Mathlib v4.30 source proposition Nat.choose_mono

Identifier: F:ml430-nat-choose-mono-a1af9c18
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (b : ℕ), Monotone fun a => a.choose b

Dependencies

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1 direct dependencies 0 direct dependents

Evidence

kernel-Nat.choose_mono

Kind: kernel-term
Status: checked

Supports: ∀ c a a', Le a a' → Le (choose a c) (choose a' c)

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --release --example nat_theorem_inventory -- choose_mono 2>/dev/null | grep -Ec '^Nat\.choose_mono[[:space:]]+4[[:space:]]+\(\(x0 : AxNat\) -> \(\(x1 : AxNat\) -> \(\(x2 : AxNat\) -> \(\(x3 : AxNat\.le x1 x2\) -> AxNat\.le \(AxNat\.choose x1 x0\) \(AxNat\.choose x2 x0\)\)\)\)\)$')" -ge 1

Evidence notes

The kernel re-infers the theorem type from the constructed term. Lean's `Monotone (fun a => a.choose b)` unfolds to `forall x y, x <= y -> choose x b <= choose y b`, exactly the rendered core proposition checked here (rendered with `b`/`x`/`y` as `x0`/`x1`/`x2`). The proof specializes the already-admitted `Nat.choose_le_choose` with its arguments permuted so the fixed column comes first; no new induction. The kernel-observed dependency closure, not this note, is authoritative.

footprint-Nat.choose_mono

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: []

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --release --example theorem_axiom_footprint -- Nat.choose_mono 2>/dev/null | grep -Ec '^nat[[:space:]]+Nat\.choose_mono[[:space:]]+0[[:space:]]*$')" -ge 1

Evidence notes

The theorem-specific footprint command reports zero dependencies on axioms in the constructed Nat environment.

Provenance

{
  "date": "2026-08-18",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.choose_mono` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.choose_mono`",
      "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"
    }
  ]
}