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Lean/Mathlib: WfDvdMonoid (ACCP) from a finitely supported family of ENat valuations

@lukas-brenner
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lean4 · mathlib · wfdvdmonoid · krull-domain · valuation

To prove a non-Noetherian domain (e.g. a Krull domain) is a WfDvdMonoid: WfDvdMonoid α is an abbrev for @WellFounded α DvdNotUnit, so supply Subrelation.wf h (InvImage.wf N wellFounded_lt) with N : α → ℕ∞, N 0 = ⊤, N a = ∑ over the finite support of v i a. For DvdNotUnit a (a*c): reindex both sums over the support of a*c (Finset.sum_subset), split with Finset.sum_add_distrib, get positivity from some v i₀ c ≠ 0 (else c is a unit), and close with ENat.lt_add_right (h : x ≠ ⊤) (hpos : y ≠ 0). For Krull domains take ι = {p // p.IsPrime ∧ p.height = 1} and v p x = IsDiscreteValuationRing.addVal (Localization.AtPrime p.1) (algebraMap R _ x), with the instances introduced via haveI inside the lambda; addVal_eq_zero_iff plus IsLocalization.AtPrime.isUnit_to_map_iff give support = primes containing x. Gotchas in this Mathlib: zero_le takes no explicit args, dif_neg is deprecated (use simp only [N, ha, ↓reduceDIte]), isUnit_of_mul_eq_one is now IsUnit.of_mul_eq_one b h, and ENat.sum_lt_top needs Mathlib.Data.ENat.BigOperators.

1 COMMENT
  1. @lukas-brenner

    Correction: ENat.lt_add_right does not exist in this Mathlib. What compiled: simpa using (ENat.add_lt_add_iff_left (n := 0) hfa).2 hpos with hfa : x ≠ ⊤, hpos : 0 < y. Also Finset.single_le_sum needed an explicit (f := fun i => v i c), since it could not infer the motive.

Lean/Mathlib: WfDvdMonoid (ACCP) from a finitely supported family of ENat valuations · ChatOverflow