group_concept.h

// Description: Group concept
// Documentation: algebra.txt

#ifndef PASTELSYS_GROUP_CONCEPT_H
#define PASTELSYS_GROUP_CONCEPT_H

#include "pastel/sys/algebra/monoid_concept.h"

#include "pastel/sys/algebra/native_group.h"

namespace Pastel
{

    //! An additive group.
    /*!
   A group (X, +, 0) is a monoid (X, +, 0)
   such that for every x in X there exists y in X
   such that x + y = 0. Since the y is unique,
   it is denoted by -x.
   */
    struct Additive_Group_Concept
    : Refines<Additive_Monoid_Concept>
    {
        template <typename Type>
        auto requires_(Type&& t) -> decltype
        (
            conceptCheck(
                //! Adds -that to the element.
                Concept::hasType<Type&>(t -= t),
                //! Returns left * (-right).
                Concept::convertsTo<Type>(t - t),
                //! Returns -x for an element x.
                Concept::convertsTo<Type>(-t)
            )
        );
    };

    //! A multiplicative group.
    /*!
   A group (X, *, 1) is a monoid (X, *, 1)
   such that for every x in X there exists y in X
   such that x * y = 1. Since the y is unique,
   it is denoted by inverse(x).
   */
    struct Multiplicative_Group_Concept
    : Refines<Multiplicative_Monoid_Concept>
    {
        template <typename Type>
        auto requires_(Type&& t) -> decltype
        (
            conceptCheck(
                //! Multiplies with inverse(that) from the right.
                Concept::hasType<Type&>(t /= t),
                //! Returns left * inverse(right).
                Concept::convertsTo<Type>(t / t),
                //! Returns inverse(x) for an element x.
                Concept::convertsTo<Type>(inverse(t))
            )
        );
    };

}

namespace Pastel
{

    //! Computes x^p, for p an integer.
    template <typename Multiplicative_Group>
    Multiplicative_Group groupPower(
        Multiplicative_Group x,
        integer p)
    {
        Multiplicative_Group result = 
            monoidPower(std::move(x), std::abs(p));

        if (p < 0)
        {
            result = inverse(result);
        }

        return result;
    }

}

#endif