A positive choice from zero is zero

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

Recorded description

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

Formal statement

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

Dependencies

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0 direct dependencies 4 direct dependents

Evidence

kernel-Nat.zero_choose_succ

Kind: kernel-term
Status: checked

Supports: Nat.zero_choose_succ 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 -- zero_choose_succ 2>/dev/null | grep -xFc $'Nat.zero_choose_succ\t1\t((x0 : AxNat) -> Eq.{1} AxNat (AxNat.choose AxNat.zero (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.zero_choose_succ

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: []

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Nat.zero_choose_succ 2>/dev/null | grep -xFc $'nat\tNat.zero_choose_succ\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"
}