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Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Unique factorization and normalization #

Main definitions #

Noncomputably determines the multiset of prime factors.

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    An arbitrary choice of factors of x : M is exactly the (unique) normalized set of factors, if M has a trivial group of units.

    @[deprecated UniqueFactorizationMonoid.prod_normalizedFactors (since := "2024-12-04")]

    Alias of UniqueFactorizationMonoid.prod_normalizedFactors.

    @[deprecated UniqueFactorizationMonoid.zero_notMem_normalizedFactors (since := "2025-05-23")]

    Alias of UniqueFactorizationMonoid.zero_notMem_normalizedFactors.

    Relatively prime elements have disjoint prime factors (as multisets).

    The multiset of normalized factors of x is nil if and only if x is a unit. The converse is true without the nonzero assumption, see normalizedFactors_of_isUnit.

    If x is a unit, then the multiset of normalized factors of x is nil. The converse is true with a nonzero assumption, see normalizedFactors_eq_zero_iff.

    If the monoid equiv f : α ≃* β commutes with normalize then, for a : α, it yields a bijection between the normalizedFactors of a and of f a.

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      theorem UniqueFactorizationMonoid.normalizedFactorsEquiv_symm_apply {α : Type u_1} [CancelCommMonoidWithZero α] [NormalizationMonoid α] [UniqueFactorizationMonoid α] {β : Type u_2} [CancelCommMonoidWithZero β] [NormalizationMonoid β] [UniqueFactorizationMonoid β] {F : Type u_3} [EquivLike F α β] [MulEquivClass F α β] {f : F} [DecidableEq α] (he : ∀ (x : α), normalize (f x) = f (normalize x)) {a : α} {q : β} (hq : q normalizedFactors (f a)) :
      ((normalizedFactorsEquiv he a).symm q, hq) = (↑f).symm q

      Noncomputably defines a normalizationMonoid structure on a UniqueFactorizationMonoid.

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