Mathlib v4.30 source proposition Nat.Prime.five_le_of_ne_two_of_ne_three

Identifier: F:ml430-nat-prime-five-le-of-ne-two-of-ne-three-c069e786
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {p : ℕ}, Nat.Prime p → p ≠ 2 → p ≠ 3 → 5 ≤ p

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

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

Evidence

kernel-Nat.five_le_of_ne_two_of_ne_three

Kind: kernel-term
Status: checked

Supports: prime_condition p -> p != 2 -> p != 3 -> 5 <= p

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --release --example nat_theorem_inventory -- five_le_of_ne_two_of_ne_three 2>/dev/null | grep -xFc 'Nat.five_le_of_ne_two_of_ne_three	6	((x0 : AxNat) -> ((x1 : And (AxNat.le (AxNat.succ (AxNat.succ AxNat.zero)) x0) (((x1 : AxNat) -> ((x2 : AxNat.dvd x1 x0) -> Or (Eq.{1} AxNat x1 (AxNat.succ AxNat.zero)) (Eq.{1} AxNat x1 x0))))) -> ((x2 : Not (Eq.{1} AxNat x0 (AxNat.succ (AxNat.succ AxNat.zero)))) -> ((x3 : Not (Eq.{1} AxNat x0 (AxNat.succ (AxNat.succ (AxNat.succ AxNat.zero))))) -> AxNat.le (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ AxNat.zero))))) x0))))')" -ge 1

Evidence notes

The kernel re-infers the theorem type from the constructed term, matching Mathlib's `Prime p -> p != 2 -> p != 3 -> 5 <= p`. Primality is spelled inline (2 <= x /\ forall c, c|x -> c=1 \/ c=x), matching this prelude's convention (no bare `Prime` predicate; see primes.rs module doc). Proved by splitting at Nat.lt_or_ge p 5 (ops::cases_lt_or_ge): the Le 5 p side is the hypothesis itself; the Lt p 5 side is a genuine 5-way case split to concrete p in {0,1,2,3,4} (ops::cases_lt_bound_absurd, a new finite-cases eliminator whose branches discharge a FIXED goal by contradiction) -- p=0,1 contradict the primality lower bound 2<=p, p=2,3 contradict the two Not hypotheses directly, p=4 is refuted as composite (4=2*2) via the already-declared not_prime_of_dvd_of_ne.

footprint-Nat.five_le_of_ne_two_of_ne_three

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: []

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --release --example theorem_axiom_footprint -- Nat.five_le_of_ne_two_of_ne_three 2>/dev/null | grep -xFc $'nat\tNat.five_le_of_ne_two_of_ne_three\t0\t')" -ge 1

Evidence notes

The theorem-specific footprint command reports zero dependencies on axioms in the constructed Nat environment.

Provenance

{
  "date": "2026-08-18",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.Prime.five_le_of_ne_two_of_ne_three` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.Prime.five_le_of_ne_two_of_ne_three`",
      "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"
    }
  ]
}