0.00/0.09	YES
0.00/0.09	We consider the system theBenchmark.
0.00/0.09	
0.00/0.09	  Alphabet:
0.00/0.09	
0.00/0.09	    cons : [a -> a * listf] --> listf 
0.00/0.09	    dapply : [a * a -> a * a -> a] --> a 
0.00/0.09	    lapply : [a * listf] --> a 
0.00/0.09	    nil : [] --> listf 
0.00/0.09	
0.00/0.09	  Rules:
0.00/0.09	
0.00/0.09	    dapply(x, f, g) => f (g x) 
0.00/0.09	    lapply(x, nil) => x 
0.00/0.09	    lapply(x, cons(f, y)) => f lapply(x, y) 
0.00/0.09	
0.00/0.09	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.00/0.09	
0.00/0.09	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]).
0.00/0.09	
0.00/0.09	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
0.00/0.09	
0.00/0.09	  Dependency Pairs P_0:
0.00/0.09	
0.00/0.09	    0] lapply#(X, cons(F, Y)) =#> lapply#(X, Y)   
0.00/0.09	
0.00/0.09	  Rules R_0:
0.00/0.09	
0.00/0.09	    dapply(X, F, G) => F (G X) 
0.00/0.09	    lapply(X, nil) => X 
0.00/0.09	    lapply(X, cons(F, Y)) => F lapply(X, Y) 
0.00/0.09	
0.00/0.09	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
0.00/0.09	
0.00/0.09	We consider the dependency pair problem (P_0, R_0, static, formative).
0.00/0.09	
0.00/0.09	We apply the subterm criterion with the following projection function:
0.00/0.09	
0.00/0.09	  nu(lapply#) = 2 
0.00/0.09	
0.00/0.09	Thus, we can orient the dependency pairs as follows:
0.00/0.09	
0.00/0.09	  nu(lapply#(X, cons(F, Y))) = cons(F, Y) |> Y = nu(lapply#(X, Y)) 
0.00/0.09	
0.00/0.09	By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], 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.
0.00/0.09	
0.00/0.09	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
0.00/0.09	
0.00/0.09	
0.00/0.09	+++ Citations +++
0.00/0.09	
0.00/0.09	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.09	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
0.00/0.09	[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.
0.00/0.09	EOF