Documentation Verification Report

Basic

πŸ“ Source: Mathlib/Order/Interval/Set/Basic.lean

Statistics

MetricCount
DefinitionsdecidableMemIcc, decidableMemIci, decidableMemIco, decidableMemIic, decidableMemIio, decidableMemIoc, decidableMemIoi, decidableMemIoo, instIccUnique
9
TheoremsIci_eq, Ici_eq, Ioi_eq, Iic_eq, Iio_eq, Iic_eq, Icc_bot, Icc_bot_top, Icc_diff_Ico_same, Icc_diff_Ioc_same, Icc_diff_Ioo_same, Icc_diff_both, Icc_diff_left, Icc_diff_right, Icc_eq_Icc_iff, Icc_eq_Ico_same_iff, Icc_eq_Ioc_same_iff, Icc_eq_Ioo_same_iff, Icc_eq_empty, Icc_eq_empty_iff, Icc_eq_empty_of_lt, Icc_eq_singleton_iff, Icc_inter_Icc, Icc_inter_Icc_eq_singleton, Icc_ofDual, Icc_prod_Icc, Icc_prod_eq, Icc_self, Icc_ssubset_Icc_left, Icc_ssubset_Icc_right, Icc_subset_Icc, Icc_subset_Icc_iff, Icc_subset_Icc_left, Icc_subset_Icc_right, Icc_subset_Ici_iff, Icc_subset_Ici_self, Icc_subset_Ico_iff, Icc_subset_Ico_right, Icc_subset_Iic_iff, Icc_subset_Iic_self, Icc_subset_Iio_iff, Icc_subset_Ioc_iff, Icc_subset_Ioi_iff, Icc_subset_Ioo, Icc_subset_Ioo_iff, Icc_toDual, Icc_top, Ici_False, Ici_True, Ici_bot, Ici_diff_Ioi_same, Ici_diff_left, Ici_inj, Ici_injective, Ici_inter_Ici, Ici_inter_Iic, Ici_inter_Iio, Ici_ofDual, Ici_prod_Ici, Ici_prod_eq, Ici_ssubset_Ici, Ici_subset_Ici, Ici_subset_Ioi, Ici_toDual, Ici_top, Ico_bot, Ico_diff_Ioo_same, Ico_diff_left, Ico_eq_Icc_same_iff, Ico_eq_Ioc_same_iff, Ico_eq_Ioo_same_iff, Ico_eq_empty, Ico_eq_empty_iff, Ico_eq_empty_of_le, Ico_insert_right, Ico_inter_Ici, Ico_ofDual, Ico_self, Ico_subset_Icc_self, Ico_subset_Ici_self, Ico_subset_Ico, Ico_subset_Ico_left, Ico_subset_Ico_right, Ico_subset_Iio_self, Ico_subset_Ioo_left, Ico_toDual, Ico_union_right, Iic_False, Iic_True, Iic_bot, Iic_diff_Iio_same, Iic_diff_right, Iic_inj, Iic_injective, Iic_inter_Ici, Iic_inter_Iic, Iic_inter_Ioc_of_le, Iic_inter_Ioi, Iic_ofDual, Iic_prod_Iic, Iic_prod_eq, Iic_ssubset_Iic, Iic_subset_Iic, Iic_subset_Iio, Iic_toDual, Iic_top, Iio_False, Iio_True, Iio_bot, Iio_eq_empty_iff, Iio_insert, Iio_inter_Ici, Iio_inter_Ioi, Iio_nonempty, Iio_ofDual, Iio_ssubset_Iic_self, Iio_ssubset_Iio, Iio_subset_Iic, Iio_subset_Iic_self, Iio_subset_Iio, Iio_toDual, Iio_top, Iio_union_right, Ioc_diff_Ioo_same, Ioc_diff_right, Ioc_eq_Icc_same_iff, Ioc_eq_Ico_same_iff, Ioc_eq_Ioo_same_iff, Ioc_eq_empty, Ioc_eq_empty_iff, Ioc_eq_empty_of_le, Ioc_insert_left, Ioc_inter_Iic, Ioc_ofDual, Ioc_self, Ioc_subset_Icc_self, Ioc_subset_Iic_self, Ioc_subset_Ioc, Ioc_subset_Ioc_left, Ioc_subset_Ioc_right, Ioc_subset_Ioi_self, Ioc_subset_Ioo_right, Ioc_toDual, Ioc_top, Ioc_union_left, Ioi_False, Ioi_True, Ioi_bot, Ioi_eq_empty_iff, Ioi_insert, Ioi_inter_Iic, Ioi_inter_Iio, Ioi_nonempty, Ioi_ofDual, Ioi_ssubset_Ici_self, Ioi_ssubset_Ioi, Ioi_subset_Ici, Ioi_subset_Ici_self, Ioi_subset_Ioi, Ioi_toDual, Ioi_top, Ioi_union_left, Ioo_eq_Icc_same_iff, Ioo_eq_Ico_same_iff, Ioo_eq_Ioc_same_iff, Ioo_eq_empty, Ioo_eq_empty_iff, Ioo_eq_empty_of_le, Ioo_insert_left, Ioo_insert_right, Ioo_ofDual, Ioo_self, Ioo_subset_Icc_self, Ioo_subset_Ico_self, Ioo_subset_Iio_self, Ioo_subset_Ioc_self, Ioo_subset_Ioi_self, Ioo_subset_Ioo, Ioo_subset_Ioo_left, Ioo_subset_Ioo_right, Ioo_toDual, Ioo_union_both, Ioo_union_left, Ioo_union_right, eq_endpoints_or_mem_Ioo_of_mem_Icc, eq_left_or_mem_Ioo_of_mem_Ico, eq_right_or_mem_Ioo_of_mem_Ioc, instNoMaxOrderElemIci, instNoMaxOrderElemIoi, instNoMinOrderElemIic, instNoMinOrderElemIio, left_mem_Icc, left_mem_Ici, left_mem_Ico, left_mem_Ioc, left_mem_Ioo, left_notMem_Ioc, left_notMem_Ioo, mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset, mem_Icc_of_Ico, mem_Icc_of_Ioc, mem_Icc_of_Ioo, mem_Ici_Ioi_of_subset_of_subset, mem_Ici_of_Ioi, mem_Ico_of_Ioo, mem_Iic_Iio_of_subset_of_subset, mem_Iic_of_Iio, mem_Ioc_of_Ioo, nonempty_Icc, nonempty_Icc_subtype, nonempty_Ici, nonempty_Ici_subtype, nonempty_Ico, nonempty_Ico_subtype, nonempty_Iic, nonempty_Iic_subtype, nonempty_Iio, nonempty_Iio_subtype, nonempty_Ioc, nonempty_Ioc_subtype, nonempty_Ioi, nonempty_Ioi_subtype, nonempty_Ioo, nonempty_Ioo_subtype, notMem_Icc_of_gt, notMem_Icc_of_lt, notMem_Ico_of_ge, notMem_Ico_of_lt, notMem_Iio_self, notMem_Ioc_of_gt, notMem_Ioc_of_le, notMem_Ioi_self, notMem_Ioo_of_ge, notMem_Ioo_of_le, right_mem_Icc, right_mem_Ico, right_mem_Iic, right_mem_Ioc, right_mem_Ioo, right_notMem_Ico, right_notMem_Ioo, self_mem_Ici, self_mem_Iic, self_notMem_Iio, self_notMem_Ioi, subsingleton_Icc_iff, subsingleton_Icc_of_ge, instNoMaxOrderElemIco, instNoMaxOrderElemIio, instNoMaxOrderElemIoo, instNoMinOrderElemIoc, instNoMinOrderElemIoi, instNoMinOrderElemIoo
253
Total262

IsBot

Theorems

NameKindAssumesProvesValidatesDepends On
Ici_eq πŸ“–mathematicalIsBot
Preorder.toLE
Set.Ici
Set.univ
β€”Set.eq_univ_of_forall

IsMax

Theorems

NameKindAssumesProvesValidatesDepends On
Ici_eq πŸ“–mathematicalIsMax
Preorder.toLE
PartialOrder.toPreorder
Set.Ici
Set
Set.instSingletonSet
β€”Set.eq_singleton_iff_unique_mem
Set.self_mem_Iic
eq_of_ge
Ioi_eq πŸ“–mathematicalIsMax
Preorder.toLE
Set.Ioi
Set
Set.instEmptyCollection
β€”Set.Ioi_eq_empty_iff

IsMin

Theorems

NameKindAssumesProvesValidatesDepends On
Iic_eq πŸ“–mathematicalIsMin
Preorder.toLE
PartialOrder.toPreorder
Set.Iic
Set
Set.instSingletonSet
β€”Set.eq_singleton_iff_unique_mem
Set.self_mem_Ici
eq_of_le
Iio_eq πŸ“–mathematicalIsMin
Preorder.toLE
Set.Iio
Set
Set.instEmptyCollection
β€”Set.Iio_eq_empty_iff

IsTop

Theorems

NameKindAssumesProvesValidatesDepends On
Iic_eq πŸ“–mathematicalIsTop
Preorder.toLE
Set.Iic
Set.univ
β€”Set.eq_univ_of_forall

Set

Definitions

NameCategoryTheorems
decidableMemIcc πŸ“–CompOpβ€”
decidableMemIci πŸ“–CompOp
2 mathmath: concaveOn_univ_piecewise_Ici_of_antitoneOn_Ici_monotoneOn_Iic, convexOn_univ_piecewise_Ici_of_monotoneOn_Ici_antitoneOn_Iic
decidableMemIco πŸ“–CompOpβ€”
decidableMemIic πŸ“–CompOp
3 mathmath: ContMDiff.piecewise_Iic, convexOn_univ_piecewise_Iic_of_antitoneOn_Iic_monotoneOn_Ici, concaveOn_univ_piecewise_Iic_of_monotoneOn_Iic_antitoneOn_Ici
decidableMemIio πŸ“–CompOpβ€”
decidableMemIoc πŸ“–CompOpβ€”
decidableMemIoi πŸ“–CompOpβ€”
decidableMemIoo πŸ“–CompOp
4 mathmath: isMIntegralCurveOn_piecewise, MvPolynomial.psum_eq_mul_esymm_sub_sum, eqOn_piecewise_of_isMIntegralCurveOn_Ioo, Polynomial.IsUnitTrinomial.irreducible_aux1
instIccUnique πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
Icc_bot πŸ“–mathematicalβ€”Icc
Bot.bot
OrderBot.toBot
Preorder.toLE
Iic
β€”Ici_bot
univ_inter
Icc_bot_top πŸ“–mathematicalβ€”Icc
Bot.bot
OrderBot.toBot
Preorder.toLE
BoundedOrder.toOrderBot
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
univ
β€”Icc_top
Ici_bot
Icc_diff_Ico_same πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Set
instSDiff
Icc
Ico
instSingletonSet
β€”Icc_diff_right
diff_diff_cancel_left
singleton_subset_iff
right_mem_Icc
Icc_diff_Ioc_same πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Set
instSDiff
Icc
Ioc
instSingletonSet
β€”Icc_diff_left
diff_diff_cancel_left
singleton_subset_iff
left_mem_Icc
Icc_diff_Ioo_same πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Set
instSDiff
Icc
Ioo
instInsert
instSingletonSet
β€”Icc_diff_both
diff_diff_cancel_left
Icc_diff_both πŸ“–mathematicalβ€”Set
instSDiff
Icc
PartialOrder.toPreorder
instInsert
instSingletonSet
Ioo
β€”insert_eq
diff_diff
Icc_diff_left
Ioc_diff_right
Icc_diff_left πŸ“–mathematicalβ€”Set
instSDiff
Icc
PartialOrder.toPreorder
instSingletonSet
Ioc
β€”ext
Icc_diff_right πŸ“–mathematicalβ€”Set
instSDiff
Icc
PartialOrder.toPreorder
instSingletonSet
Ico
β€”ext
Icc_eq_Icc_iff πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Iccβ€”Icc_eq_empty_iff
le_refl
le_antisymm
Icc_eq_Ico_same_iff πŸ“–mathematicalβ€”Icc
Ico
Preorder.toLE
β€”ext_iff
Icc_eq_empty
Ico_eq_empty
le_of_lt
Icc_eq_Ioc_same_iff πŸ“–mathematicalβ€”Icc
Ioc
Preorder.toLE
β€”ext_iff
Icc_eq_empty
Ioc_eq_empty
le_of_lt
Icc_eq_Ioo_same_iff πŸ“–mathematicalβ€”Icc
Ioo
Preorder.toLE
β€”ext_iff
Icc_eq_empty
Ioo_eq_empty
le_of_lt
Icc_eq_empty πŸ“–mathematicalPreorder.toLEIcc
Set
instEmptyCollection
β€”eq_empty_iff_forall_notMem
LE.le.trans
Icc_eq_empty_iff πŸ“–mathematicalβ€”Icc
Set
instEmptyCollection
Preorder.toLE
β€”not_nonempty_iff_eq_empty
not_iff_not
nonempty_Icc
Icc_eq_empty_of_lt πŸ“–mathematicalPreorder.toLTIcc
Set
instEmptyCollection
β€”Icc_eq_empty
LT.lt.not_ge
Icc_eq_singleton_iff πŸ“–mathematicalβ€”Icc
PartialOrder.toPreorder
Set
instSingletonSet
β€”nonempty_Icc
singleton_nonempty
eq_of_mem_singleton
left_mem_Icc
right_mem_Icc
Icc_self
Icc_inter_Icc πŸ“–mathematicalβ€”Set
instInter
Icc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
SemilatticeSup.toMax
Lattice.toSemilatticeSup
SemilatticeInf.toMin
β€”Ici_inter_Iic
Ici_inter_Ici
Iic_inter_Iic
inter_isAssoc
inter_isComm
Icc_inter_Icc_eq_singleton πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Set
instInter
Icc
instSingletonSet
β€”Ici_inter_Iic
Iic_inter_Ici
inter_inter_inter_comm
Icc_self
inter_singleton_of_mem
Icc_ofDual πŸ“–mathematicalβ€”Icc
DFunLike.coe
Equiv
OrderDual
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
preimage
OrderDual.toDual
OrderDual.instPreorder
β€”ext
Icc_prod_Icc πŸ“–mathematicalβ€”SProd.sprod
Set
instSProd
Icc
Prod.instPreorder
β€”ext
Icc_prod_eq πŸ“–mathematicalβ€”Icc
Prod.instPreorder
SProd.sprod
Set
instSProd
β€”Icc_prod_Icc
Icc_self πŸ“–mathematicalβ€”Icc
PartialOrder.toPreorder
Set
instSingletonSet
β€”ext
Icc_ssubset_Icc_left πŸ“–mathematicalPreorder.toLE
Preorder.toLT
Set
instHasSSubset
Icc
β€”ssubset_iff_of_subset
Icc_subset_Icc
le_of_lt
left_mem_Icc
lt_irrefl
LT.lt.trans_le
Icc_ssubset_Icc_right πŸ“–mathematicalPreorder.toLE
Preorder.toLT
Set
instHasSSubset
Icc
β€”ssubset_iff_of_subset
Icc_subset_Icc
le_of_lt
right_mem_Icc
lt_irrefl
LT.lt.trans_le
Icc_subset_Icc πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
β€”LE.le.trans
le_trans
Icc_subset_Icc_iff πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
β€”le_rfl
LE.le.trans
Icc_subset_Icc_left πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
β€”Icc_subset_Icc
le_rfl
Icc_subset_Icc_right πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
β€”Icc_subset_Icc
le_rfl
Icc_subset_Ici_iff πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
Ici
β€”le_rfl
LE.le.trans
Icc_subset_Ici_self πŸ“–mathematicalβ€”Set
instHasSubset
Icc
Ici
β€”β€”
Icc_subset_Ico_iff πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
Ico
Preorder.toLT
β€”le_rfl
LE.le.trans
LE.le.trans_lt
Icc_subset_Ico_right πŸ“–mathematicalPreorder.toLTSet
instHasSubset
Icc
Ico
β€”LE.le.trans_lt
Icc_subset_Iic_iff πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
Iic
β€”le_rfl
LE.le.trans
Icc_subset_Iic_self πŸ“–mathematicalβ€”Set
instHasSubset
Icc
Iic
β€”β€”
Icc_subset_Iio_iff πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
Iio
Preorder.toLT
β€”le_rfl
LE.le.trans_lt
Icc_subset_Ioc_iff πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
Ioc
Preorder.toLT
β€”le_rfl
LT.lt.trans_le
LE.le.trans
Icc_subset_Ioi_iff πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
Ioi
Preorder.toLT
β€”le_rfl
LT.lt.trans_le
Icc_subset_Ioo πŸ“–mathematicalPreorder.toLTSet
instHasSubset
Icc
Ioo
β€”LT.lt.trans_le
LE.le.trans_lt
Icc_subset_Ioo_iff πŸ“–mathematicalPreorder.toLESet
instHasSubset
Icc
Ioo
Preorder.toLT
β€”le_rfl
LT.lt.trans_le
LE.le.trans_lt
Icc_toDual πŸ“–mathematicalβ€”Icc
OrderDual
OrderDual.instPreorder
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
preimage
OrderDual.ofDual
β€”ext
Icc_top πŸ“–mathematicalβ€”Icc
Top.top
OrderTop.toTop
Preorder.toLE
Ici
β€”Iic_top
inter_univ
Ici_False πŸ“–mathematicalβ€”Ici
PartialOrder.toPreorder
Prop.partialOrder
univ
β€”ext
instIsEmptyFalse
Ici_True πŸ“–mathematicalβ€”Ici
PartialOrder.toPreorder
Prop.partialOrder
Set
instSingletonSet
β€”ext
Ici_bot πŸ“–mathematicalβ€”Ici
Bot.bot
OrderBot.toBot
Preorder.toLE
univ
β€”IsBot.Ici_eq
isBot_bot
Ici_diff_Ioi_same πŸ“–mathematicalβ€”Set
instSDiff
Ici
PartialOrder.toPreorder
Ioi
instSingletonSet
β€”Ici_diff_left
diff_diff_cancel_left
singleton_subset_iff
self_mem_Ici
Ici_diff_left πŸ“–mathematicalβ€”Set
instSDiff
Ici
PartialOrder.toPreorder
instSingletonSet
Ioi
β€”ext
mem_diff
mem_Ici
mem_singleton_iff
mem_Ioi
lt_iff_le_and_ne'
Ici_inj πŸ“–mathematicalβ€”Ici
PartialOrder.toPreorder
β€”Ici_injective
Ici_injective πŸ“–mathematicalβ€”Set
Ici
PartialOrder.toPreorder
β€”eq_of_forall_ge_iff
ext_iff
Ici_inter_Ici πŸ“–mathematicalβ€”Set
instInter
Ici
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
SemilatticeSup.toMax
β€”ext
mem_inter_iff
sup_le_iff
Ici_inter_Iic πŸ“–mathematicalβ€”Set
instInter
Ici
Iic
Icc
β€”β€”
Ici_inter_Iio πŸ“–mathematicalβ€”Set
instInter
Ici
Iio
Ico
β€”β€”
Ici_ofDual πŸ“–mathematicalβ€”Ici
DFunLike.coe
Equiv
OrderDual
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
preimage
OrderDual.toDual
Iic
OrderDual.instPreorder
β€”β€”
Ici_prod_Ici πŸ“–mathematicalβ€”SProd.sprod
Set
instSProd
Ici
Prod.instPreorder
β€”β€”
Ici_prod_eq πŸ“–mathematicalβ€”Ici
Prod.instPreorder
SProd.sprod
Set
instSProd
β€”β€”
Ici_ssubset_Ici πŸ“–mathematicalβ€”Set
instHasSSubset
Ici
Preorder.toLT
β€”ssubset_iff_exists
lt_of_le_not_ge
Ici_subset_Ici
LE.le.trans'
ssubset_iff_of_subset
LT.lt.le
self_mem_Ici
LT.lt.not_ge
Ici_subset_Ici πŸ“–mathematicalβ€”Set
instHasSubset
Ici
Preorder.toLE
β€”self_mem_Iic
LE.le.trans'
Ici_subset_Ioi πŸ“–mathematicalβ€”Set
instHasSubset
Ici
Ioi
Preorder.toLT
β€”self_mem_Ici
lt_of_le_of_lt'
Ici_toDual πŸ“–mathematicalβ€”Ici
OrderDual
OrderDual.instPreorder
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
preimage
OrderDual.ofDual
Iic
β€”β€”
Ici_top πŸ“–mathematicalβ€”Ici
PartialOrder.toPreorder
Top.top
OrderTop.toTop
Preorder.toLE
Set
instSingletonSet
β€”IsMax.Ici_eq
isMax_top
Ico_bot πŸ“–mathematicalβ€”Ico
Bot.bot
OrderBot.toBot
Preorder.toLE
Iio
β€”Ici_bot
univ_inter
Ico_diff_Ioo_same πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
Set
instSDiff
Ico
Ioo
instSingletonSet
β€”Ico_diff_left
diff_diff_cancel_left
singleton_subset_iff
left_mem_Ico
Ico_diff_left πŸ“–mathematicalβ€”Set
instSDiff
Ico
PartialOrder.toPreorder
instSingletonSet
Ioo
β€”ext
Ico_eq_Icc_same_iff πŸ“–mathematicalβ€”Ico
Icc
Preorder.toLE
β€”Icc_eq_Ico_same_iff
Ico_eq_Ioc_same_iff πŸ“–mathematicalβ€”Ico
Ioc
Preorder.toLT
β€”Ioc_eq_Ico_same_iff
Ico_eq_Ioo_same_iff πŸ“–mathematicalβ€”Ico
Ioo
Preorder.toLT
β€”Ioo_eq_Ico_same_iff
Ico_eq_empty πŸ“–mathematicalPreorder.toLTIco
Set
instEmptyCollection
β€”eq_empty_iff_forall_notMem
LE.le.trans_lt
Ico_eq_empty_iff πŸ“–mathematicalβ€”Ico
Set
instEmptyCollection
Preorder.toLT
β€”not_nonempty_iff_eq_empty
not_iff_not
nonempty_Ico
Ico_eq_empty_of_le πŸ“–mathematicalPreorder.toLEIco
Set
instEmptyCollection
β€”Ico_eq_empty
LE.le.not_gt
Ico_insert_right πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Set
instInsert
Ico
Icc
β€”insert_eq
union_comm
Ico_union_right
Ico_inter_Ici πŸ“–mathematicalβ€”Set
instInter
Ico
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Ici
SemilatticeSup.toMax
β€”Ici_inter_Iio
Ici_inter_Ici
inter_right_comm
Ico_ofDual πŸ“–mathematicalβ€”Ico
DFunLike.coe
Equiv
OrderDual
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
preimage
OrderDual.toDual
Ioc
OrderDual.instPreorder
β€”ext
Ico_self πŸ“–mathematicalβ€”Ico
Set
instEmptyCollection
β€”Ico_eq_empty
lt_irrefl
Ico_subset_Icc_self πŸ“–mathematicalβ€”Set
instHasSubset
Ico
Icc
β€”le_of_lt
Ico_subset_Ici_self πŸ“–mathematicalβ€”Set
instHasSubset
Ico
Ici
β€”β€”
Ico_subset_Ico πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ico
β€”LE.le.trans
LT.lt.trans_le
Ico_subset_Ico_left πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ico
β€”Ico_subset_Ico
le_rfl
Ico_subset_Ico_right πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ico
β€”Ico_subset_Ico
le_rfl
Ico_subset_Iio_self πŸ“–mathematicalβ€”Set
instHasSubset
Ico
Iio
β€”β€”
Ico_subset_Ioo_left πŸ“–mathematicalPreorder.toLTSet
instHasSubset
Ico
Ioo
β€”LT.lt.trans_le
Ico_toDual πŸ“–mathematicalβ€”Ico
OrderDual
OrderDual.instPreorder
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
preimage
OrderDual.ofDual
Ioc
β€”ext
Ico_union_right πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Set
instUnion
Ico
instSingletonSet
Icc
β€”Ioc_toDual
Icc_toDual
Ioc_union_left
LE.le.dual
Iic_False πŸ“–mathematicalβ€”Iic
PartialOrder.toPreorder
Prop.partialOrder
Set
instSingletonSet
β€”ext
Iic_True πŸ“–mathematicalβ€”Iic
PartialOrder.toPreorder
Prop.partialOrder
univ
β€”ext
Iic_bot πŸ“–mathematicalβ€”Iic
PartialOrder.toPreorder
Bot.bot
OrderBot.toBot
Preorder.toLE
Set
instSingletonSet
β€”IsMin.Iic_eq
isMin_bot
Iic_diff_Iio_same πŸ“–mathematicalβ€”Set
instSDiff
Iic
PartialOrder.toPreorder
Iio
instSingletonSet
β€”Iic_diff_right
diff_diff_cancel_left
singleton_subset_iff
self_mem_Iic
Iic_diff_right πŸ“–mathematicalβ€”Set
instSDiff
Iic
PartialOrder.toPreorder
instSingletonSet
Iio
β€”ext
Iic_inj πŸ“–mathematicalβ€”Iic
PartialOrder.toPreorder
β€”Iic_injective
Iic_injective πŸ“–mathematicalβ€”Set
Iic
PartialOrder.toPreorder
β€”eq_of_forall_le_iff
ext_iff
Iic_inter_Ici πŸ“–mathematicalβ€”Set
instInter
Iic
Ici
Icc
β€”inter_comm
Iic_inter_Iic πŸ“–mathematicalβ€”Set
instInter
Iic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
SemilatticeInf.toMin
β€”ext
Iic_inter_Ioc_of_le πŸ“–mathematicalPreorder.toLESet
instInter
Iic
Ioc
β€”ext
LE.le.trans
Iic_inter_Ioi πŸ“–mathematicalβ€”Set
instInter
Iic
Ioi
Ioc
β€”inter_comm
Iic_ofDual πŸ“–mathematicalβ€”Iic
DFunLike.coe
Equiv
OrderDual
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
preimage
OrderDual.toDual
Ici
OrderDual.instPreorder
β€”β€”
Iic_prod_Iic πŸ“–mathematicalβ€”SProd.sprod
Set
instSProd
Iic
Prod.instPreorder
β€”β€”
Iic_prod_eq πŸ“–mathematicalβ€”Iic
Prod.instPreorder
SProd.sprod
Set
instSProd
β€”β€”
Iic_ssubset_Iic πŸ“–mathematicalβ€”Set
instHasSSubset
Iic
Preorder.toLT
β€”ssubset_iff_exists
lt_of_le_not_ge
Iic_subset_Iic
LE.le.trans
ssubset_iff_of_subset
LT.lt.le
self_mem_Iic
LT.lt.not_ge
Iic_subset_Iic πŸ“–mathematicalβ€”Set
instHasSubset
Iic
Preorder.toLE
β€”self_mem_Ici
LE.le.trans
Iic_subset_Iio πŸ“–mathematicalβ€”Set
instHasSubset
Iic
Iio
Preorder.toLT
β€”self_mem_Iic
lt_of_le_of_lt
Iic_toDual πŸ“–mathematicalβ€”Iic
OrderDual
OrderDual.instPreorder
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
preimage
OrderDual.ofDual
Ici
β€”β€”
Iic_top πŸ“–mathematicalβ€”Iic
Top.top
OrderTop.toTop
Preorder.toLE
univ
β€”IsTop.Iic_eq
isTop_top
Iio_False πŸ“–mathematicalβ€”Iio
PartialOrder.toPreorder
Prop.partialOrder
Set
instEmptyCollection
β€”IsMin.Iio_eq
Iio_True πŸ“–mathematicalβ€”Iio
PartialOrder.toPreorder
Prop.partialOrder
Set
instSingletonSet
β€”ext
Iio_bot πŸ“–mathematicalβ€”Iio
Bot.bot
OrderBot.toBot
Preorder.toLE
Set
instEmptyCollection
β€”IsMin.Iio_eq
isMin_bot
Iio_eq_empty_iff πŸ“–mathematicalβ€”Iio
Set
instEmptyCollection
IsMin
Preorder.toLE
β€”β€”
Iio_insert πŸ“–mathematicalβ€”Set
instInsert
Iio
PartialOrder.toPreorder
Iic
β€”ext
le_iff_eq_or_lt
Iio_inter_Ici πŸ“–mathematicalβ€”Set
instInter
Iio
Ici
Ico
β€”inter_comm
Iio_inter_Ioi πŸ“–mathematicalβ€”Set
instInter
Iio
Ioi
Ioo
β€”inter_comm
Iio_nonempty πŸ“–mathematicalβ€”Nonempty
Iio
IsMin
Preorder.toLE
β€”β€”
Iio_ofDual πŸ“–mathematicalβ€”Iio
DFunLike.coe
Equiv
OrderDual
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
preimage
OrderDual.toDual
Ioi
OrderDual.instPreorder
β€”β€”
Iio_ssubset_Iic_self πŸ“–mathematicalβ€”Set
instHasSSubset
Iio
Iic
β€”Iio_subset_Iic_self
lt_irrefl
le_rfl
Iio_ssubset_Iio πŸ“–mathematicalPreorder.toLTSet
instHasSSubset
Iio
β€”ssubset_iff_of_subset
Iio_subset_Iio
LT.lt.le
lt_irrefl
Iio_subset_Iic πŸ“–mathematicalPreorder.toLESet
instHasSubset
Iio
Iic
β€”Subset.trans
Iio_subset_Iio
Iio_subset_Iic_self
Iio_subset_Iic_self πŸ“–mathematicalβ€”Set
instHasSubset
Iio
Iic
β€”le_of_lt
Iio_subset_Iio πŸ“–mathematicalPreorder.toLESet
instHasSubset
Iio
β€”lt_of_lt_of_le
Iio_toDual πŸ“–mathematicalβ€”Iio
OrderDual
OrderDual.instPreorder
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
preimage
OrderDual.ofDual
Ioi
β€”β€”
Iio_top πŸ“–mathematicalβ€”Iio
PartialOrder.toPreorder
Top.top
OrderTop.toTop
Preorder.toLE
Compl.compl
Set
instCompl
instSingletonSet
β€”ext
lt_top_iff_ne_top
Iio_union_right πŸ“–mathematicalβ€”Set
instUnion
Iio
PartialOrder.toPreorder
instSingletonSet
Iic
β€”ext
le_iff_lt_or_eq
Ioc_diff_Ioo_same πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
Set
instSDiff
Ioc
Ioo
instSingletonSet
β€”Ioc_diff_right
diff_diff_cancel_left
singleton_subset_iff
right_mem_Ioc
Ioc_diff_right πŸ“–mathematicalβ€”Set
instSDiff
Ioc
PartialOrder.toPreorder
instSingletonSet
Ioo
β€”ext
Ioc_eq_Icc_same_iff πŸ“–mathematicalβ€”Ioc
Icc
Preorder.toLE
β€”Icc_eq_Ioc_same_iff
Ioc_eq_Ico_same_iff πŸ“–mathematicalβ€”Ioc
Ico
Preorder.toLT
β€”ext_iff
Ioc_eq_empty
Ico_eq_empty
Ioc_eq_Ioo_same_iff πŸ“–mathematicalβ€”Ioc
Ioo
Preorder.toLT
β€”Ioo_eq_Ioc_same_iff
Ioc_eq_empty πŸ“–mathematicalPreorder.toLTIoc
Set
instEmptyCollection
β€”eq_empty_iff_forall_notMem
LT.lt.trans_le
Ioc_eq_empty_iff πŸ“–mathematicalβ€”Ioc
Set
instEmptyCollection
Preorder.toLT
β€”not_nonempty_iff_eq_empty
not_iff_not
nonempty_Ioc
Ioc_eq_empty_of_le πŸ“–mathematicalPreorder.toLEIoc
Set
instEmptyCollection
β€”Ioc_eq_empty
LE.le.not_gt
Ioc_insert_left πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Set
instInsert
Ioc
Icc
β€”insert_eq
union_comm
Ioc_union_left
Ioc_inter_Iic πŸ“–mathematicalβ€”Set
instInter
Ioc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Iic
SemilatticeInf.toMin
β€”Ioi_inter_Iic
inter_assoc
Iic_inter_Iic
Ioc_ofDual πŸ“–mathematicalβ€”Ioc
DFunLike.coe
Equiv
OrderDual
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
preimage
OrderDual.toDual
Ico
OrderDual.instPreorder
β€”ext
Ioc_self πŸ“–mathematicalβ€”Ioc
Set
instEmptyCollection
β€”Ioc_eq_empty
lt_irrefl
Ioc_subset_Icc_self πŸ“–mathematicalβ€”Set
instHasSubset
Ioc
Icc
β€”le_of_lt
Ioc_subset_Iic_self πŸ“–mathematicalβ€”Set
instHasSubset
Ioc
Iic
β€”β€”
Ioc_subset_Ioc πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ioc
β€”LE.le.trans_lt
LE.le.trans
Ioc_subset_Ioc_left πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ioc
β€”Ioc_subset_Ioc
le_rfl
Ioc_subset_Ioc_right πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ioc
β€”Ioc_subset_Ioc
le_rfl
Ioc_subset_Ioi_self πŸ“–mathematicalβ€”Set
instHasSubset
Ioc
Ioi
β€”β€”
Ioc_subset_Ioo_right πŸ“–mathematicalPreorder.toLTSet
instHasSubset
Ioc
Ioo
β€”LE.le.trans_lt
Ioc_toDual πŸ“–mathematicalβ€”Ioc
OrderDual
OrderDual.instPreorder
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
preimage
OrderDual.ofDual
Ico
β€”ext
Ioc_top πŸ“–mathematicalβ€”Ioc
Top.top
OrderTop.toTop
Preorder.toLE
Ioi
β€”Iic_top
inter_univ
Ioc_union_left πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Set
instUnion
Ioc
instSingletonSet
Icc
β€”Icc_diff_left
diff_union_self
union_eq_self_of_subset_right
singleton_subset_iff
left_mem_Icc
Ioi_False πŸ“–mathematicalβ€”Ioi
PartialOrder.toPreorder
Prop.partialOrder
Set
instSingletonSet
β€”instIsEmptyFalse
ext
Ioi_True πŸ“–mathematicalβ€”Ioi
PartialOrder.toPreorder
Prop.partialOrder
Set
instEmptyCollection
β€”IsMax.Ioi_eq
Ioi_bot πŸ“–mathematicalβ€”Ioi
PartialOrder.toPreorder
Bot.bot
OrderBot.toBot
Preorder.toLE
Compl.compl
Set
instCompl
instSingletonSet
β€”ext
bot_lt_iff_ne_bot
Ioi_eq_empty_iff πŸ“–mathematicalβ€”Ioi
Set
instEmptyCollection
IsMax
Preorder.toLE
β€”eq_empty_iff_forall_notMem
mem_Ioi
isMax_iff_forall_not_lt
Ioi_insert πŸ“–mathematicalβ€”Set
instInsert
Ioi
PartialOrder.toPreorder
Ici
β€”ext
le_iff_eq_or_lt'
Ioi_inter_Iic πŸ“–mathematicalβ€”Set
instInter
Ioi
Iic
Ioc
β€”β€”
Ioi_inter_Iio πŸ“–mathematicalβ€”Set
instInter
Ioi
Iio
Ioo
β€”β€”
Ioi_nonempty πŸ“–mathematicalβ€”Nonempty
Ioi
IsMax
Preorder.toLE
β€”nonempty_iff_ne_empty
Ioi_eq_empty_iff
not_isMax_iff
Ioi_ofDual πŸ“–mathematicalβ€”Ioi
DFunLike.coe
Equiv
OrderDual
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
preimage
OrderDual.toDual
Iio
OrderDual.instPreorder
β€”β€”
Ioi_ssubset_Ici_self πŸ“–mathematicalβ€”Set
instHasSSubset
Ioi
Ici
β€”Ioi_subset_Ici_self
lt_irrefl
le_rfl
Ioi_ssubset_Ioi πŸ“–mathematicalPreorder.toLTSet
instHasSSubset
Ioi
β€”ssubset_iff_of_subset
Ioi_subset_Ioi
LT.lt.le
lt_irrefl
Ioi_subset_Ici πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ioi
Ici
β€”Subset.trans
Ioi_subset_Ioi
Ioi_subset_Ici_self
Ioi_subset_Ici_self πŸ“–mathematicalβ€”Set
instHasSubset
Ioi
Ici
β€”le_of_lt
Ioi_subset_Ioi πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ioi
β€”lt_of_lt_of_le'
Ioi_toDual πŸ“–mathematicalβ€”Ioi
OrderDual
OrderDual.instPreorder
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
preimage
OrderDual.ofDual
Iio
β€”β€”
Ioi_top πŸ“–mathematicalβ€”Ioi
Top.top
OrderTop.toTop
Preorder.toLE
Set
instEmptyCollection
β€”IsMax.Ioi_eq
isMax_top
Ioi_union_left πŸ“–mathematicalβ€”Set
instUnion
Ioi
PartialOrder.toPreorder
instSingletonSet
Ici
β€”ext
le_iff_lt_or_eq'
Ioo_eq_Icc_same_iff πŸ“–mathematicalβ€”Ioo
Icc
Preorder.toLE
β€”Icc_eq_Ioo_same_iff
Ioo_eq_Ico_same_iff πŸ“–mathematicalβ€”Ioo
Ico
Preorder.toLT
β€”ext_iff
Ioo_eq_empty
Ico_eq_empty
Ioo_eq_Ioc_same_iff πŸ“–mathematicalβ€”Ioo
Ioc
Preorder.toLT
β€”ext_iff
Ioo_eq_empty
Ioc_eq_empty
Ioo_eq_empty πŸ“–mathematicalPreorder.toLTIoo
Set
instEmptyCollection
β€”eq_empty_iff_forall_notMem
LT.lt.trans
Ioo_eq_empty_iff πŸ“–mathematicalβ€”Ioo
Set
instEmptyCollection
Preorder.toLT
β€”not_nonempty_iff_eq_empty
not_iff_not
nonempty_Ioo
Ioo_eq_empty_of_le πŸ“–mathematicalPreorder.toLEIoo
Set
instEmptyCollection
β€”Ioo_eq_empty
LE.le.not_gt
Ioo_insert_left πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
Set
instInsert
Ioo
Ico
β€”insert_eq
union_comm
Ioo_union_left
Ioo_insert_right πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
Set
instInsert
Ioo
Ioc
β€”insert_eq
union_comm
Ioo_union_right
Ioo_ofDual πŸ“–mathematicalβ€”Ioo
DFunLike.coe
Equiv
OrderDual
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
preimage
OrderDual.toDual
OrderDual.instPreorder
β€”ext
Ioo_self πŸ“–mathematicalβ€”Ioo
Set
instEmptyCollection
β€”Ioo_eq_empty
lt_irrefl
Ioo_subset_Icc_self πŸ“–mathematicalβ€”Set
instHasSubset
Ioo
Icc
β€”Subset.trans
Ioo_subset_Ico_self
Ico_subset_Icc_self
Ioo_subset_Ico_self πŸ“–mathematicalβ€”Set
instHasSubset
Ioo
Ico
β€”le_of_lt
Ioo_subset_Iio_self πŸ“–mathematicalβ€”Set
instHasSubset
Ioo
Iio
β€”β€”
Ioo_subset_Ioc_self πŸ“–mathematicalβ€”Set
instHasSubset
Ioo
Ioc
β€”le_of_lt
Ioo_subset_Ioi_self πŸ“–mathematicalβ€”Set
instHasSubset
Ioo
Ioi
β€”β€”
Ioo_subset_Ioo πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ioo
β€”LE.le.trans_lt
LT.lt.trans_le
Ioo_subset_Ioo_left πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ioo
β€”Ioo_subset_Ioo
le_rfl
Ioo_subset_Ioo_right πŸ“–mathematicalPreorder.toLESet
instHasSubset
Ioo
β€”Ioo_subset_Ioo
le_rfl
Ioo_toDual πŸ“–mathematicalβ€”Ioo
OrderDual
OrderDual.instPreorder
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.toDual
preimage
OrderDual.ofDual
β€”ext
Ioo_union_both πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Set
instUnion
Ioo
instInsert
instSingletonSet
Icc
β€”diff_union_of_subset
left_mem_Icc
right_mem_Icc
Icc_diff_both
Ioo_union_left πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
Set
instUnion
Ioo
instSingletonSet
Ico
β€”Ico_diff_left
diff_union_self
union_eq_self_of_subset_right
singleton_subset_iff
left_mem_Ico
Ioo_union_right πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
Set
instUnion
Ioo
instSingletonSet
Ioc
β€”Ioo_toDual
Ico_toDual
Ioo_union_left
LT.lt.dual
eq_endpoints_or_mem_Ioo_of_mem_Icc πŸ“–mathematicalSet
instMembership
Icc
PartialOrder.toPreorder
Iooβ€”eq_right_or_mem_Ioo_of_mem_Ioc
LE.le.eq_or_lt'
eq_left_or_mem_Ioo_of_mem_Ico πŸ“–mathematicalSet
instMembership
Ico
PartialOrder.toPreorder
Iooβ€”LE.le.eq_or_lt'
eq_right_or_mem_Ioo_of_mem_Ioc πŸ“–mathematicalSet
instMembership
Ioc
PartialOrder.toPreorder
Iooβ€”LE.le.eq_or_lt
instNoMaxOrderElemIci πŸ“–mathematicalβ€”NoMaxOrder
Elem
Ici
Preorder.toLT
Set
instMembership
β€”NoMaxOrder.exists_gt
LE.le.trans'
LT.lt.le
instNoMaxOrderElemIoi πŸ“–mathematicalβ€”NoMaxOrder
Elem
Ioi
Preorder.toLT
Set
instMembership
β€”NoMaxOrder.exists_gt
gt_trans
instNoMinOrderElemIic πŸ“–mathematicalβ€”NoMinOrder
Elem
Iic
Preorder.toLT
Set
instMembership
β€”NoMinOrder.exists_lt
LE.le.trans
LT.lt.le
instNoMinOrderElemIio πŸ“–mathematicalβ€”NoMinOrder
Elem
Iio
Preorder.toLT
Set
instMembership
β€”NoMinOrder.exists_lt
lt_trans
left_mem_Icc πŸ“–mathematicalβ€”Set
instMembership
Icc
Preorder.toLE
β€”β€”
left_mem_Ici πŸ“–mathematicalβ€”Set
instMembership
Ici
β€”self_mem_Ici
left_mem_Ico πŸ“–mathematicalβ€”Set
instMembership
Ico
Preorder.toLT
β€”β€”
left_mem_Ioc πŸ“–mathematicalβ€”Set
instMembership
Ioc
β€”β€”
left_mem_Ioo πŸ“–mathematicalβ€”Set
instMembership
Ioo
β€”β€”
left_notMem_Ioc πŸ“–mathematicalβ€”Set
instMembership
Ioc
β€”β€”
left_notMem_Ioo πŸ“–mathematicalβ€”Set
instMembership
Ioo
β€”β€”
mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset πŸ“–mathematicalSet
instHasSubset
Ioo
PartialOrder.toPreorder
Icc
instMembership
instInsert
Ico
Ioc
instSingletonSet
β€”Subset.antisymm
insert_eq_of_mem
diff_singleton_subset_iff
Icc_diff_right
Ico_diff_left
subset_diff_singleton
Icc_diff_left
Ioc_diff_right
mem_Icc_of_Ico πŸ“–mathematicalSet
instMembership
Ico
Iccβ€”Ico_subset_Icc_self
mem_Icc_of_Ioc πŸ“–mathematicalSet
instMembership
Ioc
Iccβ€”Ioc_subset_Icc_self
mem_Icc_of_Ioo πŸ“–mathematicalSet
instMembership
Ioo
Iccβ€”Ioo_subset_Icc_self
mem_Ici_Ioi_of_subset_of_subset πŸ“–mathematicalSet
instHasSubset
Ioi
PartialOrder.toPreorder
Ici
instMembership
instInsert
instSingletonSet
β€”by_cases
Subset.antisymm
Ioi_union_left
union_subset_iff
singleton_subset_iff
lt_of_le_of_ne'
mem_Ici_of_Ioi πŸ“–mathematicalSet
instMembership
Ioi
Iciβ€”Ioi_subset_Ici_self
mem_Ico_of_Ioo πŸ“–mathematicalSet
instMembership
Ioo
Icoβ€”Ioo_subset_Ico_self
mem_Iic_Iio_of_subset_of_subset πŸ“–mathematicalSet
instHasSubset
Iio
PartialOrder.toPreorder
Iic
instMembership
instInsert
instSingletonSet
β€”by_cases
Subset.antisymm
Iio_union_right
union_subset_iff
lt_of_le_of_ne
mem_Iic_of_Iio πŸ“–mathematicalSet
instMembership
Iio
Iicβ€”Iio_subset_Iic_self
mem_Ioc_of_Ioo πŸ“–mathematicalSet
instMembership
Ioo
Iocβ€”Ioo_subset_Ioc_self
nonempty_Icc πŸ“–mathematicalβ€”Nonempty
Icc
Preorder.toLE
β€”LE.le.trans
left_mem_Icc
nonempty_Icc_subtype πŸ“–mathematicalPreorder.toLEElem
Icc
β€”Nonempty.to_subtype
nonempty_Icc
nonempty_Ici πŸ“–mathematicalβ€”Nonempty
Ici
β€”self_mem_Ici
nonempty_Ici_subtype πŸ“–mathematicalβ€”Elem
Ici
β€”Nonempty.to_subtype
nonempty_Ici
nonempty_Ico πŸ“–mathematicalβ€”Nonempty
Ico
Preorder.toLT
β€”LE.le.trans_lt
left_mem_Ico
nonempty_Ico_subtype πŸ“–mathematicalPreorder.toLTElem
Ico
β€”Nonempty.to_subtype
nonempty_Ico
nonempty_Iic πŸ“–mathematicalβ€”Nonempty
Iic
β€”self_mem_Iic
nonempty_Iic_subtype πŸ“–mathematicalβ€”Elem
Iic
β€”Nonempty.to_subtype
nonempty_Iic
nonempty_Iio πŸ“–mathematicalβ€”Nonempty
Iio
β€”NoMinOrder.exists_lt
nonempty_Iio_subtype πŸ“–mathematicalβ€”Elem
Iio
β€”Nonempty.to_subtype
nonempty_Iio
nonempty_Ioc πŸ“–mathematicalβ€”Nonempty
Ioc
Preorder.toLT
β€”LT.lt.trans_le
right_mem_Ioc
nonempty_Ioc_subtype πŸ“–mathematicalPreorder.toLTElem
Ioc
β€”Nonempty.to_subtype
nonempty_Ioc
nonempty_Ioi πŸ“–mathematicalβ€”Nonempty
Ioi
β€”NoMaxOrder.exists_gt
nonempty_Ioi_subtype πŸ“–mathematicalβ€”Elem
Ioi
β€”Nonempty.to_subtype
nonempty_Ioi
nonempty_Ioo πŸ“–mathematicalβ€”Nonempty
Ioo
Preorder.toLT
β€”LT.lt.trans
exists_between
nonempty_Ioo_subtype πŸ“–mathematicalPreorder.toLTElem
Ioo
β€”Nonempty.to_subtype
nonempty_Ioo
notMem_Icc_of_gt πŸ“–mathematicalPreorder.toLTSet
instMembership
Icc
β€”LT.lt.not_ge
notMem_Icc_of_lt πŸ“–mathematicalPreorder.toLTSet
instMembership
Icc
β€”LT.lt.not_ge
notMem_Ico_of_ge πŸ“–mathematicalPreorder.toLESet
instMembership
Ico
β€”lt_irrefl
LT.lt.trans_le
notMem_Ico_of_lt πŸ“–mathematicalPreorder.toLTSet
instMembership
Ico
β€”LT.lt.not_ge
notMem_Iio_self πŸ“–mathematicalβ€”Set
instMembership
Iio
β€”self_notMem_Iio
notMem_Ioc_of_gt πŸ“–mathematicalPreorder.toLTSet
instMembership
Ioc
β€”LT.lt.not_ge
notMem_Ioc_of_le πŸ“–mathematicalPreorder.toLESet
instMembership
Ioc
β€”lt_irrefl
LT.lt.trans_le
notMem_Ioi_self πŸ“–mathematicalβ€”Set
instMembership
Ioi
β€”self_notMem_Ioi
notMem_Ioo_of_ge πŸ“–mathematicalPreorder.toLESet
instMembership
Ioo
β€”lt_irrefl
LT.lt.trans_le
notMem_Ioo_of_le πŸ“–mathematicalPreorder.toLESet
instMembership
Ioo
β€”lt_irrefl
LT.lt.trans_le
right_mem_Icc πŸ“–mathematicalβ€”Set
instMembership
Icc
Preorder.toLE
β€”β€”
right_mem_Ico πŸ“–mathematicalβ€”Set
instMembership
Ico
β€”β€”
right_mem_Iic πŸ“–mathematicalβ€”Set
instMembership
Iic
β€”self_mem_Iic
right_mem_Ioc πŸ“–mathematicalβ€”Set
instMembership
Ioc
Preorder.toLT
β€”β€”
right_mem_Ioo πŸ“–mathematicalβ€”Set
instMembership
Ioo
β€”β€”
right_notMem_Ico πŸ“–mathematicalβ€”Set
instMembership
Ico
β€”β€”
right_notMem_Ioo πŸ“–mathematicalβ€”Set
instMembership
Ioo
β€”β€”
self_mem_Ici πŸ“–mathematicalβ€”Set
instMembership
Ici
β€”mem_Ici
le_refl
self_mem_Iic πŸ“–mathematicalβ€”Set
instMembership
Iic
β€”β€”
self_notMem_Iio πŸ“–mathematicalβ€”Set
instMembership
Iio
β€”β€”
self_notMem_Ioi πŸ“–mathematicalβ€”Set
instMembership
Ioi
β€”mem_Ioi
lt_self_iff_false
subsingleton_Icc_iff πŸ“–mathematicalβ€”Subsingleton
Icc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Preorder.toLE
β€”Mathlib.Tactic.Contrapose.contrapose₁
le_refl
LT.lt.le
LT.lt.ne
subsingleton_Icc_of_ge
subsingleton_Icc_of_ge πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
Subsingleton
Icc
β€”le_antisymm
le_imp_le_of_le_of_le

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
instNoMaxOrderElemIco πŸ“–mathematicalβ€”NoMaxOrder
Set.Elem
Set.Ico
Preorder.toLT
Set
Set.instMembership
β€”exists_between
LE.le.trans
LT.lt.le
instNoMaxOrderElemIio πŸ“–mathematicalβ€”NoMaxOrder
Set.Elem
Set.Iio
Preorder.toLT
Set
Set.instMembership
β€”exists_between
instNoMaxOrderElemIoo πŸ“–mathematicalβ€”NoMaxOrder
Set.Elem
Set.Ioo
Preorder.toLT
Set
Set.instMembership
β€”exists_between
LT.lt.trans
instNoMinOrderElemIoc πŸ“–mathematicalβ€”NoMinOrder
Set.Elem
Set.Ioc
Preorder.toLT
Set
Set.instMembership
β€”exists_between
LE.le.trans
LT.lt.le
instNoMinOrderElemIoi πŸ“–mathematicalβ€”NoMinOrder
Set.Elem
Set.Ioi
Preorder.toLT
Set
Set.instMembership
β€”exists_between
instNoMinOrderElemIoo πŸ“–mathematicalβ€”NoMinOrder
Set.Elem
Set.Ioo
Preorder.toLT
Set
Set.instMembership
β€”exists_between
LT.lt.trans

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