Mathlib v4.30 source proposition Int.Prime.dvd_mul'

Identifier: F:ml430-int-prime-dvd-mul-23b73e69
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Int.Prime.dvd_mul'` in the pinned Mathlib v4.30 source.

Formal statement

∀ {m n : ℤ} {p : ℕ}, Nat.Prime p → ↑p ∣ m * n → ↑p ∣ m ∨ ↑p ∣ n

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Int.prime_dvd_mul'

Kind: kernel-term
Status: checked

Supports: ∀ {m n : ℤ} {p : ℕ}, Nat.Prime p → ↑p ∣ m * n → ↑p ∣ m ∨ ↑p ∣ n

Checker command

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

Evidence notes

Direct application of gcd.rs's already-proved Int.euclid_lemma at pr := ofNat p: Int.natAbs (Int.ofNat p) reduces to p by rfl (nat_abs.rs's own doc comment), so the primality hypothesis this fact states (2 <= p and every divisor of p is 1 or p) is DEFINITIONALLY the one euclid_lemma consumes on natAbs pr -- no transport code, a bare application of the six arguments (of_p, m, n, prime_hyp, prod_hyp) plus binder wrapping. int_theorem_inventory's rendered type for Int.prime_dvd_mul' matches this fact's formal.statement exactly (verified by direct comparison of the rendered De Bruijn-indexed type against the formula above, argument by argument). New proof, lane int-prime-dvd (2026-09-01): crates/axeyum-lean-kernel/src/int_prelude/prime_dvd_mul_mirrors.rs, wired in at the end of build_int_prelude_uncached (after exists_gcd_one, before nothing -- last in the build, needing gcd.rs's Int.euclid_lemma). No Nat declaration was added: this proof consumes only already-existing Int-prelude machinery.

footprint-Int.prime_dvd_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.Prime.dvd_mul'` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.Prime.dvd_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"
    }
  ]
}