Mathlib v4.30 source proposition Nat.Coprime.of_dvd

Identifier: F:ml430-nat-coprime-of-dvd-18fcd09f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a₁ a₂ b₁ b₂ : ℕ}, a₁ ∣ a₂ → b₁ ∣ b₂ → a₂.Coprime b₂ → a₁.Coprime b₁

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.coprime_of_dvd

Kind: kernel-term
Status: checked

Supports: ∀ {a₁ a₂ b₁ b₂ : ℕ}, a₁ ∣ a₂ → b₁ ∣ b₂ → a₂.Coprime b₂ → a₁.Coprime b₁

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.coprime_of_dvd` 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. A two-step composition: `coprime_of_dvd_right` shrinks the right side from `b2` to `b1`, then `coprime_of_dvd_left` shrinks the left side from `a2` to `a1` -- no new induction or Bezout reasoning.

footprint-Nat.coprime_of_dvd

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