Documentation Verification Report

Defs

πŸ“ Source: Mathlib/Data/Nat/Factorization/Defs.lean

Statistics

MetricCount
Definitionsfactorization, factorizationEquiv, factorizationLCMLeft, factorizationLCMRight, Β«termOrdCompl[_]_Β», Β«termOrdProj[_]_Β»
6
Theoremsfactorization, factorization_pos_of_dvd, factorization_pow, eq_of_factorization_eq, eq_of_factorization_eq', factorization_def, factorization_eq_primeFactorsList_multiset, factorization_eq_zero_iff, factorization_eq_zero_of_non_prime, factorization_eq_zero_of_not_dvd, factorization_eq_zero_of_not_prime, factorization_eq_zero_of_remainder, factorization_inj, factorization_le_iff_dvd, factorization_minFac_ne_zero, factorization_mul, factorization_mul_apply_of_coprime, factorization_mul_of_coprime, factorization_one, factorization_one_right, factorization_pow, factorization_prod, factorization_prod_pow_eq_self, factorization_zero, factorization_zero_right, multiplicity_eq_factorization, ordProj_dvd, ordProj_dvd_ordProj_iff_dvd, pow_succ_factorization_not_dvd, primeFactorsList_count_eq, prod_pow_factorization_eq_self, support_factorization
32
Total38

Nat

Definitions

NameCategoryTheorems
factorization πŸ“–CompOp
132 mathmath: Prime.exists_orderOf_eq_pow_factorization_exponent, Primes.prodNatEquiv_symm_apply, multiplicity_eq_factorization, factorization_eq_zero_of_not_dvd, IsPrimePow.exists_ordCompl_eq_one, Prime.factorization_pos_of_dvd, ordCompl_le, multiplicative_factorization', IsPrimePow.minFac_pow_factorization_eq, factorization_prod_pow_eq_self, factorization_le_factorization_mul_left, factorization_eq_of_coprime_right, factorization_mul_apply_of_coprime, ordCompl_dvd_ordCompl_iff_dvd, ordProj_of_not_prime, pow_succ_factorization_not_dvd, Prime.factorization_pow, factorization_factorial, ordCompl_dvd_ordCompl_of_dvd, factorization_mul, isPrimePow_iff_factorization_eq_single, factorization_eq_card_pow_dvd_of_lt, ArithmeticFunction.IsMultiplicative.multiplicative_factorization, ordProj_dvd_ordProj_iff_dvd, exists_factorization_lt_of_lt, factorization_eq_one_of_squarefree, factorization_choose_prime_pow_add_factorization, factorization_prod_apply, Real.log_nat_eq_sum_factorization, factorization_one_right, ordProj_mul_ordCompl_eq_self, pairwise_coprime_pow_primeFactors_factorization, Real.logb_nat_eq_sum_factorization, factorization_choose_le_one, Prime.factorization, support_factorization, dvd_ordCompl_of_dvd_not_dvd, Prime.dvd_iff_one_le_factorization, eq_factorization_iff, factorization_zero, ArithmeticFunction.cardFactors_eq_sum_factorization, ceilRoot_def, dvd_iff_div_factorization_eq_tsub, exponent_eq_exponent_mul_factorization_of_prime_pow_eq_base_pow, factorization_centralBinom_eq_zero_of_two_mul_lt, Squarefree.natFactorization_le_one, factorization_choose_of_lt_three_mul, factorization_lt, not_dvd_ordCompl, factorization_lcm, factorization_ceilRoot, ArithmeticFunction.sigma_eq_prod_primeFactors_sum_range_factorization_pow_mul, primeFactorsList_count_eq, setOf_pow_dvd_eq_Icc_factorization, factorization_choose', factorization_choose_le_log, factorization_eq_zero_of_not_prime, Icc_factorization_eq_pow_dvd, factorization_le_iff_dvd, ordCompl_pos, ordCompl_mul, ordProj_le, factorization_le_factorization_choose_add, card_divisors, Prime.factorization_self, ordProj_dvd_ordProj_of_dvd, Sylow.card_eq_multiplicity, factorization_centralBinom_of_two_mul_self_lt_three_mul, ordCompl_self_pow_mul, factorization_one, prod_pow_primeFactors_factorization, ordProj_self_pow, exists_ordCompl_eq_one_iff_isPrimePow, factorization_le_factorization_mul_right, prod_factorization_eq_prod_primeFactors, factorization_eq_zero_iff_remainder, Prime.pow_dvd_iff_dvd_ordProj, prod_pow_factorization_eq_self, factorization_eq_primeFactorsList_multiset, factorization_floorRoot, floorRoot_def, totient_eq_prod_factorization, factorization_div, factorization_choose_eq_zero_of_lt, centralBinom_factorization_small, prod_pow_factorization_choose, factorization_prime_le_iff_dvd, sum_divisors, prod_primeFactors_prod_factorization, factorization_pow_self, factorization_eq_card_pow_dvd, factorization_factorial_mul_succ, factorization_ordCompl, multiplicative_factorization, factorization_eq_zero_iff', sub_one_mul_factorization_factorial, ordProj_dvd, factorization_def, coprime_ordCompl, Prime.pow_dvd_iff_le_factorization, factorization_factorial_le_div_pred, dvd_ordProj_of_dvd, factorization_pow, ordProj_pos, ordCompl_eq_self_iff_zero_or_not_dvd, prod_primeFactors_invOn_squarefree, ordProj_mul, factorization_eq_zero_of_remainder, factorization_choose_prime_pow, ArithmeticFunction.carmichael_factorization, factorization_choose, factorization_eq_zero_iff, Prime.exists_addOrderOf_eq_pow_padic_val_nat_add_exponent, factorization_eq_zero_of_non_prime, prod_pow_factorization_centralBinom, ordCompl_dvd, factorization_eq_zero_of_lt, factorization_gcd, factorization_inj, factorization_prod, factorization_mul_of_coprime, isPrimePow_iff_minFac_pow_factorization_eq, ordCompl_self_pow, factorization_zero_right, pow_factorization_choose_le, factorization_le_of_le_pow, factorization_eq_of_coprime_left, factorization_le_factorization_of_dvd_right, ordCompl_of_not_prime, factorization_factorial_mul, squarefree_iff_factorization_le_one, factorization_factorial_eq_zero_of_lt
factorizationEquiv πŸ“–CompOp
1 mathmath: factorizationEquiv_inv_apply
factorizationLCMLeft πŸ“–CompOp
9 mathmath: IsPrimitiveRoot.pow_mul_pow_lcm, factorizationLCMLeft_pos, coprime_factorizationLCMLeft_factorizationLCMRight, factorizationLCMLeft_mul_factorizationLCMRight, factorizationLCMLeft_zero_right, factorizationLCMLeft_dvd_left, AddCommute.addOrderOf_add_nsmul_eq_lcm, factorizationLCMLeft_zero_left, Commute.orderOf_mul_pow_eq_lcm
factorizationLCMRight πŸ“–CompOp
9 mathmath: IsPrimitiveRoot.pow_mul_pow_lcm, coprime_factorizationLCMLeft_factorizationLCMRight, factorizationLCMLeft_mul_factorizationLCMRight, factorizationLCMRight_dvd_right, AddCommute.addOrderOf_add_nsmul_eq_lcm, factorizationLCMRight_pos, factorizationLCRight_zero_left, factorizationLCMRight_zero_right, Commute.orderOf_mul_pow_eq_lcm
Β«termOrdCompl[_]_Β» πŸ“–CompOpβ€”
Β«termOrdProj[_]_Β» πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
eq_of_factorization_eq πŸ“–β€”DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”β€”eq_of_perm_primeFactorsList
primeFactorsList_count_eq
eq_of_factorization_eq' πŸ“–β€”factorizationβ€”β€”eq_of_factorization_eq
factorization_def πŸ“–mathematicalPrimeDFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
padicValNat
β€”β€”
factorization_eq_primeFactorsList_multiset πŸ“–mathematicalβ€”factorization
DFunLike.coe
AddEquiv
Multiset
Finsupp
MulZeroClass.toZero
instMulZeroClass
Multiset.instAdd
Finsupp.instAdd
AddMonoid.toAddZeroClass
instAddMonoid
EquivLike.toFunLike
AddEquiv.instEquivLike
Multiset.toFinsupp
Multiset.ofList
primeFactorsList
β€”Finsupp.ext
Multiset.coe_count
primeFactorsList_count_eq
factorization_eq_zero_iff πŸ“–mathematicalβ€”DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
Prime
β€”β€”
factorization_eq_zero_of_non_prime πŸ“–mathematicalPrimeDFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”factorization_eq_zero_of_not_prime
factorization_eq_zero_of_not_dvd πŸ“–mathematicalβ€”DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”β€”
factorization_eq_zero_of_not_prime πŸ“–mathematicalPrimeDFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”β€”
factorization_eq_zero_of_remainder πŸ“–mathematicalβ€”DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”factorization_eq_zero_of_not_dvd
Dvd.intro
factorization_inj πŸ“–mathematicalβ€”Set.InjOn
Finsupp
MulZeroClass.toZero
instMulZeroClass
factorization
setOf
β€”eq_of_factorization_eq
factorization_le_iff_dvd πŸ“–mathematicalβ€”Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instLE
factorization
β€”factorization_prod_pow_eq_self
Finsupp.prod_dvd_prod_of_subset_of_dvd
Finsupp.support_mono
instCanonicallyOrderedAdd
pow_dvd_pow
factorization_mul
right_ne_zero_of_mul
self_le_add_right
Finsupp.instCanonicallyOrderedAddOfAddLeftMono
instOrderedSub
IsOrderedAddMonoid.toAddLeftMono
instIsOrderedAddMonoid
factorization_minFac_ne_zero πŸ“–β€”β€”β€”β€”factorization_eq_zero_iff
minFac_prime
minFac_dvd
factorization_mul πŸ“–mathematicalβ€”factorization
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instAdd
AddMonoid.toAddZeroClass
instAddMonoid
β€”Finsupp.ext
perm_primeFactorsList_mul
factorization_mul_apply_of_coprime πŸ“–mathematicalβ€”DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”perm_primeFactorsList_mul_of_coprime
factorization_mul_of_coprime πŸ“–mathematicalβ€”factorization
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instAdd
AddMonoid.toAddZeroClass
instAddMonoid
β€”Finsupp.ext
Finsupp.add_apply
factorization_mul_apply_of_coprime
factorization_one πŸ“–mathematicalβ€”factorization
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instZero
β€”Finsupp.ext
padicValNat.one
primeFactors_one
factorization_one_right πŸ“–mathematicalβ€”DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”factorization_eq_zero_of_not_prime
not_prime_one
factorization_pow πŸ“–mathematicalβ€”factorization
Monoid.toNatPow
instMonoid
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instNatSMul
instAddMonoid
β€”pow_zero
factorization_one
zero_nsmul
eq_or_ne
zero_pow
factorization_zero
nsmul_zero
mul_comm
factorization_mul
pow_ne_zero
isReduced_of_noZeroDivisors
IsStrictOrderedRing.noZeroDivisors
instIsStrictOrderedRing
CanonicallyOrderedAdd.toExistsAddOfLE
instCanonicallyOrderedAdd
add_smul
one_smul
add_comm
factorization_prod πŸ“–mathematicalβ€”factorization
Finset.prod
instCommMonoid
Finset.sum
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instAddCommMonoid
instAddCommMonoid
β€”Finset.induction_on'
factorization_one
Finset.prod_ne_zero_iff
instNontrivial
IsStrictOrderedRing.noZeroDivisors
instIsStrictOrderedRing
CanonicallyOrderedAdd.toExistsAddOfLE
instCanonicallyOrderedAdd
Finset.prod_insert
factorization_mul
Finset.sum_insert
factorization_prod_pow_eq_self πŸ“–mathematicalβ€”Finsupp.prod
MulZeroClass.toZero
instMulZeroClass
instCommMonoid
factorization
Monoid.toNatPow
instMonoid
β€”factorization_eq_primeFactorsList_multiset
Multiset.toFinsupp_toMultiset
prod_primeFactorsList
factorization_zero πŸ“–mathematicalβ€”factorization
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instZero
β€”Finsupp.ext
padicValNat.zero
primeFactors_zero
factorization_zero_right πŸ“–mathematicalβ€”DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”factorization_eq_zero_of_not_prime
not_prime_zero
multiplicity_eq_factorization πŸ“–mathematicalPrimemultiplicity
instMonoid
DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”padicValNat_def'
Prime.ne_one
ordProj_dvd πŸ“–mathematicalβ€”Monoid.toNatPow
instMonoid
DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”primeFactorsList_count_eq
dvd_of_primeFactorsList_subperm
pow_ne_zero
isReduced_of_noZeroDivisors
IsStrictOrderedRing.noZeroDivisors
instIsStrictOrderedRing
CanonicallyOrderedAdd.toExistsAddOfLE
instCanonicallyOrderedAdd
Prime.ne_zero
Prime.primeFactorsList_pow
factorization_eq_zero_of_not_prime
pow_zero
Unique.instSubsingleton
ordProj_dvd_ordProj_iff_dvd πŸ“–mathematicalβ€”Monoid.toNatPow
instMonoid
DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”factorization_le_iff_dvd
Finsupp.le_def
factorization_zero_right
pow_zero
zero_add
factorization_one_right
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
instIsOrderedAddMonoid
contravariant_swap_add_of_contravariant_add
contravariant_lt_of_covariant_le
instNoMaxOrder
instCanonicallyOrderedAdd
pow_succ_factorization_not_dvd πŸ“–mathematicalPrimeMonoid.toNatPow
instMonoid
DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”factorization_pow
Prime.factorization
Finsupp.smul_single
mul_one
Finsupp.single_add
Finsupp.single_eq_same
IsOrderedAddMonoid.toAddLeftMono
instIsOrderedAddMonoid
IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT
AddLeftCancelSemigroup.toIsLeftCancelAdd
contravariant_lt_of_covariant_le
instCanonicallyOrderedAdd
factorization_le_iff_dvd
pow_ne_zero
isReduced_of_noZeroDivisors
IsStrictOrderedRing.noZeroDivisors
instIsStrictOrderedRing
CanonicallyOrderedAdd.toExistsAddOfLE
Prime.ne_zero
primeFactorsList_count_eq πŸ“–mathematicalβ€”primeFactorsList
DFunLike.coe
Finsupp
MulZeroClass.toZero
instMulZeroClass
Finsupp.instFunLike
factorization
β€”primeFactorsList_zero
padicValNat.zero
primeFactors_zero
factorization_def
le_antisymm
le_padicValNat_iff_replicate_subperm_primeFactorsList
LT.lt.ne'
le_rfl
lt_iff_not_ge
IsOrderedAddMonoid.toAddLeftMono
instIsOrderedAddMonoid
IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT
AddLeftCancelSemigroup.toIsLeftCancelAdd
contravariant_lt_of_covariant_le
instCanonicallyOrderedAdd
prime_of_mem_primeFactorsList
prod_pow_factorization_eq_self πŸ“–mathematicalPrimefactorization
Finsupp.prod
MulZeroClass.toZero
instMulZeroClass
instCommMonoid
Monoid.toNatPow
instMonoid
β€”Finsupp.prod.eq_1
factorization_prod
pow_ne_zero
isReduced_of_noZeroDivisors
IsStrictOrderedRing.noZeroDivisors
instIsStrictOrderedRing
CanonicallyOrderedAdd.toExistsAddOfLE
instCanonicallyOrderedAdd
Prime.ne_zero
Finset.sum_congr
Prime.factorization_pow
Finsupp.sum_single
support_factorization πŸ“–mathematicalβ€”Finsupp.support
MulZeroClass.toZero
instMulZeroClass
factorization
primeFactors
β€”β€”

Nat.Prime

Theorems

NameKindAssumesProvesValidatesDepends On
factorization πŸ“–mathematicalNat.PrimeNat.factorization
Finsupp.single
MulZeroClass.toZero
Nat.instMulZeroClass
β€”Finsupp.ext
Nat.primeFactorsList_count_eq
Nat.primeFactorsList_prime
Finsupp.single_apply
if_congr
factorization_pos_of_dvd πŸ“–mathematicalNat.PrimeDFunLike.coe
Finsupp
MulZeroClass.toZero
Nat.instMulZeroClass
Finsupp.instFunLike
Nat.factorization
β€”Nat.primeFactorsList_count_eq
Nat.mem_primeFactorsList_iff_dvd
factorization_pow πŸ“–mathematicalNat.PrimeNat.factorization
Monoid.toNatPow
Nat.instMonoid
Finsupp.single
MulZeroClass.toZero
Nat.instMulZeroClass
β€”Nat.factorization_pow
factorization
Finsupp.smul_single
mul_one

---

← Back to Index