AdicValuation.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 8 curated AI-written annotations and 1204 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Pinned FLT blueprint introduction

4 declarations, 0 contain sorry, 0 declared as axioms

Adic Valuation

Material destined for Mathlib.

Imports (1)

Mathlib.RingTheory.DedekindDomain.AdicValuation

Imported by 9 files

instance IsDedekindDomain.HeightOneSpectrum.(instance at line 23) line 23 source
lemma IsDedekindDomain.HeightOneSpectrum.intValuation_eq_coe_neg_multiplicity line 32 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, IsDedekindDomain.HeightOneSpectrum, Ne, OfNat.ofNat, Eq, WithZero, Multiplicative, Int, Valuation, IsDedekindDomain.HeightOneSpectrum.intValuation, WithZero.exp, Neg.neg, multiplicity, Ideal, IsDedekindDomain.HeightOneSpectrum.asIdeal, Ideal.span, Singleton.singleton, Set
Proof uses: Bot.bot, Eq.mpr, Not, id, congrArg, ne_eq, propext, Ideal.span_singleton_eq_bot, Associates.count, Associates.mk, Associates.factors, IsDedekindDomain.HeightOneSpectrum.intValuation_if_neg, Multiset.count, UniqueFactorizationMonoid.normalizedFactors, Nat, Ideal.count_associates_factors_eq, IsDedekindDomain.HeightOneSpectrum.ne_bot, normalize, Units, Eq.symm and 10 more
83 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
instance IsDedekindDomain.HeightOneSpectrum.liesOver_under A height one prime of B lies over its pullback to A line 46 source
Source documentation
A height one prime of B lies over its pullback to A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, IsDomain, Algebra, Algebra.IsIntegral, IsDedekindDomain.HeightOneSpectrum, Ideal.LiesOver, IsDedekindDomain.HeightOneSpectrum.asIdeal, IsDedekindDomain.HeightOneSpectrum.under
Proof uses: Ideal.LiesOver.mk, rfl, Ideal
8 instances, coercions and auxiliary constants left out.
```
instance liesOver_under {A : Type*} [CommRing A] [IsDedekindDomain A] {B : Type*} [CommRing B]
    [IsDomain B] [Algebra A B] [Algebra.IsIntegral A B] (w : HeightOneSpectrum B) :
    w.asIdeal.LiesOver (under A w).asIdeal :=
…
```
def IsDedekindDomain.HeightOneSpectrum.adicCompletion.algEquiv adicCompletion.equiv as a K-algebra isomorphism onto the underlying completion line 52 source
Source documentation
adicCompletion.equiv as a K-algebra isomorphism onto the underlying completion.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, Field, Algebra, IsFractionRing, IsDedekindDomain.HeightOneSpectrum, AlgEquiv, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Valuation.Completion, WithZero, Multiplicative, Int, IsDedekindDomain.HeightOneSpectrum.valuation, WithVal
Proof uses: AlgEquiv.ofRingEquiv, IsDedekindDomain.HeightOneSpectrum.adicCompletion.equiv
39 instances, coercions and auxiliary constants left out.
```
noncomputable def adicCompletion.algEquiv
    {A : Type*} [CommRing A] [IsDedekindDomain A] (K : Type*) [Field K] [Algebra A K]
    [IsFractionRing A K] (v : HeightOneSpectrum A) :
    v.adicCompletion K ≃ₐ[K] (v.valuation K).Completion :=
…
```