Mathlib v4.30 source proposition Nat.dvd_iff_mod_eq_zero

Identifier: F:ml430-nat-dvd-iff-mod-eq-zero-d795bfff
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, m ∣ n ↔ n % m = 0

Dependencies

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4 direct dependencies 0 direct dependents

Evidence

kernel-Nat.dvd_iff_mod_eq_zero

Kind: kernel-term
Status: checked

Supports: ∀ {m n : ℕ}, m ∣ n ↔ n % m = 0

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.dvd_iff_mod_eq_zero` 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), wired in via one `declare_gcd_dvd_mirrors` call. Case split on `m`: `m = 0` reduces both sides to `n = 0` directly (`dvd 0 n` unfolds to `n = 0`; `mod n 0 = n` via `mod_zero`); `m = succ j` specializes the existing `div_mod_remainder_eq_zero_iff_dvd` at the executable witness (`div_mod_exec`) and flips the `Iff` order to match. `nat_theorem_inventory`'s rendered type is `(x0:AxNat)->(x1:AxNat)->Iff (dvd x0 x1) (Eq (mod x1 x0) zero)`, 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); grepping a made-up name (`Nat.dvd_iff_mod_eq_zero_bogus`) exits non-zero (`nat_theorem_inventory` fails closed on an absent name -- measured: `error: no Nat theorem matches ... -- an absent theorem is a failed check, not an empty report`, exit 1).

footprint-Nat.dvd_iff_mod_eq_zero

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.dvd_iff_mod_eq_zero` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.dvd_iff_mod_eq_zero`",
      "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"
    }
  ]
}