ProofAtlas

Lean evidence record

Heron’s Formula — Coordinate Identity: 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.

noncomputable def HeronFormulaStatement : Prop :=
  ∀ p q r : Point,
    (twiceSignedArea p q r) ^ 2 =
      4 * heronRadicand (side q r) (side r p) (side p q)
Definitions used in this proposition

Point

abbrev Point : Type :=
  Fin 2 → ℝ

sqDist

def sqDist (p q : Point) : ℝ :=
  (p 0 - q 0) ^ 2 + (p 1 - q 1) ^ 2

side

noncomputable def side (p q : Point) : ℝ :=
  Real.sqrt (sqDist p q)

twiceSignedArea

def twiceSignedArea (p q r : Point) : ℝ :=
  (q 0 - p 0) * (r 1 - p 1) - (q 1 - p 1) * (r 0 - p 0)

heronRadicand

noncomputable def heronRadicand (a b c : ℝ) : ℝ :=
  let s := (a + b + c) / 2
  s * (s - a) * (s - b) * (s - c)
Lean wrapper declaration

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

theorem heronFormula :
    HeronFormulaStatement
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
heronFormula
Source commit
7ff22e517713

Mechanical evidence

Lean verification

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

Artifact ID
artifact.known-heron-formula.coordinate.v002
Accepted-result title
Accepted Result: Heron Formula Coordinate Theorem
Accepted-result status
Accepted formalization of a known theorem
Accepted-result boundary
This target records the coordinate-plane Heron formula: for three points in Fin 2 -> R, the square of twice signed area equals four times the Heron radicand of the side lengths. Non-claim: This is not a reviewed abstract triangle-area API 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.HeronFormula.heronFormula
AtlasKnownTheorems.HeronFormula.HeronFormulaStatement
AtlasKnownTheorems.HeronFormula.heron_factor_identity
AtlasKnownTheorems.HeronFormula.coordinate_heron_factor_identity
Lean build
passed
Recorded build time
2.5 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
7ff22e5177133d6e1f33590dda6ad7163487ce27
Source SHA-256
sha256:6684a874e8d475e985201e0061e0bd43aa2ba8ac62f6aad56a69f8ce7f4085fc
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