Mathlib v4.30 source proposition Nat.Prime.coprime_descFactorial_of_lt_of_le

Identifier: F:ml430-nat-prime-coprime-descfactorial-of-lt-of-le-716dffc3
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {p n k : ℕ}, Nat.Prime p → n < p → k ≤ n → p.Coprime (n.descFactorial k)

Dependencies

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

Evidence

kernel-Nat.prime_coprime_descFactorial_of_lt_of_le

Kind: kernel-term
Status: checked

Supports: ∀ {p n k : ℕ}, Nat.Prime p → n < p → k ≤ n → p.Coprime (n.descFactorial k)

Checker command

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

Evidence notes

`declare_prime_coprime_desc_factorial_of_lt_of_le` (`nat_prelude/prime_dvd_factorial_lcm.rs`) admits `Nat.prime_coprime_descFactorial_of_lt_of_le` through the trusted `Kernel::add_declaration` gate. Same induction shape as `prime_coprime_factorial_of_lt`, this time on `k` with `n` and `p` (and the `n < p` hypothesis) held fixed against `Nat.descFactorial`'s own recursion. At `k = 0`, `n.descFactorial 0 == 1` (defeq), same `gcd_dvd_right`/`eq_one_of_dvd_one` base case. At `k = succ j`, the induction hypothesis is weakened from `succ j <= n` to `j <= n` (`le_of_lt`, since `Le (succ j) n` is defeq `Lt j n`); `coprime_of_lt_prime` needs `0 < n - j` (a locally-copied `sub_pos_of_lt`, built from `sub_add_cancel`/`add_comm`/`pos_of_lt_add_left` per this crate's own per-file local-helper convention) and `n - j < p` (`sub_le` bounding `n - j <= n`, then `lt_of_le_of_lt` against `n < p`); `coprime_mul_of_coprime` combines it with the induction hypothesis, and `desc_factorial_succ` (`n.descFactorial (succ j) == (n-j) * n.descFactorial j`, defeq) identifies the product.

footprint-Nat.prime_coprime_descFactorial_of_lt_of_le

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