1.89/1.98	YES
1.97/2.02	We consider the system theBenchmark.
1.97/2.02	
1.97/2.02	  Alphabet:
1.97/2.02	
1.97/2.02	    0 : [] --> a 
1.97/2.02	    asort : [b] --> b 
1.97/2.02	    cons : [a * b] --> b 
1.97/2.02	    dsort : [b] --> b 
1.97/2.02	    insert : [a -> a -> a * a -> a -> a * b * a] --> b 
1.97/2.02	    max : [] --> a -> a -> a 
1.97/2.02	    min : [] --> a -> a -> a 
1.97/2.02	    nil : [] --> b 
1.97/2.02	    s : [a] --> a 
1.97/2.02	    sort : [a -> a -> a * a -> a -> a * b] --> b 
1.97/2.02	
1.97/2.02	  Rules:
1.97/2.02	
1.97/2.02	    sort(f, g, nil) => nil 
1.97/2.02	    sort(f, g, cons(x, y)) => insert(f, g, sort(f, g, y), x) 
1.97/2.02	    insert(f, g, nil, x) => cons(x, nil) 
1.97/2.02	    insert(f, g, cons(x, y), z) => cons(f x z, insert(f, g, y, g x z)) 
1.97/2.02	    max 0 x => x 
1.97/2.02	    max x 0 => x 
1.97/2.02	    max s(x) s(y) => max x y 
1.97/2.02	    min 0 x => 0 
1.97/2.02	    min x 0 => 0 
1.97/2.02	    min s(x) s(y) => min x y 
1.97/2.02	    asort(x) => sort(min, max, x) 
1.97/2.02	    dsort(x) => sort(max, min, x) 
1.97/2.02	
1.97/2.02	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
1.97/2.02	
1.97/2.02	We use rule removal, following [Kop12, Theorem 2.23].
1.97/2.02	
1.97/2.02	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
1.97/2.02	
1.97/2.02	  sort(F, G, nil) >? nil 
1.97/2.02	  sort(F, G, cons(X, Y)) >? insert(F, G, sort(F, G, Y), X) 
1.97/2.02	  insert(F, G, nil, X) >? cons(X, nil) 
1.97/2.02	  insert(F, G, cons(X, Y), Z) >? cons(F X Z, insert(F, G, Y, G X Z)) 
1.97/2.02	  max 0 X >? X 
1.97/2.02	  max X 0 >? X 
1.97/2.02	  max s(X) s(Y) >? max X Y 
1.97/2.02	  min 0 X >? 0 
1.97/2.02	  min X 0 >? 0 
1.97/2.02	  min s(X) s(Y) >? min X Y 
1.97/2.02	  asort(X) >? sort(min, max, X) 
1.97/2.02	  dsort(X) >? sort(max, min, X) 
1.97/2.02	
1.97/2.02	We use a recursive path ordering as defined in [Kop12, Chapter 5].
1.97/2.02	
1.97/2.02	Argument functions:
1.97/2.02	
1.97/2.02	  [[0]] = _|_ 
1.97/2.02	  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_2, x_3, x_4, x_1) 
1.97/2.02	  [[nil]] = _|_ 
1.97/2.02	
1.97/2.02	We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, asort, cons, dsort, max, min, s, sort}, and the following precedence: dsort > asort = max > min > s > sort > insert > @_{o -> o} > cons > @_{o -> o -> o}
1.97/2.02	
1.97/2.02	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
1.97/2.02	
1.97/2.02	  sort(F, G, _|_) >= _|_ 
1.97/2.02	  sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X) 
1.97/2.02	  insert(F, G, _|_, X) >= cons(X, _|_) 
1.97/2.02	  insert(F, G, cons(X, Y), Z) > cons(@_{o -> o}(@_{o -> o -> o}(F, X), Z), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z))) 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(max, _|_), X) > X 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(max, X), _|_) >= X 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y) 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(min, _|_), X) >= _|_ 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(min, X), _|_) >= _|_ 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(min, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(min, X), Y) 
1.97/2.02	  asort(X) >= sort(min, max, X) 
1.97/2.02	  dsort(X) >= sort(max, min, X) 
1.97/2.02	
1.97/2.02	With these choices, we have:
1.97/2.02	
1.97/2.02	  1] sort(F, G, _|_) >= _|_  by (Bot) 
1.97/2.02	
1.97/2.02	  2] sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because [3], by (Star) 
1.97/2.02	  3] sort*(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because sort > insert, [4], [6], [8] and [14], by (Copy) 
1.97/2.02	  4] sort*(F, G, cons(X, Y)) >= F  because [5], by (Select) 
1.97/2.02	  5] F >= F  by (Meta) 
1.97/2.02	  6] sort*(F, G, cons(X, Y)) >= G  because [7], by (Select) 
1.97/2.02	  7] G >= G  by (Meta) 
1.97/2.02	  8] sort*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort in Mul, [9], [10] and [11], by (Stat) 
1.97/2.02	  9] F >= F  by (Meta) 
1.97/2.02	  10] G >= G  by (Meta) 
1.97/2.02	  11] cons(X, Y) > Y  because [12], by definition 
1.97/2.02	  12] cons*(X, Y) >= Y  because [13], by (Select) 
1.97/2.02	  13] Y >= Y  by (Meta) 
1.97/2.02	  14] sort*(F, G, cons(X, Y)) >= X  because [15], by (Select) 
1.97/2.02	  15] cons(X, Y) >= X  because [16], by (Star) 
1.97/2.02	  16] cons*(X, Y) >= X  because [17], by (Select) 
1.97/2.02	  17] X >= X  by (Meta) 
1.97/2.02	
1.97/2.02	  18] insert(F, G, _|_, X) >= cons(X, _|_)  because [19], by (Star) 
1.97/2.02	  19] insert*(F, G, _|_, X) >= cons(X, _|_)  because insert > cons, [20] and [22], by (Copy) 
1.97/2.02	  20] insert*(F, G, _|_, X) >= X  because [21], by (Select) 
1.97/2.02	  21] X >= X  by (Meta) 
1.97/2.02	  22] insert*(F, G, _|_, X) >= _|_  by (Bot) 
1.97/2.02	
1.97/2.02	  23] insert(F, G, cons(X, Y), Z) > cons(@_{o -> o}(@_{o -> o -> o}(F, X), Z), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z)))  because [24], by definition 
1.97/2.02	  24] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, X), Z), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z)))  because insert > cons, [25] and [35], by (Copy) 
1.97/2.02	  25] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, X), Z)  because insert > @_{o -> o}, [26] and [33], by (Copy) 
1.97/2.02	  26] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, X)  because insert > @_{o -> o -> o}, [27] and [29], by (Copy) 
1.97/2.02	  27] insert*(F, G, cons(X, Y), Z) >= F  because [28], by (Select) 
1.97/2.02	  28] F >= F  by (Meta) 
1.97/2.02	  29] insert*(F, G, cons(X, Y), Z) >= X  because [30], by (Select) 
1.97/2.02	  30] cons(X, Y) >= X  because [31], by (Star) 
1.97/2.02	  31] cons*(X, Y) >= X  because [32], by (Select) 
1.97/2.02	  32] X >= X  by (Meta) 
1.97/2.02	  33] insert*(F, G, cons(X, Y), Z) >= Z  because [34], by (Select) 
1.97/2.02	  34] Z >= Z  by (Meta) 
1.97/2.02	  35] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z))  because [36], [37], [27], [40], [41] and [43], by (Stat) 
1.97/2.02	  36] G >= G  by (Meta) 
1.97/2.02	  37] cons(X, Y) > Y  because [38], by definition 
1.97/2.02	  38] cons*(X, Y) >= Y  because [39], by (Select) 
1.97/2.02	  39] Y >= Y  by (Meta) 
1.97/2.02	  40] insert*(F, G, cons(X, Y), Z) >= G  because [36], by (Select) 
1.97/2.02	  41] insert*(F, G, cons(X, Y), Z) >= Y  because [42], by (Select) 
1.97/2.02	  42] cons(X, Y) >= Y  because [38], by (Star) 
1.97/2.02	  43] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, X), Z)  because insert > @_{o -> o}, [44] and [33], by (Copy) 
1.97/2.02	  44] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, X)  because insert > @_{o -> o -> o}, [40] and [29], by (Copy) 
1.97/2.02	
1.97/2.02	  45] @_{o -> o}(@_{o -> o -> o}(max, _|_), X) > X  because [46], by definition 
1.97/2.02	  46] @_{o -> o}*(@_{o -> o -> o}(max, _|_), X) >= X  because [47], by (Select) 
1.97/2.02	  47] X >= X  by (Meta) 
1.97/2.02	
1.97/2.02	  48] @_{o -> o}(@_{o -> o -> o}(max, X), _|_) >= X  because [49], by (Star) 
1.97/2.02	  49] @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [50], by (Select) 
1.97/2.02	  50] @_{o -> o -> o}(max, X) @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [51] 
1.97/2.02	  51] @_{o -> o -> o}*(max, X, @_{o -> o}*(@_{o -> o -> o}(max, X), _|_)) >= X  because [52], by (Select) 
1.97/2.02	  52] X >= X  by (Meta) 
1.97/2.02	
1.97/2.02	  53] @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because @_{o -> o} in Mul, [54] and [59], by (Fun) 
1.97/2.02	  54] @_{o -> o -> o}(max, s(X)) >= @_{o -> o -> o}(max, X)  because @_{o -> o -> o} in Mul, [55] and [56], by (Fun) 
1.97/2.02	  55] max >= max  by (Fun) 
1.97/2.02	  56] s(X) >= X  because [57], by (Star) 
1.97/2.02	  57] s*(X) >= X  because [58], by (Select) 
1.97/2.02	  58] X >= X  by (Meta) 
1.97/2.02	  59] s(Y) >= Y  because [60], by (Star) 
1.97/2.02	  60] s*(Y) >= Y  because [61], by (Select) 
1.97/2.02	  61] Y >= Y  by (Meta) 
1.97/2.02	
1.97/2.02	  62] @_{o -> o}(@_{o -> o -> o}(min, _|_), X) >= _|_  by (Bot) 
1.97/2.02	
1.97/2.02	  63] @_{o -> o}(@_{o -> o -> o}(min, X), _|_) >= _|_  by (Bot) 
1.97/2.02	
1.97/2.02	  64] @_{o -> o}(@_{o -> o -> o}(min, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(min, X), Y)  because @_{o -> o} in Mul, [65] and [70], by (Fun) 
1.97/2.02	  65] @_{o -> o -> o}(min, s(X)) >= @_{o -> o -> o}(min, X)  because @_{o -> o -> o} in Mul, [66] and [67], by (Fun) 
1.97/2.02	  66] min >= min  by (Fun) 
1.97/2.02	  67] s(X) >= X  because [68], by (Star) 
1.97/2.02	  68] s*(X) >= X  because [69], by (Select) 
1.97/2.02	  69] X >= X  by (Meta) 
1.97/2.02	  70] s(Y) >= Y  because [71], by (Star) 
1.97/2.02	  71] s*(Y) >= Y  because [72], by (Select) 
1.97/2.02	  72] Y >= Y  by (Meta) 
1.97/2.02	
1.97/2.02	  73] asort(X) >= sort(min, max, X)  because [74], by (Star) 
1.97/2.02	  74] asort*(X) >= sort(min, max, X)  because asort > sort, [75], [76] and [77], by (Copy) 
1.97/2.02	  75] asort*(X) >= min  because asort > min, by (Copy) 
1.97/2.02	  76] asort*(X) >= max  because asort = max and asort in Mul, by (Stat) 
1.97/2.02	  77] asort*(X) >= X  because [78], by (Select) 
1.97/2.02	  78] X >= X  by (Meta) 
1.97/2.02	
1.97/2.02	  79] dsort(X) >= sort(max, min, X)  because [80], by (Star) 
1.97/2.02	  80] dsort*(X) >= sort(max, min, X)  because dsort > sort, [81], [82] and [83], by (Copy) 
1.97/2.02	  81] dsort*(X) >= max  because dsort > max, by (Copy) 
1.97/2.02	  82] dsort*(X) >= min  because dsort > min, by (Copy) 
1.97/2.02	  83] dsort*(X) >= X  because [84], by (Select) 
1.97/2.02	  84] X >= X  by (Meta) 
1.97/2.02	
1.97/2.02	We can thus remove the following rules:
1.97/2.02	
1.97/2.02	  insert(F, G, cons(X, Y), Z) => cons(F X Z, insert(F, G, Y, G X Z)) 
1.97/2.02	  max 0 X => X 
1.97/2.02	
1.97/2.02	We use rule removal, following [Kop12, Theorem 2.23].
1.97/2.02	
1.97/2.02	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
1.97/2.02	
1.97/2.02	  sort(F, G, nil) >? nil 
1.97/2.02	  sort(F, G, cons(X, Y)) >? insert(F, G, sort(F, G, Y), X) 
1.97/2.02	  insert(F, G, nil, X) >? cons(X, nil) 
1.97/2.02	  max X 0 >? X 
1.97/2.02	  max s(X) s(Y) >? max X Y 
1.97/2.02	  min 0 X >? 0 
1.97/2.02	  min X 0 >? 0 
1.97/2.02	  min s(X) s(Y) >? min X Y 
1.97/2.02	  asort(X) >? sort(min, max, X) 
1.97/2.02	  dsort(X) >? sort(max, min, X) 
1.97/2.02	
1.97/2.02	We orient these requirements with a polynomial interpretation in the natural numbers.
1.97/2.02	
1.97/2.02	The following interpretation satisfies the requirements:
1.97/2.02	
1.97/2.02	  0 = 3 
1.97/2.02	  asort = \y0.3 + 3y0 
1.97/2.02	  cons = \y0y1.3 + y0 + 2y1 
1.97/2.02	  dsort = \y0.3 + 3y0 
1.97/2.02	  insert = \G0G1y2y3.3 + y2 + y3 + G0(y3,y3) + G1(y3,y3) 
1.97/2.02	  max = \y0y1.0 
1.97/2.02	  min = \y0y1.0 
1.97/2.02	  nil = 0 
1.97/2.02	  s = \y0.3 + 3y0 
1.97/2.02	  sort = \G0G1y2.3y2 + G0(y2,y2) + 2y2y2G1(y2,y2) + 2G1(y2,y2) + y2y2G0(y2,y2) 
1.97/2.02	
1.97/2.02	Using this interpretation, the requirements translate to:
1.97/2.02	
1.97/2.02	  [[sort(_F0, _F1, nil)]] = F0(0,0) + 2F1(0,0) >= 0 = [[nil]] 
1.97/2.02	  [[sort(_F0, _F1, cons(_x2, _x3))]] = 9 + 3x2 + 6x3 + 2x2x2F1(3 + x2 + 2x3,3 + x2 + 2x3) + 4x2x3F0(3 + x2 + 2x3,3 + x2 + 2x3) + 4x3x3F0(3 + x2 + 2x3,3 + x2 + 2x3) + 6x2F0(3 + x2 + 2x3,3 + x2 + 2x3) + 8x2x3F1(3 + x2 + 2x3,3 + x2 + 2x3) + 8x3x3F1(3 + x2 + 2x3,3 + x2 + 2x3) + 10F0(3 + x2 + 2x3,3 + x2 + 2x3) + 12x2F1(3 + x2 + 2x3,3 + x2 + 2x3) + 12x3F0(3 + x2 + 2x3,3 + x2 + 2x3) + 20F1(3 + x2 + 2x3,3 + x2 + 2x3) + 24x3F1(3 + x2 + 2x3,3 + x2 + 2x3) + x2x2F0(3 + x2 + 2x3,3 + x2 + 2x3) > 3 + x2 + 3x3 + F0(x2,x2) + F0(x3,x3) + F1(x2,x2) + 2x3x3F1(x3,x3) + 2F1(x3,x3) + x3x3F0(x3,x3) = [[insert(_F0, _F1, sort(_F0, _F1, _x3), _x2)]] 
1.97/2.02	  [[insert(_F0, _F1, nil, _x2)]] = 3 + x2 + F0(x2,x2) + F1(x2,x2) >= 3 + x2 = [[cons(_x2, nil)]] 
1.97/2.02	  [[max _x0 0]] = 3 + x0 > x0 = [[_x0]] 
1.97/2.02	  [[max s(_x0) s(_x1)]] = 6 + 3x0 + 3x1 > x0 + x1 = [[max _x0 _x1]] 
1.97/2.02	  [[min 0 _x0]] = 3 + x0 >= 3 = [[0]] 
1.97/2.02	  [[min _x0 0]] = 3 + x0 >= 3 = [[0]] 
1.97/2.02	  [[min s(_x0) s(_x1)]] = 6 + 3x0 + 3x1 > x0 + x1 = [[min _x0 _x1]] 
1.97/2.02	  [[asort(_x0)]] = 3 + 3x0 > 3x0 = [[sort(min, max, _x0)]] 
1.97/2.02	  [[dsort(_x0)]] = 3 + 3x0 > 3x0 = [[sort(max, min, _x0)]] 
1.97/2.02	
1.97/2.02	We can thus remove the following rules:
1.97/2.02	
1.97/2.02	  sort(F, G, cons(X, Y)) => insert(F, G, sort(F, G, Y), X) 
1.97/2.02	  max X 0 => X 
1.97/2.02	  max s(X) s(Y) => max X Y 
1.97/2.02	  min s(X) s(Y) => min X Y 
1.97/2.02	  asort(X) => sort(min, max, X) 
1.97/2.02	  dsort(X) => sort(max, min, X) 
1.97/2.02	
1.97/2.02	We use rule removal, following [Kop12, Theorem 2.23].
1.97/2.02	
1.97/2.02	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
1.97/2.02	
1.97/2.02	  sort(F, G, nil) >? nil 
1.97/2.02	  insert(F, G, nil, X) >? cons(X, nil) 
1.97/2.02	  min(0, X) >? 0 
1.97/2.02	  min(X, 0) >? 0 
1.97/2.02	
1.97/2.02	We orient these requirements with a polynomial interpretation in the natural numbers.
1.97/2.02	
1.97/2.02	The following interpretation satisfies the requirements:
1.97/2.02	
1.97/2.02	  0 = 0 
1.97/2.02	  cons = \y0y1.y0 + y1 
1.97/2.02	  insert = \G0G1y2y3.3 + 3y2 + 3y3 + G0(0,0) + G1(0,0) 
1.97/2.02	  min = \y0y1.3 + 3y0 + 3y1 
1.97/2.02	  nil = 0 
1.97/2.02	  sort = \G0G1y2.3 + 3y2 + G0(0,0) + G1(0,0) 
1.97/2.02	
1.97/2.02	Using this interpretation, the requirements translate to:
1.97/2.02	
1.97/2.02	  [[sort(_F0, _F1, nil)]] = 3 + F0(0,0) + F1(0,0) > 0 = [[nil]] 
1.97/2.02	  [[insert(_F0, _F1, nil, _x2)]] = 3 + 3x2 + F0(0,0) + F1(0,0) > x2 = [[cons(_x2, nil)]] 
1.97/2.02	  [[min(0, _x0)]] = 3 + 3x0 > 0 = [[0]] 
1.97/2.02	  [[min(_x0, 0)]] = 3 + 3x0 > 0 = [[0]] 
1.97/2.02	
1.97/2.02	We can thus remove the following rules:
1.97/2.02	
1.97/2.02	  sort(F, G, nil) => nil 
1.97/2.02	  insert(F, G, nil, X) => cons(X, nil) 
1.97/2.02	  min(0, X) => 0 
1.97/2.02	  min(X, 0) => 0 
1.97/2.02	
1.97/2.02	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.
1.97/2.02	
1.97/2.02	
1.97/2.02	+++ Citations +++
1.97/2.02	
1.97/2.02	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
1.97/2.02	EOF