ProofAtlas

Lean evidence record

Algebraic Lemma Toward Pick’s 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

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.

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 AlgebraicPickLemma : Prop :=
  ∀ (P : LatticePolygon) (r : Nat), BoundedBy P r →
    polygonArea P = weightedLatticePointSum P r
Definitions used in this proposition

LatticePolygon

structure LatticePolygon where
  len : Nat
  vertex : Fin (len + 1) → LatticePoint

BoundedBy

def BoundedBy (P : LatticePolygon) (r : Nat) : Prop :=
  ∀ i : Fin (P.len + 1), maxAbsCoord (P.vertex i) ≤ r

polygonArea

def polygonArea (P : LatticePolygon) : Rat :=
  ∑ i : Fin (P.len + 1), trapezoidArea (P.vertex i) (P.vertex (i + 1))

edgeWeight

noncomputable def edgeWeight (u v : LatticePoint) (r : Nat) : Rat :=
  ∑ p ∈ latticeBox r, discreteAngle (pointSub u p) (pointSub v p)

weightedLatticePointSum

noncomputable def weightedLatticePointSum (P : LatticePolygon) (r : Nat) : Rat :=
  ∑ i : Fin (P.len + 1), edgeWeight (P.vertex i) (P.vertex (i + 1)) r
Lean wrapper declaration

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

theorem algebraic_pick_lemma : AlgebraicPickLemma
Open main Lean file

Technical evidence record

Source identity, checker results, and assumptions

Main Lean declaration
algebraic_pick_lemma
Source commit
0a5907176fbb

Mechanical evidence

How Lean checked the proof

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

Artifact ID
artifact.known-pick-theorem.algebraic-pick-lemma.v002
Declarations covered by recorded evidence
AtlasKnownTheorems.PickLemma.algebraic_pick_lemma
AtlasKnownTheorems.PickLemma.AlgebraicPickLemma
AtlasKnownTheorems.PickLemma.algebraicPickLemma_of_edgeWeight_eq
Lean build
passed
Recorded build time
6.3 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
0a5907176fbb498d6353ef97645a7725bb57db62
Source SHA-256
sha256:2a326a69568adc2500be407d92955420f7a98102b7eb08946e6e5b559e09f7ba
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.

Checked source + agent continuation package · 1 Lean file · 1 evidence record · Download ZIPBrowse counted source onlineDownload main Lean fileDownload checker evidence (JSON)Download Apache-2.0 license

Review bindings

Publication reviews accepted

All four required publication-review gates are accepted for the retained presentation of this exact theorem. The review bindings are recorded separately from the Lean build and do not broaden the formal statement.

Read the publication-review details