{
  "prefix": {
    "pe": "https://proofengine.info/proofs/0-999-with-9s-repeating-forever-is-strictly-less-t/",
    "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": "0.999... (with 9s repeating forever) is strictly less than 1.",
      "prov:value": "0.999... (with 9s repeating forever) is strictly less than 1."
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Algebraic proof: if x = 0.999... then x = 1",
      "pe:factId": "A1",
      "pe:evidenceType": "computed",
      "pe:method": "Algebraic: x = 0.999..., 10x - x = 9, x = 1 (verified via Fraction(9,9))",
      "pe:result": "1.0"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Geometric series proof: sum of 9/10^k for k=1..inf equals 1",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "Geometric series: a/(1-r) = (9/10)/(9/10) = 1 (verified via Fraction arithmetic)",
      "pe:result": "1.0"
    },
    "pe:evidence-A3": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Fraction proof: 1/3 = 0.333..., so 3 * (1/3) = 0.999... = 1",
      "pe:factId": "A3",
      "pe:evidenceType": "computed",
      "pe:method": "Fraction identity: 3 * (1/3) = 3/3 = 1 (verified via Fraction arithmetic)",
      "pe:result": "1.0"
    },
    "pe:evidence-A4": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Numerical convergence: partial sums approach 1 with zero gap",
      "pe:factId": "A4",
      "pe:evidenceType": "computed",
      "pe:method": "Numerical convergence: partial sums 1 - 10^(-n) -> 1 as n -> inf",
      "pe:result": "True (converges to 1)"
    },
    "pe:verdict": {
      "prov:type": "pe:Verdict",
      "prov:label": "Verdict: DISPROVED",
      "prov:value": "DISPROVED"
    }
  },
  "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"
    }
  },
  "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"
    }
  }
}