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.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 119 | 120 | theorem expSeries_apply_eq (x : πΈ) (n : β) :
(expSeries π πΈ n fun _ => x) = (n !β»ΒΉ : π) β’ x ^ n := by | simp [expSeries]
| 2,180 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 136 | 141 | theorem expSeries_apply_zero (n : β) :
(expSeries π πΈ n fun _ => (0 : πΈ)) = Pi.single (f := fun _ => πΈ) 0 1 n := by |
rw [expSeries_apply_eq]
cases' n with n
Β· rw [pow_zero, Nat.factorial_zero, Nat.cast_one, inv_one, one_smul, Pi.single_eq_same]
Β· rw [zero_pow (Nat.succ_ne_zero _), smul_zero, Pi.single_eq_of_ne n.succ_ne_zero]
| 2,180 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 145 | 146 | theorem exp_zero : exp π (0 : πΈ) = 1 := by |
simp_rw [exp_eq_tsum, β expSeries_apply_eq, expSeries_apply_zero, tsum_pi_single]
| 2,180 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 150 | 151 | theorem exp_op [T2Space πΈ] (x : πΈ) : exp π (MulOpposite.op x) = MulOpposite.op (exp π x) := by |
simp_rw [exp, expSeries_sum_eq, β MulOpposite.op_pow, β MulOpposite.op_smul, tsum_op]
| 2,180 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 155 | 157 | theorem exp_unop [T2Space πΈ] (x : πΈα΅α΅α΅) :
exp π (MulOpposite.unop x) = MulOpposite.unop (exp π x) := by |
simp_rw [exp, expSeries_sum_eq, β MulOpposite.unop_pow, β MulOpposite.unop_smul, tsum_unop]
| 2,180 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 160 | 162 | theorem star_exp [T2Space πΈ] [StarRing πΈ] [ContinuousStar πΈ] (x : πΈ) :
star (exp π x) = exp π (star x) := by |
simp_rw [exp_eq_tsum, β star_pow, β star_inv_natCast_smul, β tsum_star]
| 2,180 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 172 | 175 | theorem _root_.Commute.exp_right [T2Space πΈ] {x y : πΈ} (h : Commute x y) :
Commute x (exp π y) := by |
rw [exp_eq_tsum]
exact Commute.tsum_right x fun n => (h.pow_right n).smul_right _
| 2,180 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import analysis.special_functions.exponential from "leanprover-community/mathlib"@"e1a18cad9cd462973d760af7de36b05776b8811c"
open Filter RCLike ContinuousMultili... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 67 | 72 | theorem hasStrictFDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries π πΈ).radius) :
HasStrictFDerivAt (exp π) (1 : πΈ βL[π] πΈ) 0 := by |
convert (hasFPowerSeriesAt_exp_zero_of_radius_pos h).hasStrictFDerivAt
ext x
change x = expSeries π πΈ 1 fun _ => x
simp [expSeries_apply_eq, Nat.factorial]
| 2,181 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import analysis.special_functions.exponential from "leanprover-community/mathlib"@"e1a18cad9cd462973d760af7de36b05776b8811c"
open Filter RCLike ContinuousMultili... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 220 | 224 | theorem Complex.exp_eq_exp_β : Complex.exp = NormedSpace.exp β := by |
refine funext fun x => ?_
rw [Complex.exp, exp_eq_tsum_div]
have : CauSeq.IsComplete β norm := Complex.instIsComplete
exact tendsto_nhds_unique x.exp'.tendsto_limit (expSeries_div_summable β x).hasSum.tendsto_sum_nat
| 2,181 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import analysis.special_functions.exponential from "leanprover-community/mathlib"@"e1a18cad9cd462973d760af7de36b05776b8811c"
open Filter RCLike ContinuousMultili... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 227 | 228 | theorem Real.exp_eq_exp_β : Real.exp = NormedSpace.exp β := by |
ext x; exact mod_cast congr_fun Complex.exp_eq_exp_β x
| 2,181 |
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter NormedSpace
open scoped Topology
... | Mathlib/Combinatorics/Derangements/Exponential.lean | 24 | 52 | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : β) / n.factorial) atTop (π (Real.exp (-1))) := by |
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k β€ n always
let s : β β β := fun n => β k β Finset.range n, (-1 : β) ^ k / k.factorial
suffices β n : β, (... | 2,182 |
import Mathlib.Analysis.SpecialFunctions.Exponential
#align_import analysis.special_functions.trigonometric.series from "leanprover-community/mathlib"@"ccf84e0d918668460a34aa19d02fe2e0e2286da0"
open NormedSpace
open scoped Nat
section SinCos
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 32 | 46 | theorem Complex.hasSum_cos' (z : β) :
HasSum (fun n : β => (z * Complex.I) ^ (2 * n) / β(2 * n)!) (Complex.cos z) := by |
rw [Complex.cos, Complex.exp_eq_exp_β]
have := ((expSeries_div_hasSum_exp β (z * Complex.I)).add
(expSeries_div_hasSum_exp β (-z * Complex.I))).div_const 2
replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this
dsimp [Function.comp_def] at this
simp_rw [β mul_comm 2 _] at this
refine this.prod_fiberwi... | 2,183 |
import Mathlib.Analysis.SpecialFunctions.Exponential
#align_import analysis.special_functions.trigonometric.series from "leanprover-community/mathlib"@"ccf84e0d918668460a34aa19d02fe2e0e2286da0"
open NormedSpace
open scoped Nat
section SinCos
theorem Complex.hasSum_cos' (z : β) :
HasSum (fun n : β => (z *... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 49 | 64 | theorem Complex.hasSum_sin' (z : β) :
HasSum (fun n : β => (z * Complex.I) ^ (2 * n + 1) / β(2 * n + 1)! / Complex.I)
(Complex.sin z) := by |
rw [Complex.sin, Complex.exp_eq_exp_β]
have := (((expSeries_div_hasSum_exp β (-z * Complex.I)).sub
(expSeries_div_hasSum_exp β (z * Complex.I))).mul_right Complex.I).div_const 2
replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this
dsimp [Function.comp_def] at this
simp_rw [β mul_comm 2 _] at this
re... | 2,183 |
import Mathlib.Analysis.SpecialFunctions.Exponential
#align_import analysis.special_functions.trigonometric.series from "leanprover-community/mathlib"@"ccf84e0d918668460a34aa19d02fe2e0e2286da0"
open NormedSpace
open scoped Nat
section SinCos
theorem Complex.hasSum_cos' (z : β) :
HasSum (fun n : β => (z *... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 68 | 71 | theorem Complex.hasSum_cos (z : β) :
HasSum (fun n : β => (-1) ^ n * z ^ (2 * n) / β(2 * n)!) (Complex.cos z) := by |
convert Complex.hasSum_cos' z using 1
simp_rw [mul_pow, pow_mul, Complex.I_sq, mul_comm]
| 2,183 |
import Mathlib.Analysis.SpecialFunctions.Exponential
#align_import analysis.special_functions.trigonometric.series from "leanprover-community/mathlib"@"ccf84e0d918668460a34aa19d02fe2e0e2286da0"
open NormedSpace
open scoped Nat
section SinCos
theorem Complex.hasSum_cos' (z : β) :
HasSum (fun n : β => (z *... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 75 | 79 | theorem Complex.hasSum_sin (z : β) :
HasSum (fun n : β => (-1) ^ n * z ^ (2 * n + 1) / β(2 * n + 1)!) (Complex.sin z) := by |
convert Complex.hasSum_sin' z using 1
simp_rw [mul_pow, pow_succ, pow_mul, Complex.I_sq, β mul_assoc, mul_div_assoc, div_right_comm,
div_self Complex.I_ne_zero, mul_comm _ ((-1 : β) ^ _), mul_one_div, mul_div_assoc, mul_assoc]
| 2,183 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
variable (π : Type*) {S R M : Type*}
loca... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 83 | 88 | theorem snd_expSeries_of_smul_comm
(x : tsze R M) (hx : MulOpposite.op x.fst β’ x.snd = x.fst β’ x.snd) (n : β) :
snd (expSeries π (tsze R M) (n + 1) fun _ => x) = (expSeries π R n fun _ => x.fst) β’ x.snd := by |
simp_rw [expSeries_apply_eq, snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast π (n + 1),
smul_smul, smul_assoc, Nat.factorial_succ, Nat.pred_succ, Nat.cast_mul, mul_inv_rev,
inv_mul_cancel_rightβ ((Nat.cast_ne_zero (R := π)).mpr <| Nat.succ_ne_zero n)]
| 2,184 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
variable (π : Type*) {S R M : Type*}
loca... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 91 | 100 | theorem hasSum_snd_expSeries_of_smul_comm (x : tsze R M)
(hx : MulOpposite.op x.fst β’ x.snd = x.fst β’ x.snd) {e : R}
(h : HasSum (fun n => expSeries π R n fun _ => x.fst) e) :
HasSum (fun n => snd (expSeries π (tsze R M) n fun _ => x)) (e β’ x.snd) := by |
rw [β hasSum_nat_add_iff' 1]
simp_rw [snd_expSeries_of_smul_comm _ _ hx]
simp_rw [expSeries_apply_eq] at *
rw [Finset.range_one, Finset.sum_singleton, Nat.factorial_zero, Nat.cast_one, pow_zero,
inv_one, one_smul, snd_one, sub_zero]
exact h.smul_const _
| 2,184 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
variable (π : Type*) {S R M : Type*}
loca... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 214 | 217 | theorem norm_def (x : tsze R M) : βxβ = βfst xβ + βsnd xβ := by |
rw [WithLp.prod_norm_eq_add (by norm_num)]
simp only [ENNReal.one_toReal, Real.rpow_one, div_one]
rfl
| 2,184 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
variable (π : Type*) {S R M : Type*}
loca... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 219 | 220 | theorem nnnorm_def (x : tsze R M) : βxββ = βfst xββ + βsnd xββ := by |
ext; simp [norm_def]
| 2,184 |
import Mathlib.Algebra.DualNumber
import Mathlib.Analysis.NormedSpace.TrivSqZeroExt
#align_import analysis.normed_space.dual_number from "leanprover-community/mathlib"@"806c0bb86f6128cfa2f702285727518eb5244390"
open NormedSpace -- For `NormedSpace.exp`.
namespace DualNumber
open TrivSqZeroExt
variable (π : Typ... | Mathlib/Analysis/NormedSpace/DualNumber.lean | 38 | 39 | theorem exp_smul_eps (r : R) : exp π (r β’ eps : DualNumber R) = 1 + r β’ eps := by |
rw [eps, β inr_smul, exp_inr]
| 2,185 |
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open NormedSpace -- For `NormedSpace.exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra β A] [StarRing A] [Continu... | Mathlib/Analysis/NormedSpace/Star/Exponential.lean | 42 | 48 | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by |
ext
have hcomm : Commute (I β’ (a : A)) (I β’ (b : A)) := by
unfold Commute SemiconjBy
simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]
simpa only [expUnitary_coe, AddSubgroup.coe_add, smul_add] using exp_add_of_commute hcomm
| 2,186 |
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open NormedSpace -- For `NormedSpace.exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra β A] [StarRing A] [Continu... | Mathlib/Analysis/NormedSpace/Star/Exponential.lean | 51 | 56 | theorem Commute.expUnitary {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
Commute (expUnitary a) (expUnitary b) :=
calc
selfAdjoint.expUnitary a * selfAdjoint.expUnitary b =
selfAdjoint.expUnitary b * selfAdjoint.expUnitary a := by |
rw [β h.expUnitary_add, β h.symm.expUnitary_add, add_comm]
| 2,186 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.SpecificLimits.Normed
open Filter Finset
open scoped Topology
namespace Complex
section StolzSet
open Real
def stolzSet (M : β) : Set β := {z | βzβ < 1 β§ β1 - zβ < M * (1 - βzβ)}
def stolzCone (s : β) : Set β := {z | |z.im| < s * (1 - z.re)}
| Mathlib/Analysis/Complex/AbelLimit.lean | 47 | 54 | theorem stolzSet_empty {M : β} (hM : M β€ 1) : stolzSet M = β
:= by |
ext z
rw [stolzSet, Set.mem_setOf, Set.mem_empty_iff_false, iff_false, not_and, not_lt, β sub_pos]
intro zn
calc
_ β€ 1 * (1 - βzβ) := mul_le_mul_of_nonneg_right hM zn.le
_ = β(1 : β)β - βzβ := by rw [one_mul, norm_one]
_ β€ _ := norm_sub_norm_le _ _
| 2,187 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.SpecificLimits.Normed
open Filter Finset
open scoped Topology
namespace Complex
section StolzSet
open Real
def stolzSet (M : β) : Set β := {z | βzβ < 1 β§ β1 - zβ < M * (1 - βzβ)}
def stolzCone (s : β) : Set β := {z | |z.im| < s * (1 - z.re)}
th... | Mathlib/Analysis/Complex/AbelLimit.lean | 56 | 66 | theorem nhdsWithin_lt_le_nhdsWithin_stolzSet {M : β} (hM : 1 < M) :
(π[<] 1).map ofReal' β€ π[stolzSet M] 1 := by |
rw [β tendsto_id']
refine tendsto_map' <| tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within ofReal'
(tendsto_nhdsWithin_of_tendsto_nhds <| ofRealCLM.continuous.tendsto' 1 1 rfl) ?_
simp only [eventually_iff, norm_eq_abs, abs_ofReal, abs_lt, mem_nhdsWithin]
refine β¨Set.Ioo 0 2, isOpen_Ioo, by norm_num... | 2,187 |
import Mathlib.Analysis.Complex.AbelLimit
import Mathlib.Analysis.SpecialFunctions.Complex.Arctan
#align_import data.real.pi.leibniz from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Real
open Filter Finset
open scoped Topology
| Mathlib/Data/Real/Pi/Leibniz.lean | 21 | 57 | theorem tendsto_sum_pi_div_four :
Tendsto (fun k => β i β range k, (-1 : β) ^ i / (2 * i + 1)) atTop (π (Ο / 4)) := by |
-- The series is alternating with terms of decreasing magnitude, so it converges to some limit
obtain β¨l, hβ© :
β l, Tendsto (fun n β¦ β i β range n, (-1 : β) ^ i / (2 * i + 1)) atTop (π l) := by
apply Antitone.tendsto_alternating_series_of_tendsto_zero
Β· exact antitone_iff_forall_lt.mpr fun _ _ _ β¦ b... | 2,188 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 124 | 125 | theorem ext_iff' (v w : VectorMeasure Ξ± M) : v = w β β i : Set Ξ±, v i = w i := by |
rw [β coe_injective.eq_iff, Function.funext_iff]
| 2,189 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 128 | 136 | theorem ext_iff (v w : VectorMeasure Ξ± M) : v = w β β i : Set Ξ±, MeasurableSet i β v i = w i := by |
constructor
Β· rintro rfl _ _
rfl
Β· rw [ext_iff']
intro h i
by_cases hi : MeasurableSet i
Β· exact h i hi
Β· simp_rw [not_measurable _ hi]
| 2,189 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 146 | 178 | theorem hasSum_of_disjoint_iUnion [Countable Ξ²] {f : Ξ² β Set Ξ±} (hfβ : β i, MeasurableSet (f i))
(hfβ : Pairwise (Disjoint on f)) : HasSum (fun i => v (f i)) (v (β i, f i)) := by |
cases nonempty_encodable Ξ²
set g := fun i : β => β (b : Ξ²) (_ : b β Encodable.decodeβ Ξ² i), f b with hg
have hgβ : β i, MeasurableSet (g i) :=
fun _ => MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => hfβ b
have hgβ : Pairwise (Disjoint on g) := Encodable.iUnion_decodeβ_disjoint_on hfβ
have :=... | 2,189 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.Order.SymmDiff
#align_import measure_theory.decomposition.signed_hahn from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
noncomputable section
open scoped Classical NNReal ENNReal MeasureTheory
variable {Ξ± Ξ² : Type*} [... | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | 342 | 364 | theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by |
simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
by_contra! h
have h' : β n : β, β y : β, y β s.measureOfNegatives β§ y < -n := fun n => h (-n)
choose f hf using h'
have hf' : β n : β, β B, MeasurableSet B β§ s β€[B] 0 β§ s B < -n := by
intro n
rcases hf n with β¨β¨B, β¨hBβ, hBrβ©, hBββ©, hltβ©
exact β¨B, ... | 2,190 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 135 | 137 | theorem coe_smul (r : ββ₯0) : (r : β) β’ j = r β’ j := by |
-- Porting note: replaced `show`
rw [real_smul_def, if_pos (NNReal.coe_nonneg r), Real.toNNReal_coe]
| 2,191 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 148 | 150 | theorem real_smul_posPart_nonneg (r : β) (hr : 0 β€ r) :
(r β’ j).posPart = r.toNNReal β’ j.posPart := by |
rw [real_smul_def, β smul_posPart, if_pos hr]
| 2,191 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 153 | 155 | theorem real_smul_negPart_nonneg (r : β) (hr : 0 β€ r) :
(r β’ j).negPart = r.toNNReal β’ j.negPart := by |
rw [real_smul_def, β smul_negPart, if_pos hr]
| 2,191 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 158 | 160 | theorem real_smul_posPart_neg (r : β) (hr : r < 0) :
(r β’ j).posPart = (-r).toNNReal β’ j.negPart := by |
rw [real_smul_def, β smul_negPart, if_neg (not_le.2 hr), neg_posPart]
| 2,191 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 163 | 165 | theorem real_smul_negPart_neg (r : β) (hr : r < 0) :
(r β’ j).negPart = (-r).toNNReal β’ j.posPart := by |
rw [real_smul_def, β smul_posPart, if_neg (not_le.2 hr), neg_negPart]
| 2,191 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 242 | 248 | theorem toJordanDecomposition_spec (s : SignedMeasure Ξ±) :
β (i : Set Ξ±) (hiβ : MeasurableSet i) (hiβ : 0 β€[i] s) (hiβ : s β€[iαΆ] 0),
s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hiβ hiβ β§
s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iαΆ hiβ.compl hiβ := by |
set i := s.exists_compl_positive_negative.choose
obtain β¨hiβ, hiβ, hiββ© := s.exists_compl_positive_negative.choose_spec
exact β¨i, hiβ, hiβ, hiβ, rfl, rflβ©
| 2,191 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
#align_import measure_theory.measure.complex from "leanprover-community/mathlib"@"17b3357baa47f48697ca9c243e300eb8cdd16a15"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
variable {Ξ± Ξ² : Type*} {m : MeasurableSpace Ξ±}
namespace Measur... | Mathlib/MeasureTheory/Measure/Complex.lean | 116 | 122 | theorem absolutelyContinuous_ennreal_iff (c : ComplexMeasure Ξ±) (ΞΌ : VectorMeasure Ξ± ββ₯0β) :
c βͺα΅₯ ΞΌ β ComplexMeasure.re c βͺα΅₯ ΞΌ β§ ComplexMeasure.im c βͺα΅₯ ΞΌ := by |
constructor <;> intro h
Β· constructor <;> Β· intro i hi; simp [h hi]
Β· intro i hi
rw [β Complex.re_add_im (c i), (_ : (c i).re = 0), (_ : (c i).im = 0)]
exacts [by simp, h.2 hi, h.1 hi]
| 2,192 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Data.Complex.Determinant
#align_import analysis.complex.operator_norm from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open ContinuousLinearMap
namespace Complex
@[simp... | Mathlib/Analysis/Complex/OperatorNorm.lean | 37 | 41 | theorem reCLM_norm : βreCLMβ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
1 = βreCLM 1β := by | simp
_ β€ βreCLMβ := unit_le_opNorm _ _ (by simp)
| 2,193 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Data.Complex.Determinant
#align_import analysis.complex.operator_norm from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open ContinuousLinearMap
namespace Complex
@[simp... | Mathlib/Analysis/Complex/OperatorNorm.lean | 50 | 54 | theorem imCLM_norm : βimCLMβ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
1 = βimCLM Iβ := by | simp
_ β€ βimCLMβ := unit_le_opNorm _ _ (by simp)
| 2,193 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.FieldTheory.IntermediateField
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.instances.complex from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
section ComplexSubfield
open... | Mathlib/Topology/Instances/Complex.lean | 25 | 44 | theorem Complex.subfield_eq_of_closed {K : Subfield β} (hc : IsClosed (K : Set β)) :
K = ofReal.fieldRange β¨ K = β€ := by |
suffices range (ofReal' : β β β) β K by
rw [range_subset_iff, β coe_algebraMap] at this
have :=
(Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top
(Subfield.toIntermediateField K this).toSubalgebra
simp_rw [β SetLike.coe_set_eq, IntermediateField.coe_toSubalgebra] a... | 2,194 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.FieldTheory.IntermediateField
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.instances.complex from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
section ComplexSubfield
open... | Mathlib/Topology/Instances/Complex.lean | 50 | 116 | theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield β) {Ο : K β+* β}
(hc : UniformContinuous Ο) : Ο.toFun = K.subtype β¨ Ο.toFun = conj β K.subtype := by |
letI : TopologicalDivisionRing β := TopologicalDivisionRing.mk
letI : TopologicalRing K.topologicalClosure :=
Subring.instTopologicalRing K.topologicalClosure.toSubring
set ΞΉ : K β K.topologicalClosure := β(Subfield.inclusion K.le_topologicalClosure)
have ui : UniformInducing ΞΉ :=
β¨by
erw [unifor... | 2,194 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 94 | 95 | theorem interior_setOf_re_le (a : β) : interior { z : β | z.re β€ a } = { z | z.re < a } := by |
simpa only [interior_Iic] using interior_preimage_re (Iic a)
| 2,195 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 99 | 100 | theorem interior_setOf_im_le (a : β) : interior { z : β | z.im β€ a } = { z | z.im < a } := by |
simpa only [interior_Iic] using interior_preimage_im (Iic a)
| 2,195 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 104 | 105 | theorem interior_setOf_le_re (a : β) : interior { z : β | a β€ z.re } = { z | a < z.re } := by |
simpa only [interior_Ici] using interior_preimage_re (Ici a)
| 2,195 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 109 | 110 | theorem interior_setOf_le_im (a : β) : interior { z : β | a β€ z.im } = { z | a < z.im } := by |
simpa only [interior_Ici] using interior_preimage_im (Ici a)
| 2,195 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 114 | 115 | theorem closure_setOf_re_lt (a : β) : closure { z : β | z.re < a } = { z | z.re β€ a } := by |
simpa only [closure_Iio] using closure_preimage_re (Iio a)
| 2,195 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 119 | 120 | theorem closure_setOf_im_lt (a : β) : closure { z : β | z.im < a } = { z | z.im β€ a } := by |
simpa only [closure_Iio] using closure_preimage_im (Iio a)
| 2,195 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 124 | 125 | theorem closure_setOf_lt_re (a : β) : closure { z : β | a < z.re } = { z | a β€ z.re } := by |
simpa only [closure_Ioi] using closure_preimage_re (Ioi a)
| 2,195 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 129 | 130 | theorem closure_setOf_lt_im (a : β) : closure { z : β | a < z.im } = { z | a β€ z.im } := by |
simpa only [closure_Ioi] using closure_preimage_im (Ioi a)
| 2,195 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 134 | 135 | theorem frontier_setOf_re_le (a : β) : frontier { z : β | z.re β€ a } = { z | z.re = a } := by |
simpa only [frontier_Iic] using frontier_preimage_re (Iic a)
| 2,195 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 139 | 140 | theorem frontier_setOf_im_le (a : β) : frontier { z : β | z.im β€ a } = { z | z.im = a } := by |
simpa only [frontier_Iic] using frontier_preimage_im (Iic a)
| 2,195 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.Analysis.Convex.Contractible
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Complex
import Mathlib.Analysis.Complex.ReImTopology
import Mathlib.Topology.Homotopy.Contractible
import Mathlib.Topology.PartialHomeomorph
#align_impo... | Mathlib/Analysis/Complex/UpperHalfPlane/Topology.lean | 109 | 124 | theorem ModularGroup_T_zpow_mem_verticalStrip (z : β) {N : β} (hn : 0 < N) :
β n : β€, ModularGroup.T ^ (N * n) β’ z β verticalStrip N z.im := by |
let n := Int.floor (z.re/N)
use -n
rw [modular_T_zpow_smul z (N * -n)]
refine β¨?_, (by simp only [mul_neg, Int.cast_neg, Int.cast_mul, Int.cast_natCast, vadd_im,
le_refl])β©
have h : (N * (-n : β) +α΅₯ z).re = -N * Int.floor (z.re / N) + z.re := by
simp only [Int.cast_natCast, mul_neg, vadd_re, neg_mul]... | 2,196 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 45 | 47 | theorem sinh_half_dist (z w : β) :
sinh (dist z w / 2) = dist (z : β) w / (2 * β(z.im * w.im)) := by |
rw [dist_eq, mul_div_cancel_leftβ (arsinh _) two_ne_zero, sinh_arsinh]
| 2,197 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 50 | 57 | theorem cosh_half_dist (z w : β) :
cosh (dist z w / 2) = dist (z : β) (conj (w : β)) / (2 * β(z.im * w.im)) := by |
rw [β sq_eq_sq, cosh_sq', sinh_half_dist, div_pow, div_pow, one_add_div, mul_pow, sq_sqrt]
Β· congr 1
simp only [Complex.dist_eq, Complex.sq_abs, Complex.normSq_sub, Complex.normSq_conj,
Complex.conj_conj, Complex.mul_re, Complex.conj_re, Complex.conj_im, coe_im]
ring
all_goals positivity
| 2,197 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 60 | 63 | theorem tanh_half_dist (z w : β) :
tanh (dist z w / 2) = dist (z : β) w / dist (z : β) (conj βw) := by |
rw [tanh_eq_sinh_div_cosh, sinh_half_dist, cosh_half_dist, div_div_div_comm, div_self, div_one]
positivity
| 2,197 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 66 | 68 | theorem exp_half_dist (z w : β) :
exp (dist z w / 2) = (dist (z : β) w + dist (z : β) (conj βw)) / (2 * β(z.im * w.im)) := by |
rw [β sinh_add_cosh, sinh_half_dist, cosh_half_dist, add_div]
| 2,197 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 71 | 73 | theorem cosh_dist (z w : β) : cosh (dist z w) = 1 + dist (z : β) w ^ 2 / (2 * z.im * w.im) := by |
rw [dist_eq, cosh_two_mul, cosh_sq', add_assoc, β two_mul, sinh_arsinh, div_pow, mul_pow,
sq_sqrt, sq (2 : β), mul_assoc, β mul_div_assoc, mul_assoc, mul_div_mul_left] <;> positivity
| 2,197 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 76 | 84 | theorem sinh_half_dist_add_dist (a b c : β) : sinh ((dist a b + dist b c) / 2) =
(dist (a : β) b * dist (c : β) (conj βb) + dist (b : β) c * dist (a : β) (conj βb)) /
(2 * β(a.im * c.im) * dist (b : β) (conj βb)) := by |
simp only [add_div _ _ (2 : β), sinh_add, sinh_half_dist, cosh_half_dist, div_mul_div_comm]
rw [β add_div, Complex.dist_self_conj, coe_im, abs_of_pos b.im_pos, mul_comm (dist (b : β) _),
dist_comm (b : β), Complex.dist_conj_comm, mul_mul_mul_comm, mul_mul_mul_comm _ _ _ b.im]
congr 2
rw [sqrt_mul, sqrt_mul... | 2,197 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 91 | 93 | theorem dist_le_iff_le_sinh :
dist z w β€ r β dist (z : β) w / (2 * β(z.im * w.im)) β€ sinh (r / 2) := by |
rw [β div_le_div_right (zero_lt_two' β), β sinh_le_sinh, sinh_half_dist]
| 2,197 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 96 | 98 | theorem dist_eq_iff_eq_sinh :
dist z w = r β dist (z : β) w / (2 * β(z.im * w.im)) = sinh (r / 2) := by |
rw [β div_left_inj' (two_ne_zero' β), β sinh_inj, sinh_half_dist]
| 2,197 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 101 | 105 | theorem dist_eq_iff_eq_sq_sinh (hr : 0 β€ r) :
dist z w = r β dist (z : β) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 := by |
rw [dist_eq_iff_eq_sinh, β sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
Β· norm_num
all_goals positivity
| 2,197 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 211 | 213 | theorem inner_self_im (x : F) : im βͺx, xβ« = 0 := by |
rw [β @ofReal_inj π, im_eq_conj_sub]
simp [inner_conj_symm]
| 2,198 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 220 | 221 | theorem inner_add_right (x y z : F) : βͺx, y + zβ« = βͺx, yβ« + βͺx, zβ« := by |
rw [β inner_conj_symm, inner_add_left, RingHom.map_add]; simp only [inner_conj_symm]
| 2,198 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 224 | 226 | theorem ofReal_normSq_eq_inner_self (x : F) : (normSqF x : π) = βͺx, xβ« := by |
rw [ext_iff]
exact β¨by simp only [ofReal_re]; rfl, by simp only [inner_self_im, ofReal_im]β©
| 2,198 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 229 | 229 | theorem inner_re_symm (x y : F) : re βͺx, yβ« = re βͺy, xβ« := by | rw [β inner_conj_symm, conj_re]
| 2,198 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 232 | 232 | theorem inner_im_symm (x y : F) : im βͺx, yβ« = -im βͺy, xβ« := by | rw [β inner_conj_symm, conj_im]
| 2,198 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 239 | 241 | theorem inner_smul_right (x y : F) {r : π} : βͺx, r β’ yβ« = r * βͺx, yβ« := by |
rw [β inner_conj_symm, inner_smul_left];
simp only [conj_conj, inner_conj_symm, RingHom.map_mul]
| 2,198 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
#align_import analysis.complex.arg from "leanprover-community/mathlib"@"45a46f4f03f8ae41491bf3605e8e0e363ba192fd"
variable {x y : β}
namespace Complex
| Mathlib/Analysis/Complex/Arg.lean | 31 | 38 | theorem sameRay_iff : SameRay β x y β x = 0 β¨ y = 0 β¨ x.arg = y.arg := by |
rcases eq_or_ne x 0 with (rfl | hx)
Β· simp
rcases eq_or_ne y 0 with (rfl | hy)
Β· simp
simp only [hx, hy, false_or_iff, sameRay_iff_norm_smul_eq, arg_eq_arg_iff hx hy]
field_simp [hx, hy]
rw [mul_comm, eq_comm]
| 2,199 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
#align_import analysis.complex.arg from "leanprover-community/mathlib"@"45a46f4f03f8ae41491bf3605e8e0e363ba192fd"
variable {x y : β}
namespace Complex
theorem sameRay_iff : SameRay β x y β x = 0 β¨ y = 0 β¨ x.arg ... | Mathlib/Analysis/Complex/Arg.lean | 41 | 45 | theorem sameRay_iff_arg_div_eq_zero : SameRay β x y β arg (x / y) = 0 := by |
rw [β Real.Angle.toReal_zero, β arg_coe_angle_eq_iff_eq_toReal, sameRay_iff]
by_cases hx : x = 0; Β· simp [hx]
by_cases hy : y = 0; Β· simp [hy]
simp [hx, hy, arg_div_coe_angle, sub_eq_zero]
| 2,199 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 29 | 32 | theorem eval_one_cyclotomic_prime {R : Type*} [CommRing R] {p : β} [hn : Fact p.Prime] :
eval 1 (cyclotomic p R) = p := by |
simp only [cyclotomic_prime, eval_X, one_pow, Finset.sum_const, eval_pow, eval_finset_sum,
Finset.card_range, smul_one_eq_cast]
| 2,200 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 36 | 37 | theorem evalβ_one_cyclotomic_prime {R S : Type*} [CommRing R] [Semiring S] (f : R β+* S) {p : β}
[Fact p.Prime] : evalβ f 1 (cyclotomic p R) = p := by | simp
| 2,200 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 41 | 44 | theorem eval_one_cyclotomic_prime_pow {R : Type*} [CommRing R] {p : β} (k : β)
[hn : Fact p.Prime] : eval 1 (cyclotomic (p ^ (k + 1)) R) = p := by |
simp only [cyclotomic_prime_pow_eq_geom_sum hn.out, eval_X, one_pow, Finset.sum_const, eval_pow,
eval_finset_sum, Finset.card_range, smul_one_eq_cast]
| 2,200 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 48 | 49 | theorem evalβ_one_cyclotomic_prime_pow {R S : Type*} [CommRing R] [Semiring S] (f : R β+* S)
{p : β} (k : β) [Fact p.Prime] : evalβ f 1 (cyclotomic (p ^ (k + 1)) R) = p := by | simp
| 2,200 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 70 | 111 | theorem cyclotomic_pos {n : β} (hn : 2 < n) {R} [LinearOrderedCommRing R] (x : R) :
0 < eval x (cyclotomic n R) := by |
induction' n using Nat.strong_induction_on with n ih
have hn' : 0 < n := pos_of_gt hn
have hn'' : 1 < n := one_lt_two.trans hn
have := prod_cyclotomic_eq_geom_sum hn' R
apply_fun eval x at this
rw [β cons_self_properDivisors hn'.ne', Finset.erase_cons_of_ne _ hn''.ne', Finset.prod_cons,
eval_mul, eval_... | 2,200 |
import Mathlib.Data.PNat.Prime
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import number_theo... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 92 | 95 | theorem aeval_zeta [IsDomain B] [NeZero ((n : β) : B)] :
aeval (zeta n A B) (cyclotomic n A) = 0 := by |
rw [aeval_def, β eval_map, β IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff]
exact zeta_spec n A B
| 2,201 |
import Mathlib.Data.PNat.Prime
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import number_theo... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 98 | 100 | theorem zeta_isRoot [IsDomain B] [NeZero ((n : β) : B)] : IsRoot (cyclotomic n B) (zeta n A B) := by |
convert aeval_zeta n A B using 0
rw [IsRoot.def, aeval_def, evalβ_eq_eval_map, map_cyclotomic]
| 2,201 |
import Mathlib.Data.PNat.Prime
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import number_theo... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 128 | 131 | theorem powerBasis_gen_mem_adjoin_zeta_sub_one :
(hΞΆ.powerBasis K).gen β adjoin K ({ΞΆ - 1} : Set L) := by |
rw [powerBasis_gen, adjoin_singleton_eq_range_aeval, AlgHom.mem_range]
exact β¨X + 1, by simpβ©
| 2,201 |
import Mathlib.Data.PNat.Prime
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import number_theo... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 289 | 291 | theorem norm_eq_neg_one_pow (hΞΆ : IsPrimitiveRoot ΞΆ 2) [IsDomain L] :
norm K ΞΆ = (-1 : K) ^ finrank K L := by |
rw [hΞΆ.eq_neg_one_of_two_right, show -1 = algebraMap K L (-1) by simp, Algebra.norm_algebraMap]
| 2,201 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
#align_import number_theory.primes_congruent_one from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
namespace Nat
open Polynomial Nat Filter
open scoped Nat
| Mathlib/NumberTheory/PrimesCongruentOne.lean | 26 | 57 | theorem exists_prime_gt_modEq_one {k : β} (n : β) (hk0 : k β 0) :
β p : β, Nat.Prime p β§ n < p β§ p β‘ 1 [MOD k] := by |
rcases (one_le_iff_ne_zero.2 hk0).eq_or_lt with (rfl | hk1)
Β· rcases exists_infinite_primes (n + 1) with β¨p, hnp, hpβ©
exact β¨p, hp, hnp, modEq_oneβ©
let b := k * (n !)
have hgt : 1 < (eval (βb) (cyclotomic k β€)).natAbs := by
rcases le_iff_exists_add'.1 hk1.le with β¨k, rflβ©
have hb : 2 β€ b := le_mul_... | 2,202 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
#align_import number_theory.primes_congruent_one from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
namespace Nat
open Polynomial Nat Filter
open scoped Nat
theorem exists_prime_gt_modEq_one {k : β} (n : β) (hk0 : k β 0) :
β ... | Mathlib/NumberTheory/PrimesCongruentOne.lean | 60 | 64 | theorem frequently_atTop_modEq_one {k : β} (hk0 : k β 0) :
βαΆ p in atTop, Nat.Prime p β§ p β‘ 1 [MOD k] := by |
refine frequently_atTop.2 fun n => ?_
obtain β¨p, hpβ© := exists_prime_gt_modEq_one n hk0
exact β¨p, β¨hp.2.1.le, hp.1, hp.2.2β©β©
| 2,202 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 57 | 67 | theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace π E]
[SecondCountableTopology E] {f : Ξ± β E} (hf : β c : E, (fun x => (inner c (f x) : π)) =α΅[ΞΌ] 0) :
f =α΅[ΞΌ] 0 := by |
let s := denseSeq E
have hs : DenseRange s := denseRange_denseSeq E
have hf' : βα΅ x βΞΌ, β n : β, inner (s n) (f x) = (0 : π) := ae_all_iff.mpr fun n => hf (s n)
refine hf'.mono fun x hx => ?_
rw [Pi.zero_apply, β @inner_self_eq_zero π]
have h_closed : IsClosed {c : E | inner c (f x) = (0 : π)} :=
is... | 2,203 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 74 | 106 | theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedSpace π E]
{t : Set E} (ht : TopologicalSpace.IsSeparable t) {f : Ξ± β E}
(hf : β c : Dual π E, (fun x => βͺf x, cβ«) =α΅[ΞΌ] 0) (h't : βα΅ x βΞΌ, f x β t) : f =α΅[ΞΌ] 0 := by |
rcases ht with β¨d, d_count, hdβ©
haveI : Encodable d := d_count.toEncodable
have : β x : d, β g : E βL[π] π, βgβ β€ 1 β§ g x = β(x : E)β :=
fun x => exists_dual_vector'' π (x : E)
choose s hs using this
have A : β a : E, a β t β (β x, βͺa, s xβ« = (0 : π)) β a = 0 := by
intro a hat ha
contrapose! ... | 2,203 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 125 | 157 | theorem ae_const_le_iff_forall_lt_measure_zero {Ξ²} [LinearOrder Ξ²] [TopologicalSpace Ξ²]
[OrderTopology Ξ²] [FirstCountableTopology Ξ²] (f : Ξ± β Ξ²) (c : Ξ²) :
(βα΅ x βΞΌ, c β€ f x) β β b < c, ΞΌ {x | f x β€ b} = 0 := by |
rw [ae_iff]
push_neg
constructor
Β· intro h b hb
exact measure_mono_null (fun y hy => (lt_of_le_of_lt hy hb : _)) h
intro hc
by_cases h : β b, c β€ b
Β· have : {a : Ξ± | f a < c} = β
:= by
apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_
exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x))))... | 2,203 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 164 | 221 | theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite ΞΌ] {f g : Ξ± β ββ₯0β}
(hf : Measurable f) (hg : Measurable g)
(h : β s, MeasurableSet s β ΞΌ s < β β (β«β» x in s, f x βΞΌ) β€ β«β» x in s, g x βΞΌ) : f β€α΅[ΞΌ] g := by |
have A :
β (Ξ΅ N : ββ₯0) (p : β), 0 < Ξ΅ β ΞΌ ({x | g x + Ξ΅ β€ f x β§ g x β€ N} β© spanningSets ΞΌ p) = 0 := by
intro Ξ΅ N p Ξ΅pos
let s := {x | g x + Ξ΅ β€ f x β§ g x β€ N} β© spanningSets ΞΌ p
have s_meas : MeasurableSet s := by
have A : MeasurableSet {x | g x + Ξ΅ β€ f x} := measurableSet_le (hg.add measurable... | 2,203 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 260 | 284 | theorem ae_nonneg_of_forall_setIntegral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
(hf : Integrable f ΞΌ) (hf_zero : β s, MeasurableSet s β ΞΌ s < β β 0 β€ β« x in s, f x βΞΌ) :
0 β€α΅[ΞΌ] f := by |
simp_rw [EventuallyLE, Pi.zero_apply]
rw [ae_const_le_iff_forall_lt_measure_zero]
intro b hb_neg
let s := {x | f x β€ b}
have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
have mus : ΞΌ s < β := Integrable.measure_le_lt_top hf hb_neg
have h_int_gt : (β« x in s, f x βΞΌ) β€ b * (ΞΌ s).to... | 2,203 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 291 | 302 | theorem ae_nonneg_of_forall_setIntegral_nonneg (hf : Integrable f ΞΌ)
(hf_zero : β s, MeasurableSet s β ΞΌ s < β β 0 β€ β« x in s, f x βΞΌ) : 0 β€α΅[ΞΌ] f := by |
rcases hf.1 with β¨f', hf'_meas, hf_aeβ©
have hf'_integrable : Integrable f' ΞΌ := Integrable.congr hf hf_ae
have hf'_zero : β s, MeasurableSet s β ΞΌ s < β β 0 β€ β« x in s, f' x βΞΌ := by
intro s hs h's
rw [setIntegral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)]
exact hf_zero s hs h's
exact
(ae_... | 2,203 |
import Mathlib.Geometry.Manifold.PartitionOfUnity
import Mathlib.Geometry.Manifold.Metrizable
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
open MeasureTheory Filter Metric Function Set TopologicalSpace
open scoped Topology Manifold
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace β E] [FiniteDimen... | Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean | 41 | 112 | theorem ae_eq_zero_of_integral_smooth_smul_eq_zero (hf : LocallyIntegrable f ΞΌ)
(h : β g : M β β, Smooth I π(β) g β HasCompactSupport g β β« x, g x β’ f x βΞΌ = 0) :
βα΅ x βΞΌ, f x = 0 := by |
-- record topological properties of `M`
have := I.locallyCompactSpace
have := ChartedSpace.locallyCompactSpace H M
have := I.secondCountableTopology
have := ChartedSpace.secondCountable_of_sigma_compact H M
have := ManifoldWithCorners.metrizableSpace I M
let _ : MetricSpace M := TopologicalSpace.metrizab... | 2,204 |
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Mea... | Mathlib/Analysis/Convex/Integral.lean | 56 | 81 | theorem Convex.integral_mem [IsProbabilityMeasure ΞΌ] (hs : Convex β s) (hsc : IsClosed s)
(hf : βα΅ x βΞΌ, f x β s) (hfi : Integrable f ΞΌ) : (β« x, f x βΞΌ) β s := by |
borelize E
rcases hfi.aestronglyMeasurable with β¨g, hgm, hfgβ©
haveI : SeparableSpace (range g β© s : Set E) :=
(hgm.isSeparable_range.mono inter_subset_left).separableSpace
obtain β¨yβ, hββ© : (range g β© s).Nonempty := by
rcases (hf.and hfg).exists with β¨xβ, hββ©
exact β¨f xβ, by simp only [hβ.2, mem_ra... | 2,205 |
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Mea... | Mathlib/Analysis/Convex/Integral.lean | 87 | 90 | theorem Convex.average_mem [IsFiniteMeasure ΞΌ] [NeZero ΞΌ] (hs : Convex β s) (hsc : IsClosed s)
(hfs : βα΅ x βΞΌ, f x β s) (hfi : Integrable f ΞΌ) : (β¨ x, f x βΞΌ) β s := by |
refine hs.integral_mem hsc (ae_mono' ?_ hfs) hfi.to_average
exact AbsolutelyContinuous.smul (refl _) _
| 2,205 |
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Mea... | Mathlib/Analysis/Convex/Integral.lean | 112 | 119 | theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure ΞΌ] [NeZero ΞΌ] (hg : ConvexOn β s g)
(hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : βα΅ x βΞΌ, f x β s)
(hfi : Integrable f ΞΌ) (hgi : Integrable (g β f) ΞΌ) :
(β¨ x, f x βΞΌ, β¨ x, g (f x) βΞΌ) β {p : E Γ β | p.1 β s β§ g p.1 β€ p.2} := by |
have ht_mem : βα΅ x βΞΌ, (f x, g (f x)) β {p : E Γ β | p.1 β s β§ g p.1 β€ p.2} :=
hfs.mono fun x hx => β¨hx, le_rflβ©
exact average_pair hfi hgi βΈ
hg.convex_epigraph.average_mem (hsc.epigraph hgc) ht_mem (hfi.prod_mk hgi)
| 2,205 |
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Mea... | Mathlib/Analysis/Convex/Integral.lean | 122 | 127 | theorem ConcaveOn.average_mem_hypograph [IsFiniteMeasure ΞΌ] [NeZero ΞΌ] (hg : ConcaveOn β s g)
(hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : βα΅ x βΞΌ, f x β s)
(hfi : Integrable f ΞΌ) (hgi : Integrable (g β f) ΞΌ) :
(β¨ x, f x βΞΌ, β¨ x, g (f x) βΞΌ) β {p : E Γ β | p.1 β s β§ p.2 β€ g p.1} := by |
simpa only [mem_setOf_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
hg.neg.average_mem_epigraph hgc.neg hsc hfs hfi hgi.neg
| 2,205 |
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.decomposition.lebesgue from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open scoped MeasureTheory NNReal ENN... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 86 | 90 | theorem haveLebesgueDecomposition_spec (ΞΌ Ξ½ : Measure Ξ±) [h : HaveLebesgueDecomposition ΞΌ Ξ½] :
Measurable (ΞΌ.rnDeriv Ξ½) β§
ΞΌ.singularPart Ξ½ ββ Ξ½ β§ ΞΌ = ΞΌ.singularPart Ξ½ + Ξ½.withDensity (ΞΌ.rnDeriv Ξ½) := by |
rw [singularPart, rnDeriv, dif_pos h, dif_pos h]
exact Classical.choose_spec h.lebesgue_decomposition
| 2,206 |
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.decomposition.lebesgue from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open scoped MeasureTheory NNReal ENN... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 102 | 106 | theorem measurable_rnDeriv (ΞΌ Ξ½ : Measure Ξ±) : Measurable <| ΞΌ.rnDeriv Ξ½ := by |
by_cases h : HaveLebesgueDecomposition ΞΌ Ξ½
Β· exact (haveLebesgueDecomposition_spec ΞΌ Ξ½).1
Β· rw [rnDeriv_of_not_haveLebesgueDecomposition h]
exact measurable_zero
| 2,206 |
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.decomposition.lebesgue from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open scoped MeasureTheory NNReal ENN... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 109 | 113 | theorem mutuallySingular_singularPart (ΞΌ Ξ½ : Measure Ξ±) : ΞΌ.singularPart Ξ½ ββ Ξ½ := by |
by_cases h : HaveLebesgueDecomposition ΞΌ Ξ½
Β· exact (haveLebesgueDecomposition_spec ΞΌ Ξ½).2.1
Β· rw [singularPart_of_not_haveLebesgueDecomposition h]
exact MutuallySingular.zero_left
| 2,206 |
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Decomposition.Lebesgue
#align_import measure... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 97 | 113 | theorem ae_eventually_measure_pos [SecondCountableTopology Ξ±] :
βα΅ x βΞΌ, βαΆ a in v.filterAt x, 0 < ΞΌ a := by |
set s := {x | Β¬βαΆ a in v.filterAt x, 0 < ΞΌ a} with hs
simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
change ΞΌ s = 0
let f : Ξ± β Set (Set Ξ±) := fun _ => {a | ΞΌ a = 0}
have h : v.FineSubfamilyOn f s := by
intro x hx Ξ΅ Ξ΅pos
rw [hs] at hx
simp only [frequen... | 2,207 |
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Decomposition.Lebesgue
#align_import measure... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 125 | 149 | theorem measure_le_of_frequently_le [SecondCountableTopology Ξ±] [BorelSpace Ξ±] {Ο : Measure Ξ±}
(Ξ½ : Measure Ξ±) [IsLocallyFiniteMeasure Ξ½] (hΟ : Ο βͺ ΞΌ) (s : Set Ξ±)
(hs : β x β s, βαΆ a in v.filterAt x, Ο a β€ Ξ½ a) : Ο s β€ Ξ½ s := by |
-- this follows from a covering argument using the sets satisfying `Ο a β€ Ξ½ a`.
apply ENNReal.le_of_forall_pos_le_add fun Ξ΅ Ξ΅pos _ => ?_
obtain β¨U, sU, U_open, Ξ½Uβ© : β (U : Set Ξ±), s β U β§ IsOpen U β§ Ξ½ U β€ Ξ½ s + Ξ΅ :=
exists_isOpen_le_add s Ξ½ (ENNReal.coe_pos.2 Ξ΅pos).ne'
let f : Ξ± β Set (Set Ξ±) := fun _ => ... | 2,207 |
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Decomposition.Lebesgue
#align_import measure... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 160 | 201 | theorem ae_eventually_measure_zero_of_singular (hΟ : Ο ββ ΞΌ) :
βα΅ x βΞΌ, Tendsto (fun a => Ο a / ΞΌ a) (v.filterAt x) (π 0) := by |
have A : β Ξ΅ > (0 : ββ₯0), βα΅ x βΞΌ, βαΆ a in v.filterAt x, Ο a < Ξ΅ * ΞΌ a := by
intro Ξ΅ Ξ΅pos
set s := {x | Β¬βαΆ a in v.filterAt x, Ο a < Ξ΅ * ΞΌ a} with hs
change ΞΌ s = 0
obtain β¨o, _, Οo, ΞΌoβ© : β o : Set Ξ±, MeasurableSet o β§ Ο o = 0 β§ ΞΌ oαΆ = 0 := hΟ
apply le_antisymm _ bot_le
calc
ΞΌ s β€ ΞΌ (s... | 2,207 |
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Data.Set.Pairwise.Lat... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 187 | 192 | theorem inter' (i : Fin N.succ) : dist (a.c i) (a.c (last N)) β€ a.r i + a.r (last N) := by |
rcases lt_or_le i (last N) with (H | H)
Β· exact a.inter i H
Β· have I : i = last N := top_le_iff.1 H
have := (a.rpos (last N)).le
simp only [I, add_nonneg this this, dist_self]
| 2,208 |
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Data.Set.Pairwise.Lat... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 195 | 200 | theorem hlast' (i : Fin N.succ) (h : 1 β€ Ο) : a.r (last N) β€ Ο * a.r i := by |
rcases lt_or_le i (last N) with (H | H)
Β· exact (a.hlast i H).2
Β· have : i = last N := top_le_iff.1 H
rw [this]
exact le_mul_of_one_le_left (a.rpos _).le h
| 2,208 |
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Data.Set.Pairwise.Lat... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 278 | 281 | theorem monotone_iUnionUpTo : Monotone p.iUnionUpTo := by |
intro i j hij
simp only [iUnionUpTo]
exact iUnion_mono' fun r => β¨β¨r, r.2.trans_le hijβ©, Subset.rflβ©
| 2,208 |
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
#align_import measure_theory.covering.density_theorem from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filt... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 71 | 109 | theorem closedBall_mem_vitaliFamily_of_dist_le_mul {K : β} {x y : Ξ±} {r : β} (h : dist x y β€ K * r)
(rpos : 0 < r) : closedBall y r β (vitaliFamily ΞΌ K).setsAt x := by |
let R := scalingScaleOf ΞΌ (max (4 * K + 3) 3)
simp only [vitaliFamily, VitaliFamily.enlarge, Vitali.vitaliFamily, mem_union, mem_setOf_eq,
isClosed_ball, true_and_iff, (nonempty_ball.2 rpos).mono ball_subset_interior_closedBall,
measurableSet_closedBall]
/- The measure is doubling on scales smaller than ... | 2,209 |
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