49.18/48.43	MAYBE
49.18/48.44	We consider the system theBenchmark.
49.18/48.44	
49.18/48.44	  Alphabet:
49.18/48.44	
49.18/48.44	    0 : [] --> N 
49.18/48.44	    false : [] --> B 
49.18/48.44	    g : [] --> N -> B 
49.18/48.44	    h : [] --> N -> (N -> B) -> N -> B 
49.18/48.44	    iszero : [] --> N -> N -> B 
49.18/48.44	    rec : [] --> (N -> (N -> B) -> N -> B) -> B -> N -> B 
49.18/48.44	    s : [] --> N -> N 
49.18/48.44	    true : [] --> B 
49.18/48.44	
49.18/48.44	  Rules:
49.18/48.44	
49.18/48.44	    rec f (i 0) => i 
49.18/48.44	    rec f (i (s x)) => f x (rec f (i x)) 
49.18/48.44	    g x => true 
49.18/48.44	    h x f y => false 
49.18/48.44	    iszero x y => rec h (g x) y 
49.18/48.44	
49.18/48.44	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.
49.18/48.44	
49.18/48.44	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:
49.18/48.44	
49.18/48.44	  Alphabet:
49.18/48.44	
49.18/48.44	    0 : [] --> N 
49.18/48.44	    false : [] --> B 
49.18/48.44	    g : [] --> N -> B 
49.18/48.44	    h : [] --> N -> (N -> B) -> N -> B 
49.18/48.44	    iszero : [N] --> N -> B 
49.18/48.44	    rec : [N -> (N -> B) -> N -> B * B] --> N -> B 
49.18/48.44	    s : [N] --> N 
49.18/48.44	    true : [] --> B 
49.18/48.44	    ~AP1 : [N -> B * N] --> B 
49.18/48.44	
49.18/48.44	  Rules:
49.18/48.44	
49.18/48.44	    rec(F, ~AP1(G, 0)) => G 
49.18/48.44	    rec(F, ~AP1(G, s(X))) => F X rec(F, ~AP1(G, X)) 
49.18/48.44	    g X => true 
49.18/48.44	    h X F Y => false 
49.18/48.44	    iszero(X) Y => ~AP1(rec(h, g X), Y) 
49.18/48.44	    rec(F, g 0) => g 
49.18/48.44	    rec(F, h X G 0) => h X G 
49.18/48.44	    rec(F, iszero(X) 0) => iszero(X) 
49.18/48.44	    rec(F, g s(X)) => F X rec(F, g X) 
49.18/48.44	    rec(F, h X G s(Y)) => F Y rec(F, h X G Y) 
49.18/48.44	    rec(F, iszero(X) s(Y)) => F Y rec(F, iszero(X) Y) 
49.18/48.44	    ~AP1(F, X) => F X 
49.18/48.44	
49.18/48.44	
49.18/48.44	+++ Citations +++
49.18/48.44	
49.18/48.44	[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
49.43/48.66	EOF