{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R860",
  "content_hash": null,
  "slug": "fib-formalization-camion-criterion",
  "type": "formalization",
  "title": "Camion's total-unimodularity criterion",
  "summary": "A reusable Lean theorem reduces total unimodularity to divisibility by four for square submatrices with even row and column sums.",
  "relevance": "For fib problem determinant range; fib problem nonzero support, record fib-formalization-camion-criterion (“Camion's total-unimodularity criterion”) states a machine-checkable theorem or proof obligation. The record states: A reusable Lean theorem reduces total unimodularity to divisibility by four for square submatrices with even row and column sums.",
  "relevance_source": "recorded",
  "body": "Lean selects a least bad minor, applies the checked minimal-obstruction theorem, and derives the Camion-Gomory parity certificate.",
  "status": "draft",
  "evidence_grade": "unverified_formalization",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "formalization",
    "runtime": "lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@d575c4e2ff28345440c4f8a42bf0178bcb3f6f41b703a45d9d1cbb709036f0dc",
    "citation": {
      "url": "https://mathoverflow.net/questions/513340/is-the-determinant-of-this-fibonacci-sum-indicator-matrix-always-1-0-or/513372",
      "locator": "formal/lean/TheoremDB/Matrix/Camion.lean"
    },
    "missing": [
      "source",
      "command",
      "expected_output"
    ]
  },
  "formal_statement": "theorem isTotallyUnimodular_of_camion {m n : Type*} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] (A : Matrix m n ℤ) (hentries : ∀ i j, A i j ∈ Set.range SignType.cast) (hcamion : ∀ (k : ℕ) (f : Fin k → m) (g : Fin k → n), f.Injective → g.Injective → HasEvenRowSums (A.submatrix f g) → HasEvenColumnSums (A.submatrix f g) → (4 : ℤ) ∣ entrySum (A.submatrix f g)) : A.IsTotallyUnimodular := by\n  by_contra hA\n  rcases exists_minimal_bad_square_minor A hA with ⟨k, f, g, hf, hg, hbad, hminimal⟩\n  rcases minimal_bad_square_minor_is_camion_obstruction A hentries k f g hf hg hbad hminimal with ⟨hrows, hcols, hnot_four⟩\n  exact hnot_four (hcamion k f g hf hg hrows hcols)",
  "source": {
    "url": "https://mathoverflow.net/questions/513340/is-the-determinant-of-this-fibonacci-sum-indicator-matrix-always-1-0-or/513372",
    "locator": "formal/lean/TheoremDB/Matrix/Camion.lean"
  },
  "relations": [
    {
      "slug": "R862",
      "title": "Total unimodularity of the Fibonacci-sum matrix",
      "object_type": "formalization",
      "relation": "depends_on",
      "direction": "incoming"
    },
    {
      "slug": "R864",
      "title": "Almost-TU determinant and nonvanishing cofactors",
      "object_type": "formalization",
      "relation": "depends_on",
      "direction": "outgoing"
    }
  ]
}