Documentation Verification Report

Basic

πŸ“ Source: Mathlib/GroupTheory/Perm/Cycle/Basic.lean

Statistics

MetricCount
DefinitionsIsCycle, zpowersEquivSupport, IsCycleOn, SameCycle, setoid, instDecidableRelSameCycleInv, instDecidableRelSameCycleOfNatOfDecidableEq, subtypePermOfSupport, subtypePerm_of_support_le
9
TheoremssameCycle, commute_iff, commute_iff', conj, eq_on_support_inter_nonempty_congr, eq_swap_of_apply_apply_eq_self, exists_pow_eq, exists_zpow_eq, extendDomain, inv, isConj, isConj_iff, isCycleOn, isCycle_pow_pos_of_lt_prime_order, ne_one, nonempty_support, of_pow, of_zpow, orderOf, pow_eq_one_iff, pow_eq_one_iff', pow_eq_one_iff'', pow_eq_pow_iff, pow_iff, sameCycle, sign, support_congr, support_pow_eq_iff, support_pow_of_pos_of_lt_orderOf, swap_mul, two_le_card_support, zpowersEquivSupport_apply, zpowersEquivSupport_symm_apply, apply_mem_iff, apply_ne, conj, countable, exists_pow_eq, exists_pow_eq', extendDomain, inv, isCycle_subtypePerm, of_inv, of_pow, of_zpow, pow_apply_eq, pow_apply_eq_pow_apply, pow_card_apply, range_pow, range_zpow, subsingleton, subtypePerm, zpow_apply_eq, zpow_apply_eq_zpow_apply, isCycle, apply_eq_self_iff, apply_left, apply_right, conj, eq_of_left, eq_of_right, equivalence, exists_fin_pow_eq, exists_nat_pow_eq, exists_pow_eq', exists_pow_eq'', extendDomain, inv, of_apply_left, of_apply_right, of_inv, of_pow, of_pow_left, of_pow_right, of_symm_apply_left, of_symm_apply_right, of_zpow, of_zpow_left, of_zpow_right, pow_left, pow_right, refl, rfl, subtypePerm, symm_apply_left, symm_apply_right, trans, zpow_left, zpow_right, cycle_zpow_mem_support_iff, isCycleOn_empty, isCycleOn_inv, isCycleOn_of_subsingleton, isCycleOn_one, isCycleOn_singleton, isCycleOn_swap, isCycle_iff_exists_isCycleOn, isCycle_iff_sameCycle, isCycle_inv, isCycle_swap, isCycle_swap_mul_aux₁, isCycle_swap_mul_auxβ‚‚, nodup_of_pairwise_disjoint_cycles, not_isCycle_one, sameCycle_apply_left, sameCycle_apply_right, sameCycle_comm, sameCycle_conj, sameCycle_extendDomain, sameCycle_inv, sameCycle_inv_apply_left, sameCycle_inv_apply_right, sameCycle_one, sameCycle_pow_left, sameCycle_pow_right, sameCycle_subtypePerm, sameCycle_symm_apply_left, sameCycle_symm_apply_right, sameCycle_zpow_left, sameCycle_zpow_right, subtypePerm_apply_pow_of_mem, subtypePerm_apply_zpow_of_mem, swap_isSwap_iff, zpow_eq_ofSubtype_subtypePerm_iff, exists_cycleOn, product_self_eq_disjiUnion_perm, product_self_eq_disjiUnion_perm_aux, sum_mul_sum_eq_sum_perm, sum_smul_sum_eq_sum_perm, addLeft_one_isCycle, addRight_one_isCycle, isCycleOn_formPerm, exists_cycleOn, isCycleOn_one, prod_self_eq_iUnion_perm
135
Total144

Eq

Theorems

NameKindAssumesProvesValidatesDepends On
sameCycle πŸ“–mathematicalβ€”Equiv.Perm.SameCycleβ€”Equiv.Perm.SameCycle.refl

Equiv.Perm

Definitions

NameCategoryTheorems
IsCycle πŸ“–MathDef
29 mathmath: isCycle_cycleOf_iff, Int.addLeft_one_isCycle, isCycle_cycleOf, IsCycleOn.isCycle_subtypePerm, isCycle_of_prime_order', cycleFactorsFinset_eq_finset, cycleFactorsFinset_eq_singleton_self_iff, closure_isCycle, isCycle_inv, card_cycleType_eq_one, mem_cycleType_iff, cycleFactorsFinset_eq_singleton_iff, Int.addRight_one_isCycle, isCycle_iff_exists_isCycleOn, Fin.isCycle_cycleRange, Cycle.isCycle_formPerm, isCycle_swap, isCycle_iff_sameCycle, mem_cycleFactorsFinset_iff, IsSwap.isCycle, cycleFactorsFinset_eq_list_toFinset, isCycle_of_prime_order'', isCycle_finRotate_of_le, not_isCycle_one, IsThreeCycle.isCycle, Fin.isCycle_cycleIcc, isCycle_finRotate, isCycle_of_prime_order, List.isCycle_formPerm
IsCycleOn πŸ“–MathDef
14 mathmath: isCycleOn_of_subsingleton, isCycleOn_support_cycleOf, Set.Countable.exists_cycleOn, List.Nodup.isCycleOn_formPerm, isCycleOn_inv, isCycleOn_empty, isCycle_iff_exists_isCycleOn, isCycleOn_swap, IsCycle.isCycleOn, Set.Subsingleton.isCycleOn_one, isCycleOn_one, isCycleOn_singleton, isCycleOn_support_of_mem_cycleFactorsFinset, Finset.exists_cycleOn
SameCycle πŸ“–MathDef
29 mathmath: sameCycle_conj, mem_toList_iff, SameCycle.rfl, IsCycle.sameCycle, Basis.sameCycle, sameCycle_inv_apply_left, sameCycle_apply_right, mem_support_cycleOf_iff, sameCycle_symm_apply_right, sameCycle_comm, mem_support_cycleOf_iff', sameCycle_zpow_right, sameCycle_inv_apply_right, SameCycle.equivalence, sameCycle_inv, sameCycle_apply_left, Eq.sameCycle, SameCycle.refl, sameCycle_pow_right, cycleOf_apply, isCycle_iff_sameCycle, sameCycle_iff_cycleOf_eq_of_mem_support, sameCycle_zpow_left, sameCycle_subtypePerm, List.pairwise_sameCycle_formPerm, sameCycle_pow_left, sameCycle_one, sameCycle_extendDomain, sameCycle_symm_apply_left
instDecidableRelSameCycleInv πŸ“–CompOp
1 mathmath: cycleOf_inv
instDecidableRelSameCycleOfNatOfDecidableEq πŸ“–CompOpβ€”
subtypePermOfSupport πŸ“–CompOp
2 mathmath: IsCycle.commute_iff', subtypePerm_on_cycleFactorsFinset
subtypePerm_of_support_le πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
cycle_zpow_mem_support_iff πŸ“–mathematicalIsCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
permGroup
Finset.card
support
β€”lt_of_lt_of_le
Nat.instAtLeastTwoHAddOfNat
IsStrictOrderedRing.toIsOrderedRing
Nat.instIsStrictOrderedRing
Nat.instNontrivial
IsCycle.two_le_card_support
IsCycle.orderOf
Int.instIsStrictOrderedRing
Int.instAddLeftMono
Int.instZeroLEOneClass
Int.instCharZero
zpow_add
zpow_natCast
zpow_mul
pow_orderOf_eq_one
one_zpow
IsCycle.pow_eq_one_iff
Finite.of_fintype
pow_zero
pow_ne_one_of_lt_orderOf
isCycleOn_empty πŸ“–mathematicalβ€”IsCycleOn
Set
Set.instEmptyCollection
β€”instIsEmptyFalse
isCycleOn_inv πŸ“–mathematicalβ€”IsCycleOn
Equiv.Perm
instInv
β€”Set.BijOn.perm_inv
isCycleOn_of_subsingleton πŸ“–mathematicalβ€”IsCycleOnβ€”Set.bijOn_of_subsingleton
Eq.sameCycle
isCycleOn_one πŸ“–mathematicalβ€”IsCycleOn
Equiv.Perm
instOne
Set.Subsingleton
β€”β€”
isCycleOn_singleton πŸ“–mathematicalβ€”IsCycleOn
Set
Set.instSingletonSet
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”β€”
isCycleOn_swap πŸ“–mathematicalβ€”IsCycleOn
Equiv.swap
Set
Set.instInsert
Set.instSingletonSet
β€”Equiv.bijOn_swap
Set.mem_singleton_iff
Set.mem_insert_iff
zpow_zero
coe_one
zpow_one
Equiv.swap_apply_left
Equiv.swap_apply_right
isCycle_iff_exists_isCycleOn πŸ“–mathematicalβ€”IsCycle
Set.Nontrivial
IsCycleOn
Set
Set.instMembership
β€”Equiv.injective
IsCycle.isCycleOn
Set.Nontrivial.nonempty
IsCycleOn.apply_ne
isCycle_iff_sameCycle πŸ“–mathematicalβ€”IsCycle
SameCycle
β€”Equiv.injective
zpow_apply_eq_self_of_apply_eq_self
IsCycle.exists_zpow_eq
isCycle_inv πŸ“–mathematicalβ€”IsCycle
Equiv.Perm
instInv
β€”IsCycle.inv
isCycle_swap πŸ“–mathematicalβ€”IsCycle
Equiv.swap
β€”Equiv.swap_apply_right
zpow_one
Equiv.swap_apply_def
isCycle_swap_mul_aux₁ πŸ“–mathematicalDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
instPowNat
DivInvMonoid.toZPow
Group.toDivInvMonoid
permGroup
instMul
Equiv.swap
β€”eq_or_ne
ne_and_ne_of_swap_mul_apply_ne_self
Equiv.apply_symm_apply
Equiv.swap_apply_of_ne_of_ne
Equiv.injective
pow_succ'
add_comm
zpow_add
mul_apply
zpow_one
isCycle_swap_mul_auxβ‚‚ πŸ“–mathematicalDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
permGroup
instMul
Equiv.swap
β€”isCycle_swap_mul_aux₁
eq_or_ne
ne_and_ne_of_swap_mul_apply_ne_self
mul_apply
Equiv.swap_apply_def
Equiv.apply_symm_apply
pow_succ
pow_succ'
zpow_negSucc
mul_inv_rev
Equiv.symm_apply_apply
Equiv.injective
zpow_neg
inv_zpow
Equiv.swap_inv
Equiv.mul_swap_eq_swap_mul
inv_mul_cancel
Equiv.swap_comm
mul_assoc
mul_one
Equiv.swap_apply_left
Equiv.swap_apply_of_ne_of_ne
nodup_of_pairwise_disjoint_cycles πŸ“–β€”IsCycle
Equiv.Perm
Disjoint
β€”β€”nodup_of_pairwise_disjoint
IsCycle.ne_one
not_isCycle_one πŸ“–mathematicalβ€”IsCycle
Equiv.Perm
instOne
β€”IsCycle.ne_one
sameCycle_apply_left πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.exists_congr_left
Equiv.addRight_symm
zpow_sub
zpow_one
Equiv.symm_apply_apply
sameCycle_apply_right πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”sameCycle_comm
sameCycle_apply_left
sameCycle_comm πŸ“–mathematicalβ€”SameCycleβ€”SameCycle.symm
sameCycle_conj πŸ“–mathematicalβ€”SameCycle
Equiv.Perm
instMul
instInv
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”conj_zpow
sameCycle_extendDomain πŸ“–mathematicalβ€”SameCycle
extendDomain
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
β€”extendDomain_zpow
extendDomain_apply_image
Equiv.injective
sameCycle_inv πŸ“–mathematicalβ€”SameCycle
Equiv.Perm
instInv
β€”Equiv.exists_congr_left
zpow_neg
inv_zpow'
inv_inv
sameCycle_inv_apply_left πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”sameCycle_symm_apply_left
sameCycle_inv_apply_right πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”sameCycle_symm_apply_right
sameCycle_one πŸ“–mathematicalβ€”SameCycle
Equiv.Perm
instOne
β€”one_zpow
sameCycle_pow_left πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
instPowNat
β€”zpow_natCast
sameCycle_zpow_left
sameCycle_pow_right πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
instPowNat
β€”zpow_natCast
sameCycle_zpow_right
sameCycle_subtypePerm πŸ“–mathematicalDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
SameCycle
subtypePerm
β€”subtypePerm_zpow
sameCycle_symm_apply_left πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”sameCycle_apply_left
Equiv.apply_symm_apply
sameCycle_symm_apply_right πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”sameCycle_apply_right
Equiv.apply_symm_apply
sameCycle_zpow_left πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
permGroup
β€”Equiv.exists_congr_left
Equiv.addRight_symm
zpow_add
zpow_neg
Equiv.symm_apply_apply
sameCycle_zpow_right πŸ“–mathematicalβ€”SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
permGroup
β€”sameCycle_comm
sameCycle_zpow_left
subtypePerm_apply_pow_of_mem πŸ“–mathematicalFinset
Finset.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
instPowNat
subtypePerm
SetLike.instMembership
Finset.instSetLike
β€”subtypePerm_pow
subtypePerm_apply_zpow_of_mem πŸ“–mathematicalFinset
Finset.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
permGroup
subtypePerm
SetLike.instMembership
Finset.instSetLike
β€”subtypePerm_zpow
swap_isSwap_iff πŸ“–mathematicalβ€”IsSwap
Equiv.swap
β€”IsCycle.ne_one
IsSwap.isCycle
Equiv.swap_self
zpow_eq_ofSubtype_subtypePerm_iff πŸ“–mathematicalFinset
Finset.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Finset.instHasSubset
support
DivInvMonoid.toZPow
Group.toDivInvMonoid
permGroup
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
MonoidHom.instFunLike
ofSubtype
Finset.decidableMem
subtypePerm
isInvariant_of_support_le
β€”isInvariant_of_support_le
ext
subtypePerm_apply_zpow_of_mem
congr_fun
ofSubtype_apply_of_mem
subtypePerm_zpow
subtypePerm_apply
ofSubtype_apply_of_not_mem
notMem_support
support_zpow_le

Equiv.Perm.IsCycle

Definitions

NameCategoryTheorems
zpowersEquivSupport πŸ“–CompOp
2 mathmath: zpowersEquivSupport_apply, zpowersEquivSupport_symm_apply

Theorems

NameKindAssumesProvesValidatesDepends On
commute_iff πŸ“–mathematicalEquiv.Perm.IsCycleCommute
Equiv.Perm
Equiv.Perm.instMul
Subgroup
Equiv.Perm.permGroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.zpowers
DFunLike.coe
MonoidHom
Finset
Finset.instMembership
Equiv.Perm.support
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MonoidHom.instFunLike
Equiv.Perm.ofSubtype
Finset.decidableMem
Equiv.Perm.subtypePerm
β€”commute_iff'
Equiv.Perm.apply_mem_support
Equiv.Perm.subtypePermOfSupport.eq_1
Equiv.Perm.subtypePerm_zpow
Equiv.Perm.ofSubtype_subtypePerm_of_mem
Equiv.Perm.ofSubtype_apply_of_not_mem
Equiv.Perm.notMem_support
Finset.notMem_mono
Equiv.Perm.support_zpow_le
commute_iff' πŸ“–mathematicalEquiv.Perm.IsCycleCommute
Equiv.Perm
Equiv.Perm.instMul
Finset
Finset.instMembership
Equiv.Perm.support
Subgroup
SetLike.instMembership
Finset.instSetLike
Equiv.Perm.permGroup
Subgroup.instSetLike
Subgroup.zpowers
Equiv.Perm.subtypePermOfSupport
Equiv.Perm.subtypePerm
β€”Equiv.Perm.mem_support_iff_of_commute
nonempty_support
sameCycle
Equiv.Perm.mem_support
Equiv.Perm.ext
Equiv.Perm.apply_mem_support
Equiv.Perm.subtypePerm_apply_zpow_of_mem
Commute.eq
Commute.zpow_right
zpow_add
add_comm
EquivLike.toEmbeddingLike
Equiv.Perm.congr_fun
Equiv.Perm.notMem_support
Mathlib.Tactic.Contrapose.contraposeβ‚„
conj πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm
Equiv.Perm.instMul
Equiv.Perm.instInv
β€”Equiv.symm_apply_apply
EquivLike.toEmbeddingLike
Equiv.apply_symm_apply
Equiv.Perm.SameCycle.conj
Iff.not
Equiv.Perm.eq_inv_iff_eq
eq_on_support_inter_nonempty_congr πŸ“–β€”Equiv.Perm.IsCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Finset
Finset.instMembership
Equiv.Perm.support
β€”β€”Equiv.Perm.mem_support
exists_pow_eq
Finite.of_fintype
Equiv.Perm.pow_eq_on_of_mem_support
Finset.mem_inter_of_mem
Equiv.Perm.pow_apply_mem_support
support_congr
Finset.inter_eq_left
eq_swap_of_apply_apply_eq_self πŸ“–mathematicalEquiv.Perm.IsCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.swapβ€”Equiv.ext
Equiv.swap_apply_left
Equiv.swap_apply_right
Equiv.swap_apply_of_ne_of_ne
by_contradiction
Equiv.Perm.zpow_apply_eq_of_apply_apply_eq_self
Equiv.Perm.mul_apply
zpow_add
sub_add_cancel
exists_pow_eq πŸ“–mathematicalEquiv.Perm.IsCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”exists_zpow_eq
ne_of_gt
orderOf_pos
Equiv.finite_left
zpow_natCast
zpow_mod_orderOf
exists_zpow_eq πŸ“–mathematicalEquiv.Perm.IsCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”sameCycle
extendDomain πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm.extendDomainβ€”Equiv.Perm.extendDomain_apply_image
Subtype.coe_injective
Equiv.injective
of_not_not
Equiv.Perm.extendDomain_apply_not_subtype
Equiv.apply_symm_apply
Equiv.Perm.SameCycle.extendDomain
inv πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm
Equiv.Perm.instInv
β€”Iff.not
Equiv.Perm.inv_eq_iff_eq
Equiv.Perm.SameCycle.inv
isConj πŸ“–mathematicalEquiv.Perm.IsCycle
Finset.card
Equiv.Perm.support
IsConj
Equiv.Perm
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”Equiv.Perm.isConj_of_support_equiv
Equiv.finite_left
Finite.of_fintype
orderOf
Equiv.Perm.apply_mem_support
exists_pow_eq
Equiv.Perm.mem_support
Equiv.Perm.pow_apply_mem_support
zpow_natCast
zpowersEquivSupport_symm_apply
zpowersEquivZPowers_apply
isConj_iff πŸ“–mathematicalEquiv.Perm.IsCycleIsConj
Equiv.Perm
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
Finset.card
Equiv.Perm.support
β€”isConj_iff
Finset.card_bij
Equiv.Perm.support_conj
Equiv.symm_apply_apply
Equiv.injective
Equiv.apply_symm_apply
isConj
isCycleOn πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm.IsCycleOn
setOf
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.bijOn
Iff.not
Equiv.apply_eq_iff_eq
sameCycle
isCycle_pow_pos_of_lt_prime_order πŸ“–mathematicalEquiv.Perm.IsCycle
Nat.Prime
orderOf
Equiv.Perm
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
Equiv.Perm.instPowNatβ€”nonempty_fintype
Nat.Prime.coprime_iff_not_dvd
exists_pow_eq_self_of_coprime
of_pow
Equiv.Perm.support_pow_le
ne_one πŸ“–β€”Equiv.Perm.IsCycleβ€”β€”instIsEmptyFalse
nonempty_support πŸ“–mathematicalEquiv.Perm.IsCycleFinset.Nonempty
Equiv.Perm.support
β€”Finset.nonempty_iff_ne_empty
Equiv.Perm.support_eq_empty_iff
ne_one
of_pow πŸ“–β€”Equiv.Perm.IsCycle
Equiv.Perm
Equiv.Perm.instPowNat
Finset
Finset.instHasSubset
Equiv.Perm.support
β€”β€”LE.le.antisymm
Equiv.Perm.support_pow_le
zpow_mul
of_zpow πŸ“–β€”Equiv.Perm.IsCycle
Equiv.Perm
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
Finset
Finset.instHasSubset
Equiv.Perm.support
β€”β€”of_pow
inv
zpow_negSucc
inv_inv
Equiv.Perm.support_inv
orderOf πŸ“–mathematicalEquiv.Perm.IsCycleorderOf
Equiv.Perm
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
Finset.card
Equiv.Perm.support
β€”Fintype.card_zpowers
Fintype.card_coe
Fintype.card_congr
pow_eq_one_iff πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm
Equiv.Perm.instPowNat
Equiv.Perm.instOne
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”nonempty_fintype
Equiv.Perm.mem_support
support_pow_eq_iff
pow_mul
pow_orderOf_eq_one
one_pow
pow_eq_one_iff' πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm
Equiv.Perm.instPowNat
Equiv.Perm.instOne
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”DFunLike.congr_fun
pow_eq_one_iff
pow_eq_one_iff'' πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm
Equiv.Perm.instPowNat
Equiv.Perm.instOne
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”pow_eq_one_iff'
pow_eq_pow_iff πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm
Equiv.Perm.instPowNat
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”nonempty_fintype
pow_eq_one_iff
Equiv.Perm.mem_support
pow_sub
Equiv.symm_apply_apply
Equiv.Perm.zpow_apply_comm
Equiv.injective
zpow_one
zpow_natCast
Equiv.Perm.notMem_support
mul_inv_eq_one
le_of_not_ge
pow_iff πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm
Equiv.Perm.instPowNat
orderOf
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”nonempty_fintype
support_pow_eq_iff
ne_one
pow_mul
pow_orderOf_eq_one
one_pow
orderOf
orderOf_pow
Equiv.finite_left
LT.lt.ne'
orderOf_pos
exists_pow_eq_self_of_coprime
of_pow
Equiv.Perm.support_pow_le
sameCycle πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm.SameCycleβ€”Equiv.Perm.mul_apply
zpow_add
sub_add_cancel
sign πŸ“–mathematicalEquiv.Perm.IsCycleDFunLike.coe
MonoidHom
Equiv.Perm
Units
Int.instMonoid
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
Units.instMulOneClass
MonoidHom.instFunLike
Equiv.Perm.sign
Units.instNeg
NonUnitalNonAssocRing.toHasDistribNeg
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Int.instCommRing
Monoid.toNatPow
Units.instMonoid
Units.instOne
Finset.card
Equiv.Perm.support
β€”β€”
support_congr πŸ“–β€”Equiv.Perm.IsCycle
Finset
Finset.instHasSubset
Equiv.Perm.support
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”β€”le_antisymm
Equiv.Perm.mem_support
exists_pow_eq
Finite.of_fintype
Finset.mem_of_mem_inter_left
Equiv.Perm.pow_eq_on_of_mem_support
Finset.mem_inter_of_mem
Equiv.Perm.pow_apply_mem_support
Equiv.Perm.support_congr
support_pow_eq_iff πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm.support
Equiv.Perm
Equiv.Perm.instPowNat
orderOf
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”orderOf_dvd_iff_pow_eq_one
ne_one
Equiv.Perm.support_eq_empty_iff
Equiv.Perm.support_one
le_antisymm
Equiv.Perm.support_pow_le
Mathlib.Tactic.Contrapose.contraposeβ‚‚
Equiv.Perm.ext
Equiv.Perm.pow_apply_eq_self_of_apply_eq_self
Equiv.Perm.one_apply
exists_pow_eq
Finite.of_fintype
Equiv.Perm.mem_support
Equiv.injective
Equiv.Perm.mul_apply
Commute.eq
Commute.pow_pow_self
support_pow_of_pos_of_lt_orderOf πŸ“–mathematicalEquiv.Perm.IsCycle
orderOf
Equiv.Perm
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
Equiv.Perm.support
Equiv.Perm.instPowNat
β€”support_pow_eq_iff
swap_mul πŸ“–mathematicalEquiv.Perm.IsCycleEquiv.Perm
Equiv.Perm.instMul
Equiv.swap
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.injective
exists_zpow_eq
Equiv.Perm.ne_and_ne_of_swap_mul_apply_ne_self
Equiv.Perm.isCycle_swap_mul_auxβ‚‚
sub_add_cancel
two_le_card_support πŸ“–mathematicalEquiv.Perm.IsCycleFinset.card
Equiv.Perm.support
β€”Equiv.Perm.two_le_card_support_of_ne_one
ne_one
zpowersEquivSupport_apply πŸ“–mathematicalEquiv.Perm.IsCycleDFunLike.coe
Equiv
Equiv.Perm
Subgroup
Equiv.Perm.permGroup
SetLike.instMembership
Subgroup.instSetLike
Subgroup.zpowers
Finset
Finset.instSetLike
Equiv.Perm.support
EquivLike.toFunLike
Equiv.instEquivLike
zpowersEquivSupport
Equiv.Perm.instPowNat
Finset.instMembership
Equiv.Perm.pow_apply_mem_support
Equiv.Perm.mem_support
β€”β€”
zpowersEquivSupport_symm_apply πŸ“–mathematicalEquiv.Perm.IsCycleDFunLike.coe
Equiv
Finset
SetLike.instMembership
Finset.instSetLike
Equiv.Perm.support
Equiv.Perm
Subgroup
Equiv.Perm.permGroup
Subgroup.instSetLike
Subgroup.zpowers
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
zpowersEquivSupport
Equiv.Perm.instPowNat
Finset.instMembership
Equiv.Perm.pow_apply_mem_support
Equiv.Perm.mem_support
β€”Equiv.Perm.pow_apply_mem_support
Equiv.Perm.mem_support
Equiv.symm_apply_eq
zpowersEquivSupport_apply

Equiv.Perm.IsCycleOn

Theorems

NameKindAssumesProvesValidatesDepends On
apply_mem_iff πŸ“–mathematicalEquiv.Perm.IsCycleOnSet
Set.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.symm_apply_apply
Set.BijOn.perm_inv
Set.BijOn.mapsTo
apply_ne πŸ“–β€”Equiv.Perm.IsCycleOn
Set.Nontrivial
Set
Set.instMembership
β€”β€”Set.Nontrivial.exists_ne
Function.IsFixedPt.perm_zpow
conj πŸ“–mathematicalEquiv.Perm.IsCycleOnEquiv.Perm
Equiv.Perm.instMul
Equiv.Perm.instInv
Set.image
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”Set.BijOn.comp
Equiv.bijOn_image
Equiv.bijOn_symm_image
Equiv.apply_symm_apply
Equiv.Perm.SameCycle.conj
Equiv.image_eq_preimage_symm
countable πŸ“–mathematicalEquiv.Perm.IsCycleOnSet.Countableβ€”Set.eq_empty_or_nonempty
Set.countable_empty
Set.Countable.mono
Set.countable_range
instCountableInt
exists_pow_eq πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
Finset.instSetLike
Finset.instMembership
Finset.card
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”Int.ModEq.dvd
Int.ModEq.symm
Int.mod_modEq
Int.natMod_lt
LT.lt.ne'
Finset.Nonempty.card_pos
zpow_natCast
Int.natMod.eq_1
Nat.cast_ne_zero
Int.instCharZero
sub_eq_iff_eq_add'
zpow_add
zpow_mul
EquivLike.toEmbeddingLike
Function.IsFixedPt.perm_zpow
pow_card_apply
exists_pow_eq' πŸ“–mathematicalEquiv.Perm.IsCycleOn
Set
Set.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”CanLift.prf
Set.instCanLiftFinsetCoeFinite
exists_pow_eq
extendDomain πŸ“–mathematicalEquiv.Perm.IsCycleOnEquiv.Perm.extendDomain
Set.image
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
β€”Set.BijOn.extendDomain
Equiv.Perm.SameCycle.extendDomain
inv πŸ“–mathematicalEquiv.Perm.IsCycleOnEquiv.Perm
Equiv.Perm.instInv
β€”Equiv.Perm.isCycleOn_inv
isCycle_subtypePerm πŸ“–mathematicalEquiv.Perm.IsCycleOn
Set.Nontrivial
Equiv.Perm.IsCycle
Set
Set.instMembership
Equiv.Perm.subtypePerm
apply_mem_iff
β€”apply_mem_iff
Set.Nontrivial.nonempty
apply_ne
Equiv.Perm.SameCycle.subtypePerm
of_inv πŸ“–β€”Equiv.Perm.IsCycleOn
Equiv.Perm
Equiv.Perm.instInv
β€”β€”Equiv.Perm.isCycleOn_inv
of_pow πŸ“–β€”Equiv.Perm.IsCycleOn
Equiv.Perm
Equiv.Perm.instPowNat
Set.BijOn
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”β€”Equiv.Perm.SameCycle.of_pow
of_zpow πŸ“–β€”Equiv.Perm.IsCycleOn
Equiv.Perm
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
Set.BijOn
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”β€”Equiv.Perm.SameCycle.of_zpow
pow_apply_eq πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
Finset.instSetLike
Finset.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
Finset.card
β€”Finset.eq_singleton_or_nontrivial
Finset.card_singleton
Unique.instSubsingleton
Function.IsFixedPt.iterate
Equiv.Perm.isCycleOn_singleton
Finset.coe_singleton
apply_ne
apply_mem_iff
Equiv.Perm.IsCycle.orderOf
isCycle_subtypePerm
Equiv.Perm.support_subtypePerm
Finset.filter.congr_simp
Finset.filter_true_of_mem
Finset.card_attach
orderOf_dvd_iff_pow_eq_one
Equiv.Perm.IsCycle.pow_eq_one_iff'
Finite.of_fintype
Equiv.Perm.subtypePerm_pow
pow_apply_eq_pow_apply πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
Finset.instSetLike
Finset.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
Nat.ModEq
Finset.card
β€”Nat.modEq_iff_dvd
zpow_apply_eq
sub_eq_neg_add
zpow_add
zpow_neg
zpow_natCast
pow_card_apply πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
Finset.instSetLike
Finset.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
Finset.card
β€”pow_apply_eq
dvd_rfl
range_pow πŸ“–mathematicalEquiv.Perm.IsCycleOn
Set
Set.instMembership
Set.range
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”Set.Subset.antisymm
Set.range_subset_iff
Set.MapsTo.perm_pow
Set.BijOn.mapsTo
exists_pow_eq'
range_zpow πŸ“–mathematicalEquiv.Perm.IsCycleOn
Set
Set.instMembership
Set.range
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”Set.Subset.antisymm
Set.range_subset_iff
Set.BijOn.mapsTo
Set.BijOn.perm_zpow
subsingleton πŸ“–mathematicalEquiv.Perm.IsCycleOn
Equiv.Perm
Equiv.Perm.instOne
Set.Subsingletonβ€”Equiv.Perm.isCycleOn_one
subtypePerm πŸ“–mathematicalEquiv.Perm.IsCycleOnSet
Set.instMembership
Equiv.Perm.subtypePerm
apply_mem_iff
Set.univ
β€”apply_mem_iff
Set.subsingleton_or_nontrivial
Equiv.Perm.isCycleOn_of_subsingleton
Set.Subsingleton.coe_sort
Set.eq_univ_iff_forall
apply_ne
Equiv.Perm.IsCycle.isCycleOn
isCycle_subtypePerm
zpow_apply_eq πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
Finset.instSetLike
Finset.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
Finset.card
β€”pow_apply_eq
zpow_negSucc
inv_pow
inv
dvd_neg
zpow_apply_eq_zpow_apply πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
Finset.instSetLike
Finset.instMembership
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
Int.ModEq
Finset.card
β€”Int.modEq_iff_dvd
zpow_apply_eq
sub_eq_neg_add
zpow_add
zpow_neg

Equiv.Perm.IsSwap

Theorems

NameKindAssumesProvesValidatesDepends On
isCycle πŸ“–mathematicalEquiv.Perm.IsSwapEquiv.Perm.IsCycleβ€”Equiv.Perm.isCycle_swap

Equiv.Perm.SameCycle

Definitions

NameCategoryTheorems
setoid πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
apply_eq_self_iff πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.Perm.mul_apply
zpow_one_add
add_comm
zpow_add_one
Equiv.injective
apply_left πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.Perm.sameCycle_apply_left
apply_right πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.Perm.sameCycle_apply_right
conj πŸ“–mathematicalEquiv.Perm.SameCycleEquiv.Perm
Equiv.Perm.instMul
Equiv.Perm.instInv
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.symm_apply_apply
eq_of_left πŸ“–β€”Equiv.Perm.SameCycle
Function.IsFixedPt
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”β€”Function.IsFixedPt.eq
Function.IsFixedPt.perm_zpow
eq_of_right πŸ“–β€”Equiv.Perm.SameCycle
Function.IsFixedPt
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”β€”eq_of_left
apply_eq_self_iff
equivalence πŸ“–mathematicalβ€”Equiv.Perm.SameCycleβ€”refl
symm
trans
exists_fin_pow_eq πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
orderOf
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”exists_pow_eq'
exists_nat_pow_eq πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”exists_pow_eq'
exists_pow_eq' πŸ“–mathematicalEquiv.Perm.SameCycleorderOf
Equiv.Perm
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”orderOf_pos
Equiv.finite_left
LT.lt.ne'
zpow_natCast
zpow_mod_orderOf
exists_pow_eq'' πŸ“–mathematicalEquiv.Perm.SameCycleorderOf
Equiv.Perm
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Equiv.Perm.permGroup
DFunLike.coe
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”exists_pow_eq'
orderOf_pos
Equiv.finite_left
le_rfl
pow_orderOf_eq_one
pow_zero
LT.lt.le
extendDomain πŸ“–mathematicalEquiv.Perm.SameCycleEquiv.Perm.extendDomain
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.Perm.sameCycle_extendDomain
inv πŸ“–mathematicalEquiv.Perm.SameCycleEquiv.Perm
Equiv.Perm.instInv
β€”Equiv.Perm.sameCycle_inv
of_apply_left πŸ“–β€”Equiv.Perm.SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”β€”Equiv.Perm.sameCycle_apply_left
of_apply_right πŸ“–β€”Equiv.Perm.SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”β€”Equiv.Perm.sameCycle_apply_right
of_inv πŸ“–β€”Equiv.Perm.SameCycle
Equiv.Perm
Equiv.Perm.instInv
β€”β€”Equiv.Perm.sameCycle_inv
of_pow πŸ“–β€”Equiv.Perm.SameCycle
Equiv.Perm
Equiv.Perm.instPowNat
β€”β€”zpow_mul
zpow_natCast
of_pow_left πŸ“–β€”Equiv.Perm.SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”β€”Equiv.Perm.sameCycle_pow_left
of_pow_right πŸ“–β€”Equiv.Perm.SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”β€”Equiv.Perm.sameCycle_pow_right
of_symm_apply_left πŸ“–β€”Equiv.Perm.SameCycle
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”β€”Equiv.Perm.sameCycle_symm_apply_left
of_symm_apply_right πŸ“–β€”Equiv.Perm.SameCycle
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”β€”Equiv.Perm.sameCycle_symm_apply_right
of_zpow πŸ“–β€”Equiv.Perm.SameCycle
Equiv.Perm
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”β€”zpow_mul
of_zpow_left πŸ“–β€”Equiv.Perm.SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”β€”Equiv.Perm.sameCycle_zpow_left
of_zpow_right πŸ“–β€”Equiv.Perm.SameCycle
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”β€”Equiv.Perm.sameCycle_zpow_right
pow_left πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”Equiv.Perm.sameCycle_pow_left
pow_right πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”Equiv.Perm.sameCycle_pow_right
refl πŸ“–mathematicalβ€”Equiv.Perm.SameCycleβ€”β€”
rfl πŸ“–mathematicalβ€”Equiv.Perm.SameCycleβ€”refl
subtypePerm πŸ“–mathematicalDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.SameCycle
Equiv.Perm.subtypePermβ€”Equiv.Perm.sameCycle_subtypePerm
symm_apply_left πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”Equiv.Perm.sameCycle_symm_apply_left
symm_apply_right πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”Equiv.Perm.sameCycle_symm_apply_right
trans πŸ“–β€”Equiv.Perm.SameCycleβ€”β€”zpow_add
Equiv.Perm.mul_apply
zpow_left πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”Equiv.Perm.sameCycle_zpow_left
zpow_right πŸ“–mathematicalEquiv.Perm.SameCycleDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”Equiv.Perm.sameCycle_zpow_right

Finset

Theorems

NameKindAssumesProvesValidatesDepends On
exists_cycleOn πŸ“–mathematicalβ€”Equiv.Perm.IsCycleOn
SetLike.coe
Finset
instSetLike
instHasSubset
Equiv.Perm.support
β€”List.Nodup.isCycleOn_formPerm
nodup_toList
List.mem_of_formPerm_apply_ne
Equiv.Perm.mem_support
product_self_eq_disjiUnion_perm πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
instSetLike
SProd.sprod
instSProd
disjiUnion
range
card
map
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
product_self_eq_disjiUnion_perm_aux
β€”ext
product_self_eq_disjiUnion_perm_aux
Equiv.Perm.IsCycleOn.exists_pow_eq
Equiv.Perm.iterate_eq_pow
Set.BijOn.mapsTo
Set.BijOn.iterate
product_self_eq_disjiUnion_perm_aux πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
instSetLike
Set.PairwiseDisjoint
partialOrder
instOrderBot
range
card
map
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”Set.subsingleton_or_nontrivial
Set.Subsingleton.pairwise
card_range
Nat.ModEq.eq_of_lt_of_lt
Equiv.Perm.IsCycleOn.pow_apply_eq_pow_apply
mem_range
mem_coe
sum_mul_sum_eq_sum_perm πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
instSetLike
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
sum
NonUnitalNonAssocSemiring.toAddCommMonoid
range
card
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”sum_smul_sum_eq_sum_perm
sum_smul_sum_eq_sum_perm πŸ“–mathematicalEquiv.Perm.IsCycleOn
SetLike.coe
Finset
instSetLike
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Module.toDistribMulAction
sum
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
range
card
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.instPowNat
β€”sum_smul_sum
sum_product'
product_self_eq_disjiUnion_perm_aux
sum_congr
product_self_eq_disjiUnion_perm
sum_disjiUnion
sum_map

Int

Theorems

NameKindAssumesProvesValidatesDepends On
addLeft_one_isCycle πŸ“–mathematicalβ€”Equiv.Perm.IsCycle
Equiv.addLeft
instAddGroup
β€”one_ne_zero
Equiv.zsmul_addLeft
mul_one
add_zero
addRight_one_isCycle πŸ“–mathematicalβ€”Equiv.Perm.IsCycle
Equiv.addRight
instAddGroup
β€”one_ne_zero
Equiv.zsmul_addRight
mul_one
zero_add

List.Nodup

Theorems

NameKindAssumesProvesValidatesDepends On
isCycleOn_formPerm πŸ“–mathematicalβ€”Equiv.Perm.IsCycleOn
List.formPerm
setOf
β€”Equiv.bijOn
List.formPerm_mem_iff_mem
Set.mem_setOf
LE.le.trans_lt
sub_eq_neg_add
zpow_add
zpow_neg
zpow_natCast
List.formPerm_pow_apply_getElem
add_comm

Set

Theorems

NameKindAssumesProvesValidatesDepends On
prod_self_eq_iUnion_perm πŸ“–mathematicalEquiv.Perm.IsCycleOnSProd.sprod
Set
instSProd
iUnion
image
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
DivInvMonoid.toZPow
Group.toDivInvMonoid
Equiv.Perm.permGroup
β€”ext
BijOn.mapsTo
BijOn.perm_zpow

Set.Countable

Theorems

NameKindAssumesProvesValidatesDepends On
exists_cycleOn πŸ“–mathematicalβ€”Equiv.Perm.IsCycleOn
Set
Set.instHasSubset
setOf
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
β€”Set.finite_or_infinite
List.Nodup.isCycleOn_formPerm
Finset.nodup_toList
List.mem_of_formPerm_apply_ne
nonempty_equiv_of_countable
instCountableInt
Int.infinite
to_subtype
Set.Infinite.to_subtype
Set.image_comp
Equiv.image_eq_preimage_symm
Set.ext
AddLeftCancelSemigroup.toIsLeftCancelAdd
Set.image_univ
Subtype.range_coe_subtype
Equiv.Perm.IsCycleOn.extendDomain
Equiv.Perm.IsCycle.isCycleOn
Int.addRight_one_isCycle
of_not_not
Equiv.Perm.extendDomain_apply_not_subtype

Set.Subsingleton

Theorems

NameKindAssumesProvesValidatesDepends On
isCycleOn_one πŸ“–mathematicalSet.SubsingletonEquiv.Perm.IsCycleOn
Equiv.Perm
Equiv.Perm.instOne
β€”Equiv.Perm.isCycleOn_one

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