(fa, fd, DE)
Checks if Df/f is the derivative of an element of k(t).

    a in k(t) is the derivative of an element of k(t) if there exists b in k(t)
    such that a = Db.  Either returns (ans, u), such that Df/f == Du, or None,
    which means that Df/f is not the derivative of an element of k(t).  ans is
    a list of tuples such that Add(*[i*j for i, j in ans]) == u.  This is useful
    for seeing exactly which elements of k(t) produce u.

    This function uses the structure theorem approach, which says that for any
    f in K, Df/f is the derivative of a element of K if and only if there are ri
    in QQ such that::

            ---               ---       Dt
            \    r  * Dt   +  \    r  *   i      Df
            /     i     i     /     i   ---   =  --.
            ---               ---        t        f
         i in L            i in E         i
               K/C(x)            K/C(x)


    Where C = Const(K), L_K/C(x) = { i in {1, ..., n} such that t_i is
    transcendental over C(x)(t_1, ..., t_i-1) and Dt_i = Da_i/a_i, for some a_i
    in C(x)(t_1, ..., t_i-1)* } (i.e., the set of all indices of logarithmic
    monomials of K over C(x)), and E_K/C(x) = { i in {1, ..., n} such that t_i
    is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i/t_i = Da_i, for some
    a_i in C(x)(t_1, ..., t_i-1) } (i.e., the set of all indices of
    hyperexponential monomials of K over C(x)).  If K is an elementary extension
    over C(x), then the cardinality of L_K/C(x) U E_K/C(x) is exactly the
    transcendence degree of K over C(x).  Furthermore, because Const_D(K) ==
    Const_D(C(x)) == C, deg(Dt_i) == 1 when t_i is in E_K/C(x) and
    deg(Dt_i) == 0 when t_i is in L_K/C(x), implying in particular that E_K/C(x)
    and L_K/C(x) are disjoint.

    The sets L_K/C(x) and E_K/C(x) must, by their nature, be computed
    recursively using this same function.  Therefore, it is required to pass
    them as indices to D (or T).  E_args are the arguments of the
    hyperexponentials indexed by E_K (i.e., if i is in E_K, then T[i] ==
    exp(E_args[i])).  This is needed to compute the final answer u such that
    Df/f == Du.

    log(f) will be the same as u up to a additive constant.  This is because
    they will both behave the same as monomials. For example, both log(x) and
    log(2*x) == log(x) + log(2) satisfy Dt == 1/x, because log(2) is constant.
    Therefore, the term const is returned.  const is such that
    log(const) + f == u.  This is calculated by dividing the arguments of one
    logarithm from the other.  Therefore, it is necessary to pass the arguments
    of the logarithmic terms in L_args.

    To handle the case where we are given Df/f, not f, use is_deriv_k_in_field().
Checks if Df/f is the derivative of an element of k(t).
Checks if Df / f is the derivative of an element of k ( t ) .
def is_deriv_k(fa, fd, DE):
    """
    Checks if Df/f is the derivative of an element of k(t).

    a in k(t) is the derivative of an element of k(t) if there exists b in k(t)
    such that a = Db.  Either returns (ans, u), such that Df/f == Du, or None,
    which means that Df/f is not the derivative of an element of k(t).  ans is
    a list of tuples such that Add(*[i*j for i, j in ans]) == u.  This is useful
    for seeing exactly which elements of k(t) produce u.

    This function uses the structure theorem approach, which says that for any
    f in K, Df/f is the derivative of a element of K if and only if there are ri
    in QQ such that::

            ---               ---       Dt
            \    r  * Dt   +  \    r  *   i      Df
            /     i     i     /     i   ---   =  --.
            ---               ---        t        f
         i in L            i in E         i
               K/C(x)            K/C(x)


    Where C = Const(K), L_K/C(x) = { i in {1, ..., n} such that t_i is
    transcendental over C(x)(t_1, ..., t_i-1) and Dt_i = Da_i/a_i, for some a_i
    in C(x)(t_1, ..., t_i-1)* } (i.e., the set of all indices of logarithmic
    monomials of K over C(x)), and E_K/C(x) = { i in {1, ..., n} such that t_i
    is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i/t_i = Da_i, for some
    a_i in C(x)(t_1, ..., t_i-1) } (i.e., the set of all indices of
    hyperexponential monomials of K over C(x)).  If K is an elementary extension
    over C(x), then the cardinality of L_K/C(x) U E_K/C(x) is exactly the
    transcendence degree of K over C(x).  Furthermore, because Const_D(K) ==
    Const_D(C(x)) == C, deg(Dt_i) == 1 when t_i is in E_K/C(x) and
    deg(Dt_i) == 0 when t_i is in L_K/C(x), implying in particular that E_K/C(x)
    and L_K/C(x) are disjoint.

    The sets L_K/C(x) and E_K/C(x) must, by their nature, be computed
    recursively using this same function.  Therefore, it is required to pass
    them as indices to D (or T).  E_args are the arguments of the
    hyperexponentials indexed by E_K (i.e., if i is in E_K, then T[i] ==
    exp(E_args[i])).  This is needed to compute the final answer u such that
    Df/f == Du.

    log(f) will be the same as u up to a additive constant.  This is because
    they will both behave the same as monomials. For example, both log(x) and
    log(2*x) == log(x) + log(2) satisfy Dt == 1/x, because log(2) is constant.
    Therefore, the term const is returned.  const is such that
    log(const) + f == u.  This is calculated by dividing the arguments of one
    logarithm from the other.  Therefore, it is necessary to pass the arguments
    of the logarithmic terms in L_args.

    To handle the case where we are given Df/f, not f, use is_deriv_k_in_field().
    """
    # Compute Df/f
    dfa, dfd = fd*(fd*derivation(fa, DE) - fa*derivation(fd, DE)), fd**2*fa
    dfa, dfd = dfa.cancel(dfd, include=True)

    # Our assumption here is that each monomial is recursively transcendental
    if len(DE.L_K) + len(DE.E_K) != len(DE.D) - 1:
        if [i for i in DE.cases if i == 'tan'] or \
                set([i for i in DE.cases if i == 'primitive']) - set(DE.L_K):
            raise NotImplementedError("Real version of the structure "
                "theorems with hypertangent support is not yet implemented.")

        # TODO: What should really be done in this case?
        raise NotImplementedError("Nonelementary extensions not supported "
            "in the structure theorems.")

    E_part = [DE.D[i].quo(Poly(DE.T[i], DE.T[i])).as_expr() for i in DE.E_K]
    L_part = [DE.D[i].as_expr() for i in DE.L_K]

    lhs = Matrix([E_part + L_part])
    rhs = Matrix([dfa.as_expr()/dfd.as_expr()])

    A, u = constant_system(lhs, rhs, DE)

    if not all(derivation(i, DE, basic=True).is_zero for i in u) or not A:
        # If the elements of u are not all constant
        # Note: See comment in constant_system

        # Also note: derivation(basic=True) calls cancel()
        return None
    else:
        if not all(i.is_Rational for i in u):
            raise NotImplementedError("Cannot work with non-rational "
                "coefficients in this case.")
        else:
            terms = DE.E_args + [DE.T[i] for i in DE.L_K]
            ans = list(zip(terms, u))
            result = Add(*[Mul(i, j) for i, j in ans])
            argterms = [DE.T[i] for i in DE.E_K] + DE.L_args
            l = []
            for i, j in zip(argterms, u):
                # We need to get around things like sqrt(x**2) != x
                # and also sqrt(x**2 + 2*x + 1) != x + 1
                icoeff, iterms = sqf_list(i)
                l.append(Mul(*([Pow(icoeff, j)] + [Pow(b, e*j) for b, e in iterms])))
            const = cancel(fa.as_expr()/fd.as_expr()/Mul(*l))

            return (ans, result, const)
def is_deriv_k ( fa , fd , DE ) : # Compute Df/f dfa , dfd = fd * ( fd * derivation ( fa , DE ) - fa * derivation ( fd , DE ) ) , fd ** 2 * fa dfa , dfd = dfa . cancel ( dfd , include = True ) # Our assumption here is that each monomial is recursively transcendental if len ( DE . L_K ) + len ( DE . E_K ) != len ( DE . D ) - 1 : if [ i for i in DE . cases if i == 'tan' ] or set ( [ i for i in DE . cases if i == 'primitive' ] ) - set ( DE . L_K ) : raise NotImplementedError ( "Real version of the structure " "theorems with hypertangent support is not yet implemented." ) # TODO: What should really be done in this case? raise NotImplementedError ( "Nonelementary extensions not supported " "in the structure theorems." ) E_part = [ DE . D [ i ] . quo ( Poly ( DE . T [ i ] , DE . T [ i ] ) ) . as_expr ( ) for i in DE . E_K ] L_part = [ DE . D [ i ] . as_expr ( ) for i in DE . L_K ] lhs = Matrix ( [ E_part + L_part ] ) rhs = Matrix ( [ dfa . as_expr ( ) / dfd . as_expr ( ) ] ) A , u = constant_system ( lhs , rhs , DE ) if not all ( derivation ( i , DE , basic = True ) . is_zero for i in u ) or not A : # If the elements of u are not all constant # Note: See comment in constant_system # Also note: derivation(basic=True) calls cancel() return None else : if not all ( i . is_Rational for i in u ) : raise NotImplementedError ( "Cannot work with non-rational " "coefficients in this case." ) else : terms = DE . E_args + [ DE . T [ i ] for i in DE . L_K ] ans = list ( zip ( terms , u ) ) result = Add ( * [ Mul ( i , j ) for i , j in ans ] ) argterms = [ DE . T [ i ] for i in DE . E_K ] + DE . L_args l = [ ] for i , j in zip ( argterms , u ) : # We need to get around things like sqrt(x**2) != x # and also sqrt(x**2 + 2*x + 1) != x + 1 icoeff , iterms = sqf_list ( i ) l . append ( Mul ( * ( [ Pow ( icoeff , j ) ] + [ Pow ( b , e * j ) for b , e in iterms ] ) ) ) const = cancel ( fa . as_expr ( ) / fd . as_expr ( ) / Mul ( * l ) ) return ( ans , result , const )