Documentation Verification Report

Basic

πŸ“ Source: Mathlib/SetTheory/Cardinal/Basic.lean

Statistics

MetricCount
Definitionspowerlt, Β«term_^<_Β»
2
Theoremsaleph0_le, add_le_aleph0, add_lt_aleph0, add_lt_aleph0_iff, aleph0_add_aleph0, aleph0_add_nat, aleph0_add_ofNat, aleph0_le, aleph0_le_add_iff, aleph0_le_mk, aleph0_le_mk_iff, aleph0_le_mul_iff, aleph0_le_mul_iff', aleph0_le_of_isSuccLimit, aleph0_lt_mk, aleph0_lt_mk_iff, aleph0_mul_aleph0, aleph0_mul_nat, aleph0_mul_ofNat, bddAbove_iff_small, bddAbove_image, bddAbove_of_small, bddAbove_range, bddAbove_range_comp, canLiftCardinalNat, cantor', card_le_of, card_lt_card_of_left_finite, card_lt_card_of_right_finite, compl_nonempty_of_mk_lt_mk, denumerable_iff, diff_nonempty_of_mk_lt_mk, eq_one_iff_unique, exists_eq_natCast_of_iSup_eq, exists_finset_eq_card, exists_finset_le_card, exists_nat_eq_of_le_nat, exists_notMem_of_length_lt, finset_card_lt_aleph0, iInf_eq_zero_iff, iSup_mk_le_mk_iUnion, iSup_of_empty, infinite_iff, isStrongLimit_aleph0, isSuccLimit_aleph0, isSuccPrelimit_aleph0, le_aleph0_iff_set_countable, le_aleph0_iff_subtype_countable, le_mk_diff_add_mk, le_mk_iff_exists_subset, le_one_iff_subsingleton, le_powerlt, lift_iInf, lift_iSup, lift_iSup_le, lift_iSup_le_iff, lift_iSup_le_lift_iSup, lift_iSup_le_lift_iSup', lift_iSup_le_sum, lift_mk_le_lift_mk_of_injective, lift_mk_le_lift_mk_of_surjective, lift_mk_shrink, lift_mk_shrink', lift_mk_shrink'', lift_sInf, lift_sSup, lt_aleph0, lt_aleph0_iff_finite, lt_aleph0_iff_fintype, lt_aleph0_iff_set_finite, lt_aleph0_iff_subtype_finite, lt_aleph0_of_finite, lt_one_iff_zero, mk_addOpposite, mk_additive, mk_biUnion_le, mk_biUnion_le_lift, mk_denumerable, mk_diff_add_mk, mk_emptyCollection, mk_emptyCollection_iff, mk_eq_aleph0, mk_eq_nat_iff, mk_eq_nat_iff_finset, mk_eq_nat_iff_fintype, mk_eq_two_iff, mk_eq_two_iff', mk_finset_of_fintype, mk_iUnion_eq_sum_mk, mk_iUnion_eq_sum_mk_lift, mk_iUnion_le, mk_iUnion_le_lift, mk_iUnion_le_sum_mk, mk_iUnion_le_sum_mk_lift, mk_image2_le, mk_image_embedding, mk_image_embedding_lift, mk_image_eq, mk_image_eq_lift, mk_image_eq_of_injOn, mk_image_eq_of_injOn_lift, mk_image_le, mk_image_le_lift, mk_insert, mk_insert_le, mk_int, mk_le_aleph0, mk_le_aleph0_iff, mk_le_iff_forall_finset_subset_card_le, mk_le_mk_of_subset, mk_le_one_iff_set_subsingleton, mk_list_eq_sum_pow, mk_lt_aleph0, mk_lt_aleph0_iff, mk_monotone, mk_mulOpposite, mk_multiplicative, mk_pnat, mk_preimage_down, mk_preimage_equiv, mk_preimage_equiv_lift, mk_preimage_of_injective, mk_preimage_of_injective_lift, mk_preimage_of_injective_of_subset_range, mk_preimage_of_injective_of_subset_range_lift, mk_preimage_of_subset_range, mk_preimage_of_subset_range_lift, mk_quot_le, mk_quotient_le, mk_range_eq, mk_range_eq_lift, mk_range_eq_of_injective, mk_range_inl, mk_range_inr, mk_range_le, mk_range_le_lift, mk_sUnion_le, mk_sep, mk_setProd, mk_set_eq_nat_iff_finset, mk_set_eq_one_iff, mk_set_eq_zero_iff, mk_set_ne_zero_iff, mk_singleton, mk_strictMono, mk_strictMonoOn, mk_subset_ge_of_subset_image, mk_subset_ge_of_subset_image_lift, mk_subtype_le_of_subset, mk_subtype_mono, mk_sum_compl, mk_union_add_mk_inter, mk_union_le, mk_union_le_aleph0, mk_union_of_disjoint, mk_univ, mk_vector, mul_lt_aleph0, mul_lt_aleph0_iff, mul_lt_aleph0_iff_of_ne_zero, natCast_add_one_le_iff, natCast_le_aleph0, natCast_lt_aleph0, nat_add_aleph0, nat_lt_aleph0, nat_mul_aleph0, nat_succ, not_isSuccLimit_natCast, not_isSuccLimit_of_lt_aleph0, nsmul_lt_aleph0_iff, nsmul_lt_aleph0_iff_of_ne_zero, ofNat_add_aleph0, ofNat_le_aleph0, ofNat_lt_aleph0, ofNat_mul_aleph0, one_le_aleph0, one_le_iff_ne_zero, one_le_iff_pos, one_lt_aleph0, one_lt_iff_nontrivial, one_lt_two, power_lt_aleph0, powerlt_le, powerlt_le_powerlt_left, powerlt_max, powerlt_min, powerlt_mono_left, powerlt_succ, powerlt_zero, prod_eq_of_fintype, range_natCast, sInf_eq_zero_iff, small_Icc, small_Ico, small_Iic, small_Iio, small_Ioc, small_Ioo, succ_eq_of_lt_aleph0, succ_natCast, succ_zero, sum_le_iSup, sum_le_iSup_lift, sum_le_lift_mk_mul_iSup, sum_le_lift_mk_mul_iSup_lift, sum_le_mk_mul_iSup, sum_zero_pow, three_le, two_le_iff, two_le_iff', two_le_iff_one_lt, uncountable, zero_powerlt, le_aleph0, lt_aleph0, cardinalMk_le_one, countable_infinite_iff_nonempty_denumerable, not_small_cardinal
218
Total220

Cardinal

Definitions

NameCategoryTheorems
powerlt πŸ“–CompOp
11 mathmath: powerlt_max, powerlt_le_powerlt_left, powerlt_aleph0, powerlt_zero, powerlt_mono_left, powerlt_min, le_powerlt, powerlt_le, powerlt_succ, powerlt_aleph0_le, zero_powerlt
Β«term_^<_Β» πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
add_le_aleph0 πŸ“–mathematicalβ€”Cardinal
instLE
instAdd
aleph0
β€”LE.le.trans
le_self_add
canonicallyOrderedAdd
le_add_self
add_le_add
addLeftMono
addRightMono
aleph0_add_aleph0
add_lt_aleph0 πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
instAddβ€”lt_aleph0
Nat.cast_add
natCast_lt_aleph0
add_lt_aleph0_iff πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instAdd
aleph0
β€”LE.le.trans_lt
self_le_add_right
canonicallyOrderedAdd
self_le_add_left
add_lt_aleph0
aleph0_add_aleph0 πŸ“–mathematicalβ€”Cardinal
instAdd
aleph0
β€”mk_denumerable
aleph0_add_nat πŸ“–mathematicalβ€”Cardinal
instAdd
aleph0
instNatCast
β€”LE.le.antisymm
add_le_aleph0
le_rfl
natCast_le_aleph0
le_self_add
canonicallyOrderedAdd
aleph0_add_ofNat πŸ“–mathematicalβ€”Cardinal
instAdd
aleph0
β€”aleph0_add_nat
aleph0_le πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
instNatCast
β€”LE.le.trans
natCast_le_aleph0
le_of_not_gt
lt_aleph0
LT.lt.not_ge
Nat.cast_le
addLeftMono
IsOrderedRing.toZeroLEOneClass
isOrderedRing
instCharZero
aleph0_le_add_iff πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
instAdd
β€”β€”
aleph0_le_mk πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
β€”infinite_iff
aleph0_le_mk_iff πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
Infinite
β€”infinite_iff
aleph0_le_mul_iff πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
instMul
β€”Iff.not
mul_lt_aleph0_iff
not_lt
not_and_or
aleph0_le_mul_iff' πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
instMul
β€”ne_bot_of_le_ne_bot
aleph0_ne_zero
aleph0_le_of_isSuccLimit πŸ“–mathematicalOrder.IsSuccLimit
Cardinal
PartialOrder.toPreorder
partialOrder
instLE
aleph0
β€”Mathlib.Tactic.Contrapose.contrapose₁
not_isSuccLimit_of_lt_aleph0
aleph0_lt_mk πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
β€”aleph0_lt_mk_iff
aleph0_lt_mk_iff πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
Uncountable
β€”not_le
not_countable_iff
not_iff_not
mk_le_aleph0_iff
aleph0_mul_aleph0 πŸ“–mathematicalβ€”Cardinal
instMul
aleph0
β€”mk_denumerable
aleph0_mul_nat πŸ“–mathematicalβ€”Cardinal
instMul
aleph0
instNatCast
β€”mul_comm
nat_mul_aleph0
aleph0_mul_ofNat πŸ“–mathematicalβ€”Cardinal
instMul
aleph0
β€”aleph0_mul_nat
Nat.AtLeastTwo.toNeZero
bddAbove_iff_small πŸ“–mathematicalβ€”BddAbove
Cardinal
instLE
Small
Set.Elem
β€”small_subset
small_Iic
Equiv.symm_apply_apply
le_sum
bddAbove_image πŸ“–mathematicalBddAbove
Cardinal
instLE
Set.imageβ€”bddAbove_iff_small
small_lift
small_image
bddAbove_of_small πŸ“–mathematicalβ€”BddAbove
Cardinal
instLE
β€”bddAbove_iff_small
bddAbove_range πŸ“–mathematicalβ€”BddAbove
Cardinal
instLE
Set.range
β€”bddAbove_of_small
small_range
bddAbove_range_comp πŸ“–β€”BddAbove
Cardinal
instLE
Set.range
β€”β€”Set.range_comp
bddAbove_image
canLiftCardinalNat πŸ“–mathematicalβ€”CanLift
Cardinal
instNatCast
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
β€”lt_aleph0
cantor' πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instOne
instPowCardinalβ€”LT.lt.trans_le
Nat.instAtLeastTwoHAddOfNat
cantor
power_le_power_right
Nat.cast_one
instCharZero
Order.succ_le_iff
instNoMaxOrder
card_le_of πŸ“–mathematicalFinset.cardCardinal
instLE
instNatCast
β€”Mathlib.Tactic.Contrapose.contrapose₁
Mathlib.Tactic.Push.not_forall_eq
exists_finset_le_card
nat_succ
instNoMaxOrder
card_lt_card_of_left_finite πŸ“–mathematicalSet
Set.instHasSSubset
Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
Set.Elem
β€”finite_or_infinite
card_lt_card_of_right_finite
LT.lt.trans_le
lt_aleph0_iff_subtype_finite
aleph0_le_mk_iff
card_lt_card_of_right_finite πŸ“–mathematicalSet
Set.instHasSSubset
Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
Set.Elem
β€”Set.Finite.subset
HasSSubset.SSubset.subset
Set.instIsNonstrictStrictOrderSubsetSSubset
mk_fintype
addLeftMono
IsOrderedRing.toZeroLEOneClass
isOrderedRing
instCharZero
Set.toFinset_card
Finset.card_lt_card
Set.toFinset_ssubset_toFinset
compl_nonempty_of_mk_lt_mk πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
Set.Elem
Set.Nonempty
Compl.compl
Set
Set.instCompl
β€”Set.compl_eq_univ_diff
mk_univ
denumerable_iff πŸ“–mathematicalβ€”Denumerable
aleph0
β€”mk_congr
diff_nonempty_of_mk_lt_mk πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
Set.Elem
Set.Nonempty
Set
Set.instSDiff
β€”mk_set_ne_zero_iff
LT.lt.not_ge
LE.le.trans
zero_add
eq_one_iff_unique πŸ“–mathematicalβ€”Cardinal
instOne
β€”le_antisymm_iff
Iff.and
le_one_iff_subsingleton
one_le_iff_ne_zero
mk_ne_zero_iff
exists_eq_natCast_of_iSup_eq πŸ“–β€”BddAbove
Cardinal
instLE
Set.range
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
instNatCast
β€”β€”exists_eq_of_iSup_eq_of_not_isSuccLimit
not_isSuccLimit_natCast
exists_finset_eq_card πŸ“–mathematicalβ€”Finset.cardβ€”finite_or_infinite
Finset.exists_subset_card_eq
mk_fintype
addLeftMono
IsOrderedRing.toZeroLEOneClass
isOrderedRing
instCharZero
Infinite.exists_subset_card_eq
exists_finset_le_card πŸ“–mathematicalβ€”Finset.cardβ€”exists_finset_eq_card
Eq.le
exists_nat_eq_of_le_nat πŸ“–β€”Cardinal
instLE
instNatCast
β€”β€”CanLift.prf
canLiftCardinalNat
LE.le.trans_lt
natCast_lt_aleph0
addLeftMono
IsOrderedRing.toZeroLEOneClass
isOrderedRing
instCharZero
exists_notMem_of_length_lt πŸ“–β€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instNatCast
β€”β€”Mathlib.Tactic.Contrapose.contrapose₁
mk_univ
mk_le_mk_of_subset
List.mem_toFinset
mk_coe_finset
Nat.cast_le
addLeftMono
IsOrderedRing.toZeroLEOneClass
isOrderedRing
instCharZero
List.toFinset_card_le
finset_card_lt_aleph0 πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
Set.Elem
SetLike.coe
Finset
Finset.instSetLike
aleph0
β€”lt_aleph0_of_finite
Finite.of_fintype
iInf_eq_zero_iff πŸ“–mathematicalβ€”iInf
Cardinal
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
instZero
IsEmpty
β€”β€”
iSup_mk_le_mk_iUnion πŸ“–mathematicalβ€”Cardinal
instLE
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
Set.Elem
Set.iUnion
β€”ciSup_le'
mk_le_mk_of_subset
Set.subset_iUnion
iSup_of_empty πŸ“–mathematicalβ€”iSup
Cardinal
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
instZero
β€”ciSup_of_empty
infinite_iff πŸ“–mathematicalβ€”Infinite
Cardinal
instLE
aleph0
β€”not_lt
lt_aleph0_iff_finite
not_finite_iff_infinite
isStrongLimit_aleph0 πŸ“–mathematicalβ€”IsStrongLimit
aleph0
β€”aleph0_ne_zero
Nat.instAtLeastTwoHAddOfNat
lt_aleph0
natCast_lt_aleph0
isSuccLimit_aleph0 πŸ“–mathematicalβ€”Order.IsSuccLimit
Cardinal
PartialOrder.toPreorder
partialOrder
aleph0
β€”isSuccLimit_iff
aleph0_ne_zero
isSuccPrelimit_aleph0
isSuccPrelimit_aleph0 πŸ“–mathematicalβ€”Order.IsSuccPrelimit
Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
β€”Order.isSuccPrelimit_of_succ_lt
lt_aleph0
nat_succ
natCast_lt_aleph0
le_aleph0_iff_set_countable πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
aleph0
Set.Countable
β€”mk_le_aleph0_iff
le_aleph0_iff_subtype_countable πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
Set.Countable
setOf
β€”le_aleph0_iff_set_countable
le_mk_diff_add_mk πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
instAdd
Set
Set.instSDiff
β€”LE.le.trans
mk_le_mk_of_subset
Set.subset_diff_union
mk_union_le
le_mk_iff_exists_subset πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set
Set.instHasSubset
β€”le_mk_iff_exists_set
Subtype.exists_set_subtype
mk_image_eq
Subtype.val_injective
le_one_iff_subsingleton πŸ“–mathematicalβ€”Cardinal
instLE
instOne
β€”Function.Embedding.injective
le_powerlt πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instLE
instPowCardinal
powerlt
β€”le_ciSup
Set.image_eq_range
bddAbove_image
bddAbove_Iio
lift_iInf πŸ“–mathematicalβ€”lift
iInf
Cardinal
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”Set.range_comp
lift_sInf
lift_iSup πŸ“–mathematicalBddAbove
Cardinal
instLE
Set.range
lift
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”iSup.eq_1
lift_sSup
Set.range_comp
lift_iSup_le πŸ“–mathematicalBddAbove
Cardinal
instLE
Set.range
lift
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”lift_iSup
ciSup_le'
lift_iSup_le_iff πŸ“–mathematicalBddAbove
Cardinal
instLE
Set.range
lift
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”lift_iSup
ciSup_le_iff'
bddAbove_range_comp
lift_iSup_le_lift_iSup πŸ“–mathematicalBddAbove
Cardinal
instLE
Set.range
lift
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”lift_iSup
ciSup_mono'
bddAbove_range_comp
lift_iSup_le_lift_iSup' πŸ“–mathematicalBddAbove
Cardinal
instLE
Set.range
lift
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”lift_iSup_le_lift_iSup
lift_iSup_le_sum πŸ“–mathematicalβ€”Cardinal
instLE
lift
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
sum
β€”lift_iSup
bddAbove_of_small
small_range
ciSup_le'
lift_le_sum
lift_mk_le_lift_mk_of_injective πŸ“–mathematicalβ€”Cardinal
instLE
lift
β€”mk_range_eq_of_injective
lift_le
mk_set_le
lift_mk_le_lift_mk_of_surjective πŸ“–mathematicalβ€”Cardinal
instLE
lift
β€”lift_mk_le_lift_mk_of_injective
Function.injective_surjInv
lift_mk_shrink πŸ“–mathematicalβ€”lift
Shrink
β€”lift_mk_eq
lift_mk_shrink' πŸ“–mathematicalβ€”lift
Shrink
β€”lift_mk_shrink
lift_mk_shrink'' πŸ“–mathematicalβ€”lift
Shrink
β€”lift_umax
lift_mk_shrink
lift_id
lift_sInf πŸ“–mathematicalβ€”lift
InfSet.sInf
Cardinal
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
Set.image
β€”Set.eq_empty_or_nonempty
sInf_empty
lift_zero
Set.image_empty
Monotone.map_csInf
instWellFoundedLT
lift_monotone
lift_sSup πŸ“–mathematicalBddAbove
Cardinal
instLE
lift
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
Set.image
β€”LE.le.antisymm
le_csSup_iff'
bddAbove_image
mem_range_lift_of_le
LT.lt.le
not_le
csSup_le_iff'
lift_le
Set.mem_image_of_mem
csSup_le'
le_csSup
lt_aleph0 πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
instNatCast
β€”lt_lift_iff
le_mk_iff_exists_set
Mathlib.Tactic.Contrapose.contrapose₁
Set.Infinite.to_subtype
CanLift.prf
Set.instCanLiftFinsetCoeFinite
mk_fintype
Fintype.card_coe
lift_natCast
instCharZero
natCast_lt_aleph0
lt_aleph0_iff_finite πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
Finite
β€”β€”
lt_aleph0_iff_fintype πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
Fintype
β€”lt_aleph0_iff_finite
finite_iff_nonempty_fintype
lt_aleph0_iff_set_finite πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
Set.Elem
aleph0
Set.Finite
β€”lt_aleph0_iff_finite
Set.finite_coe_iff
lt_aleph0_iff_subtype_finite πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
Set.Finite
setOf
β€”lt_aleph0_iff_set_finite
lt_aleph0_of_finite πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
β€”lt_aleph0_iff_finite
lt_one_iff_zero πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instOne
instZero
β€”bot_eq_zero'
canonicallyOrderedAdd
succ_zero
Order.lt_succ_bot_iff
instNoMaxOrder
mk_addOpposite πŸ“–mathematicalβ€”AddOppositeβ€”mk_congr
mk_additive πŸ“–mathematicalβ€”Additiveβ€”β€”
mk_biUnion_le πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set.iUnion
Set
Set.instMembership
instMul
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”Set.biUnion_eq_iUnion
mk_iUnion_le
mk_biUnion_le_lift πŸ“–mathematicalβ€”Cardinal
instLE
lift
Set.Elem
Set.iUnion
Set
Set.instMembership
instMul
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”Set.biUnion_eq_iUnion
mk_iUnion_le_lift
mk_denumerable πŸ“–mathematicalβ€”aleph0β€”denumerable_iff
mk_diff_add_mk πŸ“–mathematicalSet
Set.instHasSubset
Cardinal
instAdd
Set.Elem
Set.instSDiff
β€”mk_union_of_disjoint
disjoint_sdiff_self_left
Set.diff_union_of_subset
mk_emptyCollection πŸ“–mathematicalβ€”Set.Elem
Set
Set.instEmptyCollection
Cardinal
instZero
β€”mk_eq_zero
Set.instIsEmptyElemEmptyCollection
mk_emptyCollection_iff πŸ“–mathematicalβ€”Set.Elem
Cardinal
instZero
Set
Set.instEmptyCollection
β€”mk_set_eq_zero_iff
mk_eq_aleph0 πŸ“–mathematicalβ€”aleph0β€”LE.le.antisymm
mk_le_aleph0
mk_eq_nat_iff πŸ“–mathematicalβ€”Cardinal
instNatCast
Equiv
β€”lift_mk_fin
lift_uzero
lift_mk_eq'
mk_eq_nat_iff_finset πŸ“–mathematicalβ€”Cardinal
instNatCast
SetLike.coe
Finset
Finset.instSetLike
Set.univ
Finset.card
β€”mk_univ
mk_set_eq_nat_iff_finset
mk_eq_nat_iff_fintype πŸ“–mathematicalβ€”Cardinal
instNatCast
Fintype.card
β€”mk_eq_nat_iff_finset
Set.eq_univ_iff_forall
mk_eq_two_iff πŸ“–mathematicalβ€”Cardinal
instOfNatAtLeastTwo
instNatCast
Nat.instAtLeastTwoHAddOfNat
Set
Set.instInsert
Set.instSingletonSet
Set.univ
β€”Nat.instAtLeastTwoHAddOfNat
Finset.coe_insert
Finset.coe_singleton
mk_eq_two_iff' πŸ“–mathematicalβ€”Cardinal
instOfNatAtLeastTwo
instNatCast
Nat.instAtLeastTwoHAddOfNat
ExistsUnique
β€”Nat.instAtLeastTwoHAddOfNat
mk_eq_two_iff
Set.eq_univ_of_forall
mk_finset_of_fintype πŸ“–mathematicalβ€”Finset
Cardinal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
commSemiring
instOfNatAtLeastTwo
instNatCast
Nat.instAtLeastTwoHAddOfNat
Fintype.card
β€”Nat.instAtLeastTwoHAddOfNat
mk_fintype
Fintype.card_finset
Nat.cast_pow
mk_iUnion_eq_sum_mk πŸ“–mathematicalPairwise
Function.onFun
Set
Disjoint
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Set.Elem
Set.iUnion
sum
β€”mk_congr
mk_sigma
mk_iUnion_eq_sum_mk_lift πŸ“–mathematicalPairwise
Function.onFun
Set
Disjoint
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
lift
Set.Elem
Set.iUnion
sum
β€”mk_congr
mk_sigma
mk_iUnion_le πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set.iUnion
instMul
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”LE.le.trans
sum_le_mk_mul_iSup
mk_iUnion_le_lift πŸ“–mathematicalβ€”Cardinal
instLE
lift
Set.Elem
Set.iUnion
instMul
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”LE.le.trans
mk_iUnion_le_sum_mk_lift
Eq.trans_le
lift_sum
lift_id'
sum_le_lift_mk_mul_iSup
mk_iUnion_le_sum_mk πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set.iUnion
sum
β€”mk_le_of_surjective
Set.sigmaToiUnion_surjective
mk_sigma
mk_iUnion_le_sum_mk_lift πŸ“–mathematicalβ€”Cardinal
instLE
lift
Set.Elem
Set.iUnion
sum
β€”mk_le_of_surjective
ULift.up_surjective
Set.sigmaToiUnion_surjective
mk_sigma
mk_image2_le πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set.image2
instMul
β€”Set.image_uncurry_prod
mk_setProd
mk_image_le
mk_image_embedding πŸ“–mathematicalβ€”Set.Elem
Set.image
DFunLike.coe
Function.Embedding
Function.instFunLikeEmbedding
β€”lift_id
mk_image_embedding_lift
mk_image_embedding_lift πŸ“–mathematicalβ€”lift
Set.Elem
Set.image
DFunLike.coe
Function.Embedding
Function.instFunLikeEmbedding
β€”mk_image_eq_lift
Function.Embedding.injective
mk_image_eq πŸ“–mathematicalβ€”Set.Elem
Set.image
β€”mk_image_eq_of_injOn
Function.Injective.injOn
mk_image_eq_lift πŸ“–mathematicalβ€”lift
Set.Elem
Set.image
β€”mk_image_eq_of_injOn_lift
Function.Injective.injOn
mk_image_eq_of_injOn πŸ“–mathematicalSet.InjOnSet.Elem
Set.image
β€”mk_congr
mk_image_eq_of_injOn_lift πŸ“–mathematicalSet.InjOnlift
Set.Elem
Set.image
β€”lift_mk_eq
mk_image_le πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set.image
β€”mk_le_of_surjective
Set.imageFactorization_surjective
mk_image_le_lift πŸ“–mathematicalβ€”Cardinal
instLE
lift
Set.Elem
Set.image
β€”lift_mk_le
Set.imageFactorization_surjective
mk_insert πŸ“–mathematicalSet
Set.instMembership
Set.Elem
Set.instInsert
Cardinal
instAdd
instOne
β€”Set.union_singleton
mk_union_of_disjoint
mk_singleton
mk_insert_le πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set
Set.instInsert
instAdd
instOne
β€”Set.insert_eq_of_mem
canonicallyOrderedAdd
mk_insert
le_refl
mk_int πŸ“–mathematicalβ€”aleph0β€”mk_denumerable
mk_le_aleph0 πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
β€”mk_le_aleph0_iff
mk_le_aleph0_iff πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
Countable
β€”countable_iff_nonempty_embedding
aleph0.eq_1
lift_uzero
lift_mk_le'
mk_le_iff_forall_finset_subset_card_le πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
instNatCast
Finset.card
β€”mk_fintype
Fintype.card_coe
addLeftMono
IsOrderedRing.toZeroLEOneClass
isOrderedRing
instCharZero
LE.le.trans
mk_le_mk_of_subset
card_le_of
Finset.card_image_of_injOn
Function.Injective.injOn
Subtype.coe_injective
mk_le_mk_of_subset πŸ“–mathematicalSet
Set.instHasSubset
Cardinal
instLE
Set.Elem
β€”β€”
mk_le_one_iff_set_subsingleton πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
instOne
Set.Subsingleton
β€”le_one_iff_subsingleton
Set.subsingleton_coe
mk_list_eq_sum_pow πŸ“–mathematicalβ€”sum
Cardinal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
commSemiring
β€”mk_congr
mk_sigma
mk_vector
mk_lt_aleph0 πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
β€”mk_lt_aleph0_iff
mk_lt_aleph0_iff πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
Finite
β€”β€”
mk_monotone πŸ“–mathematicalβ€”Monotone
Set
Cardinal
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
partialOrder
Set.Elem
β€”mk_le_mk_of_subset
mk_mulOpposite πŸ“–mathematicalβ€”MulOppositeβ€”mk_congr
mk_multiplicative πŸ“–mathematicalβ€”Multiplicativeβ€”β€”
mk_pnat πŸ“–mathematicalβ€”PNat
aleph0
β€”mk_denumerable
mk_preimage_down πŸ“–mathematicalβ€”Set.Elem
Set.preimage
lift
β€”mk_uLift
eq
Function.Bijective.comp
ULift.up_bijective
Set.restrictPreimage_bijective
ULift.down_bijective
mk_preimage_equiv πŸ“–mathematicalβ€”Set.Elem
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
β€”lift_id
mk_preimage_equiv_lift
mk_preimage_equiv_lift πŸ“–mathematicalβ€”lift
Set.Elem
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
β€”mk_preimage_of_injective_of_subset_range_lift
Equiv.injective
Equiv.range_eq_univ
mk_preimage_of_injective πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set.preimage
β€”lift_id
mk_preimage_of_injective_lift
mk_preimage_of_injective_lift πŸ“–mathematicalβ€”Cardinal
instLE
lift
Set.Elem
Set.preimage
β€”lift_mk_le
Set.mem_preimage
Subtype.coind_injective
Subtype.val_injective
mk_preimage_of_injective_of_subset_range πŸ“–mathematicalSet
Set.instHasSubset
Set.range
Set.Elem
Set.preimage
β€”lift_id
mk_preimage_of_injective_of_subset_range_lift
mk_preimage_of_injective_of_subset_range_lift πŸ“–mathematicalSet
Set.instHasSubset
Set.range
lift
Set.Elem
Set.preimage
β€”le_antisymm
mk_preimage_of_injective_lift
mk_preimage_of_subset_range_lift
mk_preimage_of_subset_range πŸ“–mathematicalSet
Set.instHasSubset
Set.range
Cardinal
instLE
Set.Elem
Set.preimage
β€”lift_id
mk_preimage_of_subset_range_lift
mk_preimage_of_subset_range_lift πŸ“–mathematicalSet
Set.instHasSubset
Set.range
Cardinal
instLE
lift
Set.Elem
Set.preimage
β€”Set.image_preimage_eq_iff
mk_image_le_lift
mk_quot_le πŸ“–mathematicalβ€”Cardinal
instLE
β€”mk_le_of_surjective
mk_quotient_le πŸ“–mathematicalβ€”Cardinal
instLE
β€”mk_quot_le
mk_range_eq πŸ“–mathematicalβ€”Set.Elem
Set.range
β€”mk_congr
mk_range_eq_lift πŸ“–mathematicalβ€”lift
Set.Elem
Set.range
β€”lift_mk_eq
mk_range_eq_of_injective πŸ“–mathematicalβ€”lift
Set.Elem
Set.range
β€”lift_mk_eq'
mk_range_inl πŸ“–mathematicalβ€”Set.Elem
Set.range
lift
β€”lift_id'
Equiv.lift_cardinal_eq
lift_umax
mk_range_inr πŸ“–mathematicalβ€”Set.Elem
Set.range
lift
β€”lift_id'
Equiv.lift_cardinal_eq
lift_umax
mk_range_le πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set.range
β€”mk_le_of_surjective
Set.rangeFactorization_surjective
mk_range_le_lift πŸ“–mathematicalβ€”Cardinal
instLE
lift
Set.Elem
Set.range
β€”lift_mk_le
Set.rangeFactorization_surjective
mk_sUnion_le πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set.sUnion
instMul
Set
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
Set.instMembership
β€”Set.sUnion_eq_iUnion
mk_iUnion_le
mk_sep πŸ“–mathematicalβ€”Set.Elem
setOf
Set
Set.instMembership
β€”mk_congr
mk_setProd πŸ“–mathematicalβ€”Set.Elem
SProd.sprod
Set
Set.instSProd
Cardinal
instMul
β€”mul_def
mk_congr
mk_set_eq_nat_iff_finset πŸ“–mathematicalβ€”Set.Elem
Cardinal
instNatCast
SetLike.coe
Finset
Finset.instSetLike
Finset.card
β€”CanLift.prf
Set.instCanLiftFinsetCoeFinite
lt_aleph0_iff_set_finite
natCast_lt_aleph0
mk_fintype
Fintype.card_coe
instCharZero
mk_coe_finset
mk_set_eq_one_iff πŸ“–mathematicalβ€”Set.Elem
Cardinal
instOne
Set
Set.instSingletonSet
β€”eq_one_iff_unique
Set.exists_eq_singleton_iff_nonempty_subsingleton
Set.nonempty_coe_sort
Set.subsingleton_coe
mk_set_eq_zero_iff πŸ“–mathematicalβ€”Set.Elem
Cardinal
instZero
Set
Set.instEmptyCollection
β€”mk_eq_zero_iff
Set.isEmpty_coe_sort
mk_set_ne_zero_iff πŸ“–mathematicalβ€”Set.Nonemptyβ€”mk_ne_zero_iff
Set.nonempty_coe_sort
mk_singleton πŸ“–mathematicalβ€”Set.Elem
Set
Set.instSingletonSet
Cardinal
instOne
β€”mk_eq_one
Unique.instSubsingleton
mk_strictMono πŸ“–mathematicalβ€”StrictMono
Set
Cardinal
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
partialOrder
Set.Elem
β€”card_lt_card_of_right_finite
Set.toFinite
Subtype.finite
mk_strictMonoOn πŸ“–mathematicalβ€”StrictMonoOn
Set
Cardinal
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
partialOrder
Set.Elem
setOf
Set.Finite
β€”card_lt_card_of_right_finite
mk_subset_ge_of_subset_image πŸ“–mathematicalSet
Set.instHasSubset
Set.image
Cardinal
instLE
Set.Elem
setOf
Set.instMembership
β€”mk_sep
mk_preimage_of_subset_range
Set.image_eq_range
mk_subset_ge_of_subset_image_lift πŸ“–mathematicalSet
Set.instHasSubset
Set.image
Cardinal
instLE
lift
Set.Elem
setOf
Set.instMembership
β€”mk_sep
mk_preimage_of_subset_range_lift
Set.image_eq_range
mk_subtype_le_of_subset πŸ“–mathematicalβ€”Cardinal
instLE
β€”β€”
mk_subtype_mono πŸ“–mathematicalβ€”Cardinal
instLE
β€”β€”
mk_sum_compl πŸ“–mathematicalβ€”Cardinal
instAdd
Set.Elem
Compl.compl
Set
Set.instCompl
β€”mk_congr
mk_union_add_mk_inter πŸ“–mathematicalβ€”Cardinal
instAdd
Set.Elem
Set
Set.instUnion
Set.instInter
β€”β€”
mk_union_le πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set
Set.instUnion
instAdd
β€”self_le_add_right
canonicallyOrderedAdd
mk_union_add_mk_inter
mk_union_le_aleph0 πŸ“–mathematicalβ€”Cardinal
instLE
Set.Elem
Set
Set.instUnion
aleph0
β€”β€”
mk_union_of_disjoint πŸ“–mathematicalDisjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Set.Elem
Set.instUnion
Cardinal
instAdd
β€”β€”
mk_univ πŸ“–mathematicalβ€”Set.Elem
Set.univ
β€”mk_congr
mk_vector πŸ“–mathematicalβ€”List.Vector
Cardinal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
commSemiring
β€”mk_congr
mk_pi
prod_const
lift_uzero
mk_fintype
Fintype.card_fin
lift_natCast
mul_lt_aleph0 πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
instMulβ€”lt_aleph0
Nat.cast_mul
natCast_lt_aleph0
mul_lt_aleph0_iff πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instMul
aleph0
instZero
β€”mul_one
LE.le.trans_lt
mul_le_mul'
CanonicallyOrderedAdd.toMulLeftMono
canonicallyOrderedAdd
covariant_swap_mul_of_covariant_mul
le_rfl
one_le_iff_ne_zero
one_mul
MulZeroClass.zero_mul
MulZeroClass.mul_zero
mul_lt_aleph0_iff_of_ne_zero πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instMul
aleph0
β€”β€”
natCast_add_one_le_iff πŸ“–mathematicalβ€”Cardinal
instLE
instAdd
instNatCast
instOne
Preorder.toLT
PartialOrder.toPreorder
partialOrder
β€”Order.succ_le_iff
instNoMaxOrder
succ_natCast
natCast_le_aleph0 πŸ“–mathematicalβ€”Cardinal
instLE
instNatCast
aleph0
β€”LT.lt.le
natCast_lt_aleph0
natCast_lt_aleph0 πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instNatCast
aleph0
β€”Order.succ_le_iff
instNoMaxOrder
nat_succ
lift_mk_fin
aleph0.eq_1
lift_mk_le
nat_add_aleph0 πŸ“–mathematicalβ€”Cardinal
instAdd
instNatCast
aleph0
β€”add_comm
aleph0_add_nat
nat_lt_aleph0 πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instNatCast
aleph0
β€”natCast_lt_aleph0
nat_mul_aleph0 πŸ“–mathematicalβ€”Cardinal
instMul
instNatCast
aleph0
β€”le_antisymm
mk_le_aleph0
instCountableProd
instCountableULift
instCountableFin
instCountableNat
lift_mk_fin
le_mul_of_one_le_left
IsOrderedRing.toMulPosMono
isOrderedRing
zero_le
canonicallyOrderedAdd
Nat.cast_one
Nat.cast_le
addLeftMono
IsOrderedRing.toZeroLEOneClass
instCharZero
nat_succ πŸ“–mathematicalβ€”Cardinal
instNatCast
Order.succ
PartialOrder.toPreorder
partialOrder
instSuccOrder
β€”Nat.cast_succ
LE.le.antisymm
add_one_le_succ
Order.succ_le_of_lt
Nat.cast_lt
addLeftMono
IsOrderedRing.toZeroLEOneClass
isOrderedRing
instCharZero
not_isSuccLimit_natCast πŸ“–mathematicalβ€”Order.IsSuccLimit
Cardinal
PartialOrder.toPreorder
partialOrder
instNatCast
β€”isMin_bot
Order.not_isSuccPrelimit_succ
instNoMaxOrder
nat_succ
not_isSuccLimit_of_lt_aleph0 πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
Order.IsSuccLimitβ€”lt_aleph0
not_isSuccLimit_natCast
nsmul_lt_aleph0_iff πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
AddMonoid.toNatSMul
AddMonoidWithOne.toAddMonoid
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
commSemiring
aleph0
β€”zero_nsmul
aleph0_pos
nsmul_eq_mul
Nat.cast_one
one_mul
succ_nsmul
add_lt_aleph0_iff
Nat.instCharZero
Nat.instAtLeastTwoHAddOfNat
nsmul_lt_aleph0_iff_of_ne_zero πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
AddMonoid.toNatSMul
AddMonoidWithOne.toAddMonoid
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
commSemiring
aleph0
β€”nsmul_lt_aleph0_iff
ofNat_add_aleph0 πŸ“–mathematicalβ€”Cardinal
instAdd
aleph0
β€”nat_add_aleph0
ofNat_le_aleph0 πŸ“–mathematicalβ€”Cardinal
instLE
aleph0
β€”natCast_le_aleph0
ofNat_lt_aleph0 πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
β€”natCast_lt_aleph0
ofNat_mul_aleph0 πŸ“–mathematicalβ€”Cardinal
instMul
aleph0
β€”nat_mul_aleph0
Nat.AtLeastTwo.toNeZero
one_le_aleph0 πŸ“–mathematicalβ€”Cardinal
instLE
instOne
aleph0
β€”LT.lt.le
one_lt_aleph0
one_le_iff_ne_zero πŸ“–mathematicalβ€”Cardinal
instLE
instOne
β€”one_le_iff_pos
pos_iff_ne_zero
canonicallyOrderedAdd
one_le_iff_pos πŸ“–mathematicalβ€”Cardinal
instLE
instOne
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instZero
β€”succ_zero
Order.succ_le_iff
instNoMaxOrder
one_lt_aleph0 πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instOne
aleph0
β€”Nat.cast_one
natCast_lt_aleph0
one_lt_iff_nontrivial πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instOne
Nontrivial
β€”not_le
le_one_iff_subsingleton
not_nontrivial_iff_subsingleton
one_lt_two πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instOne
instOfNatAtLeastTwo
instNatCast
Nat.instAtLeastTwoHAddOfNat
β€”Nat.instAtLeastTwoHAddOfNat
Nat.cast_one
addLeftMono
IsOrderedRing.toZeroLEOneClass
isOrderedRing
instCharZero
power_lt_aleph0 πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
instPowCardinalβ€”lt_aleph0
power_natCast
Nat.cast_pow
natCast_lt_aleph0
powerlt_le πŸ“–mathematicalβ€”Cardinal
instLE
powerlt
instPowCardinal
β€”powerlt.eq_1
ciSup_le_iff'
Set.image_eq_range
bddAbove_image
bddAbove_Iio
powerlt_le_powerlt_left πŸ“–mathematicalCardinal
instLE
powerltβ€”powerlt_le
le_powerlt
LT.lt.trans_le
powerlt_max πŸ“–mathematicalβ€”powerlt
Cardinal
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”Monotone.map_max
powerlt_mono_left
powerlt_min πŸ“–mathematicalβ€”powerlt
Cardinal
SemilatticeInf.toMin
Lattice.toSemilatticeInf
ConditionallyCompleteLattice.toLattice
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”Monotone.map_min
powerlt_mono_left
powerlt_mono_left πŸ“–mathematicalβ€”Monotone
Cardinal
PartialOrder.toPreorder
partialOrder
powerlt
β€”powerlt_le_powerlt_left
powerlt_succ πŸ“–mathematicalβ€”powerlt
Order.succ
Cardinal
PartialOrder.toPreorder
partialOrder
instSuccOrder
instPowCardinal
β€”LE.le.antisymm
powerlt_le
power_le_power_left
Order.le_of_lt_succ
le_powerlt
Order.lt_succ
instNoMaxOrder
powerlt_zero πŸ“–mathematicalβ€”powerlt
Cardinal
instZero
β€”iSup_of_empty
Subtype.isEmpty_of_false
Iff.not
Set.mem_Iio
LE.le.not_gt
zero_le
prod_eq_of_fintype πŸ“–mathematicalβ€”prod
lift
Finset.prod
Cardinal
instCommMonoid
Finset.univ
β€”Fintype.induction_empty_option
Equiv.prod_comp
mk_congr
Fintype.univ_pempty
Finset.prod_empty
lift_one
prod.eq_1
mk_eq_one
Unique.instSubsingleton
PEmpty.instIsEmpty
mk_prod
lift_umax
mk_out
lift_prod
Fintype.prod_option
lift_mul
lift_id
range_natCast πŸ“–mathematicalβ€”Set.range
Cardinal
instNatCast
Set.Iio
PartialOrder.toPreorder
partialOrder
aleph0
β€”Set.ext
sInf_eq_zero_iff πŸ“–mathematicalβ€”InfSet.sInf
Cardinal
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
instZero
Set
Set.instEmptyCollection
Set.instMembership
β€”Set.eq_empty_or_nonempty
csInf_mem
instWellFoundedLT
sInf_empty
eq_bot_iff
csInf_le'
small_Icc πŸ“–mathematicalβ€”Small
Set.Elem
Cardinal
Set.Icc
PartialOrder.toPreorder
partialOrder
β€”small_subset
Set.Icc_subset_Iic_self
small_Iic
small_Ico πŸ“–mathematicalβ€”Small
Set.Elem
Cardinal
Set.Ico
PartialOrder.toPreorder
partialOrder
β€”small_subset
Set.Ico_subset_Iio_self
small_Iio
small_Iic πŸ“–mathematicalβ€”Small
Set.Elem
Cardinal
Set.Iic
PartialOrder.toPreorder
partialOrder
β€”mk_out
small_of_surjective
small_set
UnivLE.small
univLE_of_max
UnivLE.self
mk_set_le
le_mk_iff_exists_set
small_Iio πŸ“–mathematicalβ€”Small
Set.Elem
Cardinal
Set.Iio
PartialOrder.toPreorder
partialOrder
β€”small_subset
Set.Iio_subset_Iic_self
small_Iic
small_Ioc πŸ“–mathematicalβ€”Small
Set.Elem
Cardinal
Set.Ioc
PartialOrder.toPreorder
partialOrder
β€”small_subset
Set.Ioc_subset_Iic_self
small_Iic
small_Ioo πŸ“–mathematicalβ€”Small
Set.Elem
Cardinal
Set.Ioo
PartialOrder.toPreorder
partialOrder
β€”small_subset
Set.Ioo_subset_Iio_self
small_Iio
succ_eq_of_lt_aleph0 πŸ“–mathematicalCardinal
Preorder.toLT
PartialOrder.toPreorder
partialOrder
aleph0
Order.succ
instSuccOrder
instAdd
instOne
β€”lt_aleph0
succ_natCast
succ_natCast πŸ“–mathematicalβ€”Order.succ
Cardinal
PartialOrder.toPreorder
partialOrder
instSuccOrder
instNatCast
instAdd
instOne
β€”nat_succ
Nat.cast_one
instCharZero
succ_zero πŸ“–mathematicalβ€”Order.succ
Cardinal
PartialOrder.toPreorder
partialOrder
instSuccOrder
instZero
instOne
β€”Nat.cast_zero
Nat.cast_one
instCharZero
sum_le_iSup πŸ“–mathematicalβ€”Cardinal
instLE
sum
instMul
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”sum_le_mk_mul_iSup
sum_le_iSup_lift πŸ“–mathematicalβ€”Cardinal
instLE
sum
instMul
lift
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”sum_le_lift_mk_mul_iSup
sum_le_lift_mk_mul_iSup πŸ“–mathematicalβ€”Cardinal
instLE
sum
instMul
lift
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”lift_id
sum_le_lift_mk_mul_iSup_lift
sum_le_lift_mk_mul_iSup_lift πŸ“–mathematicalβ€”Cardinal
instLE
sum
instMul
lift
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”lift_id
lift_sum
lift_umax
sum_const
sum_le_sum
le_ciSup
bddAbove_of_small
small_range
UnivLE.small
univLE_of_max
UnivLE.self
sum_le_mk_mul_iSup πŸ“–mathematicalβ€”Cardinal
instLE
sum
instMul
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder
instConditionallyCompleteLinearOrderBot
β€”lift_id
sum_le_lift_mk_mul_iSup_lift
sum_zero_pow πŸ“–mathematicalβ€”sum
Cardinal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
commSemiring
instZero
instOne
β€”mk_eq_zero
PEmpty.instIsEmpty
mk_list_eq_sum_pow
mk_eq_one
Unique.instSubsingleton
three_le πŸ“–β€”β€”β€”β€”Nat.instAtLeastTwoHAddOfNat
Order.succ_le_iff
instNoMaxOrder
nat_succ
exists_notMem_of_length_lt
two_le_iff πŸ“–mathematicalβ€”Cardinal
instLE
instOfNatAtLeastTwo
instNatCast
Nat.instAtLeastTwoHAddOfNat
β€”Nat.instAtLeastTwoHAddOfNat
Nat.cast_two
nat_succ
Order.succ_le_iff
instNoMaxOrder
Nat.cast_one
one_lt_iff_nontrivial
nontrivial_iff
two_le_iff' πŸ“–mathematicalβ€”Cardinal
instLE
instOfNatAtLeastTwo
instNatCast
Nat.instAtLeastTwoHAddOfNat
β€”Nat.instAtLeastTwoHAddOfNat
two_le_iff
nontrivial_iff
nontrivial_iff_exists_ne
two_le_iff_one_lt πŸ“–mathematicalβ€”Cardinal
instLE
instOfNatAtLeastTwo
instNatCast
Nat.instAtLeastTwoHAddOfNat
Preorder.toLT
PartialOrder.toPreorder
partialOrder
instOne
β€”Nat.instAtLeastTwoHAddOfNat
Nat.cast_one
instCharZero
natCast_add_one_le_iff
uncountable πŸ“–mathematicalβ€”Uncountable
Cardinal
β€”Uncountable.of_not_small
not_small_cardinal
zero_powerlt πŸ“–mathematicalβ€”powerlt
Cardinal
instZero
instOne
β€”LE.le.antisymm
powerlt_le
zero_power_le
power_zero
le_powerlt
pos_iff_ne_zero
canonicallyOrderedAdd

Cardinal.IsStrongLimit

Theorems

NameKindAssumesProvesValidatesDepends On
aleph0_le πŸ“–mathematicalCardinal.IsStrongLimitCardinal
Cardinal.instLE
Cardinal.aleph0
β€”Cardinal.aleph0_le_of_isSuccLimit
isSuccLimit

Set

Theorems

NameKindAssumesProvesValidatesDepends On
countable_infinite_iff_nonempty_denumerable πŸ“–mathematicalβ€”Countable
Infinite
Denumerable
Elem
β€”nonempty_denumerable_iff
infinite_coe_iff
countable_coe_iff

Set.Countable

Theorems

NameKindAssumesProvesValidatesDepends On
le_aleph0 πŸ“–mathematicalβ€”Cardinal
Cardinal.instLE
Set.Elem
Cardinal.aleph0
β€”Cardinal.le_aleph0_iff_set_countable

Set.Finite

Theorems

NameKindAssumesProvesValidatesDepends On
lt_aleph0 πŸ“–mathematicalβ€”Cardinal
Preorder.toLT
PartialOrder.toPreorder
Cardinal.partialOrder
Set.Elem
Cardinal.aleph0
β€”Cardinal.lt_aleph0_iff_set_finite

Set.Subsingleton

Theorems

NameKindAssumesProvesValidatesDepends On
cardinalMk_le_one πŸ“–mathematicalSet.SubsingletonCardinal
Cardinal.instLE
Set.Elem
Cardinal.instOne
β€”Cardinal.mk_le_one_iff_set_subsingleton

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
not_small_cardinal πŸ“–mathematicalβ€”Small
Cardinal
β€”small_lift
not_bddAbove_univ
instNoTopOrderOfNoMaxOrder
Cardinal.instNoMaxOrder
Cardinal.bddAbove_iff_small
small_univ_iff

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