Mathlib v4.30 source proposition Nat.Prime.mul_eq_prime_sq_iff

Identifier: F:ml430-nat-prime-mul-eq-prime-sq-iff-d3fd2e31
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.Prime.mul_eq_prime_sq_iff` in the pinned Mathlib v4.30 source.

Formal statement

∀ {x y p : ℕ}, Nat.Prime p → x ≠ 1 → y ≠ 1 → (x * y = p ^ 2 ↔ x = p ∧ y = p)

Dependencies

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

Evidence

kernel-Nat.prime_mul_eq_prime_sq_iff

Kind: kernel-term
Status: checked

Supports: ∀ {x y p : ℕ}, Nat.Prime p → x ≠ 1 → y ≠ 1 → (x * y = p ^ 2 ↔ x = p ∧ y = p)

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.prime_mul_eq_prime_sq_iff` 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 (verified against a fabricated name), and the anchored `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Primality is spelled inline (`2 ≤ p ∧ ∀ c, c ∣ p → c = 1 ∨ c = p`, matching `prime_even_iff`/`euclid_lemma`'s own convention) rather than through a named `Nat.Prime` predicate, since this prelude has none -- this repository's established precedent for flipping a Mathlib `Nat.Prime` mirror. The proof itself is constructive: `x*y = p^2 = p*p` gives `p | x*y` (witness `p`), `euclid_lemma` splits into `p|x` or `p|y`, and each branch's divisor witness `k` (from `a = p*k`) satisfies `k*b = p` (`mul_assoc` + `mul_left_cancel_of_pos`), so `k`'s own primality clause forces `k=1` (giving both factors equal `p`) or `k=p` (forcing the OTHER factor to `1`, contradicting the corresponding `x != 1`/`y != 1` hypothesis).

footprint-Nat.prime_mul_eq_prime_sq_iff

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