import { Digit, DigitList, Kind, Type } from '..';
/**
 * For each digit in the multiplicand, from right to left, we multiply the digit
 * by the multiplier, shift the result to the left by the number of zeroes
 * equivalent to the position of the digit, and add the result to the sum of the
 * partial products.
 *
 * Once we have processed all of the digits in the multiplicand, we return the
 * sum of the partial products.
 *
 * This is a type-level implementation of the traditional multiplication
 * algorithm, and is O(n^2) in the number of digits.
 */
type _$multiply2<
/**
 * The multiplicand.
 */
A extends DigitList.DigitList, 
/**
 * The multiplier.
 */
B extends DigitList.DigitList, 
/**
 * As we proceed in the traditional multiplication algorithm, there are a
 * number of zeroes that are appended to the end of the partial products.
 *
 * This type parameter represents the zeroes that are appended to the end of
 * the partial products. As we proceed in the algorithm, the number of zeroes
 * appended to the end of the partial products increases by one.
 */
SCALE extends DigitList.DigitList = [], 
/**
 * The sum of the partial products, which we add to as we proceed through each
 * digit of the multiplicand.
 */
SUM extends DigitList.DigitList = [], 
/**
 * The last digit of the multiplicand, which we multiply by the multiplier.
 * As we loop through the digits of the multiplicand, we pop the last digit
 * off of the multiplicand.
 */
LAST_A extends Digit.Digit = DigitList._$last<A>, 
/**
 * The next instance of A, with the last digit popped off.
 */
POP_A extends DigitList.DigitList = DigitList._$pop<A>, 
/**
 * The result of multiplying the last digit of the multiplicand by the
 * multiplier.
 */
MUL extends DigitList.DigitList = DigitList._$multiplyDigit<B, LAST_A>, 
/**
 * Next, we shift the result of the current multiplication by the number of
 * zeroes equivalent to the position of the digit, counted from the right.
 */
MUL_SCALE extends DigitList.DigitList = [...MUL, ...SCALE], 
/**
 * Finally, we add the result of the current multiplication, shifted
 * appropriately, to the sum of the partial products.
 */
ADD extends DigitList.DigitList = DigitList._$add<SUM, MUL_SCALE>, 
/**
 * The next instance of SCALE, with one more zero appended to the end.
 */
NEXT_SCALE extends DigitList.DigitList = [Digit.Zero, ...SCALE], 
/**
 * If A is empty, we are done, and we return the sum of the partial products.
 */
DONE extends boolean = A extends [] ? true : false> = DONE extends true ? SUM : _$multiply2<POP_A, B, NEXT_SCALE, ADD>;
/**
 * `_$multiply` is a type-level function that multiplies a digit list by another digit list.
 * It returns the result of the multiplication operation.
 *
 * @template A - The digit list.
 * @template B - A digit list.
 *
 * @example
 * For example, we can use `_$multiply` to multiply a digit list ["4", "2"] by another digit list ["1", "2"]:
 *
 * ```ts
 * import { DigitList } from "hkt-toolbelt";
 *
 * type Is504 = DigitList._$multiply<["4", "2"], ["1", "2"]>; // ["5", "0", "4"]
 * ```
 *
 * @example
 * If one of the inputs is an empty digit list or the zero digit, the result will be the zero digit.
 *
 * ```ts
 * import { DigitList } from "hkt-toolbelt";
 *
 * type IsZero = DigitList._$multiply<["4", "2"], []>; // ["0"]
 * type IsZero2 = DigitList._$multiply<["4", "2"], ["0"]>; // ["0"]
 * ```
 */
export type _$multiply<A extends DigitList.DigitList, B extends DigitList.DigitList> = DigitList._$trim<Type._$cast<_$multiply2<A, B>, DigitList.DigitList>>;
interface Multiply_T<T extends DigitList.DigitList> extends Kind.Kind {
    f(x: Type._$cast<this[Kind._], DigitList.DigitList>): _$multiply<typeof x, T>;
}
/**
 * `Multiply` is a type-level function that multiplies a digit list by another digit list.
 * It returns the result of the multiplication operation.
 *
 * @template A - The digit list.
 * @template B - The single digit.
 *
 * @example
 * For example, we can use `Multiply` to multiply a digit list ["4", "2"] by another digit list ["1", "2"]:
 *
 * ```ts
 * import { $, DigitList } from "hkt-toolbelt";
 *
 * type Is504 = $<$<DigitList.Multiply, ["1", "2"]>, ["4", "2"]>; // ["5", "0", "4"]
 * ```
 *
 * @example
 * If one of the inputs is an empty digit list or the zero digit, the result will be the zero digit.
 *
 * ```ts
 * import { DigitList } from "hkt-toolbelt";
 *
 * type IsZero = $<$<DigitList.Multiply, []>, ["4", "2"]>; // ["0"]
 * type IsZero2 = $<$<DigitList.Multiply, ["0"], ["4", "2"]>; // ["0"]
 * ```
 */
export interface Multiply extends Kind.Kind {
    f(x: Type._$cast<this[Kind._], DigitList.DigitList>): Multiply_T<typeof x>;
}
export {};