0.00/0.01	YES
0.00/0.01	We consider the system theBenchmark.
0.00/0.01	
0.00/0.01	  Alphabet:
0.00/0.01	
0.00/0.01	    a : [] --> B 
0.00/0.01	    b : [] --> B 
0.00/0.01	    c : [] --> B 
0.00/0.01	    f : [A -> B] --> B 
0.00/0.01	
0.00/0.01	  Rules:
0.00/0.01	
0.00/0.01	    f(/\x.a) => b 
0.00/0.01	    b => f(/\x.c) 
0.00/0.01	
0.00/0.01	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.00/0.01	
0.00/0.01	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.01	
0.00/0.01	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.01	
0.00/0.01	  f(/\x.a) >? b 
0.00/0.01	  b >? f(/\x.c) 
0.00/0.01	
0.00/0.01	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.01	
0.00/0.01	The following interpretation satisfies the requirements:
0.00/0.01	
0.00/0.01	  a = 3 
0.00/0.01	  b = 2 
0.00/0.01	  c = 0 
0.00/0.01	  f = \G0.3G0(0) 
0.00/0.01	
0.00/0.01	Using this interpretation, the requirements translate to:
0.00/0.01	
0.00/0.01	  [[f(/\x.a)]] = 9 > 2 = [[b]] 
0.00/0.01	  [[b]] = 2 > 0 = [[f(/\x.c)]] 
0.00/0.01	
0.00/0.01	We can thus remove the following rules:
0.00/0.01	
0.00/0.01	  f(/\x.a) => b 
0.00/0.01	  b => f(/\x.c) 
0.00/0.01	
0.00/0.01	All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.
0.00/0.01	
0.00/0.01	
0.00/0.01	+++ Citations +++
0.00/0.01	
0.00/0.01	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.01	EOF