Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 75 | 75 | |
/-
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.Logic.OpClass
import Mathlib.Order.Lattice
/-!
# `max` and `min`
This file proves basic properties about maxima and minima on a `LinearOrder`.
## Ta... | max (f a) (f b) := hf.dual.map_max ha hb
| Mathlib/Order/MinMax.lean | 196 | 197 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Derivation.Killing
import Mathlib.Algebra.Lie.Killing
import Mathlib.Algebra.Lie.Sl2
import Mathlib.Algebra.Lie.Weights.Chain
import Mathlib.Line... | have : ⁅⁅e, f'⁆, e⁆ = α h • e := lie_eq_smul_of_mem_rootSpace heα h
rw [lie_smul, smul_lie, this, ← smul_assoc, smul_eq_mul, mul_assoc, inv_mul_cancel₀ hh,
mul_one, two_smul, two_smul]
refine ⟨⁅e, f⁆, e, f, ⟨fun contra ↦ ?_, rfl, hef, ?_⟩, heα, Submodule.smul_mem _ _ hfα⟩
· rw [contra] at hef
have... | Mathlib/Algebra/Lie/Weights/Killing.lean | 498 | 520 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Module.Submodule.Map
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.RingTheory.Ideal.Quotient.Defs
/-!
# modular equivalence for submodule
-... | lemma pow {I : Ideal A} {x y : A} (n : ℕ) (hxy : x ≡ y [SMOD I]) :
x ^ n ≡ y ^ n [SMOD I] := by
simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢
rw [hxy]
| Mathlib/LinearAlgebra/SModEq.lean | 95 | 98 |
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace.Complete
/-!
# Convergence in measure
We define con... | {g : α → E} :
TendstoInMeasure μ f l g ↔
∀ ε, 0 < ε → Tendsto (fun i => μ { x | ε ≤ ‖f i x - g x‖ }) l (𝓝 0) := by
simp_rw [TendstoInMeasure, dist_eq_norm]
| Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 58 | 62 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | theorem edgeFinset_card : #G.edgeFinset = Fintype.card G.edgeSet :=
Set.toFinset_card _
| Mathlib/Combinatorics/SimpleGraph/Finite.lean | 104 | 105 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Huỳnh Trần Khanh, Stuart Presnell
-/
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
import Math... |
/-- The `diag` of `s : Finset α` is sent on a finset of `Sym2 α` of card `#s`. -/
theorem card_image_diag (s : Finset α) : #(s.diag.image Sym2.mk) = #s := by
| Mathlib/Data/Sym/Card.lean | 120 | 122 |
/-
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, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... |
namespace LinearMap
variable [Semiring R] [AddCommMonoid α] [AddCommMonoid β] [AddCommMonoid γ]
variable [Module R α] [Module R β] [Module R γ]
| Mathlib/Data/Matrix/Basic.lean | 428 | 432 |
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | (s.image f).filter p = (s.filter fun a ↦ p (f a)).image f :=
ext fun b => by
simp only [mem_filter, mem_image, exists_prop]
exact
⟨by rintro ⟨⟨x, h1, rfl⟩, h2⟩; exact ⟨x, ⟨h1, h2⟩, rfl⟩,
| Mathlib/Data/Finset/Image.lean | 398 | 402 |
/-
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.IntegralCompProd
import Mathlib.Probability.Kernel.Disintegration.StandardBorel
/-!
# Lebesgue and Bochner integrals of con... |
lemma setIntegral_condKernel {s : Set β} (hs : MeasurableSet s)
{t : Set Ω} (ht : MeasurableSet t) (hf : IntegrableOn f (s ×ˢ t) ρ) :
∫ b in s, ∫ ω in t, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x in s ×ˢ t, f x ∂ρ := by
conv_rhs => rw [← ρ.disintegrate ρ.condKernel]
| Mathlib/Probability/Kernel/Disintegration/Integral.lean | 199 | 203 |
/-
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... | @[to_additive]
lemma pow_injOn_Iio_orderOf : (Set.Iio <| orderOf x).InjOn (x ^ ·) := by
| Mathlib/GroupTheory/OrderOfElement.lean | 251 | 252 |
/-
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.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Batteries.Tactic.Congr
/-... |
/-- Alternative constructor for `isLimit`,
providing a morphism of cones rather than a morphism between the cone points
and separately the factorisation condition.
| Mathlib/CategoryTheory/Limits/IsLimit.lean | 101 | 104 |
/-
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... |
theorem gcd_eq_sum : ∃ a b : R, a * x + b * y = gcd x y :=
Ideal.mem_span_pair.mp (by rw [← span_gcd]; apply Ideal.subset_span; simp)
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 194 | 197 |
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... | Mathlib/Data/ENNReal/Basic.lean | 770 | 772 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,243 | 2,245 | |
/-
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... | (o.inner_eq_zero_of_oangle_eq_pi_div_two h) (o.left_ne_zero_of_oangle_eq_pi_div_two h)]
/-- An angle in a right-angled triangle expressed using `arctan`. -/
theorem oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arctan (‖x‖ / ‖y‖) := by
... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 73 | 79 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
/-!
# Co... | ```text
TruncatedWittVector p n (ZMod p) ----> TruncatedWittVector p m (ZMod p)
| |
| |
v v
ZMod (p^n) ----------------------------> ZMod (p^m)
| Mathlib/RingTheory/WittVector/Compare.lean | 107 | 112 |
/-
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... | theorem distortion_disjUnion (h : Disjoint π₁.iUnion π₂.iUnion) :
(π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion := by
classical exact sup_union
| Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 611 | 613 |
/-
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
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | theorem coeff_zero_X (s : σ) : coeff R (0 : σ →₀ ℕ) (X s : MvPowerSeries σ R) = 0 := by
classical
rw [coeff_X, if_neg]
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 381 | 383 |
/-
Copyright (c) 2022 Julian Berman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Berman
-/
import Mathlib.GroupTheory.PGroup
import Mathlib.LinearAlgebra.Quotient.Defs
/-!
# Torsion groups
This file defines torsion groups, i.e. groups where all elements hav... | def torsion : Submonoid G where
carrier := { x | IsOfFinOrder x }
one_mem' := IsOfFinOrder.one
mul_mem' hx hy := hx.mul hy
| Mathlib/GroupTheory/Torsion.lean | 163 | 167 |
/-
Copyright (c) 2019 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.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | @[simp]
theorem KernelFork.ι_ofι {X Y P : C} (f : X ⟶ Y) (ι : P ⟶ X) (w : ι ≫ f = 0) :
| Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 91 | 92 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... |
theorem disjoint_closedBall_of_lt_infEdist {r : ℝ≥0∞} (h : r < infEdist x s) :
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 180 | 181 |
/-
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.MeasureTheory.Measure.GiryMonad
import Mathlib.MeasureTheory.Measure.Stieltjes
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Fu... | rw [stieltjesFunctionAux_def]
/-- Extend a function `f : α → ℚ → ℝ` with property `IsMeasurableRatCDF` from `ℚ` to `ℝ`,
to a function `α → StieltjesFunction`. -/
noncomputable def IsMeasurableRatCDF.stieltjesFunction (a : α) : StieltjesFunction where
toFun := stieltjesFunctionAux f a
mono' := monotone_stieltjesF... | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | 339 | 346 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,349 | 1,350 | |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... |
theorem next_cons_concat (y : α) (hy : x ≠ y) (hx : x ∉ l)
| Mathlib/Data/List/Cycle.lean | 154 | 155 |
/-
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.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
/-!
# N... | end LinearMap
namespace Module.End
| Mathlib/RingTheory/Nilpotent/Lemmas.lean | 94 | 97 |
/-
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... | let ⟨C, hC⟩ := Finite.exists_le (‖f ·‖₊); .of_bound .of_discrete C <| .of_forall hC
@[deprecated (since := "2025-02-21")]
alias Memℒp.of_discrete := MemLp.of_discrete
@[simp] lemma eLpNorm_of_isEmpty [IsEmpty α] (f : α → ε) (p : ℝ≥0∞) : eLpNorm f p μ = 0 := by
simp [Subsingleton.elim f 0]
section MapMeasure
var... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 1,045 | 1,059 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Logic.Small.Basic
import Mathlib.RingTheory.Ideal.Defs
/-!
# Injecti... | instance : Min (ExtensionOf i f) where
min X1 X2 :=
{ X1.toLinearPMap ⊓ X2.toLinearPMap with
le := fun x hx =>
(by
rcases hx with ⟨x, rfl⟩
refine ⟨X1.le (Set.mem_range_self _), X2.le (Set.mem_range_self _), ?_⟩
rw [← X1.is_extension x, ← X2.is_extension x] :
| Mathlib/Algebra/Module/Injective.lean | 112 | 119 |
/-
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... | Mathlib/Logic/Equiv/PartialEquiv.lean | 1,040 | 1,042 | |
/-
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.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.Reflective
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.Categ... | @[simp]
theorem sheafificationAdjunction_counit_app_val (P : Sheaf J D) :
((sheafificationAdjunction J D).counit.app P).val = sheafifyLift J (𝟙 P.val) P.cond := by
unfold sheafifyLift
rw [Adjunction.homEquiv_counit]
simp
@[reassoc (attr := simp)]
theorem toSheafify_sheafifyLift {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (... | Mathlib/CategoryTheory/Sites/Sheafification.lean | 163 | 171 |
/-
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... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,473 | 1,477 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Sites.Localization
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.Whiskering
import Mathlib.Cate... | (sheafToPresheaf J B).map ((sheafComposeNatTrans J F adj₁ adj₂).app P) =
whiskerRight (adj₁.unit.app P) F := by
simp [sheafComposeNatTrans, -sheafToPresheaf_obj, -sheafToPresheaf_map,
Adjunction.homEquiv_counit]
lemma sheafComposeNatTrans_app_uniq (P : Cᵒᵖ ⥤ A)
(α : G₂.obj (P ⋙ F) ⟶ (sheafComp... | Mathlib/CategoryTheory/Sites/PreservesSheafification.lean | 170 | 176 |
/-
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.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,347 | 1,352 | |
/-
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, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... |
/-- The dimension of a quotient is bounded by the dimension of the ambient space. -/
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 104 | 105 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.Complex.Norm
/-!
# The partial order on the complex numbers
This order is defined by `z ≤ w ↔ z.re ≤ w.re ∧ z.im = w.im`.
This is a natural order o... | fun ⟨h₁, h₂⟩ ↦ by rw [← abs_re_eq_norm.2 h₂.symm, abs_of_nonneg h₁]⟩
| Mathlib/Data/Complex/Order.lean | 103 | 104 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | rw [← norm_eq_zero]
refine le_antisymm ?_ (by positivity)
exact norm_inner_le_norm _ _ |>.trans <| by simp [h]
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 440 | 442 |
/-
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... | borelize α
rintro _ ⟨a, -, b, -, -, rfl⟩
exact measurableSet_Ico
theorem Dense.borel_eq_generateFrom_Ico_mem_aux {α : Type*} [TopologicalSpace α] [LinearOrder α]
[OrderTopology α] [SecondCountableTopology α] {s : Set α} (hd : Dense s)
(hbot : ∀ x, IsBot x → x ∈ s) (hIoo : ∀ x y : α, x < y → Ioo x y = ∅ →... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 258 | 265 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | exact (extChartAt I' x.2).right_inv hy.2
apply HasFDerivWithinAt.congr_of_eventuallyEq hasFDerivWithinAt_snd this
-- Porting note: the next line was `simp only [mfld_simps]`
exact (extChartAt I' x.2).right_inv <| (extChartAt I' x.2).map_source (mem_extChartAt_source _)
theorem hasMFDerivWithinAt_snd (s : Set... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 291 | 297 |
/-
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.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | exact fun l hl =>
⟨fun hsl => (hs l hl hsl).imp fun x hx => ⟨Or.inl hx.1, hx.2⟩, fun htl =>
(ht l hl htl).imp fun x hx => ⟨Or.inr hx.1, hx.2⟩⟩
theorem isComplete_iUnion_separated {ι : Sort*} {s : ι → Set α} (hs : ∀ i, IsComplete (s i))
{U : Set (α × α)} (hU : U ∈ 𝓤 α) (hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y... | Mathlib/Topology/UniformSpace/Cauchy.lean | 335 | 343 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.V... |
theorem integral_abs_condExp_le (f : α → ℝ) : ∫ x, |(μ[f|m]) x| ∂μ ≤ ∫ x, |f x| ∂μ := by
by_cases hm : m ≤ m0
swap
· simp_rw [condExp_of_not_le hm, Pi.zero_apply, abs_zero, integral_zero]
positivity
by_cases hfint : Integrable f μ
swap
· simp only [condExp_of_not_integrable hfint, Pi.zero_apply, abs_ze... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 92 | 113 |
/-
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.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
/-!
# Quadratic characters of finite fields
Further facts relying ... | have hchar : ringChar F = ringChar (ZMod p) := by rw [hFp]; exact (ringChar_zmod_n p).symm
conv => enter [1, 1, 2]; rw [hc, Nat.cast_pow, map_pow, hchar, map_ringChar]
simp only [zero_pow n.ne_zero, mul_zero, zero_eq_neg, one_ne_zero, not_false_iff]
· rw [← Iff.not_left (@quadraticChar_neg_one_iff_not_isS... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 107 | 120 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... | intro f g (h : F.map f a = F.map g a)
haveI : IsIso (equalizer.ι f g) := by
apply IsConnected.noTrivialComponent _ (equalizer.ι f g)
exact not_initial_of_inhabited F ((fiberEqualizerEquiv F f g).symm ⟨a, h⟩)
exact eq_of_epi_equalizer
/-- The evaluation map on automorphisms is injective for connected obje... | Mathlib/CategoryTheory/Galois/Basic.lean | 303 | 311 |
/-
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` ... | theorem Continuous.comp_continuousOn' {g : β → γ} {f : α → β} {s : Set α} (hg : Continuous g)
(hf : ContinuousOn f s) : ContinuousOn (fun x ↦ g (f x)) s :=
hg.comp_continuousOn hf
| Mathlib/Topology/ContinuousOn.lean | 990 | 993 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | else by
haveI : NeZero (m * n) := ⟨hmn0⟩
haveI : NeZero m := ⟨left_ne_zero_of_mul hmn0⟩
haveI : NeZero n := ⟨right_ne_zero_of_mul hmn0⟩
simp only [← ZMod.card_units_eq_totient]
rw [Fintype.card_congr (Units.mapEquiv (ZMod.chineseRemainder h).toMulEquiv).toEquiv,
Fintype.card_congr (@MulEquiv... | Mathlib/Data/Nat/Totient.lean | 125 | 131 |
/-
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.FundThmCalculus
deprecated_module (since := "2025-04-06")
| Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,242 | 1,247 | |
/-
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.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 793 | 800 | |
/-
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 ... |
@[simp]
theorem length_tails (l : List α) : length (tails l) = length l + 1 := by
induction' l with x l IH
· simp
· simpa using IH
@[simp]
theorem length_inits (l : List α) : length (inits l) = length l + 1 := by simp [inits_eq_tails]
@[simp]
theorem getElem_tails (l : List α) (n : Nat) (h : n < (tails l).leng... | Mathlib/Data/List/Infix.lean | 225 | 237 |
/-
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 [hx]
· exact arg_real_mul (conj x) (by simp [hx])
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 313 | 315 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
-/
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# The Lucas-L... | inv := ωb
val_inv := ω_mul_ωb _
inv_val := ωb_mul_ω _
| Mathlib/NumberTheory/LucasLehmer.lean | 421 | 423 |
/-
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... | · rintro ⟨x, hx, rfl⟩
exact ⟨1, x, one_ne_zero, hx, by rw [Ideal.span_singleton_one, Ideal.top_mul, Ideal.mul_top]⟩
variable (K)
/-- Induction principle for the class group: to show something holds for all `x : ClassGroup R`,
we can choose a fraction field `K` and show it holds for the equivalence class of each... | Mathlib/RingTheory/ClassGroup.lean | 147 | 161 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | assuming the measure is volume. -/
@[simp]
theorem intervalIntegrable_log (h : (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable log μ a b :=
| Mathlib/Analysis/SpecialFunctions/Integrals.lean | 229 | 231 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | eq_comm]
/-- A transcendental real number is irrational. -/
| Mathlib/Data/Real/Irrational.lean | 36 | 38 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Ma... | theorem primeFactorsList_sublist_right {n k : ℕ} (h : k ≠ 0) :
n.primeFactorsList <+ (n * k).primeFactorsList := by
rcases n with - | hn
· simp [zero_mul]
apply sublist_of_subperm_of_sorted _ (primeFactorsList_sorted _) (primeFactorsList_sorted _)
simp only [(perm_primeFactorsList_mul (Nat.succ_ne_zero _) h... | Mathlib/Data/Nat/Factors.lean | 205 | 211 |
/-
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, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | HasFDerivAt (h₂ ∘ f) (h₂' • f') x := by
rw [hy] at hh; exact hh.comp_hasFDerivAt x hf
theorem HasDerivAt.comp_hasFDerivWithinAt {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s} (x)
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 185 | 188 |
/-
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.Enum
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
/-!
### Topology of ordinals
We prov... | rw [← mem_closure_iff_iSup, hs.closure_eq]
theorem mem_closure_iff_bsup :
a ∈ closure s ↔
∃ (o : Ordinal) (_ho : o ≠ 0) (f : ∀ a < o, Ordinal),
| Mathlib/SetTheory/Ordinal/Topology.lean | 138 | 142 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | · rw [Subtype.ext_iff_val, Subtype.ext_iff_val, hg, hg', v.1.2]
simp only [v₀, List.Vector.replicate]
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 509 | 511 |
/-
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, Patrick Massot
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Measure.Real
import Mathlib.Order.Filter.Indi... | theorem continuousAt_parametric_primitive_of_dominated [FirstCountableTopology X]
{F : X → ℝ → E} (bound : ℝ → ℝ) (a b : ℝ)
{a₀ b₀ : ℝ} {x₀ : X} (hF_meas : ∀ x, AEStronglyMeasurable (F x) (μ.restrict <| Ι a b))
(h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t ∂μ.restrict <| Ι a b, ‖F x t‖ ≤ bound t)
(bound_integrable : I... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 387 | 457 |
/-
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 | 2,766 | 2,768 | |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
/-!
# Local extrema of differentiable functions
## Main definitions
In a real normed space `E` we define `posTangentCo... | else by rw [fderivWithin_zero_of_not_differentiableWithinAt hf]; rfl
/-- If `f` has a local min on `s` at `a`, `f'` is the derivative of `f` at `a` within `s`, and
| Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | 155 | 157 |
/-
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.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
/-!
# Lie algeb... | lemma mapsTo_pow_toEnd_sub_algebraMap {φ : R} {k : ℕ} {x : L} :
MapsTo ((toEnd R L M x - algebraMap R (Module.End R M) φ) ^ k) N N := by
rw [Module.End.coe_pow]
| Mathlib/Algebra/Lie/OfAssociative.lean | 341 | 343 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Aaron Anderson
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Algebra.Order.Pi
import Mathlib.Data.Fi... | Ne, coe_tsub, Pi.sub_apply, not_imp_not, zero_le, imp_true_iff]
theorem subset_support_tsub [DecidableEq ι] {f1 f2 : ι →₀ α} :
f1.support \ f2.support ⊆ (f1 - f2).support := by
simp +contextual [subset_iff]
| Mathlib/Data/Finsupp/Order.lean | 277 | 281 |
/-
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` ... | (⨅ i ∈ I, comap (fun x => x i) (𝓝[s i] x i)) ⊓
⨅ (i) (_ : i ∉ I), comap (fun x => x i) (𝓝 (x i)) := by
simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, ← iInf_principal_finite hI, comap_inf,
comap_principal, eval]
rw [iInf_split _ fun i => i ∈ I, inf_right_comm]
simp only [iInf_inf_eq]
theo... | Mathlib/Topology/ContinuousOn.lean | 316 | 323 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Yoneda
/-!
# Ind- and ... | (coyonedaOpColimitIsoLimitCoyoneda' F).app A ≪≫ limitObjIsoLimitCompEvaluation _ _
@[reassoc (attr := simp)]
lemma colimitHomIsoLimitYoneda'_hom_comp_π [HasLimitsOfShape I (Type u₂)] (A : C) (i : I) :
(colimitHomIsoLimitYoneda' F A).hom ≫ limit.π (F.rightOp ⋙ yoneda.obj A) i =
(coyoneda.map (colimit.ι F ⟨i... | Mathlib/CategoryTheory/Limits/IndYoneda.lean | 113 | 120 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | simpa using heq
· change i ≠ k at hh
change j ≠ k at hl
exact lift_word_prod_nontrivial_of_other_i f X hXnonempty hXdisj hpp w hh.symm hl.symm
include hcard in
theorem empty_of_word_prod_eq_one {w : Word H} (h : lift f w.prod = 1) :
w = Word.empty := by
by_contra hnotempty
| Mathlib/GroupTheory/CoprodI.lean | 890 | 898 |
/-
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.Order.Filter.Lift
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.Separation.Basic
/-!
# Topology on the set of filters on a type... | refine ⟨fun h => ?_, fun h => monotone_nhds h⟩
rw [← Iic_subset_Iic, ← sInter_nhds, ← sInter_nhds]
exact sInter_subset_sInter h
| Mathlib/Topology/Filter.lean | 139 | 141 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import M... | discr ℚ (hζ.subOnePowerBasis ℚ).basis =
(-1) ^ ((p ^ k : ℕ).totient / 2) * p ^ ((p : ℕ) ^ (k - 1) * ((p - 1) * k - 1)) := by
rw [← discr_prime_pow hζ (cyclotomic.irreducible_rat (p ^ k).pos)]
exact hζ.discr_zeta_eq_discr_zeta_sub_one.symm
| Mathlib/NumberTheory/Cyclotomic/Rat.lean | 55 | 59 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | noncomputable def BinaryFan.isLimitCompRightIso {X Y Y' : C} (c : BinaryFan X Y) (f : Y ⟶ Y')
[IsIso f] (h : IsLimit c) : IsLimit (BinaryFan.mk c.fst (c.snd ≫ f)) :=
BinaryFan.isLimitFlip <| BinaryFan.isLimitCompLeftIso _ f (BinaryFan.isLimitFlip h)
/-- Binary coproducts are symmetric. -/
def BinaryCofan.isColim... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 423 | 428 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | rw [Ico_eq_locus_Ioc_eq_iUnion_Ioo]
exact isOpen_iUnion fun i => isOpen_Ioo) <|
(not_modEq_iff_toIcoMod_eq_toIocMod hp).1 <| not_modEq_iff_ne_mod_zmultiples.2 hx
theorem continuousAt_toIcoMod (hx : (x : 𝕜 ⧸ zmultiples p) ≠ a) : ContinuousAt (toIcoMod hp a) x :=
let h := toIcoMod_eventuallyEq_toI... | Mathlib/Topology/Instances/AddCircle.lean | 94 | 99 |
/-
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.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | Mathlib/Data/Finsupp/Basic.lean | 1,548 | 1,550 | |
/-
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, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... |
theorem HasDerivAt.scomp_hasDerivWithinAt_of_eq (hg : HasDerivAt g₁ g₁' y)
(hh : HasDerivWithinAt h h' s x) (hy : y = h x) :
HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x := by
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 123 | 126 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Thickened indicators
This fi... | theorem thickenedIndicator_one {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α} (x_in_E : x ∈ E) :
thickenedIndicator δ_pos E x = 1 :=
thickenedIndicator_one_of_mem_closure _ _ (subset_closure x_in_E)
theorem thickenedIndicator_zero {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α}
| Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 187 | 191 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Additive.Corner.Defs
import Mathlib.Combinatorics.SimpleGraph... | (graph <| triangleIndices A).FarFromTriangleFree (ε / 9) := by
refine farFromTriangleFree _ ?_
simp_rw [card_triangleIndices, mul_comm_div, Nat.cast_pow, Nat.cast_add]
ring_nf
simpa only [mul_comm] using hε
| Mathlib/Combinatorics/Additive/Corner/Roth.lean | 58 | 63 |
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,327 | 2,328 | |
/-
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.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... |
variable {k}
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 704 | 706 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
import Mathlib.Order.Closure
/-!
# Closed sieves
A natural closure operator on sieves is a closure operator on `Sieve X... | · intro hf
rw [J₁.covers_iff]
apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
rw [← MSNS]
apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)
· intro hf
rw [J₁.covers_iff]
apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
rw [... | Mathlib/CategoryTheory/Sites/Closed.lean | 177 | 225 |
/-
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 | 1,402 | 1,409 | |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Util... | Mathlib/Data/List/GetD.lean | 155 | 155 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | ← lcm_dvd_iff]
| Mathlib/Data/Nat/GCD/Basic.lean | 80 | 81 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | rw [neg_rayOfNeZero, Ne, ray_eq_iff]
exact mt eq_zero_of_sameRay_self_neg hx
theorem neg_units_smul (u : Rˣ) (v : Module.Ray R M) : -u • v = -(u • v) := by
| Mathlib/LinearAlgebra/Ray.lean | 415 | 418 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Schem... | PresheafedSpace.restrictStalkIso_hom_eq_germ _ U.isOpenEmbedding _ _ _
@[reassoc]
lemma germ_stalkIso_inv {X : Scheme.{u}} (U : X.Opens) (V : U.toScheme.Opens) (x : U)
(hx : x ∈ V) : X.presheaf.germ (U.ι ''ᵁ V) x ⟨x, hx, rfl⟩ ≫
(U.stalkIso x).inv = U.toScheme.presheaf.germ V x hx :=
| Mathlib/AlgebraicGeometry/Restrict.lean | 144 | 149 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDegree... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 487 | 487 | |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... | rw [fourierPowSMulRight_iteratedFDeriv_fourierIntegral L hf h'f hk hn]
apply (norm_fourierIntegral_le_integral_norm _ _ _ _ _).trans
have I p (hp : p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1)) :
Integrable (fun v ↦ ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖) μ := by
simp only [Finset.mem_product, Finse... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 610 | 643 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | Mathlib/Probability/Martingale/Basic.lean | 519 | 524 | |
/-
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.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | a ∈ Set.range (fun (i : Fin (n + 1)) => s.points i) → dist a p = r := by
obtain ⟨r, hr⟩ := hr
use r
refine Set.forall_mem_range.mpr ?_
exact hr
rw [exists_dist_eq_iff_exists_dist_orthogonalProjection_eq (subset_affineSpan ℝ _) p] at hr
obtain ⟨r, hr⟩ := hr
exact
s.eq_circumcenter_of_dist... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 371 | 387 |
/-
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.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | = (π / b) ^ (Fintype.card ι / 2 : ℂ) * cexp ((∑ i, c i ^ 2) / (4 * b)) := by
simp_rw [neg_mul, Finset.mul_sum, integral_cexp_neg_sum_mul_add (fun _ ↦ hb) c, one_div,
Finset.prod_mul_distrib, Finset.prod_const, ← cpow_nat_mul, ← Complex.exp_sum, Fintype.card,
Finset.sum_div, div_eq_mul_inv]
theorem inte... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 296 | 304 |
/-
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.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
/-!
# Diophantine functions and Matiyas... |
/-- Diophantine functions are closed under integer division. -/
theorem div_dioph : DiophFn fun v => f v / g v :=
have :
Dioph fun v : Vector3 ℕ 3 =>
v &2 = 0 ∧ v &0 = 0 ∨ v &0 * v &2 ≤ v &1 ∧ v &1 < (v &0 + 1) * v &2 :=
(D&2 D= D.0 D∧ D&0 D= D.0) D∨ D&0 D* D&2 D≤ D&1 D∧ D&1 D< (D&0 D+ D.1) D* D&2
di... | Mathlib/NumberTheory/Dioph.lean | 632 | 645 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 339 | 343 | |
/-
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.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... |
/-- The multiplicity of `a` as root of `(X - a) ^ n` is `n`. -/
theorem rootMultiplicity_X_sub_C_pow [Nontrivial R] (a : R) (n : ℕ) :
rootMultiplicity a ((X - C a) ^ n) = n := by
have := rootMultiplicity_mul_X_sub_C_pow (a := a) (n := n) C.map_one_ne_zero
rwa [rootMultiplicity_C, map_one, one_mul, zero_add] at... | Mathlib/Algebra/Polynomial/RingDivision.lean | 148 | 153 |
/-
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.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | theorem Fintype.card_subtype_eq' (y : α) [Fintype { x // y = x }] :
Fintype.card { x // y = x } = 1 :=
| Mathlib/Data/Fintype/Card.lean | 151 | 152 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Data.Finsupp.Defs
/-!
# Locus of unequal values of finitely supported functions
Let `α N` be two Types, assume that `N` has a `0` and let `f g : α →₀ N... | Mathlib/Data/Finsupp/NeLocus.lean | 159 | 160 | |
/-
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 | 731 | 736 | |
/-
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.Module.Equiv.Defs
import Mathlib.Data.DFinsupp.Module
import Mathlib.Data.Finsupp.SMul
/-!
# Conversion between `Finsupp` and homogeneous `DFinsupp`... | Mathlib/Data/Finsupp/ToDFinsupp.lean | 358 | 362 | |
/-
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 ... | (CliffordAlgebra.lift (0 : QuadraticForm R Unit) <|
⟨0, fun _ : Unit => (zero_mul (0 : R)).trans (algebraMap R _).map_zero.symm⟩)
| Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 106 | 107 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | @[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
| Mathlib/Data/Set/Lattice.lean | 1,199 | 1,201 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Ralf Stephan, Neil Strickland, Ruben Van de Velde
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.... |
@[simp]
| Mathlib/Data/PNat/Basic.lean | 33 | 34 |
/-
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.ExactSequence
import Mathlib.Algebra.Homology.ShortComplex.Limits
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The snake lemma
The ... | simp only [assoc, pullback.lift_fst_assoc, ← comp_τ₂, f.comm₁₂]
@[reassoc]
| Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean | 486 | 488 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.LSeries.Convolution
import Mathlib.NumberTheory.LSeries.Deriv
import Mathlib.NumberThe... | lemma LSeries_vonMangoldt_eq_deriv_riemannZeta_div {s : ℂ} (hs : 1 < s.re) :
L ↗Λ s = - deriv riemannZeta s / riemannZeta s := by
suffices deriv (L 1) s = deriv riemannZeta s by
rw [LSeries_vonMangoldt_eq hs, ← LSeries_one_eq_riemannZeta hs, this]
refine Filter.EventuallyEq.deriv_eq <| Filter.eventuallyEq_i... | Mathlib/NumberTheory/LSeries/Dirichlet.lean | 395 | 403 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison, Adam Topaz
-/
import Mathlib.Topology.Sheaves.SheafOfFunctions
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Topology.Sheaves.SheafCondition.UniqueG... | -- We promise to provide all the ingredients of the proof later:
let Q :
∃ (W : (OpenNhds x)ᵒᵖ) (s : ∀ w : (unop W).1, T w) (hW : P.pred s),
tU = (subsheafToTypes P).presheaf.germ _ x (unop W).2 ⟨s, hW⟩ ∧
tV = (subsheafToTypes P).presheaf.germ _ x (unop W).2 ⟨s, hW⟩ :=
?_
· choose W s hW e u... | Mathlib/Topology/Sheaves/LocalPredicate.lean | 259 | 265 |
/-
Copyright (c) 2021 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.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | have hx_nnnorm : ‖f x‖₊ ≤ n := (hs n).2.2 x hx
rw [← coe_nnnorm]
norm_cast
end StronglyMeasurable
/-! ## Finitely strongly measurable functions -/
theorem finStronglyMeasurable_zero {α β} {m : MeasurableSpace α} {μ : Measure α} [Zero β]
| Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 990 | 999 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
cyclotomic' n K = (X ^ n - 1) /ₘ ∏ i ∈ Nat.properDivisors n, cyclotomic' i K := by
rw [← prod_cyclotomic'_eq_X_pow_sub_one hpos h, ← Nat.cons_self_properDivisors hpos.ne',
| Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 159 | 161 |
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