Documentation Verification Report

Basic

πŸ“ Source: Mathlib/Data/Set/Finite/Basic.lean

Statistics

MetricCount
DefinitionsfinsetSetFinite, fintype, inhabited, subtypeEquivToFinset, toFinset, natEmbedding, fintypeIio, fintypeDiff, fintypeDiffLeft, fintypeEmpty, fintypeImage, fintypeInsert, fintypeInsert', fintypeInsertOfMem, fintypeInsertOfNotMem, fintypeInter, fintypeInterOfLeft, fintypeInterOfRight, fintypeLENat, fintypeLTNat, fintypeMap, fintypeMemFinset, fintypeOfFiniteUniv, fintypeOfFintypeImage, fintypeSep, fintypeSingleton, fintypeSubset, fintypeTop, fintypeUnion, fintypeUniv
30
Theoremsfinite_diff, finite_image, finite_insert, finite_inter_of_left, finite_inter_of_right, finite_sep, finite_union, subset, of_finite_univ, of_forall_not_lt_lt, exists, exists_card_eq, exists_notMem, exists_subset_injOn_image_eq_of_surjOn, finite_toSet, finite_toSet_toFinset, forall, finite_toSet, finite_toSet, finite_toSet_toFinset, finsetSetFinite_apply_coe, finsetSetFinite_symm_apply, finite_iff_finite, card_toFinset, coeSort_toFinset, coe_toFinset, diff, disjoint_toFinset, exists_finset, exists_finset_coe, exists_notMem, finite_of_compl, image, induction_on, induction_on_subset, inf_of_left, inf_of_right, infinite_compl, injOn_iff_bijOn_of_mapsTo, insert, inter_of_left, inter_of_right, map, mem_toFinset, nonempty_fintype, ofFinset, of_diff, of_finite_image, of_injOn, of_preimage, of_subsingleton, of_surjOn, preimage, preimage_embedding, sep, ssubset_toFinset, subset, subset_toFinset, subtypeEquivToFinset_apply_coe, subtypeEquivToFinset_symm_apply_coe, sup, surjOn_iff_bijOn_of_mapsTo, symmDiff, symmDiff_congr, toFinset_compl, toFinset_diff, toFinset_empty, toFinset_eq_empty, toFinset_eq_toFinset, toFinset_eq_univ, toFinset_image, toFinset_inj, toFinset_insert, toFinset_insert', toFinset_inter, toFinset_mono, toFinset_nonempty, toFinset_nontrivial, toFinset_range, toFinset_setOf, toFinset_singleton, toFinset_ssubset, toFinset_ssubset_toFinset, toFinset_strictMono, toFinset_subset, toFinset_subset_toFinset, toFinset_symmDiff, toFinset_union, toFinset_univ, union, diff, exists_ne_map_eq_of_mapsTo, exists_notMem_finite, exists_notMem_finset, exists_subset_card_eq, image, mono, nonempty, nontrivial, of_image, preimage, preimage', finite, card_empty, card_fintypeInsertOfNotMem, card_image_of_inj_on, card_image_of_injective, card_insert, card_le_card, card_lt_card, card_ne_eq, card_range_of_injective, card_singleton, eq_of_subset_of_card_le, exists_finite_iff_finset, exists_subset_image_finite_and, finite_def, finite_empty, finite_image_iff, finite_insert, finite_le_nat, finite_lt_nat, finite_mem_finset, finite_of_finite_preimage, finite_of_forall_not_lt_lt, finite_option, finite_preimage_inl_and_inr, finite_range_const, finite_range_findGreatest, finite_range_iff, finite_range_ite, finite_singleton, finite_union, finite_univ, finite_univ_iff, infinite_image_iff, infinite_of_finite_compl, infinite_of_injOn_mapsTo, infinite_of_injective_forall_mem, infinite_range_iff, infinite_range_of_injective, infinite_union, infinite_univ, infinite_univ_iff, instCanLiftFinsetCoeFinite, not_injOn_infinite_finite_image, seq_of_forall_finite_exists, toFinite_toFinset, univ_finite_iff_nonempty_fintype, instWellFoundedLTSubtypeSetFinite
150
Total180

Finite

Theorems

NameKindAssumesProvesValidatesDepends On
of_finite_univ πŸ“–mathematicalβ€”Finiteβ€”Set.finite_univ_iff
of_forall_not_lt_lt πŸ“–mathematicalβ€”Finiteβ€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
of_subsingleton
exists_pair_ne
of_fintype
eq_or_eq_or_eq_of_forall_not_lt_lt

Finite.Set

Theorems

NameKindAssumesProvesValidatesDepends On
finite_diff πŸ“–mathematicalβ€”Finite
Set.Elem
Set
Set.instSDiff
β€”subset
Set.diff_subset
finite_image πŸ“–mathematicalβ€”Finite
Set.Elem
Set.image
β€”nonempty_fintype
Finite.of_fintype
finite_insert πŸ“–mathematicalβ€”Finite
Set.Elem
Set
Set.instInsert
β€”finite_union
Finite.of_fintype
finite_inter_of_left πŸ“–mathematicalβ€”Finite
Set.Elem
Set
Set.instInter
β€”subset
Set.inter_subset_left
finite_inter_of_right πŸ“–mathematicalβ€”Finite
Set.Elem
Set
Set.instInter
β€”subset
Set.inter_subset_right
finite_sep πŸ“–mathematicalβ€”Finite
Set.Elem
setOf
Set
Set.instMembership
β€”nonempty_fintype
Finite.of_fintype
finite_union πŸ“–mathematicalβ€”Finite
Set.Elem
Set
Set.instUnion
β€”nonempty_fintype
Finite.of_fintype
subset πŸ“–mathematicalSet
Set.instHasSubset
Finite
Set.Elem
β€”Set.sep_eq_of_subset
finite_sep

Finset

Theorems

NameKindAssumesProvesValidatesDepends On
exists πŸ“–mathematicalβ€”Set.Finite.toFinsetβ€”finite_toSet
Set.toFinite
Finite.of_fintype
Set.toFinite_toFinset
toFinset_coe
exists_card_eq πŸ“–mathematicalβ€”cardβ€”card_empty
exists_notMem
card_insert_of_notMem
exists_notMem πŸ“–mathematicalβ€”Finset
instMembership
β€”Set.Finite.exists_notMem
finite_toSet
exists_subset_injOn_image_eq_of_surjOn πŸ“–mathematicalSet.SurjOn
SetLike.coe
Finset
instSetLike
Set
Set.instHasSubset
Set.InjOn
image
β€”Set.SurjOn.exists_subset_injOn_image_eq
Set.Finite.of_finite_image
Set.Finite.coe_toFinset
coe_image
finite_toSet πŸ“–mathematicalβ€”Set.Finite
SetLike.coe
Finset
instSetLike
β€”Set.toFinite
Finite.of_fintype
finite_toSet_toFinset πŸ“–mathematicalβ€”Set.Finite.toFinset
SetLike.coe
Finset
instSetLike
finite_toSet
β€”finite_toSet
Set.toFinite
Finite.of_fintype
Set.toFinite_toFinset
toFinset_coe
forall πŸ“–mathematicalβ€”Set.Finite.toFinsetβ€”finite_toSet
Set.toFinite
Finite.of_fintype
Set.toFinite_toFinset
toFinset_coe

List

Theorems

NameKindAssumesProvesValidatesDepends On
finite_toSet πŸ“–mathematicalβ€”Set.Finite
setOf
β€”Multiset.finite_toSet

Multiset

Theorems

NameKindAssumesProvesValidatesDepends On
finite_toSet πŸ“–mathematicalβ€”Set.Finite
setOf
Multiset
instMembership
β€”Finset.finite_toSet
finite_toSet_toFinset πŸ“–mathematicalβ€”Set.Finite.toFinset
setOf
Multiset
instMembership
finite_toSet
toFinset
β€”Finset.ext
finite_toSet

OrderIso

Definitions

NameCategoryTheorems
finsetSetFinite πŸ“–CompOp
2 mathmath: finsetSetFinite_symm_apply, finsetSetFinite_apply_coe

Theorems

NameKindAssumesProvesValidatesDepends On
finsetSetFinite_apply_coe πŸ“–mathematicalβ€”Set
Set.Finite
DFunLike.coe
RelIso
Finset
Preorder.toLE
PartialOrder.toPreorder
Finset.partialOrder
Set.instLE
RelIso.instFunLike
finsetSetFinite
SetLike.coe
Finset.instSetLike
β€”β€”
finsetSetFinite_symm_apply πŸ“–mathematicalβ€”DFunLike.coe
RelIso
Set
Set.Finite
Finset
Set.instLE
Preorder.toLE
PartialOrder.toPreorder
Finset.partialOrder
RelIso.instFunLike
RelIso.symm
finsetSetFinite
Set.Finite.toFinset
β€”β€”

Set

Definitions

NameCategoryTheorems
fintypeDiff πŸ“–CompOpβ€”
fintypeDiffLeft πŸ“–CompOpβ€”
fintypeEmpty πŸ“–CompOp
1 mathmath: card_empty
fintypeImage πŸ“–CompOpβ€”
fintypeInsert πŸ“–CompOpβ€”
fintypeInsert' πŸ“–CompOpβ€”
fintypeInsertOfMem πŸ“–CompOpβ€”
fintypeInsertOfNotMem πŸ“–CompOp
1 mathmath: card_fintypeInsertOfNotMem
fintypeInter πŸ“–CompOpβ€”
fintypeInterOfLeft πŸ“–CompOp
2 mathmath: hasSum_sum_support_of_ne_finset_zero, hasProd_prod_support_of_ne_finset_one
fintypeInterOfRight πŸ“–CompOpβ€”
fintypeLENat πŸ“–CompOpβ€”
fintypeLTNat πŸ“–CompOpβ€”
fintypeMap πŸ“–CompOpβ€”
fintypeMemFinset πŸ“–CompOpβ€”
fintypeOfFiniteUniv πŸ“–CompOpβ€”
fintypeOfFintypeImage πŸ“–CompOpβ€”
fintypeSep πŸ“–CompOpβ€”
fintypeSingleton πŸ“–CompOp
21 mathmath: UniformOnFun.isometry_restrict, IsCyclotomicExtension.Rat.nrComplexPlaces_eq_totient_div_two, IsCyclotomicExtension.Rat.absdiscr_prime_pow, lipschitzOnWith_cfc_fun_of_subset, IsCyclotomicExtension.Rat.discr, lipschitzOnWith_cfc_fun, InfiniteGalois.proj_adjoin_singleton_val, IsCyclotomicExtension.Rat.absdiscr_prime_pow_succ, IsCyclotomicExtension.Rat.discr_prime_pow_succ, lipschitzOnWith_cfcβ‚™_fun, IsCyclotomicExtension.Rat.natAbs_discr, IsCyclotomicExtension.Rat.absdiscr_prime, IsCyclotomicExtension.Rat.discr_prime_pow, lipschitzOnWith_cfcβ‚™_fun_of_subset, FiniteGaloisIntermediateField.adjoin_simple_map_algEquiv, IsCyclotomicExtension.Rat.nrRealPlaces_eq_zero, FiniteGaloisIntermediateField.adjoin_simple_map_algHom, UniformOnFun.edist_continuousRestrict_of_singleton, IsCyclotomicExtension.Rat.discr_prime, card_singleton, FiniteGaloisIntermediateField.adjoin_simple_le_iff
fintypeSubset πŸ“–CompOpβ€”
fintypeTop πŸ“–CompOpβ€”
fintypeUnion πŸ“–CompOpβ€”
fintypeUniv πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
card_empty πŸ“–mathematicalβ€”Fintype.card
Elem
Set
instEmptyCollection
fintypeEmpty
β€”β€”
card_fintypeInsertOfNotMem πŸ“–mathematicalSet
instMembership
Fintype.card
Elem
instInsert
fintypeInsertOfNotMem
β€”Multiset.nodup_cons
Fintype.card_ofFinset
Finset.card_cons
toFinset_card
card_image_of_inj_on πŸ“–mathematicalβ€”Fintype.card
Elem
image
β€”Fintype.card_of_finset'
Finset.card_image_of_injOn
mem_toFinset
card_image_of_injective πŸ“–mathematicalβ€”Fintype.card
Elem
image
β€”card_image_of_inj_on
card_insert πŸ“–mathematicalSet
instMembership
Fintype.card
Elem
instInsert
β€”card_fintypeInsertOfNotMem
Fintype.card_congr'
card_le_card πŸ“–mathematicalSet
instHasSubset
Fintype.card
Elem
β€”Fintype.card_le_of_injective
inclusion_injective
card_lt_card πŸ“–mathematicalSet
instHasSSubset
Fintype.card
Elem
β€”Fintype.card_lt_of_injective_not_surjective
inclusion_injective
ssubset_iff_subset_ne
instIsNonstrictStrictOrderSubsetSSubset
instAntisymmSubset
eq_of_inclusion_surjective
card_ne_eq πŸ“–mathematicalβ€”Fintype.card
Elem
setOf
β€”toFinset_card
toFinset_setOf
Finset.filter_ne'
Finset.card_erase_of_mem
Finset.mem_univ
Finset.card_univ
card_range_of_injective πŸ“–mathematicalβ€”Fintype.card
Elem
range
β€”Fintype.card_congr
card_singleton πŸ“–mathematicalβ€”Fintype.card
Elem
Set
instSingletonSet
fintypeSingleton
β€”β€”
eq_of_subset_of_card_le πŸ“–β€”Set
instHasSubset
Fintype.card
Elem
β€”β€”eq_or_ssubset_of_subset
instIsNonstrictStrictOrderSubsetSSubset
instAntisymmSubset
not_le_of_gt
card_lt_card
exists_finite_iff_finset πŸ“–mathematicalβ€”Finite
SetLike.coe
Finset
Finset.instSetLike
β€”Finite.coe_toFinset
Finset.finite_toSet
exists_subset_image_finite_and πŸ“–mathematicalβ€”Set
instHasSubset
image
Finite
β€”Finset.coe_image
finite_def πŸ“–mathematicalβ€”Finite
Fintype
Elem
β€”finite_iff_nonempty_fintype
finite_empty πŸ“–mathematicalβ€”Finite
Set
instEmptyCollection
β€”toFinite
Finite.of_fintype
finite_image_iff πŸ“–mathematicalInjOnFinite
image
β€”Finite.of_finite_image
Finite.image
finite_insert πŸ“–mathematicalβ€”Finite
Set
instInsert
β€”Finite.subset
subset_insert
Finite.insert
finite_le_nat πŸ“–mathematicalβ€”Finite
setOf
β€”toFinite
Finite.of_fintype
finite_lt_nat πŸ“–mathematicalβ€”Finite
setOf
β€”toFinite
Finite.of_fintype
finite_mem_finset πŸ“–mathematicalβ€”Finite
setOf
Finset
Finset.instMembership
β€”toFinite
Finite.of_fintype
finite_of_finite_preimage πŸ“–mathematicalSet
instHasSubset
range
Finiteβ€”image_preimage_eq_of_subset
Finite.image
finite_of_forall_not_lt_lt πŸ“–mathematicalβ€”Finiteβ€”toFinite
Finite.of_forall_not_lt_lt
finite_option πŸ“–mathematicalβ€”Finite
setOf
Set
instMembership
β€”Finite.preimage_embedding
Finite.subset
Finite.insert
Finite.image
mem_image_of_mem
finite_preimage_inl_and_inr πŸ“–mathematicalβ€”Finite
preimage
β€”Finite.union
Finite.image
image_preimage_inl_union_image_preimage_inr
Finite.preimage
Function.Injective.injOn
Sum.inl_injective
Sum.inr_injective
finite_range_const πŸ“–mathematicalβ€”Finite
range
β€”Finite.subset
finite_singleton
range_const_subset
finite_range_findGreatest πŸ“–mathematicalβ€”Finite
range
Nat.findGreatest
β€”Finite.subset
finite_le_nat
range_subset_iff
Nat.findGreatest_le
finite_range_iff πŸ“–mathematicalβ€”Finite
range
Finite
β€”image_univ
finite_image_iff
Function.Injective.injOn
finite_range_ite πŸ“–mathematicalβ€”Finite
range
β€”Finite.subset
Finite.union
range_ite_subset
finite_singleton πŸ“–mathematicalβ€”Finite
Set
instSingletonSet
β€”toFinite
Finite.of_fintype
finite_union πŸ“–mathematicalβ€”Finite
Set
instUnion
β€”Finite.subset
subset_union_left
subset_union_right
Finite.union
finite_univ πŸ“–mathematicalβ€”Finite
univ
β€”toFinite
Subtype.finite
finite_univ_iff πŸ“–mathematicalβ€”Finite
univ
Finite
β€”Equiv.finite_iff
infinite_image_iff πŸ“–mathematicalInjOnInfinite
image
β€”finite_image_iff
infinite_of_finite_compl πŸ“–mathematicalβ€”Infiniteβ€”infinite_univ
compl_union_self
Finite.union
infinite_of_injOn_mapsTo πŸ“–β€”InjOn
MapsTo
Infinite
β€”β€”Infinite.mono
mapsTo_iff_image_subset
infinite_image_iff
infinite_of_injective_forall_mem πŸ“–mathematicalSet
instMembership
Infiniteβ€”Infinite.mono
range_subset_iff
infinite_range_of_injective
infinite_range_iff πŸ“–mathematicalβ€”Infinite
range
Infinite
β€”Iff.not
finite_range_iff
infinite_range_of_injective πŸ“–mathematicalβ€”Infinite
range
β€”image_univ
infinite_image_iff
Function.Injective.injOn
infinite_univ
infinite_union πŸ“–mathematicalβ€”Infinite
Set
instUnion
β€”β€”
infinite_univ πŸ“–mathematicalβ€”Infinite
univ
β€”infinite_univ_iff
infinite_univ_iff πŸ“–mathematicalβ€”Infinite
univ
Infinite
β€”Infinite.eq_1
finite_univ_iff
not_finite_iff_infinite
instCanLiftFinsetCoeFinite πŸ“–mathematicalβ€”CanLift
Set
Finset
SetLike.coe
Finset.instSetLike
Finite
β€”Finite.exists_finset_coe
not_injOn_infinite_finite_image πŸ“–mathematicalInfiniteInjOnβ€”finite_coe_iff
infinite_coe_iff
not_injective_infinite_finite
Mathlib.Tactic.Contrapose.contraposeβ‚„
injective_codRestrict
injOn_iff_injective
seq_of_forall_finite_exists πŸ“–mathematicalβ€”image
Iio
Nat.instPreorder
β€”image_eq_range
Nat.strongRecOn'_beta
Finite.image
finite_lt_nat
finite_empty
Function.sometimes_spec
toFinite_toFinset πŸ“–mathematicalβ€”Finite.toFinset
toFinite
Finite.of_fintype
Elem
toFinset
β€”Finite.toFinset_eq_toFinset
toFinite
Finite.of_fintype
univ_finite_iff_nonempty_fintype πŸ“–mathematicalβ€”Finite
univ
Fintype
β€”finite_univ
Finite.of_fintype

Set.BijOn

Theorems

NameKindAssumesProvesValidatesDepends On
finite_iff_finite πŸ“–mathematicalSet.BijOnSet.Finiteβ€”Set.Finite.of_surjOn
Set.Finite.of_injOn

Set.Finite

Definitions

NameCategoryTheorems
fintype πŸ“–CompOp
3 mathmath: Equiv.finsuppUnique_symm_apply_support_val, AddEquiv.finsuppUnique_symm_apply_support_val, convexHull_eq_image
inhabited πŸ“–CompOpβ€”
subtypeEquivToFinset πŸ“–CompOp
2 mathmath: subtypeEquivToFinset_apply_coe, subtypeEquivToFinset_symm_apply_coe
toFinset πŸ“–CompOp
99 mathmath: toFinset_vsub, toFinset_setOf, toFinset_prod, einfsep, Finsupp.ofSupportFinite_support, toFinset_inj, LieModule.traceForm_eq_sum_genWeightSpaceOf, coeSort_toFinset, ENNReal.finset_card_const_le_le_of_tsum_le, Nat.nth_eq_zero, ssubset_toFinset, toFinset_eq_empty, finsum_def', toFinset_add, toFinset_insert', finsum_eq_sum_of_support_subset_of_finite, finsum_mem_eq_sum_filter, PiLp.dist_eq_card, mem_toFinset, finsum_eq_sum, infsep_of_nontrivial, Finset.finite_toSet_toFinset, toFinset_subset, infsep, OrderIso.finsetSetFinite_symm_apply, finsum_mem_eq_sum, Nat.lt_card_toFinset_of_nth_ne_zero, finsum_mem_eq_finite_toFinset_sum, HahnSeries.SummableFamily.smul_support_subset_prod, Nat.exists_lt_card_finite_nth_eq, toFinset_vadd_set, Set.toFinite_toFinset, Finset.forall, Finsupp.equivFunOnFinite_symm_apply_support, subtypeEquivToFinset_apply_coe, HahnSeries.SummableFamily.coeff_support, finprod_mem_eq_prod, finsum_cond_ne, toFinset_symmDiff, coe_toFinset, toFinset_ssubset_toFinset, toFinset_diff, finprod_def', toFinset_smul_set, card_toFinset, toFinset_ssubset, lp.norm_eq_card_dsupport, toFinset_eq_toFinset, toFinset_mono, finsum_def, finprod_cond_ne, subset_toFinset, toFinset_range, finprod_eq_prod_of_mulSupport_subset_of_finite, toFinset_offDiag, toFinset_image, Nat.exists_lt_card_nth_eq, finprod_mem_eq_prod_filter, toFinset_empty, toFinset_subset_toFinset, Nat.nth_injOn, toFinset_smul, Nat.count_le_card, exists_finite_card_le_of_finite_of_linearIndependent_of_span, toFinset_nontrivial, PiLp.edist_eq_card, toFinset_inter, toFinset_singleton, Nat.nth_strictMonoOn, Multiset.finite_toSet_toFinset, disjoint_toFinset, MeasureTheory.Measure.count_apply_finite', finprod_mem_eq_finite_toFinset_prod, encard_eq_coe_toFinset_card, Nat.nth_eq_getD_sort, Finset.exists, Nat.count_lt_card, Set.ncard_eq_toFinset_card, toFinset_vadd, finprod_def, toFinset_mul, MeasureTheory.Measure.count_apply_finite, LinearMap.trace_eq_sum_trace_restrict', toFinset_strictMono, toFinset_univ, subtypeEquivToFinset_symm_apply_coe, convexHull_eq, toFinset_union, toFinset_image2, toFinset_zero, toFinset_nonempty, toFinset_one, toFinset_insert, PiLp.norm_eq_card, toFinset_eq_univ, Nat.image_nth_Iio_card, finprod_eq_prod, toFinset_compl, Nat.card_eq_card_finite_toFinset

Theorems

NameKindAssumesProvesValidatesDepends On
card_toFinset πŸ“–mathematicalβ€”Finset.card
toFinset
Fintype.card
Set.Elem
β€”Fintype.card_of_finset'
mem_toFinset
coeSort_toFinset πŸ“–mathematicalβ€”Finset
SetLike.instMembership
Finset.instSetLike
toFinset
Set.Elem
β€”Finset.coe_sort_coe
coe_toFinset
coe_toFinset πŸ“–mathematicalβ€”SetLike.coe
Finset
Finset.instSetLike
toFinset
β€”Set.coe_toFinset
diff πŸ“–mathematicalβ€”Set.Finite
Set
Set.instSDiff
β€”subset
Set.diff_subset
disjoint_toFinset πŸ“–mathematicalβ€”Disjoint
Finset
Finset.partialOrder
Finset.instOrderBot
toFinset
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
β€”Set.disjoint_toFinset
exists_finset πŸ“–mathematicalβ€”Finset
Finset.instMembership
Set
Set.instMembership
β€”nonempty_fintype
Set.mem_toFinset
exists_finset_coe πŸ“–mathematicalβ€”SetLike.coe
Finset
Finset.instSetLike
β€”nonempty_fintype
Set.coe_toFinset
exists_notMem πŸ“–mathematicalβ€”Set
Set.instMembership
β€”Set.infinite_univ
subset
finite_of_compl πŸ“–mathematicalβ€”Finiteβ€”Set.finite_univ_iff
Set.union_compl_self
union
image πŸ“–mathematicalβ€”Set.Finite
Set.image
β€”to_subtype
Set.toFinite
Finite.Set.finite_image
induction_on πŸ“–β€”Set
Set.instEmptyCollection
Set.finite_empty
Set.instInsert
insert
β€”β€”Set.finite_empty
insert
CanLift.prf
Set.instCanLiftFinsetCoeFinite
Finset.cons_induction_on
Finset.coe_empty
Finset.coe_cons
Set.toFinite
Finite.of_fintype
induction_on_subset πŸ“–β€”Set
Set.instEmptyCollection
Set.finite_empty
Set.instInsert
insert
subset
β€”β€”Set.finite_empty
subset
insert
induction_on
Set.insert_subset_iff
Set.Subset.rfl
inf_of_left πŸ“–mathematicalβ€”Set.Finite
Set
SemilatticeInf.toMin
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
β€”inter_of_left
inf_of_right πŸ“–mathematicalβ€”Set.Finite
Set
SemilatticeInf.toMin
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
β€”inter_of_right
infinite_compl πŸ“–mathematicalβ€”Set.Infinite
Compl.compl
Set
Set.instCompl
β€”Set.infinite_univ
Set.union_compl_self
union
injOn_iff_bijOn_of_mapsTo πŸ“–mathematicalSet.MapsToSet.InjOn
Set.BijOn
β€”Set.finite_coe_iff
Set.MapsTo.restrict_surjective_iff
Finite.injective_iff_surjective
Set.BijOn.injOn
insert πŸ“–mathematicalβ€”Set.Finite
Set
Set.instInsert
β€”union
Set.finite_singleton
inter_of_left πŸ“–mathematicalβ€”Set.Finite
Set
Set.instInter
β€”subset
Set.inter_subset_left
inter_of_right πŸ“–mathematicalβ€”Set.Finite
Set
Set.instInter
β€”subset
Set.inter_subset_right
map πŸ“–mathematicalβ€”Set.Finite
Set
Set.instFunctor
β€”image
mem_toFinset πŸ“–mathematicalβ€”Finset
Finset.instMembership
toFinset
Set
Set.instMembership
β€”Set.mem_toFinset
nonempty_fintype πŸ“–mathematicalβ€”Fintype
Set.Elem
β€”Set.finite_def
ofFinset πŸ“–mathematicalFinset
Finset.instMembership
Set
Set.instMembership
Set.Finiteβ€”Set.toFinite
Finite.of_fintype
of_diff πŸ“–mathematicalβ€”Set.Finiteβ€”subset
union
Set.subset_diff_union
of_finite_image πŸ“–mathematicalSet.InjOnSet.Finiteβ€”to_subtype
Finite.of_injective
Set.BijOn.mapsTo
Set.InjOn.bijOn_image
Function.Bijective.injective
Set.BijOn.bijective
of_injOn πŸ“–mathematicalSet.MapsTo
Set.InjOn
Set.Finiteβ€”of_finite_image
subset
Set.image_subset_iff
of_preimage πŸ“–mathematicalβ€”Set.Finiteβ€”image
Function.Surjective.image_preimage
of_subsingleton πŸ“–mathematicalβ€”Set.Finiteβ€”Set.toFinite
Subtype.finite
Finite.of_subsingleton
of_surjOn πŸ“–mathematicalSet.SurjOnSet.Finiteβ€”subset
image
preimage πŸ“–mathematicalSet.InjOn
Set.preimage
Set.Finiteβ€”of_finite_image
subset
Set.image_preimage_subset
preimage_embedding πŸ“–mathematicalβ€”Set.Finite
Set.preimage
DFunLike.coe
Function.Embedding
Function.instFunLikeEmbedding
β€”preimage
Function.Embedding.injective
sep πŸ“–mathematicalβ€”Set.Finite
setOf
Set
Set.instMembership
β€”subset
Set.sep_subset
ssubset_toFinset πŸ“–mathematicalβ€”Finset
Finset.instHasSSubset
toFinset
Set
Set.instHasSSubset
SetLike.coe
Finset.instSetLike
β€”Finset.coe_ssubset
coe_toFinset
subset πŸ“–mathematicalSet
Set.instHasSubset
Set.Finiteβ€”to_subtype
Finite.Set.subset
subset_toFinset πŸ“–mathematicalβ€”Finset
Finset.instHasSubset
toFinset
Set
Set.instHasSubset
SetLike.coe
Finset.instSetLike
β€”Finset.coe_subset
coe_toFinset
subtypeEquivToFinset_apply_coe πŸ“–mathematicalβ€”Finset
Finset.instMembership
toFinset
DFunLike.coe
Equiv
Set
Set.instMembership
EquivLike.toFunLike
Equiv.instEquivLike
subtypeEquivToFinset
β€”β€”
subtypeEquivToFinset_symm_apply_coe πŸ“–mathematicalβ€”Set
Set.instMembership
DFunLike.coe
Equiv
Finset
Finset.instMembership
toFinset
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
subtypeEquivToFinset
β€”β€”
sup πŸ“–mathematicalβ€”Set.Finite
Set
SemilatticeSup.toMax
Lattice.toSemilatticeSup
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
β€”union
surjOn_iff_bijOn_of_mapsTo πŸ“–mathematicalSet.MapsToSet.SurjOn
Set.BijOn
β€”Set.finite_coe_iff
Finite.injective_iff_surjective
Set.MapsTo.restrict_surjective_iff
Set.BijOn.surjOn
symmDiff πŸ“–mathematicalβ€”Set.Finite
symmDiff
Set
SemilatticeSup.toMax
Lattice.toSemilatticeSup
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
Set.instSDiff
β€”union
diff
symmDiff_congr πŸ“–mathematicalβ€”Set.Finite
symmDiff
Set
SemilatticeSup.toMax
Lattice.toSemilatticeSup
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
Set.instSDiff
β€”subset
union
symmDiff_triangle
symmDiff_comm
toFinset_compl πŸ“–mathematicalβ€”toFinset
Compl.compl
Set
Set.instCompl
Finset
BooleanAlgebra.toCompl
Finset.booleanAlgebra
β€”Finset.ext
toFinset_diff πŸ“–mathematicalβ€”toFinset
Set
Set.instSDiff
Finset
Finset.instSDiff
β€”Finset.ext
toFinset_empty πŸ“–mathematicalβ€”toFinset
Set
Set.instEmptyCollection
Finset
Finset.instEmptyCollection
β€”β€”
toFinset_eq_empty πŸ“–mathematicalβ€”toFinset
Finset
Finset.instEmptyCollection
Set
Set.instEmptyCollection
β€”Set.toFinset_eq_empty
toFinset_eq_toFinset πŸ“–mathematicalβ€”toFinset
Set.toFinset
β€”toFinset.eq_1
Fintype.subsingleton
toFinset_eq_univ πŸ“–mathematicalβ€”toFinset
Finset.univ
Set.univ
β€”Set.toFinset_eq_univ
toFinset_image πŸ“–mathematicalβ€”toFinset
Set.image
Finset.image
β€”Finset.ext
toFinset_inj πŸ“–mathematicalβ€”toFinsetβ€”β€”
toFinset_insert πŸ“–mathematicalβ€”toFinset
Set
Set.instInsert
Finset
Finset.instInsert
subset
Set.subset_insert
β€”Finset.ext
subset
Set.subset_insert
toFinset_insert' πŸ“–mathematicalβ€”toFinset
Set
Set.instInsert
insert
Finset
Finset.instInsert
β€”toFinset_insert
insert
toFinset_inter πŸ“–mathematicalβ€”toFinset
Set
Set.instInter
Finset
Finset.instInter
β€”Finset.ext
toFinset_mono πŸ“–mathematicalSet
Set.instHasSubset
Finset
Finset.instHasSubset
toFinset
β€”toFinset_subset_toFinset
toFinset_nonempty πŸ“–mathematicalβ€”Finset.Nonempty
toFinset
Set.Nonempty
β€”Finset.coe_nonempty
coe_toFinset
toFinset_nontrivial πŸ“–mathematicalβ€”Finset.Nontrivial
toFinset
Set.Nontrivial
β€”Finset.Nontrivial.eq_1
coe_toFinset
toFinset_range πŸ“–mathematicalβ€”toFinset
Set.range
Finset.image
Finset.univ
β€”Finset.ext
toFinset_setOf πŸ“–mathematicalβ€”toFinset
setOf
Finset.filter
Finset.univ
β€”Set.toFinite
Finite.of_fintype
Set.toFinite_toFinset
Set.toFinset_setOf
toFinset_singleton πŸ“–mathematicalβ€”toFinset
Set
Set.instSingletonSet
Finset
Finset.instSingleton
β€”Set.toFinite_toFinset
toFinset_ssubset πŸ“–mathematicalβ€”Finset
Finset.instHasSSubset
toFinset
Set
Set.instHasSSubset
SetLike.coe
Finset.instSetLike
β€”Finset.coe_ssubset
coe_toFinset
toFinset_ssubset_toFinset πŸ“–mathematicalβ€”Finset
Finset.instHasSSubset
toFinset
Set
Set.instHasSSubset
β€”coe_toFinset
toFinset_strictMono πŸ“–mathematicalSet
Set.instHasSSubset
Finset
Finset.instHasSSubset
toFinset
β€”toFinset_ssubset_toFinset
toFinset_subset πŸ“–mathematicalβ€”Finset
Finset.instHasSubset
toFinset
Set
Set.instHasSubset
SetLike.coe
Finset.instSetLike
β€”Finset.coe_subset
coe_toFinset
toFinset_subset_toFinset πŸ“–mathematicalβ€”Finset
Finset.instHasSubset
toFinset
Set
Set.instHasSubset
β€”coe_toFinset
toFinset_symmDiff πŸ“–mathematicalβ€”toFinset
symmDiff
Set
SemilatticeSup.toMax
Lattice.toSemilatticeSup
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
Set.instSDiff
Finset
Finset.instLattice
Finset.instSDiff
β€”Finset.ext
toFinset_union πŸ“–mathematicalβ€”toFinset
Set
Set.instUnion
Finset
Finset.instUnion
β€”Finset.ext
toFinset_univ πŸ“–mathematicalβ€”toFinset
Set.univ
Finset.univ
β€”toFinset_setOf
Finset.filter_true
union πŸ“–mathematicalβ€”Set.Finite
Set
Set.instUnion
β€”Set.toFinite
Finite.Set.finite_union
eq_1

Set.Infinite

Definitions

NameCategoryTheorems
natEmbedding πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
diff πŸ“–mathematicalSet.InfiniteSet
Set.instSDiff
β€”Set.Finite.of_diff
exists_ne_map_eq_of_mapsTo πŸ“–mathematicalSet.Infinite
Set.MapsTo
Set
Set.instMembership
β€”Mathlib.Tactic.Contrapose.contrapose₁
Mathlib.Tactic.Push.not_and_eq
Set.infinite_of_injOn_mapsTo
not_imp_not
exists_notMem_finite πŸ“–mathematicalSet.InfiniteSet
Set.instMembership
β€”Set.Finite.subset
Mathlib.Tactic.Push.not_and_eq
exists_notMem_finset πŸ“–mathematicalSet.InfiniteSet
Set.instMembership
Finset
Finset.instMembership
β€”exists_notMem_finite
Finset.finite_toSet
exists_subset_card_eq πŸ“–mathematicalSet.InfiniteSet
Set.instHasSubset
SetLike.coe
Finset
Finset.instSetLike
Finset.card
β€”Finset.coe_map
Set.image_congr
Finset.coe_range
Subtype.coe_preimage_self
Finset.card_map
Finset.card_range
image πŸ“–mathematicalSet.InjOn
Set.Infinite
Set.imageβ€”Set.infinite_image_iff
mono πŸ“–β€”Set
Set.instHasSubset
Set.Infinite
β€”β€”Set.Finite.subset
nonempty πŸ“–mathematicalSet.InfiniteSet.Nonemptyβ€”Set.nonempty_iff_ne_empty
Set.finite_empty
nontrivial πŸ“–mathematicalSet.InfiniteSet.Nontrivialβ€”Set.not_subsingleton_iff
Set.Subsingleton.finite
of_image πŸ“–β€”Set.Infinite
Set.image
β€”β€”Set.Finite.image
preimage πŸ“–mathematicalSet.Infinite
Set
Set.instHasSubset
Set.range
Set.preimageβ€”Set.finite_of_finite_preimage
preimage' πŸ“–mathematicalSet.Infinite
Set
Set.instInter
Set.range
Set.preimageβ€”mono
Set.preimage_mono
Set.inter_subset_left
preimage
Set.inter_subset_right

Set.Nat

Definitions

NameCategoryTheorems
fintypeIio πŸ“–CompOpβ€”

Set.Subsingleton

Theorems

NameKindAssumesProvesValidatesDepends On
finite πŸ“–mathematicalSet.SubsingletonSet.Finiteβ€”induction_on
Set.finite_empty
Set.finite_singleton

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
instWellFoundedLTSubtypeSetFinite πŸ“–mathematicalβ€”WellFoundedLT
Set
Set.Finite
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
β€”OrderEmbedding.wellFoundedLT
Finset.wellFoundedLT

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