Mathlib v4.30 source proposition Int.not_prime_of_int_mul

Identifier: F:ml430-int-not-prime-of-int-mul-e3060f5d
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b : ℤ} {c : ℕ}, a.natAbs ≠ 1 → b.natAbs ≠ 1 → a * b = ↑c → ¬Nat.Prime c

Dependencies

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9 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Int.not_prime_of_int_mul

Kind: kernel-term
Status: checked

Supports: ∀ {a b : ℤ} {c : ℕ}, a.natAbs ≠ 1 → b.natAbs ≠ 1 → a * b = ↑c → ¬Nat.Prime c

Checker command

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

Evidence notes

Reduces to a Nat-level fact about x := natAbs a, y := natAbs b: x ≠ 1 → y ≠ 1 → ¬prime_condition(x*y), proved by case-splitting x via the already-proved Nat.zero_or_succ. At x = 0: x*y = 0 (Nat.zero_mul), and the already-proved Nat.prime_ne_zero applied at 0 gives Not (Eq 0 0), contradicted by Eq.refl 0. At x = succ k (so 1 <= x, via the NatOps trait's zero_lt_succ helper transported along x = succ k): x itself is the discriminating divisor for the already-proved Nat.not_prime_of_dvd_of_ne -- x ∣ x*y (Nat.dvd_mul), x ≠ 1 (given), and x ≠ x*y because x = x*y together with x*1 = x (Nat.mul_one) and 1 <= x (Nat.mul_left_cancel_of_pos) would force y = 1, contradicting the other hypothesis. The Nat-level fact is then transported to c along natAbs(a*b) = natAbs(ofNat c) ~ c (via gcd::declare_nat_abs_mul and IntDev::int_eq_rewrite on the hypothesis a*b = ofNat c). int_theorem_inventory's rendered type for Int.not_prime_of_int_mul 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 -- every Nat-level fact used (zero_or_succ, zero_mul, mul_one, mul_left_cancel_of_pos, dvd_mul, prime_ne_zero, not_prime_of_dvd_of_ne) already existed in nat_prelude, per this lane's boundary with the sibling nat-prime-factorial-lcm lane.

footprint-Int.not_prime_of_int_mul

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