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nb-23 solutions — Numbers and the standard library

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# nb-23 solutions — Numbers and the standard library

*Checkable against the standalone probe below (`law engine test
probe.lawtest --program probe.law`: 2/2 pass) and, after adding the
rule to the package, `law test packs/examples/language-demo/numerics`
(67 + 2 checked, all passed). Tool `law 0.1.0`, language `0.2`,
semantics `law.core/0.2`, std `0.2.0`. Identifiers and code as
written.*

## Exercise: late-payment split

**Answer: the four-way split fires at 470 KZT; the three-way split
carries `INEXACT_DIVISION` instead of a rounded amount.**

`1880 / 4 = 470` is exact, so `late_quarter() == 470 KZT` holds and
the rule fires (`TRUE_ONLY`). `1880 / 3` is not 10-smooth, so the
machine refuses the quiet rounding and the evaluation carries the
coded `INEXACT_DIVISION` issue — the same discipline as section 5's
`monthly`/`baddiv` pair, at a different amount. Add to the package:

```law
pure function late_quarter() -> Money = 1880 KZT / 4.0;
pure function late_third() -> Money = 1880 KZT / 3.0;
relation late_ok() kind institutional;
rule LateOk strict {
    when late_quarter() == 470 KZT;
    then late_ok();
}
```

Two tests (append after the file header, standalone shape as in the
article's Standalone B):

```law
test "late quarter is exact" {
    given {
        context { legal_time @2026-03-02; decision_time @2026-03-02T09:00:00Z; knowledge_time @2026-03-02T09:00:00Z; timezone "UTC"; }
    }
    evaluate truth(late_ok());
    expect truth_status == TRUE_ONLY;
}
test "late third is loud" {
    given {
        context { legal_time @2026-03-02; decision_time @2026-03-02T09:00:00Z; knowledge_time @2026-03-02T09:00:00Z; timezone "UTC"; }
    }
    evaluate late_third();
    expect issue(INEXACT_DIVISION);
}
```

Verified 2026-10-03 as a standalone probe: `test PASS: late quarter
is exact`, `test PASS: late third is loud`, `lawc test: 2/2 тестов
прошли`. The divisor stays `Decimal` (`/ 4`, with an `Integer`
divisor, is refused at check time with `LDC-E2108`); the mixed
`1880 KZT + 10 EUR` variant would stay `NEITHER` like the article's
`mixed()` rule and would NOT satisfy part (b), which asks for the
loud issue, not silence.

## Refusal reading

Both probes refuse with the same two codes, exit 1:
`LDC-E2105` (name resolution — the term is neither in the file
nor in the std v1-list) and `LDC-E2403` (computability proof —
without an effect descriptor for the callee, `f` cannot be
proved `PURE_DETERMINISTIC`). Switching the `ln2` probe's
return type to `RealExpr` adds a third diagnostic,
`LDC-E2101` at 4:17 (`тип "RealExpr" не разрешается`), while
the two term codes stay: the type layer refuses independently
of the term layer. Verified 2026-10-03: three `error` lines,
exit 1. If you predicted the type change would remove a code,
re-read [§4](/tutorials/northbridge/solutions/nb-04-solutions/#4-matchedcount18--value-and-what-does-it-say) — each layer reports its own missing piece.