AbelSummation

📁 Source: Mathlib/NumberTheory/AbelSummation.lean

Statistics

Metric	Count
Definitions	0
Theorems integrableOn_mul_sum_Icc, locallyIntegrableOn_mul_sum_Icc, sum_mul_eq_sub_integral_mul, sum_mul_eq_sub_integral_mul', sum_mul_eq_sub_integral_mul₀, sum_mul_eq_sub_integral_mul₀', sum_mul_eq_sub_integral_mul₁, sum_mul_eq_sub_sub_integral_mul, sum_mul_eq_sub_sub_integral_mul', summable_mul_of_bigO_atTop, summable_mul_of_bigO_atTop', tendsto_sum_mul_atTop_nhds_one_sub_integral, tendsto_sum_mul_atTop_nhds_one_sub_integral₀	13
Total	13

(root)

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
integrableOn_mul_sum_Icc 📖	mathematical	Real Real.instLE Real.instZero MeasureTheory.IntegrableOn MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing Set.Icc Real.instPreorder MeasureTheory.MeasureSpace.volume	Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing Finset.Icc Nat.instPreorder Nat.instLocallyFiniteOrder Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing	—	Real.instIsStrictOrderedRing le_or_gt eq_or_lt_of_le Nat.floor_le_floor MeasureTheory.Measure.instOuterMeasureClass Filter.mp_mem Filter.univ_mem' LE.le.trans Nat.floor_le LT.lt.le Nat.lt_floor_add_one MeasureTheory.IntegrableOn.congr_fun_ae MeasureTheory.Integrable.mul_const Filter.EventuallyEq.mul Filter.EventuallyEq.refl MeasureTheory.ae_restrict_iff' measurableSet_Icc BorelSpace.opensMeasurable Real.borelSpace HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid intervalIntegrable_iff_integrableOn_Icc_of_le PseudoEMetricSpace.pseudoMetrizableSpace Real.noAtoms_volume enorm_ne_top MeasureTheory.Integrable.congr MeasureTheory.IntegrableOn.mono_set Set.Icc_subset_Icc MeasureTheory.ae_restrict_of_ae_restrict_of_subset Nat.pos_of_floor_pos LE.le.trans_lt Nat.cast_add_one Nat.le_floor_iff le_of_lt Nat.floor_lt Nat.cast_one IsOrderedAddMonoid.toAddLeftMono Real.instZeroLEOneClass RCLike.charZero_rclike le_rfl IntervalIntegrable.trans_iterate_Ico IntervalIntegrable.trans Set.Icc_eq_empty_of_lt MeasureTheory.integrableOn_empty
locallyIntegrableOn_mul_sum_Icc 📖	mathematical	Real Real.instLE Real.instZero MeasureTheory.LocallyIntegrableOn Real.measurableSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing Set.Ici Real.instPreorder MeasureTheory.MeasureSpace.volume Real.measureSpace	Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing Finset.Icc Nat.instPreorder Nat.instLocallyFiniteOrder Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing	—	Real.instIsStrictOrderedRing MeasureTheory.locallyIntegrableOn_iff PseudoEMetricSpace.pseudoMetrizableSpace locallyCompact_of_proper instProperSpaceReal isLocallyClosed_Ici instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid IsCompact.sInf_mem instClosedIicTopology MeasureTheory.IntegrableOn.mono_set integrableOn_mul_sum_Icc LE.le.trans MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset Set.Icc_subset_Ici_iff Real.sInf_le_sSup IsCompact.bddBelow IsCompact.bddAbove CompactIccSpace.isCompact_Icc ConditionallyCompleteLinearOrder.toCompactIccSpace instOrderTopologyReal Bornology.IsBounded.subset_Icc_sInf_sSup Real.instIsOrderBornology IsCompact.isBounded Set.not_nonempty_iff_eq_empty MeasureTheory.integrableOn_empty
sum_mul_eq_sub_integral_mul 📖	mathematical	Real Real.instLE Real.instZero DifferentiableAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddCommGroup.toAddCommGroup NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing MeasureTheory.IntegrableOn MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Icc Real.instPreorder MeasureTheory.MeasureSpace.volume	Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.Icc Nat.instPreorder Nat.instLocallyFiniteOrder Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing Real.instNatCast SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup MeasureTheory.integral NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.restrict Set.Ioc	—	Real.instIsStrictOrderedRing Finset.left_notMem_Ioc Finset.Icc_eq_cons_Ioc Finset.sum_cons Nat.floor_zero sum_mul_eq_sub_sub_integral_mul le_rfl Nat.cast_zero Finset.Icc_self Finset.sum_singleton Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.mul_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.add_mul Mathlib.Tactic.Ring.mul_add Mathlib.Tactic.Ring.mul_pf_left Mathlib.Tactic.Ring.mul_pf_right Mathlib.Tactic.Ring.one_mul Mathlib.Tactic.Ring.mul_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.zero_mul Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.add_pf_add_gt Mathlib.Tactic.Ring.add_pf_add_lt Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Tactic.Ring.add_overlap_pf_zero Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_add
sum_mul_eq_sub_integral_mul' 📖	mathematical	DifferentiableAt Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddCommGroup.toAddCommGroup NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing MeasureTheory.IntegrableOn MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Icc Real.instPreorder Real.instZero Real.instNatCast MeasureTheory.MeasureSpace.volume	Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.Icc Nat.instPreorder Nat.instLocallyFiniteOrder Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup MeasureTheory.integral NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.restrict Set.Ioc Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing	—	Real.instIsStrictOrderedRing Finset.sum_congr Nat.floor_natCast sum_mul_eq_sub_integral_mul Nat.cast_nonneg Real.instIsOrderedRing
sum_mul_eq_sub_integral_mul₀ 📖	mathematical	MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField DifferentiableAt Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddCommGroup.toAddCommGroup NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing MeasureTheory.IntegrableOn MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Icc Real.instPreorder Real.instOne MeasureTheory.MeasureSpace.volume	Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.Icc Nat.instPreorder Nat.instLocallyFiniteOrder Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing Distrib.toMul NonUnitalNonAssocSemiring.toDistrib Real.instNatCast SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup MeasureTheory.integral NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.restrict Set.Ioc	—	Real.instIsStrictOrderedRing le_or_gt Nat.one_le_floor_iff Finset.left_notMem_Ioc Finset.Icc_eq_cons_Ioc Finset.sum_cons Finset.Icc_add_one_left_eq_Ioc Nat.instNoMaxOrder zero_add Nat.floor_one sum_mul_eq_sub_sub_integral_mul zero_le_one Real.instZeroLEOneClass Nat.cast_one Nat.instZeroLEOneClass Nat.Ioc_succ_singleton Finset.sum_singleton MulZeroClass.mul_zero Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.mul_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.add_mul Mathlib.Tactic.Ring.mul_add Mathlib.Tactic.Ring.mul_pf_left Mathlib.Tactic.Ring.mul_pf_right Mathlib.Tactic.Ring.one_mul Mathlib.Tactic.Ring.mul_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.zero_mul Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.cast_zero Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_zero Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.add_pf_add_gt Mathlib.Tactic.Ring.add_pf_add_lt Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Tactic.Ring.add_overlap_pf_zero Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_add Finset.sum_congr Nat.floor_eq_zero Finset.Icc_self Set.Ioc_eq_empty_of_le LT.lt.le MeasureTheory.Measure.restrict_empty MeasureTheory.integral_zero_measure sub_self
sum_mul_eq_sub_integral_mul₀' 📖	mathematical	MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField DifferentiableAt Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddCommGroup.toAddCommGroup NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing MeasureTheory.IntegrableOn MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Icc Real.instPreorder Real.instOne Real.instNatCast MeasureTheory.MeasureSpace.volume	Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.Icc Nat.instPreorder Nat.instLocallyFiniteOrder Distrib.toMul NonUnitalNonAssocSemiring.toDistrib SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup MeasureTheory.integral NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.restrict Set.Ioc Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing	—	Real.instIsStrictOrderedRing Finset.sum_congr Nat.floor_natCast sum_mul_eq_sub_integral_mul₀
sum_mul_eq_sub_integral_mul₁ 📖	mathematical	MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField DifferentiableAt Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddCommGroup.toAddCommGroup NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing MeasureTheory.IntegrableOn MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Icc Real.instPreorder instOfNatAtLeastTwo Real.instNatCast Nat.instAtLeastTwoHAddOfNat MeasureTheory.MeasureSpace.volume	Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.Icc Nat.instPreorder Nat.instLocallyFiniteOrder Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing Distrib.toMul NonUnitalNonAssocSemiring.toDistrib SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup MeasureTheory.integral NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.restrict Set.Ioc	—	Nat.instAtLeastTwoHAddOfNat Real.instIsStrictOrderedRing Finset.sum_eq_zero MulZeroClass.mul_zero Set.Ioc_eq_empty_of_le LT.lt.le MeasureTheory.Measure.restrict_empty MeasureTheory.integral_zero_measure sub_self Nat.le_floor Real.instIsOrderedRing Finset.add_sum_Ioc_eq_sum_Icc Finset.sum_union_eq_right Nat.floor_ofNat sum_mul_eq_sub_sub_integral_mul zero_le_two Real.instZeroLEOneClass IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid Finset.sum_congr Finset.sum_insert Nat.instCharZero Finset.sum_singleton zero_add
sum_mul_eq_sub_sub_integral_mul 📖	mathematical	Real Real.instLE Real.instZero DifferentiableAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddCommGroup.toAddCommGroup NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing MeasureTheory.IntegrableOn MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Icc Real.instPreorder MeasureTheory.MeasureSpace.volume	Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.Ioc Nat.instPreorder Nat.instLocallyFiniteOrder Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing Real.instNatCast SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup Finset.Icc MeasureTheory.integral NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.restrict Set.Ioc	—	Real.instIsStrictOrderedRing intervalIntegral.integral_of_le Nat.floor_le LT.lt.le Nat.lt_floor_add_one eq_or_lt_of_le Nat.floor_le_floor Finset.Ioc_eq_empty_of_le le_rfl Finset.sum_empty sub_mul sub_self Nat.cast_add_one Nat.le_floor_iff LE.le.trans Nat.floor_lt Finset.sum_congr Finset.sum_Ioc_by_parts Finset.range_eq_Ico Finset.Ico_add_one_right_eq_Icc Nat.instNoMaxOrder Finset.Ico_add_one_add_one_eq_Ioc IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid Real.instZeroLEOneClass RCLike.charZero_rclike Nat.cast_one intervalIntegral.sum_integral_adjacent_intervals_Ico intervalIntegrable_iff_integrableOn_Icc_of_le PseudoEMetricSpace.pseudoMetrizableSpace Real.noAtoms_volume enorm_ne_top MeasureTheory.IntegrableOn.mono_set integrableOn_mul_sum_Icc Set.Icc_subset_Icc_iff Nat.cast_add intervalIntegral.integral_interval_sub_left Set.Icc_subset_Icc_right intervalIntegral.integral_add_adjacent_intervals Set.Icc_subset_Icc_left Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.mul_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.add_mul Mathlib.Tactic.Ring.mul_add Mathlib.Tactic.Ring.mul_pf_left Mathlib.Tactic.Ring.mul_pf_right Mathlib.Tactic.Ring.one_mul Mathlib.Tactic.Ring.mul_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.zero_mul Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.add_pf_add_lt Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.mul_one Mathlib.Meta.NormNum.IsNat.to_raw_eq Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Tactic.Ring.add_pf_add_gt Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Tactic.Ring.add_overlap_pf_zero Mathlib.Meta.NormNum.isInt_add Mathlib.Tactic.Ring.add_congr
sum_mul_eq_sub_sub_integral_mul' 📖	mathematical	DifferentiableAt Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddCommGroup.toAddCommGroup NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing MeasureTheory.IntegrableOn MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Icc Real.instPreorder Real.instNatCast MeasureTheory.MeasureSpace.volume	Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.Ioc Nat.instPreorder Nat.instLocallyFiniteOrder Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup Finset.Icc MeasureTheory.integral NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.restrict Set.Ioc Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing	—	Real.instIsStrictOrderedRing Finset.sum_congr Nat.floor_natCast sum_mul_eq_sub_sub_integral_mul Nat.cast_nonneg Real.instIsOrderedRing Nat.cast_le IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid Real.instZeroLEOneClass RCLike.charZero_rclike
summable_mul_of_bigO_atTop 📖	mathematical	DifferentiableAt Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedField.toNorm DenselyNormedField.toNormedField RCLike.toDenselyNormedField MeasureTheory.LocallyIntegrableOn Real.measurableSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Ici Real.instPreorder Real.instZero MeasureTheory.MeasureSpace.volume Real.measureSpace Asymptotics.IsBigO Real.norm Filter.atTop Nat.instPreorder Real.instMul Real.instNatCast Finset.sum Real.instAddCommMonoid Finset.Icc Nat.instLocallyFiniteOrder Real.instOne Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing MeasureTheory.IntegrableAtFilter MeasureTheory.MeasureSpace.toMeasurableSpace	Summable ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedField.toNormedCommRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing SummationFilter.unconditional	—	Real.instIsStrictOrderedRing Nat.cast_zero sum_mul_eq_sub_integral_mul' MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset PseudoEMetricSpace.pseudoMetrizableSpace Set.Icc_subset_Ici_self CompactIccSpace.isCompact_Icc ConditionallyCompleteLinearOrder.toCompactIccSpace instOrderTopologyReal
summable_mul_of_bigO_atTop' 📖	mathematical	DifferentiableAt Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Norm.norm NormedField.toNorm DenselyNormedField.toNormedField RCLike.toDenselyNormedField MeasureTheory.LocallyIntegrableOn Real.measurableSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Ici Real.instPreorder Real.instOne MeasureTheory.MeasureSpace.volume Real.measureSpace Asymptotics.IsBigO Real.norm Filter.atTop Nat.instPreorder Real.instMul Real.instNatCast Finset.sum Real.instAddCommMonoid Finset.Icc Nat.instLocallyFiniteOrder Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing MeasureTheory.IntegrableAtFilter MeasureTheory.MeasureSpace.toMeasurableSpace	Summable ESeminormedAddCommMonoid.toAddCommMonoid SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedField.toNormedCommRing ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing SummationFilter.unconditional	—	Real.instIsStrictOrderedRing Finset.left_notMem_Ioc Finset.Icc_eq_cons_Ioc Finset.sum_cons Finset.Icc_add_one_left_eq_Ioc Nat.instNoMaxOrder zero_add Finset.sum_congr norm_zero LT.lt.ne' LT.lt.trans_le zero_lt_one Nat.instZeroLEOneClass Finset.mem_Icc Summable.congr_atTop SeminormedAddCommGroup.toIsTopologicalAddGroup Nat.cast_one Nat.cast_zero sum_mul_eq_sub_integral_mul₀' MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset PseudoEMetricSpace.pseudoMetrizableSpace Set.Icc_subset_Ici_self CompactIccSpace.isCompact_Icc ConditionallyCompleteLinearOrder.toCompactIccSpace instOrderTopologyReal Filter.mp_mem Filter.eventually_ne_atTop instNoTopOrderOfNoMaxOrder Filter.univ_mem'
tendsto_sum_mul_atTop_nhds_one_sub_integral 📖	mathematical	DifferentiableAt Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddCommGroup.toAddCommGroup NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing MeasureTheory.LocallyIntegrableOn Real.measurableSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Ici Real.instPreorder Real.instZero MeasureTheory.MeasureSpace.volume Real.measureSpace Filter.Tendsto Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing Real.instNatCast Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.Icc Nat.instPreorder Nat.instLocallyFiniteOrder Filter.atTop nhds Asymptotics.IsBigO NormedField.toNorm Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing MeasureTheory.IntegrableAtFilter MeasureTheory.MeasureSpace.toMeasurableSpace	SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup MeasureTheory.integral NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.restrict Set.Ioi	—	Real.instIsStrictOrderedRing Filter.Tendsto.congr intervalIntegral.integral_of_le Nat.cast_nonneg Real.instIsOrderedRing MeasureTheory.intervalIntegral_tendsto_integral_Ioi instIsCountablyGenerated_atTop OrderTopology.of_linearLocallyFinite instDiscreteTopologyNat TopologicalSpace.SecondCountableTopology.to_separableSpace TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace Countable.LindelofSpace instCountableNat PseudoEMetricSpace.pseudoMetrizableSpace integrableOn_Ici_iff_integrableOn_Ioi instMeasurableSingletonClassOfMeasurableEq instMeasurableEqOfSecondCountableTopologyOfT2Space BorelSpace.opensMeasurable Real.borelSpace instSecondCountableTopologyReal TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace Real.noAtoms_volume enorm_ne_top MeasureTheory.LocallyIntegrableOn.integrableOn_of_isBigO_atTop ConditionallyCompleteLinearOrder.toCompactIccSpace instOrderTopologyReal atTop_isMeasurablyGenerated instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid locallyIntegrableOn_mul_sum_Icc le_rfl tendsto_natCast_atTop_atTop Real.instArchimedean sum_mul_eq_sub_integral_mul' MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset Set.Icc_subset_Ici_self CompactIccSpace.isCompact_Icc Filter.Tendsto.sub IsTopologicalAddGroup.to_continuousSub SeminormedAddCommGroup.toIsTopologicalAddGroup
tendsto_sum_mul_atTop_nhds_one_sub_integral₀ 📖	mathematical	MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField DifferentiableAt Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.instAddCommGroup NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddCommGroup.toAddCommGroup NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing MeasureTheory.LocallyIntegrableOn Real.measurableSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup deriv Set.Ici Real.instPreorder Real.instOne MeasureTheory.MeasureSpace.volume Real.measureSpace Filter.Tendsto Distrib.toMul NonUnitalNonAssocSemiring.toDistrib Real.instNatCast Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.Icc Nat.instPreorder Nat.instLocallyFiniteOrder Filter.atTop nhds Asymptotics.IsBigO NormedField.toNorm Real.norm Nat.floor Real.semiring Real.partialOrder FloorRing.toFloorSemiring Real.instRing Real.linearOrder Real.instIsStrictOrderedRing Real.instFloorRing MeasureTheory.IntegrableAtFilter MeasureTheory.MeasureSpace.toMeasurableSpace	SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup MeasureTheory.integral NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.restrict Set.Ioi	—	Real.instIsStrictOrderedRing Filter.mp_mem Filter.eventually_ge_atTop Filter.univ_mem' intervalIntegral.integral_of_le Nat.one_le_cast IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid Real.instZeroLEOneClass RCLike.charZero_rclike Filter.Tendsto.congr' MeasureTheory.intervalIntegral_tendsto_integral_Ioi instIsCountablyGenerated_atTop OrderTopology.of_linearLocallyFinite instDiscreteTopologyNat TopologicalSpace.SecondCountableTopology.to_separableSpace TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace Countable.LindelofSpace instCountableNat PseudoEMetricSpace.pseudoMetrizableSpace integrableOn_Ici_iff_integrableOn_Ioi instMeasurableSingletonClassOfMeasurableEq instMeasurableEqOfSecondCountableTopologyOfT2Space BorelSpace.opensMeasurable Real.borelSpace instSecondCountableTopologyReal TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace Real.noAtoms_volume enorm_ne_top MeasureTheory.LocallyIntegrableOn.integrableOn_of_isBigO_atTop ConditionallyCompleteLinearOrder.toCompactIccSpace instOrderTopologyReal atTop_isMeasurablyGenerated instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal locallyIntegrableOn_mul_sum_Icc zero_le_one tendsto_natCast_atTop_atTop Real.instIsOrderedRing Real.instArchimedean Filter.Tendsto.congr sum_mul_eq_sub_integral_mul₀' MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset Set.Icc_subset_Ici_self CompactIccSpace.isCompact_Icc Filter.Tendsto.sub IsTopologicalAddGroup.to_continuousSub SeminormedAddCommGroup.toIsTopologicalAddGroup

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