Mathlib v4.30 source proposition Nat.Prime.coprime_factorial_of_lt

Identifier: F:ml430-nat-prime-coprime-factorial-of-lt-2dbea201
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {p n : ℕ}, Nat.Prime p → n < p → p.Coprime n.factorial

Dependencies

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9 direct dependencies 2 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.coprime_factorial_of_lt_prime

Kind: kernel-term
Status: checked

Supports: ∀ {p n : ℕ}, Nat.Prime p → n < p → p.Coprime n.factorial

Checker command

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

Evidence notes

`declare_coprime_factorial_of_lt_prime` (`nat_prelude/gauss_lemma.rs`) admits `Nat.coprime_factorial_of_lt_prime` through the trusted `Kernel::add_declaration` gate. REPOINTED 2026-09-02 by the daily retrieval audit: this fact previously named `Nat.prime_coprime_factorial_of_lt` (`nat_prelude/prime_dvd_factorial_lcm.rs`), a SECOND declaration of the identical proposition -- `shape_search --include-constructed --duplicates` reported the pair, and the two rendered types are byte-identical in `kernel_declaration_projection`. The later declaration was deleted and this fact repointed at the earlier one (ADR-0608's survivor rule: `gauss_lemma`'s landed first, under ADR-1070, and is already consumed by `int_prelude/gauss_factorial_coprime.rs`). The proposition proved is unchanged. `Coprime` has no separate name over `Nat` in this prelude (matching `coprime_of_lt_prime`'s own convention); it is spelled `gcd p n! = 1` directly. Induction on `n`, `p` and the primality hypothesis held fixed: at `n = 0`, `factorial 0 == 1` (defeq), and `gcd p 1 = 1` unconditionally (`gcd_dvd_right` + `eq_one_of_dvd_one`); at `n = succ k`, the induction hypothesis is weakened from `succ k < p` to `k < p` (`le_succ` + `le_trans`), `coprime_of_lt_prime` (flipped by `gcd_comm`) gives `gcd p (succ k) = 1` directly from `succ k < p`, and `coprime_mul_of_coprime` combines the two coprimality facts, with `factorial (succ k)` defeq `mul (factorial k) (succ k)`.

footprint-Nat.coprime_factorial_of_lt_prime

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