58.74/57.97	MAYBE
58.74/57.98	We consider the system theBenchmark.
58.74/57.98	
58.74/57.98	  Alphabet:
58.74/57.98	
58.74/57.98	    0 : [] --> N 
58.74/57.98	    even : [] --> N -> N -> B 
58.74/57.98	    false : [] --> B 
58.74/57.98	    g : [] --> N -> B 
58.74/57.98	    h : [] --> N -> (N -> B) -> N -> B 
58.74/57.98	    not : [] --> B -> B 
58.74/57.98	    rec : [] --> (N -> (N -> B) -> N -> B) -> B -> N -> B 
58.74/57.98	    s : [] --> N -> N 
58.74/57.98	    true : [] --> B 
58.74/57.98	
58.74/57.98	  Rules:
58.74/57.98	
58.74/57.98	    rec f (i 0) => i 
58.74/57.98	    rec f (i (s x)) => f x (rec f (i x)) 
58.74/57.98	    g x => true 
58.74/57.98	    h x f y => not (f y) 
58.74/57.98	    not true => false 
58.74/57.98	    not false => true 
58.74/57.98	    even x y => rec h (g x) y 
58.74/57.98	
58.74/57.98	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.
58.74/57.98	
58.74/57.98	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:
58.74/57.98	
58.74/57.98	  Alphabet:
58.74/57.98	
58.74/57.98	    0 : [] --> N 
58.74/57.98	    even : [N] --> N -> B 
58.74/57.98	    false : [] --> B 
58.74/57.98	    g : [] --> N -> B 
58.74/57.98	    h : [] --> N -> (N -> B) -> N -> B 
58.74/57.98	    not : [B] --> B 
58.74/57.98	    rec : [N -> (N -> B) -> N -> B * B] --> N -> B 
58.74/57.98	    s : [N] --> N 
58.74/57.98	    true : [] --> B 
58.74/57.98	    ~AP1 : [N -> B * N] --> B 
58.74/57.98	
58.74/57.98	  Rules:
58.74/57.98	
58.74/57.98	    rec(F, ~AP1(G, 0)) => G 
58.74/57.98	    rec(F, ~AP1(G, s(X))) => F X rec(F, ~AP1(G, X)) 
58.74/57.98	    g X => true 
58.74/57.98	    h X F Y => not(~AP1(F, Y)) 
58.74/57.98	    not(true) => false 
58.74/57.98	    not(false) => true 
58.74/57.98	    even(X) Y => ~AP1(rec(h, g X), Y) 
58.74/57.98	    rec(F, even(X) 0) => even(X) 
58.74/57.98	    rec(F, g 0) => g 
58.74/57.98	    rec(F, h X G 0) => h X G 
58.74/57.98	    rec(F, even(X) s(Y)) => F Y rec(F, even(X) Y) 
58.74/57.98	    rec(F, g s(X)) => F X rec(F, g X) 
58.74/57.98	    rec(F, h X G s(Y)) => F Y rec(F, h X G Y) 
58.74/57.98	    ~AP1(F, X) => F X 
58.74/57.98	
58.74/57.98	
58.74/57.98	+++ Citations +++
58.74/57.98	
58.74/57.98	[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
59.04/58.28	EOF