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Cheat sheet

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# Cheat sheet

The whole language on one page. Every `law` block below belongs to one
package: put them together in `parking.law` and it compiles; the scenarios
at the end run against it. Each section links to the page that explains the
construct in depth.

<p class="play-open"><a href="/play/?preset=cheat-sheet">Open this package in the playground →</a> — edit the rules and the scenarios of this page and run them in your browser.</p>

## Package header

```law
language "law.core" version "0.2";
package demo.parking version "0.1.0";
namespace "urn:law:demo:parking";
```

Three lines open every package: the language version, the package name and
version, and the namespace that makes every name globally unique.

## Vocabulary

```law
entity Applicant;
entity Office;

enum Zone {
    central;
    outer;
}

relation resident(a: Applicant) kind institutional;
relation vehicle_registered(a: Applicant) kind institutional;
relation permit_eligible(a: Applicant) kind institutional;
relation permit_suspended(a: Applicant) kind institutional;
relation outstanding_fines(a: Applicant) kind institutional;
relation disability_badge(a: Applicant) kind institutional;
relation fraud_flag(a: Applicant) kind institutional;
relation household_member(a: Applicant, m: Applicant) kind empirical;
relation large_household(a: Applicant) kind institutional;
relation lives_in(a: Applicant, z: Zone) kind empirical;
relation application_filed(a: Applicant, on: Date) kind empirical { key(a); }
relation decision_due(a: Applicant, due: Date) kind institutional;
relation decision_notified(a: Applicant) kind empirical;
relation resells_permit(a: Applicant) kind empirical;
relation revocation_notice(o: Office, a: Applicant) kind empirical;
relation permit_revoked(a: Applicant) kind institutional;
relation permit_office(o: Office) kind institutional;
```

| Form | Meaning |
|---|---|
| `entity Name;` | a kind of thing the norm talks about |
| `enum Name { a; b; }` | a closed list of values |
| `relation name(x: Type, …) kind …;` | a statement that can be true, false, both or unknown |
| `kind institutional` | true because a rule or an authority says so |
| `kind empirical` | true because it happened; comes from evidence |
| `{ key(a); }` | at most one record per key: a second value is a conflict |

## Values

```law
const MONTHLY_RATE: Money = 10 EUR;
const LARGE_HOUSEHOLD: Integer = 4;

function permit_fee(months: Integer) -> Money = months * MONTHLY_RATE;
```

Constants and pure functions. Money carries its currency (`10 EUR`),
percentages are written `25 percent`, dates `@2026-03-01`, periods
`30 calendar_day`.

## Rules

```law
rule PermitEligibility defeasible {
    for a: Applicant;
    when resident(a) and vehicle_registered(a);
    then permit_eligible(a);
    unless permit_suspended(a);
}
```

| Strength | When to use it |
|---|---|
| `strict` | no exception is possible; `unless` on it is an error (`LDC-E4110`) |
| `defeasible` | the general rule that exceptions may defeat |
| `defeater` | an exception that withdraws a conclusion without asserting the opposite |

`for` binds variables, `when` is the condition, `then` the conclusion,
`unless` a proviso that blocks this rule only. A conclusion `not p(x)` is a
refusal: it supports the opposite. See
[Your first rule](/language/first-rule/) and
[Exceptions and priorities](/language/exceptions-and-priorities/).

## Exceptions and priority

```law
rule FinesRefusal defeasible {
    for a: Applicant;
    when outstanding_fines(a);
    then not permit_eligible(a);
}

rule BadgeEligibility defeasible {
    for a: Applicant;
    when disability_badge(a);
    then permit_eligible(a);
}

priority RefusalOverEligibility {
    prefer FinesRefusal over PermitEligibility;
    reason lex_specialis;
}

rule FraudBlocksBadge defeater {
    for a: Applicant;
    when fraud_flag(a);
    defeat permit_eligible(a);
}
```

| You want | Write |
|---|---|
| an exception inside one rule | `unless …;` |
| a refusal | a rule with `then not p(x);` |
| one rule to beat another | `priority … { prefer A over B; reason …; }` |
| to cancel support without refusing | a `defeater` rule with `defeat p(x);` |

Two rules with opposite conclusions and no priority give `BOTH`: the
conflict is reported, not silently resolved.

## Definitions, negation, aggregates

```law
definition central_resident(a: Applicant) exact {
    when resident(a) and lives_in(a, central);
}

rule LargeHousehold strict {
    for a: Applicant;
    when resident(a)
        and count(collect m: Applicant where household_member(a, m)) >= LARGE_HOUSEHOLD;
    then large_household(a);
}
```

| In a condition | Means |
|---|---|
| `p(x)` | `p(x)` is established |
| `not p(x)` | the opposite of `p(x)` is established |
| `not_known(p(x))` | nothing establishes `p(x)` |
| `count(collect m: T where …)` | how many values satisfy the condition |
| `sum(collect all v: Money, m: T where …)` | the total of the values |

`definition` names a concept ("who counts as …"); the explanation then says
"by definition". Missing and refuted facts are different answers — see
[Missing and conflicting facts](/language/missing-and-conflicting/).

## Closed lists

```law
relation applicant_on_file(a: Applicant) kind institutional;

closure ResidentsRegister {
    predicate resident;
    domain applicant_on_file;
    snapshot "urn:snapshot:demo-parking-residents:2026";
    complete_as_of @2026-01-01T00:00:00Z;
    derive_explicit_negative true;
}
```

By default silence proves nothing: a missing record gives `NEITHER`. A
`closure` says the register is complete for a domain at a moment, so inside
that domain a missing record becomes a denial (`FALSE_ONLY`); outside it,
silence stays silence.

## Time and deadlines

```law
deadline policy CALENDAR_DAYS {
    start_count next_day;
    include_end true;
    roll no_roll;
}

rule DecisionDeadline strict {
    for a: Applicant;
    for on: Date;
    when application_filed(a, on);
    then decision_due(a, add_calendar_period(on, 30 calendar_day));
}
```

| Form | Meaning |
|---|---|
| `add_calendar_period(d, 30 calendar_day)` | calendar arithmetic; needs a deadline policy in the case |
| `add_business_days(d, n)` | working days; also needs the official calendar |
| `effective [@2026-01-01, infinity);` | inside a rule: when the rule is in force |
| `deadline policy … { … }` | how to count: start, end, rolling over holidays |

Without a policy the deadline is not guessed: the answer is
`MISSING_POLICY`. See [Sources and legal time](/language/sources-and-legal-time/).

## Sources and editions

```law
source PARKING_RULES {
    kind municipal_act;
    jurisdiction NORTHBRIDGE;
    number "2026-1";
    label en official "Northbridge Residential Parking Rules";
}

edition PARKING_RULES_2026 of PARKING_RULES {
    language en;
    officiality official;
    adopted @2025-12-01;
    in_force [@2026-01-01, infinity);
    materialization_status PINNED_OFFICIAL_BYTES;
}

publication PARKING_RULES_2026_TEXT of PARKING_RULES_2026 {
    media_type "text/plain; charset=utf-8";
    uri "urn:demo:parking:rules:2026";
    retrieved_at @2026-01-05T09:00:00Z;
    content_hash "sha256:9060e85e1fd30cd22b8a9f618afb8aabd15ae356b23b47f86ab1689b6aacb660";
    local_path "sources/parking-rules-2026.txt";
}

fragment PARKING_RULES_2026_ART2 in PARKING_RULES_2026 {
    kind article;
    locator "article/2";
    label en official "Article 2. Eligibility";
    text en official """A residential parking permit is issued to an applicant who is a resident of Northbridge and has a vehicle registered at their address.
""";
    content_hash "sha256:e309c8a75c1eab992dd0ad66ff477f9111bf99651e167ce1de6a651129a758be";
}

relation eligibility_notice(a: Applicant) kind institutional;

@source(PARKING_RULES_2026_ART2)
rule EligibilityNotice defeasible {
    effective [@2026-01-01, infinity);
    for a: Applicant;
    when permit_eligible(a);
    then eligibility_notice(a);
}
```

| Form | Meaning |
|---|---|
| `source` | the act itself: kind, jurisdiction, number |
| `edition … of …` | one version of the act, with its dates in force |
| `publication … of …` | the official bytes of an edition, pinned by hash |
| `fragment … in …` | an article or clause, with its text and hash |
| `@source(FRAGMENT)` | anchors a rule to the text it formalizes |
| `effective [from, to);` | when the rule applies; outside it the rule does not fire |

The answer cites the fragment, and the hash proves which text was used. See
[Sources and legal time](/language/sources-and-legal-time/).

## Duties, prohibitions, powers

```law
rule DecisionDuty strict {
    for a: Applicant;
    for o: Office;
    for on, due: Date;
    when application_filed(a, on) and decision_due(a, due) and permit_office(o);
    then duty NotifyDecision {
        bearer o;
        beneficiary a;
        goal achievement {
            condition decision_notified(a);
            window [on, due];
        }
    };
}

rule NoResale strict {
    for a: Applicant;
    for o: Office;
    when permit_eligible(a) and permit_office(o);
    then prohibition NoPermitResale {
        bearer a;
        beneficiary o;
        action resells_permit(a);
        window [@2026-01-01, @2026-12-31];
    };
}

rule RevocationPower strict {
    for a: Applicant;
    for o: Office;
    when permit_office(o) and permit_eligible(a);
    then power RevokePermit {
        holder o;
        over a;
        exercise revocation_notice(o, a);
        valid_when (resells_permit(a));
        effect create(permit_revoked(a));
    };
}
```

| Position | Says | Key fields |
|---|---|---|
| `duty` | the bearer must bring something about | `bearer`, `beneficiary`, `goal`, `window` |
| `prohibition` | the bearer must not do something | `bearer`, `action`, `window` |
| `power` | the holder can change the legal situation | `holder`, `over`, `exercise`, `valid_when`, `effect` |

The engine reports each position's state: `ACTIVE`, `FULFILLED`,
`VIOLATED` and others, with the facts behind it.

## Packages and imports

```law
pub relation permit_holder(a: Applicant) kind institutional;
```

Only declarations marked `pub` are visible to other packages. A package
that uses another one names it in `law.toml` and imports it; foreign names
are written with the package prefix. These lines are not part of this
page's package:

```toml
[dependencies]
"demo.parking" = "0.1.0"
```

```law
import demo.parking version "0.1.0";

rule ResidentDiscount strict {
    for a: demo.parking::Applicant;
    when demo.parking::permit_holder(a);
    then discount(a);
}
```

`law add demo.parking@0.1.0` writes the dependency and the lock for you.
Referring to a name that is not `pub` fails with `LDC-E1105`.

## Asking questions

Questions are asked in a scenario (`.lawtest`) or with `law ask`.

| Question | Form | Answer |
|---|---|---|
| is it true? | `evaluate truth(p(x));` | a truth status |
| who? | `evaluate collect a: Applicant where p(a);` | a list |
| how much? | `evaluate permit_fee(3);` | a value |
| when? | `evaluate truth(decision_due(x, @2026-04-02));` | a truth status for the date |
| who owes what? | `evaluate positions();` | positions and their states |

## Four answers

| Status | Means |
|---|---|
| `TRUE_ONLY` | established, nothing against |
| `FALSE_ONLY` | refuted, nothing for |
| `BOTH` | grounds on both sides: a conflict the package does not resolve |
| `NEITHER` | not established, not refuted: something is missing |

Next to the truth status every answer carries an `evaluation_status`:
`COMPUTED` when the engine had everything it needed, or a reason such as
`MISSING_POLICY` when it did not.

## Scenarios

```law
test "the general rule" {
    given {
        context { legal_time @2026-03-01; decision_time @2026-03-01T09:00:00Z; knowledge_time @2026-03-01T09:00:00Z; timezone "UTC"; }
        assert resident(entity_ref("urn:demo:parking:ann")) { id "resident-ann"; origin case_input; }
        assert vehicle_registered(entity_ref("urn:demo:parking:ann")) { id "vehicle_registered-ann"; origin case_input; }
    }
    evaluate truth(permit_eligible(entity_ref("urn:demo:parking:ann")));
    expect truth_status == TRUE_ONLY;
    expect applied(PermitEligibility);
}
```

| Part | Holds |
|---|---|
| `context { … }` | legal time, decision time, knowledge time, timezone, deadline policy |
| `assert p(x) { id "…"; origin case_input; }` | one fact with its identity and origin |
| `assert not p(x) { … }` | a stated denial |
| `evaluate …;` | the question |
| `expect …;` | `truth_status ==`, `evaluation_status ==`, `applied(Rule)`, `value ==`, `collected(…)`, `position(Name, STATE)` |

More in [Testing a package](/language/testing-a-package/).

## Commands

```bash
law init my-package
law engine check parking.law
law engine test tests/scenarios.lawtest --program parking.law
law engine fmt parking.law
law ask my-case --query 'evaluate truth(permit_eligible(entity_ref("urn:demo:parking:ann")));' --query-id q1
law explain LDC-E4110
law codegen demo.parking --target ts --out gen --name parking --version 0.1.0
```

| Command | Does |
|---|---|
| `law init` | a new package or case |
| `law engine check` | compile and report diagnostics |
| `law engine test` | run scenarios against the package |
| `law engine fmt` | format the source |
| `law ask` | ask a question of a case; the answer comes with its proof and hashes |
| `law explain LDC-E…` | what a diagnostic means and how to fix it |
| `law codegen` | a TypeScript, Python or Go module of the package, no engine needed |

## Common errors

Each row is checked: the page is changed as described, and the compiler
must report exactly this code.

| Code | Cause | Fix |
|---|---|---|
| `LDC-E0201` | a missing `;` or rule strength | end declarations and clauses with `;`; specify the rule strength |
| `LDC-E0203` | a keyword used as a name (`entity rule;`) | pick another name |
| `LDC-E1201` | the same name declared twice | rename one of them |
| `LDC-E1305` | a duty without `bearer` | name who owes the duty |
| `LDC-E1308` (warning) | `defeat` in a non-defeater rule | make the rule a `defeater` |
| `LDC-E1330` | a name in the conclusion that nothing binds | bind it with `for` and use it in `when` |
| `LDC-E2101` | an unknown type | declare it with `entity`, `enum` or `type` |
| `LDC-E2102` | a relation that is not declared | declare it, or fix the spelling |
| `LDC-E2103` | wrong number of arguments | match the declaration |
| `LDC-E4101` | a variable bound only under `not` | bind it by a positive condition first |
| `LDC-E4108` | a priority names a rule that does not exist | fix the rule name |
| `LDC-E4110` | `unless` on a `strict` rule | make the rule `defeasible` |

`law explain LDC-E…` prints the full card for any code.

## Types

<!-- generated: cheat-sheet types -->

| Type | What it holds |
|---|---|
| `Algebraic` | An exact algebraic real number: a minimal polynomial plus the index of its real root. |
| `Boolean` | A two-valued data type: `true` or `false`. |
| `Bounds` | A closed rational interval bundled with the derivation that produced it. |
| `Currency` | The currency of a `Money` amount — the code in a literal such as `1000000 KZT`. |
| `Date` | A calendar date, written as `@2008-03-01`. |
| `Decimal` | An exact decimal number, written as a literal such as `0.20`. |
| `Duration` | A physical duration in seconds and nanoseconds. |
| `Instant` | A point on the time axis, written with an offset: `@2026-03-31T18:00:00+05:00`. |
| `Integer` | A whole number. |
| `Jurisdiction` | The type of a jurisdictional binding: a code that names a legal order. |
| `LanguageTag` | A BCP-47 language tag such as `kk-KZ`. |
| `List` | An ordered collection that allows duplicates. |
| `Magnitude` | A value whose dimension is computed from the unit registry, with product, quotient and conversion. |
| `Money` | An amount in a named currency, written as a literal such as `1000000 KZT`. |
| `Number` | The single exact numeric domain of 0.4 programs: every numeric spelling is one value, carried as an irreducible fraction. |
| `Option` | An optional value: there is no `null`, absence is `Option<T>`. |
| `Quantity` | A number with a unit tag, written as a literal such as `300 km`. |
| `Rational` | An exact rational value: the result type of exact division, serialized as an irreducible fraction. |
| `RealExpr` | A symbolic real expression over exact leaves and named constants such as `pi()` and `ln2()`. |
| `Set` | An unordered collection without duplicates. |
| `Text` | A text value; text is NFC-normalized. |
| `Timezone` | An IANA time zone name such as `Asia/Qyzylorda`. |

<!-- /generated -->

## Standard library

<!-- generated: cheat-sheet std -->

| Function | What it does |
|---|---|
| `add_business_days(after, count)` | Steps a date by counted days under the calendar snapshot and the deadline policy. |
| `add_calendar_period(after, count)` | Steps a date by a calendar period: days, weeks, months or years. |
| `add_duration(after, duration)` | The instant that lies a physical duration after `after`. |
| `add_legal_term(date, term)` | Steps a date by a legal term, choosing the operation by the unit of the term. |
| `angle_convert(value, from, to, policy)` | Exact angle conversion under an explicit versioned policy. |
| `bounds_add(a, b)` | Interval sum: `[a.lo + b.lo, a.hi + b.hi]`. |
| `bounds_const(num, den)` | The degenerate interval `[f, f]` for the exact number `f = num/den`. |
| `bounds_div(a, b)` | Interval quotient of two bounds. |
| `bounds_mul(a, b)` | Interval product of two bounds. |
| `bounds_scale(b, num, den)` | Multiplies an interval by the exact factor `num/den`. |
| `bounds_sub(a, b)` | Interval difference: `[a.lo - b.hi, a.hi - b.lo]`. |
| `convert(value, num, den, unit)` | Converts a `Quantity` to another unit by the exact factor `num/den`. |
| `cos_bounds(x, profile)` | Certified bounds of the cosine of `x`. |
| `count(xs)` | Counts the elements of a collection; an empty collection gives 0. |
| `days_between(from, to)` | Number of calendar days from `from` to `to`; negative when `to` precedes `from`. |
| `div_round(dividend, divisor, precision, mode)` | Divides and rounds the exact quotient to a precision under a named mode. |
| `exact_sqrt(value)` | Exact square root of a rational argument; takes no precision profile. |
| `exp_bounds(x, profile)` | Certified bounds of `e` to the power `x`, for any sign of `x`. |
| `hours_between(from, to)` | Number of full hours between two instants; negative when the order is reversed. |
| `ln_bounds(x, profile)` | Certified bounds of the natural logarithm of `x`. |
| `ln2()` | The constant ln 2 as an exact symbolic value, without rounding. |
| `magnitude(q)` | Lifts a `Quantity` into the dimension algebra as a `Magnitude`. |
| `max(xs)` | The largest value of a collection. |
| `min(xs)` | The smallest value of a collection. |
| `minutes_between(from, to)` | Number of full minutes between two instants, with the sign and truncation of `hours_between`. |
| `month_end_of(d)` | The last day of the month containing the date. |
| `month_start_of(d)` | The first day of the month containing the date. |
| `pi()` | The constant π as an exact symbolic value, without rounding. |
| `pi_bounds(profile)` | Certified bounds of π at the precision of a profile. |
| `quantity_of(m, unit)` | Turns a `Magnitude` back into a `Quantity` of the named unit. |
| `round(value, precision, mode)` | Rounds an already computed value to a precision under a named mode. |
| `round_bounds(bounds, precision, mode)` | Rounds certified bounds to a single value at a precision. |
| `scalar_of(m)` | A magnitude of zero dimension as an exact `Rational`. |
| `sin_bounds(x, profile)` | Certified bounds of the sine of `x`. |
| `solve_linear(a, b)` | The root of `a·x + b = 0` over the rationals. |
| `sqrt_bounds(x, profile)` | Certified bounds of the square root of `x`. |
| `sum(xs)` | Exact sum of a collection of `Decimal`, `Money` or `Quantity` values. |
| `temp_convert(value, from, to, role, policy)` | Exact temperature conversion under an explicit versioned policy. |
| `text_length(t)` | Number of code points of a text after NFC normalization. |
| `unit_convert(m, unit)` | Converts a `Magnitude` to another unit of the same dimension. |
| `unit_exponent(m, unit)` | The exponent of a simple unit in the factors of a magnitude, as an exact `Integer`. |
| `weekday_of(d)` | Day of week of a date as an integer from 1 (Monday) to 7 (Sunday). |
| `with_unit(value, unit)` | Turns a dimensionless number into a `Quantity` of the named unit. |
| `year_end_of(d)` | December 31 of the calendar year containing the date. |
| `year_start_of(d)` | January 1 of the calendar year containing the date. |

<!-- /generated -->

## Run it

```bash
law engine check parking.law
law engine test tests/cheat-sheet.lawtest --program parking.law
```

Expected output:

```text
check OK: parking.law
```

```text
test PASS: the general rule
test PASS: unless withdraws the conclusion
test PASS: the refusal wins by priority
test PASS: no priority: the conflict is kept
test PASS: a defeater cancels support
test PASS: a definition
test PASS: an aggregate
test PASS: a value
test PASS: a deadline
test PASS: no policy, no deadline
test PASS: positions
test PASS: closure: on file without a record means not a resident
test PASS: closure: outside the domain silence stays silence
test PASS: an anchored rule applies while in force
test PASS: before it is in force the rule does not fire
```

Each section of this page has a scenario of its own; the one under
[Scenarios](#scenarios) is the first of fifteen. To try a construct, change
the facts of a scenario and run it again.