The prime-counting function is monotone

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

Recorded description

If m <= n then the number of primes strictly below m is at most the number of primes strictly below n.

Formal statement

theorem Nat.primeCounting'_mono : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.le x0 x1) -> AxNat.le (AxNat.primeCounting' x0) (AxNat.primeCounting' x1))))

Dependencies

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0 direct dependencies 2 direct dependents

Evidence

kernel-Nat.primeCounting'_mono

Kind: kernel-term
Status: checked

Supports: pi'(m) <= pi'(n) whenever m <= n

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- 'Nat.primeCounting'_mono' 2>/dev/null | grep -xFc 'Nat.primeCounting'_mono	3	((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.le x0 x1) -> AxNat.le (AxNat.primeCounting' x0) (AxNat.primeCounting' x1))))')" -ge 1

Evidence notes

`Nat.primeCounting' n` is `Nat.count Nat.isPrime n`, and `Nat.count` is definitionally `Nat.countRange`; both are `Regular` definitions with strictly decreasing delta heights (14 -> 13 -> 12), so the kernel accepts `Nat.countRange_le_of_le` at the predicate `Nat.isPrime` against this stated type by delta alone. The checker matches the FULL RENDERED TYPE rather than the name: a name-only checker is satisfied by a weakened statement, and here the weakening that matters is swapping `primeCounting'` for `primeCounting`, which is a DIFFERENT function (exclusive vs inclusive) with the same monotonicity proof.

footprint-Nat.primeCounting'_mono

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"
}