103773: [Atcoder]ABC377 D - Many Segments 2

Problem Statement

You are given two sequences of positive integers of length $N$, $L=(L_1,L_2,\ldots,L_N)$ and $R=(R_1,R_2,\ldots,R_N)$, and an integer $M$.

Find the number of pairs of integers $(l,r)$ that satisfy both of the following conditions:

• $1\le l \le r \le M$
• For every $1\le i\le N$, the interval $[l,r]$ does not completely contain the interval $[L_i,R_i]$.

Constraints

• $1\le N,M\le 2\times 10^5$
• $1\le L_i\le R_i\le M$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$
$L_1$ $R_1$
$L_2$ $R_2$
$\vdots$
$L_N$ $R_N$

Output

Print the answer.

Sample Input 1

2 4
1 2
3 4

Sample Output 1

5

The five pairs $(l,r)=(1,1),(2,2),(2,3),(3,3),(4,4)$ satisfy the conditions.

For example, $(l,r)=(1,3)$ does not satisfy the conditions because the interval $[1,3]$ completely contains the interval $[1,2]$.

Sample Input 2

6 5
1 1
2 2
3 3
4 4
5 5
1 5

Sample Output 2

0

There may be cases where no pairs of integers satisfy the conditions.

Sample Input 3

6 20
8 12
14 20
11 13
5 19
4 11
1 6

Sample Output 3

102