Implementing Generalized Algebraic Data Types (GATs) in TypeScript

This challenge asks you to implement a system in TypeScript that mimics the behavior of Generalized Algebraic Data Types (GATs), a powerful feature found in languages like Haskell. GADTs allow you to define types with more sophisticated constructor signatures, enabling more expressive and type-safe data structures, particularly useful for domain-specific languages and advanced pattern matching.

Problem Description

Your goal is to create a TypeScript mechanism that allows users to define data types with constructors that can "return" different types based on the constructor used, similar to GADTs. This involves defining a way to represent these GADTs and then implementing a mechanism for pattern matching against them, ensuring type safety throughout the process.

Key Requirements:

1. GADT Definition: Users should be able to define GADTs where constructors can return specific, distinct types. This is often achieved by associating a "tag" or type parameter with each constructor.
2. Type Safety: The system must guarantee type safety. This means that when pattern matching, the inferred types within each case should be correct according to the constructor that matched.
3. Pattern Matching: Implement a robust pattern matching function that can deconstruct GADT instances and execute different logic based on the constructor.
4. Immutability: GADT instances should be immutable once created.

Expected Behavior:

You will define a set of helper types and functions. When a user defines a GADT, they will use these helpers to construct their GADT definition. A match function will then take an instance of a GADT and a set of handlers (one for each constructor) and return the result of the executed handler. The type system should correctly infer the types within each handler.

Edge Cases:

• Handling GADTs with no constructors (though this is less common).
• Ensuring that the match function requires handlers for all possible constructors to prevent runtime errors.

Examples

Example 1: A simple expression language

Imagine a GADT representing simple arithmetic expressions.

// User defines their GADT using your provided types/functions

// Tagging the return type of each constructor
type ExprTag = 'Literal' | 'Add';

type Expr<T extends ExprTag> =
    T extends 'Literal' ? Literal :
    T extends 'Add' ? Add :
    never; // Should not happen

interface Literal {
    _tag: 'Literal'; // Explicit tag to identify the constructor
    value: number;
}

interface Add {
    _tag: 'Add';
    left: Expr<'Literal' | 'Add'>; // Can contain other expressions
    right: Expr<'Literal' | 'Add'>;
}

// Helper to create instances of Literal
const literal = (value: number): Expr<'Literal'> => ({ _tag: 'Literal', value });

// Helper to create instances of Add
const add = (left: Expr<'Literal' | 'Add'>, right: Expr<'Literal' | 'Add'>): Expr<'Add'> => ({ _tag: 'Add', left, right });

// Sample GADT instances
const expr1: Expr<'Literal'> = literal(5);
const expr2: Expr<'Add'> = add(literal(2), literal(3));
const expr3: Expr<'Add'> = add(expr1, expr2);

// Now, let's assume you have a `match` function

// The `match` function needs to correctly infer the type of the expression
// and provide the correct type for the `value` or `left`/`right` arguments.
// This is the core of the challenge.

/*
// Conceptual usage of your `match` function:
const evaluate = (e: Expr<'Literal' | 'Add'>): number => {
    return match(e, {
        Literal: (literalExpr) => {
            // Inside this handler, literalExpr is correctly typed as Expr<'Literal'>
            return literalExpr.value;
        },
        Add: (addExpr) => {
            // Inside this handler, addExpr is correctly typed as Expr<'Add'>
            return evaluate(addExpr.left) + evaluate(addExpr.right);
        }
    });
};

console.log(evaluate(expr3)); // Expected output: 10
*/

Example 2: A simple state machine

Consider a GADT representing states and transitions.

// User defines their GADT

type StateTag = 'Running' | 'Stopped' | 'Paused';

type State<T extends StateTag> =
    T extends 'Running' ? RunningState :
    T extends 'Stopped' ? StoppedState :
    T extends 'Paused' ? PausedState :
    never;

interface RunningState {
    _tag: 'Running';
    timestamp: number;
}

interface StoppedState {
    _tag: 'Stopped';
    reason: string;
}

interface PausedState {
    _tag: 'Paused';
    duration: number;
}

// Helper to create instances
const running = (timestamp: number): State<'Running'> => ({ _tag: 'Running', timestamp });
const stopped = (reason: string): State<'Stopped'> => ({ _tag: 'Stopped', reason });
const paused = (duration: number): State<'Paused'> => ({ _tag: 'Paused', duration });

const currentState: State<'Running'> = running(Date.now());

// Conceptual usage of your `match` function:
/*
const describeState = (s: State<'Running' | 'Stopped' | 'Paused'>): string => {
    return match(s, {
        Running: (runningState) => {
            // runningState is typed as State<'Running'>
            return `System is running since ${new Date(runningState.timestamp).toLocaleTimeString()}`;
        },
        Stopped: (stoppedState) => {
            // stoppedState is typed as State<'Stopped'>
            return `System stopped due to: ${stoppedState.reason}`;
        },
        Paused: (pausedState) => {
            // pausedState is typed as State<'Paused'>
            return `System paused for ${pausedState.duration} seconds.`;
        }
    });
};

console.log(describeState(currentState));
*/

Constraints

• The GADT definition mechanism should be purely TypeScript. No runtime code generation is expected beyond standard TypeScript constructs.
• The match function must be strongly typed. The compiler should enforce that all possible GADT constructors have a corresponding handler.
• Performance of the match function should be efficient, ideally with minimal runtime overhead compared to manual if/else or switch statements.
• The GADT constructors should return specific, distinct types.

Notes

• Consider how to represent the "tag" of each constructor. A common approach is a unique _tag property.
• The core of this challenge lies in creating a generic match function that leverages TypeScript's conditional types and mapped types to infer the correct types within each handler.
• Think about how to map the _tag (or equivalent identifier) to the correct handler function and the corresponding GADT instance type.
• You'll likely need to define a union of all possible GADT tags for the GADT definition.