0.00/0.08	YES
0.00/0.08	We consider the system theBenchmark.
0.00/0.08	
0.00/0.08	  Alphabet:
0.00/0.08	
0.00/0.08	    cons : [c * d] --> d 
0.00/0.08	    fcons : [b -> c * a] --> a 
0.00/0.08	    fmap : [a * b] --> d 
0.00/0.08	    fnil : [] --> a 
0.00/0.08	    nil : [] --> d 
0.00/0.08	
0.00/0.08	  Rules:
0.00/0.08	
0.00/0.08	    fmap(fnil, x) => nil 
0.00/0.08	    fmap(fcons(f, x), y) => cons(f y, fmap(x, y)) 
0.00/0.08	
0.00/0.08	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.00/0.08	
0.00/0.08	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]).
0.00/0.08	
0.00/0.08	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
0.00/0.08	
0.00/0.08	  Dependency Pairs P_0:
0.00/0.08	
0.00/0.08	    0] fmap#(fcons(F, X), Y) =#> fmap#(X, Y)   
0.00/0.08	
0.00/0.08	  Rules R_0:
0.00/0.08	
0.00/0.08	    fmap(fnil, X) => nil 
0.00/0.08	    fmap(fcons(F, X), Y) => cons(F Y, fmap(X, Y)) 
0.00/0.08	
0.00/0.08	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
0.00/0.08	
0.00/0.08	We consider the dependency pair problem (P_0, R_0, static, formative).
0.00/0.08	
0.00/0.08	We apply the subterm criterion with the following projection function:
0.00/0.08	
0.00/0.08	  nu(fmap#) = 1 
0.00/0.08	
0.00/0.08	Thus, we can orient the dependency pairs as follows:
0.00/0.08	
0.00/0.08	  nu(fmap#(fcons(F, X), Y)) = fcons(F, X) |> X = nu(fmap#(X, Y)) 
0.00/0.08	
0.00/0.08	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.
0.00/0.08	
0.00/0.08	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
0.00/0.08	
0.00/0.08	
0.00/0.08	+++ Citations +++
0.00/0.08	
0.00/0.08	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.08	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
0.00/0.08	[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.
0.00/0.08	EOF