17.25/17.34	YES
17.35/17.37	We consider the system theBenchmark.
17.35/17.37	
17.35/17.37	  Alphabet:
17.35/17.37	
17.35/17.37	    cons : [d * e] --> e 
17.35/17.37	    g : [] --> b 
17.35/17.37	    map!fac62201 : [d -> d * e] --> e 
17.35/17.37	    map!fac62202 : [d -> a -> d * a * e] --> e 
17.35/17.37	    map!fac62203 : [b -> d -> c -> d * b * c * e] --> e 
17.35/17.37	
17.35/17.37	  Rules:
17.35/17.37	
17.35/17.37	    map!fac62201(f, cons(x, y)) => cons(f x, map!fac62201(f, y)) 
17.35/17.37	    map!fac62202(f, x, cons(y, z)) => cons(f y x, map!fac62202(f, x, z)) 
17.35/17.37	    map!fac62203(f, g, x, cons(y, z)) => cons(f g y x, map!fac62203(f, g, x, z)) 
17.35/17.37	
17.35/17.37	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
17.35/17.37	
17.35/17.37	We use rule removal, following [Kop12, Theorem 2.23].
17.35/17.37	
17.35/17.37	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
17.35/17.37	
17.35/17.37	  map!fac62201(F, cons(X, Y)) >? cons(F X, map!fac62201(F, Y)) 
17.35/17.37	  map!fac62202(F, X, cons(Y, Z)) >? cons(F Y X, map!fac62202(F, X, Z)) 
17.35/17.37	  map!fac62203(F, g, X, cons(Y, Z)) >? cons(F g Y X, map!fac62203(F, g, X, Z)) 
17.35/17.37	
17.35/17.37	We use a recursive path ordering as defined in [Kop12, Chapter 5].
17.35/17.37	
17.35/17.37	Argument functions:
17.35/17.37	
17.35/17.37	  [[g]] = _|_ 
17.35/17.37	
17.35/17.37	We choose Lex = {} and Mul = {@_{o -> o -> o -> o}, @_{o -> o -> o}, @_{o -> o}, cons, map!fac62201, map!fac62202, map!fac62203}, and the following precedence: map!fac62201 > map!fac62203 > @_{o -> o -> o} = map!fac62202 > @_{o -> o} > cons > @_{o -> o -> o -> o}
17.35/17.37	
17.35/17.37	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
17.35/17.37	
17.35/17.37	  map!fac62201(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map!fac62201(F, Y)) 
17.35/17.37	  map!fac62202(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z)) 
17.35/17.37	  map!fac62203(F, _|_, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X), map!fac62203(F, _|_, X, Z)) 
17.35/17.37	
17.35/17.37	With these choices, we have:
17.35/17.37	
17.35/17.37	  1] map!fac62201(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because [2], by definition 
17.35/17.37	  2] map!fac62201*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because map!fac62201 > cons, [3] and [10], by (Copy) 
17.35/17.37	  3] map!fac62201*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map!fac62201 > @_{o -> o}, [4] and [6], by (Copy) 
17.35/17.37	  4] map!fac62201*(F, cons(X, Y)) >= F  because [5], by (Select) 
17.35/17.37	  5] F >= F  by (Meta) 
17.35/17.37	  6] map!fac62201*(F, cons(X, Y)) >= X  because [7], by (Select) 
17.35/17.37	  7] cons(X, Y) >= X  because [8], by (Star) 
17.35/17.37	  8] cons*(X, Y) >= X  because [9], by (Select) 
17.35/17.37	  9] X >= X  by (Meta) 
17.35/17.37	  10] map!fac62201*(F, cons(X, Y)) >= map!fac62201(F, Y)  because map!fac62201 in Mul, [11] and [12], by (Stat) 
17.35/17.37	  11] F >= F  by (Meta) 
17.35/17.37	  12] cons(X, Y) > Y  because [13], by definition 
17.35/17.37	  13] cons*(X, Y) >= Y  because [14], by (Select) 
17.35/17.37	  14] Y >= Y  by (Meta) 
17.35/17.37	
17.35/17.37	  15] map!fac62202(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because [16], by (Star) 
17.35/17.37	  16] map!fac62202*(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because map!fac62202 > cons, [17] and [25], by (Copy) 
17.35/17.37	  17] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because map!fac62202 > @_{o -> o}, [18] and [23], by (Copy) 
17.35/17.37	  18] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, Y)  because map!fac62202 = @_{o -> o -> o}, map!fac62202 in Mul, [19] and [20], by (Stat) 
17.35/17.37	  19] F >= F  by (Meta) 
17.35/17.37	  20] cons(Y, Z) > Y  because [21], by definition 
17.35/17.37	  21] cons*(Y, Z) >= Y  because [22], by (Select) 
17.35/17.37	  22] Y >= Y  by (Meta) 
17.35/17.37	  23] map!fac62202*(F, X, cons(Y, Z)) >= X  because [24], by (Select) 
17.35/17.37	  24] X >= X  by (Meta) 
17.35/17.37	  25] map!fac62202*(F, X, cons(Y, Z)) >= map!fac62202(F, X, Z)  because map!fac62202 in Mul, [19], [26] and [27], by (Stat) 
17.35/17.37	  26] X >= X  by (Meta) 
17.35/17.37	  27] cons(Y, Z) > Z  because [28], by definition 
17.35/17.37	  28] cons*(Y, Z) >= Z  because [29], by (Select) 
17.35/17.37	  29] Z >= Z  by (Meta) 
17.35/17.37	
17.35/17.37	  30] map!fac62203(F, _|_, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X), map!fac62203(F, _|_, X, Z))  because [31], by (Star) 
17.35/17.37	  31] map!fac62203*(F, _|_, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X), map!fac62203(F, _|_, X, Z))  because map!fac62203 > cons, [32] and [44], by (Copy) 
17.35/17.37	  32] map!fac62203*(F, _|_, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X)  because map!fac62203 > @_{o -> o}, [33] and [42], by (Copy) 
17.35/17.37	  33] map!fac62203*(F, _|_, X, cons(Y, Z)) >= @_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y)  because map!fac62203 > @_{o -> o -> o}, [34] and [38], by (Copy) 
17.35/17.37	  34] map!fac62203*(F, _|_, X, cons(Y, Z)) >= @_{o -> o -> o -> o}(F, _|_)  because map!fac62203 > @_{o -> o -> o -> o}, [35] and [37], by (Copy) 
17.35/17.37	  35] map!fac62203*(F, _|_, X, cons(Y, Z)) >= F  because [36], by (Select) 
17.35/17.37	  36] F >= F  by (Meta) 
17.35/17.37	  37] map!fac62203*(F, _|_, X, cons(Y, Z)) >= _|_  by (Bot) 
17.35/17.37	  38] map!fac62203*(F, _|_, X, cons(Y, Z)) >= Y  because [39], by (Select) 
17.35/17.37	  39] cons(Y, Z) >= Y  because [40], by (Star) 
17.35/17.37	  40] cons*(Y, Z) >= Y  because [41], by (Select) 
17.35/17.37	  41] Y >= Y  by (Meta) 
17.35/17.37	  42] map!fac62203*(F, _|_, X, cons(Y, Z)) >= X  because [43], by (Select) 
17.35/17.37	  43] X >= X  by (Meta) 
17.35/17.37	  44] map!fac62203*(F, _|_, X, cons(Y, Z)) >= map!fac62203(F, _|_, X, Z)  because map!fac62203 in Mul, [45], [46], [47] and [48], by (Stat) 
17.35/17.37	  45] F >= F  by (Meta) 
17.35/17.37	  46] _|_ >= _|_  by (Bot) 
17.35/17.37	  47] X >= X  by (Meta) 
17.35/17.37	  48] cons(Y, Z) > Z  because [49], by definition 
17.35/17.37	  49] cons*(Y, Z) >= Z  because [50], by (Select) 
17.35/17.37	  50] Z >= Z  by (Meta) 
17.35/17.37	
17.35/17.37	We can thus remove the following rules:
17.35/17.37	
17.35/17.37	  map!fac62201(F, cons(X, Y)) => cons(F X, map!fac62201(F, Y)) 
17.35/17.37	
17.35/17.37	We use rule removal, following [Kop12, Theorem 2.23].
17.35/17.37	
17.35/17.37	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
17.35/17.37	
17.35/17.37	  map!fac62202(F, X, cons(Y, Z)) >? cons(F Y X, map!fac62202(F, X, Z)) 
17.35/17.37	  map!fac62203(F, g, X, cons(Y, Z)) >? cons(F g Y X, map!fac62203(F, g, X, Z)) 
17.35/17.37	
17.35/17.37	We use a recursive path ordering as defined in [Kop12, Chapter 5].
17.35/17.37	
17.35/17.37	Argument functions:
17.35/17.37	
17.35/17.37	  [[g]] = _|_ 
17.35/17.37	
17.35/17.37	We choose Lex = {} and Mul = {@_{o -> o -> o -> o}, @_{o -> o -> o}, @_{o -> o}, cons, map!fac62202, map!fac62203}, and the following precedence: map!fac62202 > @_{o -> o -> o -> o} = map!fac62203 > @_{o -> o -> o} > @_{o -> o} > cons
17.35/17.37	
17.35/17.37	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
17.35/17.37	
17.35/17.37	  map!fac62202(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z)) 
17.35/17.37	  map!fac62203(F, _|_, X, cons(Y, Z)) > cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X), map!fac62203(F, _|_, X, Z)) 
17.35/17.37	
17.35/17.37	With these choices, we have:
17.35/17.37	
17.35/17.37	  1] map!fac62202(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because [2], by (Star) 
17.35/17.37	  2] map!fac62202*(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because map!fac62202 > cons, [3] and [13], by (Copy) 
17.35/17.37	  3] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because map!fac62202 > @_{o -> o}, [4] and [11], by (Copy) 
17.35/17.37	  4] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, Y)  because map!fac62202 > @_{o -> o -> o}, [5] and [7], by (Copy) 
17.35/17.37	  5] map!fac62202*(F, X, cons(Y, Z)) >= F  because [6], by (Select) 
17.35/17.37	  6] F >= F  by (Meta) 
17.35/17.37	  7] map!fac62202*(F, X, cons(Y, Z)) >= Y  because [8], by (Select) 
17.35/17.37	  8] cons(Y, Z) >= Y  because [9], by (Star) 
17.35/17.37	  9] cons*(Y, Z) >= Y  because [10], by (Select) 
17.35/17.37	  10] Y >= Y  by (Meta) 
17.35/17.37	  11] map!fac62202*(F, X, cons(Y, Z)) >= X  because [12], by (Select) 
17.35/17.37	  12] X >= X  by (Meta) 
17.35/17.37	  13] map!fac62202*(F, X, cons(Y, Z)) >= map!fac62202(F, X, Z)  because map!fac62202 in Mul, [14], [15] and [16], by (Stat) 
17.35/17.37	  14] F >= F  by (Meta) 
17.35/17.37	  15] X >= X  by (Meta) 
17.35/17.37	  16] cons(Y, Z) > Z  because [17], by definition 
17.35/17.37	  17] cons*(Y, Z) >= Z  because [18], by (Select) 
17.35/17.37	  18] Z >= Z  by (Meta) 
17.35/17.37	
17.35/17.37	  19] map!fac62203(F, _|_, X, cons(Y, Z)) > cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X), map!fac62203(F, _|_, X, Z))  because [20], by definition 
17.35/17.37	  20] map!fac62203*(F, _|_, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X), map!fac62203(F, _|_, X, Z))  because map!fac62203 > cons, [21] and [32], by (Copy) 
17.35/17.37	  21] map!fac62203*(F, _|_, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X)  because map!fac62203 > @_{o -> o}, [22] and [30], by (Copy) 
17.35/17.37	  22] map!fac62203*(F, _|_, X, cons(Y, Z)) >= @_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y)  because map!fac62203 > @_{o -> o -> o}, [23] and [26], by (Copy) 
17.35/17.37	  23] map!fac62203*(F, _|_, X, cons(Y, Z)) >= @_{o -> o -> o -> o}(F, _|_)  because map!fac62203 = @_{o -> o -> o -> o}, map!fac62203 in Mul, [24] and [25], by (Stat) 
17.35/17.37	  24] F >= F  by (Meta) 
17.35/17.37	  25] X >= _|_  by (Bot) 
17.35/17.37	  26] map!fac62203*(F, _|_, X, cons(Y, Z)) >= Y  because [27], by (Select) 
17.35/17.37	  27] cons(Y, Z) >= Y  because [28], by (Star) 
17.35/17.37	  28] cons*(Y, Z) >= Y  because [29], by (Select) 
17.35/17.37	  29] Y >= Y  by (Meta) 
17.35/17.37	  30] map!fac62203*(F, _|_, X, cons(Y, Z)) >= X  because [31], by (Select) 
17.35/17.37	  31] X >= X  by (Meta) 
17.35/17.37	  32] map!fac62203*(F, _|_, X, cons(Y, Z)) >= map!fac62203(F, _|_, X, Z)  because map!fac62203 in Mul, [24], [33], [34] and [35], by (Stat) 
17.35/17.37	  33] _|_ >= _|_  by (Bot) 
17.35/17.37	  34] X >= X  by (Meta) 
17.35/17.37	  35] cons(Y, Z) > Z  because [36], by definition 
17.35/17.37	  36] cons*(Y, Z) >= Z  because [37], by (Select) 
17.35/17.37	  37] Z >= Z  by (Meta) 
17.35/17.37	
17.35/17.37	We can thus remove the following rules:
17.35/17.37	
17.35/17.37	  map!fac62203(F, g, X, cons(Y, Z)) => cons(F g Y X, map!fac62203(F, g, X, Z)) 
17.35/17.37	
17.35/17.37	We use rule removal, following [Kop12, Theorem 2.23].
17.35/17.37	
17.35/17.37	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
17.35/17.37	
17.35/17.37	  map!fac62202(F, X, cons(Y, Z)) >? cons(F Y X, map!fac62202(F, X, Z)) 
17.35/17.37	
17.35/17.37	We use a recursive path ordering as defined in [Kop12, Chapter 5].
17.35/17.37	
17.35/17.37	We choose Lex = {} and Mul = {@_{o -> o -> o}, @_{o -> o}, cons, map!fac62202}, and the following precedence: @_{o -> o -> o} = map!fac62202 > @_{o -> o} > cons
17.35/17.37	
17.35/17.37	With these choices, we have:
17.35/17.37	
17.35/17.37	  1] map!fac62202(F, X, cons(Y, Z)) > cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because [2], by definition 
17.35/17.37	  2] map!fac62202*(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because map!fac62202 > cons, [3] and [11], by (Copy) 
17.35/17.37	  3] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because map!fac62202 > @_{o -> o}, [4] and [9], by (Copy) 
17.35/17.37	  4] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, Y)  because map!fac62202 = @_{o -> o -> o}, map!fac62202 in Mul, [5] and [6], by (Stat) 
17.35/17.37	  5] F >= F  by (Meta) 
17.35/17.37	  6] cons(Y, Z) > Y  because [7], by definition 
17.35/17.37	  7] cons*(Y, Z) >= Y  because [8], by (Select) 
17.35/17.37	  8] Y >= Y  by (Meta) 
17.35/17.37	  9] map!fac62202*(F, X, cons(Y, Z)) >= X  because [10], by (Select) 
17.35/17.37	  10] X >= X  by (Meta) 
17.35/17.37	  11] map!fac62202*(F, X, cons(Y, Z)) >= map!fac62202(F, X, Z)  because map!fac62202 in Mul, [5], [12] and [13], by (Stat) 
17.35/17.37	  12] X >= X  by (Meta) 
17.35/17.37	  13] cons(Y, Z) > Z  because [14], by definition 
17.35/17.37	  14] cons*(Y, Z) >= Z  because [15], by (Select) 
17.35/17.37	  15] Z >= Z  by (Meta) 
17.35/17.37	
17.35/17.37	We can thus remove the following rules:
17.35/17.37	
17.35/17.37	  map!fac62202(F, X, cons(Y, Z)) => cons(F Y X, map!fac62202(F, X, Z)) 
17.35/17.37	
17.35/17.37	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.
17.35/17.37	
17.35/17.37	
17.35/17.37	+++ Citations +++
17.35/17.37	
17.35/17.37	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
17.35/17.38	EOF