Mathlib v4.30 source proposition Nat.div_gcd_pos_of_pos_left

Identifier: F:ml430-nat-div-gcd-pos-of-pos-left-dd878a3f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a : ℕ} (b : ℕ), 0 < a → 0 < a / a.gcd b

Dependencies

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

5 direct dependencies 0 direct dependents

Evidence

kernel-Nat.div_gcd_pos_of_pos_left

Kind: kernel-term
Status: checked

Supports: ∀ {a : ℕ} (b : ℕ), 0 < a → 0 < a / a.gcd b

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.div_gcd_pos_of_pos_left` 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. Built in this session's `nat_prelude/gcd_dvd_mirrors.rs` (lane nat-gcd-dvd-mirrors), via the shared helper `pos_of_dvd_and_pos_numerator`: `div_mul_cancel` gives `(m/n)*n = m` from `dvd n m`; if `m/n` were `0` that would force `m = 0` (`zero_mul`), contradicting the hypothesis `0 < m` (via `not_lt_zero`); `Nat.zero_or_succ` on `m/n` therefore leaves only the successor case, closed directly by `zero_lt_succ`. Applied with n = gcd(a,b), m = a via `gcd_dvd_left`. `nat_theorem_inventory`'s rendered type is `(x0:AxNat)->(x1:AxNat)->(x2:lt zero x0)->lt zero (div x0 (gcd x0 x1))`, matching this fact's `formal.statement`. `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. Verified both ways: the real name greps to a count `-ge 1` (confirmed by direct run, which also confirms this exact name does not collide with its sibling `Nat.div_gcd_pos_of_pos_left_right`/`Nat.div_gcd_pos_of_pos_left_left`); grepping a made-up name (`Nat.div_gcd_pos_of_pos_left_bogus`) fails closed (`nat_theorem_inventory` exits 1 on an absent name).

footprint-Nat.div_gcd_pos_of_pos_left

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 []. `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly via `theorem_names`, and `the_build_is_deterministic`/`every_promised_name_is_admitted_with_the_expected_kind` (both re-run on every `nat_prelude::` sweep) confirm the declaration is admitted at the exact rendered type checked below.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.div_gcd_pos_of_pos_left` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.div_gcd_pos_of_pos_left`",
      "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"
    }
  ]
}