0.00/0.02	YES
0.00/0.02	We consider the system theBenchmark.
0.00/0.02	
0.00/0.02	  Alphabet:
0.00/0.02	
0.00/0.02	    pair : [nat -> nat * nat] --> nat 
0.00/0.02	    split : [nat] --> nat 
0.00/0.02	
0.00/0.02	  Rules:
0.00/0.02	
0.00/0.02	    split(f x) => pair(f, x) 
0.00/0.02	
0.00/0.02	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.
0.00/0.02	
0.00/0.02	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:
0.00/0.02	
0.00/0.02	  Alphabet:
0.00/0.02	
0.00/0.02	    pair : [nat -> nat * nat] --> nat 
0.00/0.02	    split : [nat] --> nat 
0.00/0.02	    ~AP1 : [nat -> nat * nat] --> nat 
0.00/0.02	
0.00/0.02	  Rules:
0.00/0.02	
0.00/0.02	    split(~AP1(F, X)) => pair(F, X) 
0.00/0.02	    ~AP1(F, X) => F X 
0.00/0.02	
0.00/0.02	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.02	
0.00/0.02	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.02	
0.00/0.02	  split(~AP1(F, X)) >? pair(F, X) 
0.00/0.02	  ~AP1(F, X) >? F X 
0.00/0.02	
0.00/0.02	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.02	
0.00/0.02	The following interpretation satisfies the requirements:
0.00/0.02	
0.00/0.02	  pair = \G0y1.y1 + G0(0) 
0.00/0.02	  split = \y0.3 + 3y0 
0.00/0.02	  ~AP1 = \G0y1.3 + 3y1 + 3G0(0) + 3G0(y1) 
0.00/0.02	
0.00/0.02	Using this interpretation, the requirements translate to:
0.00/0.02	
0.00/0.02	  [[split(~AP1(_F0, _x1))]] = 12 + 9x1 + 9F0(0) + 9F0(x1) > x1 + F0(0) = [[pair(_F0, _x1)]] 
0.00/0.02	  [[~AP1(_F0, _x1)]] = 3 + 3x1 + 3F0(0) + 3F0(x1) > x1 + F0(x1) = [[_F0 _x1]] 
0.00/0.02	
0.00/0.02	We can thus remove the following rules:
0.00/0.02	
0.00/0.02	  split(~AP1(F, X)) => pair(F, X) 
0.00/0.02	  ~AP1(F, X) => F X 
0.00/0.02	
0.00/0.02	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.
0.00/0.02	
0.00/0.02	
0.00/0.02	+++ Citations +++
0.00/0.02	
0.00/0.02	[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
0.00/0.02	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.02	EOF