Documentation Verification Report

Basic

📁 Source: Mathlib/Topology/Semicontinuity/Basic.lean

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Theoremscomp_lowerSemicontinuous, comp_lowerSemicontinuousOn, comp_lowerSemicontinuousOn_antitone, comp_lowerSemicontinuous_antitone, comp_upperSemicontinuous, comp_upperSemicontinuousOn, comp_upperSemicontinuousOn_antitone, comp_upperSemicontinuous_antitone, lowerSemicontinuous, upperSemicontinuous, comp_lowerSemicontinuousAt, comp_lowerSemicontinuousAt_antitone, comp_lowerSemicontinuousWithinAt, comp_lowerSemicontinuousWithinAt_antitone, comp_upperSemicontinuousAt, comp_upperSemicontinuousAt_antitone, comp_upperSemicontinuousWithinAt, comp_upperSemicontinuousWithinAt_antitone, lowerSemicontinuousAt, upperSemicontinuousAt, lowerSemicontinuousOn, upperSemicontinuousOn, lowerSemicontinuousWithinAt, upperSemicontinuousWithinAt, lowerSemicontinuousAt_indicator, lowerSemicontinuousOn_indicator, lowerSemicontinuousWithinAt_indicator, lowerSemicontinuous_indicator, upperSemicontinuousAt_indicator, upperSemicontinuousOn_indicator, upperSemicontinuousWithinAt_indicator, upperSemicontinuous_indicator, lowerSemicontinuousAt_indicator, lowerSemicontinuousOn_indicator, lowerSemicontinuousWithinAt_indicator, lowerSemicontinuous_indicator, upperSemicontinuousAt_indicator, upperSemicontinuousOn_indicator, upperSemicontinuousWithinAt_indicator, upperSemicontinuous_indicator, add, add', comp_continuous, inf, isClosed_epigraph, isClosed_preimage, isOpen_preimage, le_liminf, sup, add, add', comp_continuousAt, comp_continuousAt_of_eq, inf, le_liminf, sup, add, add', bddBelow_of_isCompact, exists_isMinOn, inf, inter_biInter_preimage_Iic_eq_empty_iff_exists_finset, isCompact_inter_preimage_Iic, le_liminf, sup, add, add', inf, le_liminf, sup, IsClosed_hypograph, add, add', comp_continuous, inf, isClosed_preimage, isOpen_preimage, limsup_le, sup, add, add', comp_continuousAt, comp_continuousAt_of_eq, inf, limsup_le, sup, add, add', bddAbove_of_isCompact, exists_isMaxOn, inf, inter_biInter_preimage_Ici_eq_empty_iff_exists_finset, isCompact_inter_preimage_Ici, limsup_le, sup, add, add', inf, limsup_le, sup, continuousAt_iff_lower_upperSemicontinuousAt, continuousOn_iff_lower_upperSemicontinuousOn, continuousWithinAt_iff_lower_upperSemicontinuousWithinAt, continuous_iff_lower_upperSemicontinuous, lowerSemicontinuousAt_biSup, lowerSemicontinuousAt_ciSup, lowerSemicontinuousAt_iSup, lowerSemicontinuousAt_iff_le_liminf, lowerSemicontinuousAt_sum, lowerSemicontinuousAt_tsum, lowerSemicontinuousOn_biSup, lowerSemicontinuousOn_ciSup, lowerSemicontinuousOn_iSup, lowerSemicontinuousOn_iff_isClosed_epigraph, lowerSemicontinuousOn_iff_le_liminf, lowerSemicontinuousOn_iff_preimage_Iic, lowerSemicontinuousOn_iff_preimage_Ioi, lowerSemicontinuousOn_sum, lowerSemicontinuousOn_tsum, lowerSemicontinuousWithinAt_biSup, lowerSemicontinuousWithinAt_ciSup, lowerSemicontinuousWithinAt_iSup, lowerSemicontinuousWithinAt_iff_le_liminf, lowerSemicontinuousWithinAt_sum, lowerSemicontinuousWithinAt_tsum, lowerSemicontinuous_biSup, lowerSemicontinuous_ciSup, lowerSemicontinuous_iSup, lowerSemicontinuous_iff_isClosed_epigraph, lowerSemicontinuous_iff_isClosed_preimage, lowerSemicontinuous_iff_isOpen_preimage, lowerSemicontinuous_iff_le_liminf, lowerSemicontinuous_sum, lowerSemicontinuous_tsum, upperSemicontinuousAt_biInf, upperSemicontinuousAt_ciInf, upperSemicontinuousAt_iInf, upperSemicontinuousAt_iff_limsup_le, upperSemicontinuousAt_sum, upperSemicontinuousOn_biInf, upperSemicontinuousOn_ciInf, upperSemicontinuousOn_iInf, upperSemicontinuousOn_iff_isClosed_hypograph, upperSemicontinuousOn_iff_limsup_le, upperSemicontinuousOn_iff_preimage_Ici, upperSemicontinuousOn_iff_preimage_Iio, upperSemicontinuousOn_sum, upperSemicontinuousWithinAt_biInf, upperSemicontinuousWithinAt_ciInf, upperSemicontinuousWithinAt_iInf, upperSemicontinuousWithinAt_iff_limsup_le, upperSemicontinuousWithinAt_sum, upperSemicontinuous_biInf, upperSemicontinuous_ciInf, upperSemicontinuous_iInf, upperSemicontinuous_iff_IsClosed_hypograph, upperSemicontinuous_iff_isClosed_preimage, upperSemicontinuous_iff_isOpen_preimage, upperSemicontinuous_iff_limsup_le, upperSemicontinuous_sum
160
Total160

Continuous

Theorems

NameKindAssumesProvesValidatesDepends On
comp_lowerSemicontinuous 📖Continuous
LowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Monotone
ContinuousAt.comp_lowerSemicontinuousAt
continuousAt
comp_lowerSemicontinuousOn 📖Continuous
LowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Monotone
ContinuousAt.comp_lowerSemicontinuousWithinAt
continuousAt
comp_lowerSemicontinuousOn_antitone 📖mathematicalContinuous
LowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Antitone
UpperSemicontinuousOnContinuousAt.comp_lowerSemicontinuousWithinAt_antitone
continuousAt
comp_lowerSemicontinuous_antitone 📖mathematicalContinuous
LowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Antitone
UpperSemicontinuousContinuousAt.comp_lowerSemicontinuousAt_antitone
continuousAt
comp_upperSemicontinuous 📖Continuous
UpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Monotone
ContinuousAt.comp_upperSemicontinuousAt
continuousAt
comp_upperSemicontinuousOn 📖Continuous
UpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Monotone
ContinuousAt.comp_upperSemicontinuousWithinAt
continuousAt
comp_upperSemicontinuousOn_antitone 📖mathematicalContinuous
UpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Antitone
LowerSemicontinuousOnContinuousAt.comp_upperSemicontinuousWithinAt_antitone
continuousAt
comp_upperSemicontinuous_antitone 📖mathematicalContinuous
UpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Antitone
LowerSemicontinuousContinuousAt.comp_upperSemicontinuousAt_antitone
continuousAt
lowerSemicontinuous 📖mathematicalContinuousLowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt.lowerSemicontinuousAt
continuousAt
upperSemicontinuous 📖mathematicalContinuousUpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt.upperSemicontinuousAt
continuousAt

ContinuousAt

Theorems

NameKindAssumesProvesValidatesDepends On
comp_lowerSemicontinuousAt 📖ContinuousAt
LowerSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Monotone
comp_lowerSemicontinuousWithinAt
comp_lowerSemicontinuousAt_antitone 📖mathematicalContinuousAt
LowerSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Antitone
UpperSemicontinuousAtcomp_lowerSemicontinuousAt
instOrderTopologyOrderDual
comp_lowerSemicontinuousWithinAt 📖ContinuousAt
LowerSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Monotone
exists_Ioc_subset_of_mem_nhds
Ioi_mem_nhds
instClosedIicTopology
OrderTopology.to_orderClosedTopology
Filter.mp_mem
Filter.univ_mem'
min_le_right
Filter.Eventually.of_forall
LT.lt.trans_le
comp_lowerSemicontinuousWithinAt_antitone 📖mathematicalContinuousAt
LowerSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Antitone
UpperSemicontinuousWithinAtcomp_lowerSemicontinuousWithinAt
instOrderTopologyOrderDual
comp_upperSemicontinuousAt 📖ContinuousAt
UpperSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Monotone
comp_lowerSemicontinuousAt
instOrderTopologyOrderDual
Monotone.dual
comp_upperSemicontinuousAt_antitone 📖mathematicalContinuousAt
UpperSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Antitone
LowerSemicontinuousAtcomp_upperSemicontinuousAt
instOrderTopologyOrderDual
comp_upperSemicontinuousWithinAt 📖ContinuousAt
UpperSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Monotone
comp_lowerSemicontinuousWithinAt
instOrderTopologyOrderDual
Monotone.dual
comp_upperSemicontinuousWithinAt_antitone 📖mathematicalContinuousAt
UpperSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Antitone
LowerSemicontinuousWithinAtcomp_upperSemicontinuousWithinAt
instOrderTopologyOrderDual
lowerSemicontinuousAt 📖mathematicalContinuousAtLowerSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Ioi_mem_nhds
instClosedIicTopology
OrderTopology.to_orderClosedTopology
upperSemicontinuousAt 📖mathematicalContinuousAtUpperSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Iio_mem_nhds
instClosedIciTopology
OrderTopology.to_orderClosedTopology

ContinuousOn

Theorems

NameKindAssumesProvesValidatesDepends On
lowerSemicontinuousOn 📖mathematicalContinuousOnLowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt.lowerSemicontinuousWithinAt
upperSemicontinuousOn 📖mathematicalContinuousOnUpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt.upperSemicontinuousWithinAt

ContinuousWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
lowerSemicontinuousWithinAt 📖mathematicalContinuousWithinAtLowerSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Ioi_mem_nhds
instClosedIicTopology
OrderTopology.to_orderClosedTopology
upperSemicontinuousWithinAt 📖mathematicalContinuousWithinAtUpperSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Iio_mem_nhds
instClosedIciTopology
OrderTopology.to_orderClosedTopology

IsClosed

Theorems

NameKindAssumesProvesValidatesDepends On
lowerSemicontinuousAt_indicator 📖mathematicalPreorder.toLELowerSemicontinuousAt
Set.indicator
LowerSemicontinuous.lowerSemicontinuousAt
lowerSemicontinuous_indicator
lowerSemicontinuousOn_indicator 📖mathematicalPreorder.toLELowerSemicontinuousOn
Set.indicator
LowerSemicontinuous.lowerSemicontinuousOn
lowerSemicontinuous_indicator
lowerSemicontinuousWithinAt_indicator 📖mathematicalPreorder.toLELowerSemicontinuousWithinAt
Set.indicator
LowerSemicontinuous.lowerSemicontinuousWithinAt
lowerSemicontinuous_indicator
lowerSemicontinuous_indicator 📖mathematicalPreorder.toLELowerSemicontinuous
Set.indicator
Filter.Eventually.of_forall
Set.indicator_of_mem
Set.indicator_of_notMem
LT.lt.trans_le
Filter.mp_mem
IsOpen.mem_nhds
isOpen_compl
Filter.univ_mem'
upperSemicontinuousAt_indicator 📖mathematicalPreorder.toLEUpperSemicontinuousAt
Set.indicator
UpperSemicontinuous.upperSemicontinuousAt
upperSemicontinuous_indicator
upperSemicontinuousOn_indicator 📖mathematicalPreorder.toLEUpperSemicontinuousOn
Set.indicator
UpperSemicontinuous.upperSemicontinuousOn
upperSemicontinuous_indicator
upperSemicontinuousWithinAt_indicator 📖mathematicalPreorder.toLEUpperSemicontinuousWithinAt
Set.indicator
UpperSemicontinuous.upperSemicontinuousWithinAt
upperSemicontinuous_indicator
upperSemicontinuous_indicator 📖mathematicalPreorder.toLEUpperSemicontinuous
Set.indicator
lowerSemicontinuous_indicator

IsOpen

Theorems

NameKindAssumesProvesValidatesDepends On
lowerSemicontinuousAt_indicator 📖mathematicalIsOpen
Preorder.toLE
LowerSemicontinuousAt
Set.indicator
LowerSemicontinuous.lowerSemicontinuousAt
lowerSemicontinuous_indicator
lowerSemicontinuousOn_indicator 📖mathematicalIsOpen
Preorder.toLE
LowerSemicontinuousOn
Set.indicator
LowerSemicontinuous.lowerSemicontinuousOn
lowerSemicontinuous_indicator
lowerSemicontinuousWithinAt_indicator 📖mathematicalIsOpen
Preorder.toLE
LowerSemicontinuousWithinAt
Set.indicator
LowerSemicontinuous.lowerSemicontinuousWithinAt
lowerSemicontinuous_indicator
lowerSemicontinuous_indicator 📖mathematicalIsOpen
Preorder.toLE
LowerSemicontinuous
Set.indicator
Filter.mp_mem
mem_nhds
Filter.univ_mem'
Set.indicator_of_mem
Filter.Eventually.of_forall
LT.lt.trans_le
Set.indicator_of_notMem
upperSemicontinuousAt_indicator 📖mathematicalIsOpen
Preorder.toLE
UpperSemicontinuousAt
Set.indicator
UpperSemicontinuous.upperSemicontinuousAt
upperSemicontinuous_indicator
upperSemicontinuousOn_indicator 📖mathematicalIsOpen
Preorder.toLE
UpperSemicontinuousOn
Set.indicator
UpperSemicontinuous.upperSemicontinuousOn
upperSemicontinuous_indicator
upperSemicontinuousWithinAt_indicator 📖mathematicalIsOpen
Preorder.toLE
UpperSemicontinuousWithinAt
Set.indicator
UpperSemicontinuous.upperSemicontinuousWithinAt
upperSemicontinuous_indicator
upperSemicontinuous_indicator 📖mathematicalIsOpen
Preorder.toLE
UpperSemicontinuous
Set.indicator
lowerSemicontinuous_indicator

LowerSemicontinuous

Theorems

NameKindAssumesProvesValidatesDepends On
add 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
add'
Continuous.continuousAt
continuous_add
add' 📖LowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt
instTopologicalSpaceProd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
LowerSemicontinuousAt.add'
comp_continuous 📖LowerSemicontinuous
Continuous
comp
inf 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinLowerSemicontinuousAt.inf
isClosed_epigraph 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
instTopologicalSpaceProd
setOf
Preorder.toLE
lowerSemicontinuous_iff_isClosed_epigraph
isClosed_preimage 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
Set.preimage
Set.Iic
lowerSemicontinuous_iff_isClosed_preimage
isOpen_preimage 📖mathematicalLowerSemicontinuousIsOpen
Set.preimage
Set.Ioi
lowerSemicontinuous_iff_isOpen_preimage
le_liminf 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.liminf
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhds
lowerSemicontinuous_iff_le_liminf
sup 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
LowerSemicontinuousAt.sup

LowerSemicontinuousAt

Theorems

NameKindAssumesProvesValidatesDepends On
add 📖mathematicalLowerSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
add'
Continuous.continuousAt
continuous_add
add' 📖LowerSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt
instTopologicalSpaceProd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
LowerSemicontinuousWithinAt.add'
comp_continuousAt 📖LowerSemicontinuousAt
ContinuousAt
comp
comp_continuousAt_of_eq 📖LowerSemicontinuousAt
ContinuousAt
comp
inf 📖mathematicalLowerSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinlowerSemicontinuousWithinAt_univ_iff
LowerSemicontinuousWithinAt.inf
le_liminf 📖mathematicalLowerSemicontinuousAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.liminf
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhds
lowerSemicontinuousAt_iff_le_liminf
sup 📖mathematicalLowerSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
lowerSemicontinuousWithinAt_univ_iff
LowerSemicontinuousWithinAt.sup

LowerSemicontinuousOn

Theorems

NameKindAssumesProvesValidatesDepends On
add 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
add'
Continuous.continuousAt
continuous_add
add' 📖LowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt
instTopologicalSpaceProd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
LowerSemicontinuousWithinAt.add'
bddBelow_of_isCompact 📖mathematicalIsCompact
LowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
BddBelow
Preorder.toLE
Set.image
Set.eq_empty_or_nonempty
Set.image_empty
bddBelow_empty
exists_isMinOn
IsMinOn.bddBelow
exists_isMinOn 📖mathematicalSet.Nonempty
IsCompact
LowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Set.instMembership
IsMinOn
Set.Nonempty.to_subtype
Filter.iInf_neBot_of_directed
Directed.mono_comp
Filter.principal_mono
Set.inter_subset_inter_right
Set.preimage_mono
Set.Iic_subset_Iic
Std.Total.directed
LE.total'
Filter.le_principal_iff
le_refl
Filter.neBot_of_le
Filter.pure_neBot
iInf_le_of_le
Set.inter_subset_left
Filter.mem_iInf_of_mem
Set.inter_subset_right
le_of_not_gt
Filter.mp_mem
Filter.Eventually.filter_mono
inf_le_right
inf_le_inf_left
Filter.univ_mem'
not_le_of_gt
Filter.eventually_const
inf 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinLowerSemicontinuousWithinAt.inf
inter_biInter_preimage_Iic_eq_empty_iff_exists_finset 📖mathematicalIsCompact
LowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Set.instInter
Set.iInter
Set.instMembership
Set.preimage
Set.Iic
Set.instEmptyCollection
Set.Elem
Finset
Finset.instMembership
Preorder.toLT
LowerSemicontinuous.isClosed_preimage
lowerSemicontinuous_restrict_iff
Set.iInter_coe_set
Set.iInter_congr_Prop
IsCompact.elim_finite_subfamily_isClosed_subtype
Set.eq_empty_iff_forall_notMem
isCompact_inter_preimage_Iic 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsCompact
Set
Set.instInter
Set.preimage
Set.Iic
lowerSemicontinuousOn_iff_preimage_Iic
IsCompact.inter_right
le_liminf 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Set
Set.instMembership
Preorder.toLE
Filter.liminf
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhdsWithin
lowerSemicontinuousOn_iff_le_liminf
sup 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
LowerSemicontinuousWithinAt.sup

LowerSemicontinuousWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
add 📖mathematicalLowerSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
add'
Continuous.continuousAt
continuous_add
add' 📖LowerSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt
instTopologicalSpaceProd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
mem_nhds_prod_iff'
IsOpen.mem_nhds
isOpen_Ioi
instClosedIicTopology
OrderTopology.to_orderClosedTopology
exists_Ioc_subset_of_mem_nhds
Filter.mp_mem
Filter.univ_mem'
inf_of_le_left
inf_of_le_right
LT.lt.le
add_le_add
IsOrderedAddMonoid.toAddLeftMono
covariant_swap_add_of_covariant_add
min_le_left
le_rfl
Filter.Eventually.of_forall
inf 📖mathematicalLowerSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinFilter.Eventually.and
le_liminf 📖mathematicalLowerSemicontinuousWithinAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.liminf
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhdsWithin
lowerSemicontinuousWithinAt_iff_le_liminf
sup 📖mathematicalLowerSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
Filter.mp_mem
Filter.univ_mem'

UpperSemicontinuous

Theorems

NameKindAssumesProvesValidatesDepends On
IsClosed_hypograph 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
instTopologicalSpaceProd
setOf
Preorder.toLE
upperSemicontinuous_iff_IsClosed_hypograph
add 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
add'
Continuous.continuousAt
continuous_add
add' 📖UpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt
instTopologicalSpaceProd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
UpperSemicontinuousAt.add'
comp_continuous 📖UpperSemicontinuous
Continuous
comp
inf 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinLowerSemicontinuous.sup
isClosed_preimage 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
Set.preimage
Set.Ici
upperSemicontinuous_iff_isClosed_preimage
isOpen_preimage 📖mathematicalUpperSemicontinuousIsOpen
Set.preimage
Set.Iio
upperSemicontinuous_iff_isOpen_preimage
limsup_le 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.limsup
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhds
upperSemicontinuous_iff_limsup_le
sup 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
LowerSemicontinuous.inf

UpperSemicontinuousAt

Theorems

NameKindAssumesProvesValidatesDepends On
add 📖mathematicalUpperSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
add'
Continuous.continuousAt
continuous_add
add' 📖UpperSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt
instTopologicalSpaceProd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
UpperSemicontinuousWithinAt.add'
comp_continuousAt 📖UpperSemicontinuousAt
ContinuousAt
comp
comp_continuousAt_of_eq 📖UpperSemicontinuousAt
ContinuousAt
comp
inf 📖mathematicalUpperSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinLowerSemicontinuousAt.sup
limsup_le 📖mathematicalUpperSemicontinuousAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.limsup
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhds
upperSemicontinuousAt_iff_limsup_le
sup 📖mathematicalUpperSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
LowerSemicontinuousAt.inf

UpperSemicontinuousOn

Theorems

NameKindAssumesProvesValidatesDepends On
add 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
add'
Continuous.continuousAt
continuous_add
add' 📖UpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt
instTopologicalSpaceProd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
UpperSemicontinuousWithinAt.add'
bddAbove_of_isCompact 📖mathematicalIsCompact
UpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
BddAbove
Preorder.toLE
Set.image
LowerSemicontinuousOn.bddBelow_of_isCompact
OrderDual.instNonempty
exists_isMaxOn 📖mathematicalSet.Nonempty
IsCompact
UpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Set.instMembership
IsMaxOn
LowerSemicontinuousOn.exists_isMinOn
inf 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinLowerSemicontinuousOn.sup
inter_biInter_preimage_Ici_eq_empty_iff_exists_finset 📖mathematicalIsCompact
UpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Set.instInter
Set.iInter
Set.instMembership
Set.preimage
Set.Ici
Set.instEmptyCollection
Set.Elem
Finset
Finset.instMembership
Preorder.toLT
LowerSemicontinuousOn.inter_biInter_preimage_Iic_eq_empty_iff_exists_finset
isCompact_inter_preimage_Ici 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsCompact
Set
Set.instInter
Set.preimage
Set.Ici
LowerSemicontinuousOn.isCompact_inter_preimage_Iic
limsup_le 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Set
Set.instMembership
Preorder.toLE
Filter.limsup
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhdsWithin
upperSemicontinuousOn_iff_limsup_le
sup 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
LowerSemicontinuousOn.inf

UpperSemicontinuousWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
add 📖mathematicalUpperSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
add'
Continuous.continuousAt
continuous_add
add' 📖UpperSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousAt
instTopologicalSpaceProd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
LowerSemicontinuousWithinAt.add'
OrderDual.isOrderedAddMonoid
instOrderTopologyOrderDual
inf 📖mathematicalUpperSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinLowerSemicontinuousWithinAt.sup
limsup_le 📖mathematicalUpperSemicontinuousWithinAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.limsup
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhdsWithin
upperSemicontinuousWithinAt_iff_limsup_le
sup 📖mathematicalUpperSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
LowerSemicontinuousWithinAt.inf

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
continuousAt_iff_lower_upperSemicontinuousAt 📖mathematicalContinuousAt
LowerSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
UpperSemicontinuousAt
continuousOn_iff_lower_upperSemicontinuousOn 📖mathematicalContinuousOn
LowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
UpperSemicontinuousOn
continuousWithinAt_iff_lower_upperSemicontinuousWithinAt 📖mathematicalContinuousWithinAt
LowerSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
UpperSemicontinuousWithinAt
ContinuousWithinAt.lowerSemicontinuousWithinAt
ContinuousWithinAt.upperSemicontinuousWithinAt
exists_Ioc_subset_of_mem_nhds
exists_Ico_subset_of_mem_nhds
Filter.mp_mem
Filter.univ_mem'
le_or_gt
le_of_lt
Filter.Eventually.of_forall
le_antisymm
mem_of_mem_nhds
continuous_iff_lower_upperSemicontinuous 📖mathematicalContinuous
LowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
UpperSemicontinuous
lowerSemicontinuousAt_biSup 📖mathematicalLowerSemicontinuousAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuousAt_iSup
lowerSemicontinuousAt_ciSup 📖mathematicalFilter.Eventually
BddAbove
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.range
nhds
LowerSemicontinuousAt
iSup
ConditionallyCompletePartialOrderSup.toSupSet
lowerSemicontinuousWithinAt_ciSup
nhdsWithin_univ
lowerSemicontinuousAt_iSup 📖mathematicalLowerSemicontinuousAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuousAt_ciSup
lowerSemicontinuousAt_iff_le_liminf 📖mathematicalLowerSemicontinuousAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.liminf
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhds
lowerSemicontinuousWithinAt_univ_iff
lowerSemicontinuousWithinAt_iff_le_liminf
nhdsWithin_univ
lowerSemicontinuousAt_sum 📖mathematicalLowerSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Finset.sumlowerSemicontinuousWithinAt_sum
lowerSemicontinuousAt_tsum 📖mathematicalLowerSemicontinuousAt
ENNReal
PartialOrder.toPreorder
ENNReal.instPartialOrder
tsum
ENNReal.instAddCommMonoid
ENNReal.instTopologicalSpace
SummationFilter.unconditional
lowerSemicontinuousWithinAt_tsum
lowerSemicontinuousOn_biSup 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuousOn_iSup
lowerSemicontinuousOn_ciSup 📖mathematicalBddAbove
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.range
LowerSemicontinuousOn
iSup
ConditionallyCompletePartialOrderSup.toSupSet
lowerSemicontinuousWithinAt_ciSup
eventually_nhdsWithin_of_forall
lowerSemicontinuousOn_iSup 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuousOn_ciSup
lowerSemicontinuousOn_iff_isClosed_epigraph 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
instTopologicalSpaceProd
setOf
Set
Set.instMembership
Preorder.toLE
LT.lt.exists_disjoint_Iio_Ioi
Filter.mp_mem
Filter.Eventually.prodMk_nhds
eventually_lt_nhds
Filter.univ_mem'
Filter.Tendsto.eventually
Continuous.tendsto
continuous_fst
IsOpen.eventually_mem
IsClosed.isOpen_compl
Continuous.prodMk_left
lowerSemicontinuousOn_iff_le_liminf 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.liminf
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhdsWithin
lowerSemicontinuousOn_iff_preimage_Iic 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
Set
Set.instInter
Set.preimage
Set.Iic
lowerSemicontinuousOn_iff_preimage_Ioi 📖mathematicalLowerSemicontinuousOn
IsOpen
Set
Set.instInter
Set.preimage
Set.Ioi
lowerSemicontinuousOn_sum 📖mathematicalLowerSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Finset.sumlowerSemicontinuousWithinAt_sum
lowerSemicontinuousOn_tsum 📖mathematicalLowerSemicontinuousOn
ENNReal
PartialOrder.toPreorder
ENNReal.instPartialOrder
tsum
ENNReal.instAddCommMonoid
ENNReal.instTopologicalSpace
SummationFilter.unconditional
lowerSemicontinuousWithinAt_tsum
lowerSemicontinuousWithinAt_biSup 📖mathematicalLowerSemicontinuousWithinAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuousWithinAt_iSup
lowerSemicontinuousWithinAt_ciSup 📖mathematicalFilter.Eventually
BddAbove
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.range
nhdsWithin
LowerSemicontinuousWithinAt
iSup
ConditionallyCompletePartialOrderSup.toSupSet
isEmpty_or_nonempty
iSup_of_empty'
lowerSemicontinuousWithinAt_const
exists_lt_of_lt_ciSup
Filter.mp_mem
Filter.univ_mem'
LT.lt.trans_le
le_ciSup
lowerSemicontinuousWithinAt_iSup 📖mathematicalLowerSemicontinuousWithinAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuousWithinAt_ciSup
lowerSemicontinuousWithinAt_iff_le_liminf 📖mathematicalLowerSemicontinuousWithinAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.liminf
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhdsWithin
LT.lt.exists_disjoint_Iio_Ioi
LT.lt.not_ge
Filter.le_liminf_of_le
Filter.isCobounded_ge_of_top
Filter.Eventually.mono
le_of_not_gt
LT.lt.false
Filter.eventually_lt_of_lt_liminf
LT.lt.trans_le
Filter.isBounded_ge_of_bot
lowerSemicontinuousWithinAt_sum 📖mathematicalLowerSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Finset.sumFinset.induction_on
lowerSemicontinuousWithinAt_const
Finset.sum_insert
LowerSemicontinuousWithinAt.add
Finset.mem_insert_self
Finset.mem_insert_of_mem
lowerSemicontinuousWithinAt_tsum 📖mathematicalLowerSemicontinuousWithinAt
ENNReal
PartialOrder.toPreorder
ENNReal.instPartialOrder
tsum
ENNReal.instAddCommMonoid
ENNReal.instTopologicalSpace
SummationFilter.unconditional
ENNReal.tsum_eq_iSup_sum
lowerSemicontinuousWithinAt_iSup
lowerSemicontinuousWithinAt_sum
ENNReal.instIsOrderedAddMonoid
ENNReal.instOrderTopology
ENNReal.instContinuousAdd
lowerSemicontinuous_biSup 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuous_iSup
lowerSemicontinuous_ciSup 📖mathematicalBddAbove
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.range
LowerSemicontinuous
iSup
ConditionallyCompletePartialOrderSup.toSupSet
lowerSemicontinuousAt_ciSup
Filter.Eventually.of_forall
lowerSemicontinuous_iSup 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuous_ciSup
lowerSemicontinuous_iff_isClosed_epigraph 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
instTopologicalSpaceProd
setOf
Preorder.toLE
lowerSemicontinuous_iff_isClosed_preimage 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
Set.preimage
Set.Iic
lowerSemicontinuous_iff_isOpen_preimage
Set.compl_Iic
lowerSemicontinuous_iff_isOpen_preimage 📖mathematicalLowerSemicontinuous
IsOpen
Set.preimage
Set.Ioi
isOpen_iff_mem_nhds
IsOpen.mem_nhds
lowerSemicontinuous_iff_le_liminf 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.liminf
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhds
lowerSemicontinuous_sum 📖mathematicalLowerSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Finset.sumlowerSemicontinuousAt_sum
lowerSemicontinuous_tsum 📖mathematicalLowerSemicontinuous
ENNReal
PartialOrder.toPreorder
ENNReal.instPartialOrder
tsum
ENNReal.instAddCommMonoid
ENNReal.instTopologicalSpace
SummationFilter.unconditional
lowerSemicontinuousAt_tsum
upperSemicontinuousAt_biInf 📖mathematicalUpperSemicontinuousAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
upperSemicontinuousAt_iInf
upperSemicontinuousAt_ciInf 📖mathematicalFilter.Eventually
BddBelow
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.range
nhds
UpperSemicontinuousAt
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
lowerSemicontinuousAt_ciSup
upperSemicontinuousAt_iInf 📖mathematicalUpperSemicontinuousAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuousAt_iSup
upperSemicontinuousAt_iff_limsup_le 📖mathematicalUpperSemicontinuousAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.limsup
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhds
lowerSemicontinuousAt_iff_le_liminf
upperSemicontinuousAt_sum 📖mathematicalUpperSemicontinuousAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Finset.sumupperSemicontinuousWithinAt_sum
upperSemicontinuousOn_biInf 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
upperSemicontinuousOn_iInf
upperSemicontinuousOn_ciInf 📖mathematicalBddBelow
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.range
UpperSemicontinuousOn
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
upperSemicontinuousWithinAt_ciInf
eventually_nhdsWithin_of_forall
upperSemicontinuousOn_iInf 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
upperSemicontinuousWithinAt_iInf
upperSemicontinuousOn_iff_isClosed_hypograph 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
instTopologicalSpaceProd
setOf
Set
Set.instMembership
Preorder.toLE
lowerSemicontinuousOn_iff_isClosed_epigraph
instClosedIciTopologyOrderDual
upperSemicontinuousOn_iff_limsup_le 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.limsup
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhdsWithin
lowerSemicontinuousOn_iff_le_liminf
upperSemicontinuousOn_iff_preimage_Ici 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
Set
Set.instInter
Set.preimage
Set.Ici
lowerSemicontinuousOn_iff_preimage_Iic
upperSemicontinuousOn_iff_preimage_Iio 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsOpen
Set
Set.instInter
Set.preimage
Set.Iio
lowerSemicontinuousOn_iff_preimage_Ioi
upperSemicontinuousOn_sum 📖mathematicalUpperSemicontinuousOn
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Finset.sumupperSemicontinuousWithinAt_sum
upperSemicontinuousWithinAt_biInf 📖mathematicalUpperSemicontinuousWithinAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
upperSemicontinuousWithinAt_iInf
upperSemicontinuousWithinAt_ciInf 📖mathematicalFilter.Eventually
BddBelow
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.range
nhdsWithin
UpperSemicontinuousWithinAt
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
lowerSemicontinuousWithinAt_ciSup
upperSemicontinuousWithinAt_iInf 📖mathematicalUpperSemicontinuousWithinAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
lowerSemicontinuousWithinAt_iSup
upperSemicontinuousWithinAt_iff_limsup_le 📖mathematicalUpperSemicontinuousWithinAt
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.limsup
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhdsWithin
lowerSemicontinuousWithinAt_iff_le_liminf
upperSemicontinuousWithinAt_sum 📖mathematicalUpperSemicontinuousWithinAt
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Finset.sumlowerSemicontinuousWithinAt_sum
OrderDual.isOrderedAddMonoid
instOrderTopologyOrderDual
instContinuousAddOrderDual
upperSemicontinuous_biInf 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
upperSemicontinuous_iInf
upperSemicontinuous_ciInf 📖mathematicalBddBelow
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.range
UpperSemicontinuous
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
upperSemicontinuousAt_ciInf
Filter.Eventually.of_forall
upperSemicontinuous_iInf 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
upperSemicontinuousAt_iInf
upperSemicontinuous_iff_IsClosed_hypograph 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
instTopologicalSpaceProd
setOf
Preorder.toLE
lowerSemicontinuous_iff_isClosed_epigraph
instClosedIciTopologyOrderDual
upperSemicontinuous_iff_isClosed_preimage 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
IsClosed
Set.preimage
Set.Ici
upperSemicontinuous_iff_isOpen_preimage
Set.compl_Ici
upperSemicontinuous_iff_isOpen_preimage 📖mathematicalUpperSemicontinuous
IsOpen
Set.preimage
Set.Iio
isOpen_iff_mem_nhds
IsOpen.mem_nhds
upperSemicontinuous_iff_limsup_le 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompletelyDistribLattice.toCompleteLattice
CompleteLinearOrder.toCompletelyDistribLattice
Preorder.toLE
Filter.limsup
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
CompleteLinearOrder.toConditionallyCompleteLinearOrderBot
nhds
lowerSemicontinuous_iff_le_liminf
upperSemicontinuous_sum 📖mathematicalUpperSemicontinuous
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Finset.sumupperSemicontinuousAt_sum

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