Documentation Verification Report

log

📁 Source: MathlibTest/Nat/log.lean

Statistics

MetricCount
Definitionslog, log, log, log, log, log, log, log, log, log
10
Theoremslog, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log, log
32
Total42

AEMeasurable

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalAEMeasurable
Real
Real.measurableSpace
Real.logMeasurable.comp_aemeasurable
Real.measurable_log

AnalyticAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalAnalyticAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
Real.instLT
Real.instZero
Real.logcomp
analyticAt_log

AnalyticOn

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalAnalyticOn
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
Real.instLT
Real.instZero
Real.logAnalyticWithinAt.comp
AnalyticAt.analyticWithinAt
analyticAt_log

AnalyticOnNhd

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalAnalyticOnNhd
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
Real.instLT
Real.instZero
Real.logAnalyticAt.comp
analyticAt_log

AnalyticWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalAnalyticWithinAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
Real.instLT
Real.instZero
Real.logAnalyticAt.comp_analyticWithinAt
analyticAt_log

ArithmeticFunction

Definitions

NameCategoryTheorems
log 📖CompOp
6 mathmath: log_apply, log_mul_moebius_eq_vonMangoldt, moebius_mul_log_eq_vonMangoldt, sum_moebius_mul_log_eq, vonMangoldt_mul_zeta, zeta_mul_vonMangoldt

Asymptotics.IsEquivalent

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalAsymptotics.IsEquivalent
Real
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Filter.Tendsto
Filter.atTop
Real.instPreorder
Real.logFilter.Tendsto.eventually_ne_atTop
instNoTopOrderOfNoMaxOrder
instNoMaxOrderOfNontrivial
Real.instIsOrderedRing
Real.instNontrivial
tendsto_atTop
Real.instIsStrictOrderedRing
instOrderTopologyReal
symm
Filter.Tendsto.congr'
Filter.mp_mem
Filter.univ_mem'
Real.log_div
Filter.Tendsto.log
Asymptotics.isEquivalent_iff_tendsto_one
Mathlib.Meta.NormNum.isNat_eq_false
RCLike.charZero_rclike
Mathlib.Meta.NormNum.isNat_ofNat
Nat.cast_one
Nat.cast_zero
Asymptotics.IsLittleO.isEquivalent
NormedDivisionRing.to_normOneClass
Real.log_one
Asymptotics.isLittleO_one_left_iff
Filter.Tendsto.comp
tendsto_norm_atTop_atTop
Real.tendsto_log_atTop

CFC

Definitions

NameCategoryTheorems
log 📖CompOp
14 mathmath: tendsto_cfc_rpow_sub_one_log, exp_log, continuousOn_log, IsSelfAdjoint.log, log_monotoneOn, log_pow', log_zero, log_smul', log_pow, log_le_log, log_one, log_exp, log_algebraMap, log_smul

Complex

Definitions

NameCategoryTheorems
log 📖CompOp
123 mathmath: log_one, log_conj_eq_ite, HasDerivWithinAt.clog_real, norm_log_one_sub_inv_add_logTaylor_neg_le, ofNat_log, tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero, log_I, log_neg_one, log_re, continuousWithinAt_log_of_re_neg_of_im_zero, HasStrictDerivAt.cpow, DifferentiableAt.clog, digamma_one_half, tendsto_mul_log_one_add_of_tendsto, norm_log_one_add_le, HasDerivAt.cpow, log_exp_eq_re_add_toIocMod, log_inv_eq_ite, hasStrictFDerivAt_cpow', ofReal_log, log_inv, natCast_log, norm_log_sub_logTaylor_le, hasDerivAt_Gammaℝ_one, HasDerivWithinAt.const_cpow, AnalyticAt.clog, contDiffAt_log, norm_log_one_add_sub_self_le, hasStrictFDerivAt_cpow, ZetaAsymptotics.tendsto_Gamma_term_aux, HasStrictDerivAt.clog_real, HasFDerivWithinAt.cpow, HasStrictFDerivAt.const_cpow, log_ofReal_mul, riemannZeta_eulerProduct_exp_log, AnalyticOn.clog, log_exp_eq_sub_toIocDiv, HasStrictFDerivAt.clog, Filter.Tendsto.clog, AnalyticOnNhd.clog, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_Gammaℝ, hasDerivAt_Gammaℂ_one, hasStrictDerivAt_const_cpow, ArithmeticFunction.convolution_vonMangoldt_const_one, HasDerivWithinAt.cpow, tendsto_nat_mul_log_one_add_of_tendsto, ArithmeticFunction.convolution_vonMangoldt_zeta, hasFDerivAt_cpow, HasFDerivAt.clog, DirichletCharacter.summable_neg_log_one_sub_mul_prime_cpow, continuousAt_clog, log_sub_self_isBigO, hasStrictDerivAt_log, HasDerivAt.clog, log_mul_eq_add_log_iff, Summable.clog_one_sub, DirichletCharacter.convolution_twist_vonMangoldt, neg_pi_lt_log_im, HasDerivWithinAt.clog, Continuous.clog, log_exp, HasFDerivWithinAt.clog, riemannZeta_one, log_div_self, hasSum_arctan_aux, deriv_const_cpow, log_im, HasStrictDerivAt.const_cpow, log_mul, completedRiemannZeta₀_one, DirichletCharacter.LSeries_eulerProduct_exp_log, HasFDerivWithinAt.const_cpow, derivWithin_const_cpow, DifferentiableWithinAt.clog, hasSum_taylorSeries_neg_log, log_eq_integral, HasStrictFDerivAt.cpow, measurable_log, log_sub_logTaylor_isBigO, summable_log_one_add_of_summable, ArithmeticFunction.LSeries_zeta_eulerProduct_exp_log, cpow_eq_nhds', cpow_def, hasDerivAt_Gamma_one_half, differentiableAt_log, norm_log_one_sub_inv_sub_self_le, hasSum_taylorSeries_log, Differentiable.clog, analyticAt_clog, DifferentiableOn.clog, norm_log_natCast_le_rpow_div, completedRiemannZeta_one, hasStrictFDerivAt_log_real, ContinuousOn.clog, log_neg_I, log_mul_ofReal, ContinuousAt.clog, cpow_eq_nhds, tendsto_log_nhdsWithin_im_nonneg_of_re_neg_of_im_zero, norm_log_one_add_half_le_self, AnalyticWithinAt.clog, cpow_def_of_ne_zero, log_im_le_pi, HasFDerivAt.cpow, exp_log, hasDerivAt_log_sub_logTaylor, Measurable.clog, ContinuousWithinAt.clog, HasDerivAt.const_cpow, hasDerivAt_log, log_zero, log_inv_eq_integral, log_ofReal_re, AEMeasurable.clog, log_conj, HasDerivAt.clog_real, log_exp_exists, Summable.hasSumUniformlyOn_log_one_add, deriv_log_comp_eq_logDeriv, EulerProduct.exp_tsum_primes_log_eq_tsum, HasFDerivAt.const_cpow, HasStrictDerivAt.clog, Summable.tendstoUniformlyOn_tsum_nat_log_one_add

ContDiff

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalContDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
NormedField.toNormedSpace
NontriviallyNormedField.toNormedField
Real.logcontDiff_iff_contDiffAt
ContDiffAt.log
contDiffAt

ContDiffAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalContDiffAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
NormedField.toNormedSpace
NontriviallyNormedField.toNormedField
Real.logcomp
Real.contDiffAt_log

ContDiffOn

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalContDiffOn
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
NormedField.toNormedSpace
NontriviallyNormedField.toNormedField
Real.logContDiffWithinAt.log

ContDiffWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalContDiffWithinAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
NormedField.toNormedSpace
NontriviallyNormedField.toNormedField
Real.logContDiffAt.comp_contDiffWithinAt
Real.contDiffAt_log

Continuous

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalContinuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.logContinuousOn.comp_continuous
Real.continuousOn_log

ContinuousAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalContinuousAt
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.logFilter.Tendsto.log

ContinuousOn

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalContinuousOn
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.logContinuousWithinAt.log

ContinuousWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalContinuousWithinAt
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.logFilter.Tendsto.log

Differentiable

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalDifferentiable
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.instAddCommGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Real.pseudoMetricSpace
Real.logDifferentiableAt.log

DifferentiableAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalDifferentiableAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.instAddCommGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Real.pseudoMetricSpace
Real.logHasFDerivAt.differentiableAt
Algebra.to_smulCommClass
UniformContinuousConstSMul.to_continuousConstSMul
Ring.uniformContinuousConstSMul
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
HasFDerivAt.log
hasFDerivAt

DifferentiableOn

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalDifferentiableOn
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.instAddCommGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Real.pseudoMetricSpace
Real.logDifferentiableWithinAt.log

DifferentiableWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalDifferentiableWithinAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.instAddCommGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Real.pseudoMetricSpace
Real.logHasFDerivWithinAt.differentiableWithinAt
Algebra.to_smulCommClass
UniformContinuousConstSMul.to_continuousConstSMul
Ring.uniformContinuousConstSMul
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
HasFDerivWithinAt.log
hasFDerivWithinAt

ENNReal

Definitions

NameCategoryTheorems
log 📖CompOp
45 mathmath: log_lt_log_iff, log_pos_real', log_eq_bot_iff, log_le_log_iff, log_lt_zero_iff, continuous_log, ExpGrowth.expGrowthSup_def, log_top, log_strictMono, log_eq_iff, log_le_log, log_of_nnreal, zero_lt_log_iff, log_pos_real, log_eq_one_iff, log_monotone, Measurable.ennreal_log, log_surjective, EReal.ENNReal.rpow_eq_exp_mul_log, exp_log, Dynamics.IsDynCoverOf.coverEntropyEntourage_le_log_card_div, log_bijective, log_pow, Dynamics.coverEntropyEntourage_le_log_coverMincard_div, log_lt_top_iff, ExpGrowth.expGrowthSup_pow, log_le_zero_iff, log_rpow, log_eq_top_iff, measurable_log, zero_le_log_iff, log_one, log_zero, ExpGrowth.expGrowthInf_def, log_injective, bot_lt_log_iff, ExpGrowth.expGrowthInf_pow, logHomeomorph_apply, log_mul_add, log_lt_log, log_ofReal_of_pos, log_inv, logOrderIso_apply, EReal.log_exp, log_ofReal

Filter.Tendsto

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalFilter.Tendsto
Real
nhds
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.logcomp
ContinuousAt.tendsto
Real.continuousAt_log

HasDerivAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalHasDerivAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.log
DivInvMonoid.toDiv
Real.instDivInvMonoid
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
IsBoundedSMul.continuousSMul
NormedSpace.toIsBoundedSMul
hasDerivWithinAt_univ
HasDerivWithinAt.log

HasDerivWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalHasDerivWithinAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.log
DivInvMonoid.toDiv
Real.instDivInvMonoid
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
div_eq_inv_mul
HasDerivAt.comp_hasDerivWithinAt
Real.hasDerivAt_log

HasFDerivAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalHasFDerivAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
SeminormedAddCommGroup.toAddCommGroup
NormedAddCommGroup.toSeminormedAddCommGroup
NormedSpace.toModule
Real.normedField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.instAddCommGroup
Semiring.toModule
Real.semiring
Real.pseudoMetricSpace
Real.log
StrongDual
ESeminormedAddCommMonoid.toAddCommMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
ContinuousLinearMap.instSMul
RingHom.id
Semiring.toNonAssocSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
NonUnitalNonAssocSemiring.toDistribSMul
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
Real.normedCommRing
Algebra.to_smulCommClass
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
UniformContinuousConstSMul.to_continuousConstSMul
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
Ring.uniformContinuousConstSMul
Real.instRing
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Real.instInv
HasDerivAt.comp_hasFDerivAt
Real.hasDerivAt_log

HasFDerivWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalHasFDerivWithinAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
SeminormedAddCommGroup.toAddCommGroup
NormedAddCommGroup.toSeminormedAddCommGroup
NormedSpace.toModule
Real.normedField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.instAddCommGroup
Semiring.toModule
Real.semiring
Real.pseudoMetricSpace
Real.log
StrongDual
ESeminormedAddCommMonoid.toAddCommMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
ContinuousLinearMap.instSMul
RingHom.id
Semiring.toNonAssocSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
NonUnitalNonAssocSemiring.toDistribSMul
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
Real.normedCommRing
Algebra.to_smulCommClass
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
UniformContinuousConstSMul.to_continuousConstSMul
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
Ring.uniformContinuousConstSMul
Real.instRing
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Real.instInv
HasDerivAt.comp_hasFDerivWithinAt
Real.hasDerivAt_log

HasStrictDerivAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalHasStrictDerivAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.log
DivInvMonoid.toDiv
Real.instDivInvMonoid
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
div_eq_inv_mul
comp
Real.hasStrictDerivAt_log

HasStrictFDerivAt

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalHasStrictFDerivAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
SeminormedAddCommGroup.toAddCommGroup
NormedAddCommGroup.toSeminormedAddCommGroup
NormedSpace.toModule
Real.normedField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.instAddCommGroup
Semiring.toModule
Real.semiring
Real.pseudoMetricSpace
Real.log
StrongDual
ESeminormedAddCommMonoid.toAddCommMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
ContinuousLinearMap.instSMul
RingHom.id
Semiring.toNonAssocSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
NonUnitalNonAssocSemiring.toDistribSMul
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
Real.normedCommRing
Algebra.to_smulCommClass
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
UniformContinuousConstSMul.to_continuousConstSMul
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
Ring.uniformContinuousConstSMul
Real.instRing
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Real.instInv
HasStrictDerivAt.comp_hasStrictFDerivAt
Real.hasStrictDerivAt_log

Int

Definitions

NameCategoryTheorems
log 📖CompOp
21 mathmath: log_inv, zpow_le_iff_le_log, lt_zpow_iff_log_lt, log_one_right, log_ofNat, log_one_left, log_of_right_le_zero, log_of_left_le_one, zpow_log_le_self, log_natCast, lt_zpow_succ_log_self, log_of_one_le_right, log_mono_right, log_zero_left, Real.floor_logb_natCast, log_zpow, log_zero_right, log_of_right_le_one, neg_log_inv_eq_clog, neg_clog_inv_eq_log, clog_inv

IntervalIntegrable

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalContinuousOn
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.uIcc
Real.lattice
IntervalIntegrable
NormedAddGroup.toENormedAddMonoid
NormedAddCommGroup.toNormedAddGroup
Real.normedAddCommGroup
Real.log
ContinuousOn.intervalIntegrable
ContinuousOn.log

IsSelfAdjoint

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalIsSelfAdjoint
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
StarRing.toStarAddMonoid
CFC.log
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
instContinuousStarReal
cfc_predicate

Measurable

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalMeasurable
Real
Real.measurableSpace
Real.logcomp
Real.measurable_log

Nat

Definitions

NameCategoryTheorems
log 📖CompOp
58 mathmath: sub_one_mul_sum_log_div_pow_eq_sub_sum_digits, Real.natFloor_logb_natCast, digits_len, log_one_right, log_of_one_lt_of_le, log_lt_of_lt_pow', max_log_padicValNat_succ_eq_log_succ, log_two_bit, log_zero_left, log_zero_right, log_lt_iff_lt_pow, log_eq_zero_iff, Int.clog_of_right_le_one, log_eq_iff, log_eq_one_iff', log_pos, log_lt_self, log_mul_base, Int.log_ofNat, log_eq_of_pow_le_of_lt_pow, log_antitone_left, padicValRat_two_harmonic, isPrimePow_nat_iff_bounded_log, log_mono, le_log_of_pow_le, factorization_choose_le_log, lt_pow_iff_log_lt, log_of_lt, log_div_base, Int.log_natCast, Real.natLog_le_logb, lt_pow_succ_log_self, Int.log_of_one_le_right, le_log_iff_pow_le, log2_eq_log_two, log_le_self, log_eq_one_iff, log_le_clog, log_pow, log_eq_log_succ_iff, log_of_left_le_one, log_div_mul_self, nat_log_eq_padicValNat_iff, log_pow_left, isPrimePow_nat_iff_bounded_log_minFac, log_pos_iff, log_anti_left, log_lt_log_succ_iff, pow_le_iff_le_log, log_one_left, log_lt_of_lt_pow, padicNorm_two_harmonic, pow_log_le_self, pow_log_le_add_one, padicValNat_le_nat_log, log_monotone, log_div_base_pow, log_mono_right

Ordinal

Definitions

NameCategoryTheorems
log 📖CompOp
35 mathmath: log_of_left_le_one, log_def, add_log_le_log_mul, lt_log_of_lt_opow, log_zero_right, log_zero_left, log_eq_zero, div_opow_log_pos, log_one_left, log_le_self, CNF_ne_zero, opow_log_le_self, log_opow_mul, succ_log_def, div_opow_log_lt, lt_opow_succ_log_self, log_opow, log_mono_right, log_opow_mul_add, mod_opow_log_lt_self, mul_eq_opow_log_succ, log_eq_iff, lt_opow_iff_log_lt', log_mod_opow_log_lt_log_self, CNF.rec_pos, CNFRec_pos, le_log_of_opow_le, CNF.ne_zero, CNF.fst_le_log, opow_le_iff_le_log, CNF_fst_le_log, log_pos, opow_le_iff_le_log', lt_opow_iff_log_lt, log_one_right

Real

Definitions

NameCategoryTheorems
log 📖CompOp
473 mathmath: Behrend.roth_lower_bound, HasStrictFDerivAt.log, le_log_one_add_of_nonneg, isBigO_log_mul_const_log_atTop, Polynomial.mahlerMeasure_def_of_ne_zero, Complex.ofNat_log, Complex.tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero, HasDerivAt.const_rpow, InformationTheory.integral_llr_add_mul_log_nonneg, not_continuousAt_inv_log_neg_one, Mathlib.Meta.Positivity.log_pos_of_isNat, log_le_self, Complex.log_re, sinh_log, Stirling.log_stirlingSeq_bounded_aux, range_log, DifferentiableWithinAt.log, HasStrictDerivAt.rpow, Continuous.log, ENNReal.log_pos_real', contDiffOn_log, hasDerivAt_sqrt_mul_log, deriv_negMulLog, deriv_binEntropy, rpow_eq_nhds_of_neg, le_log_of_zpow_le, log_div_le_sum_range_add, countingFunction_finsum_eq_finsum_add, log_sqrt, InformationTheory.mul_log_le_klDiv, posLog_norm_add_le, Chebyshev.abs_psi_sub_theta_le_sqrt_mul_log, Stirling.log_stirlingSeq_sub_log_stirlingSeq_succ, Polynomial.logMahlerMeasure_C_mul, log_neg_natCast_nonneg, ContinuousAt.log, le_pow_iff_log_le, mul_log_nonneg, summable_log_one_add_of_summable, continuous_mul_log, log_prod, log_abs, deriv_log, Behrend.roth_lower_bound_explicit, circleAverage_log_norm_sub_const₁, NumberField.Units.regOfFamily_eq_det, Height.logHeight₁_sum_le, log_exp, Height.logHeight₁_add_le, hasSum_log_one_add_inv, differentiableOn_log, isLittleO_log_rpow_rpow_atTop, integral_log_sin_zero_pi_div_two, rpow_def_of_pos, integral_inv_of_pos, log_add_one_le_harmonic, hasDerivAt_mul_log, log_mul_self_monotoneOn, Height.logHeight₁_sub_le, circleAverage_log_norm_sub_const_of_mem_closedBall, deriv_sqrt_mul_log, tendsto_mul_log_one_add_of_tendsto, hasStrictFDerivAt_rpow_of_pos, Stirling.log_stirlingSeq_formula, deriv.log, log_neg_eq_log, log_div_log, InformationTheory.klFun_apply, Behrend.log_two_mul_two_le_sqrt_log_eight, zpow_le_iff_le_log, rpow_inv_log, MeromorphicOn.extract_zeros_poles_log, isLittleO_log_id_atTop, surjOn_log', Chebyshev.psi_eq_sum_theta, Stirling.log_stirlingSeq_diff_hasSum, log_list_prod, Polynomial.logMahlerMeasure_of_degree_eq_one, zpow_lt_iff_lt_log, artanh_eq_half_log, continuousOn_log, Complex.hasDerivAt_GammaIntegral, circleAverage_log_norm_sub_const_eq_log_radius_add_posLog, not_continuousAt_deriv_mul_log_zero, ContinuousOn.log, log_two_near_10, Stirling.le_log_factorial_stirling, strictConcaveOn_log_Iio, mellin_hasDerivAt_of_isBigO_rpow, ProbabilityTheory.cgf_zero_fun, fderiv.log, UpperHalfPlane.dist_log_im_le, Complex.ofReal_log, pow_lt_iff_lt_log, AkraBazziRecurrence.eventually_deriv_one_add_smoothingFn, log_rpow, InformationTheory.hasDerivAt_klFun, continuous_log, log_neg_of_lt_zero, MeasureTheory.llr_tilted_right, ContDiff.log, AnalyticAt.log, hasSum_log_one_add, mellinInv_eq_fourierIntegralInv, strictConcaveOn_sqrt_mul_log_Ioi, Chebyshev.integral_one_div_log_sq_isBigO, Complex.natCast_log, isBigO_rpow_top_log_smul, le_zpow_iff_log_le, NumberField.Units.regulator_eq_det, InformationTheory.leftDeriv_klFun, le_exp_log, log_lt_iff_lt_exp, analyticOn_log, log_pos_iff, analyticOnNhd_log, log_nat_eq_sum_factorization, HasDerivWithinAt.const_rpow, mellinInv_eq_fourierInv, UpperHalfPlane.dist_of_re_eq, NumberField.Units.finrank_mul_regOfFamily_eq_det, intervalIntegral.intervalIntegrable_log, lt_log_iff_exp_lt, HasDerivAt.log, log_nonpos, AkraBazziRecurrence.isTheta_smoothingFn_sub_self, hasDerivAt_binEntropy, ValueDistribution.characteristic_sub_characteristic_inv_of_ne_zero, hasDerivAt_qaryEntropy, Chebyshev.sum_PrimePow_eq_sum_sum, Behrend.lower_bound_le_one, DifferentiableAt.log, Complex.log_ofReal_mul, le_log_iff_exp_le, differentiableAt_log, deriv_inv_log, log_le_rpow_div, Chebyshev.primeCounting_eq_theta_div_log_add_integral, AkraBazziRecurrence.eventually_deriv_rpow_p_mul_one_sub_smoothingFn, posLog_sub_posLog_inv, ValueDistribution.characteristic_sub_characteristic_inv_at_zero, tendsto_harmonic_sub_log, ValueDistribution.proximity_sum_top_le, circleAverage_log_norm_sub_const₀, isBigO_log_const_mul_log_atTop, Polynomial.intervalIntegrable_mahlerMeasure, ENNReal.log_of_nnreal, deriv_log', log_zpow, DifferentiableOn.log, log_of_pos, log_two_gt_d9, Mathlib.Meta.Positivity.log_pos_of_isNNRat, HasStrictFDerivAt.rpow, rpow_def_of_neg, Function.Periodic.im_invQParam, MeasureTheory.integral_llr_tilted_left, lt_log_of_rpow_lt, Mathlib.Meta.Positivity.log_pos_of_isRat_neg, contDiffAt_log, leftDeriv_mul_log, circleIntegrable_log_norm_factorizedRational, ENNReal.log_pos_real, posLog_eq_log, log_one, strictConvexOn_mul_log, Chebyshev.integrableOn_theta_div_id_mul_log_sq, hasStrictDerivAt_log, NumberField.Units.dirichletUnitTheorem.sum_logEmbedding_component, abs_log_sub_add_sum_range_le, log_pos, differentiableOn_mul_log, ValueDistribution.proximity_add_top_le, AkraBazziRecurrence.eventually_deriv_rpow_p_mul_one_add_smoothingFn, deriv_qaryEntropy, HasDerivWithinAt.rpow, ProbabilityTheory.lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self, harmonic_le_one_add_log, isBigO_logb_mul_const_log_atTop, tendsto_rpow_sub_one_log, differentiableAt_log_iff, convexOn_log_Gamma, IntervalIntegrable.log, HasFDerivWithinAt.log, log_eq_zero, ArithmeticFunction.log_apply, neg_inv_le_log, Asymptotics.IsEquivalent.log, Summable.summable_log_norm_one_add, hasStrictDerivAt_const_rpow_of_neg, log_lt_sub_one_of_pos, sum_range_le_log_div, log_lt_log, Chebyshev.primeCounting_sub_theta_div_log_isBigO, isLittleO_log_rpow_nhdsGT_zero, isBigO_logb_log, ArithmeticFunction.vonMangoldt_sum, isLittleO_log_rpow_atTop, ValueDistribution.characteristic_sub_characteristic_inv_le, ContDiffOn.log, AnalyticOnNhd.circleAverage_log_norm_of_ne_zero, le_rpow_iff_log_le, tendsto_nat_mul_log_one_add_of_tendsto, Polynomial.logMahlerMeasure_eq_log_leadingCoeff_add_sum_log_roots, ValueDistribution.characteristic_add_top_eventuallyLE, derivWithin_const_rpow, mul_log_nonpos, doublingGamma_log_convex_Ioi, le_log_of_pow_le, lt_zpow_iff_log_lt, log_le_iff_le_exp, log_div_self, ProbabilityTheory.Fernique.lintegral_closedBall_diff_exp_logRatio_mul_sq_le, hasDerivAt_half_log_one_add_div_one_sub_sub_sum_range, exp_log_of_neg, Chebyshev.theta_eq_log_primorial, NumberField.mixedEmbedding.logMap_apply_of_norm_eq_one, lt_log_of_zpow_lt, Polynomial.logMahlerMeasure_def, tendsto_mul_log_one_add_div_atTop, AEMeasurable.log, log_le_log, ZetaAsymptotics.term_one, log_div, ContDiffAt.log, Differentiable.log, integral_inv_of_neg, ProbabilityTheory.Fernique.measure_gt_normThreshold_le_exp, circleIntegrable_log_norm_meromorphicOn, MeasureTheory.log_rnDeriv_tilted_left_self, tendsto_log_nhdsGT_zero, isLittleO_abs_log_rpow_rpow_nhdsGT_zero, exp_log, NumberField.Units.dirichletUnitTheorem.mult_log_place_eq_zero, HasStrictDerivAt.log, AbsoluteValue.IsEquiv.log_div_log_pos, InformationTheory.mul_log_le_toReal_klDiv, tendsto_log_nhdsLT_zero, fderivWithin.log, NumberField.Units.dirichletUnitTheorem.log_le_of_logEmbedding_le, harmonic_floor_le_one_add_log, HasFDerivAt.log, log_div_self_antitoneOn, tendsto_rightDeriv_mul_log_atTop, posLog_def, ValueDistribution.characteristic_sub_characteristic_inv, NumberField.mixedEmbedding.fundamentalCone.expMap_symm_apply, integral_inv, continuous_log', tendsto_log_div_rpow_nhdsGT_zero, ProbabilityTheory.Fernique.logRatio_mul_normThreshold_add_one_le, Height.logHeight_eq_log_mulHeight, Chebyshev.psi_eq_theta_add_sum_theta, rightDeriv_mul_log, Measurable.log, ValueDistribution.characteristic_sum_top_le, isLittleO_const_log_atTop, log_div_sqrt_antitoneOn, HasDerivAt.rpow, integral_one_div_of_pos, ValueDistribution.characteristic_add_top_le, Polynomial.logMahlerMeasure_eq_log_MahlerMeasure, deriv_const_rpow, not_differentiableAt_inv_log_zero, deriv_mul_log, MeasureTheory.integral_rnDeriv_mul_log, integral_log_sin_zero_pi, log_natCast_nonneg, Stirling.log_stirlingSeq_bounded_by_constant, rpow_inv_log_le_exp_one, tendsto_log_comp_add_sub_log, surjOn_log, hasDerivAt_log, HasStrictFDerivAt.const_rpow, NumberField.Units.abs_det_eq_abs_det, ProbabilityTheory.cgf_zero', Stirling.log_stirlingSeq'_antitone, binEntropy_eq_log_two, ProbabilityTheory.cgf_const', integral_log, Complex.IsExpCmpFilter.isLittleO_log_norm_re, circleAverage_log_norm_add_const_eq_posLog, MeromorphicOn.circleAverage_log_norm, abs_log_mul_self_rpow_lt, ContinuousWithinAt.log, CFC.log_smul', Chebyshev.theta_eq_sum_Icc, integral_log_from_zero_of_pos, log_le_harmonic_floor, tendstoLocallyUniformlyOn_rpow_sub_one_log, AkraBazziRecurrence.eventually_deriv_one_sub_smoothingFn, Function.FactorizedRational.log_norm_meromorphicTrailingCoeffAt, Function.locallyFinsuppWithin.logCounting_divisor_eq_circleAverage_sub_const, log_le_log_iff, tendsto_deriv_mul_log_atTop, Mathlib.Meta.Positivity.log_nonneg_of_isNegNat, MeromorphicOn.log_norm_meromorphicTrailingCoeffAt_extract_zeros_poles, log_surjective, NumberField.mixedEmbedding.fundamentalCone.expMap_single_symm_apply, hasDerivAt_Gamma_one_half, rpow_def_of_nonneg, AkraBazziRecurrence.isEquivalent_smoothingFn_sub_self, intervalIntegrable_log_cos, not_DifferentiableAt_log_mul_zero, InformationTheory.rightDeriv_klFun, lt_log_of_pow_lt, continuousAt_log_iff, AkraBazziRecurrence.deriv_smoothingFn, rpow_le_iff_le_log, HasFDerivWithinAt.rpow, negMulLog_eq_neg, circleAverage_log_norm_sub_const_eq_posLog, ArithmeticFunction.vonMangoldt_apply, AkraBazziRecurrence.dist_r_b, analyticAt_log, Filter.Tendsto.log, NumberField.Units.dirichletUnitTheorem.logEmbedding_component, circleAverage_log_norm_factorizedRational, Chebyshev.theta_le_log4_mul_x, NumberField.mixedEmbedding.logMap_apply, strictMonoOn_log, Mathlib.Meta.Positivity.log_nonneg_of_isNat, AnalyticAt.harmonicAt_log_norm, log_intCast_nonneg, NumberField.mixedEmbedding.logMap_apply_of_norm_one, HasFDerivAt.const_rpow, Finsupp.logHeight_eq_log_mulHeight, Continuous.mul_log, log_nonneg, negMulLog_def, Chebyshev.psi_le, log_two_lt_d9, rpow_lt_iff_lt_log, log_multiset_prod, circleAverage_log_norm_sub_const₂, mul_log_eq_log_iff, lt_log_one_add_of_pos, MeromorphicOn.intervalIntegrable_log, MeasureTheory.llr_tilted_left, Chebyshev.integral_theta_div_log_sq_isBigO, Finsupp.log_prod, integral_one_div, cosh_log, Complex.IsExpCmpFilter.isLittleO_log_re_re, pow_le_iff_le_log, Chebyshev.integral_theta_div_log_sq_isLittleO, binEntropy_lt_log_two, log_lt_log_iff, Tactic.ComputeAsymptotics.WellFormedBasis.tail_isLittleO_head, strictAntiOn_log, rpow_eq_nhds_of_pos, Stirling.log_stirlingSeq_diff_le_geo_sum, Behrend.le_sqrt_log, measurable_log, log_pow, ValueDistribution.logCounting_zero_sub_logCounting_top_eq_circleAverage_sub_const, expPartialHomeomorph_symm_apply, tendsto_log_atTop, ProbabilityTheory.Fernique.lintegral_exp_mul_sq_norm_le_mul, log_comp_exp, log_natCast_le_rpow_div, Polynomial.logMahlerMeasure_const, Behrend.lower_bound_le_one', strictConcaveOn_log_Ioi, HasFDerivWithinAt.const_rpow, BohrMollerup.logGammaSeq_add_one, ZetaAsymptotics.term_sum_one, Polynomial.logMahlerMeasure_monomial, intervalIntegrable_log_norm_meromorphicOn, log_neg, Polynomial.logMahlerMeasure_C_mul_X_add_C, binEntropy_le_log_two, not_continuousAt_inv_log_one, sum_range_sub_log_div_le, posLog_eq_log_max_one, ArithmeticFunction.vonMangoldt_le_log, Complex.log_mul_ofReal, rpow_def_of_nonpos, circleIntegrable_log_norm_sub_const, abs_rpow_le_exp_log_mul, binEntropy_two_inv, half_mul_log_add_log_abs, ruzsaSzemerediNumberNat_lower_bound, MeasureTheory.llr_smul_right, Complex.tendsto_log_nhdsWithin_im_nonneg_of_re_neg_of_im_zero, log_inv, isBigO_logb_const_mul_log_atTop, NumberField.mixedEmbedding.fundamentalCone.sum_expMap_symm_apply, hasStrictDerivAt_const_rpow, hasSum_pow_div_log_of_abs_lt_one, continuousAt_log, CFC.log_algebraMap, le_log_of_rpow_le, tendsto_log_nat_add_one_sub_log, log_nonpos_iff, Chebyshev.intervalIntegrable_one_div_log_sq, NumberField.Units.finrank_mul_regulator_eq_det, ENNReal.log_ofReal_of_pos, log_pos_of_lt_neg_one, tendsto_pow_log_div_mul_add_atTop, log_of_nat_eq_posLog, ruzsaSzemerediNumberNat_asymptotic_lower_bound, NumberField.Units.sum_mult_mul_log, hasStrictFDerivAt_rpow_of_neg, tendsto_log_nhdsNE_zero, MeasureTheory.llr_def, isLittleO_pow_log_id_atTop, posLog_sum, ValueDistribution.characteristic_sum_top_eventuallyLE, AnalyticOnNhd.log, log_zero, MeasureTheory.integral_llr_tilted_right, ArithmeticFunction.vonMangoldt_apply_prime, tendsto_log_mul_rpow_nhdsGT_zero, Height.logHeight₁_eq_log_mulHeight₁, integral_log_from_zero, deriv_log_comp_eq_logDeriv, NumberField.Units.dirichletUnitTheorem.exists_unit, log_div_self_rpow_antitoneOn, ValueDistribution.abs_characteristic_sub_characteristic_shift_le, Complex.log_ofReal_re, log_of_ne_zero, HasDerivWithinAt.log, qaryEntropy_one, ContDiffWithinAt.log, deriv_mul_log_zero, AnalyticWithinAt.log, log_mul, integral_one_div_of_neg, exp_log_eq_abs, convexOn_mul_log, Chebyshev.eventually_primeCounting_le, Complex.IsExpCmpFilter.isTheta_cpow_exp_re_mul_log, deriv_sqrt_mul_log', Chebyshev.psi_le_const_mul_self, tendsto_log_mul_self_nhdsLT_zero, Tactic.ComputeAsymptotics.WellFormedBasis.push_log_last, abs_log_mul_self_lt, intervalIntegrable_log_sin, InformationTheory.deriv_klFun, tendsto_harmonic_sub_log_add_one, posLog_nat_mul, CFC.log_smul, HasFDerivAt.rpow, log_nonneg_iff, AnalyticOn.log, log_le_sub_one_of_pos, one_sub_inv_le_log_of_pos, MeasureTheory.llr_smul_left, intervalIntegral.intervalIntegrable_log', derivWithin.log, isBigO_rpow_zero_log_smul, ValueDistribution.proximity_sub_proximity_inv_eq_circleAverage, AbsoluteValue.IsEquiv.log_div_log_eq_log_div_log, hasSum_log_sub_log_of_abs_lt_one, posLog_norm_sum_le, lt_rpow_iff_log_lt, log_doublingGamma_eq, MeasureTheory.integrable_rnDeriv_mul_log_iff, BohrMollerup.tendsto_log_gamma, lt_pow_iff_log_lt, deriv2_sqrt_mul_log, hasStrictDerivAt_log_of_pos, log_injOn_pos, circleIntegrable_log_norm_meromorphicOn_of_nonneg, Mathlib.Meta.Positivity.log_pos_of_isNegNat, posLog_add, AkraBazziRecurrence.isLittleO_self_div_log_id, log_neg_iff, ENNReal.log_ofReal, deriv2_mul_log, hasDerivAt_negMulLog, AkraBazziRecurrence.growsPolynomially_log

Submonoid

Definitions

NameCategoryTheorems
log 📖CompOp
7 mathmath: log_mul, log_pow_eq_self, Surreal.dyadicMap_apply_pow', pow_log_eq_self, Surreal.dyadicMap_apply, log_pow_int_eq_self, powLogEquiv_symm_apply

WithZero

Definitions

NameCategoryTheorems
log 📖CompOp
21 mathmath: log_le_log, log_inv, log_le_iff_le_exp, le_log_of_exp_le, Padic.norm_eq_zpow_log_mulValuation, log_one, log_div, log_pow, log_lt_log, log_mul, log_zpow, log_exp, le_log_iff_exp_le, lt_log_iff_exp_lt, logEquiv_apply, lt_log_of_exp_lt, log_zero, log_lt_iff_lt_exp, le_exp_log, exp_log, logEquiv_unitsMk0

deriv

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalDifferentiableAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
deriv
Real.log
DivInvMonoid.toDiv
Real.instDivInvMonoid
HasDerivAt.deriv
HasDerivAt.log
DifferentiableAt.hasDerivAt

derivWithin

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalDifferentiableWithinAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
UniqueDiffWithinAt
Real.semiring
derivWithin
Real.log
DivInvMonoid.toDiv
Real.instDivInvMonoid
HasDerivWithinAt.derivWithin
HasDerivWithinAt.log
DifferentiableWithinAt.hasDerivWithinAt

fderiv

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalDifferentiableAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.instAddCommGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Real.pseudoMetricSpace
fderiv
Real.log
ContinuousLinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
NontriviallyNormedField.toNormedField
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
ContinuousLinearMap.instSMul
NonUnitalNonAssocSemiring.toDistribSMul
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
Algebra.to_smulCommClass
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
UniformContinuousConstSMul.to_continuousConstSMul
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
Ring.uniformContinuousConstSMul
Real.instRing
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Real.instInv
HasFDerivAt.fderiv
Algebra.to_smulCommClass
UniformContinuousConstSMul.to_continuousConstSMul
Ring.uniformContinuousConstSMul
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
IsTopologicalAddGroup.toContinuousAdd
SeminormedAddCommGroup.toIsTopologicalAddGroup
IsBoundedSMul.continuousSMul
NormedSpace.toIsBoundedSMul
IsTopologicalSemiring.toContinuousAdd
ContinuousMul.to_continuousSMul
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
HasFDerivAt.log
DifferentiableAt.hasFDerivAt

fderivWithin

Theorems

NameKindAssumesProvesValidatesDepends On
log 📖mathematicalDifferentiableWithinAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
Real.instAddCommGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Real.pseudoMetricSpace
UniqueDiffWithinAt
Real.semiring
fderivWithin
Real.log
ContinuousLinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
NontriviallyNormedField.toNormedField
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
ContinuousLinearMap.instSMul
NonUnitalNonAssocSemiring.toDistribSMul
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
Algebra.to_smulCommClass
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
UniformContinuousConstSMul.to_continuousConstSMul
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
Ring.uniformContinuousConstSMul
Real.instRing
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Real.instInv
HasFDerivWithinAt.fderivWithin
Algebra.to_smulCommClass
UniformContinuousConstSMul.to_continuousConstSMul
Ring.uniformContinuousConstSMul
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
IsTopologicalAddGroup.toContinuousAdd
SeminormedAddCommGroup.toIsTopologicalAddGroup
IsBoundedSMul.continuousSMul
NormedSpace.toIsBoundedSMul
IsTopologicalSemiring.toContinuousAdd
ContinuousMul.to_continuousSMul
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
HasFDerivWithinAt.log
DifferentiableWithinAt.hasFDerivWithinAt

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