Mathlib v4.30 source proposition Nat.coprime_fermatNumber_fermatNumber

Identifier: F:ml430-nat-coprime-fermatnumber-fermatnumber-161e79c7
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, m ≠ n → m.fermatNumber.Coprime n.fermatNumber

Dependencies

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

Evidence

kernel-Nat.coprime_fermatNumber_fermatNumber

Kind: kernel-term
Status: checked

Supports: `Nat.coprime_fermatNumber_fermatNumber` is admitted as a Theorem with EXACTLY the stated shape (`Coprime a b` has no separate kernel name here -- see `primes.rs` -- it is stated by `Eq (gcd a b) one` directly).

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory 2>/dev/null | awk -F'\t' '$1 == "Nat.coprime_fermatNumber_fermatNumber" && $3 == "((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : ((x2 : Eq.{1} AxNat x0 x1) -> False)) -> Eq.{1} AxNat (AxNat.gcd (AxNat.fermatNumber x0) (AxNat.fermatNumber x1)) (AxNat.succ AxNat.zero))))"' | wc -l)" -ge 1

footprint-Nat.coprime_fermatNumber_fermatNumber

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] for `Nat.coprime_fermatNumber_fermatNumber`. The `nat` prelude's trusted surface is empty (no Axiom, no Opaque, no Quotient), which bounds every declaration in it, including this one.

Checker command

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

reflexive-negative-control-Nat.coprime_fermatNumber_fermatNumber

Kind: instance-pin
Status: checked

Supports: The theorem's `Ne m n` hypothesis is load-bearing, not vacuous: at m=n=0 (excluded by the hypothesis), gcd(fermatNumber 0, fermatNumber 0) = gcd(3,3) = 3, NOT 1. Two positive concrete pairs (fermatNumber 0/1 = 3/5, gcd = 1; fermatNumber 1/2 = 5/17, gcd = 1, exercising the theorem's `Lt n m` branch and its `coprime_symmetric` swap) are checked alongside a genuinely free-variable instantiation of the theorem itself.

Checker command

cargo test -q -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::fermat_number_mirrors_apply_at_free_and_concrete_instances_with_a_reflexive_negative_control -- --exact 2>&1 | grep -c '^test result: ok'

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.coprime_fermatNumber_fermatNumber` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.coprime_fermatNumber_fermatNumber`",
      "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"
    }
  ]
}