Mathlib v4.30 source proposition Nat.Coprime.of_dvd_left

Identifier: F:ml430-nat-coprime-of-dvd-left-b0e2aa94
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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6 direct dependencies 5 direct dependents

Evidence

kernel-Nat.coprime_of_dvd_left

Kind: kernel-term
Status: checked

Supports: a1 divides a2, and gcd a2 b = 1, implies gcd a1 b = 1

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the constructed proof term against the stated type: `gcd a1 b` divides `a1` (`gcd_dvd_left`), which divides `a2` (`dvd_trans` with the `dvd a1 a2` hypothesis), and it divides `b` (`gcd_dvd_right`); `dvd_gcd` combines those into `gcd a1 b | gcd a2 b`, which is `1` by hypothesis, so `gcd a1 b | 1` and `eq_one_of_dvd_one` closes it. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` requires the admitted declaration to be printed.

footprint-Nat.coprime_of_dvd_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. 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": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane coprime-backlog",
  "source": "statement-only extraction of `Nat.Coprime.of_dvd_left` from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.coprime_of_dvd_left` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.Coprime.of_dvd_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, and this project's proof is independently constructed"
    }
  ]
}