{
  "format_version": 3,
  "claim_formal": {
    "subject": "Any triangle with sides a, b, c",
    "property": "biconditional: (a^2 + b^2 = c^2) <=> (angle opposite c = 90 degrees)",
    "operator": "==",
    "operator_note": "This is a biconditional (if and only if) claim with two sub-claims: SC1 (Forward): a^2 + b^2 = c^2 implies angle C = 90 degrees. SC2 (Converse): angle C = 90 degrees implies a^2 + b^2 = c^2. Both directions are proved via the Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C). The Law of Cosines is taken as an established theorem of Euclidean geometry. The proof is algebraic: substitution and simplification.",
    "threshold": true
  },
  "claim_natural": "For any triangle with sides a, b, c where \\(a^2 + b^2 = c^2\\), the angle opposite c is exactly 90 degrees, AND the converse also holds.",
  "evidence": {
    "A1": {
      "type": "computed",
      "label": "SC1 (Forward): a^2 + b^2 = c^2 implies angle C = 90 degrees via Law of Cosines",
      "sub_claim": "SC1",
      "method": "Algebraic: substitute a^2+b^2=c^2 into Law of Cosines, derive cos(C)=0, conclude C=90",
      "result": "True",
      "depends_on": []
    },
    "A2": {
      "type": "computed",
      "label": "SC2 (Converse): angle C = 90 degrees implies a^2 + b^2 = c^2 via Law of Cosines",
      "sub_claim": "SC2",
      "method": "Algebraic: substitute cos(90)=0 into Law of Cosines, derive c^2=a^2+b^2",
      "result": "True",
      "depends_on": []
    },
    "A3": {
      "type": "computed",
      "label": "Cross-check: numerical verification with random triangles",
      "sub_claim": null,
      "method": "Numerical: 10000 random triangles tested in both directions",
      "result": "SC1 failures: 0, SC2 failures: 0",
      "depends_on": []
    },
    "A4": {
      "type": "computed",
      "label": "Cross-check: symbolic verification using sympy",
      "sub_claim": null,
      "method": "Symbolic: sympy simplification of Law of Cosines substitution",
      "result": "SC1 symbolic: True, SC2 symbolic: True",
      "depends_on": []
    }
  },
  "cross_checks": [
    {
      "description": "Numerical: random triangle verification (10,000 trials each direction)",
      "values_compared": [
        "SC1 failures: 0",
        "SC2 failures: 0"
      ],
      "agreement": true,
      "fact_ids": []
    },
    {
      "description": "Symbolic: sympy verification of both directions",
      "values_compared": [
        "SC1 symbolic: True",
        "SC2 symbolic: True"
      ],
      "agreement": true,
      "fact_ids": []
    }
  ],
  "adversarial_checks": [
    {
      "question": "Does this proof depend on Euclidean geometry specifically?",
      "verification_performed": "Analyzed whether the Law of Cosines holds in non-Euclidean geometries. In spherical and hyperbolic geometry, the Law of Cosines takes a different form. The Pythagorean theorem as stated (a^2 + b^2 = c^2) is specific to Euclidean geometry.",
      "finding": "The claim is implicitly restricted to Euclidean geometry, which is the standard interpretation. The proof is valid in this context. In non-Euclidean geometries, the relationship between sides and angles differs, but the claim does not assert otherwise.",
      "breaks_proof": false
    },
    {
      "question": "Are there degenerate triangles where the proof fails?",
      "verification_performed": "Checked edge cases: (1) degenerate triangle where a + b = c (zero area), (2) triangle inequality violations, (3) zero-length sides. A degenerate 'triangle' with a + b = c has angle C = 180 deg and a^2 + b^2 < c^2 (by Cauchy-Schwarz), so it does not satisfy the hypothesis.",
      "finding": "Degenerate cases do not satisfy a^2 + b^2 = c^2 with positive side lengths, so they are excluded from the hypothesis. The proof requires a, b, c > 0 and the triangle inequality, which are implicit in 'for any triangle.'",
      "breaks_proof": false
    },
    {
      "question": "Is cos(C) = 0 sufficient to conclude C = 90 degrees?",
      "verification_performed": "cos(C) = 0 has solutions C = 90 + 180*k degrees for integer k. In a triangle, interior angles satisfy 0 < C < 180 degrees. The only solution in this range is C = 90 degrees.",
      "finding": "Within the valid range for triangle interior angles (0, 180), cos(C) = 0 uniquely determines C = 90 degrees. The step is valid.",
      "breaks_proof": false
    },
    {
      "question": "Does the converse require any additional conditions beyond C = 90?",
      "verification_performed": "Checked whether the converse direction assumes anything beyond C = 90 degrees. The substitution cos(90) = 0 into the Law of Cosines is direct and requires no additional conditions beyond the triangle being valid (positive sides, satisfying triangle inequality).",
      "finding": "No additional conditions needed. The converse follows directly from cos(90) = 0 in the Law of Cosines.",
      "breaks_proof": false
    }
  ],
  "verdict": {
    "value": "PROVED",
    "qualified": false,
    "qualifier": null,
    "reason": null
  },
  "key_results": {
    "sc1_forward_holds": true,
    "sc2_converse_holds": true,
    "numerical_cross_check_passed": true,
    "symbolic_cross_check_passed": true,
    "claim_holds": true
  },
  "generator": {
    "name": "proof-engine",
    "version": "0.10.0",
    "repo": "https://github.com/yaniv-golan/proof-engine",
    "generated_at": "2026-03-28"
  },
  "proof_py_url": "/proofs/for-any-triangle-with-sides-a-b-c-where-a-b-c-then/proof.py",
  "citation": {
    "doi": null,
    "concept_doi": null,
    "url": "https://proofengine.info/proofs/for-any-triangle-with-sides-a-b-c-where-a-b-c-then/",
    "author": "Proof Engine",
    "cite_bib_url": "/proofs/for-any-triangle-with-sides-a-b-c-where-a-b-c-then/cite.bib",
    "cite_ris_url": "/proofs/for-any-triangle-with-sides-a-b-c-where-a-b-c-then/cite.ris"
  },
  "depends_on": []
}