//Purpose:
//  demo "morphing" of metafunction grammar types to expression types.
//Reference:
//  http://planetmath.org/encyclopedia/HomomorphismBetweenAlgebraicSystems.html

#include <boost/proto/proto.hpp>
#include <boost/type_traits/is_same.hpp>
#include <boost/mpl/assert.hpp>

using namespace boost::proto;

  template
  < class MetaFunInst //proto metafunction instantiation forming "metafunction" grammar.
  >
struct morphism
//algebraic morphism,  Corresponds to the f in Reference.
//The A algebra in Reference corresponds to the proto metafunction grammars.
//The B algebra in Refernece corresponds to the proto expression types.
;

    typedef
  terminal<int>
metafun0_type;  

  template
  <
  >
struct morphism
  < metafun0_type //corresponds to application of nullary function in A algebra in Reference.
  >
{
        typedef
      metafun0_type::type 
    type;
}; 
   
  template
  < class Arg1
  , template<class>class MetaFun1 //corresponds to unary function in A algebra in Reference.
  >
struct morphism
  < MetaFun1<Arg1> //corresponds to application of unary function in A algebra in Reference.
  >
{
        typedef
        typename
      MetaFun1
      < typename morphism<Arg1>::type
      >::type
    type;
};    

  
    typedef 
  morphism<metafun0_type>::type
morph0_type;
    typedef
  expr
  < tag::terminal
  , argsns_::args0<int>
  >
expr0_type;
BOOST_MPL_ASSERT((::boost::is_same<morph0_type,expr0_type>));

    typedef 
  posit
  < metafun0_type
  >
metafun1_type;
    typedef
  morphism<metafun1_type>::type
morph1_type;
    typedef
  expr
  < tag::posit
  , argsns_::args1<expr0_type>
  >
expr1_type;
BOOST_MPL_ASSERT((::boost::is_same<morph1_type,expr1_type>));