EGauge

📁 Source: Mathlib/Analysis/Convex/EGauge.lean

Statistics

Metric	Count
Definitions egauge	1
Theorems egauge_le, div_le_egauge_ball, div_le_egauge_closedBall, egauge_add_add_le, egauge_anti, egauge_ball_le_of_one_lt_norm, egauge_ball_one_le_of_one_lt_norm, egauge_empty, egauge_eq_top, egauge_eq_zero_iff, egauge_le_of_mem_smul, egauge_le_of_smul_mem, egauge_le_of_smul_mem_of_ne, egauge_le_one, egauge_lt_iff, egauge_pi, egauge_pi', egauge_prod_mk, egauge_smul_left, egauge_smul_right, egauge_union, egauge_univ, egauge_univ_pi, egauge_zero_left, egauge_zero_left_eq_top, egauge_zero_right, egauge_zero_zero, le_egauge_ball_one, le_egauge_closedBall_one, le_egauge_iff, le_egauge_inter, le_egauge_of_forall_ne_zero, le_egauge_pi, le_egauge_prod, le_egauge_smul_left, le_egauge_smul_right, mem_of_egauge_lt_one, mem_smul_of_egauge_lt	38
Total	39

Set.MapsTo

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
egauge_le 📖	mathematical	Set.MapsTo DFunLike.coe	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge NNNorm.toENorm	—	iInf_mono iInf_mono' smul_set le_rfl

(root)

Definitions

Name	Category	Theorems
egauge 📖	CompOp	47 math math: egauge_prod_mk, Asymptotics.IsLittleOTVS.tendsto_div, egauge_ball_le_of_one_lt_norm, le_egauge_pi, egauge_add_add_le, egauge_univ, egauge_zero_zero, egauge_univ_pi, Filter.HasBasis.isLittleOTVS_iff, egauge_zero_left, Asymptotics.isBigOTVS_iff_smallSets, egauge_le_one, Asymptotics.IsLittleOTVS.exists_eventuallyLE_mul, le_egauge_smul_left, Asymptotics.IsBigOTVS.exists_eventuallyLE, egauge_pi, Asymptotics.IsLittleOTVS.exists_eventuallyLE_mul_ennreal, egauge_empty, Asymptotics.isLittleOTVS_iff_tendsto_div, egauge_zero_left_eq_top, le_egauge_iff, le_egauge_inter, egauge_union, le_egauge_prod, le_egauge_smul_right, egauge_ball_one_le_of_one_lt_norm, egauge_le_of_smul_mem, Asymptotics.IsBigOTVS.eventually_smallSets, Asymptotics.isBigOTVS_iff, egauge_le_of_mem_smul, egauge_eq_zero_iff, Set.MapsTo.egauge_le, Filter.HasBasis.isBigOTVS_iff, div_le_egauge_ball, Asymptotics.isLittleOTVS_iff, le_egauge_ball_one, le_egauge_of_forall_ne_zero, div_le_egauge_closedBall, egauge_anti, egauge_eq_top, egauge_smul_right, egauge_lt_iff, egauge_zero_right, le_egauge_closedBall_one, egauge_le_of_smul_mem_of_ne, egauge_pi', egauge_smul_left

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
div_le_egauge_ball 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder DivInvMonoid.toDiv ENNReal.instDivInvMonoid ENorm.enorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm ENNReal.ofNNReal egauge NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField Module.toDistribMulAction AddCommGroup.toAddCommMonoid SeminormedAddCommGroup.toAddCommGroup NormedSpace.toModule Metric.ball SeminormedAddCommGroup.toPseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NNReal.toReal	—	LE.le.trans div_le_egauge_closedBall egauge_anti Metric.ball_subset_closedBall
div_le_egauge_closedBall 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder DivInvMonoid.toDiv ENNReal.instDivInvMonoid ENorm.enorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm ENNReal.ofNNReal egauge NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField Module.toDistribMulAction AddCommGroup.toAddCommMonoid SeminormedAddCommGroup.toAddCommGroup NormedSpace.toModule Metric.closedBall SeminormedAddCommGroup.toPseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NNReal.toReal	—	le_egauge_iff enorm_smul instENormSMulClass NormedSpace.toNormSMulClass ENNReal.div_le_of_le_mul mul_le_mul_right IsOrderedMonoid.toMulLeftMono ENNReal.instIsOrderedMonoid enorm_le_coe NNReal.coe_le_coe coe_nnnorm mem_closedBall_zero_iff
egauge_add_add_le 📖	mathematical	Balanced NormedRing.toSeminormedRing NormedDivisionRing.toNormedRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing SeminormedAddGroup.toContinuousENorm Set Set.add AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ENNReal.instMax	—	Set.add_image_prod Set.MapsTo.egauge_le SemilinearMapClass.toMulActionSemiHomClass LinearMap.semilinearMapClass Set.mapsTo_image
egauge_anti 📖	mathematical	Set Set.instHasSubset	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge NNNorm.toENorm	—	Set.MapsTo.egauge_le instMulActionSemiHomClassMulActionHom
egauge_ball_le_of_one_lt_norm 📖	mathematical	Real Real.instLT Real.instOne Norm.norm NormedField.toNorm	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField Module.toDistribMulAction AddCommGroup.toAddCommMonoid SeminormedAddCommGroup.toAddCommGroup NormedSpace.toModule Metric.ball SeminormedAddCommGroup.toPseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NNReal.toReal DivInvMonoid.toDiv ENNReal.instDivInvMonoid Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENorm.enorm ENNReal.ofNNReal	—	LE.le.eq_or_lt zero_le NNReal.instCanonicallyOrderedAdd ENNReal.coe_zero ENNReal.div_zero mul_ne_zero ENNReal.instNoZeroDivisors LT.lt.trans one_pos Real.instZeroLEOneClass NeZero.charZero_one IsStrictOrderedRing.toCharZero Real.instIsStrictOrderedRing le_top eq_or_ne MulZeroClass.mul_zero ENNReal.zero_div ENNReal.instCanonicallyOrderedAdd Filter.Frequently.mono Filter.frequently_iff_neBot NormedField.nhdsNE_neBot Set.mem_smul_set_iff_inv_smul_mem₀ dist_zero_right norm_smul NormedSpace.toNormSMulClass norm_inv rescale_to_shell_semi_normed egauge_le_of_smul_mem_of_ne mem_ball_zero_iff ENNReal.coe_le_coe div_eq_inv_mul mul_assoc LE.le.trans ENNReal.coe_div_le
egauge_ball_one_le_of_one_lt_norm 📖	mathematical	Real Real.instLT Real.instOne Norm.norm NormedField.toNorm	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField Module.toDistribMulAction AddCommGroup.toAddCommMonoid SeminormedAddCommGroup.toAddCommGroup NormedSpace.toModule Metric.ball SeminormedAddCommGroup.toPseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENorm.enorm	—	div_one egauge_ball_le_of_one_lt_norm one_ne_zero NeZero.charZero_one IsStrictOrderedRing.toCharZero NNReal.instIsStrictOrderedRing
egauge_empty 📖	mathematical	—	egauge NNNorm.toENorm Set Set.instEmptyCollection Top.top ENNReal instTopENNReal	—	iInf_congr_Prop Set.smul_set_empty iInf_neg iInf_top
egauge_eq_top 📖	mathematical	—	egauge NNNorm.toENorm Top.top ENNReal instTopENNReal Set Set.instMembership Set.smulSet	—	—
egauge_eq_zero_iff 📖	mathematical	—	egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid ENNReal instZeroENNReal Filter.Frequently Set Set.instMembership Set.smulSet nhds SeminormedRing.toPseudoMetricSpace NormedRing.toSeminormedRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing	—	iInf₂_eq_bot Filter.HasBasis.frequently_iff nhds_basis_uniformity uniformity_basis_edist bot_eq_zero' ENNReal.instCanonicallyOrderedAdd edist_zero_right
egauge_le_of_mem_smul 📖	mathematical	Set Set.instMembership Set.smulSet	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge NNNorm.toENorm ENorm.enorm	—	iInf₂_le
egauge_le_of_smul_mem 📖	mathematical	Set Set.instMembership SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm ENNReal.instInv ENorm.enorm	—	eq_or_ne enorm_zero ENNReal.inv_zero LE.le.trans egauge_le_of_smul_mem_of_ne ENNReal.coe_inv_le
egauge_le_of_smul_mem_of_ne 📖	mathematical	Set Set.instMembership SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm ENNReal.ofNNReal NNReal NNReal.instInv NNNorm.nnnorm SeminormedAddGroup.toNNNorm	—	nnnorm_inv egauge_le_of_mem_smul Set.mem_inv_smul_set_iff₀
egauge_le_one 📖	mathematical	Set Set.instMembership	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal	—	enorm_one NormedDivisionRing.to_normOneClass egauge_le_of_mem_smul one_smul
egauge_lt_iff 📖	mathematical	—	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder egauge NNNorm.toENorm Set Set.instMembership Set.smulSet ENorm.enorm	—	—
egauge_pi 📖	mathematical	Balanced NormedRing.toSeminormedRing NormedDivisionRing.toNormedRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid	egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing SeminormedAddGroup.toContinuousENorm Pi.instSMul NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid Set.pi iSup ENNReal ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder CompleteLinearOrder.toConditionallyCompleteLinearOrderBot ENNReal.instCompleteLinearOrder Set Set.instMembership	—	egauge_pi'
egauge_pi' 📖	mathematical	Balanced NormedRing.toSeminormedRing NormedDivisionRing.toNormedRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid Set.univ Set Set.instMembership Filter.NeBot nhdsWithin UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonAssocRing.toNonUnitalNonAssocRing Ring.toNonAssocRing NormedRing.toRing Compl.compl Set.instCompl Set.instSingletonSet	egauge ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing SeminormedAddGroup.toContinuousENorm Pi.instSMul NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid Set.pi iSup ENNReal ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder CompleteLinearOrder.toConditionallyCompleteLinearOrderBot ENNReal.instCompleteLinearOrder	—	le_antisymm le_of_forall_gt egauge_lt_iff LE.le.trans_lt le_iSup₂ LT.lt.bot_lt Set.eq_empty_or_nonempty norm_zero instIsEmptyFalse enorm_zero exists_enorm_lt LT.lt.ne' Set.exists_max_image ENNReal.instCanonicallyOrderedAdd Set.zero_smul_set_subset Set.smul_set_pi₀' Balanced.smul_mono NormMulClass.toNormSMulClass NormedDivisionRing.toNormMulClass IsScalarTower.left Nontrivial.to_nonempty IsSimpleRing.instNontrivial DivisionRing.isSimpleRing Function.sometimes_spec iSup₂_le le_egauge_pi
egauge_prod_mk 📖	mathematical	Balanced NormedRing.toSeminormedRing NormedDivisionRing.toNormedRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid	egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing SeminormedAddGroup.toContinuousENorm Prod.instSMul SProd.sprod Set Set.instSProd ENNReal ENNReal.instMax	—	le_antisymm le_of_forall_gt Set.smul_set_prod le_total Balanced.smul_mono NormMulClass.toNormSMulClass NormedDivisionRing.toNormMulClass IsScalarTower.left le_egauge_prod
egauge_smul_left 📖	mathematical	—	egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid Set Set.smulSet ENNReal DivInvMonoid.toDiv ENNReal.instDivInvMonoid ENorm.enorm	—	le_antisymm ENNReal.le_div_iff_mul_le enorm_inv div_eq_mul_inv inv_inv le_egauge_smul_left inv_smul_smul₀
egauge_smul_right 📖	mathematical	Set.Nonempty	egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid ENNReal Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENorm.enorm	—	le_antisymm eq_or_ne zero_smul egauge_zero_right enorm_zero MulZeroClass.zero_mul mul_comm ENNReal.div_le_iff_le_mul enorm_ne_top ENNReal.div_eq_inv_mul enorm_inv LE.le.trans_eq le_egauge_smul_right inv_smul_smul₀
egauge_union 📖	mathematical	—	egauge NNNorm.toENorm Set Set.instUnion ENNReal ENNReal.instMin	—	iInf_congr_Prop Set.smul_set_union iInf_or iInf_inf_eq
egauge_univ 📖	mathematical	—	egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid Set.univ ENNReal instZeroENNReal	—	egauge_eq_zero_iff Filter.Frequently.mono Filter.frequently_iff_neBot Set.smul_set_univ₀
egauge_univ_pi 📖	mathematical	Balanced NormedRing.toSeminormedRing NormedDivisionRing.toNormedRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid	egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing SeminormedAddGroup.toContinuousENorm Pi.instSMul NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid Set.pi Set.univ iSup ENNReal ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder CompleteLinearOrder.toConditionallyCompleteLinearOrderBot ENNReal.instCompleteLinearOrder	—	egauge_pi' Set.finite_univ iSup_congr_Prop iSup_pos
egauge_zero_left 📖	mathematical	—	egauge NNNorm.toENorm SMulZeroClass.toSMul Set Set.zero Top.top ENNReal instTopENNReal	—	egauge_zero_left_eq_top
egauge_zero_left_eq_top 📖	mathematical	—	egauge NNNorm.toENorm SMulZeroClass.toSMul Set Set.zero Top.top ENNReal instTopENNReal	—	smul_zero
egauge_zero_right 📖	mathematical	Set.Nonempty	egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid ENNReal instZeroENNReal	—	Set.zero_smul_set enorm_zero ENNReal.instCanonicallyOrderedAdd egauge_le_of_mem_smul
egauge_zero_zero 📖	mathematical	—	egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid Set Set.zero NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid ENNReal instZeroENNReal	—	egauge_zero_right
le_egauge_ball_one 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm egauge NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField Module.toDistribMulAction AddCommGroup.toAddCommMonoid SeminormedAddCommGroup.toAddCommGroup NormedSpace.toModule Metric.ball SeminormedAddCommGroup.toPseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid Real Real.instOne	—	div_one div_le_egauge_ball
le_egauge_closedBall_one 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm egauge NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField Module.toDistribMulAction AddCommGroup.toAddCommMonoid SeminormedAddCommGroup.toAddCommGroup NormedSpace.toModule Metric.closedBall SeminormedAddCommGroup.toPseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid Real Real.instOne	—	div_one div_le_egauge_closedBall
le_egauge_iff 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge NNNorm.toENorm ENorm.enorm	—	le_iInf₂_iff
le_egauge_inter 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENNReal.instMax egauge NNNorm.toENorm Set Set.instInter	—	max_le egauge_anti Set.inter_subset_left Set.inter_subset_right
le_egauge_of_forall_ne_zero 📖	mathematical	Set Set.instMembership NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm	egauge SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid	—	le_egauge_iff ne_or_eq le_of_forall_gt Filter.nonempty_of_mem inter_mem_nhdsWithin Metric.eball_mem_nhds enorm_zero LE.le.trans_lt smul_zero edist_zero_right Set.mem_zero Set.mem_of_subset_of_mem Set.zero_smul_set_subset
le_egauge_pi 📖	mathematical	Set Set.instMembership	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder egauge NNNorm.toENorm Pi.instSMul Set.pi	—	Set.MapsTo.egauge_le instMulActionSemiHomClassMulActionHom
le_egauge_prod 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENNReal.instMax egauge NNNorm.toENorm Prod.instSMul SProd.sprod Set Set.instSProd	—	max_le Set.MapsTo.egauge_le instMulActionSemiHomClassMulActionHom Set.mapsTo_fst_prod Set.mapsTo_snd_prod
le_egauge_smul_left 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder DivInvMonoid.toDiv ENNReal.instDivInvMonoid egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid ENorm.enorm Set Set.smulSet	—	smul_smul ENNReal.div_le_of_le_mul enorm_mul NormedDivisionRing.toNormMulClass egauge_le_of_mem_smul Set.smul_mem_smul_set
le_egauge_smul_right 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENorm.enorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing NormedDivisionRing.toNormedRing SeminormedAddGroup.toContinuousENorm egauge SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid	—	le_egauge_iff eq_or_ne enorm_zero MulZeroClass.zero_mul ENNReal.instCanonicallyOrderedAdd ENNReal.mul_le_of_le_div' le_trans div_eq_inv_mul nnnorm_inv nnnorm_mul NormedDivisionRing.toNormMulClass egauge_le_of_mem_smul SemigroupAction.mul_smul inv_smul_smul₀ ENNReal.coe_div_le
mem_of_egauge_lt_one 📖	mathematical	Balanced NormedRing.toSeminormedRing NormedDivisionRing.toNormedRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing SeminormedAddGroup.toContinuousENorm AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal	Set Set.instMembership	—	mem_smul_of_egauge_lt enorm_one NormedDivisionRing.to_normOneClass one_smul
mem_smul_of_egauge_lt 📖	mathematical	Balanced NormedRing.toSeminormedRing NormedDivisionRing.toNormedRing SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring DivisionRing.toDivisionSemiring NormedDivisionRing.toDivisionRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder egauge ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalNormedRing.toNonUnitalSeminormedRing NormedRing.toNonUnitalNormedRing SeminormedAddGroup.toContinuousENorm ENorm.enorm	Set Set.instMembership Set.smulSet	—	egauge_lt_iff Balanced.smul_mono NormMulClass.toNormSMulClass NormedDivisionRing.toNormMulClass IsScalarTower.left LT.lt.le

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