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 : [] --> N 
0.00/0.01	    f : [N] --> N 
0.00/0.01	
0.00/0.01	  Rules:
0.00/0.01	
0.00/0.01	    g a => f(a) 
0.00/0.01	
0.00/0.01	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.
0.00/0.01	
0.00/0.01	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:
0.00/0.01	
0.00/0.01	  Alphabet:
0.00/0.01	
0.00/0.01	    a : [] --> N 
0.00/0.01	    f : [N] --> N 
0.00/0.01	    ~AP1 : [N -> N * N] --> N 
0.00/0.01	
0.00/0.01	  Rules:
0.00/0.01	
0.00/0.01	    ~AP1(F, a) => f(a) 
0.00/0.01	    ~AP1(F, X) => F X 
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	  ~AP1(F, a) >? f(a) 
0.00/0.01	  ~AP1(F, X) >? F X 
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 = 0 
0.00/0.01	  f = \y0.y0 
0.00/0.01	  ~AP1 = \G0y1.3 + 3y1 + G0(y1) 
0.00/0.01	
0.00/0.01	Using this interpretation, the requirements translate to:
0.00/0.01	
0.00/0.01	  [[~AP1(_F0, a)]] = 3 + F0(0) > 0 = [[f(a)]] 
0.00/0.01	  [[~AP1(_F0, _x1)]] = 3 + 3x1 + F0(x1) > x1 + F0(x1) = [[_F0 _x1]] 
0.00/0.01	
0.00/0.01	We can thus remove the following rules:
0.00/0.01	
0.00/0.01	  ~AP1(F, a) => f(a) 
0.00/0.01	  ~AP1(F, X) => F X 
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	[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
0.00/0.01	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.01	EOF