486451.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 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, Formal Conjectures paper, arXiv v1

2 declarations, 2 contain sorry, 0 declared as axioms

Mathoverflow 486451

Reference: mathoverflow/486451 asked by user Junyan Xu

Imports (1)

FormalConjecturesUtil

theorem Mathoverflow486451.exists_semiring_unique_left_maximal_not_unique_right_maximal There exists a semiring with a unique left maximal ideal but more than one right maximal ideals sorry line 32 source
Source documentation
There exists a semiring with a unique left maximal ideal but more than one right maximal ideals.

depends on sorry what this means
Statement uses: Exists, Semiring, And, ExistsUnique, Ideal, Ideal.IsMaximal, MulOpposite, Ne
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_semiring_unique_left_maximal_not_unique_right_maximal :
    ∃ (R : Type) (_ : Semiring R), (∃! I : Ideal R, I.IsMaximal) ∧
      ∃ I J : Ideal Rᵐᵒᵖ, I.IsMaximal ∧ J.IsMaximal ∧ I ≠ J := by
…
```
theorem Mathoverflow486451.exists_semiring_unique_left_right_maximal_ne There exists a semiring with a unique left maximal ideal and a unique right maximal ideal which are not the same as sets sorry line 45 source
Source documentation
There exists a semiring with a unique left maximal ideal and a unique right maximal ideal which are not the same as sets.
This has been shown by Goran Žužić and Moritz Firsching using an experimental pipeline: An example is the monoid algebra of the monoid of maps from $\mathbb{N}$ to $\mathbb{N}$ over $\mathbb{N}$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Semiring, ExistsUnique, Ideal, Ideal.IsMaximal, MulOpposite, Ne, Set, Exists.choose, And, Eq, Set.preimage, MulOpposite.op
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_semiring_unique_left_right_maximal_ne :
    answer(True) ↔ ∃ (R : Type) (_ : Semiring R) (hI : ∃! I : Ideal R, I.IsMaximal)
      (hJ : ∃! J : Ideal Rᵐᵒᵖ, J.IsMaximal),
        (hI.choose : Set R) ≠ MulOpposite.op ⁻¹' hJ.choose := by
…
```