<c11:model xmlns:c11="http://www.cellml.org/cellml/1.1#" xmlns:c12="http://www.cellml.org/cellml/1.2#" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">
  <c11:component name="ComplexUtilities">
    <!-- Connect up to the type you want this component to work with. -->
    <c12:variable name="base_type" type="type" public_interface="yes"/>

    <!-- The complex_type (can be computed from base_type). -->
    <c12:variable name="complex_type" type="type" public_interface="yes" />
    <c12:variable name="complex_to_base_function_type" type="type" public_interface="yes" />

    <!-- A function for fetching cartesian X co-ordinates. -->
    <c12:variable name="complex_get_cartesian_x" type="complex_to_base_function_type" public_interface="yes" />

    <c12:variable name="type_from_type_function_type" type="type" private_interface="yes" />
    <c12:variable name="complex_type_function" type="type_from_type_function_type" private_interface="yes"/>

    <!-- Local types -->
    <c12:variable name="base_type_to_dimensionless_function_type" type="type" />
    <c12:variable name="base_type_base_type_to_radians_function_type" type="type" />

    <!-- Local functions -->
    <c12:variable name="sign" type="base_type_to_dimensionless_function_type" />
    <c12:variable name="arctan2" type="base_type_base_type_to_radians_function_type" />

    <!-- Local bound variables -->
    <c12:variable name="complex" type="complex_type" />
    <c12:variable name="x" type="base_type" />
    <c12:variable name="y" type="base_type" />

    <m:math>
      <!-- An example of a units computation... -->
      <m:apply><m:eq/>
        <m:ci>complex_type</m:ci>
        <m:apply><m:ci>complex_type_function</m:ci>
          <m:ci>base_type</m:ci>
        </m:apply>
      </m:apply>

      <m:apply><m:eq/>
        <m:ci>complex_to_base_function_type</m:ci>
        <m:apply><c12:function_type/>
          <m:ci>base_type</m:ci>
          <m:ci>complex_type</m:ci>
        </m:apply>
      </m:apply>

      <m:apply><m:eq/>
        <m:ci>complex_get_real</m:ci>
        <m:lambda>
          <m:bvar><m:ci>complex</m:ci></m:bvar>
          <m:apply><c12:vector_element/>
            <m:ci>complex</m:ci>
            <m:cn c12:type="real_dimensionless">0</m:cn>
          </m:apply>
        </m:lambda>
      </m:apply>

      <m:apply><m:eq/>
        <m:ci>complex_get_imaginary</m:ci>
        <m:lambda>
          <m:bvar><m:ci>complex</m:ci></m:bvar>
          <m:apply><c12:vector_element/>
            <m:ci>complex</m:ci>
            <m:cn c12:type="real_dimensionless">1</m:cn>
          </m:apply>
        </m:lambda>
      </m:apply>

      <m:apply><m:eq/>
        <m:ci>complex_get_polar_modulus</m:ci>
        <m:lambda>
          <m:bvar><m:ci>complex</m:ci></m:bvar>
          <m:apply><m:root/>
            <m:apply><m:plus/>
              <m:apply><m:pow/>
                <m:apply><m:ci>complex_get_real</m:ci>
                  <m:ci>complex</m:ci>
                </m:apply>
                <m:cn c12:type="real_dimensionless">2</m:cn>
              </m:apply>
              <m:apply><m:pow/>
                <m:apply><m:ci>complex_get_imaginary</m:ci>
                  <m:ci>complex</m:ci>
                </m:apply>
                <m:cn c12:type="real_dimensionless">2</m:cn>
              </m:apply>
            </m:apply>
          </m:apply>
        </m:lambda>
      </m:apply>

      <m:apply><m:eq/>
        <m:ci>base_type_to_dimensionless_function_type</m:ci>
        <m:apply><c12:function_type/>
          <m:ci>real_dimensionless</m:ci>
          <m:ci>base_type</m:ci>
        </m:apply>
      </m:apply>

      <m:apply><m:eq/>
        <m:ci>base_type_base_type_to_radians_function_type</m:ci>
        <m:apply><c12:function_type/>
          <m:ci>real_radians</m:ci>
          <m:ci>base_type</m:ci>
          <m:ci>base_type</m:ci>
        </m:apply>
      </m:apply>

      <!-- This is a useful function; we could put it in another library. -->
      <m:apply><m:eq/>
        <m:ci>sign</m:ci>
        <m:lambda>
          <m:bvar>x</m:bvar>
          <m:piecewise>
            <m:piece>
              <m:cn c12:type="real_dimensionless">0</m:cn>
              <m:apply><m:eq/>
                <m:ci>x</m:ci>
                <m:cn c12:type="base_type">0</m:cn>
              </m:apply>
            </m:piece>
            <m:piece>
              <m:cn c12:type="real_dimensionless">-1</m:cn>
              <m:apply><m:lt/>
                <m:ci>x</m:ci>
                <m:cn c12:type="base_type">0</m:cn>
              </m:apply>
            </m:piece>
            <m:otherwise>
              <m:cn c12:type="real_dimensionless">1</m:cn>
            </m:otherwise>
          </m:piecewise>
        </m:lambda>
      </m:apply>

      <m:apply><m:eq/>
        <m:ci>arctan2</m:ci>
        <m:lambda>
          <m:bvar><m:ci>y</m:ci></m:bvar>
          <m:bvar><m:ci>x</m:ci></m:bvar>
          <m:apply><m:times/>
            <m:piecewise>
              <m:piece>
                <m:apply><m:divide/>
                  <m:pi/>
                  <m:cn c12:type="real_dimensionless">2</m:cn>
                </m:apply>
                <m:apply><m:eq/>
                  <m:ci>x</m:ci>
                  <m:cn c12:type="base_type">0</m:cn>
                </m:apply>
              </m:piece>
              
              <m:piece>
                <m:apply><m:arctan/>
                  <m:apply><m:abs/>
                    <m:apply><m:divide/>
                      <m:ci>y</m:ci>
                      <m:ci>x</m:ci>
                    </m:apply>
                  </m:apply>
                </m:apply>
                <m:apply><m:gt/>
                  <m:ci>x</m:ci>
                  <m:cn c12:type="base_type">0</m:cn>
                </m:apply>
              </m:piece>
              
              <m:otherwise>
                <m:apply><m:minus/>
                  <m:pi />
                  <m:apply><m:arctan/>
                    <m:apply><m:abs/>
                      <m:apply><m:divide/>
                        <m:ci>y</m:ci>
                        <m:ci>x</m:ci>
                      </m:apply>
                    </m:apply>
                  </m:apply>
                </m:apply>
              </m:otherwise>
            </m:piecewise>
            
            <m:apply><m:ci>sign</m:ci>
              <m:ci>y</m:ci>
            </m:apply>
          </m:apply>
        </m:lambda>
      </m:apply>

      <m:apply><m:eq/>
        <m:ci>complex_get_polar_argument</m:ci>
        <m:lambda>
          <m:bvar><m:ci>complex</m:ci></m:bvar>
          <m:apply><m:ci>arctan2</m:ci>
            <m:apply><m:ci>complex_get_cartesian_y</m:ci>
              <m:ci>complex</m:ci>
            </m:apply>
            <m:apply><m:ci>complex_get_cartesian_x</m:ci>
              <m:ci>complex</m:ci>
            </m:apply>
          </m:apply>
        </m:lambda>
      </m:apply>
    </m:math>
  </c11:component>
  
  <c11:import xlink:href="type_library.cellml">
    <c11:component name="TypeLibrary" component_ref="TypeLibrary"/>
  </c11:import>
  
  <c12:encapsulation>
    <c11:component_ref component_ref="ComplexUtilities">
      <c11:component_ref component_ref="TypeLibrary"/>
    </c11:component_ref>
  </c12:encapsulation>

  <c12:connection component_1="TypeLibrary" component_2="ComplexUtilities">
    <c11:map_variables variable_1="type_from_type_function_type" variable_2="type_from_type_function_type" />
    <c11:map_variables variable_1="complex_type_function" variable_2="complex_type_function" />
  </c12:connection>
</c11:model>