MAYBE
We consider the system theBenchmark.

  Alphabet:

    0 : [] --> a 
    rec : [] --> (b -> (a -> c) -> a -> c) -> c -> a -> c 
    s : [] --> b -> b 

  Rules:

    rec f (g 0) => g 
    rec f (g (s x)) => f x (rec f (g x)) 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    0 : [] --> a 
    rec : [b -> (a -> c) -> a -> c * c] --> a -> c 
    s : [b] --> b 
    ~AP1 : [a -> c * a] --> c 
    ~AP2 : [b -> c * b] --> c 

  Rules:

    rec(F, ~AP1(G, 0)) => G 
    rec(F, ~AP2(G, s(X))) => F X rec(F, ~AP2(G, X)) 
    ~AP1(F, X) => F X 
    ~AP2(F, X) => F X 


+++ Citations +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.