How to merge types with Generics

export const paginationRequest = <
  const def extends object,
  $ = {},
>(
  filter?: type.validate<def, $>,
) => {
  return type("<t>", {
    page: "number.integer > 0",
    limit: "number.integer > 0",
    filter: "t", // Also i want this field won't appear if filter value is undefined in function
  })(
    filter // Here's type error: Type string is not assignable to type validateTuple<[def], {}, bindThis<[def]>>
  );
};

export const paginationRequest = <
  const def extends object,
  $ = {},
>(
  filter?: type.validate<def, $>,
) => {
  return type("<t>", {
    page: "number.integer > 0",
    limit: "number.integer > 0",
    filter: "t", // Also i want this field won't appear if filter value is undefined in function
  })(
    filter // Here's type error: Type string is not assignable to type validateTuple<[def], {}, bindThis<[def]>>
  );
};