28.61/27.87	MAYBE
28.71/27.88	We consider the system theBenchmark.
28.71/27.88	
28.71/27.88	  Alphabet:
28.71/27.88	
28.71/27.88	    0 : [] --> N 
28.71/27.88	    false : [] --> B 
28.71/27.88	    g : [] --> N -> B 
28.71/27.88	    g2 : [] --> N -> B 
28.71/27.88	    geq : [] --> N -> N -> B 
28.71/27.88	    h : [] --> N -> (N -> B) -> N -> B 
28.71/27.88	    h2 : [] --> N -> (N -> B) -> N -> B 
28.71/27.88	    iszero : [] --> N -> N -> B 
28.71/27.88	    pred : [] --> N -> N 
28.71/27.88	    rec : [] --> (N -> (N -> B) -> N -> B) -> B -> N -> B 
28.71/27.88	    s : [] --> N -> N 
28.71/27.88	    true : [] --> B 
28.71/27.88	
28.71/27.88	  Rules:
28.71/27.88	
28.71/27.88	    rec f (i 0) => i 
28.71/27.88	    g x => true 
28.71/27.88	    h x f y => false 
28.71/27.88	    iszero x y => rec h (g x) y 
28.71/27.88	    pred 0 => 0 
28.71/27.88	    pred (s x) => x 
28.71/27.88	    g2 x => iszero x 0 
28.71/27.88	    h2 x f y => f (pred y) 
28.71/27.88	    geq x y => rec h2 (g2 x) y 
28.71/27.88	
28.71/27.88	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.
28.71/27.88	
28.71/27.88	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:
28.71/27.88	
28.71/27.88	  Alphabet:
28.71/27.88	
28.71/27.88	    0 : [] --> N 
28.71/27.88	    false : [] --> B 
28.71/27.88	    g : [] --> N -> B 
28.71/27.88	    g2 : [] --> N -> B 
28.71/27.88	    geq : [N] --> N -> B 
28.71/27.88	    h : [] --> N -> (N -> B) -> N -> B 
28.71/27.88	    h2 : [] --> N -> (N -> B) -> N -> B 
28.71/27.88	    iszero : [N] --> N -> B 
28.71/27.88	    pred : [N] --> N 
28.71/27.88	    rec : [N -> (N -> B) -> N -> B * B] --> N -> B 
28.71/27.88	    s : [N] --> N 
28.71/27.88	    true : [] --> B 
28.71/27.88	    ~AP1 : [N -> B * N] --> B 
28.71/27.88	
28.71/27.88	  Rules:
28.71/27.88	
28.71/27.88	    rec(F, ~AP1(G, 0)) => G 
28.71/27.88	    g X => true 
28.71/27.88	    h X F Y => false 
28.71/27.88	    iszero(X) Y => ~AP1(rec(h, g X), Y) 
28.71/27.88	    pred(0) => 0 
28.71/27.88	    pred(s(X)) => X 
28.71/27.88	    g2 X => iszero(X) 0 
28.71/27.88	    h2 X F Y => ~AP1(F, pred(Y)) 
28.71/27.88	    geq(X) Y => ~AP1(rec(h2, g2 X), Y) 
28.71/27.88	    rec(F, g 0) => g 
28.71/27.88	    rec(F, g2 0) => g2 
28.71/27.88	    rec(F, geq(X) 0) => geq(X) 
28.71/27.88	    rec(F, h X G 0) => h X G 
28.71/27.88	    rec(F, h2 X G 0) => h2 X G 
28.71/27.88	    rec(F, iszero(X) 0) => iszero(X) 
28.71/27.88	    ~AP1(F, X) => F X 
28.71/27.88	
28.71/27.88	
28.71/27.88	+++ Citations +++
28.71/27.88	
28.71/27.88	[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
28.85/28.05	EOF