0.00/0.25	YES
0.00/0.26	We consider the system theBenchmark.
0.00/0.26	
0.00/0.26	  Alphabet:
0.00/0.26	
0.00/0.26	    cons : [] --> a -> alist -> alist 
0.00/0.26	    map : [] --> (a -> a) -> alist -> alist 
0.00/0.26	    nil : [] --> alist 
0.00/0.26	    o : [] --> (a -> a) -> (a -> a) -> a -> a 
0.00/0.26	
0.00/0.26	  Rules:
0.00/0.26	
0.00/0.26	    map (/\x.f x) nil => nil 
0.00/0.26	    map (/\x.f x) (cons y z) => cons (f y) (map (/\u.f u) z) 
0.00/0.26	    map (/\x.f x) (map (/\y.g y) z) => map (/\u.o (/\v.f v) (/\w.g w) u) z 
0.00/0.26	    o (/\x.f x) (/\y.g y) z => f (g z) 
0.00/0.26	
0.00/0.26	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.
0.00/0.26	
0.00/0.26	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:
0.00/0.26	
0.00/0.26	  Alphabet:
0.00/0.26	
0.00/0.26	    cons : [a * alist] --> alist 
0.00/0.26	    map : [a -> a * alist] --> alist 
0.00/0.26	    nil : [] --> alist 
0.00/0.26	    o : [a -> a * a -> a * a] --> a 
0.00/0.26	    ~AP1 : [a -> a * a] --> a 
0.00/0.26	
0.00/0.26	  Rules:
0.00/0.26	
0.00/0.26	    map(/\x.~AP1(F, x), nil) => nil 
0.00/0.27	    map(/\x.~AP1(F, x), cons(X, Y)) => cons(~AP1(F, X), map(/\y.~AP1(F, y), Y)) 
0.00/0.27	    map(/\x.~AP1(F, x), map(/\y.~AP1(G, y), X)) => map(/\z.o(/\u.~AP1(F, u), /\v.~AP1(G, v), z), X) 
0.00/0.27	    o(/\x.~AP1(F, x), /\y.~AP1(G, y), X) => ~AP1(F, ~AP1(G, X)) 
0.00/0.27	    map(/\x.o(F, G, x), nil) => nil 
0.00/0.27	    map(/\x.o(F, G, x), cons(X, Y)) => cons(o(F, G, X), map(/\y.o(F, G, y), Y)) 
0.00/0.27	    map(/\x.o(F, G, x), map(/\y.~AP1(H, y), X)) => map(/\z.o(/\u.o(F, G, u), /\v.~AP1(H, v), z), X) 
0.00/0.27	    map(/\x.~AP1(F, x), map(/\y.o(G, H, y), X)) => map(/\z.o(/\u.~AP1(F, u), /\v.o(G, H, v), z), X) 
0.00/0.27	    o(/\x.o(F, G, x), /\y.~AP1(H, y), X) => o(F, G, ~AP1(H, X)) 
0.00/0.27	    o(/\x.~AP1(F, x), /\y.o(G, H, y), X) => ~AP1(F, o(G, H, X)) 
0.00/0.27	    map(/\x.o(F, G, x), map(/\y.o(H, I, y), X)) => map(/\z.o(/\u.o(F, G, u), /\v.o(H, I, v), z), X) 
0.00/0.27	    o(/\x.o(F, G, x), /\y.o(H, I, y), X) => o(F, G, o(H, I, X)) 
0.00/0.27	    ~AP1(F, X) => F X 
0.00/0.27	
0.00/0.27	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.  Additionally, we can remove some (now-)redundant rules.  This gives:
0.00/0.27	
0.00/0.27	  Alphabet:
0.00/0.27	
0.00/0.27	    cons : [a * alist] --> alist 
0.00/0.27	    map : [a -> a * alist] --> alist 
0.00/0.27	    nil : [] --> alist 
0.00/0.27	    o : [a -> a * a -> a * a] --> a 
0.00/0.27	
0.00/0.27	  Rules:
0.00/0.27	
0.00/0.27	    map(/\x.X(x), nil) => nil 
0.00/0.27	    map(/\x.X(x), cons(Y, Z)) => cons(X(Y), map(/\y.X(y), Z)) 
0.00/0.27	    map(/\x.X(x), map(/\y.Y(y), Z)) => map(/\z.o(/\u.X(u), /\v.Y(v), z), Z) 
0.00/0.27	    o(/\x.X(x), /\y.Y(y), Z) => X(Y(Z)) 
0.00/0.27	
0.00/0.27	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]).
0.00/0.27	
0.00/0.27	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
0.00/0.27	
0.00/0.27	  Dependency Pairs P_0:
0.00/0.27	
0.00/0.27	    0] map#(/\x.X(x), cons(Y, Z)) =#> map#(/\y.X(y), Z)   
0.00/0.27	    1] map#(/\x.X(x), map(/\y.Y(y), Z)) =#> map#(/\z.o(/\u.X(u), /\v.Y(v), z), Z)   
0.00/0.27	    2] map#(/\x.X(x), map(/\y.Y(y), Z)) =#> o#(/\z.X(z), /\u.Y(u), U)   
0.00/0.27	
0.00/0.27	  Rules R_0:
0.00/0.27	
0.00/0.27	    map(/\x.X(x), nil) => nil 
0.00/0.27	    map(/\x.X(x), cons(Y, Z)) => cons(X(Y), map(/\y.X(y), Z)) 
0.00/0.27	    map(/\x.X(x), map(/\y.Y(y), Z)) => map(/\z.o(/\u.X(u), /\v.Y(v), z), Z) 
0.00/0.27	    o(/\x.X(x), /\y.Y(y), Z) => X(Y(Z)) 
0.00/0.27	
0.00/0.27	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
0.00/0.27	
0.00/0.27	We consider the dependency pair problem (P_0, R_0, static, formative).
0.00/0.27	
0.00/0.27	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
0.00/0.27	
0.00/0.27	    * 0 : 0, 1, 2 
0.00/0.27	    * 1 : 0, 1, 2 
0.00/0.27	    * 2 :  
0.00/0.27	
0.00/0.27	This graph has the following strongly connected components:
0.00/0.27	
0.00/0.27	  P_1:
0.00/0.27	
0.00/0.27	    map#(/\x.X(x), cons(Y, Z)) =#> map#(/\y.X(y), Z)   
0.00/0.27	    map#(/\x.X(x), map(/\y.Y(y), Z)) =#> map#(/\z.o(/\u.X(u), /\v.Y(v), z), Z)   
0.00/0.27	
0.00/0.27	By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).
0.00/0.27	
0.00/0.27	Thus, the original system is terminating if (P_1, R_0, static, formative) is finite.
0.00/0.27	
0.00/0.27	We consider the dependency pair problem (P_1, R_0, static, formative).
0.00/0.27	
0.00/0.27	We apply the subterm criterion with the following projection function:
0.00/0.27	
0.00/0.27	  nu(map#) = 2 
0.00/0.27	
0.00/0.27	Thus, we can orient the dependency pairs as follows:
0.00/0.27	
0.00/0.27	  nu(map#(/\x.X(x), cons(Y, Z))) = cons(Y, Z) |> Z = nu(map#(/\y.X(y), Z)) 
0.00/0.27	  nu(map#(/\x.X(x), map(/\y.Y(y), Z))) = map(/\z.Y(z), Z) |> Z = nu(map#(/\y.o(/\u.X(u), /\v.Y(v), y), Z)) 
0.00/0.27	
0.00/0.27	By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_1, R_0, static, f) by ({}, R_0, static, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.
0.00/0.27	
0.00/0.27	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
0.00/0.27	
0.00/0.27	
0.00/0.27	+++ Citations +++
0.00/0.27	
0.00/0.27	[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
0.00/0.27	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.27	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
0.00/0.27	[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.
0.00/0.27	EOF