a number below minFac(n) and not zero is coprime to n

Identifier: F:nat-coprime-of-lt-minfac
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every natural n, m: if m is not 0 and m < minFac(n), then gcd(n, m) = 1.

Formal statement

theorem Nat.coprime_of_lt_min_fac : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : Not (Eq.{1} AxNat x1 AxNat.zero)) -> ((x3 : AxNat.lt x1 (AxNat.minFac x0)) -> Eq.{1} AxNat (AxNat.gcd x0 x1) (AxNat.succ AxNat.zero)))))

Dependencies

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

Evidence

kernel-Nat.coprime_of_lt_min_fac

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type. `nat_theorem_inventory` prints nothing for a name that does not exist, and `grep -Ec` consumes the pipe so the tested count decides the exit status. Anchored with `^Nat\.coprime_of_lt_min_fac[[:space:]]` so a future sibling (e.g. a helper of the same shape) cannot satisfy this row. Run `--release`: the debug build SIGABRTs building the full constructed environment (see CLAUDE.md's `prelude_theorem_inventory` gotcha).

compute-Nat.coprime_of_lt_min_fac

Kind: kernel-term
Status: checked

Supports: gcd(25, 4) = 1: 25's least prime factor is 5, and 4 < 5, 4 != 0. m=4 is a DISCRIMINATING witness -- it shares no factor with 25 or with minFac(25)=5, unlike m=2 (also coprime, but would pass even under a broken bound that let m reach minFac(n) itself).

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::coprime_of_lt_min_fac_applies_at_a_concrete_instance

Evidence notes

Applies `Nat.coprime_of_lt_min_fac` at the concrete instance `(n, m) = (25, 4)`, confirming the admitted type matches `Eq (gcd 25 4) 1`, that `minFac 25` reduces to `5`, and a NEGATIVE control that `gcd 25 5 != 1` -- `m := minFac n` itself is genuinely excluded by the strict bound, not merely untested.

footprint-Nat.coprime_of_lt_min_fac

Kind: instance-pin
Status: checked

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

Checker command

cargo run -q --release -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 rather than `Declaration::Axiom` alone. The enumeration is per-environment, not per-theorem; it bounds this theorem's footprint because a proof cannot depend on a trusted declaration the environment does not contain.

Provenance

{
  "date": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-fastfib-minfac",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}