Documentation Verification Report

Deriv

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

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Definitions0
TheoremsconcaveOn_univ_of_deriv, concaveOn_of_deriv, antitoneOn_deriv, antitoneOn_derivWithin, antitoneOn_slope_gt, antitoneOn_slope_lt, derivWithin_le_slope, deriv_le_slope, le_slope_of_hasDerivAt, le_slope_of_hasDerivWithinAt, le_slope_of_hasDerivWithinAt_Iio, leftDeriv_le_slope, slope_anti, slope_le_deriv, slope_le_derivWithin, slope_le_of_hasDerivAt, slope_le_of_hasDerivWithinAt, slope_le_of_hasDerivWithinAt_Ioi, slope_le_rightDeriv, derivWithin_le_slope, deriv_le_slope, differentiableWithinAt_Iio_of_mem_interior, differentiableWithinAt_Ioi_of_mem_interior, hasDerivWithinAt_leftDeriv_of_mem_interior, hasDerivWithinAt_rightDeriv_of_mem_interior, hasDerivWithinAt_sInf_slope_of_mem_interior, hasDerivWithinAt_sSup_slope_of_mem_interior, isMinOn_of_leftDeriv_eq_zero, isMinOn_of_leftDeriv_nonpos_of_rightDeriv_nonneg, isMinOn_of_rightDeriv_eq_zero, le_slope_of_hasDerivAt, le_slope_of_hasDerivWithinAt, le_slope_of_hasDerivWithinAt_Ioi, leftDeriv_eq_sSup_slope_of_mem_interior, leftDeriv_le_rightDeriv_of_mem_interior, monotoneOn_deriv, monotoneOn_derivWithin, monotoneOn_leftDeriv, monotoneOn_rightDeriv, monotoneOn_slope_gt, monotoneOn_slope_lt, rightDeriv_eq_sInf_slope_of_mem_interior, rightDeriv_le_slope, rightDeriv_le_slope_of_mem_interior, slope_le_deriv, slope_le_derivWithin, slope_le_leftDeriv, slope_le_leftDeriv_of_mem_interior, slope_le_of_hasDerivAt, slope_le_of_hasDerivWithinAt, slope_le_of_hasDerivWithinAt_Iio, slope_mono, convexOn_univ_of_deriv, convexOn_of_deriv, strictConcaveOn_univ_of_deriv, strictConcaveOn_of_deriv, derivWithin_lt_slope, deriv_lt_slope, leftDeriv_lt_slope, lt_slope_of_hasDerivAt, lt_slope_of_hasDerivWithinAt, lt_slope_of_hasDerivWithinAt_Iio, slope_lt_deriv, slope_lt_derivWithin, slope_lt_of_hasDerivAt, slope_lt_of_hasDerivWithinAt, slope_lt_of_hasDerivWithinAt_Ioi, slope_lt_rightDeriv, strictAntiOn_deriv, strictAntiOn_derivWithin, derivWithin_lt_slope, deriv_lt_slope, lt_slope_of_hasDerivAt, lt_slope_of_hasDerivWithinAt, lt_slope_of_hasDerivWithinAt_Ioi, rightDeriv_lt_slope, slope_lt_deriv, slope_lt_derivWithin, slope_lt_leftDeriv, slope_lt_of_hasDerivAt, slope_lt_of_hasDerivWithinAt, slope_lt_of_hasDerivWithinAt_Iio, strictMonoOn_deriv, strictMonoOn_derivWithin, strictConvexOn_univ_of_deriv, exists_deriv_lt_slope, exists_deriv_lt_slope_aux, exists_slope_lt_deriv, exists_slope_lt_deriv_aux, strictConvexOn_of_deriv, bddAbove_slope_gt_of_mem_interior, bddBelow_slope_lt_of_mem_interior, concaveOn_of_deriv2_nonpos, concaveOn_of_deriv2_nonpos', concaveOn_of_hasDerivWithinAt2_nonpos, concaveOn_univ_of_deriv2_nonpos, convexOn_of_deriv2_nonneg, convexOn_of_deriv2_nonneg', convexOn_of_hasDerivWithinAt2_nonneg, convexOn_univ_of_deriv2_nonneg, strictConcaveOn_of_deriv2_neg, strictConcaveOn_of_deriv2_neg', strictConcaveOn_univ_of_deriv2_neg, strictConvexOn_of_deriv2_pos, strictConvexOn_of_deriv2_pos', strictConvexOn_univ_of_deriv2_pos
106
Total106

Antitone

Theorems

NameKindAssumesProvesValidatesDepends On
concaveOn_univ_of_deriv 📖mathematicalDifferentiable
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Antitone
Real.instPreorder
deriv
ConcaveOn
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set.univ
AntitoneOn.concaveOn_of_deriv
convex_univ
Continuous.continuousOn
Differentiable.continuous
IsTopologicalSemiring.toContinuousAdd
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
Differentiable.differentiableOn
antitoneOn

AntitoneOn

Theorems

NameKindAssumesProvesValidatesDepends On
concaveOn_of_deriv 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
interior
AntitoneOn
Real.instPreorder
deriv
ConcaveOnneg_convexOn_iff
Real.instIsOrderedAddMonoid
MonotoneOn.convexOn_of_deriv
ContinuousOn.neg
IsTopologicalRing.toContinuousNeg
instIsTopologicalRingReal
DifferentiableOn.neg
neg
IsOrderedAddMonoid.toAddLeftMono
covariant_swap_add_of_covariant_add

ConcaveOn

Theorems

NameKindAssumesProvesValidatesDepends On
antitoneOn_deriv 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
AntitoneOn
Real.instPreorder
deriv
deriv.neg'
neg_neg
MonotoneOn.neg
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
covariant_swap_add_of_covariant_add
ConvexOn.monotoneOn_deriv
neg
DifferentiableAt.neg
antitoneOn_derivWithin 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
AntitoneOn
Real.instPreorder
derivWithin
eq_or_lt_of_le
le_refl
LE.le.trans
derivWithin_le_slope
slope_le_derivWithin
antitoneOn_slope_gt 📖mathematicalConcaveOn
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
Algebra.toSMul
Semifield.toCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
AntitoneOn
PartialOrder.toPreorder
slope
Ring.toAddCommGroup
DivisionRing.toRing
Field.toDivisionRing
Semiring.toModule
addGroupIsAddTorsor
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
setOf
Preorder.toLT
AntitoneOn.mono
slope_anti
LT.lt.ne'
antitoneOn_slope_lt 📖mathematicalConcaveOn
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
Algebra.toSMul
Semifield.toCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
AntitoneOn
PartialOrder.toPreorder
slope
Ring.toAddCommGroup
DivisionRing.toRing
Field.toDivisionRing
Semiring.toModule
addGroupIsAddTorsor
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
setOf
Preorder.toLT
AntitoneOn.mono
slope_anti
LT.lt.ne
derivWithin_le_slope 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
derivWithin
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
le_slope_of_hasDerivWithinAt
DifferentiableWithinAt.hasDerivWithinAt
deriv_le_slope 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
deriv
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
le_slope_of_hasDerivAt
DifferentiableAt.hasDerivAt
le_slope_of_hasDerivAt 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
le_slope_of_hasDerivWithinAt_Iio
HasDerivAt.hasDerivWithinAt
le_slope_of_hasDerivWithinAt 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
le_slope_of_hasDerivWithinAt_Iio
HasDerivWithinAt.mono_of_mem_nhdsWithin
Set.OrdConnected.mem_nhdsLT
instClosedIicTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
Convex.ordConnected
Real.instIsStrictOrderedRing
le_slope_of_hasDerivWithinAt_Iio 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Set.Iio
Real.instPreorder
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
neg_neg
slope_neg
neg_le_neg
Real.instIsOrderedAddMonoid
ConvexOn.slope_le_of_hasDerivWithinAt_Iio
neg
HasDerivWithinAt.neg
leftDeriv_le_slope 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Iio
Real.instPreorder
Real.instLE
derivWithin
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
le_slope_of_hasDerivWithinAt_Iio
DifferentiableWithinAt.hasDerivWithinAt
slope_anti 📖mathematicalConcaveOn
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
Algebra.toSMul
Semifield.toCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
AntitoneOn
PartialOrder.toPreorder
slope
Ring.toAddCommGroup
DivisionRing.toRing
Field.toDivisionRing
Semiring.toModule
addGroupIsAddTorsor
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
Set.instSDiff
Set.instSingletonSet
neg_neg
slope_neg_fun
MonotoneOn.neg
IsOrderedAddMonoid.toAddLeftMono
IsOrderedRing.toIsOrderedAddMonoid
IsStrictOrderedRing.toIsOrderedRing
covariant_swap_add_of_covariant_add
ConvexOn.slope_mono
neg
slope_le_deriv 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
deriv
slope_le_of_hasDerivAt
DifferentiableAt.hasDerivAt
slope_le_derivWithin 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
derivWithin
slope_le_of_hasDerivWithinAt
DifferentiableWithinAt.hasDerivWithinAt
slope_le_of_hasDerivAt 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_le_of_hasDerivWithinAt_Ioi
HasDerivAt.hasDerivWithinAt
slope_le_of_hasDerivWithinAt 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_le_of_hasDerivWithinAt_Ioi
HasDerivWithinAt.mono_of_mem_nhdsWithin
Set.OrdConnected.mem_nhdsGT
instClosedIciTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
Convex.ordConnected
Real.instIsStrictOrderedRing
slope_le_of_hasDerivWithinAt_Ioi 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Set.Ioi
Real.instPreorder
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_neg
neg_neg
neg_le_neg
Real.instIsOrderedAddMonoid
ConvexOn.le_slope_of_hasDerivWithinAt_Ioi
neg
HasDerivWithinAt.neg
slope_le_rightDeriv 📖mathematicalConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Ioi
Real.instPreorder
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
derivWithin
slope_le_of_hasDerivWithinAt_Ioi
DifferentiableWithinAt.hasDerivWithinAt

ConvexOn

Theorems

NameKindAssumesProvesValidatesDepends On
derivWithin_le_slope 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
derivWithin
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
le_slope_of_hasDerivWithinAt
DifferentiableWithinAt.hasDerivWithinAt
deriv_le_slope 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
deriv
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
le_slope_of_hasDerivAt
DifferentiableAt.hasDerivAt
differentiableWithinAt_Iio_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Iio
Real.instPreorder
HasDerivWithinAt.differentiableWithinAt
hasDerivWithinAt_sSup_slope_of_mem_interior
differentiableWithinAt_Ioi_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Ioi
Real.instPreorder
HasDerivWithinAt.differentiableWithinAt
hasDerivWithinAt_sInf_slope_of_mem_interior
hasDerivWithinAt_leftDeriv_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
derivWithin
Set.Iio
Real.instPreorder
DifferentiableWithinAt.hasDerivWithinAt
differentiableWithinAt_Iio_of_mem_interior
hasDerivWithinAt_rightDeriv_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
derivWithin
Set.Ioi
Real.instPreorder
DifferentiableWithinAt.hasDerivWithinAt
differentiableWithinAt_Ioi_of_mem_interior
hasDerivWithinAt_sInf_slope_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
InfSet.sInf
Real.instInfSet
Set.image
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
setOf
Real.instLT
Set.Ioi
Real.instPreorder
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
mem_nhds_iff_exists_Ioo_subset
instOrderTopologyReal
instNoMaxOrderOfNontrivial
Real.instIsOrderedRing
Real.instNontrivial
instNoMinOrderOfNontrivial
mem_interior_iff_mem_nhds
Set.diff_singleton_eq_self
LT.lt.trans
MonotoneOn.tendsto_nhdsWithin_Ioo_right
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
MonotoneOn.mono
monotoneOn_slope_gt
BddBelow.mono
Set.image_mono
bddBelow_slope_lt_of_mem_interior
MonotoneOn.csInf_eq_of_subset_of_forall_exists_le
exists_between
lt_min
LT.lt.trans_le
min_le_left
LE.le.trans
LT.lt.le
min_le_right
hasDerivWithinAt_sSup_slope_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
SupSet.sSup
Real.instSupSet
Set.image
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
setOf
Real.instLT
Set.Iio
Real.instPreorder
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
mem_nhds_iff_exists_Ioo_subset
instOrderTopologyReal
instNoMaxOrderOfNontrivial
Real.instIsOrderedRing
Real.instNontrivial
instNoMinOrderOfNontrivial
mem_interior_iff_mem_nhds
Set.diff_singleton_eq_self
LT.lt.trans
MonotoneOn.tendsto_nhdsWithin_Ioo_left
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
MonotoneOn.mono
monotoneOn_slope_lt
BddAbove.mono
Set.image_mono
bddAbove_slope_gt_of_mem_interior
MonotoneOn.csSup_eq_of_subset_of_forall_exists_le
exists_between
max_lt
LE.le.trans_lt
le_max_left
LE.le.trans
le_max_right
LT.lt.le
isMinOn_of_leftDeriv_eq_zero 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Iio
Real.instPreorder
Real.instZero
IsMinOnisMinOn_of_leftDeriv_nonpos_of_rightDeriv_nonneg
Eq.le
LE.le.trans
leftDeriv_le_rightDeriv_of_mem_interior
isMinOn_of_leftDeriv_nonpos_of_rightDeriv_nonneg 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Iio
Real.instPreorder
Real.instZero
Set.Ioi
IsMinOnlt_trichotomy
LE.le.trans
rightDeriv_le_slope_of_mem_interior
slope_def_field
Real.instIsStrictOrderedRing
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
AddGroup.toOrderedSub
zero_add
not_le
slope_comm
slope_le_leftDeriv_of_mem_interior
isMinOn_of_rightDeriv_eq_zero 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Ioi
Real.instPreorder
Real.instZero
IsMinOnisMinOn_of_leftDeriv_nonpos_of_rightDeriv_nonneg
LE.le.trans_eq
leftDeriv_le_rightDeriv_of_mem_interior
Eq.le
le_slope_of_hasDerivAt 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
le_slope_of_hasDerivWithinAt_Ioi
HasDerivAt.hasDerivWithinAt
le_slope_of_hasDerivWithinAt 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
le_slope_of_hasDerivWithinAt_Ioi
HasDerivWithinAt.mono_of_mem_nhdsWithin
Set.OrdConnected.mem_nhdsGT
instClosedIciTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
Convex.ordConnected
Real.instIsStrictOrderedRing
le_slope_of_hasDerivWithinAt_Ioi 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Set.Ioi
Real.instPreorder
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
le_of_tendsto
instClosedIicTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
nhdsGT_neBot
instOrderTopologyReal
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
instNoMaxOrderOfNontrivial
Real.instIsOrderedRing
Real.instNontrivial
IsBoundedSMul.continuousSMul
NormedSpace.toIsBoundedSMul
hasDerivWithinAt_iff_tendsto_slope'
Set.self_notMem_Ioi
slope_def_field
Filter.mp_mem
eventually_lt_nhds
instClosedIciTopology
Filter.univ_mem'
secant_mono
Set.OrdConnected.out
Convex.ordConnected
LT.lt.le
LT.lt.ne'
leftDeriv_eq_sSup_slope_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Iio
Real.instPreorder
SupSet.sSup
Real.instSupSet
Set.image
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
setOf
Real.instLT
HasDerivWithinAt.derivWithin
hasDerivWithinAt_sSup_slope_of_mem_interior
uniqueDiffWithinAt_Iio
leftDeriv_le_rightDeriv_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Iio
Real.instPreorder
Set.Ioi
mem_nhds_iff_exists_Ioo_subset
instOrderTopologyReal
instNoMaxOrderOfNontrivial
Real.instIsOrderedRing
Real.instNontrivial
instNoMinOrderOfNontrivial
mem_interior_iff_mem_nhds
rightDeriv_eq_sInf_slope_of_mem_interior
leftDeriv_eq_sSup_slope_of_mem_interior
csSup_le
Set.image_nonempty
exists_between
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
LT.lt.trans
le_csInf
slope_mono
interior_subset
LT.lt.ne
LT.lt.ne'
LT.lt.le
monotoneOn_deriv 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
MonotoneOn
Real.instPreorder
deriv
eq_or_lt_of_le
le_refl
LE.le.trans
deriv_le_slope
slope_le_deriv
monotoneOn_derivWithin 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
MonotoneOn
Real.instPreorder
derivWithin
eq_or_lt_of_le
le_refl
LE.le.trans
derivWithin_le_slope
slope_le_derivWithin
monotoneOn_leftDeriv 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
MonotoneOn
Real.instPreorder
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Iio
interior
eq_or_lt_of_le
le_refl
leftDeriv_eq_sSup_slope_of_mem_interior
le_csSup_of_le
bddAbove_slope_gt_of_mem_interior
Real.instIsStrictOrderedRing
instOrderTopologyReal
slope_comm
interior_subset
csSup_le
mem_nhds_iff_exists_Ioo_subset
instNoMaxOrderOfNontrivial
Real.instIsOrderedRing
Real.instNontrivial
instNoMinOrderOfNontrivial
mem_interior_iff_mem_nhds
Set.image_nonempty
exists_between
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
LT.lt.trans
slope_mono
LT.lt.ne
LT.lt.ne'
LT.lt.le
monotoneOn_rightDeriv 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
MonotoneOn
Real.instPreorder
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Ioi
interior
eq_or_lt_of_le
le_refl
rightDeriv_eq_sInf_slope_of_mem_interior
csInf_le_of_le
bddBelow_slope_lt_of_mem_interior
Real.instIsStrictOrderedRing
instOrderTopologyReal
interior_subset
le_csInf
mem_nhds_iff_exists_Ioo_subset
instNoMaxOrderOfNontrivial
Real.instIsOrderedRing
Real.instNontrivial
instNoMinOrderOfNontrivial
mem_interior_iff_mem_nhds
Set.image_nonempty
exists_between
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
LT.lt.trans
slope_comm
slope_mono
LT.lt.ne
LT.lt.ne'
LT.lt.le
monotoneOn_slope_gt 📖mathematicalConvexOn
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
Algebra.toSMul
Semifield.toCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
MonotoneOn
PartialOrder.toPreorder
slope
Ring.toAddCommGroup
DivisionRing.toRing
Field.toDivisionRing
Semiring.toModule
addGroupIsAddTorsor
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
setOf
Preorder.toLT
MonotoneOn.mono
slope_mono
LT.lt.ne'
monotoneOn_slope_lt 📖mathematicalConvexOn
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
Algebra.toSMul
Semifield.toCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
MonotoneOn
PartialOrder.toPreorder
slope
Ring.toAddCommGroup
DivisionRing.toRing
Field.toDivisionRing
Semiring.toModule
addGroupIsAddTorsor
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
setOf
Preorder.toLT
MonotoneOn.mono
slope_mono
LT.lt.ne
rightDeriv_eq_sInf_slope_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Ioi
Real.instPreorder
InfSet.sInf
Real.instInfSet
Set.image
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
setOf
Real.instLT
HasDerivWithinAt.derivWithin
hasDerivWithinAt_sInf_slope_of_mem_interior
uniqueDiffWithinAt_Ioi
rightDeriv_le_slope 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Ioi
Real.instPreorder
Real.instLE
derivWithin
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
le_slope_of_hasDerivWithinAt_Ioi
DifferentiableWithinAt.hasDerivWithinAt
rightDeriv_le_slope_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLT
Real.instLE
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Ioi
Real.instPreorder
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
rightDeriv_le_slope
interior_subset
differentiableWithinAt_Ioi_of_mem_interior
slope_le_deriv 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
deriv
slope_le_of_hasDerivAt
DifferentiableAt.hasDerivAt
slope_le_derivWithin 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
derivWithin
slope_le_of_hasDerivWithinAt
DifferentiableWithinAt.hasDerivWithinAt
slope_le_leftDeriv 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Iio
Real.instPreorder
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
derivWithin
slope_le_of_hasDerivWithinAt_Iio
DifferentiableWithinAt.hasDerivWithinAt
slope_le_leftDeriv_of_mem_interior 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLT
Real.instLE
slope
Real.instField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
derivWithin
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Set.Iio
Real.instPreorder
slope_le_leftDeriv
interior_subset
differentiableWithinAt_Iio_of_mem_interior
slope_le_of_hasDerivAt 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_le_of_hasDerivWithinAt_Iio
HasDerivAt.hasDerivWithinAt
slope_le_of_hasDerivWithinAt 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_le_of_hasDerivWithinAt_Iio
HasDerivWithinAt.mono_of_mem_nhdsWithin
Set.OrdConnected.mem_nhdsLT
instClosedIicTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
Convex.ordConnected
Real.instIsStrictOrderedRing
slope_le_of_hasDerivWithinAt_Iio 📖mathematicalConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Set.Iio
Real.instPreorder
Real.instLE
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
ge_of_tendsto
instClosedIciTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
nhdsLT_neBot
instOrderTopologyReal
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
instNoMinOrderOfNontrivial
Real.instIsOrderedRing
Real.instNontrivial
IsBoundedSMul.continuousSMul
NormedSpace.toIsBoundedSMul
hasDerivWithinAt_iff_tendsto_slope'
Set.self_notMem_Iio
slope_comm
slope_def_field
Filter.mp_mem
eventually_gt_nhds
instClosedIicTopology
Filter.univ_mem'
secant_mono
Set.OrdConnected.out
Convex.ordConnected
LT.lt.le
LT.lt.ne
slope_mono 📖mathematicalConvexOn
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
Algebra.toSMul
Semifield.toCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
MonotoneOn
PartialOrder.toPreorder
slope
Ring.toAddCommGroup
DivisionRing.toRing
Field.toDivisionRing
Semiring.toModule
addGroupIsAddTorsor
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
Set.instSDiff
Set.instSingletonSet
secant_mono
Set.mem_of_mem_diff
Set.notMem_of_mem_diff
slope_fun_def_field

Monotone

Theorems

NameKindAssumesProvesValidatesDepends On
convexOn_univ_of_deriv 📖mathematicalDifferentiable
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Monotone
Real.instPreorder
deriv
ConvexOn
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set.univ
MonotoneOn.convexOn_of_deriv
convex_univ
Continuous.continuousOn
Differentiable.continuous
IsTopologicalSemiring.toContinuousAdd
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
Differentiable.differentiableOn
monotoneOn

MonotoneOn

Theorems

NameKindAssumesProvesValidatesDepends On
convexOn_of_deriv 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
interior
MonotoneOn
Real.instPreorder
deriv
ConvexOnconvexOn_of_slope_mono_adjacent
Real.instIsStrictOrderedRing
Set.OrdConnected.out
Convex.ordConnected
HasSubset.Subset.trans
Set.instIsTransSubset
Set.Icc_subset_Icc_right
LT.lt.le
Set.subset_sUnion_of_mem
isOpen_Ioo
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
Set.Ioo_subset_Icc_self
Set.Icc_subset_Icc_left
exists_deriv_eq_slope
ContinuousOn.mono
DifferentiableOn.mono
LT.lt.trans

StrictAnti

Theorems

NameKindAssumesProvesValidatesDepends On
strictConcaveOn_univ_of_deriv 📖mathematicalContinuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
StrictAnti
Real.instPreorder
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
StrictConcaveOn
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set.univ
StrictAntiOn.strictConcaveOn_of_deriv
convex_univ
Continuous.continuousOn
strictAntiOn

StrictAntiOn

Theorems

NameKindAssumesProvesValidatesDepends On
strictConcaveOn_of_deriv 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
StrictAntiOn
Real.instPreorder
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
interior
StrictConcaveOnneg
IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono
AddLeftCancelSemigroup.toIsLeftCancelAdd
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
StrictConvexOn.neg
StrictMonoOn.strictConvexOn_of_deriv
ContinuousOn.neg
IsTopologicalRing.toContinuousNeg
instIsTopologicalRingReal
neg_neg

StrictConcaveOn

Theorems

NameKindAssumesProvesValidatesDepends On
derivWithin_lt_slope 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
derivWithin
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
lt_slope_of_hasDerivWithinAt
DifferentiableWithinAt.hasDerivWithinAt
deriv_lt_slope 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
deriv
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
lt_slope_of_hasDerivAt
DifferentiableAt.hasDerivAt
leftDeriv_lt_slope 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Iio
Real.instPreorder
derivWithin
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
lt_slope_of_hasDerivWithinAt_Iio
DifferentiableWithinAt.hasDerivWithinAt
lt_slope_of_hasDerivAt 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
lt_slope_of_hasDerivWithinAt_Iio
HasDerivAt.hasDerivWithinAt
lt_slope_of_hasDerivWithinAt 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
neg_neg
slope_neg
neg_lt_neg
Real.instIsOrderedAddMonoid
StrictConvexOn.slope_lt_of_hasDerivWithinAt
neg
HasDerivWithinAt.neg
lt_slope_of_hasDerivWithinAt_Iio 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Set.Iio
Real.instPreorder
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
neg_neg
slope_neg
neg_lt_neg
Real.instIsOrderedAddMonoid
StrictConvexOn.slope_lt_of_hasDerivWithinAt_Iio
neg
HasDerivWithinAt.neg
slope_lt_deriv 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
deriv
slope_lt_of_hasDerivAt
DifferentiableAt.hasDerivAt
slope_lt_derivWithin 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
derivWithin
slope_lt_of_hasDerivWithinAt
DifferentiableWithinAt.hasDerivWithinAt
slope_lt_of_hasDerivAt 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_neg
neg_neg
neg_lt_neg
Real.instIsOrderedAddMonoid
StrictConvexOn.lt_slope_of_hasDerivAt
neg
HasDerivAt.neg
slope_lt_of_hasDerivWithinAt 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_neg
neg_neg
neg_lt_neg
Real.instIsOrderedAddMonoid
StrictConvexOn.lt_slope_of_hasDerivWithinAt
neg
HasDerivWithinAt.neg
slope_lt_of_hasDerivWithinAt_Ioi 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Set.Ioi
Real.instPreorder
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_neg
neg_neg
neg_lt_neg
Real.instIsOrderedAddMonoid
StrictConvexOn.lt_slope_of_hasDerivWithinAt_Ioi
neg
HasDerivWithinAt.neg
slope_lt_rightDeriv 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Ioi
Real.instPreorder
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
derivWithin
slope_lt_of_hasDerivWithinAt_Ioi
DifferentiableWithinAt.hasDerivWithinAt
strictAntiOn_deriv 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
StrictAntiOn
Real.instPreorder
deriv
deriv.neg'
neg_neg
StrictMonoOn.neg
IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono
AddLeftCancelSemigroup.toIsLeftCancelAdd
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
StrictConvexOn.strictMonoOn_deriv
neg
DifferentiableAt.neg
strictAntiOn_derivWithin 📖mathematicalStrictConcaveOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
StrictAntiOn
Real.instPreorder
derivWithin
LT.lt.trans
derivWithin_lt_slope
slope_lt_derivWithin

StrictConvexOn

Theorems

NameKindAssumesProvesValidatesDepends On
derivWithin_lt_slope 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
derivWithin
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
lt_slope_of_hasDerivWithinAt
DifferentiableWithinAt.hasDerivWithinAt
deriv_lt_slope 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
deriv
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
lt_slope_of_hasDerivAt
DifferentiableAt.hasDerivAt
lt_slope_of_hasDerivAt 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
lt_slope_of_hasDerivWithinAt_Ioi
HasDerivAt.hasDerivWithinAt
lt_slope_of_hasDerivWithinAt 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
lt_slope_of_hasDerivWithinAt_Ioi
HasDerivWithinAt.mono_of_mem_nhdsWithin
Set.OrdConnected.mem_nhdsGT
instClosedIciTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
Convex.ordConnected
Real.instIsStrictOrderedRing
lt_slope_of_hasDerivWithinAt_Ioi 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Set.Ioi
Real.instPreorder
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
exists_between
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
Set.OrdConnected.out
Convex.ordConnected
LT.lt.le
secant_strict_mono
LT.lt.ne'
LE.le.trans_lt
ConvexOn.le_slope_of_hasDerivWithinAt_Ioi
convexOn
rightDeriv_lt_slope 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Ioi
Real.instPreorder
derivWithin
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
lt_slope_of_hasDerivWithinAt_Ioi
DifferentiableWithinAt.hasDerivWithinAt
slope_lt_deriv 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
deriv
slope_lt_of_hasDerivAt
DifferentiableAt.hasDerivAt
slope_lt_derivWithin 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
derivWithin
slope_lt_of_hasDerivWithinAt
DifferentiableWithinAt.hasDerivWithinAt
slope_lt_leftDeriv 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
DifferentiableWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Iio
Real.instPreorder
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
derivWithin
slope_lt_of_hasDerivWithinAt_Iio
DifferentiableWithinAt.hasDerivWithinAt
slope_lt_of_hasDerivAt 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_lt_of_hasDerivWithinAt_Iio
HasDerivAt.hasDerivWithinAt
slope_lt_of_hasDerivWithinAt 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
slope_lt_of_hasDerivWithinAt_Iio
HasDerivWithinAt.mono_of_mem_nhdsWithin
Set.OrdConnected.mem_nhdsLT
instClosedIicTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
Convex.ordConnected
Real.instIsStrictOrderedRing
slope_lt_of_hasDerivWithinAt_Iio 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
Real.instLT
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
Set.Iio
Real.instPreorder
slope
Real.instField
NormedAddTorsor.toAddTorsor
SeminormedAddCommGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toNormedAddTorsor
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
exists_between
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
Set.OrdConnected.out
Convex.ordConnected
LT.lt.le
secant_strict_mono
LT.lt.ne
LT.lt.trans_le
slope_comm
ConvexOn.slope_le_of_hasDerivWithinAt_Iio
convexOn
strictMonoOn_deriv 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
StrictMonoOn
Real.instPreorder
deriv
LT.lt.trans
deriv_lt_slope
slope_lt_deriv
strictMonoOn_derivWithin 📖mathematicalStrictConvexOn
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
StrictMonoOn
Real.instPreorder
derivWithin
LT.lt.trans
derivWithin_lt_slope
slope_lt_derivWithin

StrictMono

Theorems

NameKindAssumesProvesValidatesDepends On
strictConvexOn_univ_of_deriv 📖mathematicalContinuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
StrictMono
Real.instPreorder
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
StrictConvexOn
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set.univ
StrictMonoOn.strictConvexOn_of_deriv
convex_univ
Continuous.continuousOn
strictMonoOn

StrictMonoOn

Theorems

NameKindAssumesProvesValidatesDepends On
exists_deriv_lt_slope 📖mathematicalContinuousOn
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Icc
Real.instPreorder
Real.instLT
StrictMonoOn
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Ioo
Set
Set.instMembership
DivInvMonoid.toDiv
Real.instDivInvMonoid
Real.instSub
exists_deriv_lt_slope_aux
Mathlib.Tactic.Push.not_forall_eq
ContinuousOn.mono
Set.Icc_subset_Icc
le_rfl
LT.lt.le
mono
Set.Ioo_subset_Ioo
ne_of_lt
LT.lt.trans
ne_of_gt
MulPosReflectLE.toMulPosReflectLT
MulPosStrictMono.toMulPosReflectLE
IsStrictOrderedRing.toMulPosStrictMono
Real.instIsStrictOrderedRing
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
mul_lt_mul
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
sub_pos
lt_of_not_ge
Mathlib.Tactic.Linarith.lt_irrefl
Mathlib.Tactic.Ring.of_eq
Mathlib.Tactic.Ring.add_congr
Mathlib.Tactic.Ring.sub_congr
Mathlib.Tactic.Ring.mul_congr
Mathlib.Tactic.Ring.atom_pf
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.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.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.cast_zero
Mathlib.Meta.NormNum.isNat_ofNat
Nat.cast_zero
Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
Mathlib.Tactic.Linarith.add_neg
Mathlib.Tactic.Linarith.sub_neg_of_lt
Mathlib.Tactic.Linarith.sub_nonpos_of_le
exists_deriv_lt_slope_aux 📖mathematicalContinuousOn
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Icc
Real.instPreorder
Real.instLT
StrictMonoOn
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Ioo
Set
Set.instMembership
DivInvMonoid.toDiv
Real.instDivInvMonoid
Real.instSub
DifferentiableAt.differentiableWithinAt
differentiableAt_of_deriv_ne_zero
exists_deriv_eq_slope
Set.nonempty_Ioo
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
LT.lt.trans
exists_slope_lt_deriv 📖mathematicalContinuousOn
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Icc
Real.instPreorder
Real.instLT
StrictMonoOn
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Ioo
Set
Set.instMembership
DivInvMonoid.toDiv
Real.instDivInvMonoid
Real.instSub
exists_slope_lt_deriv_aux
Mathlib.Tactic.Push.not_forall_eq
ContinuousOn.mono
Set.Icc_subset_Icc
le_rfl
LT.lt.le
mono
Set.Ioo_subset_Ioo
ne_of_lt
LT.lt.trans
ne_of_gt
MulPosReflectLE.toMulPosReflectLT
MulPosStrictMono.toMulPosReflectLE
IsStrictOrderedRing.toMulPosStrictMono
Real.instIsStrictOrderedRing
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
mul_lt_mul
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
sub_pos
lt_of_not_ge
Mathlib.Tactic.Linarith.lt_irrefl
Mathlib.Tactic.Ring.of_eq
Mathlib.Tactic.Ring.add_congr
Mathlib.Tactic.Ring.sub_congr
Mathlib.Tactic.Ring.atom_pf
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_congr
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.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.cast_zero
Mathlib.Meta.NormNum.isNat_ofNat
Nat.cast_zero
Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
Mathlib.Tactic.Linarith.add_neg
Mathlib.Tactic.Linarith.sub_neg_of_lt
Mathlib.Tactic.Linarith.sub_nonpos_of_le
exists_slope_lt_deriv_aux 📖mathematicalContinuousOn
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Set.Icc
Real.instPreorder
Real.instLT
StrictMonoOn
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Set.Ioo
Set
Set.instMembership
DivInvMonoid.toDiv
Real.instDivInvMonoid
Real.instSub
DifferentiableAt.differentiableWithinAt
differentiableAt_of_deriv_ne_zero
exists_deriv_eq_slope
Set.nonempty_Ioo
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
LT.lt.trans
strictConvexOn_of_deriv 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
StrictMonoOn
Real.instPreorder
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
interior
StrictConvexOnstrictConvexOn_of_slope_strict_mono_adjacent
Real.instIsStrictOrderedRing
Set.OrdConnected.out
Convex.ordConnected
HasSubset.Subset.trans
Set.instIsTransSubset
Set.Icc_subset_Icc_right
LT.lt.le
Set.subset_sUnion_of_mem
isOpen_Ioo
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Real.instIsOrderedAddMonoid
Set.Ioo_subset_Icc_self
Set.Icc_subset_Icc_left
exists_slope_lt_deriv
ContinuousOn.mono
mono
exists_deriv_lt_slope
LT.lt.trans
lt_trans

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
bddAbove_slope_gt_of_mem_interior 📖mathematicalConvexOn
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
Algebra.toSMul
Semifield.toCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
BddAbove
Preorder.toLE
PartialOrder.toPreorder
Set.image
slope
Ring.toAddCommGroup
DivisionRing.toRing
Field.toDivisionRing
Semiring.toModule
addGroupIsAddTorsor
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
setOf
Preorder.toLT
Filter.Eventually.exists_gt
nhdsGT_neBot
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
instNoMaxOrderOfNontrivial
IsStrictOrderedRing.toIsOrderedRing
IsSimpleRing.instNontrivial
DivisionRing.isSimpleRing
mem_interior_iff_mem_nhds
bddAbove_iff_subset_Iic
ConvexOn.slope_mono
interior_subset
LT.lt.ne
LT.lt.ne'
LT.lt.le
LT.lt.trans
bddBelow_slope_lt_of_mem_interior 📖mathematicalConvexOn
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
Algebra.toSMul
Semifield.toCommSemiring
CommSemiring.toSemiring
Algebra.id
Set
Set.instMembership
interior
BddBelow
Preorder.toLE
PartialOrder.toPreorder
Set.image
slope
Ring.toAddCommGroup
DivisionRing.toRing
Field.toDivisionRing
Semiring.toModule
addGroupIsAddTorsor
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
setOf
Preorder.toLT
Filter.Eventually.exists_lt
nhdsLT_neBot
LinearOrderedSemiField.toDenselyOrdered
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
instNoMinOrderOfNontrivial
IsStrictOrderedRing.toIsOrderedRing
IsSimpleRing.instNontrivial
DivisionRing.isSimpleRing
mem_interior_iff_mem_nhds
bddBelow_iff_subset_Ici
ConvexOn.slope_mono
interior_subset
LT.lt.ne
LT.lt.ne'
LT.lt.le
LT.lt.trans
concaveOn_of_deriv2_nonpos 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
interior
deriv
Real.instLE
Nat.iterate
Real.instZero
ConcaveOnAntitoneOn.concaveOn_of_deriv
antitoneOn_of_deriv_nonpos
Convex.interior
instIsTopologicalAddGroupReal
UniformContinuousConstSMul.to_continuousConstSMul
Ring.uniformContinuousConstSMul
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Real.instZeroLEOneClass
DifferentiableOn.continuousOn
IsTopologicalSemiring.toContinuousAdd
ContinuousMul.to_continuousSMul
interior_interior
concaveOn_of_deriv2_nonpos' 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
deriv
Real.instLE
Nat.iterate
Real.instZero
ConcaveOnconcaveOn_of_deriv2_nonpos
DifferentiableOn.continuousOn
IsTopologicalSemiring.toContinuousAdd
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
DifferentiableOn.mono
interior_subset
concaveOn_of_hasDerivWithinAt2_nonpos 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
interior
Real.instLE
Real.instZero
ConcaveOnContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
deriv_eqOn
isOpen_interior
concaveOn_of_deriv2_nonpos
HasDerivWithinAt.differentiableWithinAt
differentiableOn_congr
HasDerivWithinAt.congr
concaveOn_univ_of_deriv2_nonpos 📖mathematicalDifferentiable
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
deriv
Real.instLE
Nat.iterate
Real.instZero
ConcaveOn
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set.univ
concaveOn_of_deriv2_nonpos'
convex_univ
Differentiable.differentiableOn
convexOn_of_deriv2_nonneg 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
interior
deriv
Real.instLE
Real.instZero
Nat.iterate
ConvexOnMonotoneOn.convexOn_of_deriv
monotoneOn_of_deriv_nonneg
Convex.interior
instIsTopologicalAddGroupReal
UniformContinuousConstSMul.to_continuousConstSMul
Ring.uniformContinuousConstSMul
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Real.instZeroLEOneClass
DifferentiableOn.continuousOn
IsTopologicalSemiring.toContinuousAdd
ContinuousMul.to_continuousSMul
interior_interior
convexOn_of_deriv2_nonneg' 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
DifferentiableOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
deriv
Real.instLE
Real.instZero
Nat.iterate
ConvexOnconvexOn_of_deriv2_nonneg
DifferentiableOn.continuousOn
IsTopologicalSemiring.toContinuousAdd
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
ContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
DifferentiableOn.mono
interior_subset
convexOn_of_hasDerivWithinAt2_nonneg 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
HasDerivWithinAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
ContinuousMul.to_continuousSMul
SeminormedRing.toPseudoMetricSpace
SeminormedCommRing.toSeminormedRing
NormedField.toNormedCommRing
NontriviallyNormedField.toNormedField
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
Semiring.toMonoidWithZero
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
IsTopologicalSemiring.toContinuousMul
NonUnitalSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonUnitalSemiring
IsTopologicalRing.toIsTopologicalSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
instIsTopologicalRingReal
interior
Real.instLE
Real.instZero
ConvexOnContinuousMul.to_continuousSMul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
deriv_eqOn
isOpen_interior
convexOn_of_deriv2_nonneg
HasDerivWithinAt.differentiableWithinAt
differentiableOn_congr
HasDerivWithinAt.congr
convexOn_univ_of_deriv2_nonneg 📖mathematicalDifferentiable
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
deriv
Real.instLE
Real.instZero
Nat.iterate
ConvexOn
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set.univ
convexOn_of_deriv2_nonneg'
convex_univ
Differentiable.differentiableOn
strictConcaveOn_of_deriv2_neg 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLT
Nat.iterate
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Real.instZero
StrictConcaveOnStrictAntiOn.strictConcaveOn_of_deriv
strictAntiOn_of_deriv_neg
Convex.interior
instIsTopologicalAddGroupReal
UniformContinuousConstSMul.to_continuousConstSMul
Ring.uniformContinuousConstSMul
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Real.instZeroLEOneClass
DifferentiableWithinAt.continuousWithinAt
IsTopologicalSemiring.toContinuousAdd
ContinuousMul.to_continuousSMul
DifferentiableAt.differentiableWithinAt
differentiableAt_of_deriv_ne_zero
LT.lt.ne
interior_interior
strictConcaveOn_of_deriv2_neg' 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLT
Nat.iterate
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Real.instZero
StrictConcaveOnstrictConcaveOn_of_deriv2_neg
interior_subset
strictConcaveOn_univ_of_deriv2_neg 📖mathematicalContinuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLT
Nat.iterate
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
Real.instZero
StrictConcaveOn
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set.univ
strictConcaveOn_of_deriv2_neg'
convex_univ
Continuous.continuousOn
strictConvexOn_of_deriv2_pos 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLT
Real.instZero
Nat.iterate
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
StrictConvexOnStrictMonoOn.strictConvexOn_of_deriv
strictMonoOn_of_deriv_pos
Convex.interior
instIsTopologicalAddGroupReal
UniformContinuousConstSMul.to_continuousConstSMul
Ring.uniformContinuousConstSMul
instIsUniformAddGroupReal
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
Real.instZeroLEOneClass
DifferentiableWithinAt.continuousWithinAt
IsTopologicalSemiring.toContinuousAdd
ContinuousMul.to_continuousSMul
DifferentiableAt.differentiableWithinAt
differentiableAt_of_deriv_ne_zero
LT.lt.ne'
interior_interior
strictConvexOn_of_deriv2_pos' 📖mathematicalConvex
Real
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
ContinuousOn
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLT
Real.instZero
Nat.iterate
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
StrictConvexOnstrictConvexOn_of_deriv2_pos
interior_subset
strictConvexOn_univ_of_deriv2_pos 📖mathematicalContinuous
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.instLT
Real.instZero
Nat.iterate
deriv
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.instAddCommGroup
NormedSpace.toModule
Real.normedField
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
NormedField.toNormedSpace
StrictConvexOn
Real.semiring
Real.partialOrder
Real.instAddCommMonoid
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Set.univ
strictConvexOn_of_deriv2_pos'
convex_univ
Continuous.continuousOn

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