MAYBE
We consider the system theBenchmark.

  Alphabet:

    0 : [] --> N 
    false : [] --> B 
    g : [] --> N -> B 
    g2 : [] --> N -> B 
    geq : [] --> N -> N -> B 
    h : [] --> N -> (N -> B) -> N -> B 
    h2 : [] --> N -> (N -> B) -> N -> B 
    iszero : [] --> N -> N -> B 
    pred : [] --> N -> N 
    rec : [] --> (N -> (N -> B) -> N -> B) -> B -> N -> B 
    s : [] --> N -> N 
    true : [] --> B 

  Rules:

    rec f (i 0) => i 
    g x => true 
    h x f y => false 
    iszero x y => rec h (g x) y 
    pred 0 => 0 
    pred (s x) => x 
    g2 x => iszero x 0 
    h2 x f y => f (pred y) 
    geq x y => rec h2 (g2 x) y 

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 
    false : [] --> B 
    g : [] --> N -> B 
    g2 : [] --> N -> B 
    geq : [N] --> N -> B 
    h : [] --> N -> (N -> B) -> N -> B 
    h2 : [] --> N -> (N -> B) -> N -> B 
    iszero : [N] --> N -> B 
    pred : [N] --> N 
    rec : [N -> (N -> B) -> N -> B * B] --> N -> B 
    s : [N] --> N 
    true : [] --> B 
    ~AP1 : [N -> B * N] --> B 

  Rules:

    rec(F, ~AP1(G, 0)) => G 
    g X => true 
    h X F Y => false 
    iszero(X) Y => ~AP1(rec(h, g X), Y) 
    pred(0) => 0 
    pred(s(X)) => X 
    g2 X => iszero(X) 0 
    h2 X F Y => ~AP1(F, pred(Y)) 
    geq(X) Y => ~AP1(rec(h2, g2 X), Y) 
    rec(F, g 0) => g 
    rec(F, g2 0) => g2 
    rec(F, geq(X) 0) => geq(X) 
    rec(F, h X G 0) => h X G 
    rec(F, h2 X G 0) => h2 X G 
    rec(F, iszero(X) 0) => iszero(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.