```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 dp = new Array(tripDays.length + 1).fill(Infinity);
  dp[0] = 0;
  const purchaseIndices = new Array(tripDays.length + 1).fill(null);

  for (let i = 1; i <= tripDays.length; i++) {
    for (const { span, cost } of passes) {
      if (i >= span && dp[i - span] !== Infinity) {
        if (dp[i] > dp[i - span] + cost) {
          dp[i] = dp[i - span] + cost;
          purchaseIndices[i] = i - span;
        }
      }
    }
  }

  const purchases: number[][] = [];
  for (let i = tripDays.length; i > 0; ) {
    if (purchaseIndices[i] !== null) {
      purchases.unshift([tripDays[purchaseIndices[i]], passes.find(p => purchaseIndices[i] + p.span === i)!.span]);
      i -= passes.find(p => purchaseIndices[i] + p.span === i)!.span;
    } else {
      throw new Error("No valid pass found to cover trip day");
    }
  }

  return { total: dp[tripDays.length], purchases };
}
```