Documentation Verification Report

Antichain

📁 Source: Mathlib/Order/Antichain.lean

Statistics

MetricCount
DefinitionsIsAntichain, IsMaxAntichain, IsStrongAntichain, IsWeakAntichain
4
Theoremsantisymm, bot_mem_iff, diff, empty, eq, eq', eq_setOf_maximal, eq_setOf_minimal, flip, greatest_iff, image, image_compl, image_embedding, image_embedding_iff, image_iso, image_iso_iff, image_relEmbedding, image_relEmbedding_iff, image_relIso, image_relIso_iff, insert, insert_of_symmetric, isAntisymm, isWeakAntichain, least_iff, maximal_mem_iff, minimal_mem_iff, mono, mono_on, not_lt, of_monotoneOn_strictAntiOn, of_strictMonoOn_antitoneOn, preimage, preimage_compl, preimage_embedding, preimage_iso, preimage_iso_iff, preimage_relEmbedding, preimage_relIso, singleton, subset, subsingleton, swap, to_dual, to_dual_iff, top_mem_iff, antichain_iff, antichain_iff, image, isAntichain, isEmpty_iff, nonempty_iff, eq, flip, image, insert, isAntichain, mono, preimage, subset, subsingleton, swap, eq, insert, subset, isAntichain, isStrongAntichain, isWeakAntichain, compl, inter_subsingleton_of_isAntichain_of_isChain, inter_subsingleton_of_isChain_of_isAntichain, isAntichain_and_greatest_iff, isAntichain_and_least_iff, isAntichain_coe_univ_iff, isAntichain_iff_forall_not_lt, isAntichain_insert, isAntichain_insert_of_symmetric, isAntichain_preimage_subtypeVal, isAntichain_singleton, isAntichain_union, isChain_and_isAntichain_iff_subsingleton, isStrongAntichain_insert, isWeakAntichain_insert, setOf_maximal_antichain, setOf_minimal_antichain, subsingleton_of_isChain_of_isAntichain
86
Total90

IsAntichain

Theorems

NameKindAssumesProvesValidatesDepends On
antisymm 📖IsAntichain
Set.univ
eq
bot_mem_iff 📖mathematicalIsAntichain
Preorder.toLE
Set
Set.instMembership
Bot.bot
OrderBot.toBot
Set.instSingletonSet
isLeast_bot_iff
least_iff
diff 📖mathematicalIsAntichainSet
Set.instSDiff
subset
Set.diff_subset
empty 📖mathematicalIsAntichain
Set
Set.instEmptyCollection
Set.pairwise_empty
eq 📖IsAntichain
Set
Set.instMembership
Set.Pairwise.eq
eq' 📖IsAntichain
Set
Set.instMembership
eq
eq_setOf_maximal 📖mathematicalIsAntichain
Preorder.toLE
Set
Set.instMembership
Maximal
setOfSet.ext
eq
eq_setOf_minimal 📖mathematicalIsAntichain
Preorder.toLE
Set
Set.instMembership
Minimal
setOfeq_setOf_maximal
to_dual
flip 📖IsAntichain
greatest_iff 📖mathematicalIsAntichain
Preorder.toLE
IsGreatest
Set
Set.instSingletonSet
isAntichain_and_greatest_iff
image 📖mathematicalIsAntichainSet.image
image_compl 📖mathematicalIsAntichain
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.image
Compl.compl
BooleanAlgebra.toCompl
flip
image_embedding
image_embedding 📖mathematicalIsAntichainSet.image
DFunLike.coe
OrderEmbedding
instFunLikeOrderEmbedding
image_relEmbedding
image_embedding_iff 📖mathematicalIsAntichain
Set.image
DFunLike.coe
OrderEmbedding
instFunLikeOrderEmbedding
image_relEmbedding_iff
image_iso 📖mathematicalIsAntichainSet.image
DFunLike.coe
OrderIso
instFunLikeOrderIso
image_relEmbedding
image_iso_iff 📖mathematicalIsAntichain
Set.image
DFunLike.coe
OrderIso
instFunLikeOrderIso
image_relEmbedding_iff
image_relEmbedding 📖mathematicalIsAntichainSet.image
DFunLike.coe
RelEmbedding
RelEmbedding.instFunLike
Set.mem_image
RelEmbedding.map_rel_iff
image_relEmbedding_iff 📖mathematicalIsAntichain
Set.image
DFunLike.coe
RelEmbedding
RelEmbedding.instFunLike
Function.Injective.preimage_image
RelEmbedding.injective
preimage_relEmbedding
image_relEmbedding
image_relIso 📖mathematicalIsAntichainSet.image
DFunLike.coe
RelIso
RelIso.instFunLike
image_relEmbedding
image_relIso_iff 📖mathematicalIsAntichain
Set.image
DFunLike.coe
RelIso
RelIso.instFunLike
image_relEmbedding_iff
insert 📖mathematicalIsAntichainSet
Set.instInsert
isAntichain_insert
insert_of_symmetric 📖mathematicalIsAntichain
Symmetric
Set
Set.instInsert
isAntichain_insert_of_symmetric
isAntisymm 📖antisymm
isWeakAntichain 📖mathematicalIsAntichain
Pi.hasLe
Preorder.toLE
IsWeakAntichainmono
le_of_strongLT
least_iff 📖mathematicalIsAntichain
Preorder.toLE
IsLeast
Set
Set.instSingletonSet
isAntichain_and_least_iff
maximal_mem_iff 📖mathematicalIsAntichain
Preorder.toLE
Maximal
Set
Set.instMembership
minimal_mem_iff
to_dual
minimal_mem_iff 📖mathematicalIsAntichain
Preorder.toLE
Minimal
Set
Set.instMembership
Minimal.prop
Eq.le
eq
mono 📖IsAntichain
Pi.hasLe
Prop.le
Set.Pairwise.mono'
compl_le_compl
mono_on 📖IsAntichain
Set.Pairwise
Set.Pairwise.imp_on
Set.Pairwise.imp
not_lt 📖mathematicalIsAntichain
Preorder.toLE
Set
Set.instMembership
Preorder.toLTLT.lt.ne
LT.lt.le
of_monotoneOn_strictAntiOn 📖mathematicalMonotoneOn
PartialOrder.toPreorder
StrictAntiOn
IsAntichain
Preorder.toLE
LE.le.not_gt
LE.le.lt_of_ne
of_strictMonoOn_antitoneOn 📖mathematicalStrictMonoOn
PartialOrder.toPreorder
AntitoneOn
IsAntichain
Preorder.toLE
LT.lt.not_ge
LE.le.lt_of_ne
preimage 📖mathematicalIsAntichainSet.preimage
preimage_compl 📖mathematicalIsAntichain
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.preimage
Compl.compl
BooleanAlgebra.toCompl
compl_inj_iff
compl_le_compl
preimage_embedding 📖mathematicalIsAntichainSet.preimage
DFunLike.coe
OrderEmbedding
instFunLikeOrderEmbedding
preimage_relEmbedding
preimage_iso 📖mathematicalIsAntichainSet.preimage
DFunLike.coe
OrderIso
instFunLikeOrderIso
preimage_relEmbedding
preimage_iso_iff 📖mathematicalIsAntichain
Set.preimage
DFunLike.coe
OrderIso
instFunLikeOrderIso
OrderIso.image_preimage
image_iso
preimage_iso
preimage_relEmbedding 📖mathematicalIsAntichainSet.preimage
DFunLike.coe
RelEmbedding
RelEmbedding.instFunLike
RelEmbedding.injective
RelEmbedding.map_rel_iff
preimage_relIso 📖mathematicalIsAntichainSet.preimage
DFunLike.coe
RelIso
RelIso.instFunLike
preimage_relEmbedding
singleton 📖mathematicalIsAntichain
Set
Set.instSingletonSet
Set.pairwise_singleton
subset 📖IsAntichain
Set
Set.instHasSubset
Set.Pairwise.mono
subsingleton 📖mathematicalIsAntichainSet.Subsingletontrichotomous_of
eq
eq'
swap 📖mathematicalIsAntichainFunction.swapflip
to_dual 📖mathematicalIsAntichainOrderDual
OrderDual.instLE
to_dual_iff 📖mathematicalIsAntichain
OrderDual
OrderDual.instLE
to_dual
top_mem_iff 📖mathematicalIsAntichain
Preorder.toLE
Set
Set.instMembership
Top.top
OrderTop.toTop
Set.instSingletonSet
isGreatest_top_iff
greatest_iff

IsGreatest

Theorems

NameKindAssumesProvesValidatesDepends On
antichain_iff 📖mathematicalIsGreatest
Preorder.toLE
IsAntichain
Set
Set.instSingletonSet
isAntichain_and_greatest_iff

IsLeast

Theorems

NameKindAssumesProvesValidatesDepends On
antichain_iff 📖mathematicalIsLeast
Preorder.toLE
IsAntichain
Set
Set.instSingletonSet
isAntichain_and_least_iff

IsMaxAntichain

Theorems

NameKindAssumesProvesValidatesDepends On
image 📖mathematicalIsMaxAntichainSet.image
DFunLike.coe
RelIso
RelIso.instFunLike
IsAntichain.image
isAntichain
RelIso.map_rel_iff'
RelIso.coe_fn_toEquiv
Equiv.eq_preimage_iff_image_eq
Equiv.image_symm_eq_preimage
RelIso.map_rel_iff
Equiv.subset_symm_image
isAntichain 📖mathematicalIsMaxAntichainIsAntichain
isEmpty_iff 📖mathematicalIsMaxAntichainIsEmpty
Set
Set.instEmptyCollection
Set.eq_empty_of_isEmpty
instIsEmptySubtype
Set.singleton_ne_empty
IsAntichain.singleton
nonempty_iff 📖mathematicalIsMaxAntichainSet.Nonemptynot_iff_not
isEmpty_iff

IsStrongAntichain

Theorems

NameKindAssumesProvesValidatesDepends On
eq 📖IsStrongAntichain
Set
Set.instMembership
Set.Pairwise.eq
flip 📖IsStrongAntichainsymm_of
image 📖mathematicalIsStrongAntichainSet.image
insert 📖mathematicalIsStrongAntichainSet
Set.instInsert
isStrongAntichain_insert
isAntichain 📖mathematicalIsStrongAntichainIsAntichainSet.Pairwise.imp
refl
mono 📖IsStrongAntichain
Pi.hasLe
Prop.le
Set.Pairwise.mono'
compl_le_compl
preimage 📖mathematicalIsStrongAntichainSet.preimage
subset 📖IsStrongAntichain
Set
Set.instHasSubset
Set.Pairwise.mono
subsingleton 📖mathematicalIsStrongAntichainSet.Subsingletondirected_of
eq
swap 📖mathematicalIsStrongAntichainFunction.swapflip

IsWeakAntichain

Theorems

NameKindAssumesProvesValidatesDepends On
eq 📖IsWeakAntichain
Set
Set.instMembership
StrongLT
Preorder.toLT
IsAntichain.eq
insert 📖mathematicalIsWeakAntichain
StrongLT
Preorder.toLT
Set
Set.instInsert
IsAntichain.insert
subset 📖IsWeakAntichain
Set
Set.instHasSubset
IsAntichain.subset

Set.Subsingleton

Theorems

NameKindAssumesProvesValidatesDepends On
isAntichain 📖mathematicalSet.SubsingletonIsAntichainpairwise
isStrongAntichain 📖mathematicalSet.SubsingletonIsStrongAntichainpairwise
isWeakAntichain 📖mathematicalSet.SubsingletonIsWeakAntichainisAntichain

Symmetric

Theorems

NameKindAssumesProvesValidatesDepends On
compl 📖mathematicalSymmetricCompl.compl
Pi.instCompl
Prop.instCompl

(root)

Definitions

NameCategoryTheorems
IsAntichain 📖MathDef
32 mathmath: isAntichain_union, IsAntichain.image_relIso_iff, IsAntichain.preimage_iso_iff, isAntichain_and_least_iff, IsAntichain.of_strictMonoOn_antitoneOn, isAntichain_singleton, IsAntichain.to_dual_iff, isChain_and_isAntichain_iff_subsingleton, Set.Subsingleton.isAntichain, IsLeast.antichain_iff, IsAntichain.singleton, IsStrongAntichain.isAntichain, isAntichain_preimage_subtypeVal, IsAntichain.of_monotoneOn_strictAntiOn, isAntichain_insert, IsGreatest.antichain_iff, isAntichain_iff_forall_not_lt, IsAntichain.image_embedding_iff, isAntichain_insert_of_symmetric, SimpleGraph.isIndepSet_iff_isAntichain_adj, IsAntichain.image_iso_iff, SimpleGraph.Coloring.color_classes_independent, isAntichain_and_greatest_iff, setOf_minimal_antichain, Matroid.isCircuit_antichain, IsAntichain.empty, setOf_maximal_antichain, SimpleGraph.Partition.independent, IsAntichain.image_relEmbedding_iff, IsMaxAntichain.isAntichain, isAntichain_coe_univ_iff, Set.Sized.isAntichain
IsMaxAntichain 📖MathDef
IsStrongAntichain 📖MathDef
2 mathmath: isStrongAntichain_insert, Set.Subsingleton.isStrongAntichain
IsWeakAntichain 📖MathDef
3 mathmath: Set.Subsingleton.isWeakAntichain, IsAntichain.isWeakAntichain, isWeakAntichain_insert

Theorems

NameKindAssumesProvesValidatesDepends On
inter_subsingleton_of_isAntichain_of_isChain 📖mathematicalIsAntichain
IsChain
Set.Subsingleton
Set
Set.instInter
inter_subsingleton_of_isChain_of_isAntichain
Set.inter_comm
inter_subsingleton_of_isChain_of_isAntichain 📖mathematicalIsChain
IsAntichain
Set.Subsingleton
Set
Set.instInter
subsingleton_of_isChain_of_isAntichain
IsChain.mono
IsAntichain.subset
isAntichain_and_greatest_iff 📖mathematicalIsAntichain
Preorder.toLE
IsGreatest
Set
Set.instSingletonSet
Set.eq_singleton_iff_unique_mem
IsAntichain.eq
IsAntichain.singleton
isGreatest_singleton
isAntichain_and_least_iff 📖mathematicalIsAntichain
Preorder.toLE
IsLeast
Set
Set.instSingletonSet
Set.eq_singleton_iff_unique_mem
IsAntichain.eq'
IsAntichain.singleton
isLeast_singleton
isAntichain_coe_univ_iff 📖mathematicalIsAntichain
Set.Elem
Set
Set.instMembership
Set.univ
Set.inter_univ
isAntichain_preimage_subtypeVal
isAntichain_iff_forall_not_lt 📖mathematicalIsAntichain
Preorder.toLE
PartialOrder.toPreorder
Preorder.toLT
IsAntichain.not_lt
LE.le.lt_of_ne
isAntichain_insert 📖mathematicalIsAntichain
Set
Set.instInsert
Set.pairwise_insert
isAntichain_insert_of_symmetric 📖mathematicalSymmetricIsAntichain
Set
Set.instInsert
Set.pairwise_insert_of_symmetric
Symmetric.compl
isAntichain_preimage_subtypeVal 📖mathematicalIsAntichain
Set.Elem
Set
Set.instMembership
Set.preimage
Set.instInter
isAntichain_singleton 📖mathematicalIsAntichain
Set
Set.instSingletonSet
IsAntichain.singleton
isAntichain_union 📖mathematicalIsAntichain
Set
Set.instUnion
Compl.compl
Pi.instCompl
Prop.instCompl
IsAntichain.eq_1
Set.pairwise_union
isChain_and_isAntichain_iff_subsingleton 📖mathematicalIsChain
IsAntichain
Set.Subsingleton
subsingleton_of_isChain_of_isAntichain
Set.Subsingleton.isChain
Set.Subsingleton.isAntichain
isStrongAntichain_insert 📖mathematicalIsStrongAntichain
Set
Set.instInsert
Set.pairwise_insert_of_symmetric
isWeakAntichain_insert 📖mathematicalIsWeakAntichain
Set
Set.instInsert
StrongLT
Preorder.toLT
isAntichain_insert
setOf_maximal_antichain 📖mathematicalIsAntichain
Preorder.toLE
PartialOrder.toPreorder
setOf
Maximal
LE.le.antisymm
setOf_minimal_antichain 📖mathematicalIsAntichain
Preorder.toLE
PartialOrder.toPreorder
setOf
Minimal
IsAntichain.swap
setOf_maximal_antichain
subsingleton_of_isChain_of_isAntichain 📖mathematicalIsChain
IsAntichain
Set.Subsingleton

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