#include<bits/stdc++.h>
#include "rect.h"
using namespace std;
using ll = long long;
// O(nlogn)
set<pair<int, int>> get_segments(vector<int> &a) {
vector<pair<int, int>> b;
set<pair<int, int>> res;
int n = (int)a.size();
for(int i = 0; i < n; i++) {
b.push_back({a[i], i});
}
sort(b.rbegin(), b.rend());
set<int> st;
for(auto [x, pos] : b) {
st.insert(pos);
auto it = st.lower_bound(pos);
if(it != st.begin()) {
auto lb = it; lb--;
if(*it - *lb > 1)
res.insert({*lb + 1, *it - 1});
}
if(it != (--st.end())) {
auto rb = it; rb++;
if(*rb - *it > 1)
res.insert({*it + 1, *rb - 1});
}
}
return res;
}
template<typename T>
void intersect(set<T> &st, set<T> st1) {
set<T> st2;
for(auto x : st1) {
if(st.count(x))
st2.insert(x);
}
st = st2;
}
ll count_rectangles(vector<vector<int>> a) {
int n = (int)a.size(), m = (int)a[0].size();
if(n < 3 || m < 3) return 0;
ll res = 0;
for(int r1 = 1; r1 < n - 1; r1++) {
set<pair<int, int>> seg;
vector<int> mx(m);
for(int r2 = r1; r2 < n - 1; r2++) {
for(int i = 0; i < m; i++)
mx[i] = max(mx[i], a[r2][i]);
vector<int> pre(m);
for(int i = 0; i < m; i++) {
pre[i] = (mx[i] < min(a[r1 - 1][i], a[r2 + 1][i]));
if(i) pre[i] += pre[i - 1];
}
if(r1 == r2) seg = get_segments(a[r2]);
else intersect(seg, get_segments(a[r2]));
for(auto [l, r] : seg) {
int x = pre[r] - (l ? pre[l - 1] : 0);
if(x == r - l + 1)
res++;
}
}
}
return res;
}
