```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;
  if (n === 0) return { total: 0, purchases: [] };

  // Sort passes by cost and then span
  passes.sort((a, b) => a.cost - b.cost || a.span - b.span);

  let dp = Array(n + 1).fill(Infinity);
  dp[0] = 0;

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

  let total = dp[n];
  if (total === Infinity) throw new Error("no valid plan found");

  const purchases: number[][] = [];
  for (let i = n; i > 0; i--) {
    if (dp[i] !== dp[i - 1]) {
      purchases.push([i - span + 1, span]);
      total -= cost;
    }
  }

  return { total, purchases };
}
```