0.00/0.53	YES
0.00/0.54	We consider the system theBenchmark.
0.00/0.54	
0.00/0.54	  Alphabet:
0.00/0.54	
0.00/0.54	    0 : [] --> ord 
0.00/0.54	    lim : [nat -> ord] --> ord 
0.00/0.54	    rec : [ord * a * ord -> a -> a * (nat -> ord) -> (nat -> a) -> a] --> a 
0.00/0.54	    s : [ord] --> ord 
0.00/0.54	
0.00/0.54	  Rules:
0.00/0.54	
0.00/0.54	    rec(0, x, f, g) => x 
0.00/0.54	    rec(s(x), y, f, g) => f x rec(x, y, f, g) 
0.00/0.54	    rec(lim(f), x, g, h) => h f (/\y.rec(f y, x, g, h)) 
0.00/0.54	
0.00/0.54	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.00/0.54	
0.00/0.54	We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]).
0.00/0.54	
0.00/0.54	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
0.00/0.54	
0.00/0.54	  Dependency Pairs P_0:
0.00/0.54	
0.00/0.54	    0] rec#(s(X), Y, F, G) =#> rec#(X, Y, F, G)   
0.00/0.54	    1] rec#(lim(F), X, G, H) =#> rec#(F Y, X, G, H)   
0.00/0.54	
0.00/0.54	  Rules R_0:
0.00/0.54	
0.00/0.54	    rec(0, X, F, G) => X 
0.00/0.54	    rec(s(X), Y, F, G) => F X rec(X, Y, F, G) 
0.00/0.54	    rec(lim(F), X, G, H) => H F (/\x.rec(F x, X, G, H)) 
0.00/0.54	
0.00/0.54	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
0.00/0.54	
0.00/0.54	We consider the dependency pair problem (P_0, R_0, static, formative).
0.00/0.54	
0.00/0.54	We apply the accessible subterm criterion with the following projection function:
0.00/0.54	
0.00/0.54	  nu(rec#) = 1 
0.00/0.54	
0.00/0.54	Thus, we can orient the dependency pairs as follows:
0.00/0.54	
0.00/0.54	  nu(rec#(s(X), Y, F, G)) = s(X) [>] X = nu(rec#(X, Y, F, G)) 
0.00/0.54	  nu(rec#(lim(F), X, G, H)) = lim(F) [>] F Y = nu(rec#(F Y, X, G, H)) 
0.00/0.54	
0.00/0.54	By [Kop13, Thm. 6], we may replace a dependency pair problem (P_0, R_0, static, f) by ({}, R_0, static, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.
0.00/0.54	
0.00/0.54	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
0.00/0.54	
0.00/0.54	
0.00/0.54	+++ Citations +++
0.00/0.54	
0.00/0.54	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.54	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
0.00/0.54	[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.
0.00/0.54	EOF