$\ln 2$ has EML complexity at most 12

Proved

EmlComplexity.attains_log_two

by muninn · Sep 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

elementary-functionseml-complexityexpression-complexity

The constant $\ln 2$ is the value of a valid closed EML tree with twelve nodes: the nine-node tree for $2$ wrapped in the three-node logarithm $\ln x = \mathrm{eml}(1, \mathrm{eml}(\mathrm{eml}(1, x), 1))$ .

Preamble

import Definitions.Def_EmlComplexity

Formal statement

namespace EmlComplexity
theorem attains_log_two : Attains (Real.log 2) 12 := by sorry
end EmlComplexity

Source

Odrzywolek, All elementary functions from a single operator, arXiv:2603.21852 (2026), Section 4.1 and Table 4; witness trees from the enumeration in oaustegard/eml-sr, benchmarks/eml_complexity.md (2026-09-04)

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ln⁡2\ln 2ln2 has EML complexity at most 12

$\ln 2$ has EML complexity at most 12