ProofAtlas

Lean evidence record

Sylvester–Gallai Theorem: Lean evidence

This technical record binds the exact theorem statement to its commit-pinned Lean source, checker results, assumptions, and publication-review status.

Read theorem page
Lean buildpassed
Unfinished proof stepsNone
Publication reviewsAccepted

Exact formal proposition

Hypotheses and conclusion

This is the meaningful proposition proved by the checked wrapper declaration. It is extracted from the same commit-pinned Lean source.

def SylvesterGallaiStatement : Prop :=
  ∀ S : Finset Point, 2 < S.card → ¬ FinsetCollinear S →
    ∃ a b : Point, OrdinaryPair S a b
Definitions used in this proposition

Point

abbrev Point : Type := ℝ × ℝ

Collinear3

def Collinear3 (p q r : Point) : Prop :=
  orientationDet p q r = 0

FinsetCollinear

def FinsetCollinear (S : Finset Point) : Prop :=
  ∃ a ∈ S, ∃ b ∈ S, a ≠ b ∧ ∀ c ∈ S, Collinear3 a b c

OrdinaryPair

def OrdinaryPair (S : Finset Point) (a b : Point) : Prop :=
  a ∈ S ∧ b ∈ S ∧ a ≠ b ∧
    ∀ c ∈ S, Collinear3 a b c → c = a ∨ c = b
Lean wrapper declaration

The checked endpoint names the expanded proposition above. Both come from the same commit-pinned source.

theorem sylvesterGallai :
    SylvesterGallaiStatement
Open main Lean file

Line counts exclude blank lines; comments and documentation count. The total is the commit-pinned first-party Lean import closure; Mathlib and other third-party dependencies are excluded.

Technical evidence record

Source identity, checker results, and assumptions

Main Lean declaration
sylvesterGallai
Source commit
164ee7b20e49

Mechanical evidence

Lean verification

These fields support the exact Lean declaration, not a broader informal claim.

Artifact ID
artifact.known-sylvester-gallai.sylvester-gallai.v002
Accepted-result title
Accepted Result: Sylvester-Gallai Theorem
Accepted-result status
Accepted formalization of a known theorem
Accepted-result boundary
This target records the concrete finite Sylvester-Gallai theorem over R x R: a finite set of more than two points that is not all on one line has an ordinary pair. Non-claim: This is not an abstract affine-plane theorem. Non-claim: This is not a projective or line-object quotient theorem. Non-claim: This accepted-result record does not establish novelty and does not by itself authorize public export or deployment.
Declarations covered by recorded evidence
AtlasKnownTheorems.SylvesterGallai.sylvesterGallai
AtlasKnownTheorems.SylvesterGallai.SylvesterGallaiStatement
AtlasKnownTheorems.SylvesterGallai.sylvesterGallai_statement
Lean build
passed
Recorded build time
4.4 s one machine-dependent evidence run, not a benchmark
Evidence collected
· clean-source provenance recorded
Unfinished proof check
passed
Lean toolchain
leanprover/lean4:v4.29.1
Recorded source commit
164ee7b20e496b563892898f787160cbdaa7ba5a
Source SHA-256
sha256:ecc13cdfd54ed7c2515844e92c7e1f0467007a7b1248ec81798e17e5ebb89a9d
Statement alignment
accepted

Lean foundations

Standard foundations used by the proof

Lean reports the logical foundations below through Mathlib. They are standard proof-system foundations, not conjectural mathematical assumptions about this theorem. The recorded closure stays within the approved classical_mathlib_standard profile, with no unexpected axiom or unfinished-proof placeholder.

  • Classical.choice
  • Quot.sound
  • propext

Files and machine-readable evidence

Reproduce or inspect the recorded check

Use the complete first-party source bundle for reconstruction, or inspect the exact main file and checker evidence separately. Mathlib and other third-party dependencies are identified but not rebundled.

Complete checked Lean source · 1 Lean file · 1 evidence record · Download ZIPBrowse counted source onlineDownload main Lean fileDownload checker evidence (JSON)Download Apache-2.0 license

Review results

Publication reviews accepted

All four required publication-review gates are accepted for the reviewed presentation of this exact theorem. The review results are separate from the Lean build and do not broaden the formal statement.

Read the publication-review details