Documentation Verification Report

Closure

📁 Source: Mathlib/Order/UpperLower/Closure.lean

Statistics

MetricCount
Definitionserase, sdiff, erase, sdiff, giLowerClosureCoe, giUpperClosureCoe, instDecidablePredMemLowerClosure, instDecidablePredMemUpperClosure, lowerClosure, upperClosure
10
TheoremslowerClosure, of_lowerClosure, of_upperClosure, upperClosure, maximal_mem_lowerClosure_iff_mem, minimal_mem_upperClosure_iff_mem, disjoint_upperClosure_left, disjoint_upperClosure_right, lowerClosure, disjoint_lowerClosure_left, disjoint_lowerClosure_right, upperClosure, Iic_sup_erase, coe_erase, coe_sdiff, erase_eq, erase_idem, erase_le, erase_lt, erase_sup_Iic, iSup_Iic, lowerClosure, lowerClosure_sup_sdiff, sdiff_eq_left, sdiff_idem, sdiff_le_left, sdiff_lt_left, sdiff_singleton, sdiff_sup_lowerClosure, upperClosure_inter_lowerClosure, Ici_inf_erase, coe_erase, coe_sdiff, erase_eq, erase_idem, erase_inf_Ici, iInf_Ici, le_erase, le_sdiff_left, lt_erase, lt_sdiff_left, sdiff_eq_left, sdiff_idem, sdiff_inf_upperClosure, sdiff_singleton, upperClosure, upperClosure_inf_sdiff, bddAbove_lowerClosure, bddBelow_upperClosure, coe_lowerClosure, coe_upperClosure, gc_lowerClosure_coe, gc_upperClosure_coe, le_upperClosure, lowerBounds_upperClosure, lowerClosure_empty, lowerClosure_eq, lowerClosure_eq_bot_iff, lowerClosure_eq_top, lowerClosure_eq_top_iff, lowerClosure_iUnion, lowerClosure_image, lowerClosure_le, lowerClosure_min, lowerClosure_mono, lowerClosure_sUnion, lowerClosure_singleton, lowerClosure_union, lowerClosure_univ, mem_lowerClosure, mem_upperClosure, ordConnected_iff_upperClosure_inter_lowerClosure, subset_lowerClosure, subset_upperClosure, upperBounds_lowerClosure, upperClosure_anti, upperClosure_empty, upperClosure_eq, upperClosure_eq_bot, upperClosure_eq_bot_iff, upperClosure_eq_top_iff, upperClosure_iUnion, upperClosure_image, upperClosure_min, upperClosure_sUnion, upperClosure_singleton, upperClosure_union, upperClosure_univ
88
Total98

BddAbove

Theorems

NameKindAssumesProvesValidatesDepends On
lowerClosure 📖mathematicalBddAbove
Preorder.toLE
SetLike.coe
LowerSet
LowerSet.instSetLike
lowerClosure
bddAbove_lowerClosure
of_lowerClosure 📖BddAbove
Preorder.toLE
SetLike.coe
LowerSet
LowerSet.instSetLike
lowerClosure
bddAbove_lowerClosure

BddBelow

Theorems

NameKindAssumesProvesValidatesDepends On
of_upperClosure 📖BddBelow
Preorder.toLE
SetLike.coe
UpperSet
UpperSet.instSetLike
upperClosure
bddBelow_upperClosure
upperClosure 📖mathematicalBddBelow
Preorder.toLE
SetLike.coe
UpperSet
UpperSet.instSetLike
upperClosure
bddBelow_upperClosure

IsAntichain

Theorems

NameKindAssumesProvesValidatesDepends On
maximal_mem_lowerClosure_iff_mem 📖mathematicalIsAntichain
Preorder.toLE
PartialOrder.toPreorder
Maximal
LowerSet
SetLike.instMembership
LowerSet.instSetLike
lowerClosure
Set
Set.instMembership
minimal_mem_upperClosure_iff_mem
to_dual
minimal_mem_upperClosure_iff_mem 📖mathematicalIsAntichain
Preorder.toLE
PartialOrder.toPreorder
Minimal
UpperSet
SetLike.instMembership
UpperSet.instSetLike
upperClosure
Set
Set.instMembership
Minimal.prop
Minimal.eq_of_ge
Eq.le
eq
LE.le.trans

IsLowerSet

Theorems

NameKindAssumesProvesValidatesDepends On
disjoint_upperClosure_left 📖mathematicalIsLowerSet
Preorder.toLE
Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
SetLike.coe
UpperSet
UpperSet.instSetLike
upperClosure
Disjoint.mono_left
subset_upperClosure
disjoint_upperClosure_right 📖mathematicalIsLowerSet
Preorder.toLE
Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
SetLike.coe
UpperSet
UpperSet.instSetLike
upperClosure
disjoint_upperClosure_left
lowerClosure 📖mathematicalIsLowerSet
Preorder.toLE
SetLike.coe
LowerSet
LowerSet.instSetLike
lowerClosure
HasSubset.Subset.antisymm
Set.instAntisymmSubset
lowerClosure_min
Set.Subset.rfl
subset_lowerClosure

IsUpperSet

Theorems

NameKindAssumesProvesValidatesDepends On
disjoint_lowerClosure_left 📖mathematicalIsUpperSet
Preorder.toLE
Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
SetLike.coe
LowerSet
LowerSet.instSetLike
lowerClosure
IsLowerSet.disjoint_upperClosure_left
toDual
disjoint_lowerClosure_right 📖mathematicalIsUpperSet
Preorder.toLE
Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
SetLike.coe
LowerSet
LowerSet.instSetLike
lowerClosure
IsLowerSet.disjoint_upperClosure_right
toDual
upperClosure 📖mathematicalIsUpperSet
Preorder.toLE
SetLike.coe
UpperSet
UpperSet.instSetLike
upperClosure
HasSubset.Subset.antisymm
Set.instAntisymmSubset
upperClosure_min
Set.Subset.rfl
subset_upperClosure

LowerSet

Definitions

NameCategoryTheorems
erase 📖CompOp
8 mathmath: erase_le, Iic_sup_erase, erase_sup_Iic, coe_erase, erase_lt, sdiff_singleton, erase_idem, erase_eq
sdiff 📖CompOp
8 mathmath: sdiff_le_left, sdiff_idem, coe_sdiff, sdiff_eq_left, sdiff_lt_left, lowerClosure_sup_sdiff, sdiff_singleton, sdiff_sup_lowerClosure

Theorems

NameKindAssumesProvesValidatesDepends On
Iic_sup_erase 📖mathematicalLowerSet
Preorder.toLE
SetLike.instMembership
instSetLike
instMax
Iic
erase
sup_comm
erase_sup_Iic
coe_erase 📖mathematicalSetLike.coe
LowerSet
Preorder.toLE
instSetLike
erase
Set
Set.instSDiff
UpperSet
UpperSet.instSetLike
UpperSet.Ici
coe_sdiff 📖mathematicalSetLike.coe
LowerSet
Preorder.toLE
instSetLike
sdiff
Set
Set.instSDiff
UpperSet
UpperSet.instSetLike
upperClosure
erase_eq 📖mathematicalerase
LowerSet
Preorder.toLE
SetLike.instMembership
instSetLike
sdiff_singleton
erase_idem 📖mathematicaleraseSetLike.coe_injective
sdiff_idem
erase_le 📖mathematicalLowerSet
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
erase
Set.diff_subset
erase_lt 📖mathematicalLowerSet
Preorder.toLE
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
erase
SetLike.instMembership
instSetLike
LE.le.lt_iff_ne
erase_le
Iff.not_left
erase_eq
erase_sup_Iic 📖mathematicalLowerSet
Preorder.toLE
SetLike.instMembership
instSetLike
instMax
erase
Iic
lowerClosure_singleton
sdiff_singleton
sdiff_sup_lowerClosure
iSup_Iic 📖mathematicaliSup
LowerSet
Preorder.toLE
instSupSet
Set
Set.instMembership
Iic
lowerClosure
ext
Set.ext
coe_iSup
Set.iUnion_congr_Prop
lowerClosure 📖mathematicallowerClosure
SetLike.coe
LowerSet
Preorder.toLE
instSetLike
SetLike.coe_injective
IsLowerSet.lowerClosure
lower'
lowerClosure_sup_sdiff 📖mathematicalSet
Set.instHasSubset
SetLike.coe
LowerSet
Preorder.toLE
instSetLike
Set.instMembership
instMax
lowerClosure
sdiff
sup_comm
sdiff_sup_lowerClosure
sdiff_eq_left 📖mathematicalsdiff
Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
SetLike.coe
LowerSet
Preorder.toLE
instSetLike
sdiff_idem 📖mathematicalsdiffSetLike.coe_injective
sdiff_idem
sdiff_le_left 📖mathematicalLowerSet
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
sdiff
Set.diff_subset
sdiff_lt_left 📖mathematicalLowerSet
Preorder.toLE
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
sdiff
Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
SetLike.coe
instSetLike
LE.le.lt_iff_ne
sdiff_le_left
Iff.not
sdiff_eq_left
sdiff_singleton 📖mathematicalsdiff
Set
Set.instSingletonSet
erase
upperClosure_singleton
sdiff_sup_lowerClosure 📖mathematicalSet
Set.instHasSubset
SetLike.coe
LowerSet
Preorder.toLE
instSetLike
Set.instMembership
instMax
sdiff
lowerClosure
le_antisymm
sup_le
sdiff_le_left
lowerClosure_le
em
Set.subset_union_right
subset_lowerClosure
Set.subset_union_left

Set.OrdConnected

Theorems

NameKindAssumesProvesValidatesDepends On
upperClosure_inter_lowerClosure 📖mathematicalSet
Set.instInter
SetLike.coe
UpperSet
Preorder.toLE
UpperSet.instSetLike
upperClosure
LowerSet
LowerSet.instSetLike
lowerClosure
HasSubset.Subset.antisymm'
Set.instAntisymmSubset
Set.subset_inter
subset_upperClosure
subset_lowerClosure
out

UpperSet

Definitions

NameCategoryTheorems
erase 📖CompOp
8 mathmath: erase_eq, Ici_inf_erase, erase_idem, erase_inf_Ici, le_erase, lt_erase, sdiff_singleton, coe_erase
sdiff 📖CompOp
8 mathmath: sdiff_idem, sdiff_eq_left, lt_sdiff_left, upperClosure_inf_sdiff, sdiff_inf_upperClosure, le_sdiff_left, coe_sdiff, sdiff_singleton

Theorems

NameKindAssumesProvesValidatesDepends On
Ici_inf_erase 📖mathematicalUpperSet
Preorder.toLE
SetLike.instMembership
instSetLike
instMin
Ici
erase
inf_comm
erase_inf_Ici
coe_erase 📖mathematicalSetLike.coe
UpperSet
Preorder.toLE
instSetLike
erase
Set
Set.instSDiff
LowerSet
LowerSet.instSetLike
LowerSet.Iic
coe_sdiff 📖mathematicalSetLike.coe
UpperSet
Preorder.toLE
instSetLike
sdiff
Set
Set.instSDiff
LowerSet
LowerSet.instSetLike
lowerClosure
erase_eq 📖mathematicalerase
UpperSet
Preorder.toLE
SetLike.instMembership
instSetLike
sdiff_singleton
erase_idem 📖mathematicaleraseSetLike.coe_injective
sdiff_idem
erase_inf_Ici 📖mathematicalUpperSet
Preorder.toLE
SetLike.instMembership
instSetLike
instMin
erase
Ici
upperClosure_singleton
sdiff_singleton
sdiff_inf_upperClosure
iInf_Ici 📖mathematicaliInf
UpperSet
Preorder.toLE
instInfSet
Set
Set.instMembership
Ici
upperClosure
ext
Set.ext
coe_iInf
Set.iUnion_congr_Prop
le_erase 📖mathematicalUpperSet
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
erase
Set.diff_subset
le_sdiff_left 📖mathematicalUpperSet
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
sdiff
Set.diff_subset
lt_erase 📖mathematicalUpperSet
Preorder.toLE
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
erase
SetLike.instMembership
instSetLike
LE.le.lt_iff_ne'
le_erase
Iff.not_left
erase_eq
lt_sdiff_left 📖mathematicalUpperSet
Preorder.toLE
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
sdiff
Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
SetLike.coe
instSetLike
LE.le.lt_iff_ne'
le_sdiff_left
Iff.not
sdiff_eq_left
sdiff_eq_left 📖mathematicalsdiff
Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
SetLike.coe
UpperSet
Preorder.toLE
instSetLike
sdiff_idem 📖mathematicalsdiffSetLike.coe_injective
sdiff_idem
sdiff_inf_upperClosure 📖mathematicalSet
Set.instHasSubset
SetLike.coe
UpperSet
Preorder.toLE
instSetLike
Set.instMembership
instMin
sdiff
upperClosure
ge_antisymm
le_inf
le_sdiff_left
le_upperClosure
em
Set.subset_union_right
subset_upperClosure
Set.subset_union_left
sdiff_singleton 📖mathematicalsdiff
Set
Set.instSingletonSet
erase
lowerClosure_singleton
upperClosure 📖mathematicalupperClosure
SetLike.coe
UpperSet
Preorder.toLE
instSetLike
SetLike.coe_injective
IsUpperSet.upperClosure
upper'
upperClosure_inf_sdiff 📖mathematicalSet
Set.instHasSubset
SetLike.coe
UpperSet
Preorder.toLE
instSetLike
Set.instMembership
instMin
upperClosure
sdiff
inf_comm
sdiff_inf_upperClosure

(root)

Definitions

NameCategoryTheorems
giLowerClosureCoe 📖CompOp
giUpperClosureCoe 📖CompOp
instDecidablePredMemLowerClosure 📖CompOp
instDecidablePredMemUpperClosure 📖CompOp
lowerClosure 📖CompOp
61 mathmath: LowerSet.iSup_Iic, lowerClosure_union, IsOpen.lowerClosure, lowerClosure_infs, lowerClosure_eq_top_iff, subset_lowerClosure, Topology.IsLowerSet.nhdsKer_eq_lowerClosure, lowerClosure_image, lowerClosure_singleton, lowerClosure_one, IsClosed.lowerClosure_pi, IsClopen.lowerClosure_pi, lowerClosure_eq_bot_iff, Topology.IsScott.lowerClosure_subset_closure, lowerClosure_univ, lowerClosure_interior_subset', Topology.IsUpper.isClosed_lowerClosure, lowerClosure_prod, lowerClosure_empty, mem_lowerClosure, LowerSet.lowerClosure_sup_sdiff, lowerClosure_eq_Iic_csSup, lowerClosure_interior_subset, Topology.IsUpperSet.closure_eq_lowerClosure, lowerClosure_zero, lowerClosure_add, coe_lowerClosure, lowerClosure_sUnion, IsAntichain.maximal_mem_lowerClosure_iff_mem, Matroid.setOf_indep_eq, IsLowerSet.lowerClosure, add_lowerClosure, ordConnected_iff_upperClosure_inter_lowerClosure, upperBounds_lowerClosure, lowerClosure_iUnion, lowerClosure_le, lowerClosure_vadd, BddAbove.lowerClosure, IsClosed.lowerClosure, Set.OrdConnected.upperClosure_inter_lowerClosure, IsUpperSet.disjoint_lowerClosure_left, Set.Finite.lowerClosure, lowerClosure_min, lowerClosure_mul, Finset.Colex.le_iff_sdiff_subset_lowerClosure, lowerClosure_smul, HasUpperLowerClosure.isOpen_lowerClosure, lowerClosure_eq_top, lowerClosure_mul_distrib, mul_lowerClosure, Topology.IsLowerSet.nhdsSet_eq_principal_lowerClosure, UpperSet.coe_sdiff, lowerClosure_add_distrib, gc_lowerClosure_coe, LowerSet.lowerClosure, IsUpperSet.disjoint_lowerClosure_right, lowerClosure_eq, closure_lowerClosure_comm_pi, lowerClosure_mono, LowerSet.sdiff_sup_lowerClosure, bddAbove_lowerClosure
upperClosure 📖CompOp
60 mathmath: upperClosure_zero, upperClosure_one, upperClosure_eq, BddBelow.upperClosure, bddBelow_upperClosure, Topology.IsUpperSet.nhdsKer_eq_upperClosure, upperClosure_image, IsClosed.upperClosure, IsUpperSet.upperClosure, LowerSet.coe_sdiff, IsClopen.upperClosure_pi, upperClosure_sups, Topology.IsLowerSet.closure_eq_upperClosure, subset_upperClosure, add_upperClosure, upperClosure_interior_subset', upperClosure_singleton, upperClosure_interior_subset, Topology.IsLower.isClosed_upperClosure, PrimitiveSpectrum.hull_finsetInf, upperClosure_mul, closure_upperClosure_comm_pi, lowerBounds_upperClosure, upperClosure_min, upperClosure_eq_top_iff, IsAntichain.minimal_mem_upperClosure_iff_mem, UpperSet.upperClosure, IsOpen.upperClosure, Topology.IsUpperSet.nhdsSet_eq_principal_upperClosure, PrimitiveSpectrum.preimage_upperClosure_compl_finset, upperClosure_empty, gc_upperClosure_coe, upperClosure_prod, ordConnected_iff_upperClosure_inter_lowerClosure, upperClosure_univ, upperClosure_eq_bot_iff, IsLowerSet.disjoint_upperClosure_left, upperClosure_add_distrib, IsLowerSet.disjoint_upperClosure_right, HasUpperLowerClosure.isOpen_upperClosure, UpperSet.upperClosure_inf_sdiff, upperClosure_sUnion, coe_upperClosure, upperClosure_add, UpperSet.sdiff_inf_upperClosure, Set.OrdConnected.upperClosure_inter_lowerClosure, upperClosure_vadd, le_upperClosure, upperClosure_mul_distrib, upperClosure_iUnion, mul_upperClosure, Set.Finite.upperClosure, IsClosed.upperClosure_pi, upperClosure_union, upperClosure_smul, mem_upperClosure, upperClosure_eq_Ici_csInf, upperClosure_eq_bot, UpperSet.iInf_Ici, upperClosure_anti

Theorems

NameKindAssumesProvesValidatesDepends On
bddAbove_lowerClosure 📖mathematicalBddAbove
Preorder.toLE
SetLike.coe
LowerSet
LowerSet.instSetLike
lowerClosure
upperBounds_lowerClosure
bddBelow_upperClosure 📖mathematicalBddBelow
Preorder.toLE
SetLike.coe
UpperSet
UpperSet.instSetLike
upperClosure
lowerBounds_upperClosure
coe_lowerClosure 📖mathematicalSetLike.coe
LowerSet
Preorder.toLE
LowerSet.instSetLike
lowerClosure
Set.iUnion
Set
Set.instMembership
Set.Iic
Set.ext
coe_upperClosure 📖mathematicalSetLike.coe
UpperSet
Preorder.toLE
UpperSet.instSetLike
upperClosure
Set.iUnion
Set
Set.instMembership
Set.Ici
Set.ext
gc_lowerClosure_coe 📖mathematicalGaloisConnection
Set
LowerSet
Preorder.toLE
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
LowerSet.instPartialOrder
lowerClosure
SetLike.coe
LowerSet.instSetLike
lowerClosure_le
gc_upperClosure_coe 📖mathematicalGaloisConnection
Set
OrderDual
UpperSet
Preorder.toLE
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
OrderDual.instPreorder
UpperSet.instPartialOrder
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
upperClosure
SetLike.coe
UpperSet.instSetLike
OrderDual.ofDual
le_upperClosure
le_upperClosure 📖mathematicalUpperSet
Preorder.toLE
PartialOrder.toPreorder
UpperSet.instPartialOrder
upperClosure
Set
Set.instHasSubset
SetLike.coe
UpperSet.instSetLike
HasSubset.Subset.trans
Set.instIsTransSubset
subset_upperClosure
UpperSet.coe_subset_coe
upperClosure_min
UpperSet.upper
lowerBounds_upperClosure 📖mathematicallowerBounds
Preorder.toLE
SetLike.coe
UpperSet
UpperSet.instSetLike
upperClosure
HasSubset.Subset.antisymm
Set.instAntisymmSubset
lowerBounds_mono_set
subset_upperClosure
LE.le.trans
lowerClosure_empty 📖mathematicallowerClosure
Set
Set.instEmptyCollection
Bot.bot
LowerSet
Preorder.toLE
LowerSet.instBot
GaloisConnection.l_bot
gc_lowerClosure_coe
lowerClosure_eq 📖mathematicalSetLike.coe
LowerSet
Preorder.toLE
LowerSet.instSetLike
lowerClosure
IsLowerSet
upperClosure_eq
lowerClosure_eq_bot_iff 📖mathematicallowerClosure
Bot.bot
LowerSet
Preorder.toLE
LowerSet.instBot
Set
Set.instEmptyCollection
GaloisConnection.l_eq_bot
gc_lowerClosure_coe
Set.subset_empty_iff
lowerClosure_eq_top 📖mathematicalBddAbove
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
lowerClosure
Top.top
LowerSet
LowerSet.instTop
SetLike.coe_injective
upperClosure_eq_bot
lowerClosure_eq_top_iff 📖mathematicallowerClosure
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Top.top
LowerSet
Preorder.toLE
LowerSet.instTop
BddAbove
instNoTopOrderOfNoMaxOrder
bddAbove_lowerClosure
lowerClosure_eq_top
lowerClosure_iUnion 📖mathematicallowerClosure
Set.iUnion
iSup
LowerSet
Preorder.toLE
LowerSet.instSupSet
GaloisConnection.l_iSup
gc_lowerClosure_coe
lowerClosure_image 📖mathematicallowerClosure
Set.image
DFunLike.coe
OrderIso
Preorder.toLE
instFunLikeOrderIso
LowerSet
PartialOrder.toPreorder
LowerSet.instPartialOrder
LowerSet.map
OrderIso.symm_symm
LowerSet.symm_map
LowerSet.ext
Set.ext
OrderIso.symm_apply_le
Set.image_congr
lowerClosure_le 📖mathematicalLowerSet
Preorder.toLE
PartialOrder.toPreorder
LowerSet.instPartialOrder
lowerClosure
Set
Set.instHasSubset
SetLike.coe
LowerSet.instSetLike
HasSubset.Subset.trans
Set.instIsTransSubset
subset_lowerClosure
LowerSet.coe_subset_coe
lowerClosure_min
LowerSet.lower
lowerClosure_min 📖mathematicalSet
Set.instHasSubset
IsLowerSet
Preorder.toLE
SetLike.coe
LowerSet
LowerSet.instSetLike
lowerClosure
lowerClosure_mono 📖mathematicalMonotone
Set
LowerSet
Preorder.toLE
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
LowerSet.instPartialOrder
lowerClosure
GaloisConnection.monotone_l
gc_lowerClosure_coe
lowerClosure_sUnion 📖mathematicallowerClosure
Set.sUnion
iSup
LowerSet
Preorder.toLE
Set
LowerSet.instSupSet
Set.instMembership
Set.sUnion_eq_biUnion
lowerClosure_iUnion
lowerClosure_singleton 📖mathematicallowerClosure
Set
Set.instSingletonSet
LowerSet.Iic
LowerSet.ext
Set.ext
lowerClosure_union 📖mathematicallowerClosure
Set
Set.instUnion
LowerSet
Preorder.toLE
LowerSet.instMax
GaloisConnection.l_sup
gc_lowerClosure_coe
lowerClosure_univ 📖mathematicallowerClosure
Set.univ
Top.top
LowerSet
Preorder.toLE
LowerSet.instTop
top_unique
subset_lowerClosure
mem_lowerClosure 📖mathematicalLowerSet
Preorder.toLE
SetLike.instMembership
LowerSet.instSetLike
lowerClosure
Set
Set.instMembership
mem_upperClosure 📖mathematicalUpperSet
Preorder.toLE
SetLike.instMembership
UpperSet.instSetLike
upperClosure
Set
Set.instMembership
ordConnected_iff_upperClosure_inter_lowerClosure 📖mathematicalSet.OrdConnected
Set
Set.instInter
SetLike.coe
UpperSet
Preorder.toLE
UpperSet.instSetLike
upperClosure
LowerSet
LowerSet.instSetLike
lowerClosure
Set.OrdConnected.upperClosure_inter_lowerClosure
Set.OrdConnected.inter
IsUpperSet.ordConnected
UpperSet.upper
IsLowerSet.ordConnected
LowerSet.lower
subset_lowerClosure 📖mathematicalSet
Set.instHasSubset
SetLike.coe
LowerSet
Preorder.toLE
LowerSet.instSetLike
lowerClosure
le_rfl
subset_upperClosure 📖mathematicalSet
Set.instHasSubset
SetLike.coe
UpperSet
Preorder.toLE
UpperSet.instSetLike
upperClosure
le_rfl
upperBounds_lowerClosure 📖mathematicalupperBounds
Preorder.toLE
SetLike.coe
LowerSet
LowerSet.instSetLike
lowerClosure
HasSubset.Subset.antisymm
Set.instAntisymmSubset
upperBounds_mono_set
subset_lowerClosure
LE.le.trans
upperClosure_anti 📖mathematicalAntitone
Set
UpperSet
Preorder.toLE
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
UpperSet.instPartialOrder
upperClosure
GaloisConnection.monotone_l
gc_upperClosure_coe
upperClosure_empty 📖mathematicalupperClosure
Set
Set.instEmptyCollection
Top.top
UpperSet
Preorder.toLE
UpperSet.instTop
GaloisConnection.l_bot
gc_upperClosure_coe
upperClosure_eq 📖mathematicalSetLike.coe
UpperSet
Preorder.toLE
UpperSet.instSetLike
upperClosure
IsUpperSet
UpperSet.upper
IsUpperSet.upperClosure
upperClosure_eq_bot 📖mathematicalBddBelow
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
upperClosure
Bot.bot
UpperSet
UpperSet.instBot
le_bot_iff
not_bddBelow_iff
le_of_lt
upperClosure_eq_bot_iff 📖mathematicalupperClosure
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Bot.bot
UpperSet
Preorder.toLE
UpperSet.instBot
BddBelow
instNoBotOrderOfNoMinOrder
bddBelow_upperClosure
upperClosure_eq_bot
upperClosure_eq_top_iff 📖mathematicalupperClosure
Top.top
UpperSet
Preorder.toLE
UpperSet.instTop
Set
Set.instEmptyCollection
GaloisConnection.l_eq_bot
gc_upperClosure_coe
Set.subset_empty_iff
upperClosure_iUnion 📖mathematicalupperClosure
Set.iUnion
iInf
UpperSet
Preorder.toLE
UpperSet.instInfSet
GaloisConnection.l_iSup
gc_upperClosure_coe
upperClosure_image 📖mathematicalupperClosure
Set.image
DFunLike.coe
OrderIso
Preorder.toLE
instFunLikeOrderIso
UpperSet
PartialOrder.toPreorder
UpperSet.instPartialOrder
UpperSet.map
OrderIso.symm_symm
UpperSet.symm_map
UpperSet.ext
Set.ext
OrderIso.le_symm_apply
Set.image_congr
upperClosure_min 📖mathematicalSet
Set.instHasSubset
IsUpperSet
Preorder.toLE
SetLike.coe
UpperSet
UpperSet.instSetLike
upperClosure
upperClosure_sUnion 📖mathematicalupperClosure
Set.sUnion
iInf
UpperSet
Preorder.toLE
Set
UpperSet.instInfSet
Set.instMembership
Set.sUnion_eq_biUnion
upperClosure_iUnion
upperClosure_singleton 📖mathematicalupperClosure
Set
Set.instSingletonSet
UpperSet.Ici
UpperSet.ext
Set.ext
upperClosure_union 📖mathematicalupperClosure
Set
Set.instUnion
UpperSet
Preorder.toLE
UpperSet.instMin
GaloisConnection.l_sup
gc_upperClosure_coe
upperClosure_univ 📖mathematicalupperClosure
Set.univ
Bot.bot
UpperSet
Preorder.toLE
UpperSet.instBot
bot_unique
subset_upperClosure

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