Mathlib v4.30 source proposition Int.gcd_ne_one_iff_gcd_mul_right_ne_one

Identifier: F:ml430-int-gcd-ne-one-iff-gcd-mul-right-ne-one-ae6099bd
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Int.gcd_ne_one_iff_gcd_mul_right_ne_one` in the pinned Mathlib v4.30 source.

Formal statement

∀ {a : ℤ} {m n : ℕ}, a.gcd (↑m * ↑n) ≠ 1 ↔ a.gcd ↑m ≠ 1 ∨ a.gcd ↑n ≠ 1

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Int.gcd_ne_one_iff_gcd_mul_right_ne_one

Kind: kernel-term
Status: checked

Supports: ∀ {a : ℤ} {m n : ℕ}, a.gcd (↑m * ↑n) ≠ 1 ↔ a.gcd ↑m ≠ 1 ∨ a.gcd ↑n ≠ 1

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example int_theorem_inventory -- 'gcd_ne_one_iff_gcd_mul_right_ne_one' 2>/dev/null | /usr/bin/grep -cE '^theorem[[:space:]]+Int\.gcd_ne_one_iff_gcd_mul_right_ne_one[[:space:]]')" -ge 1

Evidence notes

Built at Nat from x := natAbs a (Int.gcd a b reduces to Nat.gcd (natAbs a) (natAbs b) by rfl, and natAbs (ofNat m * ofNat n) reduces to mul m n by rfl): the already-proved Nat.coprime_mul_iff gives Iff (Eq (gcd x (m*n)) one) (And (Eq (gcd x m) one) (Eq (gcd x n) one)). Two purely intuitionistic Not/Iff transports (no decidability: P <-> Q gives Not P <-> Not Q for free) handle everything except one classical step, isolated to exactly where it is needed -- deciding Eq Nat (gcd x m) one via Nat.beq's already-proved soundness (Nat.eq_of_beq_eq_true) and completeness (Nat.beq_eq_true_of_eq), the SAME construction int_prelude::decide builds privately to derive Int.eq_em (this file keeps its own local copy, per this crate's per-file convention), never assumed as an axiom -- to turn Not (And q1 q2) into Or (Not q1) (Not q2). int_theorem_inventory's rendered type for Int.gcd_ne_one_iff_gcd_mul_right_ne_one matches this fact's formal.statement exactly. New proof, lane int-prime-dvd (2026-09-01): crates/axeyum-lean-kernel/src/int_prelude/prime_dvd_mul_mirrors.rs. No Nat declaration was added -- only the pre-existing Nat.coprime_mul_iff, Nat.eq_of_beq_eq_true, Nat.beq_eq_true_of_eq were consumed.

footprint-Int.gcd_ne_one_iff_gcd_mul_right_ne_one

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Int prelude's trusted surface is empty

Checker command

cargo run -q --release -p axeyum-lean-kernel --example prelude_axiom_inventory -- --require-axiom-free integer

Evidence notes

prelude_axiom_inventory --require-axiom-free integer enumerates the built Int environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: integer: axiom=0, exit 0; the axreal: axiom=30 row is the separate legacy axiomatized-reals package, unrelated to this declaration -- see CLAUDE.md's AxReal/CReal gotcha). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds this declaration's own footprint by []. int_prelude_tests::every_int_declaration_is_checked_and_axiom_free additionally checks this theorem's own Kernel::axiom_footprint directly via derived_lemmas (it is listed there).

Provenance

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