```ts
/** The cheapest set of travel passes covering every trip day. */

function validatePlan(
  tripDays: number[],
  passes: { span: number; cost: number }[],
): void {
  for (const day of tripDays) {
    if (!Number.isInteger(day) || day < 1) {
      throw new Error("trip days must be positive integers");
    }
  }
  for (let i = 1; i < tripDays.length; i += 1) {
    if (tripDays[i] <= tripDays[i - 1]) {
      throw new Error("trip days must be strictly increasing");
    }
  }
  if (passes.length === 0) {
    throw new Error("at least one pass kind is required");
  }
  for (const pass of passes) {
    if (!Number.isInteger(pass.span) || pass.span < 1) {
      throw new Error("pass span must be a positive integer");
    }
    if (!Number.isInteger(pass.cost) || pass.cost < 0) {
      throw new Error("pass cost must be a non-negative integer");
    }
  }
}

export function cheapestPassPlan(
  tripDays: number[],
  passes: { span: number; cost: number }[],
): { total: number; purchases: number[][] } {
  validatePlan(tripDays, passes);
  let total = 0;
  const purchases: number[][] = [];

  for (const day of tripDays) {
    let cheapestCost = Infinity;
    let cheapestSpan = 1;

    for (const pass of passes) {
      if (day >= pass.span - 1 && day <= day + pass.span - 1) {
        const cost = pass.cost;
        if (cost < cheapestCost) {
          cheapestCost = cost;
          cheapestSpan = pass.span;
        }
      }
    }

    if (cheapestCost !== Infinity) {
      total += cheapestCost;
      purchases.push([day, cheapestSpan]);
    }
  }

  return { total, purchases };
}
```