Lean verification record

Checked Artifact: Downward Löwenheim–Skolem Theorem (mathlib)

Proof Atlas collected build, no-sorry, axiom, and clean-source evidence directly from the pinned upstream declaration.

Verification result

The recorded Lean checks passed

Build replaypassed with retained transcript
Unfinished stepsNone found
Source identityPinned commit and file recorded
Dependency profileAxiom closure recorded

Reproducibility details

What the checker recorded

Declaration checked: FirstOrder.Language.exists_elementarySubstructure_card_eq
Module: Mathlib.ModelTheory.Skolem
Source file checked: Mathlib/ModelTheory/Skolem.lean
Package commit: 5e932f97dd25535344f80f9dd8da3aab83df0fe6
Build transcript: passed · retained with this record
Unfinished proof steps: None found by the recorded no-sorry scan
Axiom closure: Classical.choice, Quot.sound, propext
Collection provenance: Clean source state recorded

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Evidence boundary

Exact formal statement only

This target indexes Mathlib's Downward Löwenheim–Skolem theorem for a nonempty first-order L-structure M, a prescribed seed set s, and an infinite cardinal κ. After the explicit universe lifts, the seed and language cardinalities must be at most κ and κ must be at most the ambient cardinality. The endpoint gives an elementary L-substructure containing s with lifted cardinality exactly κ. It is not an unrestricted countable-submodel claim or the upward theorem.

This checker record does not establish novelty, transfer a historical acceptance decision, or authorize publication.