// Define a type for the elements of the group
type GroupElement = number;

// Define the group interface
interface Group {
  // Binary operation
  operation(x: GroupElement, y: GroupElement): GroupElement;

  // Identity element
  identity: GroupElement;

  // Inverse operation
  inverse(x: GroupElement): GroupElement;
}

// Define a function to check if an object satisfies the Group interface
function isGroup(object: any): object is Group {
  return (
    typeof object.operation === 'function' &&
    typeof object.identity !== 'undefined' &&
    typeof object.inverse === 'function'
  );
}

// Define a concrete implementation of a group: integers under addition
const integerAdditionGroup: Group = {
  operation: (x, y) => x + y,
  identity: 0,
  inverse: x => -x,
};

// Test the properties of the group
function testGroupProperties(group: Group): boolean {
  const { operation, identity, inverse } = group;
  // Closure property
  const closure = (a: GroupElement, b: GroupElement) => isGroup(operation(a, b));
  // Associativity property
  const associativity = (a: GroupElement, b: GroupElement, c: GroupElement) =>
    operation(a, operation(b, c)) === operation(operation(a, b), c);
  // Identity property
  const identityElement = (a: GroupElement) => operation(a, identity) === a && operation(identity, a) === a;
  // Inverse property
  const inverseElement = (a: GroupElement) => operation(a, inverse(a)) === identity && operation(inverse(a), a) === identity;
  // Test properties for a range of elements
  for (let i = -100; i <= 100; i++) {
    if (!closure(i, i) || !associativity(i, i, i) || !identityElement(i) || !inverseElement(i)) {
      return false;
    }
  }
  return true;
}

// Test if integers under addition form a group
console.log(testGroupProperties(integerAdditionGroup)); // Output: true

type JsonValue = string | number | boolean | null | JsonObject | JsonArray;
type JsonObject = { [key: string]: JsonValue };
type JsonArray = JsonValue[];

// Define a monoid interface for JSON values
interface JsonMonoid {
  empty: JsonValue;
  concat: (x: JsonValue, y: JsonValue) => JsonValue;
}

// Define monoids for each JSON data type
const stringMonoid: JsonMonoid = {
  empty: "",
  concat: (x, y) => `${x}${y}`,
};

const numberMonoid: JsonMonoid = {
  empty: 0,
  concat: (x, y) => x + y,
};

const booleanMonoid: JsonMonoid = {
  empty: false,
  concat: (x, y) => x || y,
};

const nullMonoid: JsonMonoid = {
  empty: null,
  concat: (_, y) => y,
};

const objectMonoid: JsonMonoid = {
  empty: {},
  concat: (x, y) => {
    const result: JsonObject = { ...x };

    for (const key in y) {
      if (y.hasOwnProperty(key)) {
        result[key] = key in result ? objectMonoid.concat(result[key], y[key]) : y[key];
      }
    }

    return result;
  },
};

const arrayMonoid: JsonMonoid = {
  empty: [],
  concat: (x, y) => [...x, ...y],
};

// Define a function to recursively apply the monoid operation based on the JSON value type
const mergeJsonValues = (x: JsonValue, y: JsonValue): JsonValue => {
  if (typeof x !== typeof y) {
    throw new Error("Incompatible types");
  }

  if (typeof x === "object" && x !== null && Array.isArray(x) === Array.isArray(y)) {
    // Recursively merge arrays and objects
    return Array.isArray(x) ? arrayMonoid.concat(x, y as JsonArray) : objectMonoid.concat(x as JsonObject, y as JsonObject);
  } else {
    // Delegate to the appropriate monoid based on the data type
    switch (typeof x) {
      case "string":
        return stringMonoid.concat(x as string, y as string);
      case "number":
        return numberMonoid.concat(x as number, y as number);
      case "boolean":
        return booleanMonoid.concat(x as boolean, y as boolean);
      case "object":
        return nullMonoid.concat(x as null, y as null);
    }
  }
};

// Test the monoid
const obj1: JsonObject = { a: { b: 1 }, c: 2 };
const obj2: JsonObject = { a: { d: 3 }, e: 4 };

const result = objectMonoid.concat(obj1, obj2);
console.log(result); // Output: { a: { b: 1, d: 3 }, c: 2, e: 4 }

class ClockGroup {
    private readonly size: number;
    private readonly hours: number[];

    constructor(size: number) {
        this.size = size;
        this.hours = Array.from({ length: size }, (_, i) => i);
    }

    add(clock1: number, clock2: number): number {
        return (clock1 + clock2) % this.size;
    }

    subtract(clock1: number, clock2: number): number {
        return (clock1 - clock2 + this.size) % this.size;
    }

    identity(): number {
        return 0;
    }

    inverse(clock: number): number {
        return (this.size - clock) % this.size;
    }

    print(): void {
        console.log("Clock Group:");
        console.log("Elements:", this.hours);
        console.log("Size:", this.size);
    }
}

// Example usage
const clockGroup = new ClockGroup(12);
clockGroup.print();

console.log("Addition (5 + 7):", clockGroup.add(5, 7)); // Output: 0 (12-hour clock wraps around)
console.log("Subtraction (7 - 5):", clockGroup.subtract(7, 5)); // Output: 2
console.log("Identity element:", clockGroup.identity()); // Output: 0
console.log("Inverse of 8:", clockGroup.inverse(8)); // Output: 4

// Define a type for the colors of the Rubik's Cube faces
type CubeColor = 'red' | 'blue' | 'green' | 'orange' | 'white' | 'yellow';

// Define a type for the Rubik's Cube face
interface CubeFace {
    color: CubeColor;
}

// Define a type for the Rubik's Cube
type RubiksCube = CubeFace[][][];

// Define a Group interface
interface Group<T> {
    operation(a: T, b: T): T;
    identity: T;
    inverse(a: T): T;
}

// Define the Rubik's Cube group interface that extends the Group interface
interface RubiksCubeGroup extends Group<CubeRotationCommand> {
    // No additional methods needed as rotation commands directly represent cube rotations
}

// Define a rotation command interface
interface CubeRotationCommand {
    axis: 'x' | 'y' | 'z'; // Axis of rotation
    layer: number; // Layer of the cube to rotate
    direction: 'clockwise' | 'counter-clockwise'; // Direction of rotation
}

// Implementation of RubiksCubeGroup
class MyRubiksCubeGroup implements RubiksCubeGroup {
    operation(command1: CubeRotationCommand, command2: CubeRotationCommand): CubeRotationCommand {
        // Combine two rotation commands
        // In practice, this could involve composing the sequences of rotations
        return command1; // Placeholder implementation
    }

    get identity(): CubeRotationCommand {
        // Identity element: no rotation
        return { axis: 'x', layer: 0, direction: 'clockwise' }; // Placeholder implementation
    }

    inverse(command: CubeRotationCommand): CubeRotationCommand {
        // Inverse of a rotation command
        // In practice, this could involve reversing the rotation
        return command; // Placeholder implementation
    }
}

// Example usage
const rubiksCubeGroup: RubiksCubeGroup = new MyRubiksCubeGroup();
console.log("Identity:", rubiksCubeGroup.identity);
console.log("Inverse of 'U':", rubiksCubeGroup.inverse('U'));
console.log("Operation of 'U' and 'R':", rubiksCubeGroup.operation('U', 'R'));