using System.Numerics;
using System.Collections.Generic;

namespace Dafny
{
  public class Set<T>
  {
    Dictionary<T, bool> dict;
    public Set() { }
    Set(Dictionary<T, bool> d) {
      dict = d;
    }
    public static Set<T> Empty {
      get {
        return new Set<T>(new Dictionary<T, bool>(0));
      }
    }
    public static Set<T> FromElements(params T[] values) {
      Dictionary<T, bool> d = new Dictionary<T, bool>(values.Length);
      foreach (T t in values)
        d.Add(t, true);
      return new Set<T>(d);
    }
    public IEnumerable<T> Elements {
      get {
        return dict.Keys;
      }
    }
    public bool Equals(Set<T> other) {
      return dict.Count == other.dict.Count && IsSubsetOf(other);
    }
    public override bool Equals(object other) {
      return other is Set<T> && Equals((Set<T>)other);
    }
    public override int GetHashCode() {
      return dict.GetHashCode();
    }
    public bool IsProperSubsetOf(Set<T> other) {
      return dict.Count < other.dict.Count && IsSubsetOf(other);
    }
    public bool IsSubsetOf(Set<T> other) {
      if (other.dict.Count < dict.Count)
        return false;
      foreach (T t in dict.Keys) {
        if (!other.dict.ContainsKey(t))
          return false;
      }
      return true;
    }
    public bool IsSupersetOf(Set<T> other) {
      return other.IsSubsetOf(this);
    }
    public bool IsProperSupersetOf(Set<T> other) {
      return other.IsProperSubsetOf(this);
    }
    public bool IsDisjointFrom(Set<T> other) {
      Dictionary<T, bool> a, b;
      if (dict.Count < other.dict.Count) {
        a = dict; b = other.dict;
      } else {
        a = other.dict; b = dict;
      }
      foreach (T t in a.Keys) {
        if (b.ContainsKey(t))
          return false;
      }
      return true;
    }
    public bool Contains(T t) {
      return dict.ContainsKey(t);
    }
    public Set<T> Union(Set<T> other) {
      if (dict.Count == 0)
        return other;
      else if (other.dict.Count == 0)
        return this;
      Dictionary<T, bool> a, b;
      if (dict.Count < other.dict.Count) {
        a = dict; b = other.dict;
      } else {
        a = other.dict; b = dict;
      }
      Dictionary<T, bool> r = new Dictionary<T, bool>();
      foreach (T t in b.Keys)
        r[t] = true;
      foreach (T t in a.Keys)
        r[t] = true;
      return new Set<T>(r);
    }
    public Set<T> Intersect(Set<T> other) {
      if (dict.Count == 0)
        return this;
      else if (other.dict.Count == 0)
        return other;
      Dictionary<T, bool> a, b;
      if (dict.Count < other.dict.Count) {
        a = dict; b = other.dict;
      } else {
        a = other.dict; b = dict;
      }
      Dictionary<T, bool> r = new Dictionary<T, bool>();
      foreach (T t in a.Keys) {
        if (b.ContainsKey(t))
          r.Add(t, true);
      }
      return new Set<T>(r);
    }
    public Set<T> Difference(Set<T> other) {
      if (dict.Count == 0)
        return this;
      else if (other.dict.Count == 0)
        return this;
      Dictionary<T, bool> r = new Dictionary<T, bool>();
      foreach (T t in dict.Keys) {
        if (!other.dict.ContainsKey(t))
          r.Add(t, true);
      }
      return new Set<T>(r);
    }
    public T Choose() {
      foreach (T t in dict.Keys) {
        // return the first one
        return t;
      }
    }
  }
  public class Sequence<T>
  {
    T[] elmts;
    public Sequence() { }
    Sequence(T[] ee) {
      elmts = ee;
    }
    public static Sequence<T> Empty {
      get {
        return new Sequence<T>(new T[0]);
      }
    }
    public static Sequence<T> FromElements(params T[] values) {
      return new Sequence<T>(values);
    }
    public BigInteger Length {
      get { return new BigInteger(elmts.Length); }
    }
    public T[] Elements {
      get {
        return elmts;
      }
    }
    public T Select(BigInteger index) {
      return elmts[(int)index];
    }
    public Sequence<T> Update(BigInteger index, T t) {
      T[] a = (T[])elmts.Clone();
      a[(int)index] = t;
      return new Sequence<T>(a);
    }
    public bool Equals(Sequence<T> other) {
      int n = elmts.Length;
      return n == other.elmts.Length && EqualUntil(other, n);
    }
    public override bool Equals(object other) {
      return other is Sequence<T> && Equals((Sequence<T>)other);
    }
    public override int GetHashCode() {
      return elmts.GetHashCode();
    }
    bool EqualUntil(Sequence<T> other, int n) {
      for (int i = 0; i < n; i++) {
        if (!elmts[i].Equals(other.elmts[i]))
          return false;
      }
      return true;
    }
    public bool IsProperPrefixOf(Sequence<T> other) {
      int n = elmts.Length;
      return n < other.elmts.Length && EqualUntil(other, n);
    }
    public bool IsPrefixOf(Sequence<T> other) {
      int n = elmts.Length;
      return n <= other.elmts.Length && EqualUntil(other, n);
    }
    public Sequence<T> Concat(Sequence<T> other) {
      if (elmts.Length == 0)
        return other;
      else if (other.elmts.Length == 0)
        return this;
      T[] a = new T[elmts.Length + other.elmts.Length];
      System.Array.Copy(elmts, 0, a, 0, elmts.Length);
      System.Array.Copy(other.elmts, 0, a, elmts.Length, other.elmts.Length);
      return new Sequence<T>(a);
    }
    public bool Contains(T t) {
      int n = elmts.Length;
      for (int i = 0; i < n; i++) {
        if (t.Equals(elmts[i]))
          return true;
      }
      return false;
    }
    public Sequence<T> Take(BigInteger n) {
      int m = (int)n;
      if (elmts.Length == m)
        return this;
      T[] a = new T[m];
      System.Array.Copy(elmts, a, m);
      return new Sequence<T>(a);
    }
    public Sequence<T> Drop(BigInteger n) {
      if (n.IsZero)
        return this;
      int m = (int)n;
      T[] a = new T[elmts.Length - m];
      System.Array.Copy(elmts, m, a, 0, elmts.Length - m);
      return new Sequence<T>(a);
    }
  }
  public struct Pair<A, B>
  {
    public readonly A Car;
    public readonly B Cdr;
    public Pair(A a, B b) {
      this.Car = a;
      this.Cdr = b;
    }
  }
  public partial class Helpers {
    public static bool QuantBool(bool frall, System.Predicate<bool> pred) {
      if (frall) {
        return pred(false) && pred(true);
      } else {
        return pred(false) || pred(true);
      }
    }
    public static bool QuantInt(BigInteger lo, BigInteger hi, bool frall, System.Predicate<BigInteger> pred) {
      for (BigInteger i = lo; i < hi; i++) {
        if (pred(i) != frall) { return !frall; }
      }
      return frall;
    }
    public static bool QuantSet<T>(Dafny.Set<T> set, bool frall, System.Predicate<T> pred) {
      foreach (var t in set.Elements) {
        if (pred(t) != frall) { return !frall; }
      }
      return frall;
    }
    public static bool QuantSeq<T>(Dafny.Sequence<T> seq, bool frall, System.Predicate<T> pred) {
      foreach (var t in seq.Elements) {
        if (pred(t) != frall) { return !frall; }
      }
      return frall;
    }
  }
}