{
  "format_version": 3,
  "claim_natural": "The binary operator eml is defined by the expression \\(\\text{eml}(a, b) = \\exp(a) - \\ln(b)\\) (where exp is the exponential function and ln is the principal branch of the natural logarithm). The expression \\(\\text{eml}(1, 1)\\) equals the base of the natural logarithm \\(e\\).",
  "claim_formal": {
    "subject": "Binary operator eml(a, b) = exp(a) - ln(b)",
    "property": "eml(1, 1) = e",
    "operator": "==",
    "operator_note": "Substituting a = 1, b = 1: eml(1, 1) = exp(1) - ln(1). exp(1) = e by definition of the exponential function. ln(1) = 0 (principal branch of the complex logarithm: Log(1) = ln|1| + i*Arg(1) = 0 + 0 = 0). Therefore eml(1, 1) = e - 0 = e. This is an exact algebraic identity, not a numerical approximation.",
    "threshold": true,
    "is_time_sensitive": false
  },
  "evidence": {
    "A1": {
      "type": "computed",
      "label": "exp(1) = e (symbolic verification)",
      "sub_claim": null,
      "method": "SymPy symbolic evaluation: simplify(exp(1) - E); verify equals 0",
      "result": "Confirmed: exp(1) = e",
      "depends_on": []
    },
    "A2": {
      "type": "computed",
      "label": "ln(1) = 0 (symbolic verification)",
      "sub_claim": null,
      "method": "SymPy symbolic evaluation of ln(1); verify equals 0",
      "result": "Confirmed: ln(1) = 0",
      "depends_on": []
    },
    "A3": {
      "type": "computed",
      "label": "eml(1, 1) - e = 0 (symbolic simplification)",
      "sub_claim": null,
      "method": "SymPy symbolic simplification of exp(1) - ln(1) - E; verify result is 0",
      "result": "Confirmed: residual = 0",
      "depends_on": [
        "A1",
        "A2"
      ]
    }
  },
  "cross_checks": [
    {
      "description": "Symbolic (A3) vs numerical: symbolic proves identity exactly; numerical confirms via Python math.e and cmath.exp/cmath.log",
      "fact_ids": [
        "A3"
      ],
      "agreement": true
    }
  ],
  "adversarial_checks": [
    {
      "question": "Is exp(1) exactly e, or just an approximation?",
      "verification_performed": "The exponential function exp is defined such that exp(1) = e by definition. In SymPy, exp(1) returns the symbolic constant E (Euler's number), not a floating-point approximation. The identity exp(1) = e is definitional, not computed.",
      "finding": "exp(1) = e is exact and definitional. No approximation is involved.",
      "breaks_proof": false
    },
    {
      "question": "Does ln(1) = 0 hold for the complex logarithm?",
      "verification_performed": "The principal branch of the complex logarithm is defined as Log(z) = ln|z| + i*Arg(z) where Arg is the principal argument in (-pi, pi]. For z = 1: |1| = 1, Arg(1) = 0, so Log(1) = ln(1) + i*0 = 0. This holds regardless of branch cut conventions since z = 1 is on the positive real axis.",
      "finding": "ln(1) = 0 holds universally for the principal branch of the complex logarithm. No branch ambiguity at z = 1.",
      "breaks_proof": false
    },
    {
      "question": "Could SymPy's simplify() fail to reduce a nonzero expression to zero?",
      "verification_performed": "For the expression exp(1) - ln(1) - E, SymPy evaluates exp(1) to E and ln(1) to 0 before simplify() is even called. The expression becomes E - 0 - E = 0, which is trivial constant arithmetic. There is no symbolic variable or complex cancellation involved. Additionally, the numerical cross-check confirms the result independently.",
      "finding": "The simplification is trivial constant folding (E - 0 - E = 0). No risk of simplify() missing a cancellation.",
      "breaks_proof": false
    }
  ],
  "verdict": {
    "value": "PROVED",
    "qualified": false,
    "qualifier": null,
    "reason": null
  },
  "key_results": {
    "exp_1_equals_e": true,
    "ln_1_equals_zero": true,
    "symbolic_residual_zero": true,
    "numerical_crosscheck_passed": true,
    "claim_holds": true
  },
  "generator": {
    "name": "proof-engine",
    "version": "1.18.0",
    "repo": "https://github.com/yaniv-golan/proof-engine",
    "generated_at": "2026-04-16"
  },
  "proof_py_url": "/proofs/the-binary-operator-eml-is-defined-by-the-expression-text-eml-a-b-exp-a-ln-b/proof.py",
  "citation": {
    "doi": "10.5281/zenodo.19635611",
    "concept_doi": "10.5281/zenodo.19626393",
    "url": "https://proofengine.info/proofs/the-binary-operator-eml-is-defined-by-the-expression-text-eml-a-b-exp-a-ln-b/",
    "author": "Proof Engine",
    "cite_bib_url": "/proofs/the-binary-operator-eml-is-defined-by-the-expression-text-eml-a-b-exp-a-ln-b/cite.bib",
    "cite_ris_url": "/proofs/the-binary-operator-eml-is-defined-by-the-expression-text-eml-a-b-exp-a-ln-b/cite.ris"
  },
  "depends_on": [
    {
      "relation": "References",
      "identifiers": [
        {
          "type": "arxiv",
          "value": "2603.21852"
        }
      ],
      "note": "Originating paper introducing the eml binary operator"
    }
  ]
}