// #include <stdio.h>
#pragma CIVL ACSL

$input int ni,nj,bi,bj;

int main(int argc, char** argv) {
  /*@ logic int UP(int up, int step) = (up % step == 1 || step == 1) ? up : up % step == 0 ?
    @                                   up + 1 : up - (up % step - 1) + step;
    @
    @ predicate moduloed(int i, int step, int offset) = 
    @             i > 0 && (i % step == offset || step == 1);
    @
    @*/ 
  int prove_choice = $choose_int(3);
  
  if (prove_choice == 0) {
    /* Proving the lemmas that only involves nj and bj : */
    $assume(ni > 1);
    $assume(ni > bi && bi > 0);
    $choose {
      $when(1) $assume(bj > 1 && (nj - bj) % bj == 1);
      $when(1) $assume(bj > 1 && (nj - bj) % bj == 0);
      $when(1) $assume(bj > 1 && (nj - bj) % bj > 1);
      $when(1) $assume(bj == 1);
    }
    // Lemmas: 
    $assert($forall (int x) (moduloed(x, bj, 1) && nj-bj > x => UP(nj-bj, bj) >= x + bj));
    $assert($forall (int x) (moduloed(x, bj, 1) && x >= nj-bj => x >= UP(nj-bj, bj)));
    $assert(nj-bj <= UP(nj-bj, bj) && UP(nj-bj, bj) < nj);
  } else if (prove_choice == 1){
    /* Proving the lemmas that only involves ni and bi : */
    $assume(nj > 1);
    $assume(nj > bj && bj > 0);
    $choose {
      $when(1) $assume(bi > 1 && (ni - bi) % bi == 1);
      $when(1) $assume(bi > 1 && (ni - bi) % bi == 0);
      $when(1) $assume(bi > 1 && (ni - bi) % bi > 1);
      $when(1) $assume(bi == 1);
    } 
    // Lemmas: 
    $assert($forall (int x) (moduloed(x, bi, 1) && ni-bi > x => UP(ni-bi, bi) >= x + bi)); 
    $assert($forall (int x) (moduloed(x, bi, 1) && x >= ni-bi => x >= UP(ni-bi, bi))); 
    $assert(ni-bi <= UP(ni-bi, bi) && UP(ni-bi, bi) < ni);
  } else {
    /* Proving the lemmas that only involves predicate 'moduloed': */
    $assume(nj > 1);
    $assume(nj > bj && bj > 0);
    $assume(ni > 1);
    $assume(ni > bi && bi > 0);
    $choose {
      $when(1) $assume(bi == 1);
      $when(1) $assume(bi > 1);
    }
    $choose {
      $when(1) $assume(bj == 1);
      $when(1) $assume(bj > 1);
    }
    // Lemmas: 
    $assert($forall (int x) moduloed(x, bi, 1) => moduloed(x+bi, bi, 1));
    $assert($forall (int x) moduloed(x, bj, 1) => moduloed(x+bj, bj, 1)); 
    $assert(moduloed(1, bj, 1)); 
    $assert(moduloed(1, bi, 1)); 
  }
}