A binomial coefficient above the diagonal is zero

Identifier: F:nat-choose-succ-self-eq-zero
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every natural n: C(n, n+1) = 0.

Formal statement

theorem Nat.choose_succ_self_eq_zero : ((x0 : AxNat) -> Eq.{1} AxNat (AxNat.choose x0 (AxNat.succ x0)) AxNat.zero)

Dependencies

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2 direct dependencies 3 direct dependents

Evidence

kernel-Nat.choose_succ_self_eq_zero

Kind: kernel-term
Status: checked

Supports: Nat.choose_succ_self_eq_zero is admitted by the trusted kernel gate with the exact type recorded in formal.statement.

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- choose_succ_self_eq_zero 2>/dev/null | grep -xFc $'Nat.choose_succ_self_eq_zero\t1\t((x0 : AxNat) -> Eq.{1} AxNat (AxNat.choose x0 (AxNat.succ x0)) AxNat.zero)')" -ge 1

Evidence notes

The inventory builds the Nat prelude and prints Kernel::render_lean of the admitted type; the exact-line check fails on a missing declaration or type drift.

footprint-Nat.choose_succ_self_eq_zero

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: []

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Nat.choose_succ_self_eq_zero 2>/dev/null | grep -xFc $'nat\tNat.choose_succ_self_eq_zero\t0\t')" -ge 1

Evidence notes

The theorem-specific kernel footprint enumeration reports no trusted dependency.

Provenance

{
  "date": "2026-08-26",
  "established_by": "axeyum-lean-kernel build_nat_prelude",
  "source": "theorem name, canonical type, direct dependencies, and footprint read from kernel inventory examples after the ranked compatibility census exposed the missing fact link"
}