Mathlib v4.30 source proposition Nat.pow_of_pow_add_prime

Identifier: F:ml430-nat-pow-of-pow-add-prime-ab61d0d3
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a n : ℕ}, 1 < a → n ≠ 0 → Nat.Prime (a ^ n + 1) → ∃ m, n = 2 ^ m

Dependencies

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

Evidence

kernel-Nat.pow_of_pow_add_prime

Kind: kernel-term
Status: checked

Supports: ∀ {a n : ℕ}, 1 < a → n ≠ 0 → Nat.Prime (a ^ n + 1) → ∃ m, n = 2 ^ m

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.pow_of_pow_add_prime : forall a n, 1 < a -> n != 0 -> PrimeCond (a^n+1) -> exists m, n = 2^m` through the trusted `Kernel::add_declaration` gate (`nat_prelude/pow_add_prime.rs`, `declare_pow_of_pow_add_prime`). Primality is spelled inline (`2 <= x /\ forall c, c | x -> c = 1 \/ c = x`), matching this prelude's `exists_prime_dvd`/`exists_prime_factorization` convention -- there is no `Nat.Prime` predicate. Route: the odd-factor witnesses `(e, t)` from `Nat.pow_two_or_has_odd_factor` feed `Nat.dvd_pow_add_one_of_odd_mul_exp` (a prior lane's odd-factor divisibility step), exhibiting `a^e+1 | a^n+1`; primality then forces that divisor to be `1` or `a^n+1`, and both are excluded (`a^e+1 >= 2` from `pow_pos` since `a > 1`; `a^e+1 != a^n+1` from `e < n`, which needs `exponent_of(t) > 1` from `t != 0` via `mul_lt_mul_left`, combined with `pow_injective` and `lt_irrefl`). Checked against a genuinely FREE `a, n` (via `Kernel::infer_in` in a real `LocalContext`, `nat_prelude_tests.rs`) in addition to this named-declaration check; `nat_theorem_inventory` exits non-zero for a name that does not exist, and the anchored `grep -cE` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Verified against a fabricated name (`pow_of_pow_add_prime_bogus_xyz`) that the same command reports count 0.

kernel-Nat.pow_two_or_has_odd_factor

Kind: kernel-term
Status: checked

Supports: ∀ n, n ≠ 0 → (∃ m, n = 2^m) ∨ (∃ e t, n = e·(2t+1) ∧ t ≠ 0)

Checker command

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

Evidence notes

The odd-factor extraction lemma `Nat.pow_two_or_has_odd_factor : forall n, n != 0 -> (exists m, n = 2^m) \/ (exists e t, n = e*(succ(2t)) /\ t != 0)` this fact's proof route depends on. NOT a `WellFounded.fix` construction: ordinary `Nat.rec` on a FUEL BOUND (`Le n fuel`, instantiated at `fuel := n` via `le_refl`) gives the induction hypothesis for every `n' <= fuel-1`, which is the strong-induction shape needed to recurse on `half := div n 2` rather than `n`'s predecessor. Splits on `Nat.even_or_odd` then on `half` via `cases_zero_succ` (`Nat`'s own constructors). Checked against a genuinely free `n` (`Kernel::infer_in`) and at the concrete instance `n=6`, where the construction's own recursion (traced by hand) produces witness `e=2, t=1` (`6=2*3`), matched against an independently-built statement of the `n=6` disjunction via `declare_theorem` (which checks the EXACT type, not merely some provable type) in `nat_prelude_tests.rs`. Verified against a fabricated name that the same anchored `grep -cE` count is 0.

footprint-Nat.pow_of_pow_add_prime

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. A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds `Nat.pow_of_pow_add_prime` and `Nat.pow_two_or_has_odd_factor`'s own footprints by `[]`. `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks both by name, in-tree, via a coverage assertion derived from `kernel.environment()` itself (not a hand-maintained list read in isolation). Full `nat_prelude::` sweep: 222 passed, 0 failed.

Provenance

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