Lemmas.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

Lemmas

Material destined for Mathlib.

Imports (5)

Imported by 2 files

def RingEquiv.heightOneSpectrumComap Pulling back elements of HeightOneSpectrum along a ring isomorphism line 26 source
Source documentation
Pulling back elements of HeightOneSpectrum along a ring isomorphism.

standard axioms: Quot.sound, propext what this means
Statement uses: CommRing, RingEquiv, IsDedekindDomain.HeightOneSpectrum
Proof uses: IsDedekindDomain.HeightOneSpectrum.mk, Ideal.comap, IsDedekindDomain.HeightOneSpectrum.asIdeal
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
def RingEquiv.heightOneSpectrumComap {A B : Type*} [CommRing A] [CommRing B] (e : A ≃+* B)
    (P : HeightOneSpectrum B) : HeightOneSpectrum A :=
…
```
def RingEquiv.heightOneSpectrum The bijection HeightOneSpectrum A ≃ HeightOneSpectrum B induced by a ring isomorphism between A and B line 38 source
Source documentation
The bijection HeightOneSpectrum A ≃ HeightOneSpectrum B induced by a ring isomorphism between A and B.

standard axioms: Quot.sound, propext what this means
Statement uses: CommRing, RingEquiv, Equiv, IsDedekindDomain.HeightOneSpectrum
Proof uses: Equiv.mk, RingEquiv.heightOneSpectrumComap, RingEquiv.symm
12 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
def RingEquiv.heightOneSpectrum {A B : Type*} [CommRing A] [CommRing B] (e : A ≃+* B) :
    HeightOneSpectrum A ≃ HeightOneSpectrum B where
…
```
def Nat.Prime.toHeightOneSpectrumInt The element of HeightOneSpectrum ℤ associated to a prime number line 52 source
Source documentation
The element of HeightOneSpectrum ℤ associated to a prime number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, IsDedekindDomain.HeightOneSpectrum, Int
Proof uses: IsDedekindDomain.HeightOneSpectrum.mk, Ideal.span, Singleton.singleton, Set
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
def Nat.Prime.toHeightOneSpectrumInt {p : ℕ} (hp : p.Prime) : HeightOneSpectrum ℤ where
…
```
def Nat.Prime.toHeightOneSpectrumRingOfIntegersRat The element of HeightOneSpectrum (𝓞 ℚ) associated to a prime number line 59 source
Source documentation
The element of HeightOneSpectrum (𝓞 ℚ) associated to a prime number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, Rat
Proof uses: Equiv, Int, RingEquiv.heightOneSpectrum, RingEquiv.symm, Rat.ringOfIntegersEquiv, Nat.Prime.toHeightOneSpectrumInt
15 instances, coercions and auxiliary constants left out.
```
noncomputable def Nat.Prime.toHeightOneSpectrumRingOfIntegersRat {p : ℕ} (hp : p.Prime) :
    IsDedekindDomain.HeightOneSpectrum (𝓞 ℚ) :=
…
```