The prime-counting function is unbounded

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

Recorded description

For every k there is an n with k <= pi'(n): the prime-counting function takes arbitrarily large values. This is Euclid's theorem in counting form.

Formal statement

theorem Nat.primeCounting'_unbounded : ((x0 : AxNat) -> Exists.{1} AxNat (fun (x1 : AxNat) => AxNat.le x0 (AxNat.primeCounting' x1)))

Dependencies

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6 direct dependencies 0 direct dependents

Evidence

kernel-Nat.primeCounting'_unbounded

Kind: kernel-term
Status: checked

Supports: pi' takes arbitrarily large values

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- 'Nat.primeCounting'_unbounded' 2>/dev/null | grep -xFc 'Nat.primeCounting'_unbounded	1	((x0 : AxNat) -> Exists.{1} AxNat (fun (x1 : AxNat) => AxNat.le x0 (AxNat.primeCounting' x1)))')" -ge 1

Evidence notes

Induction on k. The base takes the witness 0 and `Nat.zero_le`. The step eliminates the hypothesis' witness n, takes a prime p > n from `Nat.exists_prime_gt`, and uses the witness succ p: pi'(succ p) = succ (pi'(p)) >= succ (pi'(n)) >= succ k. The first equality is `countRange_succ_of_true`, which needs `Eq Bool (isPrime p) true` -- and that is exactly where `F:nat-is-prime-eq-true-of-prime` is spent. Without it the two halves do not meet and this statement is not reachable from anything else in the prelude.

footprint-Nat.primeCounting'_unbounded

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` bounds the footprint by enumerating the whole trusted surface: a theorem cannot depend on a trusted declaration the environment does not contain. All three kinds are covered -- counting `Declaration::Axiom` alone would not suffice, because `Opaque` has no proof body and `Quotient` admits `Quot.sound`. The module's own test `the_prime_counting_shelf_is_admitted_and_axiom_free` additionally asserts `Environment::contains` BEFORE reading the footprint, because `axiom_footprint` returns an empty set for a name that was never declared.

Provenance

{
  "date": "2026-09-06",
  "established_by": "lane chebyshev-pi-bounds (ADR-1655), roadmap W3-11",
  "source": "statement authored here against this kernel's own vocabulary; nothing copied",
  "prior_art": [
    {
      "who": "Euclid",
      "what": "Elements, Book IX, Proposition 20 (the set form)",
      "year": -300,
      "attribution": "standard textbook attribution; this lane did not consult the primary source"
    }
  ]
}