Documentation Verification Report

Countable

πŸ“ Source: Mathlib/Data/Set/Countable.lean

Statistics

MetricCount
DefinitionsCountable, toEncodable, enumerateCountable
3
Theoremsto_set, countable_toSet, biUnion, biUnion_iff, exists_eq_range, exists_surjective, image, image2, insert, mono, nonempty_encodable, of_diff, of_preimage_singleton, of_subsingleton, preimage, preimage_of_injOn, prod, sUnion, sUnion_iff, setOf_finite, to_subtype, union, countable, countable_of_injOn, countable, countable_coe_iff, countable_empty, countable_iUnion, countable_iUnion_iff, countable_iff_exists_injOn, countable_iff_exists_injective, countable_iff_exists_subset_range, countable_iff_exists_surjective, countable_iff_nonempty_encodable, countable_insert, countable_isBot, countable_isTop, countable_of_injective_of_countable_image, countable_pi, countable_range, countable_setOf_finite_subset, countable_setOf_nonempty_of_disjoint, countable_singleton, countable_union, countable_univ, countable_univ_iff, countable_univ_pi, enumerateCountable_mem, exists_seq_cover_iff_countable, exists_seq_iSup_eq_top_iff_countable, not_countable_univ, not_countable_univ_iff, range_enumerateCountable_of_mem, range_enumerateCountable_subset, subset_range_enumerate, to_countable
56
Total59

Countable

Theorems

NameKindAssumesProvesValidatesDepends On
to_set πŸ“–mathematicalβ€”Set.Countableβ€”Set.countable_coe_iff

Finset

Theorems

NameKindAssumesProvesValidatesDepends On
countable_toSet πŸ“–mathematicalβ€”Set.Countable
SetLike.coe
Finset
instSetLike
β€”Set.Finite.countable
finite_toSet

Set

Definitions

NameCategoryTheorems
Countable πŸ“–MathDef
187 mathmath: countable_image_lt_image_Ioi_within, MonotoneOn.countable_not_continuousWithinAt_Ioi, FirstOrder.Language.IsFraisse.is_essentially_countable, countable_image_gt_image_Ioi_within, Dense.exists_countable_dense_subset_bot_top, IsLindelof.countable, IsLindelof.elim_countable_subcover_image, MeasureTheory.exists_decomposition_of_monotoneOn_hasDerivWithinAt, MapsTo.countable_of_injOn, AntitoneOn.countable_not_continuousWithinAt, Filter.mem_countableGenerate_iff, countable_iff_exists_surjective, Summable.countable_support_nnreal, Antitone.countable_setOf_two_preimages, Cardinal.le_aleph0_iff_set_countable, Countable.image2, LindelofSpace.elim_nhds_subcover, exists_countable_separating, Cardinal.Real.Ioc_countable_iff, TopologicalSpace.countable_countableBasis, FirstOrder.Language.Substructure.cg_def, ConvexOn.univ_sSup_of_countable_affine_eq, Besicovitch.exists_closedBall_covering_tsum_measure_le, TopologicalSpace.isOpen_iUnion_countable, TopologicalSpace.countable_cover_nhdsWithin, not_countable_univ_iff, countable_iff_exists_subset_range, exists_countable_union_perfect_of_isClosed, countable_setOf_nonempty_of_disjoint, countable_insert, Countable.preimage_of_injOn, Countable.sUnion_iff, IsLindelof.countable_of_isDiscrete, PairwiseDisjoint.countable_of_isOpen, Equiv.Perm.IsCycleOn.countable, countable_setOf_isolated_right, Filter.HasCountableBasis.countable, Meromorphic.MeromorphicOn.countable_compl_analyticAt, exists_nonempty_countable_separating, countable_cover_nhds_interior, MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnionβ‚€, EMetric.countable_closure_of_compact, FiniteExhaustion.Set.nonempty_finiteExhaustion_iff, AntitoneOn.countable_setOf_two_preimages, isLindelof_iff_countable_subcover, ConvexOn.sSup_of_countable_affine_eq, Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux, Monotone.countable_not_continuousAt, Meromorphic.countable_compl_analyticAt, Countable.to_set, ConvexOn.real_univ_sSup_of_countable_affine_eq, Countable.setOf_finite, MeasureTheory.Measure.countable_meas_level_set_pos, countable_coe_iff, Countable.prod, StieltjesFunction.countable_leftLim_ne, FirstOrder.Language.exists_countable_is_age_of_iff, Dense.exists_countable_dense_subset_no_bot_top, Multipliable.countable_mulSupport, TopologicalSpace.countable_cover_nhds, countable_image_lt_image_Iio_within, Antitone.countable_not_continuousAt, countable_cover_nhdsWithin_of_sigmaCompact, not_countable_univ, countable_cover_nhds, eq_open_union_countable, IsLindelof.elim_nhds_subcover', exists_countable_upperSemicontinuous_isGLB, countable_image_gt_image_Iio_within, countable_union, MeasureTheory.LocallyIntegrableOn.exists_countable_integrableOn, countable_setOf_covBy_left, MonotoneOn.countable_not_continuousWithinAt, Countable.image, MeasureTheory.countable_generateSetAlgebra, Cardinal.Real.Icc_countable_iff, countable_range, PMF.support_countable, Countable.union, TopologicalSpace.exists_countable_dense, countable_of_injective_of_countable_image, FirstOrder.Language.age.countable_quotient, TopologicalSpace.exists_countable_of_generateFrom, MeasureTheory.Measure.countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top, Filter.Tendsto.countable_compl_preimage_ker, VitaliFamily.FineSubfamilyOn.index_countable, PairwiseDisjoint.countable_of_Ioo, exists_seq_cover_iff_countable, countable_iff_exists_injective, countable_univ, MeromorphicOn.countable_compl_analyticAt_inter, MonotoneOn.countable_setOf_two_preimages, TopologicalSpace.exists_countable_basis, Filter.mem_cocountable, TopologicalSpace.IsSeparable.exists_countable_dense_subset, countable_iff_exists_injOn, MeasurableSpace.CountablyGenerated.isCountablyGenerated, IsLindelof.elim_nhds_subcover, Vitali.exists_disjoint_covering_ae, countable_infinite_iff_nonempty_denumerable, Besicovitch.exists_disjoint_closedBall_covering_ae, FirstOrder.Language.Structure.cg_iff, Filter.IsCountablyGenerated.out, IsLindelof.elim_countable_subfamily_closed, Countable.of_diff, not_countable_complex, ConvexOn.real_sSup_of_countable_affine_eq, countable_univ_iff, TopologicalSpace.isOpen_sUnion_countable, LocallyFinite.countable_univ, MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion, to_countable, countable_isTop, Countable.mono, countable_image_lt_image_Ioi, Countable.insert, Filter.countable_compl_ker, countable_of_isolated_left', Countable.preimage_cexp, Cardinal.le_aleph0_iff_subtype_countable, Countable.preimage_circleMap, exists_seq_iSup_eq_top_iff_countable, TopologicalSpace.isOpen_biUnion_countable, isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis, countable_setOf_isolated_left_within, isMeagre_iff_countable_union_isNowhereDense, countable_singleton, IsLindelof.countable_of_discrete, Finset.countable_toSet, HasCountableSeparatingOn.exists_countable_separating, IsLindelof.elim_countable_subcover, countable_image_gt_image_Ioi, exists_countable_generateFrom_Ioi_Iio, Monotone.countable_setOf_two_preimages, countable_cover_nhds_of_sigmaCompact, isLindelof_iff_countable_subfamily_closed, Summable.countable_support_ennreal, SecondCountableTopology.is_open_generated_countable, countable_iUnion_iff, MeasureTheory.countable_meas_le_ne_meas_lt, MeasureTheory.exists_countable_measureDense, Cardinal.Real.Ico_countable_iff, countable_isBot, MeasureTheory.Measure.countable_meas_level_set_posβ‚€, Countable.biUnion_iff, MeasurableSpace.countable_countableGeneratingSet, Cardinal.countable_iff_lt_aleph_one, Algebraic.countable, isLindelof_iff_countable, exists_countable_lowerSemicontinuous_isLUB, MeasureTheory.Measure.exists_ae_subset_biUnion_countable, exists_countable_dense_bot_top, Besicovitch.exists_disjoint_closedBall_covering_ae_aux, Dense.exists_countable_dense_subset, Cardinal.Real.Ioo_countable_iff, MonotoneOn.countable_not_continuousWithinAt_Iio, TopologicalSpace.IsTopologicalBasis.exists_countable, countable_setOf_isolated_left, MeromorphicOn.countable_compl_analyticAt, TopologicalSpace.separableSpace_iff, exists_countable_dense_no_bot_top, Cardinal.not_countable_real, Countable.preimage, Filter.CountableFilterBasis.countable, MeasureTheory.Measure.countable_meas_pos_of_disjoint_of_meas_iUnion_ne_topβ‚€, countable_iff_nonempty_encodable, eq_closed_inter_countable, EMetric.subset_countable_closure_of_compact, mem_residual_iff, MeasureTheory.IsSeparable.exists_countable_measureDense, PairwiseDisjoint.countable_of_nonempty_interior, countable_setOf_covBy_right, TopologicalSpace.SeparableSpace.exists_countable_dense, countable_empty, Filter.IsCountableBasis.countable, TopologicalSpace.IsTopologicalBasis.exists_countable_biUnion_of_isOpen, Countable.of_subsingleton, Vitali.exists_disjoint_covering_ae', countable_image_gt_image_Iio, Finite.countable, Subsingleton.countable, countable_setOf_finite_subset, countable_setOf_isolated_right_within, ENNReal.exists_countable_dense_no_zero_top, countable_image_lt_image_Iio, Complex.countable_preimage_exp, Summable.countable_support
enumerateCountable πŸ“–CompOp
5 mathmath: subset_range_enumerate, MeasurableSpace.measurableSet_enumerateCountable_countableGeneratingSet, range_enumerateCountable_of_mem, enumerateCountable_mem, range_enumerateCountable_subset

Theorems

NameKindAssumesProvesValidatesDepends On
countable_coe_iff πŸ“–mathematicalβ€”Countable
Elem
Countable
β€”β€”
countable_empty πŸ“–mathematicalβ€”Countable
Set
instEmptyCollection
β€”to_countable
Encodable.countable
instIsEmptyElemEmptyCollection
countable_iUnion πŸ“–mathematicalCountableiUnionβ€”Countable.to_subtype
iUnion_eq_range_psigma
countable_range
instCountablePSigma
countable_iUnion_iff πŸ“–mathematicalβ€”Countable
iUnion
β€”Countable.mono
subset_iUnion
countable_iUnion
countable_iff_exists_injOn πŸ“–mathematicalβ€”Countable
InjOn
β€”countable_iff_exists_injective
exists_injOn_iff_injective
countable_iff_exists_injective πŸ“–mathematicalβ€”Countable
Elem
β€”countable_iff_exists_injective
countable_iff_exists_subset_range πŸ“–mathematicalβ€”Countable
Set
instHasSubset
range
β€”subset_range_enumerate
Countable.mono
countable_range
instCountableNat
countable_iff_exists_surjective πŸ“–mathematicalNonemptyCountable
Elem
β€”countable_iff_exists_surjective
Nonempty.to_subtype
countable_iff_nonempty_encodable πŸ“–mathematicalβ€”Countable
Encodable
Elem
β€”Encodable.nonempty_encodable
countable_insert πŸ“–mathematicalβ€”Countable
Set
instInsert
β€”β€”
countable_isBot πŸ“–mathematicalβ€”Countable
setOf
IsBot
Preorder.toLE
PartialOrder.toPreorder
β€”Finite.countable
finite_isBot
countable_isTop πŸ“–mathematicalβ€”Countable
setOf
IsTop
Preorder.toLE
PartialOrder.toPreorder
β€”Finite.countable
finite_isTop
countable_of_injective_of_countable_image πŸ“–mathematicalInjOnCountableβ€”MapsTo.countable_of_injOn
mapsTo_image
countable_pi πŸ“–mathematicalCountablesetOfβ€”countable_univ_pi
countable_range πŸ“–mathematicalβ€”Countable
range
β€”Countable.to_set
Function.Surjective.countable
rangeFactorization_surjective
countable_setOf_finite_subset πŸ“–mathematicalβ€”Countable
Set
setOf
Finite
instHasSubset
β€”Countable.to_subtype
Countable.mono
CanLift.prf
canLift
Subtype.canLift
instCanLiftFinsetCoeFinite
Finite.of_finite_image
Function.Injective.injOn
Subtype.val_injective
mem_range_self
countable_range
Finset.countable
countable_setOf_nonempty_of_disjoint πŸ“–mathematicalPairwise
Function.onFun
Set
Disjoint
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
instHasSubset
Countable
setOf
Nonempty
β€”countable_coe_iff
not_disjoint_iff_nonempty_inter
Function.Injective.countable
countable_singleton πŸ“–mathematicalβ€”Countable
Set
instSingletonSet
β€”to_countable
Finite.to_countable
Finite.of_fintype
countable_union πŸ“–mathematicalβ€”Countable
Set
instUnion
β€”union_eq_iUnion
Bool.countable
countable_univ πŸ“–mathematicalβ€”Countable
univ
β€”to_countable
SetCoe.countable
countable_univ_iff πŸ“–mathematicalβ€”Countable
univ
Countable
β€”countable_coe_iff
Equiv.countable_iff
countable_univ_pi πŸ“–mathematicalCountablepi
univ
β€”Countable.to_subtype
Countable.of_equiv
instCountableForallOfFinite
enumerateCountable_mem πŸ“–mathematicalSet
instMembership
enumerateCountableβ€”range_enumerateCountable_of_mem
mem_range_self
exists_seq_cover_iff_countable πŸ“–mathematicalβ€”iUnion
univ
Countable
Set
sUnion
β€”exists_seq_iSup_eq_top_iff_countable
exists_seq_iSup_eq_top_iff_countable πŸ“–mathematicalβ€”iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
Top.top
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
Countable
SupSet.sSup
β€”countable_range
instCountableNat
forall_mem_range
sSup_range
eq_empty_or_nonempty
subsingleton_of_bot_eq_top
sSup_empty
countable_iff_exists_surjective
Subtype.coe_prop
Function.Surjective.iSup_comp
sSup_eq_iSup'
not_countable_univ πŸ“–mathematicalβ€”Countable
univ
β€”not_countable_univ_iff
not_countable_univ_iff πŸ“–mathematicalβ€”Countable
univ
Uncountable
β€”countable_univ_iff
not_countable_iff
range_enumerateCountable_of_mem πŸ“–mathematicalSet
instMembership
range
enumerateCountable
β€”subset_antisymm
instAntisymmSubset
LE.le.trans_eq
range_enumerateCountable_subset
insert_eq_of_mem
subset_range_enumerate
range_enumerateCountable_subset πŸ“–mathematicalβ€”Set
instHasSubset
range
enumerateCountable
instInsert
β€”range_subset_iff
enumerateCountable.eq_1
mem_insert
subset_range_enumerate πŸ“–mathematicalβ€”Set
instHasSubset
range
enumerateCountable
β€”Encodable.encodek
to_countable πŸ“–mathematicalβ€”Countableβ€”β€”

Set.Countable

Definitions

NameCategoryTheorems
toEncodable πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
biUnion πŸ“–mathematicalSet.CountableSet.iUnion
Set
Set.instMembership
β€”biUnion_iff
biUnion_iff πŸ“–mathematicalβ€”Set.Countable
Set.iUnion
Set
Set.instMembership
β€”to_subtype
Set.biUnion_eq_iUnion
Set.countable_iUnion_iff
SetCoe.forall'
exists_eq_range πŸ“–mathematicalSet.NonemptySet.rangeβ€”exists_surjective
Function.Surjective.range_comp
Subtype.range_coe
exists_surjective πŸ“–mathematicalSet.NonemptySet.Elemβ€”Set.countable_iff_exists_surjective
image πŸ“–mathematicalβ€”Set.Countable
Set.image
β€”Set.image_eq_range
to_subtype
Set.countable_range
image2 πŸ“–mathematicalβ€”Set.Countable
Set.image2
β€”Set.image_prod
image
prod
insert πŸ“–mathematicalβ€”Set.Countable
Set
Set.instInsert
β€”Set.countable_insert
mono πŸ“–mathematicalSet
Set.instHasSubset
Set.Countableβ€”to_subtype
Function.Injective.countable
Set.inclusion_injective
nonempty_encodable πŸ“–mathematicalβ€”Encodable
Set.Elem
β€”Set.countable_iff_nonempty_encodable
of_diff πŸ“–mathematicalβ€”Set.Countableβ€”mono
Set.subset_diff_union
union
of_preimage_singleton πŸ“–mathematicalSet.Countable
Set.preimage
Set
Set.instSingletonSet
Countableβ€”Set.preimage_iUnion
Set.countable_iUnion
of_subsingleton πŸ“–mathematicalβ€”Set.Countableβ€”Set.Finite.countable
Set.Finite.of_subsingleton
preimage πŸ“–mathematicalβ€”Set.Countable
Set.preimage
β€”preimage_of_injOn
Function.Injective.injOn
preimage_of_injOn πŸ“–mathematicalSet.InjOn
Set.preimage
Set.Countableβ€”Set.MapsTo.countable_of_injOn
Set.mapsTo_preimage
prod πŸ“–mathematicalβ€”Set.Countable
SProd.sprod
Set
Set.instSProd
β€”to_subtype
Countable.of_equiv
instCountableProd
sUnion πŸ“–mathematicalSet.CountableSet.sUnionβ€”sUnion_iff
sUnion_iff πŸ“–mathematicalβ€”Set.Countable
Set.sUnion
β€”Set.sUnion_eq_biUnion
biUnion_iff
setOf_finite πŸ“–mathematicalβ€”Set.Countable
Set
setOf
Set.Finite
β€”Set.countable_setOf_finite_subset
Set.countable_univ
to_subtype πŸ“–mathematicalβ€”Countable
Set.Elem
β€”Set.countable_coe_iff
union πŸ“–mathematicalβ€”Set.Countable
Set
Set.instUnion
β€”Set.countable_union

Set.Finite

Theorems

NameKindAssumesProvesValidatesDepends On
countable πŸ“–mathematicalβ€”Set.Countableβ€”to_subtype
Set.to_countable
Finite.to_countable

Set.MapsTo

Theorems

NameKindAssumesProvesValidatesDepends On
countable_of_injOn πŸ“–mathematicalSet.MapsTo
Set.InjOn
Set.Countableβ€”Set.Countable.to_subtype
Function.Injective.codRestrict
Set.injOn_iff_injective
Function.Injective.countable

Set.Subsingleton

Theorems

NameKindAssumesProvesValidatesDepends On
countable πŸ“–mathematicalSet.SubsingletonSet.Countableβ€”Set.Finite.countable
finite

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