MAYBE
We consider the system theBenchmark.

  Alphabet:

    0 : [] --> N 
    rec : [] --> (N -> (N -> B) -> N -> B) -> B -> N -> B 
    s : [] --> N -> N 

  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 : [] --> N 
    rec : [N -> (N -> B) -> N -> B * B] --> N -> B 
    s : [N] --> N 
    ~AP1 : [N -> B * N] --> B 

  Rules:

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