Rational addition renormalises and negation is an additive inverse

Identifier: F:rat-add-neg-inverse
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

Rat.add returns the reduced representative of a sum (1/6 + 1/3 is 1/2, reached as 9/18), Rat.neg is an involution that moves its argument, and x + (-x) is 0. Negation alone needs no renormalisation, because negating a numerator does not change the magnitude the reducedness field constrains.

Formal statement

def Rat.add : ((x0 : Rat) -> ((x1 : Rat) -> Rat))

Dependencies

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1 direct dependencies 1 direct dependents

Evidence

kernel-rat-add-neg-inverse

Kind: kernel-term
Status: checked

Supports: 1/6 + 1/3 is 1/2 and is not 1/3; neg is an involution on 1/2 and does move it; 1/2 + (-1/2) is 0.

Checker command

test "$(cargo test -p axeyum-lean-kernel --lib rat_add_renormalises_and_neg_is_an_involution 2>&1 | grep -Ec '^test result: ok\. 1 passed')" -ge 1

Evidence notes

Bound to this test by name, with `1 passed` required so an empty filter cannot pass. The `1/2 + (-1/2) = 0` case is the one that matters most: it is the only check here that drives a genuinely NEGATIVE numerator through Rat.normalize's negSucc branch, which the ofNat-only cases never reach.

footprint-rat-add-neg-inverse

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- Rat is built in the integer environment, whose trusted surface is empty

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free integer

Evidence notes

Rat and its operations are admitted into the same environment as the integer development, so an empty trusted surface bounds their footprints. Environment-wide on purpose: a declaration cannot depend on a trusted declaration the environment does not contain.

Provenance

{
  "date": "2026-08-16",
  "established_by": "axeyum-lean-kernel int_prelude::rat (rational-scope lane)",
  "source": "hand-built foundational library"
}