peano.rb

class True
  class << self
    def true?
      self
    end

    def !@
      False
    end

    def &(other)
      other.true?
    end

    def |(other)
      self
    end

    def inspect
      "True"
    end
  end
end

class False
  class << self
    def !@
      True
    end

    def &(other)
      self
    end

    def |(other)
      other.true?
    end

    def true?
      self
    end

    def inspect
      "False"
    end
  end
end

class SBottom
  def zero?
    False
  end

  def greater_than_or_equal_to_zero?
    False
  end

  def ==(other)
    False
  end
end

class Zero
  def zero?
    True
  end

  def greater_than_or_equal_to_zero?
    True
  end

  def succ
    S(self)
  end

  def pred
    SBottom.new
  end

  def inspect
    "0"
  end

  def +(other)
    other
  end

  def *(other)
    self
  end

  def ==(other)
    other.zero?
  end

  def <=(other)
    other.greater_than_or_equal_to_zero?
  end

  def <(other)
    self.succ <= other
  end
end

class S
  def initialize(pred)
    @pred = pred
  end

  def pred
    @pred
  end

  def succ
    S(self)
  end

  def zero?
    False
  end

  def greater_than_or_equal_to_zero?
    True
  end

  def ==(other)
    pred == other.pred
  end

  def +(other)
    pred + other.succ
  end

  def *(other)
    self + (other.pred * self)
  end

  def <=(other)
    self.pred <= other.pred
  end

  def <(other)
    self.succ <= other
  end

  def inspect
    "S(#{pred.inspect})"
  end
end

def S(s)
  S.new(s)
end