sqrt(1) = 1 — Axeyum

Identifier: F:nat-sqrt-one
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The floor square root of one is one.

Formal statement

theorem Nat.sqrt_one : Eq.{1} AxNat (AxNat.sqrt (AxNat.succ AxNat.zero)) (AxNat.succ AxNat.zero)

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-Nat.sqrt_one

Kind: kernel-term
Status: checked

Supports: sqrt(1) = 1

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- sqrt_one 2>/dev/null | grep -Ec '^Nat\.sqrt_one[[: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 this row at all is a machine-checked proof. `Nat.sqrt 1` is `Nat.sqrtAux 1 1`: the one fuel step checks `(0 + 1) * (0 + 1) <= 1`, which is `true`, so the accumulator becomes `1` and the (now exhausted) fuel returns it. Every operand is a concrete numeral, so the whole term reduces without any free variable, and the proof is `Eq.refl`. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe and the count is tested, so the exit status depends on what the run found. Controls run 2026-08-28: `sqrt_zero` and `sqrt_one` each exit 0, and `sqrt_oneXX` exits 1.

compute-Nat.sqrt_one

Kind: kernel-term
Status: checked

Supports: the fuel recursion terminates with the right value at n = 1, not with an exhausted fuel

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::sqrt_computes_and_its_boundary_equations_apply

Evidence notes

A fuel-based definition is only correct if the fuel always suffices, and an EXHAUSTED fuel returns whatever accumulator it stopped at. So the computation test is the fuel-sufficiency check: `Kernel::def_eq` reduces `sqrt 0` through `sqrt 9` to `0, 1, 1, 1, 2, 2, 3`, with two negative controls differing by ONE (`sqrt 8 = 3` and `sqrt 9 = 2` must both be rejected) so that `def_eq` cannot pass vacuously.

footprint-Nat.sqrt_one

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 --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration. This matters more than usual here: Mathlib's `Nat.sqrt` is defined by a WELL-FOUNDED Newton's-method iteration, and the Lean equation compiler's route to that carries `Quot.sound`/`propext`. The fuel-based linear-search definition in `nat_prelude/sqrt.rs` was chosen so that this row stays empty, and this command is what makes the claim falsifiable rather than asserted.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane 203-nat-sqrt",
  "source": "one of the two boundary equations `declare_sqrt_all` (`crates/axeyum-lean-kernel/src/nat_prelude/sqrt.rs`) admits alongside the `Nat.sqrtAux`/`Nat.sqrt` definitions"
}