#include "rect.h"
#include <bits/stdc++.h>
using namespace std;
#define sz(x) ((int)(x).size())
vector<int> fen;
void add(int i, int x){
for(i++; i < sz(fen); i+=i&-i) fen[i] += x;
}
int qry(int i){
int ans = 0;
for(i++; i > 0; i-=i&-i) ans += fen[i];
return ans;
}
int qry(int i, int j){
if(i > 0) return qry(j)-qry(i-1);
else return qry(j);
}
long long count_rectangles(vector<vector<int>> a) {
int n = sz(a), m = sz(a[0]);
vector<vector<int>> r1(n, vector<int>(m, -1)), r2(n, vector<int>(m, -1));
for(int i = 1; i < n-1; i++){
stack<pair<int, int>> st({{a[i][0], 0}});
for(int j = 1; j < m; j++){
while(!st.empty() && a[i][j] > st.top().first){
auto [h, k] = st.top();
st.pop();
r2[i][k] = j-1;
}
if(!st.empty()) {
if(st.top().first > a[i][j]) r1[i][j] = st.top().second+1;
else r1[i][j] = r1[i][st.top().second];
}
st.emplace(a[i][j], j);
}
}
vector<vector<int>> c1(n, vector<int>(m, -1)), c2(n, vector<int>(m, -1));
for(int j = 1; j < m-1; j++){
stack<pair<int, int>> st({{a[0][j], 0}});
for(int i = 1; i < n; i++){
while(!st.empty() && a[i][j] > st.top().first){
auto [h, k] = st.top(); st.pop();
c2[k][j] = i-1;
}
if(!st.empty()){
if(st.top().first > a[i][j]) c1[i][j] = st.top().second+1;
else c1[i][j] = c1[st.top().second][j];
}
st.emplace(a[i][j], i);
}
}
int mx = max(n, m);
fen.resize(mx+1);
vector<bool> valid;
unordered_set<long long> noDupl;
unordered_map<int, vector<tuple<short, short, int>>> byI, byJ;
unordered_map<int, vector<short>> goodI, goodJ;
int q = 0;
for(int i = 1; i < n-1; i++){
for(int j = 1; j < m-1; j++){
// cerr << "\n" << i << " " << j << " " << r1[i][j] << " " << r2[i][j] << " " << c1[i][j] << " " << c2[i][j] << "\n";
if(r1[i][j] != -1 && r2[i][j] != -1 && c1[i][j] != -1 && c2[i][j] != -1){
int x = c1[i][j]*mx + c2[i][j], y = r1[i][j]*mx + r2[i][j];
goodJ[x].push_back(j);
goodI[y].push_back(i);
long long t = (long long)x*(mx*mx)+y;
if(noDupl.count(t)) continue;
noDupl.insert(t);
valid.push_back(true);
byI[x].emplace_back(r1[i][j], r2[i][j], q);
byJ[y].emplace_back(c1[i][j], c2[i][j], q);
q++;
}
}
}
for(const auto& [id, v] : byI){
int last = -1;
sort(goodJ[id].begin(), goodJ[id].end());
for(int j : goodJ[id]){
if(j != last) add(j, 1);
last = j;
}
for(auto [a, b, ind] : v){
if(qry(a, b) != b-a+1) valid[ind] = false;
}
last = -1;
for(int j : goodJ[id]){
if(j != last) add(j, -1);
last = j;
}
}
for(const auto& [id, v] : byJ){
int last = -1;
for(int i : goodI[id]){
if(i != last) add(i, 1);
last = i;
}
for(auto [a, b, ind] : v){
if(qry(a, b) != b-a+1) valid[ind] = false;
}
last = -1;
for(int i : goodI[id]){
if(i != last) add(i, -1);
last = i;
}
}
int ans = 0;
for(bool b : valid) ans += b;
return ans;
}
