Defs.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 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, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

abbrev MvRatFunc Documented declaration: MvRatFunc line 25 source
Source documentation
The field of multivariate rational functions in variables indexed by σ over a commutative ring K, defined as the fraction field of the multivariate polynomial ring MvPolynomial σ K.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing
Proof uses: FractionRing, MvPolynomial, Finsupp, Nat
6 instances, coercions and auxiliary constants left out.
```
abbrev MvRatFunc (σ K : Type*) [CommRing K] := FractionRing (MvPolynomial σ K)
```
def IsDefined A rational function $g$ is defined at a point $x_0$ if $g$ can be written as $p / q$ where $p, q$ are polynomials and $q(x_0) \neq 0$ line 31 source
Source documentation
A rational function $g$ is defined at a point $x_0$ if $g$ can be written as $p / q$ where $p, q$ are polynomials and $q(x_0) \neq 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDomain, MvRatFunc
Proof uses: Exists, MvPolynomial, And, Eq, HDiv.hDiv, Finsupp, Nat, nonZeroDivisors, RingHom, Algebra.algebraMap, Ne, MvPolynomial.eval, OfNat.ofNat
35 instances, coercions and auxiliary constants left out.
```
def IsDefined (σ K : Type*) [CommRing K] [IsDomain K] (g : MvRatFunc σ K) (x₀ : σ → K) : Prop :=
…
```
Mentions: MvRatFunc