Documentation Verification Report

SupClosed

📁 Source: Mathlib/Order/SupClosed.lean

Statistics

MetricCount
DefinitionsInfClosed, IsSublattice, toCompleteSemilatticeInf, toCompleteSemilatticeSup, SupClosed, infClosure, latticeClosure, supClosure
8
TheoremsbiInf_mem, biInf_mem_of_nonempty, codirectedOn, dual, finsetInf'_mem, finsetInf_mem, iInf_mem, iInf_mem_of_nonempty, image, infClosure_eq, insert_lowerBounds, insert_upperBounds, inter, preimage, prod, sInf_mem, sInf_mem_of_nonempty, supClosure, infClosed, dual, image, infClosed, inter, latticeClosure_eq, of_dual, preimage, prod, supClosed, IsSublattice_range, supClosed, infClosed, isSublattice, supClosed, infClosure, latticeClosure, supClosure, biSup_mem, biSup_mem_of_nonempty, directedOn, dual, finsetSup'_mem, finsetSup_mem, iSup_mem, iSup_mem_of_nonempty, image, infClosure, insert_lowerBounds, insert_upperBounds, inter, preimage, prod, sSup_mem, sSup_mem_of_nonempty, supClosure_eq, compl_image_latticeClosure, compl_image_latticeClosure_eq_of_compl_image_eq_self, finsetInf'_mem_infClosure, finsetSup'_mem_supClosure, image_latticeClosure, image_latticeClosure', infClosed_empty, infClosed_iInter, infClosed_infClosure, infClosed_pi, infClosed_preimage_ofDual, infClosed_preimage_toDual, infClosed_range, infClosed_sInter, infClosed_singleton, infClosed_univ, infClosure_empty, infClosure_eq_self, infClosure_idem, infClosure_isClosed, infClosure_min, infClosure_mono, infClosure_prod, infClosure_singleton, infClosure_supClosure, infClosure_univ, inf_mem_infClosure, isGLB_infClosure, isLUB_supClosure, isSublattice_empty, isSublattice_iInter, isSublattice_latticeClosure, isSublattice_pi, isSublattice_preimage_ofDual, isSublattice_preimage_toDual, isSublattice_sInter, isSublattice_singleton, isSublattice_univ, latticeClosure_empty, latticeClosure_eq_self, latticeClosure_idem, latticeClosure_isClosed, latticeClosure_min, latticeClosure_mono, latticeClosure_prod, latticeClosure_singleton, latticeClosure_sup_inf_induction, latticeClosure_univ, lowerBounds_infClosure, ofDual_preimage_latticeClosure, subset_infClosure, subset_latticeClosure, subset_supClosure, supClosed_empty, supClosed_iInter, supClosed_pi, supClosed_preimage_ofDual, supClosed_preimage_toDual, supClosed_range, supClosed_sInter, supClosed_singleton, supClosed_supClosure, supClosed_univ, supClosure_empty, supClosure_eq_self, supClosure_idem, supClosure_infClosure, supClosure_isClosed, supClosure_min, supClosure_mono, supClosure_prod, supClosure_singleton, supClosure_univ, sup_mem_supClosure, upperBounds_supClosure
129
Total137

InfClosed

Theorems

NameKindAssumesProvesValidatesDepends On
biInf_mem 📖mathematicalInfClosed
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
Set
Set.instMembership
Top.top
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
SupClosed.biSup_mem
dual
biInf_mem_of_nonempty 📖mathematicalInfClosed
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
Set.Nonempty
Set
Set.instMembership
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
SupClosed.biSup_mem_of_nonempty
dual
codirectedOn 📖mathematicalInfClosedDirectedOn
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
inf_le_left
inf_le_right
dual 📖mathematicalInfClosed
Lattice.toSemilatticeInf
SupClosed
OrderDual
OrderDual.instSemilatticeSup
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
supClosed_preimage_ofDual
finsetInf'_mem 📖mathematicalInfClosed
Finset.Nonempty
Set
Set.instMembership
Finset.inf'Finset.inf'_induction
finsetInf_mem 📖mathematicalInfClosed
Finset.Nonempty
Set
Set.instMembership
Finset.inffinsetInf'_mem
Finset.inf'_eq_inf
iInf_mem 📖mathematicalInfClosed
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
Set
Set.instMembership
Top.top
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
SupClosed.iSup_mem
dual
iInf_mem_of_nonempty 📖mathematicalInfClosed
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
Set
Set.instMembership
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
SupClosed.iSup_mem_of_nonempty
dual
image 📖mathematicalInfClosedSet.image
DFunLike.coe
InfHomClass.map_inf
Set.mem_image_of_mem
infClosure_eq 📖mathematicalInfClosedDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
infClosure_eq_self
insert_lowerBounds 📖mathematicalInfClosed
Set
Set.instMembership
lowerBounds
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Set.instInserteq_1
inf_of_le_left
insert_upperBounds 📖mathematicalInfClosed
Set
Set.instMembership
upperBounds
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Set.instInserteq_1
inf_of_le_left
inf_of_le_right
inter 📖mathematicalInfClosedSet
Set.instInter
preimage 📖mathematicalInfClosedSet.preimage
DFunLike.coe
InfHomClass.map_inf
prod 📖mathematicalInfClosedProd.instSemilatticeInf
SProd.sprod
Set
Set.instSProd
sInf_mem 📖mathematicalInfClosed
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
Set
Set.instMembership
Top.top
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
Set.instHasSubset
InfSet.sInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
SupClosed.sSup_mem
dual
sInf_mem_of_nonempty 📖mathematicalInfClosed
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
Set.Nonempty
Set
Set.instHasSubset
Set.instMembership
InfSet.sInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
SupClosed.sSup_mem_of_nonempty
dual
supClosure 📖mathematicalInfClosed
Lattice.toSemilatticeInf
DistribLattice.toLattice
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
Lattice.toSemilatticeSup
Finset.Nonempty.product
Finset.sup'_inf_sup'
finsetSup'_mem_supClosure
Finset.mem_product

IsLowerSet

Theorems

NameKindAssumesProvesValidatesDepends On
infClosed 📖mathematicalIsLowerSet
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
InfClosedinf_le_right

IsSublattice

Theorems

NameKindAssumesProvesValidatesDepends On
dual 📖mathematicalIsSublatticeOrderDual
OrderDual.instLattice
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
isSublattice_preimage_ofDual
image 📖mathematicalIsSublatticeSet.image
DFunLike.coe
SupClosed.image
LatticeHomClass.toSupHomClass
supClosed
InfClosed.image
LatticeHomClass.toInfHomClass
infClosed
infClosed 📖mathematicalIsSublatticeInfClosed
Lattice.toSemilatticeInf
inter 📖mathematicalIsSublatticeSet
Set.instInter
SupClosed.inter
supClosed
InfClosed.inter
infClosed
latticeClosure_eq 📖mathematicalIsSublatticeDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
latticeClosure_eq_self
of_dual 📖mathematicalIsSublattice
OrderDual
OrderDual.instLattice
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
isSublattice_preimage_toDual
preimage 📖mathematicalIsSublatticeSet.preimage
DFunLike.coe
SupClosed.preimage
LatticeHomClass.toSupHomClass
supClosed
InfClosed.preimage
LatticeHomClass.toInfHomClass
infClosed
prod 📖mathematicalIsSublatticeProd.instLattice
SProd.sprod
Set
Set.instSProd
SupClosed.prod
supClosed
InfClosed.prod
infClosed
supClosed 📖mathematicalIsSublatticeSupClosed
Lattice.toSemilatticeSup

IsUpperSet

Theorems

NameKindAssumesProvesValidatesDepends On
supClosed 📖mathematicalIsUpperSet
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
SupClosedle_sup_right

LinearOrder

Theorems

NameKindAssumesProvesValidatesDepends On
infClosed 📖mathematicalInfClosed
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
le_total
inf_of_le_left
inf_of_le_right
isSublattice 📖mathematicalIsSublattice
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
supClosed
infClosed
supClosed 📖mathematicalSupClosed
Lattice.toSemilatticeSup
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
le_total
sup_of_le_right
sup_of_le_left

SemilatticeInf

Definitions

NameCategoryTheorems
toCompleteSemilatticeInf 📖CompOp

SemilatticeSup

Definitions

NameCategoryTheorems
toCompleteSemilatticeSup 📖CompOp

Set.Finite

Theorems

NameKindAssumesProvesValidatesDepends On
infClosure 📖mathematicalSet.Finite
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
CanLift.prf
Set.instCanLiftFinsetCoeFinite
subset
Finset.mem_filter
Finset.finite_toSet
Finset.inf'_congr
Finset.coe_image
latticeClosure 📖mathematicalSet.Finite
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
DistribLattice.toLattice
supClosure_infClosure
supClosure
infClosure
supClosure 📖mathematicalSet.Finite
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
CanLift.prf
Set.instCanLiftFinsetCoeFinite
subset
Finset.mem_filter
Finset.finite_toSet
Finset.sup'_congr
Finset.coe_image

SupClosed

Theorems

NameKindAssumesProvesValidatesDepends On
biSup_mem 📖mathematicalSupClosed
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
Set
Set.instMembership
Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
CompleteLattice.toLattice
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
sSup_image
sSup_mem
Set.Finite.image
biSup_mem_of_nonempty 📖mathematicalSupClosed
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
Set.Nonempty
Set
Set.instMembership
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
sSup_image
sSup_mem_of_nonempty
Set.Finite.image
directedOn 📖mathematicalSupClosedDirectedOn
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
le_sup_left
le_sup_right
dual 📖mathematicalSupClosed
Lattice.toSemilatticeSup
InfClosed
OrderDual
OrderDual.instSemilatticeInf
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
infClosed_preimage_ofDual
finsetSup'_mem 📖mathematicalSupClosed
Finset.Nonempty
Set
Set.instMembership
Finset.sup'Finset.sup'_induction
finsetSup_mem 📖mathematicalSupClosed
Finset.Nonempty
Set
Set.instMembership
Finset.supfinsetSup'_mem
Finset.sup'_eq_sup
iSup_mem 📖mathematicalSupClosed
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
Set
Set.instMembership
Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
CompleteLattice.toLattice
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
isEmpty_or_nonempty
iSup_of_empty
iSup_mem_of_nonempty
iSup_mem_of_nonempty 📖mathematicalSupClosed
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
Set
Set.instMembership
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
nonempty_fintype
instFinitePLift
iSup_plift_down
Finset.univ_nonempty
PLift.instNonempty_mathlib
Finset.sup'_univ_eq_ciSup
finsetSup'_mem
image 📖mathematicalSupClosedSet.image
DFunLike.coe
SupHomClass.map_sup
Set.mem_image_of_mem
infClosure 📖mathematicalSupClosed
Lattice.toSemilatticeSup
DistribLattice.toLattice
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
Lattice.toSemilatticeInf
Finset.Nonempty.product
Finset.inf'_sup_inf'
finsetInf'_mem_infClosure
Finset.mem_product
insert_lowerBounds 📖mathematicalSupClosed
Set
Set.instMembership
lowerBounds
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Set.instInserteq_1
sup_of_le_left
sup_of_le_right
insert_upperBounds 📖mathematicalSupClosed
Set
Set.instMembership
upperBounds
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Set.instInserteq_1
sup_of_le_left
inter 📖mathematicalSupClosedSet
Set.instInter
preimage 📖mathematicalSupClosedSet.preimage
DFunLike.coe
SupHomClass.map_sup
prod 📖mathematicalSupClosedProd.instSemilatticeSup
SProd.sprod
Set
Set.instSProd
sSup_mem 📖mathematicalSupClosed
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
Set
Set.instMembership
Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
CompleteLattice.toLattice
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
Set.instHasSubset
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
Set.Finite.to_subtype
sSup_eq_iSup'
iSup_mem
sSup_mem_of_nonempty 📖mathematicalSupClosed
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
Set.Nonempty
Set
Set.instHasSubset
Set.instMembership
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
Set.Finite.to_subtype
Set.Nonempty.to_subtype
sSup_eq_iSup'
iSup_mem_of_nonempty
supClosure_eq 📖mathematicalSupClosedDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
supClosure_eq_self

(root)

Definitions

NameCategoryTheorems
InfClosed 📖MathDef
23 mathmath: supClosed_preimage_ofDual, Sublattice.infClosed', infClosed_univ, infClosed_preimage_toDual, Set.infs_self_subset, infClosed_infClosure, SupClosed.dual, Sublattice.infClosed, infClosure_eq_self, IsLowerSet.infClosed, Finset.infs_self, BooleanSubalgebra.infClosed, LinearOrder.infClosed, Set.infs_self, infClosed_preimage_ofDual, infClosed_singleton, infClosed_range, infClosed_empty, supClosed_preimage_toDual, IsSublattice.infClosed, Sublocale.infClosed, infClosure_isClosed, Finset.infs_self_subset
IsSublattice 📖CompData
11 mathmath: isSublattice_preimage_ofDual, Sublattice.isSublattice, isSublattice_latticeClosure, isSublattice_univ, LinearOrder.isSublattice, latticeClosure_eq_self, isSublattice_singleton, isSublattice_empty, IsSublattice_range, isSublattice_preimage_toDual, latticeClosure_isClosed
SupClosed 📖MathDef
22 mathmath: supClosed_preimage_ofDual, supClosure_eq_self, supClosed_empty, infClosed_preimage_toDual, Sublattice.supClosed, IsUpperSet.supClosed, supClosure_isClosed, Finset.sups_subset_self, infClosed_preimage_ofDual, supClosed_singleton, IsSublattice.supClosed, supClosed_range, supClosed_preimage_toDual, Set.sups_eq_self, Sublattice.supClosed', LinearOrder.supClosed, Finset.sups_eq_self, InfClosed.dual, Set.sups_subset_self, BooleanSubalgebra.supClosed, supClosed_supClosure, supClosed_univ
infClosure 📖CompOp
20 mathmath: inf_mem_infClosure, isGLB_infClosure, Set.Finite.infClosure, SupClosed.infClosure, infClosed_infClosure, infClosure_singleton, InfClosed.infClosure_eq, infClosure_eq_self, infClosure_univ, subset_infClosure, lowerBounds_infClosure, infClosure_min, infClosure_prod, supClosure_infClosure, infClosure_mono, infClosure_supClosure, infClosure_idem, infClosure_isClosed, finsetInf'_mem_infClosure, infClosure_empty
latticeClosure 📖CompOp
22 mathmath: isSublattice_latticeClosure, latticeClosure_univ, IsSublattice.latticeClosure_eq, ofDual_preimage_latticeClosure, latticeClosure_mono, image_latticeClosure', compl_image_latticeClosure, latticeClosure_prod, compl_image_latticeClosure_eq_of_compl_image_eq_self, latticeClosure_singleton, subset_latticeClosure, BooleanSubalgebra.latticeClosure_subset_closure, BooleanSubalgebra.closure_latticeClosure, image_latticeClosure, latticeClosure_eq_self, latticeClosure_empty, latticeClosure_idem, Set.Finite.latticeClosure, supClosure_infClosure, infClosure_supClosure, latticeClosure_isClosed, latticeClosure_min
supClosure 📖CompOp
28 mathmath: upperBounds_supClosure, supClosure_eq_self, MeasureTheory.IsSetSemiring.isSetRing_supClosure, MeasureTheory.AddContent.supClosure_apply_finpartition, MeasureTheory.IsSetSemiring.mem_supClosure_iff, MeasureTheory.dense_of_generateFrom_isSetSemiring, supClosure_empty, MeasureTheory.IsSetSemiring.diff_mem_supClosure, Set.Finite.supClosure, sup_mem_supClosure, supClosure_isClosed, supClosure_singleton, supClosure_mono, SupClosed.supClosure_eq, supClosure_infClosure, MeasureTheory.AddContent.supClosure_apply_of_mem, InfClosed.supClosure, infClosure_supClosure, supClosure_idem, MeasureTheory.exists_measure_symmDiff_lt_of_generateFrom_isSetSemiring, subset_supClosure, isLUB_supClosure, supClosure_prod, finsetSup'_mem_supClosure, MeasureTheory.AddContent.supClosure_apply, supClosed_supClosure, supClosure_univ, supClosure_min

Theorems

NameKindAssumesProvesValidatesDepends On
IsSublattice_range 📖mathematicalIsSublattice
Set.range
DFunLike.coe
supClosed_range
LatticeHomClass.toSupHomClass
infClosed_range
LatticeHomClass.toInfHomClass
compl_image_latticeClosure 📖mathematicalSet.image
Compl.compl
BooleanAlgebra.toCompl
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
image_latticeClosure'
compl_sup_distrib
compl_inf
compl_image_latticeClosure_eq_of_compl_image_eq_self 📖mathematicalSet.image
Compl.compl
BooleanAlgebra.toCompl
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
compl_image_latticeClosure
finsetInf'_mem_infClosure 📖mathematicalFinset.Nonempty
Set
Set.instMembership
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
Finset.inf'
InfClosed.finsetInf'_mem
infClosed_infClosure
subset_infClosure
finsetSup'_mem_supClosure 📖mathematicalFinset.Nonempty
Set
Set.instMembership
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
Finset.sup'
SupClosed.finsetSup'_mem
supClosed_supClosure
subset_supClosure
image_latticeClosure 📖mathematicalSemilatticeSup.toMax
Lattice.toSemilatticeSup
SemilatticeInf.toMin
Lattice.toSemilatticeInf
Set.image
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
Set.instReflSubset
Set.instAntisymmSubset
latticeClosure_sup_inf_induction
subset_latticeClosure
Set.mem_image_of_mem
IsSublattice.supClosed
isSublattice_latticeClosure
IsSublattice.infClosed
Set.image_mono
image_latticeClosure' 📖mathematicalSemilatticeSup.toMax
Lattice.toSemilatticeSup
SemilatticeInf.toMin
Lattice.toSemilatticeInf
Set.image
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
Set.image_comp
Equiv.image_symm_eq_preimage
image_latticeClosure
infClosed_empty 📖mathematicalInfClosed
Set
Set.instEmptyCollection
infClosed_iInter 📖mathematicalInfClosedSet.iInterinfClosed_sInter
Set.forall_mem_range
infClosed_infClosure 📖mathematicalInfClosed
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
ClosureOperator.isClosed_closure
infClosed_pi 📖mathematicalInfClosedPi.instSemilatticeInf
Set.pi
infClosed_preimage_ofDual 📖mathematicalInfClosed
OrderDual
OrderDual.instSemilatticeInf
Lattice.toSemilatticeSup
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
SupClosed
infClosed_preimage_toDual 📖mathematicalInfClosed
Lattice.toSemilatticeInf
Set.preimage
OrderDual
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
SupClosed
OrderDual.instSemilatticeSup
infClosed_range 📖mathematicalInfClosed
Set.range
DFunLike.coe
Set.image_univ
InfClosed.image
infClosed_univ
infClosed_sInter 📖mathematicalInfClosedSet.sInter
infClosed_singleton 📖mathematicalInfClosed
Set
Set.instSingletonSet
infClosed_univ 📖mathematicalInfClosed
Set.univ
infClosure_empty 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
Set.instEmptyCollection
infClosure_eq_self 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
InfClosed
ClosureOperator.isClosed_iff
infClosure_idem 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
ClosureOperator.idempotent
infClosure_isClosed 📖mathematicalClosureOperator.IsClosed
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
infClosure
InfClosed
infClosure_min 📖mathematicalSet
Set.instHasSubset
InfClosed
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
ClosureOperator.closure_min
infClosure_mono 📖mathematicalMonotone
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
DFunLike.coe
ClosureOperator
ClosureOperator.instFunLike
infClosure
ClosureOperator.monotone
infClosure_prod 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
Prod.instSemilatticeInf
SProd.sprod
Set.instSProd
le_antisymm
infClosure_min
Set.prod_mono
subset_infClosure
InfClosed.prod
infClosed_infClosure
Finset.Nonempty.product
Finset.coe_product
Finset.inf'_congr
Finset.prodMk_inf'_inf'
infClosure_singleton 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
Set.instSingletonSet
infClosure_supClosure 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
Lattice.toSemilatticeInf
DistribLattice.toLattice
supClosure
Lattice.toSemilatticeSup
latticeClosure
le_antisymm
infClosure_min
supClosure_min
subset_latticeClosure
IsSublattice.supClosed
isSublattice_latticeClosure
IsSublattice.infClosed
latticeClosure_min
HasSubset.Subset.trans
Set.instIsTransSubset
subset_supClosure
subset_infClosure
SupClosed.infClosure
supClosed_supClosure
infClosed_infClosure
infClosure_univ 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
Set.univ
inf_mem_infClosure 📖mathematicalSet
Set.instMembership
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
SemilatticeInf.toMin
infClosed_infClosure
subset_infClosure
isGLB_infClosure 📖mathematicalIsGLB
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
DFunLike.coe
ClosureOperator
Set
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
lowerBounds_infClosure
isLUB_supClosure 📖mathematicalIsLUB
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
DFunLike.coe
ClosureOperator
Set
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
upperBounds_supClosure
isSublattice_empty 📖mathematicalIsSublattice
Set
Set.instEmptyCollection
supClosed_empty
infClosed_empty
isSublattice_iInter 📖mathematicalIsSublatticeSet.iIntersupClosed_iInter
IsSublattice.supClosed
infClosed_iInter
IsSublattice.infClosed
isSublattice_latticeClosure 📖mathematicalIsSublattice
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
ClosureOperator.isClosed_closure
isSublattice_pi 📖mathematicalIsSublatticePi.instLattice
Set.pi
supClosed_pi
IsSublattice.supClosed
infClosed_pi
IsSublattice.infClosed
isSublattice_preimage_ofDual 📖mathematicalIsSublattice
OrderDual
OrderDual.instLattice
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
IsSublattice.infClosed
IsSublattice.supClosed
isSublattice_preimage_toDual 📖mathematicalIsSublattice
Set.preimage
OrderDual
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
OrderDual.instLattice
IsSublattice.infClosed
IsSublattice.supClosed
isSublattice_sInter 📖mathematicalIsSublatticeSet.sIntersupClosed_sInter
IsSublattice.supClosed
infClosed_sInter
IsSublattice.infClosed
isSublattice_singleton 📖mathematicalIsSublattice
Set
Set.instSingletonSet
supClosed_singleton
infClosed_singleton
isSublattice_univ 📖mathematicalIsSublattice
Set.univ
supClosed_univ
infClosed_univ
latticeClosure_empty 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
Set.instEmptyCollection
latticeClosure_eq_self 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
IsSublattice
ClosureOperator.isClosed_iff
latticeClosure_idem 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
ClosureOperator.idempotent
latticeClosure_isClosed 📖mathematicalClosureOperator.IsClosed
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
latticeClosure
IsSublattice
latticeClosure_min 📖mathematicalSet
Set.instHasSubset
IsSublattice
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
ClosureOperator.closure_min
latticeClosure_mono 📖mathematicalMonotone
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
DFunLike.coe
ClosureOperator
ClosureOperator.instFunLike
latticeClosure
ClosureOperator.monotone
latticeClosure_prod 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
Prod.instLattice
DistribLattice.toLattice
SProd.sprod
Set.instSProd
infClosure_prod
supClosure_prod
supClosure_infClosure
latticeClosure_singleton 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
Set.instSingletonSet
latticeClosure_sup_inf_induction 📖subset_latticeClosure
SemilatticeSup.toMax
Lattice.toSemilatticeSup
IsSublattice.supClosed
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
isSublattice_latticeClosure
SemilatticeInf.toMin
Lattice.toSemilatticeInf
IsSublattice.infClosed
Set.instMembership
subset_latticeClosure
IsSublattice.supClosed
isSublattice_latticeClosure
IsSublattice.infClosed
latticeClosure_min
latticeClosure_univ 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
Set.univ
lowerBounds_infClosure 📖mathematicallowerBounds
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
DFunLike.coe
ClosureOperator
Set
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
HasSubset.Subset.antisymm
Set.instAntisymmSubset
lowerBounds_mono_set
subset_infClosure
Finset.le_inf'
ofDual_preimage_latticeClosure 📖mathematicalSet.preimage
OrderDual
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
OrderDual.instLattice
isSublattice_sInter
IsSublattice.infClosed
IsSublattice.supClosed
subset_infClosure 📖mathematicalSet
Set.instHasSubset
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
infClosure
ClosureOperator.le_closure
subset_latticeClosure 📖mathematicalSet
Set.instHasSubset
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
latticeClosure
ClosureOperator.le_closure
subset_supClosure 📖mathematicalSet
Set.instHasSubset
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
ClosureOperator.le_closure
supClosed_empty 📖mathematicalSupClosed
Set
Set.instEmptyCollection
supClosed_iInter 📖mathematicalSupClosedSet.iIntersupClosed_sInter
Set.forall_mem_range
supClosed_pi 📖mathematicalSupClosedPi.instSemilatticeSup
Set.pi
supClosed_preimage_ofDual 📖mathematicalSupClosed
OrderDual
OrderDual.instSemilatticeSup
Lattice.toSemilatticeInf
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
InfClosed
supClosed_preimage_toDual 📖mathematicalSupClosed
Lattice.toSemilatticeSup
Set.preimage
OrderDual
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
InfClosed
OrderDual.instSemilatticeInf
supClosed_range 📖mathematicalSupClosed
Set.range
DFunLike.coe
Set.image_univ
SupClosed.image
supClosed_univ
supClosed_sInter 📖mathematicalSupClosedSet.sInter
supClosed_singleton 📖mathematicalSupClosed
Set
Set.instSingletonSet
supClosed_supClosure 📖mathematicalSupClosed
DFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
ClosureOperator.isClosed_closure
supClosed_univ 📖mathematicalSupClosed
Set.univ
supClosure_empty 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
Set.instEmptyCollection
supClosure_eq_self 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
SupClosed
ClosureOperator.isClosed_iff
supClosure_idem 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
ClosureOperator.idempotent
supClosure_infClosure 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
Lattice.toSemilatticeSup
DistribLattice.toLattice
infClosure
Lattice.toSemilatticeInf
latticeClosure
le_antisymm
supClosure_min
infClosure_min
subset_latticeClosure
IsSublattice.infClosed
isSublattice_latticeClosure
IsSublattice.supClosed
latticeClosure_min
HasSubset.Subset.trans
Set.instIsTransSubset
subset_infClosure
subset_supClosure
supClosed_supClosure
InfClosed.supClosure
infClosed_infClosure
supClosure_isClosed 📖mathematicalClosureOperator.IsClosed
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
supClosure
SupClosed
supClosure_min 📖mathematicalSet
Set.instHasSubset
SupClosed
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
ClosureOperator.closure_min
supClosure_mono 📖mathematicalMonotone
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
DFunLike.coe
ClosureOperator
ClosureOperator.instFunLike
supClosure
ClosureOperator.monotone
supClosure_prod 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
Prod.instSemilatticeSup
SProd.sprod
Set.instSProd
le_antisymm
supClosure_min
Set.prod_mono
subset_supClosure
SupClosed.prod
supClosed_supClosure
Finset.Nonempty.product
Finset.coe_product
Finset.sup'_congr
Finset.prodMk_sup'_sup'
supClosure_singleton 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
Set.instSingletonSet
supClosure_univ 📖mathematicalDFunLike.coe
ClosureOperator
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
Set.univ
sup_mem_supClosure 📖mathematicalSet
Set.instMembership
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
SemilatticeSup.toMax
supClosed_supClosure
subset_supClosure
upperBounds_supClosure 📖mathematicalupperBounds
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
DFunLike.coe
ClosureOperator
Set
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
ClosureOperator.instFunLike
supClosure
HasSubset.Subset.antisymm
Set.instAntisymmSubset
upperBounds_mono_set
subset_supClosure
Finset.sup'_le

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