```ts
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);
  if (tripDays.length === 0) return { total: 0, purchases: [] };

  const sortedPasses = passes.sort((a, b) => a.span - b.span);

  let totalCost = 0;
  let currentStartDay = tripDays[0];
  const purchases: number[][] = [];

  for (let i = 0; i < tripDays.length; i++) {
    const day = tripDays[i];
    if (day > currentStartDay) {
      // Buy the cheapest pass that covers the gap
      const cheapestPass = sortedPasses.find(pass => pass.span >= day - currentStartDay);
      if (!cheapestPass) throw new Error("no pass can cover the gap");
      totalCost += cheapestPass.cost;
      purchases.push([currentStartDay, cheapestPass.span]);
      currentStartDay = day + cheapestPass.span - 1;
    }
  }

  return { total: totalCost, purchases };
}
```