Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
#align_import analysis.convex.cone.dual from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
open scoped Classical
open Pointwise
variable {π E F G : Type*}
section Dua... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 144 | 161 | theorem ConvexCone.pointed_of_nonempty_of_isClosed (K : ConvexCone β H) (ne : (K : Set H).Nonempty)
(hc : IsClosed (K : Set H)) : K.Pointed := by |
obtain β¨x, hxβ© := ne
let f : β β H := (Β· β’ x)
-- f (0, β) is a subset of K
have fI : f '' Set.Ioi 0 β (K : Set H) := by
rintro _ β¨_, h, rflβ©
exact K.smul_mem (Set.mem_Ioi.1 h) hx
-- closure of f (0, β) is a subset of K
have clf : closure (f '' Set.Ioi 0) β (K : Set H) := hc.closure_subset_iff.2 fI
... | 2,234 |
import Mathlib.Analysis.Convex.Cone.InnerDual
import Mathlib.Algebra.Order.Nonneg.Module
import Mathlib.Algebra.Module.Submodule.Basic
variable {π E F G : Type*}
local notation3 "πβ₯0" => {c : π // 0 β€ c}
abbrev PointedCone (π E) [OrderedSemiring π] [AddCommMonoid E] [Module π E] :=
Submodule {c : π // 0... | Mathlib/Analysis/Convex/Cone/Pointed.lean | 51 | 52 | theorem toConvexCone_pointed (S : PointedCone π E) : (S : ConvexCone π E).Pointed := by |
simp [toConvexCone, ConvexCone.Pointed]
| 2,235 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 114 | 116 | theorem EuclideanSpace.norm_eq {π : Type*} [RCLike π] {n : Type*} [Fintype n]
(x : EuclideanSpace π n) : βxβ = β(β i, βx iβ ^ 2) := by |
simpa only [Real.coe_sqrt, NNReal.coe_sum] using congr_arg ((β) : ββ₯0 β β) x.nnnorm_eq
| 2,236 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 134 | 138 | theorem EuclideanSpace.ball_zero_eq {n : Type*} [Fintype n] (r : β) (hr : 0 β€ r) :
Metric.ball (0 : EuclideanSpace β n) r = {x | β i, x i ^ 2 < r ^ 2} := by |
ext x
have : (0 : β) β€ β i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _
simp_rw [mem_setOf, mem_ball_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_lt this hr]
| 2,236 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 140 | 143 | theorem EuclideanSpace.closedBall_zero_eq {n : Type*} [Fintype n] (r : β) (hr : 0 β€ r) :
Metric.closedBall (0 : EuclideanSpace β n) r = {x | β i, x i ^ 2 β€ r ^ 2} := by |
ext
simp_rw [mem_setOf, mem_closedBall_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_le_left hr]
| 2,236 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 145 | 150 | theorem EuclideanSpace.sphere_zero_eq {n : Type*} [Fintype n] (r : β) (hr : 0 β€ r) :
Metric.sphere (0 : EuclideanSpace β n) r = {x | β i, x i ^ 2 = r ^ 2} := by |
ext x
have : (0 : β) β€ β i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _
simp_rw [mem_setOf, mem_sphere_zero_iff_norm, norm_eq, norm_eq_abs, sq_abs,
Real.sqrt_eq_iff_sq_eq this hr, eq_comm]
| 2,236 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
#align_import linear_algebra.matrix.hermitian from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
namespace Matrix
variable {Ξ± Ξ² : Type*} {m n : Type*} {A : Matrix n n Ξ±}
open scoped Matrix
local ... | Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 56 | 57 | theorem IsHermitian.ext {A : Matrix n n Ξ±} : (β i j, star (A j i) = A i j) β A.IsHermitian := by |
intro h; ext i j; exact h i j
| 2,237 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
#align_import linear_algebra.matrix.hermitian from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
namespace Matrix
variable {Ξ± Ξ² : Type*} {m n : Type*} {A : Matrix n n Ξ±}
open scoped Matrix
local ... | Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 74 | 76 | theorem IsHermitian.transpose {A : Matrix n n Ξ±} (h : A.IsHermitian) : Aα΅.IsHermitian := by |
rw [IsHermitian, conjTranspose, transpose_map]
exact congr_arg Matrix.transpose h
| 2,237 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.l2_space from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
open RCLike Submodule Filter
open scop... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 106 | 112 | theorem summable_inner (f g : lp G 2) : Summable fun i => βͺf i, g iβ« := by |
-- Apply the Direct Comparison Test, comparing with β' i, βf iβ * βg iβ (summable by HΓΆlder)
refine .of_norm_bounded (fun i => βf iβ * βg iβ) (lp.summable_mul ?_ f g) ?_
Β· rw [Real.isConjExponent_iff]; norm_num
intro i
-- Then apply Cauchy-Schwarz pointwise
exact norm_inner_le_norm (π := π) _ _
| 2,238 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.l2_space from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
open RCLike Submodule Filter
open scop... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 164 | 171 | theorem inner_single_left (i : ΞΉ) (a : G i) (f : lp G 2) : βͺlp.single 2 i a, fβ« = βͺa, f iβ« := by |
refine (hasSum_inner (lp.single 2 i a) f).unique ?_
convert hasSum_ite_eq i βͺa, f iβ« using 1
ext j
rw [lp.single_apply]
split_ifs with h
Β· subst h; rfl
Β· simp
| 2,238 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.l2_space from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
open RCLike Submodule Filter
open scop... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 174 | 175 | theorem inner_single_right (i : ΞΉ) (a : G i) (f : lp G 2) : βͺf, lp.single 2 i aβ« = βͺf i, aβ« := by |
simpa [inner_conj_symm] using congr_arg conj (@inner_single_left _ π _ _ _ _ i a f)
| 2,238 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 52 | 54 | theorem mem_parallelepiped_iff (v : ΞΉ β E) (x : E) :
x β parallelepiped v β β t β Icc (0 : ΞΉ β β) 1, x = β i, t i β’ v i := by |
simp [parallelepiped, eq_comm]
| 2,239 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 57 | 65 | theorem parallelepiped_basis_eq (b : Basis ΞΉ β E) :
parallelepiped b = {x | β i, b.repr x i β Set.Icc 0 1} := by |
classical
ext x
simp_rw [mem_parallelepiped_iff, mem_setOf_eq, b.ext_elem_iff, _root_.map_sum,
_root_.map_smul, Finset.sum_apply', Basis.repr_self, Finsupp.smul_single, smul_eq_mul,
mul_one, Finsupp.single_apply, Finset.sum_ite_eq', Finset.mem_univ, ite_true, mem_Icc,
Pi.le_def, Pi.zero_apply, Pi.one... | 2,239 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 67 | 71 | theorem image_parallelepiped (f : E ββ[β] F) (v : ΞΉ β E) :
f '' parallelepiped v = parallelepiped (f β v) := by |
simp only [parallelepiped, β image_comp]
congr 1 with t
simp only [Function.comp_apply, _root_.map_sum, LinearMap.map_smulββ, RingHom.id_apply]
| 2,239 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 76 | 94 | theorem parallelepiped_comp_equiv (v : ΞΉ β E) (e : ΞΉ' β ΞΉ) :
parallelepiped (v β e) = parallelepiped v := by |
simp only [parallelepiped]
let K : (ΞΉ' β β) β (ΞΉ β β) := Equiv.piCongrLeft' (fun _a : ΞΉ' => β) e
have : Icc (0 : ΞΉ β β) 1 = K '' Icc (0 : ΞΉ' β β) 1 := by
rw [β Equiv.preimage_eq_iff_eq_image]
ext x
simp only [K, mem_preimage, mem_Icc, Pi.le_def, Pi.zero_apply, Equiv.piCongrLeft'_apply,
Pi.one_a... | 2,239 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 98 | 125 | theorem parallelepiped_orthonormalBasis_one_dim (b : OrthonormalBasis ΞΉ β β) :
parallelepiped b = Icc 0 1 β¨ parallelepiped b = Icc (-1) 0 := by |
have e : ΞΉ β Fin 1 := by
apply Fintype.equivFinOfCardEq
simp only [β finrank_eq_card_basis b.toBasis, finrank_self]
have B : parallelepiped (b.reindex e) = parallelepiped b := by
convert parallelepiped_comp_equiv b e.symm
ext i
simp only [OrthonormalBasis.coe_reindex]
rw [β B]
let F : β β F... | 2,239 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 128 | 138 | theorem parallelepiped_eq_sum_segment (v : ΞΉ β E) : parallelepiped v = β i, segment β 0 (v i) := by |
ext
simp only [mem_parallelepiped_iff, Set.mem_finset_sum, Finset.mem_univ, forall_true_left,
segment_eq_image, smul_zero, zero_add, β Set.pi_univ_Icc, Set.mem_univ_pi]
constructor
Β· rintro β¨t, ht, rflβ©
exact β¨t β’ v, fun {i} => β¨t i, ht _, by simpβ©, rflβ©
rintro β¨g, hg, rflβ©
choose t ht hg using @hg... | 2,239 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 82 | 86 | theorem hasPDF_iff_of_aemeasurable {X : Ξ© β E} {β : Measure Ξ©}
{ΞΌ : Measure E} (hX : AEMeasurable X β) :
HasPDF X β ΞΌ β (map X β).HaveLebesgueDecomposition ΞΌ β§ map X β βͺ ΞΌ := by |
rw [hasPDF_iff]
simp only [hX, true_and]
| 2,240 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 122 | 128 | theorem hasPDF_of_map_eq_withDensity {X : Ξ© β E} {β : Measure Ξ©} {ΞΌ : Measure E}
(hX : AEMeasurable X β) (f : E β ββ₯0β) (hf : AEMeasurable f ΞΌ) (h : map X β = ΞΌ.withDensity f) :
HasPDF X β ΞΌ := by |
refine β¨hX, ?_, ?_β© <;> rw [h]
Β· rw [withDensity_congr_ae hf.ae_eq_mk]
exact haveLebesgueDecomposition_withDensity ΞΌ hf.measurable_mk
Β· exact withDensity_absolutelyContinuous ΞΌ f
| 2,240 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 142 | 145 | theorem pdf_of_not_aemeasurable {_ : MeasurableSpace Ξ©} {β : Measure Ξ©} {ΞΌ : Measure E}
{X : Ξ© β E} (hX : Β¬AEMeasurable X β) : pdf X β ΞΌ =α΅[ΞΌ] 0 := by |
rw [pdf_def, map_of_not_aemeasurable hX]
exact rnDeriv_zero ΞΌ
| 2,240 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 152 | 155 | theorem aemeasurable_of_pdf_ne_zero {m : MeasurableSpace Ξ©} {β : Measure Ξ©} {ΞΌ : Measure E}
(X : Ξ© β E) (h : Β¬pdf X β ΞΌ =α΅[ΞΌ] 0) : AEMeasurable X β := by |
contrapose! h
exact pdf_of_not_aemeasurable h
| 2,240 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 158 | 164 | theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ξ©} {β : Measure Ξ©} {ΞΌ : Measure E} {X : Ξ© β E}
(hac : map X β βͺ ΞΌ) (hpdf : Β¬pdf X β ΞΌ =α΅[ΞΌ] 0) : HasPDF X β ΞΌ := by |
refine β¨?_, ?_, hacβ©
Β· exact aemeasurable_of_pdf_ne_zero X hpdf
Β· contrapose! hpdf
have := pdf_of_not_haveLebesgueDecomposition hpdf
filter_upwards using congrFun this
| 2,240 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 168 | 170 | theorem measurable_pdf {m : MeasurableSpace Ξ©} (X : Ξ© β E) (β : Measure Ξ©)
(ΞΌ : Measure E := by | volume_tac) : Measurable (pdf X β ΞΌ) := by
exact measurable_rnDeriv _ _
| 2,240 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 177 | 181 | theorem set_lintegral_pdf_le_map {m : MeasurableSpace Ξ©} (X : Ξ© β E) (β : Measure Ξ©)
(ΞΌ : Measure E := by | volume_tac) (s : Set E) :
β«β» x in s, pdf X β ΞΌ x βΞΌ β€ map X β s := by
apply (withDensity_apply_le _ s).trans
exact withDensity_pdf_le_map _ _ _ s
| 2,240 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 66 | 75 | theorem aemeasurable {X : Ξ© β E} {s : Set E} (hns : ΞΌ s β 0) (hnt : ΞΌ s β β)
(hu : IsUniform X s β ΞΌ) : AEMeasurable X β := by |
dsimp [IsUniform, ProbabilityTheory.cond] at hu
by_contra h
rw [map_of_not_aemeasurable h] at hu
apply zero_ne_one' ββ₯0β
calc
0 = (0 : Measure E) Set.univ := rfl
_ = _ := by rw [hu, smul_apply, restrict_apply MeasurableSet.univ,
Set.univ_inter, smul_eq_mul, ENNReal.inv_mul_cancel hns hnt]
| 2,241 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 77 | 78 | theorem absolutelyContinuous {X : Ξ© β E} {s : Set E} (hu : IsUniform X s β ΞΌ) : map X β βͺ ΞΌ := by |
rw [hu]; exact ProbabilityTheory.cond_absolutelyContinuous
| 2,241 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 80 | 84 | theorem measure_preimage {X : Ξ© β E} {s : Set E} (hns : ΞΌ s β 0) (hnt : ΞΌ s β β)
(hu : IsUniform X s β ΞΌ) {A : Set E} (hA : MeasurableSet A) :
β (X β»ΒΉ' A) = ΞΌ (s β© A) / ΞΌ s := by |
rwa [β map_apply_of_aemeasurable (hu.aemeasurable hns hnt) hA, hu, ProbabilityTheory.cond_apply',
ENNReal.div_eq_inv_mul]
| 2,241 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 95 | 98 | theorem toMeasurable_iff {X : Ξ© β E} {s : Set E} :
IsUniform X (toMeasurable ΞΌ s) β ΞΌ β IsUniform X s β ΞΌ := by |
unfold IsUniform
rw [ProbabilityTheory.cond_toMeasurable_eq]
| 2,241 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 105 | 111 | theorem hasPDF {X : Ξ© β E} {s : Set E} (hns : ΞΌ s β 0) (hnt : ΞΌ s β β)
(hu : IsUniform X s β ΞΌ) : HasPDF X β ΞΌ := by |
let t := toMeasurable ΞΌ s
apply hasPDF_of_map_eq_withDensity (hu.aemeasurable hns hnt) (t.indicator ((ΞΌ t)β»ΒΉ β’ 1)) <|
(measurable_one.aemeasurable.const_smul (ΞΌ t)β»ΒΉ).indicator (measurableSet_toMeasurable ΞΌ s)
rw [hu, withDensity_indicator (measurableSet_toMeasurable ΞΌ s), withDensity_smul _ measurable_one,
... | 2,241 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 114 | 121 | theorem pdf_eq_zero_of_measure_eq_zero_or_top {X : Ξ© β E} {s : Set E}
(hu : IsUniform X s β ΞΌ) (hΞΌs : ΞΌ s = 0 β¨ ΞΌ s = β) : pdf X β ΞΌ =α΅[ΞΌ] 0 := by |
rcases hΞΌs with H|H
Β· simp only [IsUniform, ProbabilityTheory.cond, H, ENNReal.inv_zero, restrict_eq_zero.mpr H,
smul_zero] at hu
simp [pdf, hu]
Β· simp only [IsUniform, ProbabilityTheory.cond, H, ENNReal.inv_top, zero_smul] at hu
simp [pdf, hu]
| 2,241 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 123 | 136 | theorem pdf_eq {X : Ξ© β E} {s : Set E} (hms : MeasurableSet s)
(hu : IsUniform X s β ΞΌ) : pdf X β ΞΌ =α΅[ΞΌ] s.indicator ((ΞΌ s)β»ΒΉ β’ (1 : E β ββ₯0β)) := by |
by_cases hnt : ΞΌ s = β
Β· simp [pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inr hnt), hnt]
by_cases hns : ΞΌ s = 0
Β· filter_upwards [measure_zero_iff_ae_nmem.mp hns,
pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inl hns)] with x hx h'x
simp [hx, h'x, hns]
have : HasPDF X β ΞΌ := hasPDF hns hnt hu
ha... | 2,241 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 56 | 72 | theorem volume_eq_stieltjes_id : (volume : Measure β) = StieltjesFunction.id.measure := by |
haveI : IsAddLeftInvariant StieltjesFunction.id.measure :=
β¨fun a =>
Eq.symm <|
Real.measure_ext_Ioo_rat fun p q => by
simp only [Measure.map_apply (measurable_const_add a) measurableSet_Ioo,
sub_sub_sub_cancel_right, StieltjesFunction.measure_Ioo, StieltjesFunction.id_leftLim... | 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 75 | 76 | theorem volume_val (s) : volume s = StieltjesFunction.id.measure s := by |
simp [volume_eq_stieltjes_id]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 80 | 80 | theorem volume_Ico {a b : β} : volume (Ico a b) = ofReal (b - a) := by | simp [volume_val]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 84 | 84 | theorem volume_Icc {a b : β} : volume (Icc a b) = ofReal (b - a) := by | simp [volume_val]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 88 | 88 | theorem volume_Ioo {a b : β} : volume (Ioo a b) = ofReal (b - a) := by | simp [volume_val]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 92 | 92 | theorem volume_Ioc {a b : β} : volume (Ioc a b) = ofReal (b - a) := by | simp [volume_val]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 96 | 96 | theorem volume_singleton {a : β} : volume ({a} : Set β) = 0 := by | simp [volume_val]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 100 | 104 | theorem volume_univ : volume (univ : Set β) = β :=
ENNReal.eq_top_of_forall_nnreal_le fun r =>
calc
(r : ββ₯0β) = volume (Icc (0 : β) r) := by | simp
_ β€ volume univ := measure_mono (subset_univ _)
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 108 | 109 | theorem volume_ball (a r : β) : volume (Metric.ball a r) = ofReal (2 * r) := by |
rw [ball_eq_Ioo, volume_Ioo, β sub_add, add_sub_cancel_left, two_mul]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 113 | 114 | theorem volume_closedBall (a r : β) : volume (Metric.closedBall a r) = ofReal (2 * r) := by |
rw [closedBall_eq_Icc, volume_Icc, β sub_add, add_sub_cancel_left, two_mul]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 118 | 123 | theorem volume_emetric_ball (a : β) (r : ββ₯0β) : volume (EMetric.ball a r) = 2 * r := by |
rcases eq_or_ne r β with (rfl | hr)
Β· rw [Metric.emetric_ball_top, volume_univ, two_mul, _root_.top_add]
Β· lift r to ββ₯0 using hr
rw [Metric.emetric_ball_nnreal, volume_ball, two_mul, β NNReal.coe_add,
ENNReal.ofReal_coe_nnreal, ENNReal.coe_add, two_mul]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 127 | 132 | theorem volume_emetric_closedBall (a : β) (r : ββ₯0β) : volume (EMetric.closedBall a r) = 2 * r := by |
rcases eq_or_ne r β with (rfl | hr)
Β· rw [EMetric.closedBall_top, volume_univ, two_mul, _root_.top_add]
Β· lift r to ββ₯0 using hr
rw [Metric.emetric_closedBall_nnreal, volume_closedBall, two_mul, β NNReal.coe_add,
ENNReal.ofReal_coe_nnreal, ENNReal.coe_add, two_mul]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 140 | 141 | theorem volume_interval {a b : β} : volume (uIcc a b) = ofReal |b - a| := by |
rw [β Icc_min_max, volume_Icc, max_sub_min_eq_abs]
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 145 | 150 | theorem volume_Ioi {a : β} : volume (Ioi a) = β :=
top_unique <|
le_of_tendsto' ENNReal.tendsto_nat_nhds_top fun n =>
calc
(n : ββ₯0β) = volume (Ioo a (a + n)) := by | simp
_ β€ volume (Ioi a) := measure_mono Ioo_subset_Ioi_self
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 449 | 450 | theorem regionBetween_subset (f g : Ξ± β β) (s : Set Ξ±) : regionBetween f g s β s ΓΛ’ univ := by |
simpa only [prod_univ, regionBetween, Set.preimage, setOf_subset_setOf] using fun a => And.left
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 456 | 463 | theorem measurableSet_regionBetween (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s) :
MeasurableSet (regionBetween f g s) := by |
dsimp only [regionBetween, Ioo, mem_setOf_eq, setOf_and]
refine
MeasurableSet.inter ?_
((measurableSet_lt (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_lt measurable_snd (hg.comp measurable_fst)))
exact measurable_fst hs
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 468 | 476 | theorem measurableSet_region_between_oc (hf : Measurable f) (hg : Measurable g)
(hs : MeasurableSet s) :
MeasurableSet { p : Ξ± Γ β | p.fst β s β§ p.snd β Ioc (f p.fst) (g p.fst) } := by |
dsimp only [regionBetween, Ioc, mem_setOf_eq, setOf_and]
refine
MeasurableSet.inter ?_
((measurableSet_lt (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_le measurable_snd (hg.comp measurable_fst)))
exact measurable_fst hs
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 481 | 489 | theorem measurableSet_region_between_co (hf : Measurable f) (hg : Measurable g)
(hs : MeasurableSet s) :
MeasurableSet { p : Ξ± Γ β | p.fst β s β§ p.snd β Ico (f p.fst) (g p.fst) } := by |
dsimp only [regionBetween, Ico, mem_setOf_eq, setOf_and]
refine
MeasurableSet.inter ?_
((measurableSet_le (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_lt measurable_snd (hg.comp measurable_fst)))
exact measurable_fst hs
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 494 | 502 | theorem measurableSet_region_between_cc (hf : Measurable f) (hg : Measurable g)
(hs : MeasurableSet s) :
MeasurableSet { p : Ξ± Γ β | p.fst β s β§ p.snd β Icc (f p.fst) (g p.fst) } := by |
dsimp only [regionBetween, Icc, mem_setOf_eq, setOf_and]
refine
MeasurableSet.inter ?_
((measurableSet_le (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_le measurable_snd (hg.comp measurable_fst)))
exact measurable_fst hs
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 506 | 508 | theorem measurableSet_graph (hf : Measurable f) :
MeasurableSet { p : Ξ± Γ β | p.snd = f p.fst } := by |
simpa using measurableSet_region_between_cc hf hf MeasurableSet.univ
| 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 624 | 653 | theorem ae_restrict_of_ae_restrict_inter_Ioo {ΞΌ : Measure β} [NoAtoms ΞΌ] {s : Set β} {p : β β Prop}
(h : β a b, a β s β b β s β a < b β βα΅ x βΞΌ.restrict (s β© Ioo a b), p x) :
βα΅ x βΞΌ.restrict s, p x := by |
/- By second-countability, we cover `s` by countably many intervals `(a, b)` (except maybe for
two endpoints, which don't matter since `ΞΌ` does not have any atom). -/
let T : s Γ s β Set β := fun p => Ioo p.1 p.2
let u := β i : β₯s Γ β₯s, T i
have hfinite : (s \ u).Finite := s.finite_diff_iUnion_Ioo'
obtai... | 2,242 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 659 | 686 | theorem ae_of_mem_of_ae_of_mem_inter_Ioo {ΞΌ : Measure β} [NoAtoms ΞΌ] {s : Set β} {p : β β Prop}
(h : β a b, a β s β b β s β a < b β βα΅ x βΞΌ, x β s β© Ioo a b β p x) :
βα΅ x βΞΌ, x β s β p x := by |
/- By second-countability, we cover `s` by countably many intervals `(a, b)` (except maybe for
two endpoints, which don't matter since `ΞΌ` does not have any atom). -/
let T : s Γ s β Set β := fun p => Ioo p.1 p.2
let u := β i : β₯s Γ β₯s, T i
have hfinite : (s \ u).Finite := s.finite_diff_iUnion_Ioo'
obtai... | 2,242 |
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.NumberTheory.Liouville.Residual
import Mathlib.NumberTheory.Liouville.LiouvilleWith
import Mathlib.Analysis.PSeries
#align_import number_theory.liouville.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open sc... | Mathlib/NumberTheory/Liouville/Measure.lean | 34 | 71 | theorem setOf_liouvilleWith_subset_aux :
{ x : β | β p > 2, LiouvilleWith p x } β
β m : β€, (Β· + (m : β)) β»ΒΉ' β n > (0 : β),
{ x : β | βαΆ b : β in atTop, β a β Finset.Icc (0 : β€) b,
|x - (a : β€) / b| < 1 / (b : β) ^ (2 + 1 / n : β) } := by |
rintro x β¨p, hp, hxpβ©
rcases exists_nat_one_div_lt (sub_pos.2 hp) with β¨n, hnβ©
rw [lt_sub_iff_add_lt'] at hn
suffices β y : β, LiouvilleWith p y β y β Ico (0 : β) 1 β βαΆ b : β in atTop,
β a β Finset.Icc (0 : β€) b, |y - a / b| < 1 / (b : β) ^ (2 + 1 / (n + 1 : β) : β) by
simp only [mem_iUnion, mem_pre... | 2,243 |
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.NumberTheory.Liouville.Residual
import Mathlib.NumberTheory.Liouville.LiouvilleWith
import Mathlib.Analysis.PSeries
#align_import number_theory.liouville.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open sc... | Mathlib/NumberTheory/Liouville/Measure.lean | 77 | 106 | theorem volume_iUnion_setOf_liouvilleWith :
volume (β (p : β) (_hp : 2 < p), { x : β | LiouvilleWith p x }) = 0 := by |
simp only [β setOf_exists, exists_prop]
refine measure_mono_null setOf_liouvilleWith_subset_aux ?_
rw [measure_iUnion_null_iff]; intro m; rw [measure_preimage_add_right]; clear m
refine (measure_biUnion_null_iff <| to_countable _).2 fun n (hn : 1 β€ n) => ?_
generalize hr : (2 + 1 / n : β) = r
replace hr : ... | 2,243 |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 22 | 31 | theorem volume_regionBetween_eq_integral' [SigmaFinite ΞΌ] (f_int : IntegrableOn f s ΞΌ)
(g_int : IntegrableOn g s ΞΌ) (hs : MeasurableSet s) (hfg : f β€α΅[ΞΌ.restrict s] g) :
ΞΌ.prod volume (regionBetween f g s) = ENNReal.ofReal (β« y in s, (g - f) y βΞΌ) := by |
have h : g - f =α΅[ΞΌ.restrict s] fun x => Real.toNNReal (g x - f x) :=
hfg.mono fun x hx => (Real.coe_toNNReal _ <| sub_nonneg.2 hx).symm
rw [volume_regionBetween_eq_lintegral f_int.aemeasurable g_int.aemeasurable hs,
integral_congr_ae h, lintegral_congr_ae,
lintegral_coe_eq_integral _ ((integrable_cong... | 2,244 |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 55 | 69 | theorem Real.integrable_of_summable_norm_Icc {E : Type*} [NormedAddCommGroup E] {f : C(β, E)}
(hf : Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict (Icc 0 1)β) :
Integrable f := by |
refine integrable_of_summable_norm_restrict (.of_nonneg_of_le
(fun n : β€ => mul_nonneg (norm_nonneg
(f.restrict (β¨Icc (n : β) ((n : β) + 1), isCompact_Iccβ© : Compacts β)))
ENNReal.toReal_nonneg) (fun n => ?_) hf) ?_
Β· simp only [Compacts.coe_mk, Real.volume_Icc, add_sub_cancel_left,
ENNReal... | 2,244 |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 85 | 91 | theorem integral_comp_neg_Iic {E : Type*} [NormedAddCommGroup E] [NormedSpace β E]
(c : β) (f : β β E) : (β« x in Iic c, f (-x)) = β« x in Ioi (-c), f x := by |
have A : MeasurableEmbedding fun x : β => -x :=
(Homeomorph.neg β).closedEmbedding.measurableEmbedding
have := MeasurableEmbedding.setIntegral_map (ΞΌ := volume) A f (Ici (-c))
rw [Measure.map_neg_eq_self (volume : Measure β)] at this
simp_rw [β integral_Ici_eq_integral_Ioi, this, neg_preimage, preimage_neg... | 2,244 |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 96 | 99 | theorem integral_comp_neg_Ioi {E : Type*} [NormedAddCommGroup E] [NormedSpace β E]
(c : β) (f : β β E) : (β« x in Ioi c, f (-x)) = β« x in Iic (-c), f x := by |
rw [β neg_neg c, β integral_comp_neg_Iic]
simp only [neg_neg]
| 2,244 |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 102 | 127 | theorem integral_comp_abs {f : β β β} :
β« x, f |x| = 2 * β« x in Ioi (0:β), f x := by |
have eq : β« (x : β) in Ioi 0, f |x| = β« (x : β) in Ioi 0, f x := by
refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)
rw [abs_eq_self.mpr (le_of_lt (by exact hx))]
by_cases hf : IntegrableOn (fun x => f |x|) (Ioi 0)
Β· have int_Iic : IntegrableOn (fun x β¦ f |x|) (Iic 0) := by
rw [β Measure... | 2,244 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 84 | 91 | theorem Basis.parallelepiped_eq_map {ΞΉ E : Type*} [Fintype ΞΉ] [NormedAddCommGroup E]
[NormedSpace β E] (b : Basis ΞΉ β E) :
b.parallelepiped = (PositiveCompacts.piIcc01 ΞΉ).map b.equivFun.symm
b.equivFunL.symm.continuous b.equivFunL.symm.isOpenMap := by |
classical
rw [β Basis.parallelepiped_basisFun, β Basis.parallelepiped_map]
congr with x
simp
| 2,245 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 95 | 104 | theorem Basis.map_addHaar {ΞΉ E F : Type*} [Fintype ΞΉ] [NormedAddCommGroup E] [NormedAddCommGroup F]
[NormedSpace β E] [NormedSpace β F] [MeasurableSpace E] [MeasurableSpace F] [BorelSpace E]
[BorelSpace F] [SecondCountableTopology F] [SigmaCompactSpace F]
(b : Basis ΞΉ β E) (f : E βL[β] F) :
map f b.addH... |
have : IsAddHaarMeasure (map f b.addHaar) :=
AddEquiv.isAddHaarMeasure_map b.addHaar f.toAddEquiv f.continuous f.symm.continuous
rw [eq_comm, Basis.addHaar_eq_iff, Measure.map_apply f.continuous.measurable
(PositiveCompacts.isCompact _).measurableSet, Basis.coe_parallelepiped, Basis.coe_map]
erw [β image... | 2,245 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 115 | 116 | theorem addHaarMeasure_eq_volume : addHaarMeasure Icc01 = volume := by |
convert (addHaarMeasure_unique volume Icc01).symm; simp [Icc01]
| 2,245 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 120 | 124 | theorem addHaarMeasure_eq_volume_pi (ΞΉ : Type*) [Fintype ΞΉ] :
addHaarMeasure (piIcc01 ΞΉ) = volume := by |
convert (addHaarMeasure_unique volume (piIcc01 ΞΉ)).symm
simp only [piIcc01, volume_pi_pi fun _ => Icc (0 : β) 1, PositiveCompacts.coe_mk,
Compacts.coe_mk, Finset.prod_const_one, ENNReal.ofReal_one, Real.volume_Icc, one_smul, sub_zero]
| 2,245 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 142 | 155 | theorem addHaar_eq_zero_of_disjoint_translates_aux {E : Type*} [NormedAddCommGroup E]
[NormedSpace β E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional β E] (ΞΌ : Measure E)
[IsAddHaarMeasure ΞΌ] {s : Set E} (u : β β E) (sb : IsBounded s) (hu : IsBounded (range u))
(hs : Pairwise (Disjoint on fun n => {... |
by_contra h
apply lt_irrefl β
calc
β = β' _ : β, ΞΌ s := (ENNReal.tsum_const_eq_top_of_ne_zero h).symm
_ = β' n : β, ΞΌ ({u n} + s) := by
congr 1; ext1 n; simp only [image_add_left, measure_preimage_add, singleton_add]
_ = ΞΌ (β n, {u n} + s) := Eq.symm <| measure_iUnion hs fun n => by
simpa... | 2,245 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.AddTorsorBases
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.convex.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory MeasureTheory.Measure Set Metric F... | Mathlib/Analysis/Convex/Measure.lean | 33 | 80 | theorem addHaar_frontier (hs : Convex β s) : ΞΌ (frontier s) = 0 := by |
/- If `s` is included in a hyperplane, then `frontier s β closure s` is included in the same
hyperplane, hence it has measure zero. -/
cases' ne_or_eq (affineSpan β s) β€ with hspan hspan
Β· refine measure_mono_null ?_ (addHaar_affineSubspace _ _ hspan)
exact frontier_subset_closure.trans
(closure_mi... | 2,246 |
import Mathlib.Analysis.Convex.Body
import Mathlib.Analysis.Convex.Measure
import Mathlib.MeasureTheory.Group.FundamentalDomain
#align_import measure_theory.group.geometry_of_numbers from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
namespace MeasureTheory
open ENNReal FiniteDimensio... | Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean | 50 | 58 | theorem exists_pair_mem_lattice_not_disjoint_vadd [AddCommGroup L] [Countable L] [AddAction L E]
[MeasurableSpace L] [MeasurableVAdd L E] [VAddInvariantMeasure L E ΞΌ]
(fund : IsAddFundamentalDomain L F ΞΌ) (hS : NullMeasurableSet s ΞΌ) (h : ΞΌ F < ΞΌ s) :
β x y : L, x β y β§ Β¬Disjoint (x +α΅₯ s) (y +α΅₯ s) := by |
contrapose! h
exact ((fund.measure_eq_tsum _).trans (measure_iUnionβ
(Pairwise.mono h fun i j hij => (hij.mono inf_le_left inf_le_left).aedisjoint)
fun _ => (hS.vadd _).inter fund.nullMeasurableSet).symm).trans_le
(measure_mono <| Set.iUnion_subset fun _ => Set.inter_subset_right)
| 2,247 |
import Mathlib.Analysis.Convex.Body
import Mathlib.Analysis.Convex.Measure
import Mathlib.MeasureTheory.Group.FundamentalDomain
#align_import measure_theory.group.geometry_of_numbers from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
namespace MeasureTheory
open ENNReal FiniteDimensio... | Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean | 64 | 83 | theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
[NormedSpace β E] [BorelSpace E] [FiniteDimensional β E] [IsAddHaarMeasure ΞΌ]
{L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F ΞΌ)
(h_symm : β x β s, -x β s) (h_conv : Convex β s) (h : ΞΌ F * 2 ^ fi... |
have h_vol : ΞΌ F < ΞΌ ((2β»ΒΉ : β) β’ s) := by
rw [addHaar_smul_of_nonneg ΞΌ (by norm_num : 0 β€ (2 : β)β»ΒΉ) s, β
mul_lt_mul_right (pow_ne_zero (finrank β E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
mul_right_comm, ofReal_pow (by norm_num : 0 β€ (2 : β)β»ΒΉ), ofReal_inv_of_pos zero_lt_two]
norm_num
r... | 2,247 |
import Mathlib.Analysis.Convex.Body
import Mathlib.Analysis.Convex.Measure
import Mathlib.MeasureTheory.Group.FundamentalDomain
#align_import measure_theory.group.geometry_of_numbers from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
namespace MeasureTheory
open ENNReal FiniteDimensio... | Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean | 92 | 142 | theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure [NormedAddCommGroup E]
[NormedSpace β E] [BorelSpace E] [FiniteDimensional β E] [Nontrivial E] [IsAddHaarMeasure ΞΌ]
{L : AddSubgroup E} [Countable L] [DiscreteTopology L] (fund : IsAddFundamentalDomain L F ΞΌ)
(h_symm : β x β s, -x β s) (h_... |
have h_mes : ΞΌ s β 0 := by
intro hΞΌ
suffices ΞΌ F = 0 from fund.measure_ne_zero (NeZero.ne ΞΌ) this
rw [hΞΌ, le_zero_iff, mul_eq_zero] at h
exact h.resolve_right <| pow_ne_zero _ two_ne_zero
have h_nemp : s.Nonempty := nonempty_of_measure_ne_zero h_mes
let u : β β ββ₯0 := (exists_seq_strictAnti_tends... | 2,247 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 83 | 84 | theorem centerAndRescale_center : a.centerAndRescale.c (last N) = 0 := by |
simp [SatelliteConfig.centerAndRescale]
| 2,248 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 87 | 89 | theorem centerAndRescale_radius {N : β} {Ο : β} (a : SatelliteConfig E N Ο) :
a.centerAndRescale.r (last N) = 1 := by |
simp [SatelliteConfig.centerAndRescale, inv_mul_cancel (a.rpos _).ne']
| 2,248 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 110 | 150 | theorem card_le_of_separated (s : Finset E) (hs : β c β s, βcβ β€ 2)
(h : β c β s, β d β s, c β d β 1 β€ βc - dβ) : s.card β€ 5 ^ finrank β E := by |
/- We consider balls of radius `1/2` around the points in `s`. They are disjoint, and all
contained in the ball of radius `5/2`. A volume argument gives `s.card * (1/2)^dim β€ (5/2)^dim`,
i.e., `s.card β€ 5^dim`. -/
borelize E
let ΞΌ : Measure E := Measure.addHaar
let Ξ΄ : β := (1 : β) / 2
let Ο : β := (... | 2,248 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 153 | 157 | theorem multiplicity_le : multiplicity E β€ 5 ^ finrank β E := by |
apply csSup_le
Β· refine β¨0, β¨β
, by simpβ©β©
Β· rintro _ β¨s, β¨rfl, hβ©β©
exact Besicovitch.card_le_of_separated s h.1 h.2
| 2,248 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 160 | 167 | theorem card_le_multiplicity {s : Finset E} (hs : β c β s, βcβ β€ 2)
(h's : β c β s, β d β s, c β d β 1 β€ βc - dβ) : s.card β€ multiplicity E := by |
apply le_csSup
Β· refine β¨5 ^ finrank β E, ?_β©
rintro _ β¨s, β¨rfl, hβ©β©
exact Besicovitch.card_le_of_separated s h.1 h.2
Β· simp only [mem_setOf_eq, Ne]
exact β¨s, rfl, hs, h'sβ©
| 2,248 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 174 | 246 | theorem exists_goodΞ΄ :
β Ξ΄ : β, 0 < Ξ΄ β§ Ξ΄ < 1 β§ β s : Finset E, (β c β s, βcβ β€ 2) β
(β c β s, β d β s, c β d β 1 - Ξ΄ β€ βc - dβ) β s.card β€ multiplicity E := by |
classical
/- This follows from a compactness argument: otherwise, one could extract a converging
subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
`N = multiplicity E + 1`. To formalize this, we work with functions `Fin N β E`.
-/
by_contra! h
set N := multiplic... | 2,248 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 26 | 30 | theorem Icc_mem_vitaliFamily_at_right {x y : β} (hxy : x < y) :
Icc x y β (vitaliFamily (volume : Measure β) 1).setsAt x := by |
rw [Icc_eq_closedBall]
refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)
rw [dist_comm, Real.dist_eq, abs_of_nonneg] <;> linarith
| 2,249 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 33 | 41 | theorem tendsto_Icc_vitaliFamily_right (x : β) :
Tendsto (fun y => Icc x y) (π[>] x) ((vitaliFamily (volume : Measure β) 1).filterAt x) := by |
refine (VitaliFamily.tendsto_filterAt_iff _).2 β¨?_, ?_β©
Β· filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy
Β· intro Ξ΅ Ξ΅pos
have : x β Ico x (x + Ξ΅) := β¨le_refl _, by linarithβ©
filter_upwards [Icc_mem_nhdsWithin_Ioi this] with y hy
rw [closedBall_eq_Icc]
exact I... | 2,249 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 44 | 48 | theorem Icc_mem_vitaliFamily_at_left {x y : β} (hxy : x < y) :
Icc x y β (vitaliFamily (volume : Measure β) 1).setsAt y := by |
rw [Icc_eq_closedBall]
refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)
rw [Real.dist_eq, abs_of_nonneg] <;> linarith
| 2,249 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 51 | 59 | theorem tendsto_Icc_vitaliFamily_left (x : β) :
Tendsto (fun y => Icc y x) (π[<] x) ((vitaliFamily (volume : Measure β) 1).filterAt x) := by |
refine (VitaliFamily.tendsto_filterAt_iff _).2 β¨?_, ?_β©
Β· filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy
Β· intro Ξ΅ Ξ΅pos
have : x β Ioc (x - Ξ΅) x := β¨by linarith, le_refl _β©
filter_upwards [Icc_mem_nhdsWithin_Iio this] with y hy
rw [closedBall_eq_Icc]
exact Ic... | 2,249 |
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.MeasureTheory.Covering.OneDim
import Mathlib.Order.Monotone.Extension
#align_import analysis.calculus.monotone from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Function Metric MeasureTheory MeasureTheory.Meas... | Mathlib/Analysis/Calculus/Monotone.lean | 44 | 62 | theorem tendsto_apply_add_mul_sq_div_sub {f : β β β} {x a c d : β} {l : Filter β} (hl : l β€ π[β ] x)
(hf : Tendsto (fun y => (f y - d) / (y - x)) l (π a))
(h' : Tendsto (fun y => y + c * (y - x) ^ 2) l l) :
Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (π a) := by |
have L : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (π 1) := by
have : Tendsto (fun y => 1 + c * (y - x)) l (π (1 + c * (x - x))) := by
apply Tendsto.mono_left _ (hl.trans nhdsWithin_le_nhds)
exact ((tendsto_id.sub_const x).const_mul c).const_add 1
simp only [_root_.sub_self, add_... | 2,250 |
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.MeasureTheory.Covering.OneDim
import Mathlib.Order.Monotone.Extension
#align_import analysis.calculus.monotone from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Function Metric MeasureTheory MeasureTheory.Meas... | Mathlib/Analysis/Calculus/Monotone.lean | 67 | 131 | theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
βα΅ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x := by |
/- Denote by `ΞΌ` the Stieltjes measure associated to `f`.
The general theorem `VitaliFamily.ae_tendsto_rnDeriv` ensures that `ΞΌ [x, y] / (y - x)` tends
to the Radon-Nikodym derivative as `y` tends to `x` from the right. As `ΞΌ [x,y] = f y - f (x^-)`
and `f (x^-) = f x` almost everywhere, this gives differ... | 2,250 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 83 | 86 | theorem nonempty_monotone_mem {s : Set Ξ±} (hs : s.Nonempty) :
Nonempty { u // Monotone u β§ β i : β, u i β s } := by |
obtain β¨x, hxβ© := hs
exact β¨β¨fun _ => x, fun i j _ => le_rfl, fun _ => hxβ©β©
| 2,251 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 89 | 94 | theorem eq_of_edist_zero_on {f f' : Ξ± β E} {s : Set Ξ±} (h : β β¦xβ¦, x β s β edist (f x) (f' x) = 0) :
eVariationOn f s = eVariationOn f' s := by |
dsimp only [eVariationOn]
congr 1 with p : 1
congr 1 with i : 1
rw [edist_congr_right (h <| p.snd.prop.2 (i + 1)), edist_congr_left (h <| p.snd.prop.2 i)]
| 2,251 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 107 | 124 | theorem sum_le_of_monotoneOn_Icc (f : Ξ± β E) {s : Set Ξ±} {m n : β} {u : β β Ξ±}
(hu : MonotoneOn u (Icc m n)) (us : β i β Icc m n, u i β s) :
(β i β Finset.Ico m n, edist (f (u (i + 1))) (f (u i))) β€ eVariationOn f s := by |
rcases le_total n m with hnm | hmn
Β· simp [Finset.Ico_eq_empty_of_le hnm]
let Ο := projIcc m n hmn
let v i := u (Ο i)
calc
β i β Finset.Ico m n, edist (f (u (i + 1))) (f (u i))
= β i β Finset.Ico m n, edist (f (v (i + 1))) (f (v i)) :=
Finset.sum_congr rfl fun i hi β¦ by
rw [Finset.m... | 2,251 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 127 | 130 | theorem sum_le_of_monotoneOn_Iic (f : Ξ± β E) {s : Set Ξ±} {n : β} {u : β β Ξ±}
(hu : MonotoneOn u (Iic n)) (us : β i β€ n, u i β s) :
(β i β Finset.range n, edist (f (u (i + 1))) (f (u i))) β€ eVariationOn f s := by |
simpa using sum_le_of_monotoneOn_Icc f (m := 0) (hu.mono Icc_subset_Iic_self) fun i hi β¦ us i hi.2
| 2,251 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 133 | 136 | theorem mono (f : Ξ± β E) {s t : Set Ξ±} (hst : t β s) : eVariationOn f t β€ eVariationOn f s := by |
apply iSup_le _
rintro β¨n, β¨u, hu, utβ©β©
exact sum_le f n hu fun i => hst (ut i)
| 2,251 |
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {Ξ± : Type*} [LinearOrder Ξ±] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 64 | 68 | theorem hasConstantSpeedOnWith_of_subsingleton (f : β β E) {s : Set β} (hs : s.Subsingleton)
(l : ββ₯0) : HasConstantSpeedOnWith f s l := by |
rintro x hx y hy; cases hs hx hy
rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s β© Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero]
| 2,252 |
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {Ξ± : Type*} [LinearOrder Ξ±] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 71 | 82 | theorem hasConstantSpeedOnWith_iff_ordered :
HasConstantSpeedOnWith f s l β β β¦xβ¦ (_ : x β s) β¦yβ¦ (_ : y β s),
x β€ y β eVariationOn f (s β© Icc x y) = ENNReal.ofReal (l * (y - x)) := by |
refine β¨fun h x xs y ys _ => h xs ys, fun h x xs y ys => ?_β©
rcases le_total x y with (xy | yx)
Β· exact h xs ys xy
Β· rw [eVariationOn.subsingleton, ENNReal.ofReal_of_nonpos]
Β· exact mul_nonpos_of_nonneg_of_nonpos l.prop (sub_nonpos_of_le yx)
Β· rintro z β¨zs, xz, zyβ© w β¨ws, xw, wyβ©
cases le_antisym... | 2,252 |
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {Ξ± : Type*} [LinearOrder Ξ±] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 85 | 99 | theorem hasConstantSpeedOnWith_iff_variationOnFromTo_eq :
HasConstantSpeedOnWith f s l β LocallyBoundedVariationOn f s β§
β β¦xβ¦ (_ : x β s) β¦yβ¦ (_ : y β s), variationOnFromTo f s x y = l * (y - x) := by |
constructor
Β· rintro h; refine β¨h.hasLocallyBoundedVariationOn, fun x xs y ys => ?_β©
rw [hasConstantSpeedOnWith_iff_ordered] at h
rcases le_total x y with (xy | yx)
Β· rw [variationOnFromTo.eq_of_le f s xy, h xs ys xy]
exact ENNReal.toReal_ofReal (mul_nonneg l.prop (sub_nonneg.mpr xy))
Β· rw [v... | 2,252 |
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {Ξ± : Type*} [LinearOrder Ξ±] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 102 | 137 | theorem HasConstantSpeedOnWith.union {t : Set β} (hfs : HasConstantSpeedOnWith f s l)
(hft : HasConstantSpeedOnWith f t l) {x : β} (hs : IsGreatest s x) (ht : IsLeast t x) :
HasConstantSpeedOnWith f (s βͺ t) l := by |
rw [hasConstantSpeedOnWith_iff_ordered] at hfs hft β’
rintro z (zs | zt) y (ys | yt) zy
Β· have : (s βͺ t) β© Icc z y = s β© Icc z y := by
ext w; constructor
Β· rintro β¨ws | wt, zw, wyβ©
Β· exact β¨ws, zw, wyβ©
Β· exact β¨(le_antisymm (wy.trans (hs.2 ys)) (ht.2 wt)).symm βΈ hs.1, zw, wyβ©
Β· r... | 2,252 |
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.BoundedVariation
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff
import Mathlib.... | Mathlib/Analysis/Calculus/Rademacher.lean | 63 | 77 | theorem ae_lineDifferentiableAt (hf : LipschitzWith C f) (v : E) :
βα΅ p βΞΌ, LineDifferentiableAt β f p v := by |
let L : β βL[β] E := ContinuousLinearMap.smulRight (1 : β βL[β] β) v
suffices A : β p, βα΅ (t : β) βvolume, LineDifferentiableAt β f (p + t β’ v) v from
ae_mem_of_ae_add_linearMap_mem L.toLinearMap volume ΞΌ
(measurableSet_lineDifferentiableAt hf.continuous) A
intro p
have : βα΅ (s : β), DifferentiableAt... | 2,253 |
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.BoundedVariation
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff
import Mathlib.... | Mathlib/Analysis/Calculus/Rademacher.lean | 97 | 117 | theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
(hf : LipschitzWith C f) (hg : Integrable g ΞΌ) (v : E) :
Tendsto (fun (t : β) β¦ β« x, (tβ»ΒΉ β’ (f (x + t β’ v) - f x)) * g x βΞΌ) (π[>] 0)
(π (β« x, lineDeriv β f x v * g x βΞΌ)) := by |
apply tendsto_integral_filter_of_dominated_convergence (fun x β¦ (C * βvβ) * βg xβ)
Β· filter_upwards with t
apply AEStronglyMeasurable.mul ?_ hg.aestronglyMeasurable
apply aestronglyMeasurable_const.smul
apply AEStronglyMeasurable.sub _ hf.continuous.measurable.aestronglyMeasurable
apply AEMeasurabl... | 2,253 |
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.BoundedVariation
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff
import Mathlib.... | Mathlib/Analysis/Calculus/Rademacher.lean | 119 | 160 | theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul'
(hf : LipschitzWith C f) (h'f : HasCompactSupport f) (hg : Continuous g) (v : E) :
Tendsto (fun (t : β) β¦ β« x, (tβ»ΒΉ β’ (f (x + t β’ v) - f x)) * g x βΞΌ) (π[>] 0)
(π (β« x, lineDeriv β f x v * g x βΞΌ)) := by |
let K := cthickening (βvβ) (tsupport f)
have K_compact : IsCompact K := IsCompact.cthickening h'f
apply tendsto_integral_filter_of_dominated_convergence
(K.indicator (fun x β¦ (C * βvβ) * βg xβ))
Β· filter_upwards with t
apply AEStronglyMeasurable.mul ?_ hg.aestronglyMeasurable
apply aestronglyMeas... | 2,253 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 49 | 50 | theorem normed_sub (x : E) : f.normed ΞΌ (c - x) = f.normed ΞΌ (c + x) := by |
simp_rw [f.normed_def, f.sub]
| 2,254 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 53 | 54 | theorem normed_neg (f : ContDiffBump (0 : E)) (x : E) : f.normed ΞΌ (-x) = f.normed ΞΌ x := by |
simp_rw [f.normed_def, f.neg]
| 2,254 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 69 | 72 | theorem integral_pos : 0 < β« x, f x βΞΌ := by |
refine (integral_pos_iff_support_of_nonneg f.nonneg' f.integrable).mpr ?_
rw [f.support_eq]
exact measure_ball_pos ΞΌ c f.rOut_pos
| 2,254 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 75 | 77 | theorem integral_normed : β« x, f.normed ΞΌ x βΞΌ = 1 := by |
simp_rw [ContDiffBump.normed, div_eq_mul_inv, mul_comm (f _), β smul_eq_mul, integral_smul]
exact inv_mul_cancel f.integral_pos.ne'
| 2,254 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 80 | 82 | theorem support_normed_eq : Function.support (f.normed ΞΌ) = Metric.ball c f.rOut := by |
unfold ContDiffBump.normed
rw [support_div, f.support_eq, support_const f.integral_pos.ne', inter_univ]
| 2,254 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 85 | 86 | theorem tsupport_normed_eq : tsupport (f.normed ΞΌ) = Metric.closedBall c f.rOut := by |
rw [tsupport, f.support_normed_eq, closure_ball _ f.rOut_pos.ne']
| 2,254 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 89 | 90 | theorem hasCompactSupport_normed : HasCompactSupport (f.normed ΞΌ) := by |
simp only [HasCompactSupport, f.tsupport_normed_eq (ΞΌ := ΞΌ), isCompact_closedBall]
| 2,254 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.