0.00/0.01	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	    lim : [] --> (N -> O) -> O 
0.00/0.02	    plus : [] --> O -> O -> O 
0.00/0.02	    s : [] --> O -> O 
0.00/0.02	    z : [] --> O 
0.00/0.02	
0.00/0.02	  Rules:
0.00/0.02	
0.00/0.02	    plus z x => x 
0.00/0.02	    plus (s x) y => s (plus x y) 
0.00/0.02	    plus (lim (/\x.f x)) y => lim (/\u.plus (f u) y) 
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	    lim : [N -> O] --> O 
0.00/0.02	    plus : [O * O] --> O 
0.00/0.02	    s : [O] --> O 
0.00/0.02	    z : [] --> O 
0.00/0.02	    ~AP1 : [N -> O * N] --> O 
0.00/0.02	
0.00/0.02	  Rules:
0.00/0.02	
0.00/0.02	    plus(z, X) => X 
0.00/0.02	    plus(s(X), Y) => s(plus(X, Y)) 
0.00/0.02	    plus(lim(/\x.~AP1(F, x)), X) => lim(/\y.plus(~AP1(F, y), X)) 
0.00/0.02	    ~AP1(F, X) => F X 
0.00/0.02	
0.00/0.02	Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:
0.00/0.02	
0.00/0.02	  Alphabet:
0.00/0.02	
0.00/0.02	    lim : [N -> O] --> O 
0.00/0.02	    plus : [O * O] --> O 
0.00/0.02	    s : [O] --> O 
0.00/0.02	    z : [] --> O 
0.00/0.02	
0.00/0.02	  Rules:
0.00/0.02	
0.00/0.02	    plus(z, X) => X 
0.00/0.02	    plus(s(X), Y) => s(plus(X, Y)) 
0.00/0.02	    plus(lim(/\x.X(x)), Y) => lim(/\y.plus(X(y), Y)) 
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	  plus(z, X) >? X 
0.00/0.02	  plus(s(X), Y) >? s(plus(X, Y)) 
0.00/0.02	  plus(lim(/\x.X(x)), Y) >? lim(/\y.plus(X(y), Y)) 
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	  lim = \G0.3 + G0(0) 
0.00/0.02	  plus = \y0y1.3 + y1 + 3y0 
0.00/0.02	  s = \y0.3 + y0 
0.00/0.02	  z = 3 
0.00/0.02	
0.00/0.02	Using this interpretation, the requirements translate to:
0.00/0.02	
0.00/0.02	  [[plus(z, _x0)]] = 12 + x0 > x0 = [[_x0]] 
0.00/0.02	  [[plus(s(_x0), _x1)]] = 12 + x1 + 3x0 > 6 + x1 + 3x0 = [[s(plus(_x0, _x1))]] 
0.00/0.02	  [[plus(lim(/\x._x0(x)), _x1)]] = 12 + x1 + 3F0(0) > 6 + x1 + 3F0(0) = [[lim(/\x.plus(_x0(x), _x1))]] 
0.00/0.02	
0.00/0.02	We can thus remove the following rules:
0.00/0.02	
0.00/0.02	  plus(z, X) => X 
0.00/0.02	  plus(s(X), Y) => s(plus(X, Y)) 
0.00/0.02	  plus(lim(/\x.X(x)), Y) => lim(/\y.plus(X(y), Y)) 
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