```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);

  const n = tripDays.length;
  const dp: number[] = new Array(n + 1).fill(Infinity);
  dp[0] = 0;

  for (let i = 1; i <= n; i += 1) {
    for (const pass of passes) {
      if (i >= pass.span) {
        dp[i] = Math.min(dp[i], dp[i - pass.span] + pass.cost);
      }
    }
  }

  const purchases: number[][] = [];
  let j = n;
  while (j > 0) {
    for (const pass of passes) {
      if (j >= pass.span && dp[j] === dp[j - pass.span] + pass.cost) {
        purchases.unshift([tripDays[j - pass.span], pass.span]);
        j -= pass.span;
        break;
      }
    }
  }

  return { total: dp[n], purchases };
}
```