Mathlib v4.30 source proposition Nat.ModEq.gcd_eq

Identifier: F:ml430-nat-modeq-gcd-eq-5167ff4f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m a b : ℕ}, a ≡ b [MOD m] → a.gcd m = b.gcd m

Dependencies

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14 direct dependencies 1 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.mod_eq_gcd_eq

Kind: kernel-term
Status: checked

Supports: ∀ m a b, Nat.modEq m a b -> Nat.gcd a m = Nat.gcd b m

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.mod_eq_gcd_eq` 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. Confirmed kernel type: `∀ m a b, modEq m a b -> gcd a m = gcd b m`, matching the formal statement with `a`/`b` in Mathlib's order. Proved directly via `declare_modeq_gcd_eq` (`nat_prelude/gcd.rs`): eliminate the balanced-witness `modEq m a b := ∃ u v, a+m*u=b+m*v` twice; `gcd a m` divides `a` and `m` (`gcd_dvd_left`/`gcd_dvd_right`), hence `m*u` (`dvd_mul_right_of_dvd`) and so `a+m*u` (`dvd_add`); transport along the witness equation to `b+m*v`; reorder to `m*v+b` (`add_comm`) and peel `m*v` (`dvd_add_iff_right`, reversed) to land on `gcd a m ∣ b`; close with `dvd_gcd`. The mirror argument over the symmetric equation gives `gcd b m ∣ gcd a m`; `dvd_antisymm` finishes. No `Nat.ModEq` remainder-characterization route (`mod_eq_dvd_iff`/`div_mod`) was needed -- the balanced-witness definition sufficed directly.

footprint-Nat.mod_eq_gcd_eq

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