Documentation Verification Report

Atoms

📁 Source: Mathlib/Order/Atoms.lean

Statistics

MetricCount
DefinitionstoSetOfIsAtom, toCompleteAtomicBooleanAlgebra, IsAtom, IsAtomic, IsAtomistic, IsCoatom, IsCoatomic, IsCoatomistic, booleanAlgebra, completeBooleanAlgebra, completeLattice, distribLattice, equivBool, lattice, linearOrder, orderIsoBool, preorder, IsStronglyAtomic, IsStronglyCoatomic
19
Theoremseq_iff_atom_le_iff, le_iff_atom_le_imp, isStronglyAtomic, isStronglyAtomic', isStronglyCoatomic, isStronglyCoatomic', eq_setOf_le_sSup_and_isAtom, exists_mem_le_of_le_sSup_of_isAtom, instIsAtomistic, instIsCoatomistic, isAtomistic_iff, isCoatomistic_iff, isCoatom, is_atom, isAtom_iff, isAtom_of_image, isCoatom_iff, isCoatom_iff', isCoatom_of_l_top, isAtom_iff, isAtom_iff', isAtom_of_u_bot, isCoatom_iff, isCoatom_of_image, Iic, Iic_eq, bot_covBy, bot_lt, compl, disjoint_of_ne, dual, le_iSup, le_iff, le_iff_eq, le_sSup, lt_iff, ne_bot, ne_bot_iff_eq, ne_iff_eq_bot, not_disjoint_iff_le, not_le_iff_disjoint, of_compl, of_isAtom_coe_Iic, isAtomic, eq_bot_or_exists_atom_le, exists_atom, instIsAtomic, isLUB_atoms, Ici, Ici_eq, codisjoint_of_ne, compl, covBy_top, dual, iInf_le, le_iff, le_iff_eq, lt_iff, lt_top, ne_iff_eq_top, ne_top, ne_top_iff_eq, not_codisjoint_iff_le, not_le_iff_codisjoint, of_compl, of_isCoatom_coe_Ici, sInf_le, sup_eq_top_of_ne, isCoatomic, eq_top_or_exists_le_coatom, exists_coatom, instIsCoatomic, isGLB_coatoms, isAtom_iff_isCoatom, isCoatom_iff_isAtom, bot_lt_iff_eq_top, bot_ne_top, eq_bot_of_lt, eq_bot_or_eq_top, eq_top_of_lt, equivBool_apply, equivBool_symm_apply, instComplementedLattice, instIsAtomic, instIsAtomistic, instIsCoatomic, instIsCoatomistic, lt_top_iff_eq_bot, of_forall_eq_top, toNontrivial, exists_covBy_le_of_lt, isAtomic, of_wellFounded_lt, exists_le_covBy_of_lt, of_wellFounded_gt, toIsCoatomic, eq_bot, eq_top, exists_covby_le, exists_le_covby, isStronglyAtomic, isStronglyCoatomic, instIsAtomic, instIsAtomistic, instIsCoatomic, instIsCoatomistic, instIsSimpleOrder, instIsStronglyCoatomic, isAtom_of_map_bot_of_image, isCoatom_of_map_top_of_image, isAtom_iff, isAtomic_iff, isCoatom_iff, isCoatomic_iff, isSimpleOrder, isSimpleOrder_iff, eq_bot_iff, isAtom_iff, isAtom_iff_eq_single, isAtom_single, isAtomic, isAtomistic, isCoatomic, isCoatomistic, instIsSimpleOrderProp, isAtom_iff, isCoatom_iff, isAtom_iff, isCoatom_iff, isStronglyAtomic, isStronglyCoatomic, instIsAtomistic, instIsCoatomistic, isAtom_iff, isAtom_singleton, isCoatom_iff, isCoatom_singleton_compl, isSimpleOrder_Ici_iff_isCoatom, isSimpleOrder_Iic_iff_isAtom, covBy_iff, covBy_iff', isAtom_iff, isCoatom_iff, toIsStronglyAtomic, bot_covBy_iff, bot_covBy_top, covBy_iff_atom_Ici, covBy_iff_coatom_Iic, covBy_top_iff, eq_iff_atom_le_iff, eq_sInf_coatoms, eq_sSup_atoms, exists_covBy_le_of_lt, exists_le_covBy_of_lt, instIsStronglyAtomicElemOfOrdConnected, instIsStronglyAtomicOfWellFoundedLT, instIsStronglyAtomicOrderDualOfIsStronglyCoatomic, instIsStronglyCoatomicElemOfOrdConnected, instIsStronglyCoatomicOfPredOrder, instIsStronglyCoatomicOfWellFoundedGT, isAtom_compl, isAtom_dual_iff_isCoatom, isAtom_iff_eq_top, isAtom_iff_le_of_ge, isAtom_top, isAtomic_dual_iff_isCoatomic, isAtomic_iff, isAtomic_iff_forall_isAtomic_Iic, isAtomic_iff_isCoatomic, isAtomic_of_isCoatomic_of_complementedLattice_of_isModular, isAtomic_of_orderBot_wellFounded_lt, isAtomistic_dual_iff_isCoatomistic, isAtomistic_iff, isCoatom_bot, isCoatom_compl, isCoatom_dual_iff_isAtom, isCoatom_iff_eq_bot, isCoatom_iff_ge_of_le, isCoatomic_dual_iff_isAtomic, isCoatomic_iff, isCoatomic_iff_forall_isCoatomic_Ici, isCoatomic_of_isAtomic_of_complementedLattice_of_isModular, isCoatomic_of_orderTop_gt_wellFounded, isCoatomistic_dual_iff_isAtomistic, isCoatomistic_iff, isLUB_atoms_le, isLUB_atoms_top, isSimpleOrder_iff, isSimpleOrder_iff_isAtom_top, isSimpleOrder_iff_isCoatom_bot, isSimpleOrder_iff_isSimpleOrder_orderDual, isStronglyAtomic_dual_iff_is_stronglyCoatomic, isStronglyAtomic_iff, isStronglyCoatomic_dual_iff_is_stronglyAtomic, isStronglyCoatomic_iff, le_iff_atom_le_imp, sSup_atoms_eq_top, sSup_atoms_le_eq
198
Total217

BooleanAlgebra

Theorems

NameKindAssumesProvesValidatesDepends On
eq_iff_atom_le_iff 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
toBiheytingAlgebra
le_antisymm
le_iff_atom_le_imp
le_iff_atom_le_imp 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
toBiheytingAlgebra
le_trans
of_not_not
IsAtomic.eq_bot_or_exists_atom_le
le_inf_iff
inf_compl_self
IsCompl.inf_right_eq_bot_iff
eq_compl_iff_isCompl
compl_compl

ComplementedLattice

Theorems

NameKindAssumesProvesValidatesDepends On
isStronglyAtomic 📖mathematicalIsStronglyAtomic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
LT.lt.le
exists_isCompl
IsModularLattice.complementedLattice_Iic
IsAtomic.eq_bot_or_exists_atom_le
LT.lt.ne
Subtype.mk_bot
IsCompl.codisjoint
IsUpperModularLattice.covBy_sup_of_inf_covBy
IsModularLattice.to_isUpperModularLattice
le_bot_iff
IsCompl.inf_eq_bot
inf_comm
inf_le_inf_left
IsAtom.bot_covBy
sup_le
LE.le.trans
isStronglyAtomic' 📖mathematicalIsStronglyCoatomic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
isStronglyCoatomic
isAtomic_iff_isCoatomic
isStronglyCoatomic 📖mathematicalIsStronglyCoatomic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
isStronglyAtomic_dual_iff_is_stronglyCoatomic
isStronglyAtomic
instIsModularLatticeOrderDual
instOrderDual
OrderDual.instIsAtomic
isStronglyCoatomic' 📖mathematicalIsStronglyAtomic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
isStronglyAtomic
isAtomic_iff_isCoatomic

CompleteAtomicBooleanAlgebra

Definitions

NameCategoryTheorems
toSetOfIsAtom 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
eq_setOf_le_sSup_and_isAtom 📖mathematicalIsAtom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
toCompleteBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
setOf
Preorder.toLE
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
Set.ext
CompleteLattice.le_sSup
IsAtom.le_sSup
IsAtom.le_iff
exists_mem_le_of_le_sSup_of_isAtom 📖mathematicalIsAtom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
toCompleteBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Preorder.toLE
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
Set
Set.instMembership
IsAtom.le_sSup
instIsAtomistic 📖mathematicalIsAtomistic
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
toCompleteBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
CompleteLattice.isAtomistic_iff
le_antisymm
iSup_bool_eq
sup_inf_inf_compl
ciInf_const
bot_nonempty
iInf_iSup_eq
le_sSup
lt_of_lt_of_le
le_trans
iInf_le
inf_le_right
Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
instIsEmptyFalse
le_refl
le_iSup
sSup_le
instIsCoatomistic 📖mathematicalIsCoatomistic
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
toCompleteBooleanAlgebra
CoheytingAlgebra.toOrderTop
Order.Coframe.toCoheytingAlgebra
CompleteDistribLattice.toCoframe
CompleteBooleanAlgebra.toCompleteDistribLattice
isAtomistic_dual_iff_isCoatomistic
instIsAtomistic

CompleteBooleanAlgebra

Definitions

NameCategoryTheorems
toCompleteAtomicBooleanAlgebra 📖CompOp

CompleteLattice

Theorems

NameKindAssumesProvesValidatesDepends On
isAtomistic_iff 📖mathematicalIsAtomistic
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
toConditionallyCompleteLattice
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
toBoundedOrder
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
IsAtom
isCoatomistic_iff 📖mathematicalIsCoatomistic
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
toConditionallyCompleteLattice
BoundedOrder.toOrderTop
Preorder.toLE
PartialOrder.toPreorder
toBoundedOrder
InfSet.sInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
IsCoatom

CovBy

Theorems

NameKindAssumesProvesValidatesDepends On
isCoatom 📖mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
Top.top
OrderTop.toTop
Preorder.toLE
IsCoatomcovBy_top_iff
is_atom 📖mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
Bot.bot
OrderBot.toBot
Preorder.toLE
IsAtombot_covBy_iff

GaloisCoinsertion

Theorems

NameKindAssumesProvesValidatesDepends On
isAtom_iff 📖mathematicalIsAtom
PartialOrder.toPreorder
GaloisInsertion.isCoatom_iff
OrderDual.instIsCoatomic
isAtom_of_image 📖IsAtom
PartialOrder.toPreorder
GaloisInsertion.isCoatom_of_image
IsAtom.dual
isCoatom_iff 📖mathematicalTop.top
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
IsCoatomGaloisInsertion.isAtom_iff
OrderDual.instIsAtomic
isCoatom_iff' 📖mathematicalTop.top
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
IsCoatomGaloisInsertion.isAtom_iff'
OrderDual.instIsAtomic
isCoatom_of_l_top 📖Top.top
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
IsCoatom
GaloisInsertion.isAtom_of_u_bot
IsCoatom.dual

GaloisInsertion

Theorems

NameKindAssumesProvesValidatesDepends On
isAtom_iff 📖mathematicalBot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
IsAtomIsAtomic.eq_bot_or_exists_atom_le
u_injective
IsAtom.le_iff
GaloisConnection.monotone_l
gc
l_u_eq
LE.le.trans_eq
GaloisConnection.le_u_l
GaloisConnection.l_bot
isAtom_of_u_bot
isAtom_iff' 📖mathematicalBot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
IsAtomisAtom_iff
l_u_eq
isAtom_of_u_bot 📖Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
IsAtom
OrderEmbedding.isAtom_of_map_bot_of_image
u_injective
u_le_u_iff
isCoatom_iff 📖mathematicalIsCoatom
PartialOrder.toPreorder
isCoatom_of_image
IsCoatomic.eq_top_or_exists_le_coatom
l_u_eq
GaloisConnection.u_top
gc
IsCoatom.le_iff
GaloisConnection.monotone_l
isCoatom_of_image 📖IsCoatom
PartialOrder.toPreorder
OrderEmbedding.isCoatom_of_map_top_of_image
u_injective
u_le_u_iff
GaloisConnection.u_top
gc

IsAtom

Theorems

NameKindAssumesProvesValidatesDepends On
Iic 📖mathematicalIsAtom
Preorder.toLE
Set
Set.instMembership
Set.Iic
Subtype.preorder
Set.Iic.orderBot
Iic_eq 📖mathematicalIsAtom
PartialOrder.toPreorder
Set.Iic
Set
Set.instInsert
Bot.bot
OrderBot.toBot
Preorder.toLE
Set.instSingletonSet
Set.ext
le_iff
bot_covBy 📖mathematicalIsAtom
PartialOrder.toPreorder
CovBy
Preorder.toLT
Bot.bot
OrderBot.toBot
Preorder.toLE
bot_covBy_iff
bot_lt 📖mathematicalIsAtom
PartialOrder.toPreorder
Preorder.toLT
Bot.bot
OrderBot.toBot
Preorder.toLE
lt_iff
compl 📖mathematicalIsAtom
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
IsCoatom
CoheytingAlgebra.toOrderTop
Compl.compl
BooleanAlgebra.toCompl
isCoatom_compl
disjoint_of_ne 📖mathematicalIsAtom
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Disjointnot_le_iff_disjoint
le_iff
ne_bot
dual 📖mathematicalIsAtomIsCoatom
OrderDual
OrderDual.instPreorder
OrderDual.instOrderTop
Preorder.toLE
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
isCoatom_dual_iff_isAtom
le_iSup 📖mathematicalIsAtom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
Order.Frame.toCompleteLattice
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
Preorder.toLE
iSup
ConditionallyCompletePartialOrderSup.toSupSet
of_not_not
disjoint_iSup_iff
le_bot_iff
le_iff_eq_or_lt
le_rfl
le_trans
le_iSup
le_iff 📖mathematicalIsAtom
PartialOrder.toPreorder
Preorder.toLE
Bot.bot
OrderBot.toBot
le_iff_lt_or_eq
lt_iff
le_iff_eq 📖mathematicalIsAtom
PartialOrder.toPreorder
Preorder.toLEle_iff
le_sSup 📖mathematicalIsAtom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
Order.Frame.toCompleteLattice
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
Preorder.toLE
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
Set
Set.instMembership
sSup_eq_iSup'
le_iSup
lt_iff 📖mathematicalIsAtom
PartialOrder.toPreorder
Preorder.toLT
Bot.bot
OrderBot.toBot
Preorder.toLE
Ne.bot_lt
ne_bot 📖IsAtom
ne_bot_iff_eq 📖IsAtom
PartialOrder.toPreorder
Preorder.toLE
Iff.not_right
ne_iff_eq_bot
ne_iff_eq_bot 📖mathematicalIsAtom
PartialOrder.toPreorder
Preorder.toLE
Bot.bot
OrderBot.toBot
le_iff
ne_bot
not_disjoint_iff_le 📖mathematicalIsAtom
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Disjoint
Preorder.toLE
disjoint_iff
inf_eq_left
ne_bot_iff_eq
inf_le_left
not_le_iff_disjoint 📖mathematicalIsAtom
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Preorder.toLE
Disjoint
Iff.not_right
not_disjoint_iff_le
of_compl 📖mathematicalIsAtom
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
Compl.compl
BooleanAlgebra.toCompl
IsCoatom
CoheytingAlgebra.toOrderTop
isAtom_compl
of_isAtom_coe_Iic 📖IsAtom
Set.Elem
Set.Iic
Subtype.preorder
Set
Set.instMembership
Set.Iic.orderBot
LE.le.trans
LT.lt.le
Subtype.prop

IsAtomic

Theorems

NameKindAssumesProvesValidatesDepends On
eq_bot_or_exists_atom_le 📖mathematicalBot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
IsAtom
exists_atom 📖mathematicalIsAtom
PartialOrder.toPreorder
exists_ne
eq_bot_or_exists_atom_le

IsAtomic.Set.Iic

Theorems

NameKindAssumesProvesValidatesDepends On
isAtomic 📖mathematicalIsAtomic
Set.Elem
Set.Iic
PartialOrder.toPreorder
Subtype.partialOrder
Set
Set.instMembership
Set.Iic.orderBot
LE.le.trans
IsAtom.Iic
IsAtomic.eq_bot_or_exists_atom_le

IsAtomistic

Theorems

NameKindAssumesProvesValidatesDepends On
instIsAtomic 📖mathematicalIsAtomicisLUB_atoms
Set.eq_empty_or_nonempty
isLUB_atoms 📖mathematicalIsLUB
Preorder.toLE
PartialOrder.toPreorder
IsAtom

IsCoatom

Theorems

NameKindAssumesProvesValidatesDepends On
Ici 📖mathematicalIsCoatom
Preorder.toLE
Set
Set.instMembership
Set.Ici
Subtype.preorder
Set.Ici.orderTop
IsAtom.Iic
dual
Ici_eq 📖mathematicalIsCoatom
PartialOrder.toPreorder
Set.Ici
Set
Set.instInsert
Top.top
OrderTop.toTop
Preorder.toLE
Set.instSingletonSet
IsAtom.Iic_eq
dual
codisjoint_of_ne 📖mathematicalIsCoatom
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Codisjointnot_le_iff_codisjoint
le_iff
ne_top
compl 📖mathematicalIsCoatom
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
CoheytingAlgebra.toOrderTop
IsAtom
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
Compl.compl
BooleanAlgebra.toCompl
isAtom_compl
covBy_top 📖mathematicalIsCoatom
PartialOrder.toPreorder
CovBy
Preorder.toLT
Top.top
OrderTop.toTop
Preorder.toLE
covBy_top_iff
dual 📖mathematicalIsCoatomIsAtom
OrderDual
OrderDual.instPreorder
OrderDual.instOrderBot
Preorder.toLE
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
isAtom_dual_iff_isCoatom
iInf_le 📖mathematicalIsCoatom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
Order.Coframe.toCompleteLattice
CoheytingAlgebra.toOrderTop
Order.Coframe.toCoheytingAlgebra
Preorder.toLE
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
IsAtom.le_iSup
le_iff 📖mathematicalIsCoatom
PartialOrder.toPreorder
Preorder.toLE
Top.top
OrderTop.toTop
IsAtom.le_iff
dual
le_iff_eq 📖mathematicalIsCoatom
PartialOrder.toPreorder
Preorder.toLEIsAtom.le_iff_eq
dual
lt_iff 📖mathematicalIsCoatom
PartialOrder.toPreorder
Preorder.toLT
Top.top
OrderTop.toTop
Preorder.toLE
IsAtom.lt_iff
dual
lt_top 📖mathematicalIsCoatom
PartialOrder.toPreorder
Preorder.toLT
Top.top
OrderTop.toTop
Preorder.toLE
lt_iff
ne_iff_eq_top 📖mathematicalIsCoatom
PartialOrder.toPreorder
Preorder.toLE
Top.top
OrderTop.toTop
le_iff
ne_top
ne_top 📖IsCoatom
ne_top_iff_eq 📖IsCoatom
PartialOrder.toPreorder
Preorder.toLE
Iff.not_right
ne_iff_eq_top
not_codisjoint_iff_le 📖mathematicalIsCoatom
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Codisjoint
Preorder.toLE
codisjoint_iff
sup_eq_left
ne_top_iff_eq
le_sup_left
not_le_iff_codisjoint 📖mathematicalIsCoatom
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Preorder.toLE
Codisjoint
Iff.not_right
not_codisjoint_iff_le
of_compl 📖mathematicalIsCoatom
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
CoheytingAlgebra.toOrderTop
Compl.compl
BooleanAlgebra.toCompl
IsAtom
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
isCoatom_compl
of_isCoatom_coe_Ici 📖IsCoatom
Set.Elem
Set.Ici
Subtype.preorder
Set
Set.instMembership
Set.Ici.orderTop
IsAtom.of_isAtom_coe_Iic
sInf_le 📖mathematicalIsCoatom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
Order.Coframe.toCompleteLattice
CoheytingAlgebra.toOrderTop
Order.Coframe.toCoheytingAlgebra
Preorder.toLE
InfSet.sInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
Set
Set.instMembership
sInf_eq_iInf'
iInf_le
sup_eq_top_of_ne 📖mathematicalIsCoatom
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
SemilatticeSup.toMax
Top.top
OrderTop.toTop
Preorder.toLE
codisjoint_iff
codisjoint_of_ne

IsCoatomic

Theorems

NameKindAssumesProvesValidatesDepends On
eq_top_or_exists_le_coatom 📖mathematicalTop.top
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
IsCoatom
exists_coatom 📖mathematicalIsCoatom
PartialOrder.toPreorder
exists_ne
eq_top_or_exists_le_coatom

IsCoatomic.Set.Ici

Theorems

NameKindAssumesProvesValidatesDepends On
isCoatomic 📖mathematicalIsCoatomic
Set.Elem
Set.Ici
PartialOrder.toPreorder
Subtype.partialOrder
Set
Set.instMembership
Set.Ici.orderTop
le_trans
IsCoatom.Ici
IsCoatomic.eq_top_or_exists_le_coatom

IsCoatomistic

Theorems

NameKindAssumesProvesValidatesDepends On
instIsCoatomic 📖mathematicalIsCoatomicisGLB_coatoms
Set.eq_empty_or_nonempty
isGLB_coatoms 📖mathematicalIsGLB
Preorder.toLE
PartialOrder.toPreorder
IsCoatom

IsCompl

Theorems

NameKindAssumesProvesValidatesDepends On
isAtom_iff_isCoatom 📖mathematicalIsCompl
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
IsAtom
PartialOrder.toPreorder
BoundedOrder.toOrderBot
Preorder.toLE
IsCoatom
BoundedOrder.toOrderTop
Set.isSimpleOrder_Iic_iff_isAtom
OrderIso.isSimpleOrder_iff
Set.isSimpleOrder_Ici_iff_isCoatom
isCoatom_iff_isAtom 📖mathematicalIsCompl
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
IsCoatom
PartialOrder.toPreorder
BoundedOrder.toOrderTop
Preorder.toLE
IsAtom
BoundedOrder.toOrderBot
isAtom_iff_isCoatom
symm

IsSimpleOrder

Definitions

NameCategoryTheorems
booleanAlgebra 📖CompOp
completeBooleanAlgebra 📖CompOp
completeLattice 📖CompOp
distribLattice 📖CompOp
equivBool 📖CompOp
2 mathmath: equivBool_symm_apply, equivBool_apply
lattice 📖CompOp
linearOrder 📖CompOp
orderIsoBool 📖CompOp
preorder 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
bot_lt_iff_eq_top 📖mathematicalPreorder.toLT
PartialOrder.toPreorder
Bot.bot
OrderBot.toBot
Preorder.toLE
BoundedOrder.toOrderBot
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
eq_top_of_lt
bot_lt_top
toNontrivial
bot_ne_top 📖exists_pair_ne
toNontrivial
eq_bot_or_eq_top
eq_bot_of_lt 📖mathematicalPreorder.toLTBot.bot
OrderBot.toBot
Preorder.toLE
BoundedOrder.toOrderBot
eq_bot_or_eq_top
LT.lt.ne_top
eq_bot_or_eq_top 📖mathematicalBot.bot
OrderBot.toBot
BoundedOrder.toOrderBot
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
eq_top_of_lt 📖mathematicalPreorder.toLTTop.top
OrderTop.toTop
Preorder.toLE
BoundedOrder.toOrderTop
eq_bot_or_eq_top
LT.lt.ne_bot
equivBool_apply 📖mathematicalDFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
equivBool
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
equivBool_symm_apply 📖mathematicalDFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
equivBool
Bot.bot
OrderBot.toBot
BoundedOrder.toOrderBot
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
instComplementedLattice 📖mathematicalComplementedLatticeBooleanAlgebra.toComplementedLattice
instIsAtomic 📖mathematicalIsAtomic
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
isAtom_top
ge_of_eq
eq_bot_or_eq_top
instIsAtomistic 📖mathematicalIsAtomistic
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
eq_bot_or_eq_top
instIsEmptyFalse
instIsCoatomic 📖mathematicalIsCoatomic
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
BoundedOrder.toOrderTop
Preorder.toLE
PartialOrder.toPreorder
isAtomic_dual_iff_isCoatomic
instIsAtomic
OrderDual.instIsSimpleOrder
instIsCoatomistic 📖mathematicalIsCoatomistic
BoundedOrder.toOrderTop
Preorder.toLE
PartialOrder.toPreorder
isAtomistic_dual_iff_isCoatomistic
instIsAtomistic
OrderDual.instIsSimpleOrder
lt_top_iff_eq_bot 📖mathematicalPreorder.toLT
PartialOrder.toPreorder
Top.top
OrderTop.toTop
Preorder.toLE
BoundedOrder.toOrderTop
Bot.bot
OrderBot.toBot
BoundedOrder.toOrderBot
eq_bot_of_lt
bot_lt_top
toNontrivial
of_forall_eq_top 📖mathematicalTop.top
OrderTop.toTop
BoundedOrder.toOrderTop
IsSimpleOrder
toNontrivial 📖mathematicalNontrivial

IsStronglyAtomic

Theorems

NameKindAssumesProvesValidatesDepends On
exists_covBy_le_of_lt 📖mathematicalPreorder.toLTCovBy
Preorder.toLE
isAtomic 📖mathematicalIsAtomicbot_lt_iff_ne_bot
LT.lt.exists_covby_le
bot_covBy_iff
of_wellFounded_lt 📖mathematicalPreorder.toLT
PartialOrder.toPreorder
IsStronglyAtomicEq.le
WellFounded.min_mem
WellFounded.not_lt_min
LE.le.trans
LT.lt.le

IsStronglyCoatomic

Theorems

NameKindAssumesProvesValidatesDepends On
exists_le_covBy_of_lt 📖mathematicalPreorder.toLTPreorder.toLE
CovBy
of_wellFounded_gt 📖mathematicalPreorder.toLT
PartialOrder.toPreorder
IsStronglyCoatomicisStronglyAtomic_dual_iff_is_stronglyCoatomic
IsStronglyAtomic.of_wellFounded_lt
toIsCoatomic 📖mathematicalIsCoatomicisAtomic_dual_iff_isCoatomic
IsStronglyAtomic.isAtomic
instIsStronglyAtomicOrderDualOfIsStronglyCoatomic

LT.lt

Theorems

NameKindAssumesProvesValidatesDepends On
eq_bot 📖mathematicalPreorder.toLTBot.bot
OrderBot.toBot
Preorder.toLE
BoundedOrder.toOrderBot
IsSimpleOrder.eq_bot_of_lt
eq_top 📖mathematicalPreorder.toLTTop.top
OrderTop.toTop
Preorder.toLE
BoundedOrder.toOrderTop
IsSimpleOrder.eq_top_of_lt
exists_covby_le 📖mathematicalPreorder.toLTCovBy
Preorder.toLE
exists_covBy_le_of_lt
exists_le_covby 📖mathematicalPreorder.toLTPreorder.toLE
CovBy
exists_le_covBy_of_lt

Lattice

Theorems

NameKindAssumesProvesValidatesDepends On
isStronglyAtomic 📖mathematicalIsStronglyAtomic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
toSemilatticeInf
IsAtomistic.isLUB_atoms
by_contra
LT.lt.not_ge
isLUB_le_iff
CovBy.eq_or_eq
IsAtom.bot_covBy
sup_le
LT.lt.le
IsUpperModularLattice.covBy_sup_of_inf_covBy
inf_eq_left
isStronglyCoatomic 📖mathematicalIsStronglyCoatomic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
toSemilatticeInf
isStronglyAtomic_dual_iff_is_stronglyCoatomic
isStronglyAtomic
instIsUpperModularLatticeOrderDual
OrderDual.instIsAtomistic

OrderDual

Theorems

NameKindAssumesProvesValidatesDepends On
instIsAtomic 📖mathematicalIsAtomic
OrderDual
instPartialOrder
instOrderBot
Preorder.toLE
PartialOrder.toPreorder
isAtomic_dual_iff_isCoatomic
instIsAtomistic 📖mathematicalIsAtomistic
OrderDual
instPartialOrder
instOrderBot
Preorder.toLE
PartialOrder.toPreorder
isAtomistic_dual_iff_isCoatomistic
instIsCoatomic 📖mathematicalIsCoatomic
OrderDual
instPartialOrder
instOrderTop
Preorder.toLE
PartialOrder.toPreorder
isCoatomic_dual_iff_isAtomic
instIsCoatomistic 📖mathematicalIsCoatomistic
OrderDual
instPartialOrder
instOrderTop
Preorder.toLE
PartialOrder.toPreorder
isCoatomistic_dual_iff_isAtomistic
instIsSimpleOrder 📖mathematicalIsSimpleOrder
OrderDual
instLE
instBoundedOrder
isSimpleOrder_iff_isSimpleOrder_orderDual
instIsStronglyCoatomic 📖mathematicalIsStronglyCoatomic
OrderDual
instPreorder
isStronglyCoatomic_dual_iff_is_stronglyAtomic

OrderEmbedding

Theorems

NameKindAssumesProvesValidatesDepends On
isAtom_of_map_bot_of_image 📖DFunLike.coe
OrderEmbedding
Preorder.toLE
PartialOrder.toPreorder
instFunLikeOrderEmbedding
Bot.bot
OrderBot.toBot
IsAtom
CovBy.of_image
isCoatom_of_map_top_of_image 📖DFunLike.coe
OrderEmbedding
Preorder.toLE
PartialOrder.toPreorder
instFunLikeOrderEmbedding
Top.top
OrderTop.toTop
IsCoatom
isAtom_of_map_bot_of_image

OrderIso

Theorems

NameKindAssumesProvesValidatesDepends On
isAtom_iff 📖mathematicalIsAtom
PartialOrder.toPreorder
DFunLike.coe
OrderIso
Preorder.toLE
instFunLikeOrderIso
GaloisCoinsertion.isAtom_of_image
GaloisInsertion.isAtom_of_u_bot
map_bot
symm_apply_apply
isAtomic_iff 📖mathematicalIsAtomicmap_bot
Function.Surjective.exists
surjective
isAtom_iff
Function.Surjective.forall
RelIso.eq_iff_eq
le_iff_le
isCoatom_iff 📖mathematicalIsCoatom
PartialOrder.toPreorder
DFunLike.coe
OrderIso
Preorder.toLE
instFunLikeOrderIso
isAtom_iff
isCoatomic_iff 📖mathematicalIsCoatomicisAtomic_iff
isSimpleOrder 📖mathematicalIsSimpleOrder
Preorder.toLE
PartialOrder.toPreorder
isSimpleOrder_iff
isSimpleOrder_iff 📖mathematicalIsSimpleOrder
Preorder.toLE
PartialOrder.toPreorder
isSimpleOrder_iff_isAtom_top
isAtom_iff
map_top

Pi

Theorems

NameKindAssumesProvesValidatesDepends On
eq_bot_iff 📖mathematicalBot.bot
instBotForall
isAtom_iff 📖mathematicalIsAtom
preorder
PartialOrder.toPreorder
instOrderBot
Preorder.toLE
Bot.bot
OrderBot.toBot
isAtom_iff_eq_single 📖mathematicalIsAtom
preorder
PartialOrder.toPreorder
instOrderBot
Preorder.toLE
Function.update
Bot.bot
instBotForall
OrderBot.toBot
isAtom_single 📖mathematicalIsAtom
PartialOrder.toPreorder
preorder
instOrderBot
Preorder.toLE
Function.update
Bot.bot
instBotForall
OrderBot.toBot
isAtom_iff
Function.update_self
Function.update_of_ne
isAtomic 📖mathematicalIsAtomicpartialOrder
instOrderBot
Preorder.toLE
PartialOrder.toPreorder
eq_bot_iff
IsAtomic.eq_bot_or_exists_atom_le
isAtom_single
update_le_iff
isAtomistic 📖mathematicalIsAtomisticpartialOrder
instOrderBot
Preorder.toLE
PartialOrder.toPreorder
isLUB_pi
isLUB_atoms_le
Function.update_self
isCoatomic 📖mathematicalIsCoatomicpartialOrder
instOrderTop
Preorder.toLE
PartialOrder.toPreorder
isAtomic_dual_iff_isCoatomic
isAtomic
OrderDual.instIsAtomic
isCoatomistic 📖mathematicalIsCoatomistic
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
BoundedOrder.toOrderTop
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
partialOrder
instOrderTop
isAtomistic_dual_iff_isCoatomistic
isAtomistic
OrderDual.instIsAtomistic

Prop

Theorems

NameKindAssumesProvesValidatesDepends On
instIsSimpleOrderProp 📖mathematicalIsSimpleOrder
le
instBoundedOrder
instNontrivialProp
isAtom_iff 📖mathematicalIsAtom
PartialOrder.toPreorder
partialOrder
HeytingAlgebra.toOrderBot
instHeytingAlgebra
instIsSimpleOrderProp
isCoatom_iff 📖mathematicalIsCoatom
PartialOrder.toPreorder
partialOrder
CoheytingAlgebra.toOrderTop
Order.Coframe.toCoheytingAlgebra
CompleteDistribLattice.toCoframe
CompleteBooleanAlgebra.toCompleteDistribLattice
instCompleteBooleanAlgebra
instIsSimpleOrderProp

Set

Theorems

NameKindAssumesProvesValidatesDepends On
instIsAtomistic 📖mathematicalIsAtomistic
Set
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
CompleteAtomicBooleanAlgebra.instIsAtomistic
instIsCoatomistic 📖mathematicalIsCoatomistic
Set
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
instOrderTop
CompleteAtomicBooleanAlgebra.instIsCoatomistic
isAtom_iff 📖mathematicalIsAtom
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
instSingletonSet
isAtom_iff_le_of_ge
bot_eq_empty
nonempty_iff_ne_empty
eq_singleton_iff_unique_mem
singleton_ne_empty
singleton_subset_iff
isAtom_singleton
isAtom_singleton 📖mathematicalIsAtom
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
instSingletonSet
singleton_ne_empty
ssubset_singleton_iff
isCoatom_iff 📖mathematicalIsCoatom
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
instOrderTop
Compl.compl
instCompl
instSingletonSet
IsCompl.isCoatom_iff_isAtom
DistribLattice.instIsModularLattice
isCompl_compl
isAtom_iff
isCoatom_singleton_compl 📖mathematicalIsCoatom
Set
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
instOrderTop
Compl.compl
instCompl
instSingletonSet
isCoatom_iff
isSimpleOrder_Ici_iff_isCoatom 📖mathematicalIsSimpleOrder
Elem
Ici
PartialOrder.toPreorder
Preorder.toLE
Set
instMembership
Ici.boundedOrder
IsCoatom
isSimpleOrder_iff_isCoatom_bot
le_refl
le_of_lt
isSimpleOrder_Iic_iff_isAtom 📖mathematicalIsSimpleOrder
Elem
Iic
PartialOrder.toPreorder
Preorder.toLE
Set
instMembership
Iic.instBoundedOrderElemOfOrderBot
IsAtom
isSimpleOrder_iff_isAtom_top
le_refl
le_of_lt

Set.Ici

Theorems

NameKindAssumesProvesValidatesDepends On
isAtom_iff 📖mathematicalIsAtom
Set.Elem
Set.Ici
PartialOrder.toPreorder
Subtype.preorder
Set
Set.instMembership
orderBot
CovBy
Preorder.toLT
bot_covBy_iff
Set.OrdConnected.apply_covBy_apply_iff
Subtype.range_coe_subtype
Set.ordConnected_Ici

Set.Iic

Theorems

NameKindAssumesProvesValidatesDepends On
isCoatom_iff 📖mathematicalIsCoatom
Set.Elem
Set.Iic
PartialOrder.toPreorder
Subtype.preorder
Set
Set.instMembership
orderTop
CovBy
Preorder.toLT
covBy_top_iff
Set.OrdConnected.apply_covBy_apply_iff
Subtype.range_coe_subtype
Set.ordConnected_Iic

Set.OrdConnected

Theorems

NameKindAssumesProvesValidatesDepends On
isStronglyAtomic 📖mathematicalIsStronglyAtomic
Set.Elem
Subtype.preorder
Set
Set.instMembership
LT.lt.exists_covby_le
out'
CovBy.le
CovBy.lt
isStronglyCoatomic 📖mathematicalIsStronglyCoatomic
Set.Elem
Subtype.preorder
Set
Set.instMembership
isStronglyAtomic_dual_iff_is_stronglyCoatomic
isStronglyAtomic
instIsStronglyAtomicOrderDualOfIsStronglyCoatomic
dual

SetLike

Theorems

NameKindAssumesProvesValidatesDepends On
covBy_iff 📖mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
Mathlib.Tactic.Contrapose.contrapose_iff₁
Mathlib.Tactic.Push.not_forall_eq
lt_iff_le_not_ge
lt_iff_le_and_ne
covBy_iff' 📖mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
not_iff_not
Mathlib.Tactic.Push.not_forall_eq
lt_iff_le_and_ne
lt_iff_le_not_ge
isAtom_iff 📖mathematicalIsAtom
PartialOrder.toPreorder
Bot.bot
OrderBot.toBot
Preorder.toLE
isCoatom_iff 📖mathematicalIsCoatom
PartialOrder.toPreorder
Top.top
OrderTop.toTop
Preorder.toLE

SuccOrder

Theorems

NameKindAssumesProvesValidatesDepends On
toIsStronglyAtomic 📖mathematicalIsStronglyAtomicOrder.covBy_succ_of_not_isMax
IsMax.not_lt
succ_le_of_lt

(root)

Definitions

NameCategoryTheorems
IsAtom 📖MathDef
64 mathmath: Associates.isAtom_iff, IsPredArchimedean.isAtom_findAtom, isSimpleOrder_iff_isAtom_top, Finset.isAtom_iff, GaloisInsertion.isAtom_iff, IsCompactlyGenerated.BooleanGenerators.isAtom, LieAlgebra.IsSemisimple.booleanGenerators, Multiset.isAtom_singleton, IsSimpleRing.tfae, IsAtomic.eq_bot_or_exists_atom_le, isSimpleModule_iff_isAtom, isAtomistic_iff, sSup_atoms_le_eq, Set.Ici.isAtom_iff, isCoatom_compl, IsAtomistic.isLUB_atoms, IsSimpleModule.isAtom, Filter.isAtom_pure, Set.isAtom_singleton, atom_iff_nonzero_span, LieAlgebra.IsSemisimple.sSup_atoms_eq_top, OrderIso.isAtom_iff, LieAlgebra.IsSimple.isAtom_top, nonzero_span_atom, CategoryTheory.subobject_simple_iff_isAtom, isAtomic_iff, covBy_iff_atom_Ici, isAtom_dual_iff_isCoatom, sSup_atoms_eq_top, IsCompl.isCoatom_iff_isAtom, Set.isSimpleOrder_Iic_iff_isAtom, TopologicalSpace.Closeds.isAtom_iff, isLUB_atoms_top, Prop.isAtom_iff, Set.isAtom_iff, bot_covBy_iff, Pi.isAtom_iff_eq_single, CovBy.is_atom, Finset.isAtom_singleton, Pi.isAtom_iff, IsAtomic.exists_atom, TopologicalSpace.Closeds.isAtom_coe, GaloisCoinsertion.isAtom_iff, isAtom_iff_eq_top, isAtom_top, IsCompactlyGenerated.BooleanGenerators.eq_atoms_of_sSup_eq_top, CompleteLattice.isAtomistic_iff, IsCoatom.dual, IsCompl.isAtom_iff_isCoatom, LieAlgebra.InvariantForm.atomistic, LieAlgebra.IsSemisimple.sSupIndep_isAtom, IsPredArchimedean.isAtom_findAtom_iff, isLUB_atoms_le, isCoatom_dual_iff_isAtom, Multiset.isAtom_iff, eq_sSup_atoms, LieAlgebra.IsSimple.isAtom_iff_eq_top, SetLike.isAtom_iff, IsCoatom.of_compl, IsCoatom.compl, isAtom_iff_le_of_ge, isAtom_compl, GaloisInsertion.isAtom_iff', Ultrafilter.isAtom
IsAtomic 📖CompData
19 mathmath: isCoatomic_dual_iff_isAtomic, isAtomic_iff_forall_isAtomic_Iic, OrderDual.instIsAtomic, IsStronglyAtomic.isAtomic, Finite.to_isAtomic, IsAtomic.Set.Iic.isAtomic, isAtomic_iff, isAtomic_of_complementedLattice, IsPredArchimedean.instIsAtomic, isAtomic_of_isCoatomic_of_complementedLattice_of_isModular, LieSubmodule.instIsAtomicOfIsArtinian, isAtomic_iff_isCoatomic, isAtomic_dual_iff_isCoatomic, IsAtomic.of_isChain_bounded, isAtomic_of_orderBot_wellFounded_lt, IsAtomistic.instIsAtomic, OrderIso.isAtomic_iff, instIsAtomicFilter, IsSimpleOrder.instIsAtomic
IsAtomistic 📖CompData
12 mathmath: isAtomistic_iff, OrderDual.instIsAtomistic, isAtomistic_dual_iff_isCoatomistic, isCoatomistic_dual_iff_isAtomistic, IsCompactlyGenerated.BooleanGenerators.isAtomistic_of_sSup_eq_top, isAtomistic_of_complementedLattice, CompleteAtomicBooleanAlgebra.instIsAtomistic, complementedLattice_iff_isAtomistic, instIsAtomisticSubmodule, CompleteLattice.isAtomistic_iff, Set.instIsAtomistic, IsSimpleOrder.instIsAtomistic
IsCoatom 📖MathDef
56 mathmath: eq_sInf_coatoms, GaloisCoinsertion.isCoatom_iff, OrderIso.isCoatom_iff, Subgroup.isCoatom_map, IsCoatomic.exists_coatom, SetLike.isCoatom_iff, TopologicalSpace.Opens.isCoatom_iff, covBy_iff_coatom_Iic, Prop.isCoatom_iff, isCoatom_compl, alternatingGroup.isCoatom_stabilizer_singleton, GaloisInsertion.isCoatom_iff, Order.Ideal.isMaximal_iff_isCoatom, Set.isCoatom_iff, isCoatomic_iff, isCoatom_bot, MulAction.isCoatom_stabilizer_iff_preprimitive, Finset.isCoatom_compl_singleton, Submodule.isCoatom_comap_iff, MulAction.IsPreprimitive.isCoatom_stabilizer_of_isPreprimitive, IsCoatomistic.isGLB_coatoms, isCoatom_iff_ge_of_le, isCoatomistic_iff, IsCoatomic.eq_top_or_exists_le_coatom, isAtom_dual_iff_isCoatom, IsCompl.isCoatom_iff_isAtom, isCoatom_iff_eq_bot, Order.Ideal.IsMaximal.isCoatom', Set.isCoatom_singleton_compl, CovBy.isCoatom, CompleteLattice.isCoatomistic_iff, covBy_top_iff, Ideal.IsMaximal.out, IsAtom.dual, IsAtom.of_compl, isSimpleOrder_iff_isCoatom_bot, alternatingGroup.isCoatom_stabilizer, LinearMap.isCoatom_ker_of_surjective, Equiv.Perm.isCoatom_stabilizer, IsAtom.compl, Finset.isCoatom_iff, isSimpleModule_iff_isCoatom, Equiv.Perm.isCoatom_stabilizer_of_ncard_lt_ncard_compl, IsCompl.isAtom_iff_isCoatom, GaloisCoinsertion.isCoatom_iff', Set.isSimpleOrder_Ici_iff_isCoatom, isCoatom_dual_iff_isAtom, Set.Iic.isCoatom_iff, Order.Ideal.IsMaximal.isCoatom, ContinuousMap.idealOfSet_isMaximal_iff, isAtom_compl, AddAction.isCoatom_stabilizer_iff_preprimitive, AddAction.IsPreprimitive.isCoatom_stabilizer_of_isPreprimitive, alternatingGroup.isCoatom_stabilizer_of_ncard_lt_ncard_compl, Subgroup.isCoatom_comap, Ideal.isMaximal_def
IsCoatomic 📖CompData
20 mathmath: isCoatomic_dual_iff_isAtomic, CompleteLattice.Iic_coatomic_of_compact_element, isCoatomic_iff, isCoatomic_iff_forall_isCoatomic_Ici, IsStronglyCoatomic.toIsCoatomic, Finite.to_isCoatomic, IsCoatomic.Set.Ici.isCoatomic, OrderIso.isCoatomic_iff, Module.Finite.instIsCoatomicSubmodule, CompleteLattice.coatomic_of_top_compact, isAtomic_iff_isCoatomic, isCoatomic_of_orderTop_gt_wellFounded, IsSemisimpleModule.isCoatomic_submodule, OrderDual.instIsCoatomic, isAtomic_dual_iff_isCoatomic, IsCoatomistic.instIsCoatomic, isCoatomic_of_isAtomic_of_complementedLattice_of_isModular, Ideal.instIsCoatomic, IsSimpleOrder.instIsCoatomic, IsCoatomic.of_isChain_bounded
IsCoatomistic 📖CompData
8 mathmath: OrderDual.instIsCoatomistic, CompleteAtomicBooleanAlgebra.instIsCoatomistic, isAtomistic_dual_iff_isCoatomistic, isCoatomistic_iff, CompleteLattice.isCoatomistic_iff, isCoatomistic_dual_iff_isAtomistic, IsSimpleOrder.instIsCoatomistic, Set.instIsCoatomistic
IsStronglyAtomic 📖CompData
13 mathmath: isStronglyAtomic_iff, isStronglyCoatomic_dual_iff_is_stronglyAtomic, instIsStronglyAtomic, isStronglyAtomic_dual_iff_is_stronglyCoatomic, ComplementedLattice.isStronglyCoatomic', IsStronglyAtomic.of_wellFounded_lt, Set.OrdConnected.isStronglyAtomic, Lattice.isStronglyAtomic, SuccOrder.toIsStronglyAtomic, instIsStronglyAtomicElemOfOrdConnected, ComplementedLattice.isStronglyAtomic, instIsStronglyAtomicOrderDualOfIsStronglyCoatomic, instIsStronglyAtomicOfWellFoundedLT
IsStronglyCoatomic 📖CompData
13 mathmath: isStronglyCoatomic_dual_iff_is_stronglyAtomic, isStronglyAtomic_dual_iff_is_stronglyCoatomic, instIsStronglyCoatomicElemOfOrdConnected, instIsStronglyCoatomicOfPredOrder, isStronglyCoatomic_iff, ComplementedLattice.isStronglyCoatomic, IsStronglyCoatomic.of_wellFounded_gt, ComplementedLattice.isStronglyAtomic', instIsStronglyCoatomic, Set.OrdConnected.isStronglyCoatomic, instIsStronglyCoatomicOfWellFoundedGT, Lattice.isStronglyCoatomic, OrderDual.instIsStronglyCoatomic

Theorems

NameKindAssumesProvesValidatesDepends On
bot_covBy_iff 📖mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
Bot.bot
OrderBot.toBot
Preorder.toLE
IsAtom
bot_covBy_top 📖mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
Bot.bot
OrderBot.toBot
Preorder.toLE
BoundedOrder.toOrderBot
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
IsAtom.bot_covBy
isAtom_top
covBy_iff_atom_Ici 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
CovBy
Preorder.toLT
IsAtom
Set
Set.instMembership
Set.Ici
Subtype.preorder
Set.Ici.orderBot
covBy_iff_coatom_Iic 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
CovBy
Preorder.toLT
IsCoatom
Set
Set.instMembership
Set.Iic
Subtype.preorder
Set.Iic.orderTop
covBy_top_iff 📖mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
Top.top
OrderTop.toTop
Preorder.toLE
IsCoatom
toDual_covBy_toDual_iff
bot_covBy_iff
eq_iff_atom_le_iff 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
le_antisymm_iff
le_iff_atom_le_imp
eq_sInf_coatoms 📖mathematicalInfSet.sInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
IsCoatom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
BoundedOrder.toOrderTop
Preorder.toLE
CompleteLattice.toBoundedOrder
CompleteLattice.isCoatomistic_iff
eq_sSup_atoms 📖mathematicalSupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
IsAtom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
CompleteLattice.toBoundedOrder
CompleteLattice.isAtomistic_iff
exists_covBy_le_of_lt 📖mathematicalPreorder.toLTCovBy
Preorder.toLE
IsStronglyAtomic.exists_covBy_le_of_lt
exists_le_covBy_of_lt 📖mathematicalPreorder.toLTPreorder.toLE
CovBy
IsStronglyCoatomic.exists_le_covBy_of_lt
instIsStronglyAtomicElemOfOrdConnected 📖mathematicalIsStronglyAtomic
Set.Elem
Subtype.preorder
Set
Set.instMembership
Set.OrdConnected.isStronglyAtomic
instIsStronglyAtomicOfWellFoundedLT 📖mathematicalIsStronglyAtomic
PartialOrder.toPreorder
IsStronglyAtomic.of_wellFounded_lt
wellFounded_lt
instIsStronglyAtomicOrderDualOfIsStronglyCoatomic 📖mathematicalIsStronglyAtomic
OrderDual
OrderDual.instPreorder
isStronglyAtomic_dual_iff_is_stronglyCoatomic
instIsStronglyCoatomicElemOfOrdConnected 📖mathematicalIsStronglyCoatomic
Set.Elem
Subtype.preorder
Set
Set.instMembership
Set.OrdConnected.isStronglyCoatomic
instIsStronglyCoatomicOfPredOrder 📖mathematicalIsStronglyCoatomicisStronglyAtomic_dual_iff_is_stronglyCoatomic
SuccOrder.toIsStronglyAtomic
instIsStronglyCoatomicOfWellFoundedGT 📖mathematicalIsStronglyCoatomic
PartialOrder.toPreorder
IsStronglyCoatomic.of_wellFounded_gt
wellFounded_gt
isAtom_compl 📖mathematicalIsAtom
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
Compl.compl
BooleanAlgebra.toCompl
IsCoatom
CoheytingAlgebra.toOrderTop
IsCompl.isAtom_iff_isCoatom
DistribLattice.instIsModularLattice
IsCompl.symm
isCompl_compl
isAtom_dual_iff_isCoatom 📖mathematicalIsAtom
OrderDual
OrderDual.instPreorder
OrderDual.instOrderBot
Preorder.toLE
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
IsCoatom
isAtom_iff_eq_top 📖mathematicalIsAtom
PartialOrder.toPreorder
BoundedOrder.toOrderBot
Preorder.toLE
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
IsSimpleOrder.eq_bot_or_eq_top
isAtom_top
isAtom_iff_le_of_ge 📖mathematicalIsAtom
Preorder.toLE
not_imp_comm
isAtom_top 📖mathematicalIsAtom
PartialOrder.toPreorder
BoundedOrder.toOrderBot
Preorder.toLE
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
top_ne_bot
IsSimpleOrder.toNontrivial
IsSimpleOrder.eq_bot_or_eq_top
ne_of_lt
isAtomic_dual_iff_isCoatomic 📖mathematicalIsAtomic
OrderDual
OrderDual.instPartialOrder
OrderDual.instOrderBot
Preorder.toLE
PartialOrder.toPreorder
IsCoatomic
IsAtomic.eq_bot_or_exists_atom_le
IsCoatomic.eq_top_or_exists_le_coatom
isAtomic_iff 📖mathematicalIsAtomic
Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
IsAtom
isAtomic_iff_forall_isAtomic_Iic 📖mathematicalIsAtomic
Set.Elem
Set.Iic
PartialOrder.toPreorder
Subtype.partialOrder
Set
Set.instMembership
Set.Iic.orderBot
IsAtomic.Set.Iic.isAtomic
le_refl
IsAtom.of_isAtom_coe_Iic
IsAtomic.eq_bot_or_exists_atom_le
isAtomic_iff_isCoatomic 📖mathematicalIsAtomic
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
IsCoatomic
BoundedOrder.toOrderTop
isCoatomic_of_isAtomic_of_complementedLattice_of_isModular
isAtomic_of_isCoatomic_of_complementedLattice_of_isModular
isAtomic_of_isCoatomic_of_complementedLattice_of_isModular 📖mathematicalIsAtomic
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
isCoatomic_dual_iff_isAtomic
isCoatomic_of_isAtomic_of_complementedLattice_of_isModular
instIsModularLatticeOrderDual
ComplementedLattice.instOrderDual
OrderDual.instIsAtomic
isAtomic_of_orderBot_wellFounded_lt 📖mathematicalPreorder.toLT
PartialOrder.toPreorder
IsAtomicIsStronglyAtomic.isAtomic
IsStronglyAtomic.of_wellFounded_lt
isAtomistic_dual_iff_isCoatomistic 📖mathematicalIsAtomistic
OrderDual
OrderDual.instPartialOrder
OrderDual.instOrderBot
Preorder.toLE
PartialOrder.toPreorder
IsCoatomistic
IsAtomistic.isLUB_atoms
IsCoatomistic.isGLB_coatoms
isAtomistic_iff 📖mathematicalIsAtomistic
IsLUB
Preorder.toLE
PartialOrder.toPreorder
IsAtom
isCoatom_bot 📖mathematicalIsCoatom
PartialOrder.toPreorder
BoundedOrder.toOrderTop
Preorder.toLE
Bot.bot
OrderBot.toBot
BoundedOrder.toOrderBot
isAtom_dual_iff_isCoatom
isAtom_top
OrderDual.instIsSimpleOrder
isCoatom_compl 📖mathematicalIsCoatom
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
CoheytingAlgebra.toOrderTop
Compl.compl
BooleanAlgebra.toCompl
IsAtom
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
IsCompl.isCoatom_iff_isAtom
DistribLattice.instIsModularLattice
IsCompl.symm
isCompl_compl
isCoatom_dual_iff_isAtom 📖mathematicalIsCoatom
OrderDual
OrderDual.instPreorder
OrderDual.instOrderTop
Preorder.toLE
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
IsAtom
isCoatom_iff_eq_bot 📖mathematicalIsCoatom
PartialOrder.toPreorder
BoundedOrder.toOrderTop
Preorder.toLE
Bot.bot
OrderBot.toBot
BoundedOrder.toOrderBot
IsSimpleOrder.eq_bot_or_eq_top
isCoatom_bot
isCoatom_iff_ge_of_le 📖mathematicalIsCoatom
Preorder.toLE
isAtom_iff_le_of_ge
isCoatomic_dual_iff_isAtomic 📖mathematicalIsCoatomic
OrderDual
OrderDual.instPartialOrder
OrderDual.instOrderTop
Preorder.toLE
PartialOrder.toPreorder
IsAtomic
IsCoatomic.eq_top_or_exists_le_coatom
IsAtomic.eq_bot_or_exists_atom_le
isCoatomic_iff 📖mathematicalIsCoatomic
Top.top
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
IsCoatom
isCoatomic_iff_forall_isCoatomic_Ici 📖mathematicalIsCoatomic
Set.Elem
Set.Ici
PartialOrder.toPreorder
Subtype.partialOrder
Set
Set.instMembership
Set.Ici.orderTop
isAtomic_dual_iff_isCoatomic
isAtomic_iff_forall_isAtomic_Iic
isCoatomic_dual_iff_isAtomic
isCoatomic_of_isAtomic_of_complementedLattice_of_isModular 📖mathematicalIsCoatomic
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
BoundedOrder.toOrderTop
Preorder.toLE
PartialOrder.toPreorder
ComplementedLattice.exists_isCompl
eq_top_of_isCompl_bot
IsCompl.symm
IsModularLattice.complementedLattice_Ici
IsCoatom.of_isCoatom_coe_Ici
IsCompl.isAtom_iff_isCoatom
IsModularLattice.isModularLattice_Ici
OrderIso.isAtom_iff
IsAtom.Iic
IsAtomic.eq_bot_or_exists_atom_le
isCoatomic_of_orderTop_gt_wellFounded 📖mathematicalPreorder.toLT
PartialOrder.toPreorder
IsCoatomicisAtomic_dual_iff_isCoatomic
isAtomic_of_orderBot_wellFounded_lt
isCoatomistic_dual_iff_isAtomistic 📖mathematicalIsCoatomistic
OrderDual
OrderDual.instPartialOrder
OrderDual.instOrderTop
Preorder.toLE
PartialOrder.toPreorder
IsAtomistic
IsCoatomistic.isGLB_coatoms
IsAtomistic.isLUB_atoms
isCoatomistic_iff 📖mathematicalIsCoatomistic
IsGLB
Preorder.toLE
PartialOrder.toPreorder
IsCoatom
isLUB_atoms_le 📖mathematicalIsLUB
Preorder.toLE
PartialOrder.toPreorder
setOf
IsAtom
IsAtomistic.isLUB_atoms
isLUB_atoms_top 📖mathematicalIsLUB
Preorder.toLE
PartialOrder.toPreorder
setOf
IsAtom
Top.top
OrderTop.toTop
isLUB_atoms_le
isSimpleOrder_iff 📖mathematicalIsSimpleOrder
Nontrivial
Bot.bot
OrderBot.toBot
BoundedOrder.toOrderBot
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
isSimpleOrder_iff_isAtom_top 📖mathematicalIsSimpleOrder
Preorder.toLE
PartialOrder.toPreorder
IsAtom
BoundedOrder.toOrderBot
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
isAtom_top
eq_or_lt_of_le
le_top
isSimpleOrder_iff_isCoatom_bot 📖mathematicalIsSimpleOrder
Preorder.toLE
PartialOrder.toPreorder
IsCoatom
BoundedOrder.toOrderTop
Bot.bot
OrderBot.toBot
BoundedOrder.toOrderBot
isSimpleOrder_iff_isSimpleOrder_orderDual
isSimpleOrder_iff_isAtom_top
isSimpleOrder_iff_isSimpleOrder_orderDual 📖mathematicalIsSimpleOrder
OrderDual
OrderDual.instLE
OrderDual.instBoundedOrder
OrderDual.instNontrivial
IsSimpleOrder.toNontrivial
IsSimpleOrder.eq_bot_or_eq_top
exists_pair_ne
isStronglyAtomic_dual_iff_is_stronglyCoatomic 📖mathematicalIsStronglyAtomic
OrderDual
OrderDual.instPreorder
IsStronglyCoatomic
isStronglyAtomic_iff 📖mathematicalIsStronglyAtomic
CovBy
Preorder.toLT
Preorder.toLE
isStronglyCoatomic_dual_iff_is_stronglyAtomic 📖mathematicalIsStronglyCoatomic
OrderDual
OrderDual.instPreorder
IsStronglyAtomic
isStronglyAtomic_dual_iff_is_stronglyCoatomic
isStronglyCoatomic_iff 📖mathematicalIsStronglyCoatomic
Preorder.toLE
CovBy
Preorder.toLT
le_iff_atom_le_imp 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
LE.le.trans
IsLUB.mono
isLUB_atoms_le
sSup_atoms_eq_top 📖mathematicalSupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
setOf
IsAtom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
CompleteLattice.toBoundedOrder
Top.top
OrderTop.toTop
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
BoundedOrder.toOrderTop
IsLUB.sSup_eq
isLUB_atoms_top
sSup_atoms_le_eq 📖mathematicalSupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
setOf
IsAtom
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
CompleteLattice.toBoundedOrder
IsLUB.sSup_eq
isLUB_atoms_le

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