Mathlib v4.30 source proposition Nat.log2_eq_log_two

Identifier: F:ml430-nat-log2-eq-log-two-28085932
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.log2_eq_log_two` in the pinned Mathlib v4.30 source.

Formal statement

∀ {n : ℕ}, n.log2 = Nat.log 2 n

Dependencies

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1 direct dependencies 0 direct dependents

Evidence

kernel-Nat.log2_eq_log_two

Kind: kernel-term
Status: checked

Supports: `Nat.log2_eq_log_two` is admitted as a Theorem with EXACTLY the stated shape.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory 2>/dev/null | awk -F'\t' '$1 == "Nat.log2_eq_log_two" && $3 == "((x0 : AxNat) -> Eq.{1} AxNat (AxNat.log2 x0) (AxNat.log (AxNat.succ (AxNat.succ AxNat.zero)) x0))"' | wc -l)" -ge 1

footprint-Nat.log2_eq_log_two

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] for `Nat.log2_eq_log_two` (and for the `Nat.log2` definition it is about). The `nat` prelude's trusted surface is empty (no Axiom, no Opaque, no Quotient), which bounds every declaration in it.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Provenance

{
  "date": "2026-08-30",
  "established_by": "lane clog-log2-finish (2026-08-30), new kernel construction",
  "source": "statement-only extraction of `Nat.log2_eq_log_two` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.log2_eq_log_two`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    },
    {
      "who": "the Lean 4 core developers (Gabriel Ebner, Robin Arnez)",
      "what": "`Nat.log2`'s own definition, `Init/Data/Nat/Log2.lean`",
      "where": "leanprover/lean4 toolchain v4.30.0 (commit d024af09…, resolved by scripts/check-lean-gate.sh)",
      "year": 2026,
      "attribution": "read directly (not inferred) before writing any code, to apply the mirror-flip criterion: is `Nat.log2` a Mathlib THEOREM about a structurally different `def`, or does Lean core DEFINE it the way this prelude does?"
    }
  ]
}