0.00/0.43	YES
0.00/0.45	We consider the system theBenchmark.
0.00/0.45	
0.00/0.45	  Alphabet:
0.00/0.45	
0.00/0.45	    cons : [nat * list] --> list 
0.00/0.45	    explode : [list * nat -> nat * nat] --> nat 
0.00/0.45	    implode : [list * nat -> nat * nat] --> nat 
0.00/0.45	    nil : [] --> list 
0.00/0.45	    op : [nat -> nat * nat -> nat] --> nat -> nat 
0.00/0.45	
0.00/0.45	  Rules:
0.00/0.45	
0.00/0.45	    op(f, g) x => f (g x) 
0.00/0.45	    implode(nil, f, x) => x 
0.00/0.45	    implode(cons(x, y), f, z) => implode(y, f, f z) 
0.00/0.45	    explode(nil, f, x) => x 
0.00/0.45	    explode(cons(x, y), f, z) => explode(y, op(f, f), f z) 
0.00/0.45	
0.00/0.45	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.00/0.45	
0.00/0.45	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.45	
0.00/0.45	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.45	
0.00/0.45	  op(F, G) X >? F (G X) 
0.00/0.45	  implode(nil, F, X) >? X 
0.00/0.45	  implode(cons(X, Y), F, Z) >? implode(Y, F, F Z) 
0.00/0.45	  explode(nil, F, X) >? X 
0.00/0.45	  explode(cons(X, Y), F, Z) >? explode(Y, op(F, F), F Z) 
0.00/0.45	
0.00/0.45	We use a recursive path ordering as defined in [Kop12, Chapter 5].
0.00/0.45	
0.00/0.45	Argument functions:
0.00/0.45	
0.00/0.45	  [[explode(x_1, x_2, x_3)]] = explode(x_1, x_3, x_2) 
0.00/0.45	
0.00/0.45	We choose Lex = {explode, implode} and Mul = {@_{o -> o}, cons, nil, op}, and the following precedence: cons > explode > nil > implode > op > @_{o -> o}
0.00/0.45	
0.00/0.45	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
0.00/0.45	
0.00/0.45	  @_{o -> o}(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X)) 
0.00/0.45	  implode(nil, F, X) >= X 
0.00/0.45	  implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z)) 
0.00/0.45	  explode(nil, F, X) > X 
0.00/0.45	  explode(cons(X, Y), F, Z) > explode(Y, op(F, F), @_{o -> o}(F, Z)) 
0.00/0.45	
0.00/0.45	With these choices, we have:
0.00/0.45	
0.00/0.45	  1] @_{o -> o}(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [2], by (Star) 
0.00/0.45	  2] @_{o -> o}*(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [3], by (Select) 
0.00/0.45	  3] op(F, G) @_{o -> o}*(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [4] 
0.00/0.45	  4] op*(F, G, @_{o -> o}*(op(F, G), X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because op > @_{o -> o}, [5] and [7], by (Copy) 
0.00/0.45	  5] op*(F, G, @_{o -> o}*(op(F, G), X)) >= F  because [6], by (Select) 
0.00/0.45	  6] F >= F  by (Meta) 
0.00/0.45	  7] op*(F, G, @_{o -> o}*(op(F, G), X)) >= @_{o -> o}(G, X)  because op > @_{o -> o}, [8] and [10], by (Copy) 
0.00/0.45	  8] op*(F, G, @_{o -> o}*(op(F, G), X)) >= G  because [9], by (Select) 
0.00/0.45	  9] G >= G  by (Meta) 
0.00/0.45	  10] op*(F, G, @_{o -> o}*(op(F, G), X)) >= X  because [11], by (Select) 
0.00/0.45	  11] @_{o -> o}*(op(F, G), X) >= X  because [12], by (Select) 
0.00/0.45	  12] op(F, G) @_{o -> o}*(op(F, G), X) >= X  because [13] 
0.00/0.45	  13] op*(F, G, @_{o -> o}*(op(F, G), X)) >= X  because [14], by (Select) 
0.00/0.45	  14] @_{o -> o}*(op(F, G), X) >= X  because [15], by (Select) 
0.00/0.45	  15] X >= X  by (Meta) 
0.00/0.45	
0.00/0.45	  16] implode(nil, F, X) >= X  because [17], by (Star) 
0.00/0.45	  17] implode*(nil, F, X) >= X  because [18], by (Select) 
0.00/0.45	  18] X >= X  by (Meta) 
0.00/0.45	
0.00/0.45	  19] implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [20], by (Star) 
0.00/0.45	  20] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [21], [24], [26] and [28], by (Stat) 
0.00/0.45	  21] cons(X, Y) > Y  because [22], by definition 
0.00/0.45	  22] cons*(X, Y) >= Y  because [23], by (Select) 
0.00/0.45	  23] Y >= Y  by (Meta) 
0.00/0.45	  24] implode*(cons(X, Y), F, Z) >= Y  because [25], by (Select) 
0.00/0.45	  25] cons(X, Y) >= Y  because [22], by (Star) 
0.00/0.45	  26] implode*(cons(X, Y), F, Z) >= F  because [27], by (Select) 
0.00/0.45	  27] F >= F  by (Meta) 
0.00/0.45	  28] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode > @_{o -> o}, [26] and [29], by (Copy) 
0.00/0.45	  29] implode*(cons(X, Y), F, Z) >= Z  because [30], by (Select) 
0.00/0.45	  30] Z >= Z  by (Meta) 
0.00/0.45	
0.00/0.45	  31] explode(nil, F, X) > X  because [32], by definition 
0.00/0.45	  32] explode*(nil, F, X) >= X  because [33], by (Select) 
0.00/0.45	  33] X >= X  by (Meta) 
0.00/0.45	
0.00/0.45	  34] explode(cons(X, Y), F, Z) > explode(Y, op(F, F), @_{o -> o}(F, Z))  because [35], by definition 
0.00/0.45	  35] explode*(cons(X, Y), F, Z) >= explode(Y, op(F, F), @_{o -> o}(F, Z))  because [36], [39], [41] and [44], by (Stat) 
0.00/0.45	  36] cons(X, Y) > Y  because [37], by definition 
0.00/0.45	  37] cons*(X, Y) >= Y  because [38], by (Select) 
0.00/0.45	  38] Y >= Y  by (Meta) 
0.00/0.45	  39] explode*(cons(X, Y), F, Z) >= Y  because [40], by (Select) 
0.00/0.45	  40] cons(X, Y) >= Y  because [37], by (Star) 
0.00/0.45	  41] explode*(cons(X, Y), F, Z) >= op(F, F)  because explode > op, [42] and [42], by (Copy) 
0.00/0.45	  42] explode*(cons(X, Y), F, Z) >= F  because [43], by (Select) 
0.00/0.45	  43] F >= F  by (Meta) 
0.00/0.45	  44] explode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because explode > @_{o -> o}, [42] and [45], by (Copy) 
0.00/0.45	  45] explode*(cons(X, Y), F, Z) >= Z  because [46], by (Select) 
0.00/0.45	  46] Z >= Z  by (Meta) 
0.00/0.45	
0.00/0.45	We can thus remove the following rules:
0.00/0.45	
0.00/0.45	  explode(nil, F, X) => X 
0.00/0.45	  explode(cons(X, Y), F, Z) => explode(Y, op(F, F), F Z) 
0.00/0.45	
0.00/0.45	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.45	
0.00/0.45	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.45	
0.00/0.45	  op(F, G, X) >? F (G X) 
0.00/0.45	  implode(nil, F, X) >? X 
0.00/0.45	  implode(cons(X, Y), F, Z) >? implode(Y, F, F Z) 
0.00/0.45	
0.00/0.45	We use a recursive path ordering as defined in [Kop12, Chapter 5].
0.00/0.45	
0.00/0.45	Argument functions:
0.00/0.45	
0.00/0.45	  [[implode(x_1, x_2, x_3)]] = implode(x_1, x_3, x_2) 
0.00/0.45	
0.00/0.45	We choose Lex = {implode} and Mul = {@_{o -> o}, cons, nil, op}, and the following precedence: op > nil > implode > cons > @_{o -> o}
0.00/0.45	
0.00/0.45	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
0.00/0.45	
0.00/0.45	  op(F, G, X) > @_{o -> o}(F, @_{o -> o}(G, X)) 
0.00/0.45	  implode(nil, F, X) >= X 
0.00/0.45	  implode(cons(X, Y), F, Z) > implode(Y, F, @_{o -> o}(F, Z)) 
0.00/0.45	
0.00/0.45	With these choices, we have:
0.00/0.45	
0.00/0.45	  1] op(F, G, X) > @_{o -> o}(F, @_{o -> o}(G, X))  because [2], by definition 
0.00/0.45	  2] op*(F, G, X) >= @_{o -> o}(F, @_{o -> o}(G, X))  because op > @_{o -> o}, [3] and [5], by (Copy) 
0.00/0.45	  3] op*(F, G, X) >= F  because [4], by (Select) 
0.00/0.45	  4] F >= F  by (Meta) 
0.00/0.45	  5] op*(F, G, X) >= @_{o -> o}(G, X)  because op > @_{o -> o}, [6] and [8], by (Copy) 
0.00/0.45	  6] op*(F, G, X) >= G  because [7], by (Select) 
0.00/0.45	  7] G >= G  by (Meta) 
0.00/0.45	  8] op*(F, G, X) >= X  because [9], by (Select) 
0.00/0.45	  9] X >= X  by (Meta) 
0.00/0.45	
0.00/0.45	  10] implode(nil, F, X) >= X  because [11], by (Star) 
0.00/0.45	  11] implode*(nil, F, X) >= X  because [12], by (Select) 
0.00/0.45	  12] X >= X  by (Meta) 
0.00/0.45	
0.00/0.45	  13] implode(cons(X, Y), F, Z) > implode(Y, F, @_{o -> o}(F, Z))  because [14], by definition 
0.00/0.45	  14] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [15], [18], [20] and [22], by (Stat) 
0.00/0.45	  15] cons(X, Y) > Y  because [16], by definition 
0.00/0.45	  16] cons*(X, Y) >= Y  because [17], by (Select) 
0.00/0.45	  17] Y >= Y  by (Meta) 
0.00/0.45	  18] implode*(cons(X, Y), F, Z) >= Y  because [19], by (Select) 
0.00/0.45	  19] cons(X, Y) >= Y  because [16], by (Star) 
0.00/0.45	  20] implode*(cons(X, Y), F, Z) >= F  because [21], by (Select) 
0.00/0.45	  21] F >= F  by (Meta) 
0.00/0.45	  22] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode > @_{o -> o}, [20] and [23], by (Copy) 
0.00/0.45	  23] implode*(cons(X, Y), F, Z) >= Z  because [24], by (Select) 
0.00/0.45	  24] Z >= Z  by (Meta) 
0.00/0.45	
0.00/0.45	We can thus remove the following rules:
0.00/0.45	
0.00/0.45	  op(F, G, X) => F (G X) 
0.00/0.45	  implode(cons(X, Y), F, Z) => implode(Y, F, F Z) 
0.00/0.45	
0.00/0.45	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.45	
0.00/0.45	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.45	
0.00/0.45	  implode(nil, F, X) >? X 
0.00/0.45	
0.00/0.45	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.45	
0.00/0.45	The following interpretation satisfies the requirements:
0.00/0.45	
0.00/0.45	  implode = \y0G1y2.3 + y0 + y2 + G1(0) 
0.00/0.45	  nil = 3 
0.00/0.45	
0.00/0.45	Using this interpretation, the requirements translate to:
0.00/0.45	
0.00/0.45	  [[implode(nil, _F0, _x1)]] = 6 + x1 + F0(0) > x1 = [[_x1]] 
0.00/0.45	
0.00/0.45	We can thus remove the following rules:
0.00/0.45	
0.00/0.45	  implode(nil, F, X) => X 
0.00/0.45	
0.00/0.45	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.
0.00/0.45	
0.00/0.45	
0.00/0.45	+++ Citations +++
0.00/0.45	
0.00/0.45	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.45	EOF