{
  "prefix": {
    "pe": "https://proofengine.info/proofs/for-any-triangle-with-sides-a-b-c-where-a-b-c-then/",
    "prov": "http://www.w3.org/ns/prov#",
    "xsd": "http://www.w3.org/2001/XMLSchema#",
    "schema": "http://schema.org/"
  },
  "entity": {
    "pe:claim": {
      "prov:type": "pe:Claim",
      "prov:label": "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.",
      "prov:value": "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."
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "SC1 (Forward): a^2 + b^2 = c^2 implies angle C = 90 degrees via Law of Cosines",
      "pe:factId": "A1",
      "pe:evidenceType": "computed",
      "pe:method": "Algebraic: substitute a^2+b^2=c^2 into Law of Cosines, derive cos(C)=0, conclude C=90",
      "pe:result": "True"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "SC2 (Converse): angle C = 90 degrees implies a^2 + b^2 = c^2 via Law of Cosines",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "Algebraic: substitute cos(90)=0 into Law of Cosines, derive c^2=a^2+b^2",
      "pe:result": "True"
    },
    "pe:evidence-A3": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Cross-check: numerical verification with random triangles",
      "pe:factId": "A3",
      "pe:evidenceType": "computed",
      "pe:method": "Numerical: 10000 random triangles tested in both directions",
      "pe:result": "SC1 failures: 0, SC2 failures: 0"
    },
    "pe:evidence-A4": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Cross-check: symbolic verification using sympy",
      "pe:factId": "A4",
      "pe:evidenceType": "computed",
      "pe:method": "Symbolic: sympy simplification of Law of Cosines substitution",
      "pe:result": "SC1 symbolic: True, SC2 symbolic: True"
    },
    "pe:verdict": {
      "prov:type": "pe:Verdict",
      "prov:label": "Verdict: PROVED",
      "prov:value": "PROVED"
    }
  },
  "activity": {
    "pe:determine-verdict": {
      "prov:type": "pe:VerdictDetermination",
      "prov:label": "Determine verdict from evidence",
      "prov:endTime": "2026-03-28"
    }
  },
  "agent": {
    "pe:proof-engine": {
      "prov:type": "prov:SoftwareAgent",
      "prov:label": "Proof Engine",
      "schema:version": "0.10.0",
      "schema:url": "https://github.com/yaniv-golan/proof-engine"
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  "wasGeneratedBy": {
    "pe:gen-verdict": {
      "prov:entity": "pe:verdict",
      "prov:activity": "pe:determine-verdict"
    }
  },
  "wasDerivedFrom": {},
  "wasAttributedTo": {
    "pe:attr-verdict": {
      "prov:entity": "pe:verdict",
      "prov:agent": "pe:proof-engine"
    }
  },
  "used": {
    "pe:used-A1": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A1"
    },
    "pe:used-A2": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A2"
    },
    "pe:used-A3": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A3"
    },
    "pe:used-A4": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A4"
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}