Documentation Verification Report

Closure

πŸ“ Source: Mathlib/Order/Closure.lean

Statistics

MetricCount
DefinitionsClosureOperator, IsClosed, conjBy, gi, id, instFunLike, instInhabited, mk', mkβ‚‚, ofCompletePred, ofPred, toCloseds, toOrderHom, closureOperator, lowerAdjoint, LowerAdjoint, closed, closureOperator, id, instCoeFunForall, instInhabitedId, toClosed, toFun, equivClosureOperator
24
Theoremsclosure_eq, closure_le_iff, closure_iSup_closure, closure_iSupβ‚‚_closure, closure_inf_le, closure_isGLB, closure_min, closure_sup_closure, closure_sup_closure_le, closure_sup_closure_left, closure_sup_closure_right, closure_top, conjBy_apply, conjBy_refl, conjBy_trans, eq_ofPred_closed, ext, ext_iff, ext_isClosed, id_apply, id_isClosed, idempotent, idempotent', instOrderHomClass, isClosed_closure, isClosed_iff, isClosed_iff_closure_le, isClosed_top, le_closure, le_closure', le_closure_iff, mk'_apply, mk'_isClosed, mkβ‚‚_apply, mkβ‚‚_isClosed, monotone, ofCompletePred_apply, ofCompletePred_isClosed, ofPred_apply, ofPred_isClosed, sInf_isClosed, setOf_isClosed_eq_range_closure, closureOperator_apply, closureOperator_isClosed, lowerAdjoint_toFun, closed_eq_range_close, closureOperator_apply, closureOperator_isClosed, closure_eq_self_of_mem_closed, closure_iSup_closure, closure_iSupβ‚‚_closure, closure_iUnion_closure, closure_iUnionβ‚‚_closure, closure_inf_le, closure_is_closed, closure_le_closed_iff_le, closure_sup_closure, closure_sup_closure_le, closure_sup_closure_left, closure_sup_closure_right, closure_top, closure_union_closure, closure_union_closure_left, closure_union_closure_right, closure_union_closure_subset, eq_of_le, ext, ext_iff, gc, gc', id_toFun, idempotent, le_closure, le_closure_iff, le_iff_subset, mem_closed_iff, mem_closed_iff_closure_le, mem_iff, monotone, notMem_of_notMem_closure, subset_closure, equivClosureOperator_apply, equivClosureOperator_symm_apply, closureOperator_gi_self
84
Total108

ClosureOperator

Definitions

NameCategoryTheorems
IsClosed πŸ“–MathDef
27 mathmath: setOf_isClosed_eq_range_closure, ofPred_isClosed, Matroid.isClosed_iff_isFlat, GaloisConnection.closureOperator_isClosed, CategoryTheory.GrothendieckTopology.closureOperator_isClosed, ofCompletePred_isClosed, supClosure_isClosed, mkβ‚‚_isClosed, isClosed_iff, IntermediateField.normalClosureOperator_isClosed, isClosed_iff_closure_le, absConvexHull_isClosed, convexHull_isClosed, Matroid.isFlat_iff_isClosed, LowerAdjoint.closureOperator_isClosed, isClosed_restrictScalars_separableClosure, infClosure_isClosed, closedAbsConvexHull_isClosed, latticeClosure_isClosed, eq_ofPred_closed, closureOperator_gi_self, mk'_isClosed, isClosed_top, isClosed_closure, closure_isGLB, closedConvexHull_isClosed, id_isClosed
conjBy πŸ“–CompOp
5 mathmath: conjBy_apply, OrderIso.equivClosureOperator_apply, OrderIso.equivClosureOperator_symm_apply, conjBy_trans, conjBy_refl
gi πŸ“–CompOp
1 mathmath: closureOperator_gi_self
id πŸ“–CompOp
2 mathmath: id_apply, id_isClosed
instFunLike πŸ“–CompOp
245 mathmath: upperBounds_supClosure, setOf_isClosed_eq_range_closure, convexIndependent_set_iff_inter_convexHull_subset, IsLinearMap.image_convexHull, convexHull_insert, subset_closedConvexHull, Geometry.SimplicialComplex.vertex_mem_convexHull_iff, closedConvexHull_min, convexHull_multiset_sum, Geometry.SimplicialComplex.convexHull_subset_space, ofCompletePred_apply, closure_iSupβ‚‚_closure, supClosure_eq_self, Geometry.SimplicialComplex.mem_space_iff, isSublattice_latticeClosure, Convex.convex_remove_iff_notMem_convexHull_remove, inf_mem_infClosure, convexHull_eq_iInter, latticeClosure_univ, Complex.convexHull_reProdIm, isGLB_infClosure, convexJoin_segments, IsSublattice.latticeClosure_eq, convexHull_union_neg_eq_absConvexHull, interior_convexHull_nonempty_iff_affineSpan_eq_top, mem_absConvexHull_iff, conjBy_apply, Geometry.SimplicialComplex.convexHull_inter_convexHull, AffineBasis.convexHull_eq_nonneg_coord, segment_subset_convexHull, convexHull_toCone_eq_sInf, Balanced.convexHull, subset_absConvexHull, balancedHull_subset_convexHull_union_neg, Convex.convexHull_union, closedConvexHull_eq_closure_convexHull, Finset.mem_convexHull', subset_convexHull, Convex.convexHull_subset_iff, AffineBasis.interior_convexHull, Set.Finite.infClosure, ofDual_preimage_latticeClosure, Geometry.SimplicialComplex.inter_subset_convexHull, convexJoin_segment_singleton, le_closure_iff, convexJoin_singleton_segment, convexHull_diam, subset_closedAbsConvexHull, Finset.centerMass_id_mem_convexHull, latticeClosure_mono, Set.Finite.isCompact_convexHull, image_latticeClosure', Finset.centerMass_id_mem_convexHull_of_nonpos, convexHull_add, affineSpan_convexHull, MeasureTheory.IsSetSemiring.isSetRing_supClosure, convexHull_basis_eq_stdSimplex, MeasureTheory.AddContent.supClosure_apply_finpartition, Convex.radon_partition, ConvexOn.bddAbove_convexHull, MeasureTheory.IsSetSemiring.mem_supClosure_iff, SupClosed.infClosure, Matroid.closure_eq_subtypeClosure, infClosed_infClosure, ConvexIndependent.mem_convexHull_iff, not_disjoint_segment_convexHull_triple, convexHull_vadd, MeasureTheory.dense_of_generateFrom_isSetSemiring, exists_mem_interior_convexHull_affineBasis, supClosure_empty, compl_image_latticeClosure, closedAbsConvexHull_min, ofPred_apply, PrimitiveSpectrum.gc_closureOperator, totallyBounded_absConvexHull, ext_iff, infClosure_singleton, absConvexHull_nonempty, absConvexHull_empty, absConvexHull_eq_self, Convex.mem_extremePoints_iff_mem_diff_convexHull_diff, latticeClosure_prod, InfClosed.infClosure_eq, le_closure, convexHull_add_subset, MeasureTheory.IsSetSemiring.diff_mem_supClosure, infClosure_eq_self, closedConvexHull_closure_eq_closedConvexHull, compl_image_latticeClosure_eq_of_compl_image_eq_self, Polynomial.rootSet_derivative_subset_convexHull_rootSet, latticeClosure_singleton, closure_convexHull_extremePoints, closure_sup_closure_right, Set.Finite.isClosed_convexHull, convexHull_empty, absConvexHull_add_subset, Finset.centerMass_mem_convexHull, convexHull_pair, infClosure_univ, subset_latticeClosure, convexJoin_subset_convexHull, Geometry.SimplicialComplex.disjoint_or_exists_inter_eq_convexHull, subset_infClosure, convexHull_eq_empty, convexHull_singleton, BooleanSubalgebra.latticeClosure_subset_closure, convex_convexHull, convexHull_sum, isBounded_convexHull, Finset.mem_convexHull, BooleanSubalgebra.closure_latticeClosure, convexHull_neg, convexIndependent_set_iff_notMem_convexHull_diff, convexHull_sphere_eq_closedBall, Set.Finite.supClosure, Convex.exists_subset_interior_convexHull_finset_of_isCompact, IntermediateField.normalClosureOperator_apply, convexHull_mono, convex_absConvexHull, convexHullAddMonoidHom_apply, absConvexHull_mono, Finset.convexHull_eq, totallyBounded_convexHull, Set.Nonempty.absConvexHull, sup_mem_supClosure, closure_sup_closure_left, lowerBounds_infClosure, absConvex_convexClosedHull, closure_sup_closure, convexHull_eq_singleton, convexHull_eq_zero, closure_iSup_closure, isCompact_closedAbsConvexHull_of_totallyBounded, absConvexHull_subset_closedAbsConvexHull, toWeakSpace_closedConvexHull_eq, convexHull_pi, closure_min, supClosure_singleton, image_latticeClosure, doublyStochastic_eq_convexHull_permMatrix, latticeClosure_eq_self, absConvexHull_eq_iInter, latticeClosure_empty, Finset.centroid_mem_convexHull, supClosure_mono, LowerAdjoint.closureOperator_apply, AbsConvex.absConvexHull_subset_iff, SupClosed.supClosure_eq, PrimitiveSpectrum.closedsGC_closureOperator, absConvex_absConvexHull, LinearMap.image_convexHull, IsClosed.closure_le_iff, convexHull_univ, convexHull_ediam, infClosure_min, absConvexHull_min, AffineIndependent.convexHull_inter', mk'_apply, latticeClosure_idem, closure_top, convexHull_zero, isClosed_iff_closure_le, convexHull_eq_union_convexHull_finite_subsets, zero_mem_absConvexHull, infClosure_prod, affineCombination_mem_convexHull, mem_convexHull_iff_exists_fintype, absConvexHull_univ, convexHull_toCone_isLeast, Set.Finite.latticeClosure, extremePoints_convexHull_subset, Geometry.SimplicialComplex.face_subset_face_iff, Convex.convexHull_eq, convexHull_convexHull_union_right, mkβ‚‚_apply, supClosure_infClosure, closure_inf_le, infClosure_mono, convexHull_convexHull_union_left, convexHull_min, MeasureTheory.AddContent.supClosure_apply_of_mem, InfClosed.supClosure, closedAbsConvexHull_eq_closure_absConvexHull, convexHull_prod, infClosure_supClosure, convexHull_eq_union, convexHull_eq, monotone, supClosure_idem, convexHull_smul, Set.Nonempty.convexHull, absConvexHull_eq_convexHull_balancedHull, Set.Finite.convexHull_eq_image, infClosure_idem, GaloisConnection.closureOperator_apply, mem_convexHull_of_exists_fintype, MeasureTheory.exists_measure_symmDiff_lt_of_generateFrom_isSetSemiring, convexHull_rangle_single_eq_stdSimplex, AffineBasis.centroid_mem_interior_convexHull, AffineMap.image_convexHull, convexHull_union, convexHull_sub, closure_subset_closedConvexHull, ConcaveOn.bddBelow_convexHull, convexHull_list_sum, convexHull_nonempty_iff, subset_supClosure, finsetInf'_mem_infClosure, mem_convexHull_iff, parallelepiped_eq_convexHull, instOrderHomClass, eq_ofPred_closed, absConvexHull_eq_empty, isLUB_supClosure, isClosed_closedAbsConvexHull, supClosure_prod, AffineIndependent.convexHull_inter, finsetSup'_mem_supClosure, infClosure_empty, convex_closedConvexHull, closure_subset_closedAbsConvexHull, MeasureTheory.AddContent.supClosure_apply, IsClosed.closure_eq, closure_sup_closure_le, Set.Finite.convexHull_eq, supClosed_supClosure, isClosed_closedConvexHull, balanced_absConvexHull, convexHull_subset_affineSpan, idempotent, convexHull_eq_self, isClosed_closure, supClosure_univ, id_apply, balancedHull_convexHull_subseteq_absConvexHull, latticeClosure_min, convexHull_range_eq_exists_affineCombination, convexIndependent_iff_notMem_convexHull_diff, closure_isGLB, AbsConvex.absConvexHull_eq, supClosure_min, closedAbsConvexHull_closure_eq_closedAbsConvexHull, Complex.rectangle_eq_convexHull, convexHull_subset_closedConvexHull, Finset.centerMass_mem_convexHull_of_nonpos
instInhabited πŸ“–CompOpβ€”
mk' πŸ“–CompOp
2 mathmath: mk'_apply, mk'_isClosed
mkβ‚‚ πŸ“–CompOp
2 mathmath: mkβ‚‚_isClosed, mkβ‚‚_apply
ofCompletePred πŸ“–CompOp
2 mathmath: ofCompletePred_apply, ofCompletePred_isClosed
ofPred πŸ“–CompOp
3 mathmath: ofPred_isClosed, ofPred_apply, eq_ofPred_closed
toCloseds πŸ“–CompOp
1 mathmath: closureOperator_gi_self
toOrderHom πŸ“–CompOp
3 mathmath: idempotent', isClosed_iff, le_closure'

Theorems

NameKindAssumesProvesValidatesDepends On
closure_iSup_closure πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
instFunLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
β€”le_antisymm
le_closure_iff
iSup_le
monotone
le_iSup
iSup_mono
le_closure
closure_iSupβ‚‚_closure πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
instFunLike
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
β€”le_antisymm
le_closure_iff
iSupβ‚‚_le
monotone
le_iSupβ‚‚
iSupβ‚‚_mono
le_closure
closure_inf_le πŸ“–mathematicalβ€”Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
DFunLike.coe
ClosureOperator
instFunLike
SemilatticeInf.toMin
β€”Monotone.map_inf_le
monotone
closure_isGLB πŸ“–mathematicalβ€”IsGLB
Preorder.toLE
PartialOrder.toPreorder
setOf
IsClosed
DFunLike.coe
ClosureOperator
instFunLike
β€”closure_min
le_closure
isClosed_closure
closure_min πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
IsClosed
DFunLike.coe
ClosureOperator
instFunLike
β€”IsClosed.closure_le_iff
closure_sup_closure πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
instFunLike
SemilatticeSup.toMax
β€”closure_sup_closure_left
closure_sup_closure_right
closure_sup_closure_le πŸ“–mathematicalβ€”Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
SemilatticeSup.toMax
DFunLike.coe
ClosureOperator
instFunLike
β€”Monotone.le_map_sup
monotone
closure_sup_closure_left πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
instFunLike
SemilatticeSup.toMax
β€”le_antisymm
le_closure_iff
sup_le
monotone
le_sup_left
LE.le.trans
le_sup_right
le_closure
le_imp_le_of_le_of_le
le_refl
OrderHomClass.mono
instOrderHomClass
sup_le_sup
closure_sup_closure_right πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
instFunLike
SemilatticeSup.toMax
β€”sup_comm
closure_sup_closure_left
closure_top πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
instFunLike
Top.top
OrderTop.toTop
Preorder.toLE
β€”LE.le.antisymm
le_top
le_closure
conjBy_apply πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
instFunLike
conjBy
OrderHom
OrderHom.instFunLike
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderIso.conj
OrderHomClass.toOrderHom
instOrderHomClass
β€”β€”
conjBy_refl πŸ“–mathematicalβ€”conjBy
OrderIso.refl
Preorder.toLE
β€”β€”
conjBy_trans πŸ“–mathematicalβ€”conjBy
OrderIso.trans
Preorder.toLE
β€”β€”
eq_ofPred_closed πŸ“–mathematicalβ€”ofPred
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
instFunLike
IsClosed
le_closure
isClosed_closure
closure_min
β€”ext
le_closure
isClosed_closure
closure_min
ext πŸ“–β€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
instFunLike
β€”β€”DFunLike.ext
ext_iff πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
instFunLike
β€”ext
ext_isClosed πŸ“–β€”IsClosed
PartialOrder.toPreorder
β€”β€”ext
IsGLB.unique
closure_isGLB
Set.ext
id_apply πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
instFunLike
id
β€”β€”
id_isClosed πŸ“–mathematicalβ€”IsClosed
PartialOrder.toPreorder
id
β€”β€”
idempotent πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
instFunLike
β€”idempotent'
idempotent' πŸ“–mathematicalβ€”OrderHom.toFun
toOrderHom
β€”β€”
instOrderHomClass πŸ“–mathematicalβ€”OrderHomClass
ClosureOperator
Preorder.toLE
instFunLike
β€”OrderHom.mono
isClosed_closure πŸ“–mathematicalβ€”IsClosed
PartialOrder.toPreorder
DFunLike.coe
ClosureOperator
instFunLike
β€”isClosed_iff
idempotent
isClosed_iff πŸ“–mathematicalβ€”IsClosed
OrderHom.toFun
toOrderHom
β€”β€”
isClosed_iff_closure_le πŸ“–mathematicalβ€”IsClosed
PartialOrder.toPreorder
Preorder.toLE
DFunLike.coe
ClosureOperator
instFunLike
β€”Eq.le
IsClosed.closure_eq
isClosed_iff
LE.le.antisymm
le_closure
isClosed_top πŸ“–mathematicalβ€”IsClosed
PartialOrder.toPreorder
Top.top
OrderTop.toTop
Preorder.toLE
β€”isClosed_iff
closure_top
le_closure πŸ“–mathematicalβ€”Preorder.toLE
PartialOrder.toPreorder
DFunLike.coe
ClosureOperator
instFunLike
β€”le_closure'
le_closure' πŸ“–mathematicalβ€”Preorder.toLE
OrderHom.toFun
toOrderHom
β€”β€”
le_closure_iff πŸ“–mathematicalβ€”Preorder.toLE
PartialOrder.toPreorder
DFunLike.coe
ClosureOperator
instFunLike
β€”monotone
idempotent
LE.le.trans
le_closure
mk'_apply πŸ“–mathematicalMonotone
PartialOrder.toPreorder
Preorder.toLE
DFunLike.coe
ClosureOperator
instFunLike
mk'
β€”β€”
mk'_isClosed πŸ“–mathematicalMonotone
PartialOrder.toPreorder
Preorder.toLE
IsClosed
mk'
β€”β€”
mkβ‚‚_apply πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
DFunLike.coe
ClosureOperator
instFunLike
mkβ‚‚
β€”β€”
mkβ‚‚_isClosed πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
IsClosed
mkβ‚‚
β€”β€”
monotone πŸ“–mathematicalβ€”Monotone
PartialOrder.toPreorder
DFunLike.coe
ClosureOperator
instFunLike
β€”OrderHom.monotone'
ofCompletePred_apply πŸ“–mathematicalInfSet.sInf
CompleteSemilatticeInf.toInfSet
CompleteLattice.toCompleteSemilatticeInf
DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
instFunLike
ofCompletePred
iInf
Preorder.toLE
β€”β€”
ofCompletePred_isClosed πŸ“–mathematicalInfSet.sInf
CompleteSemilatticeInf.toInfSet
CompleteLattice.toCompleteSemilatticeInf
IsClosed
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
ofCompletePred
β€”β€”
ofPred_apply πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
DFunLike.coe
ClosureOperator
instFunLike
ofPred
β€”β€”
ofPred_isClosed πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
IsClosed
ofPred
β€”β€”
sInf_isClosed πŸ“–mathematicalIsClosed
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
InfSet.sInf
CompleteSemilatticeInf.toInfSet
β€”isClosed_iff_closure_le
Monotone.map_sInf_le
monotone
biInf_congr
isClosed_iff
sInf_eq_iInf
setOf_isClosed_eq_range_closure πŸ“–mathematicalβ€”setOf
IsClosed
PartialOrder.toPreorder
Set.range
DFunLike.coe
ClosureOperator
instFunLike
β€”Set.ext
IsClosed.closure_eq
isClosed_closure

ClosureOperator.IsClosed

Theorems

NameKindAssumesProvesValidatesDepends On
closure_eq πŸ“–mathematicalClosureOperator.IsClosed
PartialOrder.toPreorder
DFunLike.coe
ClosureOperator
ClosureOperator.instFunLike
β€”ClosureOperator.isClosed_iff
closure_le_iff πŸ“–mathematicalClosureOperator.IsClosed
PartialOrder.toPreorder
Preorder.toLE
DFunLike.coe
ClosureOperator
ClosureOperator.instFunLike
β€”closure_eq
ClosureOperator.le_closure_iff

GaloisConnection

Definitions

NameCategoryTheorems
closureOperator πŸ“–CompOp
5 mathmath: closureOperator_isClosed, PrimitiveSpectrum.gc_closureOperator, PrimitiveSpectrum.closedsGC_closureOperator, closureOperator_apply, closureOperator_gi_self
lowerAdjoint πŸ“–CompOp
1 mathmath: lowerAdjoint_toFun

Theorems

NameKindAssumesProvesValidatesDepends On
closureOperator_apply πŸ“–mathematicalGaloisConnection
PartialOrder.toPreorder
DFunLike.coe
ClosureOperator
ClosureOperator.instFunLike
closureOperator
β€”β€”
closureOperator_isClosed πŸ“–mathematicalGaloisConnection
PartialOrder.toPreorder
ClosureOperator.IsClosed
closureOperator
β€”β€”
lowerAdjoint_toFun πŸ“–mathematicalGaloisConnectionLowerAdjoint.toFun
lowerAdjoint
β€”β€”

LowerAdjoint

Definitions

NameCategoryTheorems
closed πŸ“–CompOp
7 mathmath: closed_eq_range_close, mem_closed_iff_closure_le, FirstOrder.Language.Substructure.mem_closed_of_isRelational, mem_closed_iff, FirstOrder.Language.Substructure.closed, FirstOrder.Language.Substructure.mem_closed_iff, closure_is_closed
closureOperator πŸ“–CompOp
2 mathmath: closureOperator_apply, closureOperator_isClosed
id πŸ“–CompOp
1 mathmath: id_toFun
instCoeFunForall πŸ“–CompOpβ€”
instInhabitedId πŸ“–CompOpβ€”
toClosed πŸ“–CompOpβ€”
toFun πŸ“–CompOp
68 mathmath: closure_iUnionβ‚‚_closure, FirstOrder.Language.Substructure.closure_eq_of_isRelational, mem_iff, closed_eq_range_close, closure_union_closure, FirstOrder.Language.Substructure.fg_closure_singleton, closure_le_closed_iff_le, le_iff_subset, FirstOrder.Language.Substructure.small_closure, FirstOrder.Language.Substructure.cg_def, FirstOrder.Language.Substructure.mem_closure, idempotent, closure_sup_closure_right, FirstOrder.Language.Substructure.fg_closure, mem_closed_iff_closure_le, gc, le_closure, FirstOrder.Language.Substructure.fg_def, ext_iff, FirstOrder.Language.Substructure.lift_card_closure_le_card_term, FirstOrder.Language.Substructure.mem_closure_iff_exists_term, closure_sup_closure_left, id_toFun, gc', closure_iSupβ‚‚_closure, FirstOrder.Language.Substructure.fg_iff_exists_fin_generating_family, FirstOrder.Language.isExtensionPair_iff_exists_embedding_closure_singleton_sup, closure_sup_closure, closure_union_closure_subset, FirstOrder.Language.Substructure.closure_insert, subset_closure, GaloisConnection.lowerAdjoint_toFun, closure_union_closure_right, FirstOrder.Language.Substructure.closure_empty, FirstOrder.Language.Substructure.iSup_eq_closure, le_closure_iff, monotone, closureOperator_apply, FirstOrder.Language.Substructure.subset_closure, FirstOrder.Language.Substructure.closure_iUnion, FirstOrder.Language.Substructure.closure_union, FirstOrder.Language.Structure.cg_iff, closure_iSup_closure, closure_top, mem_closed_iff, closure_inf_le, FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize, closure_eq_self_of_mem_closed, FirstOrder.Language.Substructure.closure_withConstants_eq, FirstOrder.Language.Substructure.map_closure, FirstOrder.Language.Substructure.closure_le, Set.Countable.substructure_closure, FirstOrder.Language.Substructure.subset_closure_withConstants, closure_sup_closure_le, FirstOrder.Language.Substructure.cg_iff_empty_or_exists_nat_generating_family, closure_is_closed, FirstOrder.Language.Hom.eqOn_closure, FirstOrder.Language.Substructure.lift_card_closure_le, FirstOrder.Language.Substructure.closure_eq, FirstOrder.Language.Substructure.mem_closure_iff_of_isRelational, FirstOrder.Language.Substructure.closure_mono, FirstOrder.Language.Substructure.closure_univ, closure_union_closure_left, FirstOrder.Language.Substructure.closure_image, closure_iUnion_closure, FirstOrder.Language.Substructure.cg_closure_singleton, FirstOrder.Language.Structure.fg_iff, FirstOrder.Language.Substructure.cg_closure

Theorems

NameKindAssumesProvesValidatesDepends On
closed_eq_range_close πŸ“–mathematicalβ€”closed
PartialOrder.toPreorder
Set.range
toFun
β€”ClosureOperator.setOf_isClosed_eq_range_closure
closureOperator_apply πŸ“–mathematicalβ€”DFunLike.coe
ClosureOperator
PartialOrder.toPreorder
ClosureOperator.instFunLike
closureOperator
toFun
β€”β€”
closureOperator_isClosed πŸ“–mathematicalβ€”ClosureOperator.IsClosed
PartialOrder.toPreorder
closureOperator
β€”β€”
closure_eq_self_of_mem_closed πŸ“–mathematicalSet
Set.instMembership
closed
toFunβ€”β€”
closure_iSup_closure πŸ“–mathematicalβ€”toFun
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
β€”ClosureOperator.closure_iSup_closure
closure_iSupβ‚‚_closure πŸ“–mathematicalβ€”toFun
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
β€”ClosureOperator.closure_iSupβ‚‚_closure
closure_iUnion_closure πŸ“–mathematicalβ€”toFun
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
SetLike.coe
Set.iUnion
β€”SetLike.coe_injective
closure_iSup_closure
closure_iUnionβ‚‚_closure πŸ“–mathematicalβ€”toFun
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
SetLike.coe
Set.iUnion
β€”SetLike.coe_injective
closure_iSupβ‚‚_closure
closure_inf_le πŸ“–mathematicalβ€”Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
toFun
SemilatticeInf.toMin
β€”ClosureOperator.closure_inf_le
closure_is_closed πŸ“–mathematicalβ€”Set
Set.instMembership
closed
PartialOrder.toPreorder
toFun
β€”idempotent
closure_le_closed_iff_le πŸ“–mathematicalclosed
PartialOrder.toPreorder
Preorder.toLE
toFun
β€”ClosureOperator.IsClosed.closure_le_iff
closure_sup_closure πŸ“–mathematicalβ€”toFun
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
SemilatticeSup.toMax
β€”ClosureOperator.closure_sup_closure
closure_sup_closure_le πŸ“–mathematicalβ€”Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
SemilatticeSup.toMax
toFun
β€”ClosureOperator.closure_sup_closure_le
closure_sup_closure_left πŸ“–mathematicalβ€”toFun
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
SemilatticeSup.toMax
β€”ClosureOperator.closure_sup_closure_left
closure_sup_closure_right πŸ“–mathematicalβ€”toFun
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
SemilatticeSup.toMax
β€”ClosureOperator.closure_sup_closure_right
closure_top πŸ“–mathematicalβ€”toFun
PartialOrder.toPreorder
Top.top
OrderTop.toTop
Preorder.toLE
β€”ClosureOperator.closure_top
closure_union_closure πŸ“–mathematicalβ€”toFun
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
SetLike.coe
Set.instUnion
β€”closure_union_closure_right
closure_union_closure_left
closure_union_closure_left πŸ“–mathematicalβ€”toFun
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
SetLike.coe
Set.instUnion
β€”SetLike.coe_injective
closure_sup_closure_left
closure_union_closure_right πŸ“–mathematicalβ€”toFun
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
SetLike.coe
Set.instUnion
β€”SetLike.coe_injective
closure_sup_closure_right
closure_union_closure_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
Set.instUnion
SetLike.coe
toFun
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
β€”closure_sup_closure_le
eq_of_le πŸ“–β€”Set
Set.instHasSubset
SetLike.coe
Preorder.toLE
PartialOrder.toPreorder
toFun
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
β€”β€”LE.le.antisymm
le_iff_subset
ext πŸ“–β€”toFunβ€”β€”β€”
ext_iff πŸ“–mathematicalβ€”toFunβ€”ext
gc πŸ“–mathematicalβ€”GaloisConnection
toFun
β€”gc'
gc' πŸ“–mathematicalβ€”GaloisConnection
toFun
β€”β€”
id_toFun πŸ“–mathematicalβ€”toFun
id
β€”β€”
idempotent πŸ“–mathematicalβ€”toFun
PartialOrder.toPreorder
β€”ClosureOperator.idempotent
le_closure πŸ“–mathematicalβ€”Preorder.toLE
toFun
β€”GaloisConnection.le_u_l
gc
le_closure_iff πŸ“–mathematicalβ€”Preorder.toLE
PartialOrder.toPreorder
toFun
β€”ClosureOperator.le_closure_iff
le_iff_subset πŸ“–mathematicalβ€”Preorder.toLE
toFun
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
SetLike.coe
Set.instHasSubset
β€”gc
mem_closed_iff πŸ“–mathematicalβ€”Set
Set.instMembership
closed
toFun
β€”β€”
mem_closed_iff_closure_le πŸ“–mathematicalβ€”Set
Set.instMembership
closed
PartialOrder.toPreorder
Preorder.toLE
toFun
β€”ClosureOperator.isClosed_iff_closure_le
mem_iff πŸ“–mathematicalβ€”SetLike.instMembership
toFun
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
SetLike.coe
β€”le_iff_subset
LE.le.trans
le_rfl
monotone πŸ“–mathematicalβ€”Monotone
toFun
β€”Monotone.comp
GaloisConnection.monotone_u
gc
GaloisConnection.monotone_l
notMem_of_notMem_closure πŸ“–mathematicalSetLike.instMembership
toFun
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
SetLike.coe
Set.instMembershipβ€”subset_closure
subset_closure πŸ“–mathematicalβ€”Set
Set.instHasSubset
SetLike.coe
toFun
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
β€”le_closure

OrderIso

Definitions

NameCategoryTheorems
equivClosureOperator πŸ“–CompOp
2 mathmath: equivClosureOperator_apply, equivClosureOperator_symm_apply

Theorems

NameKindAssumesProvesValidatesDepends On
equivClosureOperator_apply πŸ“–mathematicalβ€”DFunLike.coe
Equiv
ClosureOperator
EquivLike.toFunLike
Equiv.instEquivLike
equivClosureOperator
ClosureOperator.conjBy
β€”β€”
equivClosureOperator_symm_apply πŸ“–mathematicalβ€”DFunLike.coe
Equiv
ClosureOperator
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
equivClosureOperator
ClosureOperator.conjBy
symm
Preorder.toLE
β€”β€”

(root)

Definitions

NameCategoryTheorems
ClosureOperator πŸ“–CompData
247 mathmath: upperBounds_supClosure, ClosureOperator.setOf_isClosed_eq_range_closure, convexIndependent_set_iff_inter_convexHull_subset, IsLinearMap.image_convexHull, convexHull_insert, subset_closedConvexHull, Geometry.SimplicialComplex.vertex_mem_convexHull_iff, closedConvexHull_min, convexHull_multiset_sum, Geometry.SimplicialComplex.convexHull_subset_space, ClosureOperator.ofCompletePred_apply, ClosureOperator.closure_iSupβ‚‚_closure, supClosure_eq_self, Geometry.SimplicialComplex.mem_space_iff, isSublattice_latticeClosure, Convex.convex_remove_iff_notMem_convexHull_remove, inf_mem_infClosure, convexHull_eq_iInter, latticeClosure_univ, Complex.convexHull_reProdIm, isGLB_infClosure, convexJoin_segments, IsSublattice.latticeClosure_eq, convexHull_union_neg_eq_absConvexHull, interior_convexHull_nonempty_iff_affineSpan_eq_top, mem_absConvexHull_iff, ClosureOperator.conjBy_apply, Geometry.SimplicialComplex.convexHull_inter_convexHull, AffineBasis.convexHull_eq_nonneg_coord, segment_subset_convexHull, convexHull_toCone_eq_sInf, Balanced.convexHull, subset_absConvexHull, balancedHull_subset_convexHull_union_neg, Convex.convexHull_union, closedConvexHull_eq_closure_convexHull, Finset.mem_convexHull', subset_convexHull, Convex.convexHull_subset_iff, AffineBasis.interior_convexHull, Set.Finite.infClosure, ofDual_preimage_latticeClosure, Geometry.SimplicialComplex.inter_subset_convexHull, convexJoin_segment_singleton, ClosureOperator.le_closure_iff, convexJoin_singleton_segment, convexHull_diam, subset_closedAbsConvexHull, Finset.centerMass_id_mem_convexHull, latticeClosure_mono, Set.Finite.isCompact_convexHull, image_latticeClosure', Finset.centerMass_id_mem_convexHull_of_nonpos, convexHull_add, affineSpan_convexHull, MeasureTheory.IsSetSemiring.isSetRing_supClosure, convexHull_basis_eq_stdSimplex, MeasureTheory.AddContent.supClosure_apply_finpartition, Convex.radon_partition, ConvexOn.bddAbove_convexHull, MeasureTheory.IsSetSemiring.mem_supClosure_iff, SupClosed.infClosure, Matroid.closure_eq_subtypeClosure, infClosed_infClosure, ConvexIndependent.mem_convexHull_iff, not_disjoint_segment_convexHull_triple, convexHull_vadd, MeasureTheory.dense_of_generateFrom_isSetSemiring, exists_mem_interior_convexHull_affineBasis, supClosure_empty, compl_image_latticeClosure, closedAbsConvexHull_min, ClosureOperator.ofPred_apply, PrimitiveSpectrum.gc_closureOperator, totallyBounded_absConvexHull, ClosureOperator.ext_iff, infClosure_singleton, absConvexHull_nonempty, absConvexHull_empty, absConvexHull_eq_self, OrderIso.equivClosureOperator_apply, Convex.mem_extremePoints_iff_mem_diff_convexHull_diff, latticeClosure_prod, InfClosed.infClosure_eq, ClosureOperator.le_closure, convexHull_add_subset, MeasureTheory.IsSetSemiring.diff_mem_supClosure, infClosure_eq_self, closedConvexHull_closure_eq_closedConvexHull, compl_image_latticeClosure_eq_of_compl_image_eq_self, Polynomial.rootSet_derivative_subset_convexHull_rootSet, latticeClosure_singleton, closure_convexHull_extremePoints, ClosureOperator.closure_sup_closure_right, Set.Finite.isClosed_convexHull, convexHull_empty, absConvexHull_add_subset, Finset.centerMass_mem_convexHull, convexHull_pair, infClosure_univ, subset_latticeClosure, convexJoin_subset_convexHull, Geometry.SimplicialComplex.disjoint_or_exists_inter_eq_convexHull, subset_infClosure, convexHull_eq_empty, convexHull_singleton, BooleanSubalgebra.latticeClosure_subset_closure, convex_convexHull, convexHull_sum, isBounded_convexHull, Finset.mem_convexHull, BooleanSubalgebra.closure_latticeClosure, convexHull_neg, convexIndependent_set_iff_notMem_convexHull_diff, convexHull_sphere_eq_closedBall, Set.Finite.supClosure, Convex.exists_subset_interior_convexHull_finset_of_isCompact, IntermediateField.normalClosureOperator_apply, convexHull_mono, convex_absConvexHull, convexHullAddMonoidHom_apply, absConvexHull_mono, Finset.convexHull_eq, totallyBounded_convexHull, Set.Nonempty.absConvexHull, sup_mem_supClosure, ClosureOperator.closure_sup_closure_left, lowerBounds_infClosure, absConvex_convexClosedHull, ClosureOperator.closure_sup_closure, convexHull_eq_singleton, convexHull_eq_zero, ClosureOperator.closure_iSup_closure, isCompact_closedAbsConvexHull_of_totallyBounded, absConvexHull_subset_closedAbsConvexHull, toWeakSpace_closedConvexHull_eq, convexHull_pi, ClosureOperator.closure_min, supClosure_singleton, image_latticeClosure, doublyStochastic_eq_convexHull_permMatrix, latticeClosure_eq_self, absConvexHull_eq_iInter, latticeClosure_empty, Finset.centroid_mem_convexHull, supClosure_mono, LowerAdjoint.closureOperator_apply, AbsConvex.absConvexHull_subset_iff, SupClosed.supClosure_eq, PrimitiveSpectrum.closedsGC_closureOperator, absConvex_absConvexHull, LinearMap.image_convexHull, ClosureOperator.IsClosed.closure_le_iff, convexHull_univ, convexHull_ediam, infClosure_min, absConvexHull_min, AffineIndependent.convexHull_inter', ClosureOperator.mk'_apply, latticeClosure_idem, ClosureOperator.closure_top, convexHull_zero, ClosureOperator.isClosed_iff_closure_le, convexHull_eq_union_convexHull_finite_subsets, zero_mem_absConvexHull, infClosure_prod, affineCombination_mem_convexHull, mem_convexHull_iff_exists_fintype, absConvexHull_univ, convexHull_toCone_isLeast, OrderIso.equivClosureOperator_symm_apply, Set.Finite.latticeClosure, extremePoints_convexHull_subset, Geometry.SimplicialComplex.face_subset_face_iff, Convex.convexHull_eq, convexHull_convexHull_union_right, ClosureOperator.mkβ‚‚_apply, supClosure_infClosure, ClosureOperator.closure_inf_le, infClosure_mono, convexHull_convexHull_union_left, convexHull_min, MeasureTheory.AddContent.supClosure_apply_of_mem, InfClosed.supClosure, closedAbsConvexHull_eq_closure_absConvexHull, convexHull_prod, infClosure_supClosure, convexHull_eq_union, convexHull_eq, ClosureOperator.monotone, supClosure_idem, convexHull_smul, Set.Nonempty.convexHull, absConvexHull_eq_convexHull_balancedHull, Set.Finite.convexHull_eq_image, infClosure_idem, GaloisConnection.closureOperator_apply, mem_convexHull_of_exists_fintype, MeasureTheory.exists_measure_symmDiff_lt_of_generateFrom_isSetSemiring, convexHull_rangle_single_eq_stdSimplex, AffineBasis.centroid_mem_interior_convexHull, AffineMap.image_convexHull, convexHull_union, convexHull_sub, closure_subset_closedConvexHull, ConcaveOn.bddBelow_convexHull, convexHull_list_sum, convexHull_nonempty_iff, subset_supClosure, finsetInf'_mem_infClosure, mem_convexHull_iff, parallelepiped_eq_convexHull, ClosureOperator.instOrderHomClass, ClosureOperator.eq_ofPred_closed, absConvexHull_eq_empty, isLUB_supClosure, isClosed_closedAbsConvexHull, supClosure_prod, AffineIndependent.convexHull_inter, finsetSup'_mem_supClosure, infClosure_empty, convex_closedConvexHull, closure_subset_closedAbsConvexHull, MeasureTheory.AddContent.supClosure_apply, ClosureOperator.IsClosed.closure_eq, ClosureOperator.closure_sup_closure_le, Set.Finite.convexHull_eq, supClosed_supClosure, isClosed_closedConvexHull, balanced_absConvexHull, convexHull_subset_affineSpan, ClosureOperator.idempotent, convexHull_eq_self, ClosureOperator.isClosed_closure, supClosure_univ, ClosureOperator.id_apply, balancedHull_convexHull_subseteq_absConvexHull, latticeClosure_min, convexHull_range_eq_exists_affineCombination, convexIndependent_iff_notMem_convexHull_diff, ClosureOperator.closure_isGLB, AbsConvex.absConvexHull_eq, supClosure_min, closedAbsConvexHull_closure_eq_closedAbsConvexHull, Complex.rectangle_eq_convexHull, convexHull_subset_closedConvexHull, Finset.centerMass_mem_convexHull_of_nonpos
LowerAdjoint πŸ“–CompDataβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
closureOperator_gi_self πŸ“–mathematicalβ€”GaloisConnection.closureOperator
ClosureOperator.Closeds
Subtype.preorder
PartialOrder.toPreorder
ClosureOperator.IsClosed
ClosureOperator.toCloseds
GaloisInsertion.gc
ClosureOperator.gi
β€”ClosureOperator.ext
GaloisInsertion.gc

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