refactor(Data/Nat/Factorization/Defs): redefine Nat.factorization in terms of primeFactorsList instead of padicValNat

[tb65536]

This PR redefines Nat.factorization in terms of primeFactorsList instead of padicValNat. This will allow Nat.factorization to remain computable even as padicValNat is deprecated in favor of the non-computable multiplicity.

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PR summary c52578998d

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[jcommelin]

Do you want to now rename this, and deprecate this _def version?

Suggested change

	theorem factorization_def (n : ℕ) {p : ℕ} (pp : p.Prime) : n.factorization p = padicValNat p n := by
	theorem factorization_eq_padicValNat (n : ℕ) {p : ℕ} (pp : p.Prime) : n.factorization p = padicValNat p n := by

[tb65536]

Well, we want both statements (connecting Nat.factorization with both Nat.primeFactorsList and padicValNat (which will eventually be replaced by multiplicity)), and I thought that renaming might be a bit confusing deprecation-wise for now.

[b-mehta]

computable even as padicValNat is deprecated in favor of the non-computable multiplicity.

Hold on, can't we define Nat.factorization in terms of Nat.maxPowDvdDiv, which is the efficient one used by padicValNat anyway? Then we get computability, efficiency, and we can safely kill multiplicity.

[b-mehta]

If you don't like my previous idea, then maybe Multiset.toFinsupp should be used, since you're building the finsupp out of a list (via a multiset) anyway

[SnirBroshi]

Multiset.toFinsupp is noncomputable

[tb65536]

Hold on, can't we define Nat.factorization in terms of Nat.maxPowDvdDiv, which is the efficient one used by padicValNat anyway? Then we get computability, efficiency, and we can safely kill multiplicity.

It's not any more efficient unfortunately, since you still need to compute n.primeFactors.

[b-mehta]

Multiset.toFinsupp is noncomputable

that's so annoying - it's just because it's bundled. the .toFun field is computable :/

[SnirBroshi]

Yeah, maybe it should be split to toFinsupp and toFinsuppAddEquiv