#include <civlc.cvh>

#define N 4
$input float x[N];
$input float epsilon;

typedef struct Interval {
  float left;
  float right;
} Interval;

Interval emptyInterval;

Interval makeInterval(float left, float right) {
  Interval result;

  if (left <= right) {
    result.left = left;
    result.right = right;
  } else {
    result.left = 0;
    result.right = -1;
  }
  return result;
}

int empty(Interval a) {
  return a.left > a.right;
}

int subset(Interval a, Interval b) {
  return a.left >= b.left && a.right <= b.right;
}

Interval round(Interval a) {
  if (empty(a))
    return a;
  else {
    float left, right;
  
    if (a.left>=0) {
      left = (1-epsilon)*a.left;
      right = (1+epsilon)*a.right;
    } else {
      left = (1+epsilon)*a.left;
      if (a.right>=0)
        right = (1+epsilon)*a.right;
      else
        right = (1-epsilon)*a.right;
    }
    return makeInterval(left, right);
  }
}

Interval plus(Interval a, Interval b) {
  if (empty(a))
    return a;
  if (empty(b))
    return b;
  else
    return makeInterval(a.left + b.left, a.right + b.right);
}

Interval fplus(Interval a, Interval b) {
  return round(plus(a,b));
}

void main() {
  Interval a[N];
  Interval r1, r2;

  emptyInterval = makeInterval(0,-1);
  $assume(epsilon>0 && epsilon<0.1);
  for (int i=0; i<N; i++) {
    a[i].left = x[i];
    a[i].right = x[i];
  }
  r1 = fplus(fplus(fplus(a[0],a[1]),a[2]),a[3]);
  r2 = fplus(fplus(a[0],a[1]), fplus(a[2],a[3]));
  $assert(subset(r2, r1));
}