Mathlib v4.30 source proposition Nat.choose_le_add

Identifier: F:ml430-nat-choose-le-add-9c463139
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (a b c : ℕ), a.choose c ≤ (a + b).choose c

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

2 direct dependencies 1 direct dependents

Evidence

kernel-Nat.choose_le_add

Kind: kernel-term
Status: checked

Supports: ∀ (a b c : ℕ), a.choose c ≤ (a + b).choose c

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- choose_le_add 2>/dev/null | grep -Ec '^Nat\.choose_le_add[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.choose_le_add` through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. By induction on `b`: `b = 0` closes by `le_refl` since `add a zero` reduces to `a` definitionally; `b = succ b'` chains the induction hypothesis with `choose_le_succ (a+b') c` via `le_trans`, since `add a (succ b')` reduces to `succ (add a b')` definitionally.

footprint-Nat.choose_le_add

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

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

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by [].

Provenance

{
  "date": "2026-08-18",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-cascade",
  "source": "statement-only extraction of `Nat.choose_le_add` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.choose_le_add`",
      "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"
    }
  ]
}