Bekić's theorem
In computability theory, Bekić's theorem or Bekić's lemma is a theorem about fixed-points which allows splitting a mutual recursion into recursions on one variable at a time. It was created by Austrian Hans Bekić (1936-1982) in 1969, and published posthumously in a book by Cliff Jones in 1984. The theorem is set up as follows. Consider two operators  and  on pointed directed-complete partial orders  and , continuous in each component. Then define the operator . This is monotone with respect to the product order (componentwise order). By the Kleene fixed-point theorem, it has a least fixed point , a pair  in  such that  and . Bekić's theorem (called the "bisection lemma" in his notes) is that the simultaneous least fixed point  can be separated into a series of least fixed points on  and , in particular:

In this presentation  is defined in terms of . It can instead be defined in a symmetric presentation:

Proof (Bekić):
 since it is the fixed point. Similarly . Hence  is a fixed point of . Conversely, if there is a pre-fixed point  with , then  and ; hence  and  is the minimal fixed point. Variants

In a complete lattice 
A variant of the theorem strengthens the conditions on  and  to be that they are complete lattices, and finds the least fixed point using the Knaster–Tarski theorem. The requirement for continuity of  and  can then be weakened to only requiring them to be monotonic functions. Categorical formulation 
Bekić's lemma has been generalized to fix-points of endofunctors of categories (initial -algebras). Given two functors  and  such that all  and  exist, the fix-point  is carried by the pair:

Usage 

Bekić's theorem can be applied repeatedly to find the least fixed point of a tuple in terms of least fixed points of single variables. Although the resulting expression might become rather complex, it can be easier to reason about fixed points of single variables when designing an automated theorem prover. References 

Order theory
Fixed-point theorems