Documentation Verification Report

Defs

📁 Source: Mathlib/Topology/Algebra/InfiniteSum/Defs.lean

Statistics

MetricCount
DefinitionsHasProd, HasSum, Multipliable, tprod, tsum, «term∏'[_]_,_», «term∏'_,_», «term∑'[_]_,_», «term∑'_,_»
9
TheoremshasProd, hasProd_support, hasSum, hasSum_support, hasProd_comap_iff, hasProd_comap_iff_of_hasSupport, hasProd_iff, hasProd_map_iff, hasSum_comap_iff, hasSum_comap_iff_of_hasSupport, hasSum_iff, hasSum_map_iff, multipliable, tprod_eq, unique, summable, tsum_eq, unique, hasProd, hasProd_iff, mono_filter, hasSum, hasSum_iff, mono_filter, hasProd_bot, hasProd_fintype, hasProd_fintype_support, hasProd_prod_of_ne_finset_one, hasProd_prod_support_of_ne_finset_one, hasProd_subtype_comap_iff_of_mulSupport_subset, hasProd_subtype_iff_of_mulSupport_subset, hasSum_bot, hasSum_fintype, hasSum_fintype_support, hasSum_subtype_comap_iff_of_support_subset, hasSum_subtype_iff_of_support_subset, hasSum_sum_of_ne_finset_zero, hasSum_sum_support_of_ne_finset_zero, multipliable_bot, multipliable_of_ne_finset_one, summable_bot, summable_of_ne_finset_zero, tprod_bot, tprod_def, tprod_eq_of_filter_le, tprod_eq_of_multipliable_unconditional, tprod_eq_one_of_not_multipliable, tsum_bot, tsum_def, tsum_eq_of_filter_le, tsum_eq_of_summable_unconditional, tsum_eq_zero_of_not_summable
52
Total61

Finset

Theorems

NameKindAssumesProvesValidatesDepends On
hasProd 📖mathematicalHasProd
Set.Elem
SetLike.coe
Finset
instSetLike
Set
Set.instMembership
prod
prod_congr
Set.toFinset_congr
SummationFilter.support_eq_univ
Set.toFinset_univ
prod_map
prod_attach
hasProd_support
SummationFilter.instHasSupportOfLeAtTop
hasProd_support 📖mathematicalHasProd
Set.Elem
SetLike.coe
Finset
instSetLike
Set
Set.instMembership
prod
map
Function.Embedding.subtype
Set.toFinset
SummationFilter.support
Subtype.fintype
FinsetCoe.fintype
prod_map
prod_congr
hasProd_fintype_support
hasSum 📖mathematicalHasSum
Set.Elem
SetLike.coe
Finset
instSetLike
Set
Set.instMembership
sum
sum_congr
Set.toFinset_congr
SummationFilter.support_eq_univ
Set.toFinset_univ
sum_map
sum_attach
hasSum_support
SummationFilter.instHasSupportOfLeAtTop
hasSum_support 📖mathematicalHasSum
Set.Elem
SetLike.coe
Finset
instSetLike
Set
Set.instMembership
sum
map
Function.Embedding.subtype
Set.toFinset
SummationFilter.support
Subtype.fintype
FinsetCoe.fintype
sum_map
sum_congr
hasSum_fintype_support

Function.Injective

Theorems

NameKindAssumesProvesValidatesDepends On
hasProd_comap_iff 📖mathematicalMulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
HasProd
SummationFilter.comap
Finset.prod_congr
Filter.tendsto_congr
Finset.prod_preimage
hasProd_comap_iff_of_hasSupport 📖mathematicalMulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
HasProd
SummationFilter.comap
Finset.prod_congr
Filter.tendsto_congr'
Filter.mp_mem
SummationFilter.HasSupport.eventually_le_support
Filter.univ_mem'
Finset.prod_preimage
hasProd_iff 📖mathematicalMulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
HasProd
SummationFilter.unconditional
hasProd_comap_iff
SummationFilter.comap_unconditional
hasProd_map_iff 📖mathematicalHasProd
SummationFilter.map
Finset.prod_map
Finset.prod_congr
hasSum_comap_iff 📖mathematicalAddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
HasSum
SummationFilter.comap
Finset.sum_congr
Filter.tendsto_map'_iff
Filter.tendsto_congr
Finset.sum_preimage
hasSum_comap_iff_of_hasSupport 📖mathematicalAddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
HasSum
SummationFilter.comap
Finset.sum_congr
Filter.tendsto_map'_iff
Filter.tendsto_congr'
Filter.mp_mem
SummationFilter.HasSupport.eventually_le_support
Filter.univ_mem'
Finset.sum_preimage
hasSum_iff 📖mathematicalAddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
HasSum
SummationFilter.unconditional
hasSum_comap_iff
SummationFilter.comap_unconditional
hasSum_map_iff 📖mathematicalHasSum
SummationFilter.map
Filter.tendsto_map'_iff
Finset.sum_map
Finset.sum_congr

HasProd

Theorems

NameKindAssumesProvesValidatesDepends On
multipliable 📖mathematicalHasProdMultipliable
tprod_eq 📖mathematicalHasProdtprodunique
Multipliable.hasProd
unique 📖HasProdtendsto_nhds_unique
SummationFilter.instNeBotFinsetFilterOfNeBot

HasSum

Theorems

NameKindAssumesProvesValidatesDepends On
summable 📖mathematicalHasSumSummable
tsum_eq 📖mathematicalHasSumtsumunique
Summable.hasSum
unique 📖HasSumtendsto_nhds_unique
SummationFilter.instNeBotFinsetFilterOfNeBot

Multipliable

Theorems

NameKindAssumesProvesValidatesDepends On
hasProd 📖mathematicalMultipliableHasProd
tprod
tprod_def
Finset.prod_congr
Set.toFinset_congr
Set.inter_eq_left
Set.Finite.coe_toFinset
Finset.toFinset_coe
finprod_eq_prod_of_mulSupport_subset
Set.mulSupport_mulIndicator
Set.mulIndicator_of_mem
hasProd_prod_support_of_ne_finset_one
hasProd_iff 📖mathematicalMultipliableHasProd
tprod
HasProd.tprod_eq
hasProd
mono_filter 📖Multipliable
Filter
Finset
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
SummationFilter.filter
Filter.Tendsto.mono_left

Summable

Theorems

NameKindAssumesProvesValidatesDepends On
hasSum 📖mathematicalSummableHasSum
tsum
tsum_def
Finset.sum_congr
Set.toFinset_congr
Set.inter_eq_left
Set.Finite.coe_toFinset
Set.inter_subset_right
Finset.toFinset_coe
finsum_eq_sum_of_support_subset
Set.support_indicator
Set.subset_inter_iff
Set.inter_subset_left
Set.indicator_of_mem
Set.Finite.mem_toFinset
Set.mem_inter_iff
Function.mem_support
hasSum_sum_support_of_ne_finset_zero
hasSum_iff 📖mathematicalSummableHasSum
tsum
HasSum.tsum_eq
hasSum
mono_filter 📖Summable
Filter
Finset
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
SummationFilter.filter
Filter.Tendsto.mono_left

(root)

Definitions

NameCategoryTheorems
HasProd 📖MathDef
68 mathmath: HasSum.exp, Pi.hasProd, Topology.IsInducing.hasProd_iff, HasProdUniformly.hasProd, Function.Injective.hasProd_comap_iff, hasProd_nat_add_iff, hasProd_ite_eq, hasProd_fintype, hasProd_prod_support_of_ne_finset_one, Multipliable.hasProd_iff_tendsto_nat, hasProd_fintype_support, EulerProduct.eulerProduct_hasProd, HasSum.cexp, ArithmeticFunction.IsMultiplicative.eulerProduct_hasProd, HasProd.of_subsingleton_cod, Nat.Partition.hasProd_powerSeriesMk_card_countRestricted, HasProdLocallyUniformly.hasProd, hasProd_subtype_iff_mulIndicator, hasProd_nat_add_iff', hasProd_one_add_of_hasSum_prod, hasProd_prod_of_ne_finset_one, Equiv.hasProd_iff_of_mulSupport, Equiv.hasProd_iff, Function.Injective.hasProd_comap_iff_of_hasSupport, hasProd_zero_of_exists_eq_zero, hasProd_one_iff_of_one_le, EulerProduct.eulerProduct_completely_multiplicative_hasProd, hasProd_pi_single, hasProd_bot, Multipliable.hasProd_iff, Nat.Partition.hasProd_powerSeriesMk_card_restricted, EulerProduct.eulerProduct_hasProd_mulIndicator, Multipliable.hasProd, SummationFilter.hasProd_symmetricIco_int_iff, Finset.hasProd_support, DirichletCharacter.LSeries_eulerProduct_hasProd, hasProd_iff_hasProd, hasProd_one, hasProd_of_isGLB_of_le_one, hasProd_subtype_iff_of_mulSupport_subset, Function.Injective.hasProd_iff, Function.Injective.hasProd_map_iff, HasProdLocallyUniformlyOn.hasProd, Finset.hasProd_compl_iff, tprod_def, hasProd_of_isLUB, Complex.hasProd_of_hasSum_log, Finset.hasProd, hasProd_single, Nat.Partition.hasProd_genFun, hasProd_subtype_comap_iff_of_mulSupport_subset, ENNReal.hasProd_iInf_prod, hasProd_of_isLUB_of_one_le, SummationFilter.hasProd_symmetricIcc_iff, hasProd_unique, HasSum.rexp, hasProd_empty, Function.Injective.hasProd_range_iff, hasProd_subtype_mulSupport, riemannZeta_eulerProduct_hasProd, Real.hasProd_of_hasSum_log, hasProd_one_iff, SummationFilter.hasProd_symmetricIoc_int_iff, hasProd_iff_hasProd_of_ne_one_bij, Finset.hasProd_iff_compl, hasProd_extend_one, HasProdUniformlyOn.hasProd, hasProd_singleton
HasSum 📖MathDef
246 mathmath: HurwitzZeta.hasSum_int_hurwitzZetaOdd, aux_hasSum_of_le_geometric, PowerSeries.WithPiTopology.hasSum_iff_hasSum_coeff, PowerSeries.hasSum_aeval, hasSum_mul_left_iff, hasSum_geometric_of_norm_lt_one, hasSum_div_const_iff, EisensteinSeries.hasSum_e2Summand_symmetricIcc, hasSum_iff_tendsto_nat_of_summable_norm, HasSumLocallyUniformlyOn.hasSum, HurwitzZeta.hasSum_int_cosKernel₀, HurwitzZeta.hasSum_int_hurwitzZetaEven, Complex.hasSum_deriv_of_summable_norm, has_pointwise_sum_fourier_series_of_summable, HasFPowerSeriesWithinOnBall.hasSum_sub, hasSum_choose_mul_geometric_of_norm_lt_one, hasSum_pi_single, HasSumUniformly.hasSum, Complex.hasSum_taylorSeries_of_entire, MeasureTheory.Integrable.hasSum_intervalIntegral, hasSum_one_div_nat_pow_mul_fourier, ContinuousLinearEquiv.hasSum', hasSum_iff_hasSum, Set.Countable.exists_pos_hasSum_le, hasSum_coe_mul_geometric_of_norm_lt_one', HasFPowerSeriesAt.eventually_hasSum, HasSumLocallyUniformly.hasSum, Real.hasSum_log_one_add_inv, FormalMultilinearSeries.hasSum_of_finite, Complex.hasSum_taylorSeries_on_ball, NormedSpace.exp_series_hasSum_exp', hasSum_subtype_iff_of_support_subset, RCLike.hasSum_ofReal, hasSum_sum_support_of_ne_finset_zero, HasSum.hasSum_at_zero, hasSum_nat_jacobiTheta, exists_dist_slope_lt_pairwiseDisjoint_hasSum, PeriodPair.hasSum_derivWeierstrassPExcept, hasSum_empty, HasFPowerSeriesOnBall.hasSum_iteratedFDeriv, Real.hasSum_sinh, IsHilbertSum.hasSum_linearIsometryEquiv_symm, Stirling.log_stirlingSeq_diff_hasSum, ENNReal.hasSum_toReal, HurwitzZeta.hasSum_nat_sinZeta, Real.hasSum_cosh, Function.Injective.hasSum_comap_iff_of_hasSupport, hasSum_single, MvPowerSeries.hasSum_aeval, PeriodPair.hasSum_derivWeierstrassP, hasSum_zeta_four, Finset.hasSum_support, MeasureTheory.VectorMeasure.hasSum_of_disjoint_iUnion, HurwitzZeta.hasSum_hurwitzZeta_of_one_lt_re, hasSum_unique, hasSum_subtype_iff_indicator, intervalIntegral.hasSum_intervalIntegral_of_summable_norm, Real.hasSum_cos, MeasureTheory.hasSum_lintegral_measure, Equiv.hasSum_iff, lp.hasSum_single, Real.hasSum_log_one_add, hasSum_jacobiTheta₂_term, PowerSeries.hasSum_of_monomials_self, HurwitzZeta.hasSum_nat_completedCosZeta, HurwitzZeta.hasSum_nat_hurwitzZetaEven, HurwitzZeta.hasSum_int_cosKernel, NormedSpace.expSeries_hasSum_exp, FormalMultilinearSeries.hasSum, Real.hasSum_ofDigitsTerm_digits, hasSum_fourier_series_L2, NNReal.hasSum_real_toNNReal_of_nonneg, HilbertBasis.hasSum_orthogonalProjection, SummationFilter.hasSum_symmetricIoc_int_iff, HilbertBasis.hasSum_repr, hasSum_zero_iff_of_nonneg, Function.Injective.hasSum_map_iff, SummationFilter.hasSum_symmetricIcc_iff, HurwitzZeta.hasSum_nat_sinKernel, MvPowerSeries.WithPiTopology.hasSum_iff_hasSum_coeff, Complex.hasSum_ofReal, hasSum_const_div_iff, Summable.hasSum_iff_tendsto_nat, hasSum_sum_range_mul_of_summable_norm', HurwitzZeta.hasSum_int_cosZeta, hasSum_jacobiTheta₂_term_fderiv, hasSum_fintype_support, Complex.hasSum_sin, MeasureTheory.VectorMeasure.m_iUnion', EisensteinSeries.hasSum_e2Summand_symmetricIco, hasSum_jacobiTheta₂'_term, Function.Injective.hasSum_comap_iff, HurwitzZeta.hasSum_nat_cosZeta, hasSum_cauchyPowerSeries_integral, NormedSpace.expSeries_div_hasSum_exp_of_mem_ball, HasFPowerSeriesWithinOnBall.hasSum_derivSeries_of_hasFDerivWithinAt, hasSum_unop, HasFPowerSeriesOnBall.hasSum, HurwitzZeta.hasSum_expZeta_of_one_lt_re, HurwitzZeta.hasSum_int_evenKernel, OrthogonalFamily.hasSum_linearIsometry, hasSum_one_div_nat_pow_mul_cos, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, hasSum_fintype, hasSum_two_pi_I_cauchyPowerSeries_integral, EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_geometric, hasSum_nat_add_iff', hasSum_sq_fourierCoeffOn, hasSum_geometric_two, Finset.hasSum_iff_compl, MeasureTheory.preVariation.iUnion, hasSum_ite_eq, ContinuousLinearEquiv.hasSum, HurwitzZeta.hasSum_int_sinKernel, hasSum_extend_zero, RCLike.hasSum_conj, ZetaAsymptotics.term_tsum_one, Finset.hasSum, hasSum_mul_right_iff, EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_geometric, NNReal.exists_pos_sum_of_countable, hasSum_op, NNReal.hasSum_geometric, UnitAddTorus.hasSum_mFourier_series_L2, hasSum_zero, HurwitzZeta.hasSum_nat_completedSinZeta, Complex.hasSum_taylorSeries_on_emetric_ball, SummationFilter.hasSum_symmetricIco_int_iff, hasSum_geom_series_inverse, hasSum_coe_mul_geometric_of_norm_lt_one, NNReal.hasSum_iff_tendsto_nat, hasSum_choose_mul_geometric_of_norm_lt_one', PadicInt.hasSum_mahlerSeries, PeriodPair.hasSum_weierstrassP, hasSum_geometric_two', Complex.hasSum_cos', hasSum_sum_range_mul_of_summable_norm, HasFPowerSeriesOnBall.eventually_hasSum, HasSum.of_subsingleton_cod, UnitAddTorus.hasSum_mFourier_series_of_summable, MeasureTheory.hasSum_integral_measure, HurwitzZeta.hasSum_nat_hurwitzZetaOdd, PadicInt.hasSum_mahler, Summable.hasSum, hasSum_iff_hasSum_of_ne_zero_bij, ENNReal.hasSum, lp.hasSum_norm, PeriodPair.weierstrassPSeries_hasSum, ModularFormClass.hasSum_qExpansion_of_abs_lt, Complex.hasSum_conj, MvPowerSeries.WithPiTopology.hasSum_of_monomials_self, HurwitzZeta.hasSum_int_oddKernel, intervalIntegral.hasSum_integral_of_dominated_convergence, ENNReal.hasSum_iff_tendsto_nat, Summable.hasSum_iff, HasFPowerSeriesAt.eventually_hasSum_of_comp, RCLike.hasSum_iff, HurwitzZeta.hasSum_nat_hurwitzZetaEven_of_mem_Icc, Pi.hasSum, MeasureTheory.hasSum_integral_iUnion, lp.hasSum_inner, HasFPowerSeriesAt.eventually_hasSum_sub, UnitAddTorus.hasSum_prod_mFourierCoeff, hasSum_one_div_nat_pow_mul_sin, hasSum_bot, Complex.hasSum_conj', HasFPowerSeriesOnBall.eventually_hasSum_sub, hasSum_fourier_series_of_summable, Function.Injective.hasSum_iff, Complex.hasSum_taylorSeries_neg_log, hasSum_nat_add_iff, hasSum_zeta_two, hasSum_sum_of_ne_finset_zero, NormedSpace.expSeries_div_hasSum_exp, Function.Injective.hasSum_range_iff, EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum, MeasureTheory.hasSum_integral_iUnion_ae, PeriodPair.hasSum_weierstrassPExcept, hasSum_of_isLUB, Real.hasSum_sin, hasSum_geometric_of_abs_lt_one, PMF.hasSum_coe_one, HurwitzZeta.hasSum_int_sinZeta, ModularFormClass.hasSum_qExpansion_of_norm_lt, hasSum_sq_fourierCoeff, Real.hasSum_arctan, PeriodPair.weierstrassPExceptSeries_hasSum, Topology.IsInducing.hasSum_iff, UnitAddTorus.hasSum_mFourier_series_apply_of_summable, UnitAddTorus.hasSum_sq_mFourierCoeff, Complex.hasSum_taylorSeries_log, Equiv.hasSum_iff_of_support, PeriodPair.hasSum_sumInvPow, hasSum_zero_iff, HasFPowerSeriesOnBall.hasSum_sub, Complex.hasSum_sinh, Complex.hasSum_sin', ProbabilityTheory.geometricPMFRealSum, hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, NNReal.hasSum_coe, NNReal.hasSum_nat_add_iff, ENNReal.hasSum_coe, Real.hasSum_pow_div_log_of_abs_lt_one, NormedSpace.expSeries_hasSum_exp_of_mem_ball', HasFPowerSeriesWithinOnBall.hasSum, MeasureTheory.VectorMeasure.m_iUnion, Complex.hasSum_arctan, HasSumUniformlyOn.hasSum, EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum, HurwitzZeta.hasSum_int_completedCosZeta, Complex.hasSum_cosh, hasSum_of_isLUB_of_nonneg, hasSum_zeta_nat, Complex.hasSum_cos, MvPowerSeries.hasSum_eval₂, HurwitzZeta.hasSum_int_completedSinZeta, Quaternion.hasSum_coe, HurwitzZeta.hasSum_nat_hurwitzZetaOdd_of_mem_Icc, hasSum_iff_tendsto_nat_of_nonneg, MeasureTheory.Integrable.hasSum_intervalIntegral_comp_add_int, Finset.hasSum_compl_iff, HurwitzZeta.hasSum_nat_cosKernel₀, ProbabilityTheory.poissonPMFRealSum, HurwitzZeta.hasSum_int_completedHurwitzZetaEven, MeasureTheory.hasSum_integral_of_summable_integral_norm, RCLike.hasSum_conj', hasSum_subtype_comap_iff_of_support_subset, tsum_def, Complex.hasSum_iff, ModularFormClass.hasSum_qExpansion, HilbertBasis.hasSum_inner_mul_inner, hasSum_of_subseq_of_summable, hasFPowerSeriesAt_iff', hasSum_of_isGLB_of_nonpos, NormedSpace.expSeries_hasSum_exp_of_mem_ball, hasSum_singleton, hasFPowerSeriesAt_iff, Real.hasSum_log_sub_log_of_abs_lt_one, hasSum_subtype_support, hasSum_L_function_mod_four_eval_three, HilbertBasis.hasSum_repr_symm, hasSum_geometric_of_lt_one, PowerSeries.hasSum_eval₂, HurwitzZeta.hasSum_int_evenKernel₀, hasSum_iff_hasSum_compl, Complex.hasSum_taylorSeries_on_eball
Multipliable 📖MathDef
64 mathmath: NonarchimedeanGroup.multipliable_of_tendsto_cofinite_one, Finset.multipliable, Set.Finite.multipliable_compl_iff, MultipliableLocallyUniformlyOn.multipliable, multipliable_int_iff_multipliable_nat_and_neg_add_one, multipliable_pnat_iff_multipliable_succ, multipliable_one, multipliable_nat_add_iff, multipliable_one_add_of_summable, MvPowerSeries.WithPiTopology.multipliable_one_add_of_tendsto_order_atTop_nhds_top, multipliable_norm_one_add_of_summable_norm, ENNReal.multipliable_of_le_one, multipliable_subtype_and_compl, multipliable_subtype_iff_mulIndicator, Complex.multipliable_of_summable_log, multipliable_const_iff, Multipliable.of_subsingleton_cod, multipliable_iff_cauchySeq_finset, Equiv.multipliable_iff_of_mulSupport, Topology.IsInducing.multipliable_iff_tprod_comp_mem_range, Set.Finite.multipliable, Multipliable.of_finite, Real.multipliable_one_add_of_summable, MultipliableUniformly.multipliable, multipliable_int_iff_multipliable_nat_and_neg, Multipliable.map_iff_of_leftInverse, multipliable_one_add_of_summable_prod, multipliable_iff_nat_tprod_vanishing, multipliable_of_exists_eq_zero, Multipliable.map_iff_of_equiv, multipliable_iff_tprod_vanishing, multipliable_congr, multipliable_sineTerm, Function.Injective.multipliable_iff, multipliable_empty, pnat_multipliable_iff_multipliable_succ, multipliable_of_ne_finset_one, multipliable_iff_vanishing, Finset.multipliable_compl_iff, multipliable_congr_cofinite, MultipliableLocallyUniformly.multipliable, Nat.Partition.multipliable_powerSeriesMk_card_restricted, multipliable_mabs_iff, tprod_def, MultipliableUniformlyOn.multipliable, HasProd.multipliable, Function.Surjective.multipliable_iff_of_hasProd_iff, PowerSeries.WithPiTopology.multipliable_one_add_of_tendsto_order_atTop_nhds_top, multipliable_of_finite_mulSupport, multipliable_congr_atTop, multipliable_inv_iff, multipliable_extend_one, Real.multipliable_of_summable_log', NonarchimedeanGroup.multipliable_iff_tendsto_cofinite_one, Nat.Partition.multipliable_powerSeriesMk_card_countRestricted, PowerSeries.WithPiTopology.multipliable_one_sub_X_pow, Equiv.multipliable_iff, Nat.Partition.multipliable_genFun, multipliable_pnat_iff_multipliable_nat, Complex.multipliable_one_add_of_summable, Pi.multipliable, multipliable_bot, Real.multipliable_of_summable_log, MvPowerSeries.WithPiTopology.multipliable_one_add_of_tendsto_weightedOrder_atTop_nhds_top
tprod 📖CompOp
168 mathmath: logDeriv_tprod_eq_tsum, Multipliable.tprod_sigma, tprod_eq_zero_mul', tprod_int_eq_zero_mul_tprod_pnat_sq, Multipliable.tprod_mul_tprod_compl, tprod_subtype_mulSupport, DirichletCharacter.LSeries_eulerProduct_tprod, tprod_image, IsUltrametricDist.nnnorm_tprod_le_of_forall_le, MultipliableUniformlyOn.hasProdUniformlyOn, tprod_of_add_one_of_neg_add_one, tprod_int_rec, HasProdUniformlyOn.tprod_eqOn, tprod_extend_one, prod_eq_tprod_mulIndicator, multipliableUniformlyOn_iff_hasProdUniformlyOn, Equiv.tprod_eq_tprod_of_mulSupport, HasProd.tprod_fiberwise, ContinuousMap.tprod_apply, Multipliable.tprod_le_tprod, Complex.cexp_tsum_eq_tprod, cauchySeq_finset_iff_nat_tprod_vanishing, tprod_tprod_eq_mulSingle, tprod_setProd_singleton_left, IsUltrametricDist.norm_tprod_le, Multipliable.tprod_eq_mul_tprod_ite', Topology.IsClosedEmbedding.map_tprod, Multipliable.one_lt_tprod, one_le_tprod, IsUltrametricDist.norm_tprod_le_of_forall_le_of_nonneg, tprod_nat_mul_neg_add_one, tprod_pnat_eq_tprod_succ, tprod_empty, Multipliable.tprod_vanishing, Multipliable.tprod_lt_tprod, Finset.tprod_subtype', tprod_apply, Function.Injective.tprod_eq, tprod_nat_mul_neg, EulerProduct.eulerProduct_tprod, tprod_zero_pnat_eq_tprod_nat, multipliableLocallyUniformlyOn_iff_hasProdLocallyUniformlyOn, Multipliable.tprod_eq_mul_tprod_ite, Multipliable.prod_mul_tprod_nat_add, ArithmeticFunction.IsMultiplicative.eulerProduct_tprod, Multipliable.tprod_eq_zero_mul, multipliableUniformly_iff_hasProdUniformly, Summable.hasProdUniformlyOn_nat_one_add, EulerProduct.eulerProduct_completely_multiplicative_tprod, Nat.Partition.powerSeriesMk_card_restricted_eq_tprod, tprod_fintype, ModularForm.differentiableAt_eta_tprod, tprod_singleton, MultipliableLocallyUniformly.hasProdLocallyUniformly, Multipliable.tprod_div, tprod_eq_tprod_of_ne_one_bij, tprod_one, tprod_even_mul_odd, HasProd.tprod_eq, Topology.IsInducing.multipliable_iff_tprod_comp_mem_range, tprod_iSup_decode₂, Multipliable.tprod_prod, Multipliable.norm_tprod, tprod_univ, tprod_one_add, Multipliable.tprod_le_of_prod_range_le, Summable.hasProdLocallyUniformlyOn_one_add, tprod_setProd_singleton_right, Multipliable.prod_mul_tprod_nat_mul', Multipliable.hasProd_iff, tprod_eq_of_filter_le, Multipliable.prod_mul_tprod_subtype_compl, Multipliable.tprod_finset_bUnion_disjoint, cauchySeq_finset_iff_tprod_vanishing, tprod_eq_tprod_primes_mul_tprod_primes_of_mulSupport_subset_prime_powers, Multipliable.nat_tprod_vanishing, tprod_congr, tprod_const, Multipliable.prod_le_tprod, Multipliable.tprod_prod_uncurry, tprod_eq_tprod_of_hasProd_iff_hasProd, Multipliable.hasProd, Equiv.tprod_eq, SummationFilter.tprod_symmetricIcc_eq_tprod_symmetricIco, tprod_eq_one_of_not_multipliable, multipliable_iff_nat_tprod_vanishing, tprod_eq_prod', Multipliable.tprod_sigma', tprod_subtype_eq_of_mulSupport_subset, Nat.Partition.powerSeriesMk_card_countRestricted_eq_tprod, tprod_dite_right, Multipliable.sigma', tprod_eq_mulSingle, tprod_one_add_ordered, multipliable_iff_tprod_vanishing, tprod_of_exists_eq_zero, tprod_le_one, Multipliable.tprod_le_of_prod_le, tprod_subtype, tprod_ite_eq, Multipliable.prod_mul_tprod_compl, Multipliable.tprod_mono, Multipliable.abs_tprod, tprod_congr₂, tprod_of_nat_of_neg_add_one, tprod_eq_of_multipliable_unconditional, Function.Surjective.tprod_eq_tprod_of_hasProd_iff_hasProd, euler_sineTerm_tprod, HasProdUniformlyOn.tprod_eq, tprod_eq_tprod_primes_of_mulSupport_subset_prime_powers, IsUltrametricDist.norm_tprod_le_of_forall_le, IsUltrametricDist.nnnorm_tprod_le, Multipliable.tprod_prod', Multipliable.tprod_mul, tprod_int_eq_zero_mul_tprod_pnat, Multipliable.tendsto_prod_tprod_nat, tprod_eq_finprod, tprod_dite_left, Summable.hasProdLocallyUniformlyOn_nat_one_add, Multipliable.prod, MultipliableUniformly.hasProdUniformly, tprod_congr_set_coe, Function.LeftInverse.map_tprod, Multipliable.tprod_subtype_le, tprod_def, tprod_setElem_eq_tprod_setElem_diff, tprod_bool, MeasureTheory.Measure.infinitePi_singleton, tprod_iUnion_decode₂, HasProdLocallyUniformlyOn.tprod_eqOn, Multipliable.le_tprod', MultipliableLocallyUniformlyOn.hasProdLocallyUniformlyOn, Multipliable.tprod_subtype_mul_tprod_subtype_compl, multipliableLocallyUniformly_iff_hasProdLocallyUniformly, tprod_inv, tprod_comp_neg, Finset.tprod_subtype, Multipliable.tprod_of_nat_of_neg, Summable.hasProdUniformlyOn_one_add, Multipliable.tprod_le_tprod_of_inj, riemannZeta_eulerProduct_tprod, tprod_eq_tprod_diff_singleton, Multipliable.tprod_strict_mono, Multipliable.tprod_finsetProd, tprod_eq_prod, hasProdUniformlyOn_of_clog, Multipliable.tsum_congr_cofinite₀, Real.rexp_tsum_eq_tprod, tprod_one_sub_ordered, Multipliable.le_tprod, tprod_bot, Multipliable.tprod_union_disjoint, Multipliable.tprod_sum, Multipliable.tprod_comm, Multipliable.tprod_eq_one_iff, ENNReal.tprod_eq_iInf_prod, tendsto_tprod_compl_atTop_one, tprod_pi_single, Multipliable.tprod_comm', Multipliable.sigma, MeasureTheory.Measure.infinitePi_pi_univ, tprod_range, Multipliable.map_tprod, MeasureTheory.Measure.infinitePi_pi_of_countable, Nat.Partition.genFun_eq_tprod, tprod_le_of_prod_le', tprod_congr_subtype, tendsto_prod_nat_add
tsum 📖CompOp
674 mathmath: logDeriv_tprod_eq_tsum, MeasureTheory.hasFiniteIntegral_count_iff_enorm, Complex.cos_eq_tsum, PMF.tsum_coe, tsum_setProd_singleton_right, tsum_congr₂, ordinaryHypergeometric_sum_eq, summable_sigma_of_nonneg, Summable.sum_add_tsum_subtype_compl, completedZeta_eq_tsum_of_one_lt_re, EisensteinSeries.hasSum_e2Summand_symmetricIcc, Real.fourierCoeff_tsum_comp_add, tendsto_tsum_div_pow_atTop_integral, Summable.sum_add_tsum_compl, Real.tsum_eq_tsum_fourier, MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum'', PeriodPair.coeff_weierstrassPExceptSeries, Complex.hasSum_deriv_of_summable_norm, MeasureTheory.IsAddFundamentalDomain.integral_eq_tsum, intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm, geom_series_mul_shift, LSeries.term_convolution', tsum_eisSummand_eq_riemannZeta_mul_eisensteinSeries, continuousOn_tsum, ENNReal.tsum_set_const, SummationFilter.tsum_symmetricIcc_eq_tsum_symmetricIco, tsum_coe_mul_geometric_of_norm_lt_one', lp.inner_eq_tsum, PMF.toMeasure_bind_apply, Nat.sumByResidueClasses, Complex.tsum_exp_neg_mul_int_sq, PMF.toMeasure_ofFinset_apply, Summable.tsum_add_tsum_compl, MeasureTheory.lintegral_iUnion₀, MeasureTheory.SimpleFunc.lintegral_sum, MeasureTheory.measure_biUnion, tsum_nat_add_neg, ContinuousAlternatingMap.tsum_eval, tsum_geometric_encode_lt_top, ENNReal.tendsto_nat_tsum, Summable.tsum_mul_tsum, cauchySeq_finset_iff_nat_tsum_vanishing, Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable, ContinuousLinearEquiv.map_tsum, two_mul_riemannZeta_eq_tsum_int_inv_pow_of_even, MeasureTheory.measure_preimage_fst_singleton_eq_tsum, ENNReal.tsum_le_tsum, ENNReal.tsum_iUnion_le_tsum, tsum_const_smul', MeasureTheory.measure_biUnion_le, ENNReal.exists_pos_sum_of_countable', tsum_choose_mul_geometric_of_norm_lt_one', ENNReal.tsum_mul_left, Function.LeftInverse.map_tsum, IsHilbertSum.linearIsometryEquiv_symm_apply, tsum_setProd_singleton_left, tsum_coe_mul_geometric_of_norm_lt_one, tsum_geometric_lt_top, Real.tendsto_tsum_powerSeries_nhdsWithin_lt, Summable.map_tsum, iteratedDerivWithin_tsum_cexp_eq, ENNReal.tsum_sub, tsum_int_eq_zero_add_tsum_pnat, ENNReal.tsum_eq_top_of_eq_top, Matrix.blockDiagonal_tsum, tsum_nonpos, Complex.taylorSeries_eq_of_entire, NNReal.summable_sigma, tsum_mul_tsum_of_nonarchimedean, ENNReal.tsum_lt_tsum, tsum_range, Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzCone, MeasureTheory.Measure.le_sum_apply, Besicovitch.exists_closedBall_covering_tsum_measure_le, MeasureTheory.lintegral_biUnion, Summable.nat_tsum_vanishing, tendstoUniformly_tsum_nat, PMF.toMeasure_ofFintype_apply, MeasureTheory.lintegral_countable, MeasureTheory.Measure.mkMetric_apply, MeasureTheory.Measure.sum_apply_of_countable, tsum_geometric_of_norm_lt_one, PowerSeries.aeval_eq_sum, Summable.sigma, PMF.toMeasure_apply, PMF.toMeasure_ofMultiset_apply, ModularForm.tsum_logDeriv_eta_q, Metric.PiNatEmbed.edist_def, ENNReal.tsum_toNNReal_eq, PMF.toOuterMeasure_ofFinset_apply, ENNReal.tsum_add, tendstoUniformly_tsum_of_cofinite_eventually, tsum_congr, IsUltrametricDist.norm_tsum_le, ENNReal.tsum_biUnion_le, Complex.cexp_tsum_eq_tprod, ENNReal.hasSum_toReal, NNReal.tsum_mul_right, BoxIntegral.unitPartition.integralSum_eq_tsum_div, Summable.tsum_pos, Complex.taylorSeries_eq_on_ball', lp.norm_rpow_eq_tsum, tsum_eq_tsum_diff_singleton, NormedSpace.exp_eq_tsum, FormalMultilinearSeries.nnnorm_changeOrigin_le, MeasureTheory.OuterMeasure.iInf_apply', VitaliFamily.FineSubfamilyOn.measure_le_tsum, geom_series_succ, EulerProduct.eulerProduct_completely_multiplicative, tsum_geometric_le_of_norm_lt_one, ZLattice.tsum_norm_rpow_le, NormedSpace.exp_eq_tsum_div, FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum, PiCountable.edist_eq_tsum, ENNReal.tsum_biUnion, MeasureTheory.Measure.mkMetric_le_liminf_tsum, Metric.PiNatEmbed.dist_def, tsum_subtype, ENNReal.tsum_const_eq_top_of_ne_zero, tendsto_tsum_of_dominated_convergence, Matrix.conjTranspose_tsum, Summable.tsum_const_smul, ContMDiff.sum_section_of_locallyFinite, intervalIntegral.hasSum_intervalIntegral_of_summable_norm, EulerProduct.eulerProduct_hasProd, Summable.tsum_sigma, Real.tsum_exp_neg_mul_int_sq, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, edist_le_tsum_of_edist_le_of_tendsto₀, PadicInt.mahlerSeries_apply, ProbabilityTheory.Kernel.compProd_eq_tsum_compProd, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, NormedSpace.expSeries_sum_eq, Summable.tsum_prod', tsum_mul_tsum_of_summable_norm, hasDerivAt_tsum_of_isPreconnected, lowerSemicontinuousWithinAt_tsum, Summable.tsum_mul_tsum_eq_tsum_sum_range, EisensteinSeries.G2_eq_tsum_symmetricIco, tsum_eq_finsum, Summable.tendsto_alternating_series_tsum, Complex.cos_eq_tsum', ENNReal.tendsto_tsum_compl_atTop_zero, Summable.tsum_eq_zero_add, cot_series_rep', MeasureTheory.Content.innerContent_iSup_nat, ArithmeticFunction.IsMultiplicative.eulerProduct_hasProd, edist_le_tsum_of_edist_le_of_tendsto, tsum_eq_of_summable_unconditional, tsum_eq_tsum_primes_of_support_subset_prime_powers, EulerProduct.eulerProduct, MeasureTheory.lintegral_iUnion_le, mul_neg_geom_series, ZLattice.tsumNormRPowBound_spec, MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum_of_ac, tendsto_sub_mul_tsum_nat_cpow, MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum', MeasureTheory.IsFundamentalDomain.setIntegral_eq_tsum', ENNReal.coe_tsum, ENNReal.tsum_set_one, MvPowerSeries.WithPiTopology.as_tsum, MDifferentiableOn.sum_section_of_locallyFinite, tsum_zero_pnat_eq_tsum_nat, summableUniformly_iff_hasSumUniformly, ZetaAsymptotics.term_tsum_of_lt, NNReal.hasSum_real_toNNReal_of_nonneg, MeasureTheory.IsFundamentalDomain.setLIntegral_eq_tsum, Real.inner_le_Lp_mul_Lq_tsum_of_nonneg', tsum_pow_div_one_sub_eq_tsum_sigma, Equiv.tsum_eq, PowerSeries.eval₂_eq_tsum, ENNReal.tsum_schlomilch_le, riemannZeta_eulerProduct_exp_log, ProbabilityTheory.tsum_prob_mem_Ioi_lt_top, tsum_pnat_eq_tsum_succ, tsum_image, tsum_subtype_eq_of_support_subset, Summable.tsum_of_nat_of_neg, tsum_geometric_of_lt_one, zeta_eq_tsum_one_div_nat_cpow, ENNReal.tsum_iSup_eq, tendstoUniformlyOn_tsum, EulerProduct.eulerProduct_tprod, Function.Surjective.tsum_eq_tsum_of_hasSum_iff_hasSum, MeasureTheory.Measure.sum_apply, ENNReal.tsum_top, StieltjesFunction.length_subadditive_Icc_Ioo, IsUltrametricDist.norm_tsum_le_of_forall_le, PeriodPair.weierstrassPExcept_eq_tsum, ENNReal.tsum_union_le, Summable.tsum_mul_tsum_eq_tsum_sum_antidiagonal, ENNReal.tsum_le_of_sum_range_le, hasDerivAt_tsum, Real.tsum_eq_tsum_fourier_of_rpow_decay_of_summable, ENNReal.tsum_geometric_add_one, ENNReal.tsum_eq_iSup_nat', ENNReal.tsum_sigma', MeasureTheory.OuterMeasure.iInf_apply, MeasureTheory.withDensity_tsum, iteratedFDeriv_tsum, hasSum_sum_range_mul_of_summable_norm', ContinuousMap.periodic_tsum_comp_add_zsmul, tsum_mul_left, ArithmeticFunction.IsMultiplicative.eulerProduct_tprod, tsum_comp_neg, MeasureTheory.IsAddFundamentalDomain.setIntegral_eq_tsum, EulerProduct.prod_filter_prime_geometric_eq_tsum_factoredNumbers, tsum_congr_subtype, tendstoUniformlyOn_tsum_nat, tsum_le_of_sum_le', MeasureTheory.IsFundamentalDomain.setIntegral_eq_tsum, ENNReal.ofReal_tsum_of_nonneg, MeasureTheory.IsAddFundamentalDomain.integral_eq_tsum'', EisensteinSeries.hasSum_e2Summand_symmetricIco, Real.sin_eq_tsum, MeasureTheory.tsum_measure_le_measure_univ, ENNReal.tsum_eq_iSup_sum, MeasureTheory.Measure.restrict_iUnion_apply, EisensteinSeries.q_expansion_riemannZeta, EulerProduct.eulerProduct_completely_multiplicative_tprod, EulerProduct.prod_primesBelow_tsum_eq_tsum_smoothNumbers, FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum, MeasureTheory.OuterMeasure.biInf_apply, ContinuousLinearMap.map_tsum, NNReal.tsum_eq_add_tsum_ite, Real.sinh_eq_tsum, Nat.Partition.powerSeriesMk_card_restricted_eq_tprod, Real.cos_eq_tsum, ProbabilityTheory.lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self, MeasureTheory.IsAddFundamentalDomain.setLIntegral_eq_tsum', tsum_iSup_decode₂, Commute.tsum_right, ContinuousMap.tsum_apply, Summable.tsum_eq_add_tsum_ite', tsum_geometric_two, PMF.toOuterMeasure_ofMultiset_apply, Summable.tsum_le_tsum, lowerSemicontinuousOn_tsum, MeasureTheory.IsAddFundamentalDomain.integral_eq_tsum', MeasureTheory.FiniteMeasure.apply_iUnion_le, MeasureTheory.Measure.sum_apply₀, MeasureTheory.measure_iUnion₀, MeasureTheory.Measure.m_iUnion, MvPowerSeries.eval₂_eq_tsum, ENNReal.tsum_eq_iSup_sum', ENNReal.tsum_prod', MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum, MeasureTheory.lintegral_count, summableUniformlyOn_iff_hasSumUniformlyOn, tsum_div_const, tsum_riemannZetaSummand, tsum_extend_zero, tsum_int_eq_zero_add_two_mul_tsum_pnat, Real.Lp_add_le_tsum_of_nonneg, MeasureTheory.OuterMeasure.ofFunction_eq_iInf_mem, ENNReal.tsum_two_zpow_neg_add_one, EulerProduct.prod_primesBelow_geometric_eq_tsum_smoothNumbers, tsum_choose_mul_geometric_of_norm_lt_one, ordinaryHypergeometric_eq_tsum, NNReal.tsum_comp_le_tsum_of_inj, Summable.tsum_mul_left, Summable.tsum_prod, ContinuousMultilinearMap.tsum_eval, tsum_geometric_of_abs_lt_one, ProbabilityTheory.Kernel.withDensity_tsum, Complex.tsum_exp_neg_quadratic, tsum_apply, MeasureTheory.lintegral_countable', PMF.integral_eq_tsum, Real.tsum_le_of_sum_range_le, Complex.cosh_eq_tsum, PMF.seq_apply, NNReal.coe_tsum_of_nonneg, ENNReal.tsum_mul_right, MeasureTheory.IsFundamentalDomain.integral_eq_tsum'', tsum_of_enorm_bounded, PMF.toOuterMeasure_ofFintype_apply, Summable.tendsto_sum_tsum_nat, Summable.le_tsum', fderiv_tsum, NNReal.tendsto_sum_nat_add, EisensteinSeries.norm_le_tsum_norm, PowerSeries.WithPiTopology.one_sub_mul_tsum_pow_of_constantCoeff_eq_zero, MvPowerSeries.WithPiTopology.one_sub_mul_tsum_pow_of_constantCoeff_eq_zero, EulerProduct.eulerProduct_completely_multiplicative_hasProd, cot_series_rep, MeasureTheory.IsFundamentalDomain.setLIntegral_eq_tsum', dist_le_tsum_of_dist_le_of_tendsto₀, tsum_const_smul'', tprod_one_add, ENNReal.tsum_geometric_two, MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint, continuous_tsum, ENNReal.tsum_le_tsum_comp_of_surjective, Complex.abel_aux, tsum_mul_right, MDifferentiable.sum_section_of_locallyFinite, MeasureTheory.measure_iUnion, alternating_series_error_bound, MeasureTheory.integral_tsum_of_summable_integral_norm, tsum_congr_set_coe, MDifferentiableWithinAt.sum_section_of_locallyFinite, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum, ENNReal.tsum_const_smul, MeasureTheory.OuterMeasure.iUnion_nat, tsum_geometric_two', Summable.one_sub_mul_tsum_pow, ArithmeticFunction.IsMultiplicative.eulerProduct, Nat.Partition.hasProd_powerSeriesMk_card_restricted, Summable.prod, EulerProduct.prod_filter_prime_tsum_eq_tsum_factoredNumbers, ENNReal.tsum_sigma, ENNReal.tsum_eq_limsup_sum_nat, FormalMultilinearSeries.ofScalarsSum_eq_tsum, ZetaAsymptotics.zeta_limit_aux1, tsum_of_nat_of_neg_add_one, RCLike.conj_tsum, IsUltrametricDist.nnnorm_tsum_le_of_forall_le, tsum_zero, PMF.toOuterMeasure_bindOnSupport_apply, Summable.tsum_le_of_sum_range_le, EulerProduct.eulerProduct_hasProd_mulIndicator, Summable.toCompl_tsum, ENNReal.le_tsum, Summable.tsum_subtype_le, Summable.tsum_subtype_add_tsum_subtype_compl, contDiffOn_tsum_cexp, dist_le_tsum_dist_of_tendsto₀, ENNReal.exists_pos_sum_of_countable, ENNReal.le_tsum_of_forall_lt_exists_sum, MeasureTheory.VectorMeasure.of_disjoint_iUnion, sum_eq_tsum_indicator, MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum'', EisensteinSeries.qExpansion_identity_pnat, ENNReal.tsum_condensed_le, zeta_eq_tsum_one_div_nat_add_one_cpow, IsUltrametricDist.norm_tsum_le_of_forall_le_of_nonneg, MeasureTheory.IsFundamentalDomain.measure_eq_tsum_of_ac, tsum_eq_tsum_of_hasSum_iff_hasSum, MeasureTheory.Measure.hausdorffMeasure_le_liminf_tsum, Summable.tsum_le_of_sum_le, Topology.IsClosedEmbedding.map_tsum, Summable.tsum_lt_tsum, PeriodPair.coeff_weierstrassPSeries, dist_le_tsum_dist_of_tendsto, HasSumUniformlyOn.tsum_eq, MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum_of_ac, MvPowerSeries.WithPiTopology.tsum_pow_mul_one_sub_of_constantCoeff_eq_zero, ENNReal.exists_pos_tsum_mul_lt_of_countable, HasSum.tsum_eq, MeasureTheory.IsAddFundamentalDomain.measure_eq_tsum, tsum_geometric_inv_two_ge, MDifferentiableAt.sum_section_of_locallyFinite, tendstoUniformlyOn_tsum_of_cofinite_eventually, MeasureTheory.measure_iUnion_le, hasSum_sum_range_mul_of_summable_norm, tsum_dite_left, NNReal.tsum_lt_tsum, MeasureTheory.ProbabilityMeasure.apply_iUnion_le, MeasureTheory.integral_sum_measure, NNReal.tsum_eq_toNNReal_tsum, Summable.sum_le_tsum, ENNReal.tsum_coe_eq, tsum_nat_add_neg_add_one, tsum_unop, tsum_tsum_eq_single, RCLike.im_tsum, Quaternion.tsum_coe, tsum_iUnion_decode₂, deriv_tsum, MeasureTheory.IsFundamentalDomain.measure_eq_tsum', lowerSemicontinuousAt_tsum, MeasureTheory.OuterMeasure.sInf_apply, Equiv.tsum_eq_tsum_of_support, PowerSeries.as_tsum, tsum_even_add_odd, geom_series_mul_one_add, Summable.hasSum, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, Matrix.transpose_tsum, ENNReal.tsum_comp_le_tsum_of_injective, ENNReal.sum_le_tsum, tprod_one_add_ordered, Complex.taylorSeries_eq_on_eball, tsum_eq_sum, measure_eq_measure_preimage_add_measure_tsum_Ico_zpow, Complex.re_tsum, lp.tsum_mul_le_mul_norm', nnnorm_tsum_le, EisensteinSeries.q_expansion_bernoulli, tendsto_tsum_compl_atTop_zero, tsum_prod_pow_eq_tsum_sigma, norm_tsum_le_tsum_norm, MeasureTheory.Content.innerContent_iUnion_nat, ContMDiffWithinAt.sum_section_of_locallyFinite, MeasureTheory.OuterMeasure.boundedBy_apply, MeasureTheory.IsAddFundamentalDomain.setLIntegral_eq_tsum, tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm', ENNReal.le_tsum_condensed, tsum_dirichletSummand, Real.inner_le_Lp_mul_Lq_tsum_of_nonneg, Topology.IsInducing.summable_iff_tsum_comp_mem_range, MeasureTheory.addContent_iUnion_eq_sum_of_tendsto_zero, ArithmeticFunction.vonMangoldt.LSeries_residueClass_lower_bound, Measurable.ennreal_tsum, tsum_geometric_inv_two, HasSumUniformlyOn.tsum_eqOn, tsum_empty, EulerProduct.norm_tsum_factoredNumbers_sub_tsum_lt, DirichletCharacter.LSeries_eulerProduct_exp_log, SummableLocallyUniformlyOn.differentiableOn, Summable.hasSum_iff, Summable.tsum_pow_mul_one_sub, tsum_eq_tsum_of_ne_zero_bij, tsum_neg, ENNReal.sum_add_tsum_compl, MeasureTheory.lintegral_count', MeasureTheory.lintegral_iUnion, ENNReal.tsum_biUnion', tsum_sq_fourierCoeffOn, ENNReal.tendsto_sum_nat_add, PowerSeries.WithPiTopology.tsum_pow_mul_one_sub_of_constantCoeff_eq_zero, Summable.tsum_union_disjoint, Summable.tsum_finset_bUnion_disjoint, EisensteinSeries.tendsto_double_sum_S_act, Complex.sinh_eq_tsum, MeasureTheory.addContent_iUnion_eq_tsum_of_disjoint_of_IsSigmaSubadditive, MeasureTheory.IsFundamentalDomain.integral_eq_tsum_of_ac, tsum_fintype, ENNReal.tsum_prod, NNReal.Lp_add_le_tsum', MeasureTheory.tsum_measure_preimage_singleton, Summable.tsum_sigma', Complex.taylorSeries_eq_on_emetric_ball, dist_le_tsum_of_dist_le_of_tendsto, MeasureTheory.IsAddFundamentalDomain.integral_eq_tsum_of_ac, hasFDerivAt_tsum, NNReal.sum_add_tsum_nat_add, tsum_eq_single, Complex.sin_eq_tsum, Real.tsum_eq_tsum_fourier_of_rpow_decay, deriv_tsum_apply, MeasureTheory.IsAddFundamentalDomain.measure_eq_tsum', enorm_tsum_le_tsum_enorm, Summable.tsum_lt_tsum_of_nonneg, Nat.Partition.multipliable_powerSeriesMk_card_restricted, Function.Injective.tsum_eq, Summable.tsum_eq_add_tsum_ite, Real.tsum_le_of_sum_le, PeriodPair.hasSumLocallyUniformly_aux, ENNReal.tsum_coe_eq_top_iff_not_summable_coe, MeasureTheory.Measure.le_count_apply, Complex.taylorSeries_eq_on_ball, ContinuousLinearEquiv.tsum_eq_iff, tendsto_sub_mul_tsum_nat_rpow, NormedSpace.expSeries_sum_eq_div, Summable.tsum_smul_const, ENNReal.tsum_eq_iSup_nat, tsum_dite_right, EulerProduct.one_sub_inv_eq_geometric_of_summable_norm, MeasureTheory.Measure.restrict_iUnion_apply_ae, MeasureTheory.integral_countable, MeasureTheory.measure_sUnion₀, tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm, EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum, summable_iff_tsum_vanishing, derivWithin_tsum, tsum_eq_zero_of_not_summable, NNReal.tsum_strict_mono, ENNReal.tsum_eq_liminf_sum_nat, ContMDiffOn.sum_section_of_locallyFinite, Summable.sum_add_tsum_nat_add', RCLike.ofReal_tsum, MeasureTheory.Measure.hausdorffMeasure_apply, tsum_mul_tsum_of_summable_norm', Summable.tsum_comm', EisensteinSeries.G2_eq_tsum_cexp, tsum_pi_single, summable_iff_cauchySeq_finset_and_tsum_mem, tsum_int_rec, Summable.tsum_sum, EisensteinSeries.qExpansion_identity, VitaliFamily.FineSubfamilyOn.measure_le_tsum_of_absolutelyContinuous, NNReal.inner_le_Lp_mul_Lq_tsum', tsum_eisSummand_eq_tsum_sigma_mul_cexp_pow, AEMeasurable.ennreal_tsum, tsum_setElem_eq_tsum_setElem_diff, ContMDiffAt.sum_section_of_locallyFinite, MeasureTheory.OuterMeasure.sInf_apply', tsum_ite_eq, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, ArithmeticFunction.LSeries_zeta_eulerProduct_exp_log, ENNReal.tsum_geometric_two_encode_le_two, jacobiTheta_eq_tsum_nat, MvPowerSeries.aeval_eq_sum, PMF.toMeasure_bindOnSupport_apply, ENNReal.tsum_comm, Complex.taylorSeries_eq_on_eball', tendsto_sum_nat_add, MeasureTheory.OuterMeasure.sum_apply, FormalMultilinearSeries.ofScalars_sum_eq, tsum_star, Summable.tsum_eq_zero_iff, PMF.toOuterMeasure_apply, HasSumLocallyUniformlyOn.tsum_eqOn, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, HasSumUniformlyOn.of_norm_le_summable, ProbabilityTheory.Fernique.lintegral_exp_mul_sq_norm_le_mul, tsum_smul_tsum, summableLocallyUniformlyOn_iff_hasSumLocallyUniformlyOn, Summable.le_tsum, tsum_comp_le_tsum_of_inj, MeasureTheory.preVariation.sum_le_preVariationFun_iUnion, Summable.sigma', SummableLocallyUniformly.hasSumLocallyUniformly, geom_series_mul_neg, contDiff_tsum, Summable.tsum_strict_mono, MeasureTheory.Measure.tsum_indicator_apply_singleton, MeasureTheory.integral_tsum, MeasureTheory.integral_iUnion, lp.norm_eq_tsum_rpow, Summable.tsum_le_tsum_of_inj, differentiable_tsum', ENNReal.tsum_fiberwise, contDiff_tsum_of_eventually, Summable.tsum_mono, NNReal.tsum_mul_left, SummableLocallyUniformlyOn.hasSumLocallyUniformlyOn, ENNReal.tsum_apply, MeasureTheory.SimpleFunc.tsum_eapproxDiff, Nat.Partition.hasProd_genFun, NormedSpace.exp_eq_tsum_rat, Summable.tsum_mul_right, Summable.tsum_sub, MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum', Complex.sin_eq_tsum', Summable.tsum_prod_uncurry, NNReal.tsum_le_of_sum_range_le, tendstoUniformly_tsum, SummableUniformlyOn.hasSumUniformlyOn, tsum_of_nnnorm_bounded, tsum_const, tsum_sq_fourierCoeff, MeasureTheory.measure_sUnion, PMF.filter_apply, ENNReal.tsum_mono_subtype, MeasureTheory.IsAddFundamentalDomain.measure_eq_tsum_of_ac, Complex.ofReal_tsum, PiCountable.dist_eq_tsum, iteratedDerivWithin_tsum, NNReal.inner_le_Lp_mul_Lq_tsum, Commute.tsum_left, AEMeasurable.nnreal_tsum, NNReal.tendsto_tsum_compl_atTop_zero, MeasureTheory.OuterMeasureClass.measure_iUnion_nat_le, summable_iff_summable_compl_and_tsum_mem, Matrix.diagonal_tsum, ProbabilityTheory.Kernel.sum_apply', MeasureTheory.OuterMeasure.biInf_apply', tsum_nonneg, ENNReal.tsum_geometric, NNReal.coe_tsum, tsum_singleton, Real.tsum_eq_tsum_fourierIntegral, SchwartzMap.tsum_eq_tsum_fourierIntegral, ENNReal.tsum_const, PMF.bind_apply, EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum, SummableUniformly.hasSumUniformly, Complex.taylorSeries_eq_of_entire', Complex.conj_tsum, Real.cosh_eq_tsum, MeasureTheory.OuterMeasure.f_iUnion, MeasureTheory.lintegral_tsum, Real.rexp_tsum_eq_tprod, Matrix.blockDiagonal'_tsum, tprod_one_sub_ordered, ZLattice.sum_piFinset_Icc_rpow_le, lowerSemicontinuous_tsum, HilbertBasis.tsum_inner_mul_inner, tsum_subtype_support, MeasureTheory.IsAddFundamentalDomain.setIntegral_eq_tsum', MeasureTheory.OuterMeasure.iUnion_eq_of_caratheodory, MeasureTheory.lintegral_biUnion₀, Summable.tsum_vanishing, zeta_nat_eq_tsum_of_gt_one, OrthogonalFamily.linearIsometry_apply, PMF.map_apply, tsum_of_norm_bounded, Finset.tsum_subtype, Summable.sum_add_tsum_nat_add, MeasureTheory.measure_preimage_snd_singleton_eq_tsum, tsum_op, MeasureTheory.lintegral_sum_measure, NNReal.tsum_pos, PMF.toOuterMeasure_bind_apply, tendstoUniformlyOn_tsum_nat_eventually, Summable.tsum_finsetSum, MeasureTheory.hasSum_integral_of_summable_integral_norm, Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay, ENNReal.tsum_iUnion_le, tsum_def, differentiable_tsum, MeasureTheory.OuterMeasure.ofFunction_apply, iteratedFDeriv_tsum_apply, ENNReal.tsum_one, PMF.bindOnSupport_apply, NNReal.Lp_add_le_tsum, tsum_of_add_one_of_neg_add_one, MeasureTheory.IsFundamentalDomain.integral_eq_tsum', Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet, ProbabilityTheory.mgf_sum_measure, HasSumUniformlyOn.of_norm_le_summable_eventually, summableLocallyUniformly_iff_hasSumLocallyUniformly, Nat.Partition.multipliable_genFun, summable_partition, tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm', hasFDerivAt_tsum_of_isPreconnected, tsum_bot, ENNReal.tsum_biUnion_le_tsum, Measurable.ennreal_tsum', tsum_univ, cauchySeq_finset_iff_tsum_vanishing, geom_series_eq_inverse, tsum_bool, tsum_geometric_nnreal, Summable.tsum_comm, MeasureTheory.integral_iUnion_ae, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, ENNReal.tsum_eq_zero, fderiv_tsum_apply, Measurable.nnreal_tsum, Summable.hasSumUniformlyOn_log_one_add, ENNReal.tsum_eq_add_tsum_ite, RCLike.re_tsum, summable_prod_of_nonneg, Summable.tsum_ofReal_lt_top, MeasureTheory.IsFundamentalDomain.measure_eq_tsum, MeasureTheory.integral_countable', Complex.im_tsum, tsum_eq_tsum_primes_add_tsum_primes_of_support_subset_prime_powers, Complex.taylorSeries_eq_on_emetric_ball', ENNReal.tsum_toReal_eq, Complex.differentiableOn_tsum_of_summable_norm, summable_iff_nat_tsum_vanishing, Nat.Partition.genFun_eq_tprod, EulerProduct.norm_tsum_smoothNumbers_sub_tsum_lt, MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint₀, tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm, tsum_eq_zero_add', EulerProduct.exp_tsum_primes_log_eq_tsum, ENNReal.le_tsum_schlomilch, Complex.tendsto_tsum_powerSeries_nhdsWithin_lt, Summable.tsum_add, MeasureTheory.IsFundamentalDomain.integral_eq_tsum, IsUltrametricDist.nnnorm_tsum_le, tsum_eq_sum', pi_mul_cot_pi_q_exp, SchwartzMap.tsum_eq_tsum_fourier, ZLattice.exists_finsetSum_norm_rpow_le_tsum, Summable.tendstoUniformlyOn_tsum_nat_log_one_add, lp.tsum_mul_le_mul_norm, tsum_eq_of_filter_le, Orthonormal.tsum_inner_products_le, MeasureTheory.measure_biUnion₀, PMF.toMeasure_apply_eq_tsum, tendsto_zero_geometric_tsum_pnat, Finset.tsum_subtype', Real.Lp_add_le_tsum_of_nonneg', HasSum.tsum_fiberwise, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq, PMF.normalize_apply
«term∏'[_]_,_» 📖CompOp
«term∏'_,_» 📖CompOp
«term∑'[_]_,_» 📖CompOp
«term∑'_,_» 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
hasProd_bot 📖mathematicalSummationFilter.NeBotHasProdMathlib.Tactic.Contrapose.contrapose₂
HasProd.eq_1
Filter.tendsto_bot
hasProd_fintype 📖mathematicalHasProd
Finset.prod
Finset.univ
Finset.prod_congr
Set.toFinset_congr
SummationFilter.support_eq_univ
Set.toFinset_univ
hasProd_fintype_support
SummationFilter.instHasSupportOfLeAtTop
hasProd_fintype_support 📖mathematicalHasProd
Finset.prod
Set.toFinset
SummationFilter.support
Subtype.fintype
Set
Set.instMembership
tendsto_nhds_of_eventually_eq
Filter.biInter_mem
Set.toFinite
Subtype.finite
Finite.of_fintype
SummationFilter.eventually_mem_or_not_mem
Filter.mp_mem
Filter.univ_mem'
Set.iInter_congr_Prop
hasProd_prod_of_ne_finset_one 📖mathematicalMulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
HasProd
Finset.prod
Filter.Tendsto.mono_left
hasProd_subtype_iff_of_mulSupport_subset
Function.mulSupport_subset_iff'
Finset.hasProd
SummationFilter.instLeAtTopUnconditional
SummationFilter.LeAtTop.le_atTop
hasProd_prod_support_of_ne_finset_one 📖mathematicalMulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
HasProd
Finset.prod
Set.toFinset
Set
Set.instInter
SetLike.coe
Finset
Finset.instSetLike
SummationFilter.support
Set.fintypeInterOfLeft
FinsetCoe.fintype
tendsto_nhds_of_eventually_eq
Filter.biInter_mem
Set.toFinite
Finite.Set.finite_inter_of_left
Finite.of_fintype
Filter.mp_mem
SummationFilter.HasSupport.eventually_le_support
Filter.univ_mem'
Finset.prod_congr_of_eq_on_inter
hasProd_subtype_comap_iff_of_mulSupport_subset 📖mathematicalSet
Set.instHasSubset
Function.mulSupport
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
HasProd
Set.Elem
Set.instMembership
SummationFilter.comap
Function.Embedding.subtype
Function.Injective.hasProd_comap_iff
Subtype.coe_injective
Subtype.range_coe_subtype
Function.mulSupport_subset_iff'
hasProd_subtype_iff_of_mulSupport_subset 📖mathematicalSet
Set.instHasSubset
Function.mulSupport
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
HasProd
Set.Elem
Set.instMembership
SummationFilter.unconditional
SummationFilter.comap_unconditional
hasProd_subtype_comap_iff_of_mulSupport_subset
hasSum_bot 📖mathematicalSummationFilter.NeBotHasSumMathlib.Tactic.Contrapose.contrapose₂
HasSum.eq_1
Filter.tendsto_bot
hasSum_fintype 📖mathematicalHasSum
Finset.sum
Finset.univ
Finset.sum_congr
Set.toFinset_congr
SummationFilter.support_eq_univ
Set.toFinset_univ
hasSum_fintype_support
SummationFilter.instHasSupportOfLeAtTop
hasSum_fintype_support 📖mathematicalHasSum
Finset.sum
Set.toFinset
SummationFilter.support
Subtype.fintype
Set
Set.instMembership
tendsto_nhds_of_eventually_eq
Filter.biInter_mem
Set.toFinite
Subtype.finite
Finite.of_fintype
SummationFilter.eventually_mem_or_not_mem
Filter.mp_mem
Filter.univ_mem'
Set.mem_iInter
Set.iInter_congr_Prop
Set.mem_compl_iff
hasSum_subtype_comap_iff_of_support_subset 📖mathematicalSet
Set.instHasSubset
Function.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
HasSum
Set.Elem
Set.instMembership
SummationFilter.comap
Function.Embedding.subtype
Function.Injective.hasSum_comap_iff
Subtype.coe_injective
Subtype.range_coe_subtype
Function.support_subset_iff'
hasSum_subtype_iff_of_support_subset 📖mathematicalSet
Set.instHasSubset
Function.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
HasSum
Set.Elem
Set.instMembership
SummationFilter.unconditional
SummationFilter.comap_unconditional
hasSum_subtype_comap_iff_of_support_subset
hasSum_sum_of_ne_finset_zero 📖mathematicalAddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
HasSum
Finset.sum
Filter.Tendsto.mono_left
hasSum_subtype_iff_of_support_subset
Function.support_subset_iff'
Finset.hasSum
SummationFilter.instLeAtTopUnconditional
SummationFilter.LeAtTop.le_atTop
hasSum_sum_support_of_ne_finset_zero 📖mathematicalAddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
HasSum
Finset.sum
Set.toFinset
Set
Set.instInter
SetLike.coe
Finset
Finset.instSetLike
SummationFilter.support
Set.fintypeInterOfLeft
FinsetCoe.fintype
tendsto_nhds_of_eventually_eq
Filter.biInter_mem
Set.toFinite
Finite.Set.finite_inter_of_left
Finite.of_fintype
Filter.mp_mem
SummationFilter.HasSupport.eventually_le_support
Filter.univ_mem'
Finset.sum_congr_of_eq_on_inter
Set.mem_iInter
multipliable_bot 📖mathematicalSummationFilter.NeBotMultipliablehasProd_bot
multipliable_of_ne_finset_one 📖mathematicalMulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
MultipliableHasProd.multipliable
hasProd_prod_support_of_ne_finset_one
summable_bot 📖mathematicalSummationFilter.NeBotSummablehasSum_bot
summable_of_ne_finset_zero 📖mathematicalAddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SummableHasSum.summable
hasSum_sum_support_of_ne_finset_zero
tprod_bot 📖mathematicalSummationFilter.NeBottprod
finprod
multipliable_bot
tprod_def
SummationFilter.support_eq_univ
SummationFilter.leAtTop_of_not_NeBot
Set.inter_univ
Set.mulIndicator_univ
eq_true_intro
SummationFilter.instHasSupportOfLeAtTop
hasProd_bot
finprod_of_infinite_mulSupport
tprod_def 📖mathematicaltprod
Multipliable
SummationFilter.HasSupport
Set.Finite
Set
Set.instInter
Function.mulSupport
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
SummationFilter.support
finprod
Set.mulIndicator
HasProd
tprod_eq_of_filter_le 📖mathematicalFilter
Finset
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
SummationFilter.filter
Multipliable
tprodMultipliable.hasProd_iff
Multipliable.mono_filter
Filter.Tendsto.mono_left
Multipliable.hasProd
tprod_eq_of_multipliable_unconditional 📖mathematicalMultipliable
SummationFilter.unconditional
tprodtprod_eq_of_filter_le
SummationFilter.LeAtTop.le_atTop
tprod_eq_one_of_not_multipliable 📖mathematicalMultipliabletprod
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
tprod_def
tsum_bot 📖mathematicalSummationFilter.NeBottsum
finsum
summable_bot
tsum_def
SummationFilter.support_eq_univ
SummationFilter.leAtTop_of_not_NeBot
Set.inter_univ
Set.indicator_univ
eq_true_intro
SummationFilter.instHasSupportOfLeAtTop
hasSum_bot
finsum_of_infinite_support
tsum_def 📖mathematicaltsum
Summable
SummationFilter.HasSupport
Set.Finite
Set
Set.instInter
Function.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SummationFilter.support
finsum
Set.indicator
HasSum
tsum_eq_of_filter_le 📖mathematicalFilter
Finset
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
SummationFilter.filter
Summable
tsumSummable.hasSum_iff
Summable.mono_filter
Filter.Tendsto.mono_left
Summable.hasSum
tsum_eq_of_summable_unconditional 📖mathematicalSummable
SummationFilter.unconditional
tsumtsum_eq_of_filter_le
SummationFilter.LeAtTop.le_atTop
tsum_eq_zero_of_not_summable 📖mathematicalSummabletsum
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
tsum_def

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