The setoid ordered ring's additive unit law is right-handed, so the left one is a theorem

Identifier: F:algs-orderedring-zero-add
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

Let R be an AlgS.OrderedRing and a an element of R. Then 0 + a is equivalent to a. The record carries addZero (a + 0 ~ a) as a field and not this; it follows from addComm against addZero in two steps. It is needed once per summand by ADR-1631's Bernoulli computation, because every sum that reduces out of AlgS.OrderedRing.sumRange begins with a literal zero on the LEFT.

Formal statement

theorem AlgS.OrderedRing.zero_add : forall (R : AlgS.OrderedRing) (a : R.carrier), R.equiv (R.add R.zero a) a

Dependencies

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0 direct dependencies 3 direct dependents

Evidence

kernel-axiom-free

Kind: kernel-term
Status: checked

Supports: The declaration is admitted by the trusted kernel gate with an empty Kernel::axiom_footprint, together with every other declaration of ADR-1631's generic Bernoulli layer. The test asserts Environment::contains FIRST, because an absent name has an empty footprint too, and it pins its own name-list length, so a declaration silently dropped from the layer fails it.

Checker command

cargo test --release -p axeyum-lean-kernel --lib rat_prelude::binomial_s::binomial_s_tests::binomial_s_declarations_are_axiom_free -- --exact 2>&1 | grep -c '^test result: ok. 1 passed'

Evidence notes

grep -c, not grep -q: -q would SIGPIPE the producer under pipefail, and -c exits 1 printing 0 when the line is absent, so a suite that compiled to zero tests and exited 0 is caught rather than counted as a pass. --release is mandatory: the rational prelude SIGABRTs on the default debug thread stack.

Provenance

{
  "date": "2026-09-05",
  "established_by": "lane binomial-hoeffding (roadmap W3-12, ADR-1631)"
}