_o... | Mathlib/Analysis/BoxIntegral/Basic.lean | 718 | 789 |
/-
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.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
/-!
# Ackermann function
In this file, we def... | induction' n with n IH
-- The base case is easy.
· apply (ha m).trans (ack_strictMono_left m <| (le_max_left a b).trans_lt _)
omega
· -- We get rid of the first `pair`.
simp only
apply (hb _).trans ((ack_pair_lt _ _ _).trans_le _)
-- If m is the maximum, we get a ... | Mathlib/Computability/Ackermann.lean | 311 | 376 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | theorem piMap_image_pi {f : ∀ i, α i → β i} (hf : ∀ i ∉ s, Surjective (f i)) (t : ∀ i, Set (α i)) :
Pi.map f '' s.pi t = s.pi fun i ↦ f i '' t i := by
| Mathlib/Data/Set/Prod.lean | 823 | 824 |
/-
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.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | intro i j hij
rw [dite_eq_right_iff]
intro hij'
exfalso
exact hij hij'
/-! This lemma and the following ones can be used in order to compute
| Mathlib/Algebra/Homology/Homotopy.lean | 333 | 339 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | refine lintegral_mono_ae (h.mono fun x hx => ?_)
dsimp [enorm]
gcongr
theorem eLpNorm'_mono_ae {f : α → F} {g : α → G} (hq : 0 ≤ q) (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ ‖g x‖) :
eLpNorm' f q μ ≤ eLpNorm' g q μ :=
eLpNorm'_mono_enorm_ae hq (by simpa only [enorm_le_iff_norm_le] using h)
theorem eLpNorm'_congr_enorm_ae {f ... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 328 | 344 |
/-
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.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import ... | namespace CliffordAlgebraDualNumber
open scoped DualNumber
open DualNumber TrivSqZeroExt
variable {R : Type*} [CommRing R]
theorem ι_mul_ι (r₁ r₂) : ι (0 : QuadraticForm R R) r₁ * ι (0 : QuadraticForm R R) r₂ = 0 := by
rw [← mul_one r₁, ← mul_one r₂, ← smul_eq_mul r₁, ← smul_eq_mul r₂, LinearMap.map_smul,
| Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 339 | 348 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
/-!
# A `norm_num` extension for Jacobi and Legendre symbols
We extend the `norm_num` tactic so that it can be... | theorem jacobiSymNat.qr₁_mod (a b ab : ℕ) (r : ℤ) (ha : a % 4 = 1) (hb : b % 2 = 1)
(hab : b % a = ab) (hr : jacobiSymNat ab a = r) : jacobiSymNat a b = r :=
jacobiSymNat.qr₁ _ _ _ ha hb <| jacobiSymNat.mod_left _ _ ab r hab hr
theorem jacobiSymNat.qr₁' (a b : ℕ) (r : ℤ) (ha : a % 2 = 1) (hb : b % 4 = 1)
| Mathlib/Tactic/NormNum/LegendreSymbol.lean | 158 | 162 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | algebra over `ℝ`, e.g., `ℂ`).
TODO: introduce a typeclass saying that `1 / n` tends to 0 at top, making it possible to get this
statement simultaneously on `ℚ`, `ℝ` and `ℂ`. -/
theorem tendsto_natCast_div_add_atTop {𝕜 : Type*} [DivisionRing 𝕜] [TopologicalSpace 𝕜]
[CharZero 𝕜] [Algebra ℝ 𝕜] [ContinuousSMul ℝ ... | Mathlib/Analysis/SpecificLimits/Basic.lean | 74 | 79 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... |
lemma map_preΨ (n : ℤ) : (W.map f).preΨ n = (W.preΨ n).map f := by
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 566 | 567 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.PInfty
/-!
# Decomposition of the Q endomorphisms
In this file, we obtain a lemma `decomposition_Q` which expresses
explicitly the p... | @[simps]
def postComp : MorphComponents X n Z' where
a := f.a ≫ h
b i := f.b i ≫ h
| Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 120 | 124 |
/-
Copyright (c) 2021 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.Group.Measure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this file we show propertie... | @[to_additive]
theorem quasiMeasurePreserving_div_of_right_invariant [IsMulRightInvariant μ] :
QuasiMeasurePreserving (fun p : G × G => p.1 / p.2) (μ.prod ν) μ := by
refine QuasiMeasurePreserving.prod_of_left measurable_div (Eventually.of_forall fun y => ?_)
exact (measurePreserving_div_right μ y).quasiMeasureP... | Mathlib/MeasureTheory/Group/Prod.lean | 424 | 429 |
/-
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 ... |
alias ⟨HasStrictDerivAt.hasStrictFDerivAt, _⟩ := hasStrictDerivAt_iff_hasStrictFDerivAt
| Mathlib/Analysis/Calculus/Deriv/Basic.lean | 201 | 203 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | (h : Pairwise (Disjoint on f)) :
lift.{v} #(⋃ i, f i) = sum fun i => #(f i) :=
calc
lift.{v} #(⋃ i, f i) = #(Σi, f i) :=
| Mathlib/SetTheory/Cardinal/Basic.lean | 722 | 725 |
/-
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.Ring.Nat
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Order.Partition.Fi... |
theorem IsEquipartition.average_le_card_part (hP : P.IsEquipartition) (ht : t ∈ P.parts) :
#s / #P.parts ≤ #t := by
rw [← P.sum_card_parts]
exact Finset.EquitableOn.le hP ht
| Mathlib/Order/Partition/Equipartition.lean | 61 | 66 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Floris van Doorn
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# `norm_num` plugin for ... | rw [pn.out, Nat.cast_id, pn', Multiset.range_succ]
/-- Either show the expression `s : Q(Multiset α)` is Zero, or remove one element from it.
| Mathlib/Tactic/NormNum/BigOperators.lean | 181 | 183 |
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... | calc
‖exp (x + z) - exp x - z * exp x‖ = ‖exp x * (exp z - 1 - z)‖ := by
congr
rw [exp_add]
ring
_ = ‖exp x‖ * ‖exp z - 1 - z‖ := norm_mul _ _
_ ≤ ‖exp x‖ * ‖z‖ ^ 2 :=
mul_le_mul_of_nonneg_left (norm_exp_sub_one_sub_id_le hz) (norm_nonneg _)
theorem locally_lipschitz_exp {r : ℝ} (... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 33 | 42 |
/-
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.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Tauto
/-!
# More basic logic properties
A few more logic le... | Mathlib/Logic/Lemmas.lean | 24 | 24 | |
/-
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.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,716 | 1,719 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 137 | 139 | |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | haveI := Classical.decEq R
Multiset.toFinset (nthRoots n a)
| Mathlib/Algebra/Polynomial/Roots.lean | 315 | 316 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunc... | linear map from `TangentSpace (𝓡 n) v` to `E`. The range of this map is the orthogonal complement
of `v` in `E`.
Note that there is an abuse here of the defeq between `E` and the tangent space to `E` at `(v:E`).
In general this defeq is not canonical, but in this case (the tangent space of a vector space) it is
cano... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 465 | 486 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | ContinuousWithinAt f (s \ {x}) x ↔ ContinuousWithinAt f s x :=
continuousWithinAt_singleton.diff_iff
| Mathlib/Topology/ContinuousOn.lean | 771 | 773 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... |
@[simp]
| Mathlib/Algebra/Polynomial/EraseLead.lean | 55 | 56 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... |
/-- If distinct points `p₂` and `p₃` lie in `s` but `p₁` does not, the three points are affinely
independent. -/
theorem affineIndependent_of_ne_of_not_mem_of_mem_of_mem {s : AffineSubspace k P} {p₁ p₂ p₃ : P}
(hp₂p₃ : p₂ ≠ p₃) (hp₁ : p₁ ∉ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) :
AffineIndependent k ![p₁, p₂, p₃] :=... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 700 | 713 |
/-
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... | rcases lt_trichotomy r 0 with (hr | hr | hr)
· rw [← neg_smul]
exact Or.inr ⟨hx, smul_ne_zero hr.ne hx,
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 398 | 400 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Top... | obtain ⟨y, hy, hxy⟩ := hx l
exact ((hxy.trans_lt (hlA y hy)).trans hl).not_le le_rfl
@[deprecated (since := "2024-11-12")] alias iInter_halfspaces_eq := iInter_halfSpaces_eq
end
namespace RCLike
| Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | 207 | 214 |
/-
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
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.MeasurableSpace.Prod
import Mathlib.MeasureTheory.Measure.Ty... | measurable_ereal_toReal.comp_aemeasurable hf
@[measurability]
theorem measurable_coe_ennreal_ereal : Measurable ((↑) : ℝ≥0∞ → EReal) :=
continuous_coe_ennreal_ereal.measurable
| Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 417 | 421 |
/-
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... | end GroupWithZero
section CommGroupWithZero
| Mathlib/Data/Int/GCD.lean | 406 | 408 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,363 | 1,380 | |
/-
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.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | rfl)
(by
ext x
simp_rw [ringQuotToIdealQuotient, lift_def, preLift_def, mkRingHom_def]
| Mathlib/Algebra/RingQuot.lean | 518 | 521 |
/-
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.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | · exact measurableSet_preimage (measurable_singularPart_fun_right κ η a)
(measurableSet_singleton _)
· exact measurable_singularPart_fun_right κ η a
· exact measurable_singularPart_fun κ η
refine ae_of_all _ (fun x hx ↦ ?_)
simp only [mem_compl_iff, mutuallySingularSetSlice, mem_setOf, not_le] at hx
s... | Mathlib/Probability/Kernel/RadonNikodym.lean | 282 | 288 |
/-
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.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | apply le_antisymm
· intro p hp
| Mathlib/RingTheory/Polynomial/Basic.lean | 67 | 68 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | alias norm_le_insert := norm_le_norm_add_norm_sub
| Mathlib/Analysis/Normed/Group/Basic.lean | 533 | 534 |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | (h : xn a1 i ≡ xn a1 j [MOD xn a1 n])
(ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)) : i = j :=
if npos : n = 0 then by simp_all
else
(lt_or_eq_of_le ij).resolve_left fun ij' =>
if jn : j = n then by
refine _root_.ne_of_gt ?_ h
rw [jn, Nat.mod_self]
have x0 : 0 < xn a1 0 % xn a... | Mathlib/NumberTheory/PellMatiyasevic.lean | 655 | 663 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | lemma affineCombination_mem_interior_iff {n : ℕ} {s : Simplex k P n} {w : Fin (n + 1) → k}
(hw : ∑ i, w i = 1) :
Finset.univ.affineCombination k s.points w ∈ s.interior ↔ ∀ i, w i ∈ Set.Ioo 0 1 := by
refine ⟨fun ⟨w', hw', hw'01, hww'⟩ ↦ ?_, fun h ↦ ⟨w, hw, h, rfl⟩⟩
simp_rw [← (affineIndependent_iff_eq_of_fi... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 975 | 980 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.DiscreteValuationRing.Basi... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 378 | 380 | |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.DSlope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathl... | variable {a : ℕ → E}
theorem hasSum_at_zero (a : ℕ → E) : HasSum (fun n => (0 : 𝕜) ^ n • a n) (a 0) := by
convert hasSum_single (α := E) 0 fun b h ↦ _ <;> simp [*]
theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m) s)
(ha : ∀ k < n, a k = 0) : ∃ t : E, z ^ n • t = s ∧ HasSum (fun m... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 48 | 62 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,676 | 1,677 | |
/-
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 | 1,755 | 1,761 | |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,692 | 1,693 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | exact tendsto_bot
/-!
### `ContinuousOn`
-/
| Mathlib/Topology/ContinuousOn.lean | 571 | 575 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Equalizers and coequalizers
This file defines (co)equalizers a... |
@[simp]
| Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 126 | 127 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.SetTheory.Cardinal.Regular
import Mathlib.SetTheory.Game.Birthday
/-!
# Short games
A combinatorial game is `Short` [C... |
instance short0 : Short 0 :=
Short.ofIsEmpty
instance short1 : Short 1 :=
Short.mk (fun i => by cases i; infer_instance) fun j => by cases j
/-- Evidence that every `PGame` in a list is `Short`. -/
class inductive ListShort : List PGame.{u} → Type (u + 1)
| nil : ListShort []
-- Porting note: We introduce `c... | Mathlib/SetTheory/Game/Short.lean | 153 | 168 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 774 | 783 | |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | rcases h _ hc with ⟨M, hM⟩
refine ⟨M / ‖c‖, forall_mem_range.mpr fun i ↦ (le_div_iff₀' (norm_pos_iff.2 hc₀)).2 ?_⟩
exact hM ⟨i, map_smul_eq_mul ..⟩
end NontriviallyNormedField
end Seminorm
/-! ### The norm as a seminorm -/
section normSeminorm
variable (𝕜) (E) [NormedField 𝕜] [SeminormedAddCommGroup E] [N... | Mathlib/Analysis/Seminorm.lean | 1,271 | 1,284 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.RingThe... | (IsLocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting note: need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
| Mathlib/RingTheory/Localization/AtPrime.lean | 166 | 170 |
/-
Copyright (c) 2023 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak
-/
import Mathlib.Computability.Language
/-!
# Context-Free Grammars
This file contains the definition of a context-free grammar, which is a grammar that has a single
no... | /-- Rule `r` rewrites string `u` is to string `v` iff they share both a prefix `p` and postfix `q`
such that the remaining middle part of `u` is the input of `r` and the remaining middle part
of `u` is the output of `r`. -/
theorem rewrites_iff :
r.Rewrites u v ↔ ∃ p q : List (Symbol T N),
u = p ++ [Symbol.no... | Mathlib/Computability/ContextFreeGrammar.lean | 82 | 88 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.... | theorem ofMkLE_arrow {B A : C} {f : A ⟶ B} [Mono f] {X : Subobject B} (h : mk f ≤ X) :
ofMkLE f X h ≫ X.arrow = f := by simp [ofMkLE]
/-- An inequality of subobjects is witnessed by some morphism between the corresponding objects. -/
def ofMkLEMk {B A₁ A₂ : C} (f : A₁ ⟶ B) (g : A₂ ⟶ B) [Mono f] [Mono g] (h : mk f ... | Mathlib/CategoryTheory/Subobject/Basic.lean | 331 | 335 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | · cases w
simp only [heq_eq_eq, forall_true_left] at h
simp [biproduct.ι_π_ne _ h]
· simp [biproduct.ι_π_ne_assoc _ w] }
isBilimit :=
{ isLimit := mkFanLimit _
(fun s => biproduct.lift fun b => biproduct.lift fun c => s.proj ⟨b, c⟩)
isCol... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 684 | 698 |
/-
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.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | /-- The function `evenKernel a - L` has exponential decay at `+∞`, where `L = 1` if
`a = 0` and `L = 0` otherwise. -/
lemma isBigO_atTop_evenKernel_sub (a : UnitAddCircle) : ∃ p : ℝ, 0 < p ∧
(evenKernel a · - (if a = 0 then 1 else 0)) =O[atTop] (rexp <| -p * ·) := by
induction a using QuotientAddGroup.induction_o... | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 218 | 231 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | open ENNReal
| Mathlib/Topology/MetricSpace/Thickening.lean | 381 | 382 |
/-
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.RingTheory.DedekindDomain.Ideal
/-!
# The ideal class group
This file defines the ideal class group `ClassGroup R` of fractional ideals of `R`
inside its f... | rw [hx, IsFractionRing.mk'_eq_div, map_zero, zero_div]
· exact (FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal _ hy).mp h
· rintro ⟨x, y, hx, hy, h⟩
have hy' : y ∈ R⁰ := mem_nonZeroDivisors_iff_ne_zero.mpr hy
refine ⟨IsLocalization.mk' _ x ⟨y, hy'⟩, ?_, ?_⟩
· contrapose! hx
rwa [mk'_eq_iff_e... | Mathlib/RingTheory/ClassGroup.lean | 281 | 294 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGrou... |
theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
| Mathlib/NumberTheory/ModularForms/SlashActions.lean | 184 | 189 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... | (toSubgroupEquiv φ u g'.1, g'.2)
theorem unitsSMulGroup_snd (u : ℤˣ) (g : G) :
| Mathlib/GroupTheory/HNNExtension.lean | 344 | 346 |
/-
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
... |
end IsRefl
end Reflexive
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 179 | 182 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | exact AlgHom.congr_fun (map_comp_map _ _) x
@[simp]
theorem equivOfIsometry_refl :
(equivOfIsometry <| QuadraticMap.IsometryEquiv.refl Q₁) = AlgEquiv.refl := by
ext x
exact AlgHom.congr_fun (map_id Q₁) x
end Map
| Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 361 | 369 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Mario Carneiro
-/
import Mathlib.Computability.Halting
/-!
# Strong reducibility and degrees.
This file defines the notions of computable many-one reduction and one-one
reduc... | open Computable
theorem computable_of_manyOneReducible {p : α → Prop} {q : β → Prop} (h₁ : p ≤₀ q)
| Mathlib/Computability/Reduce.lean | 111 | 113 |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... | simp only [mapIdxM.go, mapIdxMAuxSpec_cons, map_eq_pure_bind, seq_eq_bind_map,
LawfulMonad.bind_assoc, pure_bind]
congr
conv => { lhs; intro x; rw [ih _ _ h]; }
funext x
| Mathlib/Data/List/Indexes.lean | 216 | 220 |
/-
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... |
/-- The right whiskering of an isomorphism is an isomorphism. -/
@[simps!]
| Mathlib/CategoryTheory/Monoidal/Category.lean | 346 | 348 |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... |
lemma infClosure_mono : Monotone (infClosure : Set α → Set α) := infClosure.monotone
@[simp] lemma infClosure_eq_self : infClosure s = s ↔ InfClosed s := infClosure.isClosed_iff.symm
alias ⟨_, InfClosed.infClosure_eq⟩ := infClosure_eq_self
lemma infClosure_idem (s : Set α) : infClosure (infClosure s) = infClosure s... | Mathlib/Order/SupClosed.lean | 370 | 379 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... |
@[simp]
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 265 | 266 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Cast.Field
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.CardComm... | else
n % 4 * n :: reciprocalFactors (n / 4 + 1)
| Mathlib/GroupTheory/CommutingProbability.lean | 152 | 154 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... |
theorem prod_univ {s : Set α} : s ×ˢ (univ : Set β) = Prod.fst ⁻¹' s := by simp [prod_eq]
| Mathlib/Data/Set/Prod.lean | 90 | 92 |
/-
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.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... |
@[deprecated (since := "2024-10-28")] alias inducing_subtype_val := IsInducing.subtypeVal
| Mathlib/Topology/Constructions.lean | 317 | 318 |
/-
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 | 1,264 | 1,266 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | exact ⟨a, hat, funext hab⟩
theorem piMap_image_univ_pi (f : ∀ i, α i → β i) (t : ∀ i, Set (α i)) :
Pi.map f '' univ.pi t = univ.pi fun i ↦ f i '' t i :=
piMap_image_pi (by simp) t
@[simp]
theorem range_piMap (f : ∀ i, α i → β i) : range (Pi.map f) = pi univ fun i ↦ range (f i) := by
simp only [← image_univ,... | Mathlib/Data/Set/Prod.lean | 832 | 841 |
/-
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 | 1,669 | 1,676 | |
/-
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_tmul (m : M) (p : P) (q : Q) :
rightComm R A M P Q ((m ⊗ₜ p) ⊗ₜ q) = (m ⊗ₜ q) ⊗ₜ p :=
rfl
| Mathlib/LinearAlgebra/TensorProduct/Tower.lean | 490 | 493 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Pointwise.Set.Lattice
import Mathlib.Algebra.Group.Subgroup.MulOppositeLemmas
import Mathlib.Algebra.Gr... | @[to_additive]
theorem set_mul_normal_comm (S : Set G) (N : Subgroup G) [hN : N.Normal] :
S * (N : Set G) = (N : Set G) * S := set_mul_normalizer_comm S N subset_normalizer_of_normal
/-- The carrier of `H ⊔ N` is just `↑H * ↑N` (pointwise set product)
when `H` is a subgroup of the normalizer of `N` in `G`. -/
@[to... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 238 | 249 |
/-
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.CategoryTheory.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | variable [L.HasTotal c₁₂]
@[reassoc (attr := simp)]
lemma ιTotal_map (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂) (h : ComplexShape.π c₁ c₂ c₁₂ (i₁, i₂) = i₁₂) :
K.ιTotal c₁₂ i₁ i₂ i₁₂ h ≫ (total.map φ c₁₂).f i₁₂ =
| Mathlib/Algebra/Homology/TotalComplex.lean | 428 | 432 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 277 | 277 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | rintro _ ⟨J₁, h₁, rfl⟩ _ ⟨J₂, h₂, rfl⟩ Hne
have : J₁ ≠ J₂ := by
rintro rfl
exact Hne rfl
exact ((Box.disjoint_coe.2 <| π.disjoint_coe_of_mem h₁ h₂ this).inf_left' _).inf_right' _)
@[simp]
theorem mem_restrict : J₁ ∈ π.restrict J ↔ ∃ J' ∈ π, (J₁ : WithBot (Box ι)) = ↑J ⊓ ↑J' := by
| Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 426 | 433 |
/-
Copyright (c) 2023 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# D... |
lemma doubleFactorial_le_factorial : ∀ n, n‼ ≤ n !
| 0 => le_rfl
| n + 1 => by
rw [factorial_eq_mul_doubleFactorial]; exact Nat.le_mul_of_pos_right _ n.doubleFactorial_pos
| Mathlib/Data/Nat/Factorial/DoubleFactorial.lean | 51 | 55 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preser... | /-- Define the equalizing cone -/
abbrev equalizerCone (F : WalkingParallelPair ⥤ C) : Cone F :=
Cone.ofFork
(Fork.ofι (pullbackFst F)
| Mathlib/CategoryTheory/Limits/Constructions/Equalizers.lean | 51 | 54 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... | | .nterm e .. => e
instance : Coe NormalExpr Expr where coe := NormalExpr.e
| Mathlib/Tactic/Abel.lean | 149 | 151 |
/-
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... | · rw [s.eq_empty_of_isEmpty]; simp
rw [← (f.invFunOn_injOn_image s).encard_image]
apply encard_le_encard
exact f.invFunOn_image_image_subset s
theorem Finite.injOn_of_encard_image_eq (hs : s.Finite) (h : (f '' s).encard = s.encard) :
InjOn f s := by
obtain (h' | hne) := isEmpty_or_nonempty α
· rw [s.eq... | Mathlib/Data/Set/Card.lean | 418 | 443 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
/-!
# Multiplicative... | @[simp]
theorem one_apply_coe (a : Rˣ) : (1 : MulChar R R') a = 1 := by classical exact dif_pos a.isUnit
| Mathlib/NumberTheory/MulChar/Basic.lean | 245 | 246 |
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,833 | 1,836 | |
/-
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.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... |
lemma threeAPFree_sphere {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[StrictConvexSpace ℝ E] (x : E) (r : ℝ) : ThreeAPFree (sphere x r) := by
obtain rfl | hr := eq_or_ne r 0
· rw [sphere_zero]
exact threeAPFree_singleton _
· convert threeAPFree_frontier isClosed_closedBall (strictConvex_closedBa... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 68 | 74 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... |
/-- If `t : Cone F.op` is a limit cone, then `t.unop : Cocone F` is a colimit cocone.
-/
def IsLimit.unop {t : Cone F.op} (P : IsLimit t) : IsColimit t.unop where
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 1,130 | 1,133 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.Normal.Defs
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
/-!
# Algebraica... | rcases exists_pow_nat_eq x zero_lt_two with ⟨z, rfl⟩
exact ⟨z, sq z⟩
| Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 99 | 101 |
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