81.24/81.30	MAYBE
81.24/81.31	We consider the system theBenchmark.
81.24/81.31	
81.24/81.31	  Alphabet:
81.24/81.31	
81.24/81.31	    cons : [] --> a -> alist -> alist 
81.24/81.31	    map : [] --> (a -> a) -> alist -> alist 
81.24/81.31	    nil : [] --> alist 
81.24/81.31	    o : [] --> (a -> a) -> (a -> a) -> a -> a 
81.24/81.31	    xap : [] --> (a -> a) -> a -> a 
81.24/81.31	
81.24/81.31	  Rules:
81.24/81.31	
81.24/81.31	    map (/\x.xap f x) nil => nil 
81.24/81.31	    map (/\x.xap f x) (cons y z) => cons (xap f y) (map (/\u.xap f u) z) 
81.24/81.31	    map (/\x.xap f x) (map (/\y.xap g y) z) => map (o (/\u.xap f u) (/\v.xap g v)) z 
81.24/81.31	    o (/\x.f x) (/\y.g y) z => f (g z) 
81.24/81.31	    xap f x => f x 
81.24/81.31	
81.24/81.31	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.
81.24/81.31	
81.24/81.31	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:
81.24/81.31	
81.24/81.31	  Alphabet:
81.24/81.31	
81.24/81.31	    cons : [a * alist] --> alist 
81.24/81.31	    map : [a -> a * alist] --> alist 
81.24/81.31	    nil : [] --> alist 
81.24/81.31	    o : [a -> a * a -> a] --> a -> a 
81.24/81.31	    xap : [a -> a * a] --> a 
81.24/81.31	    ~AP1 : [a -> a * a] --> a 
81.24/81.31	
81.24/81.31	  Rules:
81.24/81.31	
81.24/81.31	    map(/\x.xap(F, x), nil) => nil 
81.24/81.31	    map(/\x.xap(F, x), cons(X, Y)) => cons(xap(F, X), map(/\y.xap(F, y), Y)) 
81.24/81.31	    map(/\x.xap(F, x), map(/\y.xap(G, y), X)) => map(o(/\z.xap(F, z), /\u.xap(G, u)), X) 
81.24/81.31	    o(/\x.~AP1(F, x), /\y.~AP1(G, y)) X => ~AP1(F, ~AP1(G, X)) 
81.24/81.31	    xap(F, X) => ~AP1(F, X) 
81.24/81.31	    o(/\x.o(F, G) x, /\y.~AP1(H, y)) X => o(F, G) ~AP1(H, X) 
81.24/81.31	    o(/\x.xap(F, x), /\y.~AP1(G, y)) X => xap(F, ~AP1(G, X)) 
81.24/81.31	    o(/\x.~AP1(F, x), /\y.o(G, H) y) X => ~AP1(F, o(G, H) X) 
81.24/81.31	    o(/\x.~AP1(F, x), /\y.xap(G, y)) X => ~AP1(F, xap(G, X)) 
81.24/81.31	    o(/\x.o(F, G) x, /\y.o(H, I) y) X => o(F, G) (o(H, I) X) 
81.24/81.31	    o(/\x.o(F, G) x, /\y.xap(H, y)) X => o(F, G) xap(H, X) 
81.24/81.31	    o(/\x.xap(F, x), /\y.o(G, H) y) X => xap(F, o(G, H) X) 
81.24/81.31	    o(/\x.xap(F, x), /\y.xap(G, y)) X => xap(F, xap(G, X)) 
81.24/81.31	    ~AP1(F, X) => F X 
81.24/81.31	
81.24/81.31	
81.24/81.31	+++ Citations +++
81.24/81.31	
81.24/81.31	[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
81.32/81.40	EOF