Every element of the computed prime factorization is prime

Identifier: F:nat-factorization-prime
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every natural n and every x with positive multiplicity in Nat.factorization n, x is prime. Each step of the trial division emits minFac of the current cofactor, which is prime.

Formal statement

theorem Nat.factorization_prime : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.lt AxNat.zero (AxNat.Multiset.count (AxNat.factorization x0) x1)) -> And (AxNat.le (AxNat.succ (AxNat.succ AxNat.zero)) x1) (((x3 : AxNat) -> ((x4 : AxNat.dvd x3 x1) -> Or (Eq.{1} AxNat x3 (AxNat.succ AxNat.zero)) (Eq.{1} AxNat x3 x1)))))))

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

nat-factorization-prime-1

Kind: kernel-term
Status: checked

Supports: Nat.factorization_prime is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

out=$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- factorization_prime 2>&1) && test "$(printf "%s\n" "$out" | grep -cE '^Nat\.factorization_prime.5.\(\(x0 : AxNat\) -> \(\(x1 : AxNat\) -> \(\(x2 : AxNat\.lt AxNat\.zero \(AxNat\.Multiset\.count \(AxNat\.factorization x0\) x1\)\) -> And \(AxNat\.le \(AxNat\.succ \(AxNat\.succ AxNat\.zero\)\) x1\) \(\(\(x3 : AxNat\) -> \(\(x4 : AxNat\.dvd x3 x1\) -> Or \(Eq\.\{1\} AxNat x3 \(AxNat\.succ AxNat\.zero\)\) \(Eq\.\{1\} AxNat x3 x1\)\)\)\)\)\)\)$')" = 1

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted kernel gate, so a successful build IS the type-check. The pattern pins the ARITY and the FULL rendered type, not the theorem's presence: `scripts/new-fact.py` rejected a name-only pattern as population-only, because it catches the row disappearing but not the statement changing inside a row that is still there.

footprint-nat-factorization-prime

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- the Nat environment admits no trusted declaration

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

Reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0` over the FULL trusted surface (Axiom, Opaque and Quotient), not `Declaration::Axiom` alone, and exits non-zero unless the surface is empty. The enumeration is per-environment rather than per-theorem; it bounds this theorem's footprint because a proof cannot depend on a trusted declaration the environment does not contain. Read directly from `Kernel::axiom_footprint` via `theorem_axiom_footprint`, this theorem's row prints footprint size 0 and an empty axiom column, and all 904 `nat` rows do. (Count the tool's OWN summary line, not `grep -c '^nat'` -- the summary starts with `nat:` too, so the naive count is 905.)

Provenance

{
  "date": "2026-09-02",
  "established_by": "axeyum-lean-kernel build_nat_prelude",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type; declared by `nat_prelude/factorization_multiset.rs` (lane nat-factorization)."
}