Documentation Verification Report

Lattice

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

Statistics

MetricCount
DefinitionssUnionPowersetGI, sigmaEquiv, sigmaToiUnion, unionEqSigmaOfDisjoint
4
TheoremsiInter_nat_add, iInter_comp, iUnion_comp, iUnion_nat_add, iInter_comp, iInter_congr, iUnion_comp, iUnion_congr, Ici_iSup, Ici_iSupβ‚‚, Ici_sSup, Iic_iInf, Iic_iInfβ‚‚, Iic_sInf, of_sUnion, of_sUnion_eq_univ, univ, biInter_and, biInter_and', biInter_const, biInter_empty, biInter_eq_iInter, biInter_ge, biInter_ge_eq_iInf, biInter_gt_eq_iInf, biInter_iUnion, biInter_insert, biInter_inter, biInter_le, biInter_le_eq_iInter, biInter_lt_eq_iInter, biInter_mono, biInter_pair, biInter_singleton, biInter_subset_biInter_left, biInter_subset_biUnion, biInter_subset_of_mem, biInter_union, biInter_univ, biUnion_and, biUnion_and', biUnion_compl_eq_of_pairwise_disjoint_of_iUnion_eq_univ, biUnion_const, biUnion_diff_biUnion_subset, biUnion_empty, biUnion_eq_iUnion, biUnion_ge, biUnion_ge_eq_iUnion, biUnion_gt_eq_iUnion, biUnion_iUnion, biUnion_insert, biUnion_le, biUnion_le_eq_iUnion, biUnion_lt_eq_iUnion, biUnion_mono, biUnion_of_singleton, biUnion_pair, biUnion_self, biUnion_singleton, biUnion_subset_biUnion_left, biUnion_union, biUnion_univ, biUnion_univ_pi, coe_snd_unionEqSigmaOfDisjoint, coe_unionEqSigmaOfDisjoint_symm_apply, compl_iInter, compl_iInterβ‚‚, compl_iUnion, compl_iUnionβ‚‚, compl_sInter, compl_sUnion, diff_iInter, diff_iUnion, directedOn_iUnion, directedOn_sUnion, disjoint_iUnion_left, disjoint_iUnion_right, disjoint_iUnionβ‚‚_left, disjoint_iUnionβ‚‚_right, disjoint_sUnion_left, disjoint_sUnion_right, exists_set_mem_of_union_eq_top, iInf_eq_dif, iInter_and, iInter_coe_set, iInter_comm, iInter_congr, iInter_congr_Prop, iInter_congr_of_surjective, iInter_const, iInter_dite, iInter_eq_compl_iUnion_compl, iInter_eq_const, iInter_eq_empty_iff, iInter_eq_if, iInter_eq_univ, iInter_exists, iInter_false, iInter_ge_eq_iInter_nat_add, iInter_iInter_eq_left, iInter_iInter_eq_or_left, iInter_iInter_eq_right, iInter_inter, iInter_inter_distrib, iInter_ite, iInter_mono, iInter_mono', iInter_mono'', iInter_of_empty, iInter_option, iInter_or, iInter_plift_down, iInter_plift_up, iInter_psigma, iInter_psigma', iInter_setOf, iInter_sigma, iInter_sigma', iInter_subset, iInter_subset_iInterβ‚‚, iInter_subset_iUnion, iInter_subset_of_subset, iInter_subtype, iInter_sum, iInter_true, iInter_union, iInter_union_of_antitone, iInter_union_of_monotone, iInter_univ, iInterβ‚‚_comm, iInterβ‚‚_congr, iInterβ‚‚_eq_empty_iff, iInterβ‚‚_mono, iInterβ‚‚_mono', iInterβ‚‚_subset, iInterβ‚‚_subset_of_subset, iInterβ‚‚_union, iUnion_and, iUnion_coe_set, iUnion_comm, iUnion_congr, iUnion_congr_Prop, iUnion_congr_of_surjective, iUnion_const, iUnion_diff, iUnion_dite, iUnion_empty, iUnion_eq_compl_iInter_compl, iUnion_eq_const, iUnion_eq_dif, iUnion_eq_empty, iUnion_eq_if, iUnion_eq_range_psigma, iUnion_eq_range_sigma, iUnion_eq_univ_iff, iUnion_exists, iUnion_false, iUnion_ge_eq_iUnion_nat_add, iUnion_iInter_ge_nat_add, iUnion_iInter_subset, iUnion_iUnion_eq_left, iUnion_iUnion_eq_or_left, iUnion_iUnion_eq_right, iUnion_image_preimage_sigma_mk_eq_self, iUnion_insert_eq_range_union_iUnion, iUnion_inter, iUnion_inter_of_antitone, iUnion_inter_of_monotone, iUnion_inter_subset, iUnion_ite, iUnion_le_nat, iUnion_mono, iUnion_mono', iUnion_mono'', iUnion_nonempty_index, iUnion_nonempty_self, iUnion_of_empty, iUnion_of_singleton, iUnion_of_singleton_coe, iUnion_option, iUnion_or, iUnion_plift_down, iUnion_plift_up, iUnion_psigma, iUnion_psigma', iUnion_range_eq_iUnion, iUnion_range_eq_sUnion, iUnion_setOf, iUnion_sigma, iUnion_sigma', iUnion_singleton_eq_range, iUnion_subset, iUnion_subset_iUnion_const, iUnion_subset_iff, iUnion_subtype, iUnion_sum, iUnion_true, iUnion_union, iUnion_union_distrib, iUnion_univ_pi, iUnionβ‚‚_comm, iUnionβ‚‚_congr, iUnionβ‚‚_eq_univ_iff, iUnionβ‚‚_inter, iUnionβ‚‚_mono, iUnionβ‚‚_mono', iUnionβ‚‚_subset, iUnionβ‚‚_subset_iUnion, iUnionβ‚‚_subset_iff, insert_iInter, insert_iUnion, inter_biInter, inter_empty_of_inter_sUnion_empty, inter_eq_iInter, inter_iInter, inter_iInter_nat_succ, inter_iUnion, inter_iUnionβ‚‚, mem_biInter, mem_biUnion, mem_iInter_of_mem, mem_iInterβ‚‚, mem_iInterβ‚‚_of_mem, mem_iUnion_of_mem, mem_iUnionβ‚‚, mem_iUnionβ‚‚_of_mem, mem_sUnion_of_mem, nonempty_biUnion, nonempty_iInter, nonempty_iInter_Ici_iff, nonempty_iInter_Iic_iff, nonempty_iInterβ‚‚, nonempty_iUnion, nonempty_of_nonempty_iUnion, nonempty_of_nonempty_iUnion_eq_univ, nonempty_of_union_eq_top_of_nonempty, nonempty_sInter, nonempty_sUnion, notMem_of_notMem_sUnion, pi_def, pi_diff_pi_subset, pi_iUnion_eq_iInter_pi, range_sigma_eq_iUnion_range, sInter_diff_singleton_univ, sInter_empty, sInter_eq_biInter, sInter_eq_compl_sUnion_compl, sInter_eq_empty_iff, sInter_eq_iInter, sInter_eq_univ, sInter_iUnion, sInter_image, sInter_image2, sInter_insert, sInter_mono, sInter_pair, sInter_range, sInter_singleton, sInter_subset_of_mem, sInter_subset_sInter, sInter_union, sInter_union_sInter, sUnion_diff_singleton_empty, sUnion_empty, sUnion_eq_biUnion, sUnion_eq_compl_sInter_compl, sUnion_eq_empty, sUnion_eq_iUnion, sUnion_eq_univ_iff, sUnion_iUnion, sUnion_image, sUnion_image2, sUnion_insert, sUnion_inter_sUnion, sUnion_mem_empty_univ, sUnion_mono, sUnion_mono_subsets, sUnion_mono_supsets, sUnion_pair, sUnion_powerset_gc, sUnion_range, sUnion_singleton, sUnion_subset, sUnion_subset_iff, sUnion_subset_sUnion, sUnion_union, setOf_exists, setOf_forall, sigmaToiUnion_bijective, sigmaToiUnion_injective, sigmaToiUnion_surjective, subset_biUnion_of_mem, subset_iInter, subset_iInter_iff, subset_iInterβ‚‚, subset_iInterβ‚‚_iff, subset_iUnion, subset_iUnion_of_subset, subset_iUnionβ‚‚, subset_iUnionβ‚‚_of_subset, subset_powerset_iff, subset_sInter, subset_sInter_iff, subset_sUnion_of_mem, subset_sUnion_of_subset, union_distrib_iInter_left, union_distrib_iInter_right, union_distrib_iInterβ‚‚_left, union_distrib_iInterβ‚‚_right, union_eq_iUnion, union_iInter, union_iInterβ‚‚, union_iUnion, union_iUnion_nat_succ, univ_pi_eq_iInter, exists_sUnion, forall_sUnion, iInf_iUnion, iInf_sUnion, iSup_iUnion, iSup_sUnion, sInf_sUnion, sSup_iUnion, sSup_sUnion
324
Total328

Antitone

Theorems

NameKindAssumesProvesValidatesDepends On
iInter_nat_add πŸ“–mathematicalAntitone
Set
Nat.instPreorder
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
Set.iInterβ€”iInf_nat_add

Function.Surjective

Theorems

NameKindAssumesProvesValidatesDepends On
iInter_comp πŸ“–mathematicalβ€”Set.iInterβ€”iInf_comp
iUnion_comp πŸ“–mathematicalβ€”Set.iUnionβ€”iSup_comp

Monotone

Theorems

NameKindAssumesProvesValidatesDepends On
iUnion_nat_add πŸ“–mathematicalMonotone
Set
Nat.instPreorder
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
Set.iUnionβ€”iSup_nat_add

Set

Definitions

NameCategoryTheorems
sUnionPowersetGI πŸ“–CompOpβ€”
sigmaEquiv πŸ“–CompOpβ€”
sigmaToiUnion πŸ“–CompOp
3 mathmath: sigmaToiUnion_bijective, sigmaToiUnion_injective, sigmaToiUnion_surjective
unionEqSigmaOfDisjoint πŸ“–CompOp
2 mathmath: coe_snd_unionEqSigmaOfDisjoint, coe_unionEqSigmaOfDisjoint_symm_apply

Theorems

NameKindAssumesProvesValidatesDepends On
Ici_iSup πŸ“–mathematicalβ€”Ici
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
iInter
β€”ext
Ici_iSupβ‚‚ πŸ“–mathematicalβ€”Ici
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
iInter
β€”Ici_iSup
Ici_sSup πŸ“–mathematicalβ€”Ici
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
SupSet.sSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
iInter
Set
instMembership
β€”sSup_eq_iSup
Ici_iSupβ‚‚
Iic_iInf πŸ“–mathematicalβ€”Iic
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
iInf
CompleteSemilatticeInf.toInfSet
iInter
β€”ext
Iic_iInfβ‚‚ πŸ“–mathematicalβ€”Iic
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
iInf
CompleteSemilatticeInf.toInfSet
iInter
β€”Iic_iInf
Iic_sInf πŸ“–mathematicalβ€”Iic
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
InfSet.sInf
CompleteSemilatticeInf.toInfSet
iInter
Set
instMembership
β€”sInf_eq_iInf
Iic_iInfβ‚‚
biInter_and πŸ“–mathematicalβ€”iInterβ€”iInter_and
iInter_comm
biInter_and' πŸ“–mathematicalβ€”iInterβ€”iInter_and
iInter_comm
biInter_const πŸ“–mathematicalNonemptyiInter
Set
instMembership
β€”biInf_const
biInter_empty πŸ“–mathematicalβ€”iInter
Set
instMembership
instEmptyCollection
univ
β€”iInf_emptyset
biInter_eq_iInter πŸ“–mathematicalβ€”iInter
Set
instMembership
Elem
β€”iInf_subtype'
biInter_ge πŸ“–mathematicalβ€”iInter
Preorder.toLE
PartialOrder.toPreorder
Set
instInter
Preorder.toLT
β€”biInf_ge_eq_inf
biInter_ge_eq_iInf πŸ“–mathematicalβ€”iInter
Preorder.toLE
β€”biInf_ge_eq_iInf
biInter_gt_eq_iInf πŸ“–mathematicalβ€”iInterβ€”biInf_gt_eq_iInf
biInter_iUnion πŸ“–mathematicalβ€”iInter
Set
instMembership
iUnion
β€”iInter_congr_Prop
iInter_exists
iInter_comm
biInter_insert πŸ“–mathematicalβ€”iInter
Set
instMembership
instInsert
instInter
β€”iInter_congr_Prop
iInter_iInter_eq_or_left
biInter_inter πŸ“–mathematicalNonemptyiInter
Set
instMembership
instInter
β€”biInter_eq_iInter
Nonempty.to_subtype
biInter_le πŸ“–mathematicalβ€”iInter
Preorder.toLE
PartialOrder.toPreorder
Set
instInter
Preorder.toLT
β€”biInf_le_eq_inf
biInter_le_eq_iInter πŸ“–mathematicalβ€”iInter
Preorder.toLE
β€”biInf_le_eq_iInf
biInter_lt_eq_iInter πŸ“–mathematicalβ€”iInterβ€”biInf_lt_eq_iInf
biInter_mono πŸ“–mathematicalSet
instHasSubset
iInter
instMembership
β€”HasSubset.Subset.trans
instIsTransSubset
biInter_subset_biInter_left
iInterβ‚‚_mono
biInter_pair πŸ“–mathematicalβ€”iInter
Set
instMembership
instInsert
instSingletonSet
instInter
β€”biInter_insert
biInter_singleton
biInter_singleton πŸ“–mathematicalβ€”iInter
Set
instMembership
instSingletonSet
β€”iInf_singleton
biInter_subset_biInter_left πŸ“–mathematicalSet
instHasSubset
iInter
instMembership
β€”subset_iInterβ‚‚
biInter_subset_of_mem
biInter_subset_biUnion πŸ“–mathematicalNonemptySet
instHasSubset
iInter
instMembership
iUnion
β€”biInf_le_biSup
biInter_subset_of_mem πŸ“–mathematicalSet
instMembership
instHasSubset
iInter
β€”iInterβ‚‚_subset
biInter_union πŸ“–mathematicalβ€”iInter
Set
instMembership
instUnion
instInter
β€”iInf_union
biInter_univ πŸ“–mathematicalβ€”iInter
Set
instMembership
univ
β€”iInf_univ
biUnion_and πŸ“–mathematicalβ€”iUnionβ€”iUnion_and
iUnion_comm
biUnion_and' πŸ“–mathematicalβ€”iUnionβ€”iUnion_and
iUnion_comm
biUnion_compl_eq_of_pairwise_disjoint_of_iUnion_eq_univ πŸ“–mathematicaliUnion
univ
Pairwise
Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Compl.compl
instCompl
instMembership
β€”ext
Disjoint.ne_of_mem
mem_compl_iff
biUnion_const πŸ“–mathematicalNonemptyiUnion
Set
instMembership
β€”biSup_const
biUnion_diff_biUnion_subset πŸ“–mathematicalβ€”Set
instHasSubset
instSDiff
iUnion
instMembership
β€”biUnion_subset_biUnion_left
union_diff_self
subset_union_right
biUnion_empty πŸ“–mathematicalβ€”iUnion
Set
instMembership
instEmptyCollection
β€”iSup_emptyset
biUnion_eq_iUnion πŸ“–mathematicalβ€”iUnion
Set
instMembership
Elem
β€”iSup_subtype'
biUnion_ge πŸ“–mathematicalβ€”iUnion
Preorder.toLE
PartialOrder.toPreorder
Set
instUnion
Preorder.toLT
β€”biSup_ge_eq_sup
biUnion_ge_eq_iUnion πŸ“–mathematicalβ€”iUnion
Preorder.toLE
β€”biSup_ge_eq_iSup
biUnion_gt_eq_iUnion πŸ“–mathematicalβ€”iUnionβ€”biSup_gt_eq_iSup
biUnion_iUnion πŸ“–mathematicalβ€”iUnion
Set
instMembership
β€”iUnion_congr_Prop
iUnion_exists
iUnion_comm
biUnion_insert πŸ“–mathematicalβ€”iUnion
Set
instMembership
instInsert
instUnion
β€”iUnion_congr_Prop
iUnion_iUnion_eq_or_left
biUnion_le πŸ“–mathematicalβ€”iUnion
Preorder.toLE
PartialOrder.toPreorder
Set
instUnion
Preorder.toLT
β€”biSup_le_eq_sup
biUnion_le_eq_iUnion πŸ“–mathematicalβ€”iUnion
Preorder.toLE
β€”biSup_le_eq_iSup
biUnion_lt_eq_iUnion πŸ“–mathematicalβ€”iUnionβ€”biSup_lt_eq_iSup
biUnion_mono πŸ“–mathematicalSet
instHasSubset
iUnion
instMembership
β€”HasSubset.Subset.trans
instIsTransSubset
biUnion_subset_biUnion_left
iUnionβ‚‚_mono
biUnion_of_singleton πŸ“–mathematicalβ€”iUnion
Set
instMembership
instSingletonSet
β€”ext
biUnion_pair πŸ“–mathematicalβ€”iUnion
Set
instMembership
instInsert
instSingletonSet
instUnion
β€”iUnion_congr_Prop
iUnion_iUnion_eq_or_left
iUnion_iUnion_eq_left
biUnion_self πŸ“–mathematicalβ€”iUnion
Set
instMembership
β€”Subset.antisymm
iUnionβ‚‚_subset
Subset.refl
mem_biUnion
biUnion_singleton πŸ“–mathematicalβ€”iUnion
Set
instMembership
instSingletonSet
β€”iSup_singleton
biUnion_subset_biUnion_left πŸ“–mathematicalSet
instHasSubset
iUnion
instMembership
β€”iUnionβ‚‚_subset
subset_biUnion_of_mem
biUnion_union πŸ“–mathematicalβ€”iUnion
Set
instMembership
instUnion
β€”iSup_union
biUnion_univ πŸ“–mathematicalβ€”iUnion
Set
instMembership
univ
β€”iSup_univ
biUnion_univ_pi πŸ“–mathematicalβ€”iUnion
Set
instMembership
pi
univ
β€”ext
iUnion_congr_Prop
coe_snd_unionEqSigmaOfDisjoint πŸ“–mathematicalPairwise
Function.onFun
Set
Disjoint
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
instMembership
Elem
DFunLike.coe
Equiv
iUnion
EquivLike.toFunLike
Equiv.instEquivLike
unionEqSigmaOfDisjoint
β€”Equiv.symm_apply_apply
coe_unionEqSigmaOfDisjoint_symm_apply πŸ“–mathematicalPairwise
Function.onFun
Set
Disjoint
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
instMembership
iUnion
DFunLike.coe
Equiv
Elem
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
unionEqSigmaOfDisjoint
β€”β€”
compl_iInter πŸ“–mathematicalβ€”Compl.compl
Set
instCompl
iInter
iUnion
β€”compl_iInf
compl_iInterβ‚‚ πŸ“–mathematicalβ€”Compl.compl
Set
instCompl
iInter
iUnion
β€”compl_iInter
compl_iUnion πŸ“–mathematicalβ€”Compl.compl
Set
instCompl
iUnion
iInter
β€”compl_iSup
compl_iUnionβ‚‚ πŸ“–mathematicalβ€”Compl.compl
Set
instCompl
iUnion
iInter
β€”compl_iUnion
compl_sInter πŸ“–mathematicalβ€”Compl.compl
Set
instCompl
sInter
sUnion
image
β€”sUnion_eq_compl_sInter_compl
compl_compl_image
compl_sUnion πŸ“–mathematicalβ€”Compl.compl
Set
instCompl
sUnion
sInter
image
β€”ext
sInter_image
diff_iInter πŸ“–mathematicalβ€”Set
instSDiff
iInter
iUnion
β€”diff_eq
compl_iInter
inter_iUnion
diff_iUnion πŸ“–mathematicalβ€”Set
instSDiff
iUnion
iInter
β€”diff_eq
compl_iUnion
inter_iInter
directedOn_iUnion πŸ“–mathematicalDirected
Set
instHasSubset
DirectedOn
iUnionβ€”β€”
directedOn_sUnion πŸ“–mathematicalDirectedOn
Set
instHasSubset
sUnionβ€”sUnion_eq_iUnion
directedOn_iUnion
directedOn_iff_directed
disjoint_iUnion_left πŸ“–mathematicalβ€”Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
iUnion
β€”iSup_disjoint_iff
disjoint_iUnion_right πŸ“–mathematicalβ€”Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
iUnion
β€”disjoint_iSup_iff
disjoint_iUnionβ‚‚_left πŸ“–mathematicalβ€”Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
iUnion
β€”iSupβ‚‚_disjoint_iff
disjoint_iUnionβ‚‚_right πŸ“–mathematicalβ€”Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
iUnion
β€”disjoint_iSupβ‚‚_iff
disjoint_sUnion_left πŸ“–mathematicalβ€”Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
sUnion
β€”sSup_disjoint_iff
disjoint_sUnion_right πŸ“–mathematicalβ€”Disjoint
Set
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
sUnion
β€”disjoint_sSup_iff
exists_set_mem_of_union_eq_top πŸ“–β€”iUnion
Set
instMembership
Top.top
BooleanAlgebra.toTop
instBooleanAlgebra
β€”β€”mem_univ
mem_iUnion
iInf_eq_dif πŸ“–mathematicalβ€”iInter
Set
univ
β€”iInf_eq_dif
iInter_and πŸ“–mathematicalβ€”iInterβ€”iInf_and
iInter_coe_set πŸ“–mathematicalβ€”iInter
Elem
Set
instMembership
β€”iInter_subtype
iInter_comm πŸ“–mathematicalβ€”iInterβ€”iInf_comm
iInter_congr πŸ“–mathematicalβ€”iInterβ€”iInf_congr
iInter_congr_Prop πŸ“–mathematicalβ€”iInterβ€”iInf_congr_Prop
iInter_congr_of_surjective πŸ“–mathematicalβ€”iInterβ€”Function.Surjective.iInf_congr
iInter_const πŸ“–mathematicalβ€”iInterβ€”iInf_const
iInter_dite πŸ“–mathematicalβ€”iInter
Set
instInter
β€”iInf_dite
iInter_eq_compl_iUnion_compl πŸ“–mathematicalβ€”iInter
Compl.compl
Set
instCompl
iUnion
β€”compl_iUnion
compl_compl
iInter_eq_const πŸ“–mathematicalβ€”iInterβ€”iInter_congr
iInter_const
iInter_eq_empty_iff πŸ“–mathematicalβ€”iInter
Set
instEmptyCollection
instMembership
β€”β€”
iInter_eq_if πŸ“–mathematicalβ€”iInter
Set
univ
β€”iInf_eq_if
iInter_eq_univ πŸ“–mathematicalβ€”iInter
univ
β€”iInf_eq_top
iInter_exists πŸ“–mathematicalβ€”iInterβ€”iInf_exists
iInter_false πŸ“–mathematicalβ€”iInter
univ
β€”iInf_false
iInter_ge_eq_iInter_nat_add πŸ“–mathematicalβ€”iInterβ€”iInf_ge_eq_iInf_nat_add
iInter_iInter_eq_left πŸ“–mathematicalβ€”iInterβ€”iInf_iInf_eq_left
iInter_iInter_eq_or_left πŸ“–mathematicalβ€”iInter
Set
instInter
β€”iInter_or
iInter_inter_distrib
iInter_iInter_eq_left
iInter_iInter_eq_right πŸ“–mathematicalβ€”iInterβ€”iInf_iInf_eq_right
iInter_inter πŸ“–mathematicalβ€”Set
instInter
iInter
β€”iInf_inf
iInter_inter_distrib πŸ“–mathematicalβ€”iInter
Set
instInter
β€”iInf_inf_eq
iInter_ite πŸ“–mathematicalβ€”iInter
Set
instInter
β€”iInter_dite
iInter_mono πŸ“–mathematicalSet
instHasSubset
iInterβ€”iInf_mono
iInter_mono' πŸ“–mathematicalSet
instHasSubset
iInterβ€”subset_iInter
iInter_subset_of_subset
iInter_mono'' πŸ“–mathematicalSet
instHasSubset
iInterβ€”iInf_mono
iInter_of_empty πŸ“–mathematicalβ€”iInter
univ
β€”iInf_of_empty
iInter_option πŸ“–mathematicalβ€”iInter
Set
instInter
β€”iInf_option
iInter_or πŸ“–mathematicalβ€”iInter
Set
instInter
β€”iInf_or
iInter_plift_down πŸ“–mathematicalβ€”iInterβ€”iInf_plift_down
iInter_plift_up πŸ“–mathematicalβ€”iInterβ€”iInf_plift_up
iInter_psigma πŸ“–mathematicalβ€”iInterβ€”iInf_psigma
iInter_psigma' πŸ“–mathematicalβ€”iInterβ€”iInf_psigma'
iInter_setOf πŸ“–mathematicalβ€”iInter
setOf
β€”ext
mem_iInter
iInter_sigma πŸ“–mathematicalβ€”iInterβ€”iInf_sigma
iInter_sigma' πŸ“–mathematicalβ€”iInterβ€”iInf_sigma'
iInter_subset πŸ“–mathematicalβ€”Set
instHasSubset
iInter
β€”iInf_le
iInter_subset_iInterβ‚‚ πŸ“–mathematicalβ€”Set
instHasSubset
iInter
β€”iInter_mono
subset_iInter
Subset.rfl
iInter_subset_iUnion πŸ“–mathematicalβ€”Set
instHasSubset
iInter
iUnion
β€”iInf_le_iSup
iInter_subset_of_subset πŸ“–mathematicalSet
instHasSubset
iInterβ€”iInf_le_of_le
iInter_subtype πŸ“–mathematicalβ€”iInterβ€”iInf_subtype
iInter_sum πŸ“–mathematicalβ€”iInter
Set
instInter
β€”iInf_sum
iInter_true πŸ“–mathematicalβ€”iInterβ€”iInf_true
iInter_union πŸ“–mathematicalβ€”Set
instUnion
iInter
β€”iInf_sup_eq
iInter_union_of_antitone πŸ“–mathematicalAntitone
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
iInter
instUnion
β€”iInf_sup_of_antitone
iInter_union_of_monotone πŸ“–mathematicalMonotone
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
iInter
instUnion
β€”iInf_sup_of_monotone
iInter_univ πŸ“–mathematicalβ€”iInter
univ
β€”iInf_top
iInterβ‚‚_comm πŸ“–mathematicalβ€”iInterβ€”iInfβ‚‚_comm
iInterβ‚‚_congr πŸ“–mathematicalβ€”iInterβ€”iInter_congr
iInterβ‚‚_eq_empty_iff πŸ“–mathematicalβ€”iInter
Set
instEmptyCollection
instMembership
β€”β€”
iInterβ‚‚_mono πŸ“–mathematicalSet
instHasSubset
iInterβ€”iInfβ‚‚_mono
iInterβ‚‚_mono' πŸ“–mathematicalSet
instHasSubset
iInterβ€”subset_iInterβ‚‚_iff
HasSubset.Subset.trans
instIsTransSubset
iInterβ‚‚_subset
iInterβ‚‚_subset πŸ“–mathematicalβ€”Set
instHasSubset
iInter
β€”iInfβ‚‚_le
iInterβ‚‚_subset_of_subset πŸ“–mathematicalSet
instHasSubset
iInterβ€”iInfβ‚‚_le_of_le
iInterβ‚‚_union πŸ“–mathematicalβ€”Set
instUnion
iInter
β€”iInter_union
iUnion_and πŸ“–mathematicalβ€”iUnionβ€”iSup_and
iUnion_coe_set πŸ“–mathematicalβ€”iUnion
Elem
Set
instMembership
β€”iUnion_subtype
iUnion_comm πŸ“–mathematicalβ€”iUnionβ€”iSup_comm
iUnion_congr πŸ“–mathematicalβ€”iUnionβ€”iSup_congr
iUnion_congr_Prop πŸ“–mathematicalβ€”iUnionβ€”iSup_congr_Prop
iUnion_congr_of_surjective πŸ“–mathematicalβ€”iUnionβ€”Function.Surjective.iSup_congr
iUnion_const πŸ“–mathematicalβ€”iUnionβ€”iSup_const
iUnion_diff πŸ“–mathematicalβ€”Set
instSDiff
iUnion
β€”iUnion_inter
iUnion_dite πŸ“–mathematicalβ€”iUnion
Set
instUnion
β€”iSup_dite
iUnion_empty πŸ“–mathematicalβ€”iUnion
Set
instEmptyCollection
β€”iSup_bot
iUnion_eq_compl_iInter_compl πŸ“–mathematicalβ€”iUnion
Compl.compl
Set
instCompl
iInter
β€”compl_iInter
compl_compl
iUnion_eq_const πŸ“–mathematicalβ€”iUnionβ€”iUnion_congr
iUnion_const
iUnion_eq_dif πŸ“–mathematicalβ€”iUnion
Set
instEmptyCollection
β€”iSup_eq_dif
iUnion_eq_empty πŸ“–mathematicalβ€”iUnion
Set
instEmptyCollection
β€”iSup_eq_bot
iUnion_eq_if πŸ“–mathematicalβ€”iUnion
Set
instEmptyCollection
β€”iSup_eq_if
iUnion_eq_range_psigma πŸ“–mathematicalβ€”iUnion
range
Elem
Set
instMembership
β€”β€”
iUnion_eq_range_sigma πŸ“–mathematicalβ€”iUnion
range
Elem
Set
instMembership
β€”β€”
iUnion_eq_univ_iff πŸ“–mathematicalβ€”iUnion
univ
Set
instMembership
β€”β€”
iUnion_exists πŸ“–mathematicalβ€”iUnionβ€”iSup_exists
iUnion_false πŸ“–mathematicalβ€”iUnion
Set
instEmptyCollection
β€”iSup_false
iUnion_ge_eq_iUnion_nat_add πŸ“–mathematicalβ€”iUnionβ€”iSup_ge_eq_iSup_nat_add
iUnion_iInter_ge_nat_add πŸ“–mathematicalβ€”iUnion
iInter
β€”iSup_iInf_ge_nat_add
iUnion_iInter_subset πŸ“–mathematicalβ€”Set
instHasSubset
iUnion
iInter
β€”iSup_iInf_le_iInf_iSup
iUnion_iUnion_eq_left πŸ“–mathematicalβ€”iUnionβ€”iSup_iSup_eq_left
iUnion_iUnion_eq_or_left πŸ“–mathematicalβ€”iUnion
Set
instUnion
β€”iUnion_or
iUnion_union_distrib
iUnion_iUnion_eq_left
iUnion_iUnion_eq_right πŸ“–mathematicalβ€”iUnionβ€”iSup_iSup_eq_right
iUnion_image_preimage_sigma_mk_eq_self πŸ“–mathematicalβ€”iUnion
image
preimage
β€”ext
iUnion_insert_eq_range_union_iUnion πŸ“–mathematicalβ€”iUnion
Set
instInsert
instUnion
range
β€”iUnion_union_distrib
union_comm
iUnion_singleton_eq_range
iUnion_inter πŸ“–mathematicalβ€”Set
instInter
iUnion
β€”iSup_inf_eq
iUnion_inter_of_antitone πŸ“–mathematicalAntitone
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
iUnion
instInter
β€”iSup_inf_of_antitone
iUnion_inter_of_monotone πŸ“–mathematicalMonotone
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
iUnion
instInter
β€”iSup_inf_of_monotone
iUnion_inter_subset πŸ“–mathematicalβ€”Set
instHasSubset
iUnion
instInter
β€”le_iSup_inf_iSup
iUnion_ite πŸ“–mathematicalβ€”iUnion
Set
instUnion
β€”iUnion_dite
iUnion_le_nat πŸ“–mathematicalβ€”iUnion
setOf
univ
β€”subset_antisymm
instAntisymmSubset
subset_univ
mem_iUnion_of_mem
mem_setOf
le_refl
iUnion_mono πŸ“–mathematicalSet
instHasSubset
iUnionβ€”iSup_mono
iUnion_mono' πŸ“–mathematicalSet
instHasSubset
iUnionβ€”iSup_mono'
iUnion_mono'' πŸ“–mathematicalSet
instHasSubset
iUnionβ€”iSup_mono
iUnion_nonempty_index πŸ“–mathematicalβ€”iUnion
Nonempty
Set
instMembership
β€”iSup_exists
iUnion_nonempty_self πŸ“–mathematicalβ€”iUnion
Nonempty
β€”iUnion_nonempty_index
biUnion_self
iUnion_of_empty πŸ“–mathematicalβ€”iUnion
Set
instEmptyCollection
β€”iSup_of_empty
iUnion_of_singleton πŸ“–mathematicalβ€”iUnion
Set
instSingletonSet
univ
β€”β€”
iUnion_of_singleton_coe πŸ“–mathematicalβ€”iUnion
Elem
Set
instSingletonSet
instMembership
β€”iUnion_singleton_eq_range
Subtype.range_coe_subtype
iUnion_option πŸ“–mathematicalβ€”iUnion
Set
instUnion
β€”iSup_option
iUnion_or πŸ“–mathematicalβ€”iUnion
Set
instUnion
β€”iSup_or
iUnion_plift_down πŸ“–mathematicalβ€”iUnionβ€”iSup_plift_down
iUnion_plift_up πŸ“–mathematicalβ€”iUnionβ€”iSup_plift_up
iUnion_psigma πŸ“–mathematicalβ€”iUnionβ€”iSup_psigma
iUnion_psigma' πŸ“–mathematicalβ€”iUnionβ€”iSup_psigma'
iUnion_range_eq_iUnion πŸ“–mathematicalElemiUnion
range
Set
instMembership
β€”ext
mem_iUnion
iUnion_range_eq_sUnion πŸ“–mathematicalElem
Set
instMembership
iUnion
range
sUnion
β€”ext
iUnion_setOf πŸ“–mathematicalβ€”iUnion
setOf
β€”ext
mem_iUnion
iUnion_sigma πŸ“–mathematicalβ€”iUnionβ€”iSup_sigma
iUnion_sigma' πŸ“–mathematicalβ€”iUnionβ€”iSup_sigma'
iUnion_singleton_eq_range πŸ“–mathematicalβ€”iUnion
Set
instSingletonSet
range
β€”ext
iUnion_subset πŸ“–mathematicalSet
instHasSubset
iUnionβ€”iSup_le
iUnion_subset_iUnion_const πŸ“–mathematicalβ€”Set
instHasSubset
iUnion
β€”iSup_const_mono
iUnion_subset_iff πŸ“–mathematicalβ€”Set
instHasSubset
iUnion
β€”Subset.trans
le_iSup
iUnion_subset
iUnion_subtype πŸ“–mathematicalβ€”iUnionβ€”iSup_subtype
iUnion_sum πŸ“–mathematicalβ€”iUnion
Set
instUnion
β€”iSup_sum
iUnion_true πŸ“–mathematicalβ€”iUnionβ€”iSup_true
iUnion_union πŸ“–mathematicalβ€”Set
instUnion
iUnion
β€”iSup_sup
iUnion_union_distrib πŸ“–mathematicalβ€”iUnion
Set
instUnion
β€”iSup_sup_eq
iUnion_univ_pi πŸ“–mathematicalβ€”iUnion
pi
univ
β€”ext
iUnionβ‚‚_comm πŸ“–mathematicalβ€”iUnionβ€”iSupβ‚‚_comm
iUnionβ‚‚_congr πŸ“–mathematicalβ€”iUnionβ€”iUnion_congr
iUnionβ‚‚_eq_univ_iff πŸ“–mathematicalβ€”iUnion
univ
Set
instMembership
β€”β€”
iUnionβ‚‚_inter πŸ“–mathematicalβ€”Set
instInter
iUnion
β€”iUnion_inter
iUnionβ‚‚_mono πŸ“–mathematicalSet
instHasSubset
iUnionβ€”iSupβ‚‚_mono
iUnionβ‚‚_mono' πŸ“–mathematicalSet
instHasSubset
iUnionβ€”iSupβ‚‚_mono'
iUnionβ‚‚_subset πŸ“–mathematicalSet
instHasSubset
iUnionβ€”iUnion_subset
iUnionβ‚‚_subset_iUnion πŸ“–mathematicalβ€”Set
instHasSubset
iUnion
β€”iUnion_mono
iUnion_subset
Subset.rfl
iUnionβ‚‚_subset_iff πŸ“–mathematicalβ€”Set
instHasSubset
iUnion
β€”β€”
insert_iInter πŸ“–mathematicalβ€”Set
instInsert
iInter
β€”iInter_union
insert_iUnion πŸ“–mathematicalβ€”Set
instInsert
iUnion
β€”iUnion_union
inter_biInter πŸ“–mathematicalNonemptyiInter
Set
instMembership
instInter
β€”inter_comm
biInter_inter
iInter_congr_Prop
inter_empty_of_inter_sUnion_empty πŸ“–β€”Set
instMembership
instInter
sUnion
instEmptyCollection
β€”β€”eq_empty_of_subset_empty
inter_subset_inter_right
subset_sUnion_of_mem
inter_eq_iInter πŸ“–mathematicalβ€”Set
instInter
iInter
β€”inf_eq_iInf
inter_iInter πŸ“–mathematicalβ€”Set
instInter
iInter
β€”inf_iInf
inter_iInter_nat_succ πŸ“–mathematicalβ€”Set
instInter
iInter
β€”inf_iInf_nat_succ
inter_iUnion πŸ“–mathematicalβ€”Set
instInter
iUnion
β€”inf_iSup_eq
inter_iUnionβ‚‚ πŸ“–mathematicalβ€”Set
instInter
iUnion
β€”inter_iUnion
mem_biInter πŸ“–mathematicalSet
instMembership
iInterβ€”mem_iInterβ‚‚_of_mem
mem_biUnion πŸ“–mathematicalSet
instMembership
iUnionβ€”mem_iUnionβ‚‚_of_mem
mem_iInter_of_mem πŸ“–mathematicalSet
instMembership
iInterβ€”mem_iInter
mem_iInterβ‚‚ πŸ“–mathematicalβ€”Set
instMembership
iInter
β€”β€”
mem_iInterβ‚‚_of_mem πŸ“–mathematicalSet
instMembership
iInterβ€”mem_iInterβ‚‚
mem_iUnion_of_mem πŸ“–mathematicalSet
instMembership
iUnionβ€”mem_iUnion
mem_iUnionβ‚‚ πŸ“–mathematicalβ€”Set
instMembership
iUnion
β€”β€”
mem_iUnionβ‚‚_of_mem πŸ“–mathematicalSet
instMembership
iUnionβ€”mem_iUnionβ‚‚
mem_sUnion_of_mem πŸ“–mathematicalSet
instMembership
sUnionβ€”β€”
nonempty_biUnion πŸ“–mathematicalβ€”Nonempty
iUnion
Set
instMembership
β€”β€”
nonempty_iInter πŸ“–mathematicalβ€”Nonempty
iInter
Set
instMembership
β€”β€”
nonempty_iInter_Ici_iff πŸ“–mathematicalβ€”Nonempty
iInter
Ici
BddAbove
Preorder.toLE
range
β€”nonempty_iInter_Iic_iff
nonempty_iInter_Iic_iff πŸ“–mathematicalβ€”Nonempty
iInter
Iic
BddBelow
Preorder.toLE
range
β€”ext
nonempty_iInterβ‚‚ πŸ“–mathematicalβ€”Nonempty
iInter
Set
instMembership
β€”β€”
nonempty_iUnion πŸ“–mathematicalβ€”Nonempty
iUnion
β€”β€”
nonempty_of_nonempty_iUnion πŸ“–β€”Nonempty
iUnion
β€”β€”mem_iUnion
nonempty_of_nonempty_iUnion_eq_univ πŸ“–β€”iUnion
univ
β€”β€”nonempty_of_nonempty_iUnion
univ_nonempty
nonempty_of_union_eq_top_of_nonempty πŸ“–mathematicaliUnion
Set
instMembership
Top.top
BooleanAlgebra.toTop
instBooleanAlgebra
Nonemptyβ€”exists_set_mem_of_union_eq_top
nonempty_sInter πŸ“–mathematicalβ€”Nonempty
sInter
Set
instMembership
β€”β€”
nonempty_sUnion πŸ“–mathematicalβ€”Nonempty
sUnion
Set
instMembership
β€”β€”
notMem_of_notMem_sUnion πŸ“–β€”Set
instMembership
sUnion
β€”β€”β€”
pi_def πŸ“–mathematicalβ€”pi
iInter
Set
instMembership
preimage
Function.eval
β€”ext
pi_diff_pi_subset πŸ“–mathematicalβ€”Set
instHasSubset
instSDiff
pi
iUnion
instMembership
preimage
Function.eval
β€”diff_subset_comm
pi_iUnion_eq_iInter_pi πŸ“–mathematicalβ€”pi
iUnion
iInter
β€”ext
range_sigma_eq_iUnion_range πŸ“–mathematicalβ€”range
iUnion
β€”ext
sInter_diff_singleton_univ πŸ“–mathematicalβ€”sInter
Set
instSDiff
instSingletonSet
univ
β€”sInf_diff_singleton_top
sInter_empty πŸ“–mathematicalβ€”sInter
Set
instEmptyCollection
univ
β€”sInf_empty
sInter_eq_biInter πŸ“–mathematicalβ€”sInter
iInter
Set
instMembership
β€”sInter_image
image_id'
sInter_eq_compl_sUnion_compl πŸ“–mathematicalβ€”sInter
Compl.compl
Set
instCompl
sUnion
image
β€”compl_compl
compl_sInter
sInter_eq_empty_iff πŸ“–mathematicalβ€”sInter
Set
instEmptyCollection
instMembership
β€”β€”
sInter_eq_iInter πŸ“–mathematicalβ€”sInter
iInter
Elem
Set
instMembership
β€”Subtype.range_coe
sInter_eq_univ πŸ“–mathematicalβ€”sInter
univ
β€”sInf_eq_top
sInter_iUnion πŸ“–mathematicalβ€”sInter
iUnion
Set
iInter
β€”sInter_eq_biInter
biInter_iUnion
sInter_image πŸ“–mathematicalβ€”sInter
image
Set
iInter
instMembership
β€”sInf_image
sInter_image2 πŸ“–mathematicalβ€”sInter
image2
Set
iInter
instMembership
β€”sInf_image2
sInter_insert πŸ“–mathematicalβ€”sInter
Set
instInsert
instInter
β€”sInf_insert
sInter_mono πŸ“–mathematicalSet
instHasSubset
sInterβ€”sInter_subset_sInter
sInter_pair πŸ“–mathematicalβ€”sInter
Set
instInsert
instSingletonSet
instInter
β€”sInf_pair
sInter_range πŸ“–mathematicalβ€”sInter
range
Set
iInter
β€”β€”
sInter_singleton πŸ“–mathematicalβ€”sInter
Set
instSingletonSet
β€”sInf_singleton
sInter_subset_of_mem πŸ“–mathematicalSet
instMembership
instHasSubset
sInter
β€”sInf_le
sInter_subset_sInter πŸ“–mathematicalSet
instHasSubset
sInterβ€”subset_sInter
sInter_subset_of_mem
sInter_union πŸ“–mathematicalβ€”sInter
Set
instUnion
instInter
β€”sInf_union
sInter_union_sInter πŸ“–mathematicalβ€”Set
instUnion
sInter
iInter
instMembership
SProd.sprod
instSProd
β€”sInf_sup_sInf
sUnion_diff_singleton_empty πŸ“–mathematicalβ€”sUnion
Set
instSDiff
instSingletonSet
instEmptyCollection
β€”sSup_diff_singleton_bot
sUnion_empty πŸ“–mathematicalβ€”sUnion
Set
instEmptyCollection
β€”sSup_empty
sUnion_eq_biUnion πŸ“–mathematicalβ€”sUnion
iUnion
Set
instMembership
β€”sUnion_image
image_id'
sUnion_eq_compl_sInter_compl πŸ“–mathematicalβ€”sUnion
Compl.compl
Set
instCompl
sInter
image
β€”compl_compl
compl_sUnion
sUnion_eq_empty πŸ“–mathematicalβ€”sUnion
Set
instEmptyCollection
β€”sSup_eq_bot
sUnion_eq_iUnion πŸ“–mathematicalβ€”sUnion
iUnion
Elem
Set
instMembership
β€”Subtype.range_coe
sUnion_eq_univ_iff πŸ“–mathematicalβ€”sUnion
univ
Set
instMembership
β€”β€”
sUnion_iUnion πŸ“–mathematicalβ€”sUnion
iUnion
Set
β€”sUnion_eq_biUnion
biUnion_iUnion
sUnion_image πŸ“–mathematicalβ€”sUnion
image
Set
iUnion
instMembership
β€”sSup_image
sUnion_image2 πŸ“–mathematicalβ€”sUnion
image2
Set
iUnion
instMembership
β€”sSup_image2
sUnion_insert πŸ“–mathematicalβ€”sUnion
Set
instInsert
instUnion
β€”sSup_insert
sUnion_inter_sUnion πŸ“–mathematicalβ€”Set
instInter
sUnion
iUnion
instMembership
SProd.sprod
instSProd
β€”sSup_inf_sSup
sUnion_mem_empty_univ πŸ“–mathematicalSet
instHasSubset
instInsert
instEmptyCollection
instSingletonSet
univ
instMembership
sUnion
β€”β€”
sUnion_mono πŸ“–mathematicalSet
instHasSubset
sUnionβ€”sUnion_subset_sUnion
sUnion_mono_subsets πŸ“–mathematicalSet
instHasSubset
sUnion
image
β€”mem_image_of_mem
sUnion_mono_supsets πŸ“–mathematicalSet
instHasSubset
sUnion
image
β€”β€”
sUnion_pair πŸ“–mathematicalβ€”sUnion
Set
instInsert
instSingletonSet
instUnion
β€”sSup_pair
sUnion_powerset_gc πŸ“–mathematicalβ€”GaloisConnection
Set
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
sUnion
powerset
β€”gc_sSup_Iic
sUnion_range πŸ“–mathematicalβ€”sUnion
range
Set
iUnion
β€”β€”
sUnion_singleton πŸ“–mathematicalβ€”sUnion
Set
instSingletonSet
β€”sSup_singleton
sUnion_subset πŸ“–mathematicalSet
instHasSubset
sUnionβ€”sSup_le
sUnion_subset_iff πŸ“–mathematicalβ€”Set
instHasSubset
sUnion
β€”sSup_le_iff
sUnion_subset_sUnion πŸ“–mathematicalSet
instHasSubset
sUnionβ€”sUnion_subset
subset_sUnion_of_mem
sUnion_union πŸ“–mathematicalβ€”sUnion
Set
instUnion
β€”sSup_union
setOf_exists πŸ“–mathematicalβ€”setOf
iUnion
β€”ext
mem_iUnion
setOf_forall πŸ“–mathematicalβ€”setOf
iInter
β€”ext
mem_iInter
sigmaToiUnion_bijective πŸ“–mathematicalPairwise
Function.onFun
Set
Disjoint
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Function.Bijective
Elem
iUnion
sigmaToiUnion
β€”sigmaToiUnion_injective
sigmaToiUnion_surjective
sigmaToiUnion_injective πŸ“–mathematicalPairwise
Function.onFun
Set
Disjoint
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Elem
iUnion
sigmaToiUnion
β€”by_contradiction
Disjoint.le_bot
Sigma.eq
sigmaToiUnion_surjective πŸ“–mathematicalβ€”Elem
iUnion
sigmaToiUnion
β€”β€”
subset_biUnion_of_mem πŸ“–mathematicalSet
instMembership
instHasSubset
iUnion
β€”subset_iUnionβ‚‚
subset_iInter πŸ“–mathematicalSet
instHasSubset
iInterβ€”le_iInf
subset_iInter_iff πŸ“–mathematicalβ€”Set
instHasSubset
iInter
β€”le_iInf_iff
subset_iInterβ‚‚ πŸ“–mathematicalSet
instHasSubset
iInterβ€”subset_iInter
subset_iInterβ‚‚_iff πŸ“–mathematicalβ€”Set
instHasSubset
iInter
β€”β€”
subset_iUnion πŸ“–mathematicalβ€”Set
instHasSubset
iUnion
β€”le_iSup
subset_iUnion_of_subset πŸ“–mathematicalSet
instHasSubset
iUnionβ€”le_iSup_of_le
subset_iUnionβ‚‚ πŸ“–mathematicalβ€”Set
instHasSubset
iUnion
β€”le_iSupβ‚‚
subset_iUnionβ‚‚_of_subset πŸ“–mathematicalSet
instHasSubset
iUnionβ€”le_iSupβ‚‚_of_le
subset_powerset_iff πŸ“–mathematicalβ€”Set
instHasSubset
powerset
sUnion
β€”sUnion_subset_iff
subset_sInter πŸ“–mathematicalSet
instHasSubset
sInterβ€”le_sInf
subset_sInter_iff πŸ“–mathematicalβ€”Set
instHasSubset
sInter
β€”le_sInf_iff
subset_sUnion_of_mem πŸ“–mathematicalSet
instMembership
instHasSubset
sUnion
β€”le_sSup
subset_sUnion_of_subset πŸ“–mathematicalSet
instHasSubset
instMembership
sUnionβ€”Subset.trans
subset_sUnion_of_mem
union_distrib_iInter_left πŸ“–mathematicalβ€”Set
instUnion
iInter
β€”sup_iInf_eq
union_distrib_iInter_right πŸ“–mathematicalβ€”Set
instUnion
iInter
β€”iInf_sup_eq
union_distrib_iInterβ‚‚_left πŸ“–mathematicalβ€”Set
instUnion
iInter
β€”union_distrib_iInter_left
union_distrib_iInterβ‚‚_right πŸ“–mathematicalβ€”Set
instUnion
iInter
β€”union_distrib_iInter_right
union_eq_iUnion πŸ“–mathematicalβ€”Set
instUnion
iUnion
β€”sup_eq_iSup
union_iInter πŸ“–mathematicalβ€”Set
instUnion
iInter
β€”sup_iInf_eq
union_iInterβ‚‚ πŸ“–mathematicalβ€”Set
instUnion
iInter
β€”union_iInter
union_iUnion πŸ“–mathematicalβ€”Set
instUnion
iUnion
β€”sup_iSup
union_iUnion_nat_succ πŸ“–mathematicalβ€”Set
instUnion
iUnion
β€”sup_iSup_nat_succ
univ_pi_eq_iInter πŸ“–mathematicalβ€”pi
univ
iInter
preimage
Function.eval
β€”pi_def
iInter_congr_Prop
iInter_true

Set.BijOn

Theorems

NameKindAssumesProvesValidatesDepends On
iInter_comp πŸ“–mathematicalSet.BijOnSet.iInter
Set
Set.instMembership
β€”iInf_comp
iInter_congr πŸ“–mathematicalSet.BijOnSet.iInter
Set
Set.instMembership
β€”iInf_congr
iUnion_comp πŸ“–mathematicalSet.BijOnSet.iUnion
Set
Set.instMembership
β€”iSup_comp
iUnion_congr πŸ“–mathematicalSet.BijOnSet.iUnion
Set
Set.instMembership
β€”iSup_congr

Set.Nonempty

Theorems

NameKindAssumesProvesValidatesDepends On
of_sUnion πŸ“–mathematicalSet.Nonempty
Set.sUnion
Setβ€”Set.nonempty_sUnion
of_sUnion_eq_univ πŸ“–mathematicalSet.sUnion
Set.univ
Set.Nonempty
Set
β€”of_sUnion
Set.univ_nonempty

Set.Sigma

Theorems

NameKindAssumesProvesValidatesDepends On
univ πŸ“–mathematicalβ€”Set.univ
Set.iUnion
Set.range
β€”Set.ext
Set.mem_range_self
Sigma.eta

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
exists_sUnion πŸ“–mathematicalβ€”Set
Set.instMembership
Set.sUnion
β€”iSup_sUnion
forall_sUnion πŸ“–β€”β€”β€”β€”iInf_sUnion
iInf_iUnion πŸ“–mathematicalβ€”iInf
CompleteSemilatticeInf.toInfSet
CompleteLattice.toCompleteSemilatticeInf
Set
Set.instMembership
Set.iUnion
β€”iSup_iUnion
iInf_sUnion πŸ“–mathematicalβ€”iInf
CompleteSemilatticeInf.toInfSet
CompleteLattice.toCompleteSemilatticeInf
Set
Set.instMembership
Set.sUnion
β€”Set.sUnion_eq_iUnion
iInf_iUnion
iInf_subtype''
iSup_iUnion πŸ“–mathematicalβ€”iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
Set
Set.instMembership
Set.iUnion
β€”iSup_comm
iSup_congr_Prop
iSup_exists
iSup_sUnion πŸ“–mathematicalβ€”iSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
Set
Set.instMembership
Set.sUnion
β€”Set.sUnion_eq_iUnion
iSup_iUnion
iSup_subtype''
sInf_sUnion πŸ“–mathematicalβ€”InfSet.sInf
CompleteSemilatticeInf.toInfSet
CompleteLattice.toCompleteSemilatticeInf
Set.sUnion
iInf
Set
Set.instMembership
β€”sSup_sUnion
sSup_iUnion πŸ“–mathematicalβ€”SupSet.sSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
Set.iUnion
iSup
β€”sSup_eq_iSup
iSup_iUnion
sSup_sUnion πŸ“–mathematicalβ€”SupSet.sSup
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
Set.sUnion
iSup
Set
Set.instMembership
β€”Set.sUnion_eq_biUnion
sSup_eq_iSup
iSup_iUnion
iSup_congr_Prop

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