```typescript
/** 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");
    }
  }
}

function cheapestPassPlan(
  tripDays: number[],
  passes: { span: number; cost: number }[],
): { total: number; purchases: number[][] } {
  validatePlan(tripDays, passes);

  // Sort trips and pass spans in ascending order
  const sortedTrips = [...tripDays].sort((a, b) => a - b);
  const sortedPasses = passes.sort((a, b) => a.span - b.span);

  let totalCost = 0;
  let purchases: number[][] = [];
  let currentTripIndex = 0;

  for (const pass of sortedPasses) {
    while (
      currentTripIndex < sortedTrips.length &&
      sortedTrips[currentTripIndex] <= sortedTrips[sortedTrips.length - 1] + pass.span - 1
    ) {
      totalCost += pass.cost;
      purchases.push([sortedTrips[currentTripIndex], pass.span]);
      currentTripIndex++;
    }
  }

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