Documentation Verification Report

Associated

πŸ“ Source: Mathlib/Algebra/GroupWithZero/Associated.lean

Statistics

MetricCount
Definitionssetoid, Associates, instBot, instBoundedOrder, instCommMonoid, instCommMonoidWithZero, instDecidableRelDvd, instInhabited, instMul, instOne, instOrderBot, instOrderTop, instPartialOrder, instPreorder, instTopOfZero, instUniqueOfSubsingleton, instZero, mk, mkMonoidHom, uniqueUnits, instDecidableRelAssociatedOfIsLeftCancelMulZeroOfDvd
21
Theoremscomm, dvd, dvd', dvd_dvd, dvd_iff_dvd_left, dvd_iff_dvd_right, eq_zero_iff, instIsTrans, instRefl, instSymm, irreducible, irreducible_iff, isUnit, isUnit_iff, map, mul_left, mul_mul, mul_right, ne_zero_iff, of_eq, of_mul_left, of_mul_right, of_pow_associated_of_prime, of_pow_associated_of_prime', pow_pow, prime, prime_iff, refl, rfl, trans, le_or_le, associated_map_mk, bot_eq_one, coe_unit_eq_one, decompositionMonoid_iff, dvdNotUnit_iff_lt, dvdNotUnit_of_lt, dvd_eq_le, dvd_of_mk_le_mk, exists_non_zero_rep, exists_rep, forall_associated, instDecompositionMonoid, instIsCancelMulZero, instNoZeroDivisors, instNontrivial, irreducible_iff_prime_iff, irreducible_mk, isPrimal_mk, isUnit_iff_eq_bot, isUnit_iff_eq_one, isUnit_mk, le_mul_left, le_mul_right, le_of_mul_le_mul_left, le_one_iff, le_zero, mkMonoidHom_apply, mk_dvdNotUnit_mk_iff, mk_dvd_mk, mk_eq_mk_iff_associated, mk_eq_one, mk_eq_zero, mk_injective, mk_isRelPrime_iff, mk_le_mk_iff_dvd, mk_le_mk_of_dvd, mk_mul_mk, mk_ne_zero, mk_one, mk_pow, mk_quot_out, mk_surjective, mk_zero, mul_mono, one_eq_mk_one, one_le, one_or_eq_of_le_of_prime, prime_mk, quot_mk_eq_mk, quot_out, quot_out_zero, quotient_mk_eq_mk, not_associated, associated_of_dvd, dvd_iff, dvd_irreducible_iff_associated, dvd_or_isRelPrime, isRelPrime_iff_not_dvd, isUnit_iff_not_associated_of_dvd, associated_of_dvd, dvd_prime_iff_associated, associated_eq_eq, associated_iff_eq, associated_isUnit_mul_left_iff, associated_isUnit_mul_right_iff, associated_mul_isUnit_left_iff, associated_mul_isUnit_right_iff, associated_mul_unit_left, associated_mul_unit_left_iff, associated_mul_unit_right, associated_mul_unit_right_iff, associated_of_dvd_dvd, associated_one_iff_isUnit, associated_one_of_associated_mul_one, associated_one_of_mul_eq_one, associated_unit_mul_left, associated_unit_mul_left_iff, associated_unit_mul_right, associated_unit_mul_right_iff, associated_zero_iff_eq_zero, associates_irreducible_iff_prime, dvdNotUnit_of_dvdNotUnit_associated, dvd_dvd_iff_associated, dvd_prime_pow, eq_of_prime_pow_eq, eq_of_prime_pow_eq', isUnit_of_associated_mul, prime_dvd_prime_iff_eq, prime_mul_iff, prime_pow_iff, unit_associated_one
122
Total143

Associated

Definitions

NameCategoryTheorems
setoid πŸ“–CompOp
24 mathmath: FractionalIdeal.finprod_heightOneSpectrum_factorization_principal, Associates.finite_factors, IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, FractionalIdeal.count_well_defined, IsDedekindDomain.HeightOneSpectrum.intValuationDef_if_neg, NumberField.mixedEmbedding.fundamentalCone.integerSetQuotEquivAssociates_apply, FractionalIdeal.finprod_heightOneSpectrum_factorization, FractionalIdeal.count_ne_zero, Associates.quotient_mk_eq_mk, associatesNonZeroDivisorsEquiv_symm_mk_mk, NumberField.mixedEmbedding.fundamentalCone.integerSetToAssociates_apply, Ideal.finite_mulSupport_coe, IsDedekindDomain.HeightOneSpectrum.intValuation_def, Associates.quot_mk_eq_mk, Associates.mk_quot_out, Ideal.finprod_count, Ideal.finprod_heightOneSpectrum_factorization_coe, FractionalIdeal.finite_factors', Associates.quot_out_zero, FractionalIdeal.finprod_heightOneSpectrum_factorization_principal_fraction, Ideal.finprod_not_dvd, Ideal.finite_mulSupport_inv, Associates.quot_out, FractionalIdeal.count_coe

Theorems

NameKindAssumesProvesValidatesDepends On
comm πŸ“–mathematicalβ€”Associatedβ€”symm
dvd πŸ“–mathematicalAssociatedsemigroupDvd
Monoid.toSemigroup
β€”β€”
dvd' πŸ“–mathematicalAssociatedsemigroupDvd
Monoid.toSemigroup
β€”dvd
symm
dvd_dvd πŸ“–mathematicalAssociatedsemigroupDvd
Monoid.toSemigroup
β€”dvd
symm
dvd_iff_dvd_left πŸ“–mathematicalAssociatedsemigroupDvd
Monoid.toSemigroup
β€”Units.mul_right_dvd
dvd_iff_dvd_right πŸ“–mathematicalAssociatedsemigroupDvd
Monoid.toSemigroup
β€”Units.dvd_mul_right
eq_zero_iff πŸ“–mathematicalAssociated
MonoidWithZero.toMonoid
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
β€”Units.eq_mul_inv_iff_mul_eq
MulZeroClass.zero_mul
instIsTrans πŸ“–mathematicalβ€”IsTrans
Associated
β€”trans
instRefl πŸ“–mathematicalβ€”Associatedβ€”refl
instSymm πŸ“–mathematicalβ€”Associatedβ€”symm
irreducible πŸ“–β€”Associated
Irreducible
β€”β€”isUnit
symm
Irreducible.not_isUnit
Units.mul_inv_cancel_right
mul_assoc
Irreducible.isUnit_or_isUnit
Units.inv_mul_cancel_right
irreducible_iff πŸ“–mathematicalAssociatedIrreducibleβ€”irreducible
symm
isUnit πŸ“–β€”Associated
IsUnit
β€”β€”β€”
isUnit_iff πŸ“–mathematicalAssociatedIsUnitβ€”isUnit
symm
map πŸ“–mathematicalAssociatedDFunLike.coeβ€”map_mul
MonoidHomClass.toMulHomClass
Units.coe_map
MonoidHom.coe_coe
mul_left πŸ“–mathematicalAssociatedMulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”mul_assoc
mul_mul πŸ“–mathematicalAssociated
CommMonoid.toMonoid
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”trans
mul_right
mul_left
mul_right πŸ“–mathematicalAssociated
CommMonoid.toMonoid
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”mul_right_comm
ne_zero_iff πŸ“–β€”Associated
MonoidWithZero.toMonoid
β€”β€”eq_zero_iff
of_eq πŸ“–mathematicalβ€”Associatedβ€”Units.val_one
mul_one
of_mul_left πŸ“–β€”Associated
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
β€”β€”symm
mul_left_cancelβ‚€
IsCancelMulZero.toIsLeftCancelMulZero
mul_assoc
mul_left_comm
mul_comm
of_mul_right πŸ“–β€”Associated
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
β€”β€”mul_comm
of_mul_left
of_pow_associated_of_prime πŸ“–β€”Prime
Associated
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Monoid.toNatPow
β€”β€”dvd_iff_dvd_right
dvd_pow_self
LT.lt.ne'
Prime.dvd_prime_iff_associated
Prime.dvd_of_dvd_pow
of_pow_associated_of_prime' πŸ“–β€”Prime
Associated
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Monoid.toNatPow
β€”β€”symm
of_pow_associated_of_prime
pow_pow πŸ“–mathematicalAssociated
CommMonoid.toMonoid
Monoid.toNatPowβ€”pow_zero
pow_succ'
mul_mul
prime πŸ“–β€”Associated
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Prime
β€”β€”ne_zero_iff
Prime.ne_zero
Prime.not_unit
Units.mul_inv_cancel_right
Prime.dvd_or_dvd
prime_iff πŸ“–mathematicalAssociated
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Primeβ€”prime
symm
refl πŸ“–mathematicalβ€”Associatedβ€”mul_one
rfl πŸ“–mathematicalβ€”Associatedβ€”refl
trans πŸ“–β€”Associatedβ€”β€”Units.val_mul
mul_assoc

Associates

Definitions

NameCategoryTheorems
instBot πŸ“–CompOp
2 mathmath: bot_eq_one, isUnit_iff_eq_bot
instBoundedOrder πŸ“–CompOpβ€”
instCommMonoid πŸ“–CompOp
20 mathmath: prod_le_prod, decompositionMonoid_iff, emultiplicity_mk_eq_emultiplicity, prod_le_prod_iff_le, instIsOrderedMonoid, mkMonoidHom_apply, dvd_eq_le, finset_prod_mk, isPrimal_mk, mk_pow, prod_mk, instDecompositionMonoid, isUnit_iff_eq_bot, coe_unit_eq_one, isUnit_iff_eq_one, mk_isRelPrime_iff, isUnit_mk, mk_dvd_mk, prod_eq_one_iff, prod_coe
instCommMonoidWithZero πŸ“–CompOp
62 mathmath: isAtom_iff, FractionalIdeal.finprod_heightOneSpectrum_factorization_principal, factors_mul, prod_top, finite_factors, IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, FractionalIdeal.count_well_defined, irreducible_mk, IsDedekindDomain.HeightOneSpectrum.intValuationDef_if_neg, mk_dvdNotUnit_mk_iff, pow_factors, FractionalIdeal.finprod_heightOneSpectrum_factorization, dvdNotUnit_of_lt, lcm_mk_mk, DivisorChain.element_of_chain_not_isUnit_of_index_ne_zero, factors_zero, FractionalIdeal.count_ne_zero, factors_subsingleton, factors_mk, le_singleton_iff, associatesNonZeroDivisorsEquiv_symm_mk_mk, prod_add, factors_le, DivisorChain.isPrimePow_of_has_chain, factors_mono, factors_eq_some_iff_ne_zero, map_subtype_coe_factors', mem_factors'_iff_dvd, mk_mem_nonZeroDivisors_associates, WfDvdMonoid.wfDvdMonoid_associates, is_pow_of_dvd_count, dvdNotUnit_iff_lt, Ideal.finite_mulSupport_coe, ufm, prime_mk, factors_eq_top_iff_zero, IsDedekindDomain.HeightOneSpectrum.intValuation_def, irreducible_of_mem_factorSet, DivisorChain.second_of_chain_is_irreducible, irreducible_iff_prime_iff, DivisorChain.eq_pow_second_of_chain_of_has_chain, Ideal.finprod_count, FactorSet.coe_add, Ideal.finprod_heightOneSpectrum_factorization_coe, factor_orderIso_map_one_eq_bot, FractionalIdeal.finite_factors', Ideal.associatesNonZeroDivisorsEquivIsPrincipal_coe, dvd_of_mem_factors, associates_irreducible_iff_prime, prod_le, DivisorChain.first_of_chain_isUnit, FractionalIdeal.finprod_heightOneSpectrum_factorization_principal_fraction, Ideal.finprod_not_dvd, factors_one, IsDedekindDomain.HeightOneSpectrum.associates_irreducible, FactorSet.sup_add_inf_eq_add, Ideal.finite_mulSupport_inv, mem_factors'_of_dvd, prod_coe, gcd_mk_mk, FractionalIdeal.count_coe, FactorSet.prod_eq_zero_iff
instDecidableRelDvd πŸ“–CompOpβ€”
instInhabited πŸ“–CompOpβ€”
instMul πŸ“–CompOp
23 mathmath: factors_mul, associatesEquivOfUniqueUnits_symm_apply, associatesNonZeroDivisorsEquiv_mk_mk, associatesNonZeroDivisorsEquiv_symm_mk_mk, prod_add, mkFactorOrderIsoOfFactorDvdEquiv_apply_coe, instNoZeroDivisors, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_mul, count_mul_of_coprime', mk_mul_mk, sup_mul_inf, associatesEquivOfUniqueUnits_apply, instIsCancelMulZero, le_mul_left, mkFactorOrderIsoOfFactorDvdEquiv_symm_apply_coe, count_mul, Ideal.associatesEquivIsPrincipal_mul, out_mul, instCanonicallyOrderedMul, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_coe, count_mul_of_coprime, le_mul_right, mul_mono
instOne πŸ“–CompOp
17 mathmath: coe_factor_orderIso_map_eq_one_iff, out_one, bot_eq_one, one_or_eq_of_le_of_prime, mk_one, one_eq_mk_one, mk_eq_one, coe_unit_eq_one, one_le, isUnit_iff_eq_one, le_one_iff, factor_orderIso_map_one_eq_bot, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_map_one, coprime_iff_inf_one, prod_eq_one_iff, factors_one, Ideal.associatesEquivIsPrincipal_map_one
instOrderBot πŸ“–CompOp
1 mathmath: isAtom_iff
instOrderTop πŸ“–CompOpβ€”
instPartialOrder πŸ“–CompOp
1 mathmath: instIsOrderedMonoid
instPreorder πŸ“–CompOp
32 mathmath: isAtom_iff, prime_pow_dvd_iff_le, le_of_count_ne_zero, prod_le_prod, prod_le_prod_iff_le, out_dvd_iff, mk_le_mk_iff_dvd, dvd_eq_le, le_singleton_iff, factors_le, mkFactorOrderIsoOfFactorDvdEquiv_apply_coe, dvdNotUnit_iff_lt, prod_mono, emultiplicity_prime_eq_emultiplicity_image_by_factor_orderIso, one_le, emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso, mk_le_mk_of_dvd, le_mul_left, mkFactorOrderIsoOfFactorDvdEquiv_symm_apply_coe, WfDvdMonoid.iff_wellFounded_associates, le_one_iff, prime_pow_le_iff_le_bcount, factor_orderIso_map_one_eq_bot, instCanonicallyOrderedMul, dvd_out_iff, map_prime_of_factor_orderIso, WfDvdMonoid.wellFoundedLT_associates, prod_le, mem_normalizedFactors_factor_orderIso_of_mem_normalizedFactors, DivisorChain.exists_chain_of_prime_pow, le_mul_right, le_zero
instTopOfZero πŸ“–CompOp
1 mathmath: out_top
instUniqueOfSubsingleton πŸ“–CompOpβ€”
instZero πŸ“–CompOp
13 mathmath: prod_top, mk_zero, factors_zero, instNoZeroDivisors, out_eq_zero_iff, factors_eq_top_iff_zero, out_zero, instIsCancelMulZero, mk_eq_zero, Ideal.associatesEquivIsPrincipal_map_zero, quot_out_zero, FactorSet.prod_eq_zero_iff, le_zero
mk πŸ“–CompOpβ€”
mkMonoidHom πŸ“–CompOp
1 mathmath: mkMonoidHom_apply
uniqueUnits πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
associated_map_mk πŸ“–mathematicalAssociates
CommMonoid.toMonoid
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
instCommMonoid
MonoidHom.instFunLike
Associatedβ€”mk_eq_mk_iff_associated
bot_eq_one πŸ“–mathematicalβ€”Bot.bot
Associates
instBot
instOne
β€”β€”
coe_unit_eq_one πŸ“–mathematicalβ€”Units.val
Associates
CommMonoid.toMonoid
instCommMonoid
instOne
β€”Unique.instSubsingleton
decompositionMonoid_iff πŸ“–mathematicalβ€”DecompositionMonoid
Associates
CommMonoid.toMonoid
Monoid.toSemigroup
instCommMonoid
β€”β€”
dvdNotUnit_iff_lt πŸ“–mathematicalβ€”DvdNotUnit
Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
instCommMonoidWithZero
Preorder.toLT
instPreorder
CommMonoidWithZero.toCommMonoid
β€”dvd_and_not_dvd_iff
instIsCancelMulZero
dvdNotUnit_of_lt πŸ“–mathematicalAssociates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Preorder.toLT
instPreorder
CommMonoidWithZero.toCommMonoid
DvdNotUnit
instCommMonoidWithZero
β€”not_lt_of_ge
dvd_zero
Mathlib.Tactic.Contrapose.contraposeβ‚„
coe_unit_eq_one
mul_one
dvd_eq_le πŸ“–mathematicalβ€”Associates
CommMonoid.toMonoid
semigroupDvd
Monoid.toSemigroup
instCommMonoid
Preorder.toLE
instPreorder
β€”β€”
dvd_of_mk_le_mk πŸ“–mathematicalAssociates
CommMonoid.toMonoid
Preorder.toLE
instPreorder
semigroupDvd
Monoid.toSemigroup
β€”β€”
exists_non_zero_rep πŸ“–mathematicalβ€”MonoidWithZero.toMonoidβ€”β€”
exists_rep πŸ“–β€”β€”β€”β€”β€”
forall_associated πŸ“–β€”β€”β€”β€”β€”
instDecompositionMonoid πŸ“–mathematicalβ€”DecompositionMonoid
Associates
CommMonoid.toMonoid
Monoid.toSemigroup
instCommMonoid
β€”decompositionMonoid_iff
instIsCancelMulZero πŸ“–mathematicalβ€”IsCancelMulZero
Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
instMul
CommMonoidWithZero.toCommMonoid
instZero
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
β€”IsLeftCancelMulZero.to_isCancelMulZero
Quotient.exact'
mul_assoc
Quotient.sound'
mul_left_cancelβ‚€
IsCancelMulZero.toIsLeftCancelMulZero
mk_ne_zero
instNoZeroDivisors πŸ“–mathematicalβ€”NoZeroDivisors
Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
instMul
CommMonoidWithZero.toCommMonoid
instZero
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
β€”IsRightCancelMulZero.to_noZeroDivisors
IsCancelMulZero.toIsRightCancelMulZero
instIsCancelMulZero
instNontrivial πŸ“–mathematicalβ€”Nontrivial
Associates
MonoidWithZero.toMonoid
β€”mk_ne_zero
one_ne_zero
NeZero.one
irreducible_iff_prime_iff πŸ“–mathematicalβ€”Irreducible
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Prime
Associates
instCommMonoidWithZero
β€”β€”
irreducible_mk πŸ“–mathematicalβ€”Irreducible
Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
instCommMonoidWithZero
β€”Associated.comm
Iff.and
Associated.refl
instIsDedekindFiniteMonoid
mul_assoc
isPrimal_mk πŸ“–mathematicalβ€”IsPrimal
Associates
CommMonoid.toMonoid
Monoid.toSemigroup
instCommMonoid
β€”Function.Surjective.exists
mk_surjective
mk_eq_mk_iff_associated
Units.mul_right_dvd
mul_assoc
isUnit_iff_eq_bot πŸ“–mathematicalβ€”IsUnit
Associates
CommMonoid.toMonoid
instCommMonoid
Bot.bot
instBot
β€”isUnit_iff_eq_one
bot_eq_one
isUnit_iff_eq_one πŸ“–mathematicalβ€”IsUnit
Associates
CommMonoid.toMonoid
instCommMonoid
instOne
β€”coe_unit_eq_one
isUnit_one
isUnit_mk πŸ“–mathematicalβ€”IsUnit
Associates
CommMonoid.toMonoid
instCommMonoid
β€”isUnit_iff_eq_one
one_eq_mk_one
mk_eq_mk_iff_associated
associated_one_iff_isUnit
le_mul_left πŸ“–mathematicalβ€”Associates
CommMonoid.toMonoid
Preorder.toLE
instPreorder
instMul
β€”mul_comm
le_mul_right
le_mul_right πŸ“–mathematicalβ€”Associates
CommMonoid.toMonoid
Preorder.toLE
instPreorder
instMul
β€”β€”
le_of_mul_le_mul_left πŸ“–β€”Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Preorder.toLE
instPreorder
CommMonoidWithZero.toCommMonoid
instMul
β€”β€”mul_left_cancelβ‚€
IsCancelMulZero.toIsLeftCancelMulZero
instIsCancelMulZero
mul_assoc
le_one_iff πŸ“–mathematicalβ€”Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Preorder.toLE
instPreorder
CommMonoidWithZero.toCommMonoid
instOne
β€”bot_eq_one
le_bot_iff
le_zero πŸ“–mathematicalβ€”Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Preorder.toLE
instPreorder
CommMonoidWithZero.toCommMonoid
instZero
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
β€”le_top
mkMonoidHom_apply πŸ“–mathematicalβ€”DFunLike.coe
MonoidHom
Associates
CommMonoid.toMonoid
MulOneClass.toMulOne
Monoid.toMulOneClass
instCommMonoid
MonoidHom.instFunLike
mkMonoidHom
β€”β€”
mk_dvdNotUnit_mk_iff πŸ“–mathematicalβ€”DvdNotUnit
Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
instCommMonoidWithZero
β€”Function.Surjective.exists
mk_surjective
Associated.comm
Iff.and
instIsDedekindFiniteMonoid
mul_assoc
Associated.refl
mk_dvd_mk πŸ“–mathematicalβ€”Associates
CommMonoid.toMonoid
semigroupDvd
Monoid.toSemigroup
instCommMonoid
β€”Function.Surjective.exists
mk_surjective
Associated.comm
mul_assoc
Associated.refl
mk_eq_mk_iff_associated πŸ“–mathematicalβ€”Associatedβ€”β€”
mk_eq_one πŸ“–mathematicalβ€”Associates
instOne
IsUnit
β€”mk_one
mk_eq_mk_iff_associated
associated_one_iff_isUnit
mk_eq_zero πŸ“–mathematicalβ€”MonoidWithZero.toMonoid
Associates
instZero
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
β€”associated_zero_iff_eq_zero
mk_injective πŸ“–mathematicalβ€”Associatesβ€”associated_iff_eq
mk_eq_mk_iff_associated
mk_isRelPrime_iff πŸ“–mathematicalβ€”IsRelPrime
Associates
CommMonoid.toMonoid
instCommMonoid
β€”β€”
mk_le_mk_iff_dvd πŸ“–mathematicalβ€”Associates
CommMonoid.toMonoid
Preorder.toLE
instPreorder
semigroupDvd
Monoid.toSemigroup
β€”β€”
mk_le_mk_of_dvd πŸ“–mathematicalsemigroupDvd
Monoid.toSemigroup
CommMonoid.toMonoid
Associates
Preorder.toLE
instPreorder
β€”β€”
mk_mul_mk πŸ“–mathematicalβ€”Associates
CommMonoid.toMonoid
instMul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”β€”
mk_ne_zero πŸ“–β€”β€”β€”β€”mk_eq_zero
mk_one πŸ“–mathematicalβ€”MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Associates
instOne
β€”β€”
mk_pow πŸ“–mathematicalβ€”CommMonoid.toMonoid
Monoid.toNatPow
Associates
instCommMonoid
β€”pow_zero
pow_succ
mk_quot_out πŸ“–mathematicalβ€”Associated
Quot.out
Associated.setoid
β€”mk_eq_mk_iff_associated
quot_out
mk_surjective πŸ“–mathematicalβ€”Associatesβ€”forall_associated
mk_zero πŸ“–mathematicalβ€”Associates
instZero
β€”β€”
mul_mono πŸ“–mathematicalAssociates
CommMonoid.toMonoid
Preorder.toLE
instPreorder
instMulβ€”mul_left_comm
mul_comm
one_eq_mk_one πŸ“–mathematicalβ€”Associates
instOne
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”β€”
one_le πŸ“–mathematicalβ€”Associates
CommMonoid.toMonoid
Preorder.toLE
instPreorder
instOne
β€”Dvd.intro
one_mul
one_or_eq_of_le_of_prime πŸ“–mathematicalPrime
Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
instCommMonoidWithZero
Preorder.toLE
instPreorder
CommMonoidWithZero.toCommMonoid
instOneβ€”mk_surjective
Irreducible.dvd_iff
Prime.irreducible
mk_le_mk_iff_dvd
prime_mk πŸ“–mathematicalβ€”Prime
Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
instCommMonoidWithZero
β€”Prime.eq_1
forall_associated
quot_mk_eq_mk πŸ“–mathematicalβ€”Associated.setoidβ€”β€”
quot_out πŸ“–mathematicalβ€”Quot.out
Associated.setoid
β€”Quot.out_eq
quot_out_zero πŸ“–mathematicalβ€”Quot.out
Associated.setoid
MonoidWithZero.toMonoid
Associates
instZero
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
β€”mk_eq_zero
quot_out
quotient_mk_eq_mk πŸ“–mathematicalβ€”Associated.setoidβ€”β€”

Associates.Prime

Theorems

NameKindAssumesProvesValidatesDepends On
le_or_le πŸ“–β€”Prime
Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Associates.instCommMonoidWithZero
Preorder.toLE
Associates.instPreorder
CommMonoidWithZero.toCommMonoid
Associates.instMul
β€”β€”β€”

DvdNotUnit

Theorems

NameKindAssumesProvesValidatesDepends On
not_associated πŸ“–mathematicalDvdNotUnitAssociated
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
β€”Units.isUnit
mul_right_inj'
IsCancelMulZero.toIsLeftCancelMulZero

Irreducible

Theorems

NameKindAssumesProvesValidatesDepends On
associated_of_dvd πŸ“–mathematicalIrreducible
semigroupDvd
Monoid.toSemigroup
Associatedβ€”Associated.symm
dvd_iff
not_isUnit
dvd_iff πŸ“–mathematicalIrreduciblesemigroupDvd
Monoid.toSemigroup
IsUnit
Associated
β€”isUnit_or_isUnit
associated_mul_unit_left
IsUnit.dvd
Associated.dvd
Associated.symm
dvd_irreducible_iff_associated πŸ“–mathematicalIrreduciblesemigroupDvd
Monoid.toSemigroup
Associated
β€”associated_of_dvd
Associated.dvd
dvd_or_isRelPrime πŸ“–mathematicalIrreduciblesemigroupDvd
Monoid.toSemigroup
IsRelPrime
β€”isRelPrime_iff_not_dvd
isRelPrime_iff_not_dvd πŸ“–mathematicalIrreducibleIsRelPrime
semigroupDvd
Monoid.toSemigroup
β€”not_isUnit
dvd_rfl
Mathlib.Tactic.Contrapose.contraposeβ‚‚
dvd_iff
Dvd.dvd.trans
Associated.dvd
isUnit_iff_not_associated_of_dvd πŸ“–mathematicalIrreducible
semigroupDvd
Monoid.toSemigroup
IsUnit
Associated
β€”not_isUnit
Associated.isUnit
Associated.symm
dvd_iff

Prime

Theorems

NameKindAssumesProvesValidatesDepends On
associated_of_dvd πŸ“–mathematicalPrime
semigroupDvd
SemigroupWithZero.toSemigroup
MonoidWithZero.toSemigroupWithZero
CommMonoidWithZero.toMonoidWithZero
Associated
MonoidWithZero.toMonoid
β€”Irreducible.associated_of_dvd
irreducible
dvd_prime_iff_associated πŸ“–mathematicalPrimesemigroupDvd
SemigroupWithZero.toSemigroup
MonoidWithZero.toSemigroupWithZero
CommMonoidWithZero.toMonoidWithZero
Associated
MonoidWithZero.toMonoid
β€”Irreducible.dvd_irreducible_iff_associated
irreducible

(root)

Definitions

NameCategoryTheorems
Associates πŸ“–CompOp
121 mathmath: Associates.isAtom_iff, FractionalIdeal.finprod_heightOneSpectrum_factorization_principal, Associates.factors_mul, Associates.prod_top, Associates.finite_factors, associatesEquivOfUniqueUnits_symm_apply, IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, FractionalIdeal.count_well_defined, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_apply, Associates.irreducible_mk, Associates.instNontrivial, Associates.out_one, IsDedekindDomain.HeightOneSpectrum.intValuationDef_if_neg, Associates.mk_dvdNotUnit_mk_iff, Associates.decompositionMonoid_iff, Associates.mk_zero, NumberField.mixedEmbedding.fundamentalCone.integerSetQuotEquivAssociates_apply, Associates.pow_factors, emultiplicity_mk_eq_emultiplicity, Associates.instIsOrderedMonoid, Associates.out_dvd_iff, Associates.mk_le_mk_iff_dvd, Associates.factorSetMem_eq_mem, Associates.mkMonoidHom_apply, FractionalIdeal.finprod_heightOneSpectrum_factorization, Associates.bot_eq_one, Associates.lcm_mk_mk, Associates.factors_zero, FractionalIdeal.count_ne_zero, Associates.dvd_eq_le, Associates.finset_prod_mk, Associates.factors_subsingleton, Associates.factors_mk, Associates.le_singleton_iff, Associates.mk_one, associatesNonZeroDivisorsEquiv_symm_mk_mk, Associates.one_eq_mk_one, Associates.prod_add, Associates.factors_le, mkFactorOrderIsoOfFactorDvdEquiv_apply_coe, Associates.isPrimal_mk, Associates.mk_pow, Associates.prod_mk, Associates.mk_eq_one, Associates.instDecompositionMonoid, Associates.factors_eq_some_iff_ne_zero, Associates.map_subtype_coe_factors', Associates.mem_factors'_iff_dvd, Associates.isUnit_iff_eq_bot, Associates.mk_surjective, mk_mem_nonZeroDivisors_associates, Associates.instNoZeroDivisors, WfDvdMonoid.wfDvdMonoid_associates, Associates.rel_associated_iff_map_eq_map, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_mul, Associates.is_pow_of_dvd_count, Associates.dvdNotUnit_iff_lt, Ideal.finite_mulSupport_coe, Associates.ufm, Associates.coe_unit_eq_one, Associates.one_le, Ideal.associatesEquivIsPrincipal_symm_apply, Associates.out_eq_zero_iff, Associates.mem_factors_iff_dvd, Associates.prime_mk, Associates.mk_mul_mk, Associates.factors_eq_top_iff_zero, Associates.mk_le_mk_of_dvd, Associates.out_zero, IsDedekindDomain.HeightOneSpectrum.intValuation_def, Associates.sup_mul_inf, Ideal.associatesEquivIsPrincipal_apply, associatesEquivOfUniqueUnits_apply, Associates.mk_injective, Associates.instIsCancelMulZero, Associates.le_mul_left, mkFactorOrderIsoOfFactorDvdEquiv_symm_apply_coe, Associates.irreducible_iff_prime_iff, Ideal.associatesEquivIsPrincipal_mul, Ideal.finprod_count, Associates.isUnit_iff_eq_one, WfDvdMonoid.iff_wellFounded_associates, Associates.le_one_iff, Associates.out_mul, Associates.mk_eq_zero, Associates.FactorSet.coe_add, Associates.mk_isRelPrime_iff, Associates.isUnit_mk, Ideal.finprod_heightOneSpectrum_factorization_coe, factor_orderIso_map_one_eq_bot, Ideal.associatesEquivIsPrincipal_map_zero, Associates.mk_dvd_mk, FractionalIdeal.finite_factors', Associates.mem_factors_of_dvd, Associates.instCanonicallyOrderedMul, Associates.dvd_out_iff, Associates.out_injective, NumberField.mixedEmbedding.fundamentalCone.integerSetToAssociates_surjective, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_map_one, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_coe, Associates.quot_out_zero, WfDvdMonoid.wellFoundedLT_associates, Associates.coprime_iff_inf_one, associates_irreducible_iff_prime, Associates.prod_eq_one_iff, Associates.prod_le, FractionalIdeal.finprod_heightOneSpectrum_factorization_principal_fraction, Ideal.finprod_not_dvd, Associates.factors_one, IsDedekindDomain.HeightOneSpectrum.associates_irreducible, Associates.FactorSet.sup_add_inf_eq_add, Ideal.finite_mulSupport_inv, Associates.le_mul_right, Associates.mem_factors'_of_dvd, Ideal.associatesEquivIsPrincipal_map_one, Associates.prod_coe, Associates.out_top, Associates.gcd_mk_mk, FractionalIdeal.count_coe, Associates.FactorSet.prod_eq_zero_iff, Associates.le_zero
instDecidableRelAssociatedOfIsLeftCancelMulZeroOfDvd πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
associated_eq_eq πŸ“–mathematicalβ€”Associatedβ€”associated_iff_eq
associated_iff_eq πŸ“–mathematicalβ€”Associatedβ€”Units.eq_one
mul_one
associated_isUnit_mul_left_iff πŸ“–mathematicalIsUnit
CommMonoid.toMonoid
Associated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”mul_comm
associated_mul_isUnit_left_iff
associated_isUnit_mul_right_iff πŸ“–mathematicalIsUnit
CommMonoid.toMonoid
Associated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”Associated.comm
associated_isUnit_mul_left_iff
associated_mul_isUnit_left_iff πŸ“–mathematicalIsUnitAssociated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”Associated.trans
associated_mul_unit_right
associated_mul_unit_left
associated_mul_isUnit_right_iff πŸ“–mathematicalIsUnitAssociated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”Associated.comm
associated_mul_isUnit_left_iff
associated_mul_unit_left πŸ“–mathematicalIsUnitAssociated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”Units.mul_inv_cancel_right
associated_mul_unit_left_iff πŸ“–mathematicalβ€”Associated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Units.val
β€”associated_mul_isUnit_left_iff
Units.isUnit
associated_mul_unit_right πŸ“–mathematicalIsUnitAssociated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”Associated.symm
associated_mul_unit_left
associated_mul_unit_right_iff πŸ“–mathematicalβ€”Associated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Units.val
β€”associated_mul_isUnit_right_iff
Units.isUnit
associated_of_dvd_dvd πŸ“–mathematicalsemigroupDvd
SemigroupWithZero.toSemigroup
MonoidWithZero.toSemigroupWithZero
Associated
MonoidWithZero.toMonoid
β€”MulZeroClass.zero_mul
mul_assoc
mul_one
mul_left_cancelβ‚€
associated_one_iff_isUnit πŸ“–mathematicalβ€”Associated
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
IsUnit
β€”Associated.symm
one_mul
associated_one_of_associated_mul_one πŸ“–β€”Associated
CommMonoid.toMonoid
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MulOne.toOne
β€”β€”associated_one_of_mul_eq_one
mul_assoc
associated_one_of_mul_eq_one πŸ“–mathematicalMulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
MulOne.toOne
Associatedβ€”instIsDedekindFiniteMonoid
unit_associated_one
associated_unit_mul_left πŸ“–mathematicalIsUnit
CommMonoid.toMonoid
Associated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”mul_comm
associated_mul_unit_left
associated_unit_mul_left_iff πŸ“–mathematicalβ€”Associated
CommMonoid.toMonoid
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Units.val
β€”associated_isUnit_mul_left_iff
Units.isUnit
associated_unit_mul_right πŸ“–mathematicalIsUnit
CommMonoid.toMonoid
Associated
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”Associated.symm
associated_unit_mul_left
associated_unit_mul_right_iff πŸ“–mathematicalβ€”Associated
CommMonoid.toMonoid
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Units.val
β€”associated_isUnit_mul_right_iff
Units.isUnit
associated_zero_iff_eq_zero πŸ“–mathematicalβ€”Associated
MonoidWithZero.toMonoid
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
β€”Associated.symm
MulZeroClass.zero_mul
Associated.refl
associates_irreducible_iff_prime πŸ“–mathematicalβ€”Irreducible
Associates
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
Associates.instCommMonoidWithZero
Prime
β€”irreducible_iff_prime
Associates.instIsCancelMulZero
Associates.instDecompositionMonoid
dvdNotUnit_of_dvdNotUnit_associated πŸ“–β€”DvdNotUnit
Associated
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
β€”β€”Associated.symm
DvdNotUnit.not_unit
Units.ne_zero
mul_comm
mul_assoc
Units.mul_inv
mul_one
dvd_dvd_iff_associated πŸ“–mathematicalβ€”semigroupDvd
SemigroupWithZero.toSemigroup
MonoidWithZero.toSemigroupWithZero
Associated
MonoidWithZero.toMonoid
β€”associated_of_dvd_dvd
Associated.dvd_dvd
dvd_prime_pow πŸ“–mathematicalPrimesemigroupDvd
SemigroupWithZero.toSemigroup
MonoidWithZero.toSemigroupWithZero
CommMonoidWithZero.toMonoidWithZero
Monoid.toNatPow
MonoidWithZero.toMonoid
Associated
β€”pow_zero
Prime.left_dvd_or_dvd_right_of_dvd_mul
pow_succ'
mul_dvd_mul_iff_left
IsCancelMulZero.toIsLeftCancelMulZero
Prime.ne_zero
Associated.trans
Associated.mul_left
Associated.refl
LE.le.trans
Dvd.dvd.trans
Associated.dvd
pow_dvd_pow
eq_of_prime_pow_eq πŸ“–β€”Prime
Monoid.toNatPow
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
β€”β€”associated_iff_eq
Associated.of_pow_associated_of_prime
eq_of_prime_pow_eq' πŸ“–β€”Prime
Monoid.toNatPow
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
β€”β€”associated_iff_eq
Associated.of_pow_associated_of_prime'
isUnit_of_associated_mul πŸ“–mathematicalAssociated
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
IsUnitβ€”IsUnit.of_mul_eq_one
instIsDedekindFiniteMonoid
mul_right_inj'
IsCancelMulZero.toIsLeftCancelMulZero
mul_assoc
mul_one
prime_dvd_prime_iff_eq πŸ“–mathematicalPrimesemigroupDvd
SemigroupWithZero.toSemigroup
MonoidWithZero.toSemigroupWithZero
CommMonoidWithZero.toMonoidWithZero
β€”Prime.dvd_prime_iff_associated
associated_eq_eq
prime_mul_iff πŸ“–mathematicalβ€”Prime
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
CommMonoidWithZero.toMonoidWithZero
IsUnit
MonoidWithZero.toMonoid
β€”of_irreducible_mul
Prime.irreducible
Associated.prime
associated_unit_mul_left
associated_mul_unit_left
Associated.symm
associated_unit_mul_right
prime_pow_iff πŸ“–mathematicalβ€”Prime
Monoid.toNatPow
MonoidWithZero.toMonoid
CommMonoidWithZero.toMonoidWithZero
β€”pow_one
unit_associated_one πŸ“–mathematicalβ€”Associated
Units.val
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
β€”Units.mul_inv

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