Theorem. Fixed point theorem for endofunctors on \textbf {Cpo}^\textbf {R} [dt-005B]

Every cocontinuous endofunctor \mathcal {F} on \mathbf {Cpo}^\textbf {R} has a least fixed point, given by the colimit of the ascending \omega -chain:

Where the retraction pairs (f_i, g_i) are defined by: \begin {array}{lcl} (f_0,g_0) & \triangleq & (\lambda x. \bot , \lambda x. \bot )\\ (f_{i+1},g_{i+1}) & \triangleq & \mathcal {F}(f_i,g_i) \\ \end {array}