13.06/11.11	MAYBE
13.06/11.11	We consider the system theBenchmark.
13.06/11.11	
13.06/11.11	  Alphabet:
13.06/11.11	
13.06/11.11	    !plus : [nat * nat] --> nat 
13.06/11.11	    0 : [] --> nat 
13.06/11.11	    dec : [nat * nat] --> nat 
13.06/11.11	    p : [nat * nat] --> nat 
13.06/11.11	    quad : [nat] --> nat 
13.06/11.11	    rec : [nat * nat * nat * nat -> nat -> nat] --> nat 
13.06/11.11	    s : [nat] --> nat 
13.06/11.11	    sqr : [nat] --> nat 
13.06/11.11	    sumsqr : [nat] --> nat 
13.06/11.11	
13.06/11.11	  Rules:
13.06/11.11	
13.06/11.11	    dec(0, x) => 0 
13.06/11.11	    dec(x, 0) => x 
13.06/11.11	    dec(s(x), s(y)) => dec(x, y) 
13.06/11.11	    rec(0, x, y, f) => y 
13.06/11.11	    rec(s(x), s(y), z, f) => rec(dec(x, y), s(y), f z x, f) 
13.06/11.11	    sumsqr(x) => rec(x, s(0), 0, /\y./\z.!plus(sqr(p(s(z), 0)), y)) 
13.06/11.11	    !plus(0, x) => x 
13.06/11.11	    !plus(s(x), y) => s(!plus(x, y)) 
13.06/11.11	    quad(0) => 0 
13.06/11.11	    quad(s(x)) => s(s(s(s(quad(x))))) 
13.06/11.11	    sqr(p(0, 0)) => 0 
13.06/11.11	    sqr(p(s(s(x)), y)) => sqr(p(x, s(y))) 
13.06/11.11	    sqr(p(s(0), x)) => !plus(quad(sqr(p(x, 0))), s(quad(x))) 
13.06/11.11	    sqr(p(0, s(x))) => quad(sqr(p(s(x), 0))) 
13.06/11.11	
13.06/11.11	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
13.06/11.11	
13.06/11.13	EOF