Division with remainder always exists for a positive divisor

Identifier: F:nat-div-mod-exists
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every natural number n and every divisor d with 1 <= d, there exist a quotient q and a remainder r such that n = d*q + r and r < d.

Formal statement

theorem Nat.div_mod_exists : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.le (AxNat.succ AxNat.zero) x0) -> Exists.{1} AxNat (fun (x3 : AxNat) => Exists.{1} AxNat (fun (x4 : AxNat) => AxNat.divMod x0 x1 x3 x4)))))

Dependencies

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1 direct dependencies 5 direct dependents

Evidence

kernel-Nat.div_mod_exists

Kind: kernel-term
Status: checked

Supports: For every natural number n and every divisor d with 1 <= d, there exist a quotient q and a remainder r such that n = d*q + r and r < d.

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing the row at all is a machine-checked proof. TIGHTENED 2026-08-16: the command was `cargo test -p axeyum-lean-kernel --lib nat_prelude`, a whole-suite run that passes or fails identically for every fact citing it and would stay green if THIS theorem were deleted. It now names its own subject twice over -- `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -q` requires the admitted declaration to be printed.

footprint-Nat.div_mod_exists

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` enumerates the built Nat environment and reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0` (re-run 2026-08-14 by this lane). That is the whole environment, not this theorem, and that is what makes it sound as a footprint: a theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual footprint by []. Counting `Declaration::Axiom` alone would NOT have sufficed -- `Opaque` has no proof body and `Quotient` admits `Quot.sound`, one of Lean's own axioms -- so the enumeration covers all three.

Provenance

{
  "date": "2026-08-14",
  "established_by": "axeyum-lean-kernel build_nat_prelude, re-measured by lane facts-number",
  "source": "proposition extracted from the S:number / S:arithmetic-structure strands of the math-education concept graph; statement authored here, nothing copied"
}