#include "rect.h"
#include "bits/stdc++.h"
using namespace std;
#define FAST ios_base::sync_with_stdio(false); cin.tie(0);
#define pb push_back
#define eb emplace_back
#define ins insert
#define f first
#define s second
#define cbr cerr<<"hi\n"
#define mmst(x, v) memset((x), v, sizeof ((x)))
#define siz(x) ll(x.size())
#define all(x) (x).begin(), (x).end()
#define lbd(x,y) (lower_bound(all(x),y)-x.begin())
#define ubd(x,y) (upper_bound(all(x),y)-x.begin())
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count()); //can be used by calling rng() or shuffle(A, A+n, rng)
inline long long rand(long long x, long long y) { return rng() % (y+1-x) + x; } //inclusivesss
string inline to_string(char c) {string s(1,c);return s;} template<typename T> inline T gcd(T a,T b){ return a==0?llabs(b):gcd(b%a,a); }
using ll=long long;
using ld=long double;
#define FOR(i,s,e) for(int i=s;i<=ll(e);++i)
#define DEC(i,s,e) for(int i=s;i>=ll(e);--i)
using pi=pair<int,int>; using spi=pair<ll,pi>; using dpi=pair<pi,pi>;
#define LLINF ((long long)1e18)
#define INF int(1e9+1e6)
#define MAXN (2502)
ll n, m, ans;
vector<int> r[MAXN][MAXN], c[MAXN][MAXN];
vector<dpi> e;
long long count_rectangles(std::vector<std::vector<int>> A) {
n = siz(A), m=siz(A[0]);
FOR(i,0,n-1){
vector<int> v;
DEC(j,m-1,0){
DEC(k,siz(v)-1,0) { if(abs(v[k]-j)>1) e.pb({{i,j},{i,v[k]}}); if(A[i][v[k]]>=A[i][j]) break; }
while(v.size()&&A[i][v.back()]<=A[i][j])v.pop_back();
v.eb(j);
}
}
FOR(j,0,m-1){
vector<int> v;
FOR(i,0,n-1){
DEC(k,siz(v)-1,0) { if(abs(v[k]-i)>1) e.pb({{v[k],j},{i,j}}); if(A[v[k]][j]>=A[i][j]) break; }
while(v.size()&&A[v.back()][j]<=A[i][j])v.pop_back();
v.eb(i);
}
}
vector<int> len(siz(e), 0);
ll ind=0;
for(auto i:e)if(i.f.f==i.s.f){
c[i.f.s][i.s.s].eb(ind++);//.f.f
}else{
r[i.f.f][i.s.f].eb(ind++);//.f.s
}
FOR(i,0,n-1)FOR(j,0,n-1){
DEC(k,siz(r[i][j])-1,0){
if(k == siz(r[i][j])-1 || e[r[i][j][k+1]].f.s-1 != e[r[i][j][k]].f.s) len[r[i][j][k]]=1;
else len[r[i][j][k]]=len[r[i][j][k+1]]+1;
}
}
FOR(i,0,m-1)FOR(j,0,m-1){
DEC(k,siz(c[i][j])-1,0){
if(k == siz(c[i][j])-1 || e[c[i][j][k+1]].f.f-1 != e[c[i][j][k]].f.f) len[c[i][j][k]]=1;
else len[c[i][j][k]]=len[c[i][j][k+1]]+1;
}
}
FOR(i,0,n-1)FOR(j,0,n-1)r[i][j].clear();
FOR(i,0,m-1)FOR(j,0,m-1)c[i][j].clear();
FOR(i,0,siz(e)-1){
if(e[i].f.s==e[i].s.s) r[e[i].f.f][e[i].f.s].eb(i);
else c[e[i].f.f][e[i].f.s].eb(i);
}
FOR(i,0,n-3)FOR(j,1,m-2){
for(auto i1:r[i][j]){
for(auto i2:c[i+1][j-1]){
if(len[i1]<e[i2].s.s-(j-1)-1)break;
if(len[i2]>=e[i1].s.f-i-1)++ ans;
}
}
}
return ans;
}
