We consider the system Applicative_05__Ex5Sorting.

  Alphabet:

    0 : [] --> a 
    ascending!fac6220sort : [b] --> b 
    cons : [a * b] --> b 
    descending!fac6220sort : [b] --> b 
    insert : [a -> a -> a * a -> a -> a * b * a] --> b 
    max : [] --> a -> a -> a 
    min : [] --> a -> a -> a 
    nil : [] --> b 
    s : [a] --> a 
    sort : [a -> a -> a * a -> a -> a * b] --> b 

  Rules:

    max 0 x => x 
    max x 0 => x 
    max s(x) s(y) => max x y 
    min 0 x => 0 
    min x 0 => 0 
    min s(x) s(y) => min x y 
    insert(f, g, nil, x) => cons(x, nil) 
    insert(f, g, cons(x, y), z) => cons(f z x, insert(f, g, y, g z x)) 
    sort(f, g, nil) => nil 
    sort(f, g, cons(x, y)) => insert(f, g, sort(f, g, y), x) 
    ascending!fac6220sort(x) => sort(min, max, x) 
    descending!fac6220sort(x) => sort(max, min, x) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  max 0 X >? X 
  max X 0 >? X 
  max s(X) s(Y) >? max X Y 
  min 0 X >? 0 
  min X 0 >? 0 
  min s(X) s(Y) >? min X Y 
  insert(F, G, nil, X) >? cons(X, nil) 
  insert(F, G, cons(X, Y), Z) >? cons(F Z X, insert(F, G, Y, G Z X)) 
  sort(F, G, nil) >? nil 
  sort(F, G, cons(X, Y)) >? insert(F, G, sort(F, G, Y), X) 
  ascending!fac6220sort(X) >? sort(min, max, X) 
  descending!fac6220sort(X) >? sort(max, min, X) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_3, x_2, x_4, x_1) 
  [[nil]] = _|_ 

We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, ascending!fac6220sort, cons, descending!fac6220sort, max, min, s, sort}, and the following precedence: descending!fac6220sort > ascending!fac6220sort = max > min > s > sort > insert > @_{o -> o} > cons > @_{o -> o -> o}

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  @_{o -> o}(@_{o -> o -> o}(max, _|_), X) >= X 
  @_{o -> o}(@_{o -> o -> o}(max, X), _|_) >= X 
  @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y) 
  @_{o -> o}(@_{o -> o -> o}(min, _|_), X) >= _|_ 
  @_{o -> o}(@_{o -> o -> o}(min, X), _|_) >= _|_ 
  @_{o -> o}(@_{o -> o -> o}(min, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(min, X), Y) 
  insert(F, G, _|_, X) > cons(X, _|_) 
  insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))) 
  sort(F, G, _|_) >= _|_ 
  sort(F, G, cons(X, Y)) > insert(F, G, sort(F, G, Y), X) 
  ascending!fac6220sort(X) >= sort(min, max, X) 
  descending!fac6220sort(X) >= sort(max, min, X) 

With these choices, we have:

  1] @_{o -> o}(@_{o -> o -> o}(max, _|_), X) >= X  because [2], by (Star) 
  2] @_{o -> o}*(@_{o -> o -> o}(max, _|_), X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] @_{o -> o}(@_{o -> o -> o}(max, X), _|_) >= X  because [5], by (Star) 
  5] @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [6], by (Select) 
  6] @_{o -> o -> o}(max, X) @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [7] 
  7] @_{o -> o -> o}*(max, X, @_{o -> o}*(@_{o -> o -> o}(max, X), _|_)) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

  9] @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because @_{o -> o} in Mul, [10] and [16], by (Fun) 
  10] @_{o -> o -> o}(max, s(X)) >= @_{o -> o -> o}(max, X)  because [11], by (Star) 
  11] @_{o -> o -> o}*(max, s(X)) >= @_{o -> o -> o}(max, X)  because @_{o -> o -> o} in Mul, [12] and [13], by (Stat) 
  12] max >= max  by (Fun) 
  13] s(X) > X  because [14], by definition 
  14] s*(X) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 
  16] s(Y) >= Y  because [17], by (Star) 
  17] s*(Y) >= Y  because [18], by (Select) 
  18] Y >= Y  by (Meta) 

  19] @_{o -> o}(@_{o -> o -> o}(min, _|_), X) >= _|_  by (Bot) 

  20] @_{o -> o}(@_{o -> o -> o}(min, X), _|_) >= _|_  by (Bot) 

  21] @_{o -> o}(@_{o -> o -> o}(min, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(min, X), Y)  because @_{o -> o} in Mul, [22] and [27], by (Fun) 
  22] @_{o -> o -> o}(min, s(X)) >= @_{o -> o -> o}(min, X)  because @_{o -> o -> o} in Mul, [23] and [24], by (Fun) 
  23] min >= min  by (Fun) 
  24] s(X) >= X  because [25], by (Star) 
  25] s*(X) >= X  because [26], by (Select) 
  26] X >= X  by (Meta) 
  27] s(Y) >= Y  because [28], by (Star) 
  28] s*(Y) >= Y  because [29], by (Select) 
  29] Y >= Y  by (Meta) 

  30] insert(F, G, _|_, X) > cons(X, _|_)  because [31], by definition 
  31] insert*(F, G, _|_, X) >= cons(X, _|_)  because insert > cons, [32] and [34], by (Copy) 
  32] insert*(F, G, _|_, X) >= X  because [33], by (Select) 
  33] X >= X  by (Meta) 
  34] insert*(F, G, _|_, X) >= _|_  by (Bot) 

  35] insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because [36], by (Star) 
  36] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because insert > cons, [37] and [47], by (Copy) 
  37] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [38] and [43], by (Copy) 
  38] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [39] and [41], by (Copy) 
  39] insert*(F, G, cons(X, Y), Z) >= F  because [40], by (Select) 
  40] F >= F  by (Meta) 
  41] insert*(F, G, cons(X, Y), Z) >= Z  because [42], by (Select) 
  42] Z >= Z  by (Meta) 
  43] insert*(F, G, cons(X, Y), Z) >= X  because [44], by (Select) 
  44] cons(X, Y) >= X  because [45], by (Star) 
  45] cons*(X, Y) >= X  because [46], by (Select) 
  46] X >= X  by (Meta) 
  47] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [48], [39], [51], [53] and [55], by (Stat) 
  48] cons(X, Y) > Y  because [49], by definition 
  49] cons*(X, Y) >= Y  because [50], by (Select) 
  50] Y >= Y  by (Meta) 
  51] insert*(F, G, cons(X, Y), Z) >= G  because [52], by (Select) 
  52] G >= G  by (Meta) 
  53] insert*(F, G, cons(X, Y), Z) >= Y  because [54], by (Select) 
  54] cons(X, Y) >= Y  because [49], by (Star) 
  55] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [56] and [43], by (Copy) 
  56] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [51] and [41], by (Copy) 

  57] sort(F, G, _|_) >= _|_  by (Bot) 

  58] sort(F, G, cons(X, Y)) > insert(F, G, sort(F, G, Y), X)  because [59], by definition 
  59] sort*(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because sort > insert, [60], [62], [64] and [70], by (Copy) 
  60] sort*(F, G, cons(X, Y)) >= F  because [61], by (Select) 
  61] F >= F  by (Meta) 
  62] sort*(F, G, cons(X, Y)) >= G  because [63], by (Select) 
  63] G >= G  by (Meta) 
  64] sort*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort in Mul, [65], [66] and [67], by (Stat) 
  65] F >= F  by (Meta) 
  66] G >= G  by (Meta) 
  67] cons(X, Y) > Y  because [68], by definition 
  68] cons*(X, Y) >= Y  because [69], by (Select) 
  69] Y >= Y  by (Meta) 
  70] sort*(F, G, cons(X, Y)) >= X  because [71], by (Select) 
  71] cons(X, Y) >= X  because [72], by (Star) 
  72] cons*(X, Y) >= X  because [73], by (Select) 
  73] X >= X  by (Meta) 

  74] ascending!fac6220sort(X) >= sort(min, max, X)  because [75], by (Star) 
  75] ascending!fac6220sort*(X) >= sort(min, max, X)  because ascending!fac6220sort > sort, [76], [77] and [78], by (Copy) 
  76] ascending!fac6220sort*(X) >= min  because ascending!fac6220sort > min, by (Copy) 
  77] ascending!fac6220sort*(X) >= max  because ascending!fac6220sort = max and ascending!fac6220sort in Mul, by (Stat) 
  78] ascending!fac6220sort*(X) >= X  because [79], by (Select) 
  79] X >= X  by (Meta) 

  80] descending!fac6220sort(X) >= sort(max, min, X)  because [81], by (Star) 
  81] descending!fac6220sort*(X) >= sort(max, min, X)  because descending!fac6220sort > sort, [82], [83] and [84], by (Copy) 
  82] descending!fac6220sort*(X) >= max  because descending!fac6220sort > max, by (Copy) 
  83] descending!fac6220sort*(X) >= min  because descending!fac6220sort > min, by (Copy) 
  84] descending!fac6220sort*(X) >= X  because [85], by (Select) 
  85] X >= X  by (Meta) 

We can thus remove the following rules:

  insert(F, G, nil, X) => cons(X, nil) 
  sort(F, G, cons(X, Y)) => insert(F, G, sort(F, G, Y), X) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  max 0 X >? X 
  max X 0 >? X 
  max s(X) s(Y) >? max X Y 
  min 0 X >? 0 
  min X 0 >? 0 
  min s(X) s(Y) >? min X Y 
  insert(F, G, cons(X, Y), Z) >? cons(F Z X, insert(F, G, Y, G Z X)) 
  sort(F, G, nil) >? nil 
  ascending!fac6220sort(X) >? sort(min, max, X) 
  descending!fac6220sort(X) >? sort(max, min, X) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_3, x_4, x_2, x_1) 
  [[nil]] = _|_ 

We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, ascending!fac6220sort, cons, descending!fac6220sort, max, min, s, sort}, and the following precedence: descending!fac6220sort > insert > @_{o -> o -> o} > @_{o -> o} > cons > ascending!fac6220sort = min > max > s > sort

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  @_{o -> o}(@_{o -> o -> o}(max, _|_), X) > X 
  @_{o -> o}(@_{o -> o -> o}(max, X), _|_) >= X 
  @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y) 
  @_{o -> o}(@_{o -> o -> o}(min, _|_), X) >= _|_ 
  @_{o -> o}(@_{o -> o -> o}(min, X), _|_) >= _|_ 
  @_{o -> o}(@_{o -> o -> o}(min, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(min, X), Y) 
  insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))) 
  sort(F, G, _|_) >= _|_ 
  ascending!fac6220sort(X) >= sort(min, max, X) 
  descending!fac6220sort(X) >= sort(max, min, X) 

With these choices, we have:

  1] @_{o -> o}(@_{o -> o -> o}(max, _|_), X) > X  because [2], by definition 
  2] @_{o -> o}*(@_{o -> o -> o}(max, _|_), X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] @_{o -> o}(@_{o -> o -> o}(max, X), _|_) >= X  because [5], by (Star) 
  5] @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [6], by (Select) 
  6] @_{o -> o -> o}(max, X) @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [7] 
  7] @_{o -> o -> o}*(max, X, @_{o -> o}*(@_{o -> o -> o}(max, X), _|_)) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

  9] @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because @_{o -> o} in Mul, [10] and [15], by (Fun) 
  10] @_{o -> o -> o}(max, s(X)) >= @_{o -> o -> o}(max, X)  because @_{o -> o -> o} in Mul, [11] and [12], by (Fun) 
  11] max >= max  by (Fun) 
  12] s(X) >= X  because [13], by (Star) 
  13] s*(X) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 
  15] s(Y) >= Y  because [16], by (Star) 
  16] s*(Y) >= Y  because [17], by (Select) 
  17] Y >= Y  by (Meta) 

  18] @_{o -> o}(@_{o -> o -> o}(min, _|_), X) >= _|_  by (Bot) 

  19] @_{o -> o}(@_{o -> o -> o}(min, X), _|_) >= _|_  by (Bot) 

  20] @_{o -> o}(@_{o -> o -> o}(min, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(min, X), Y)  because @_{o -> o} in Mul, [21] and [27], by (Fun) 
  21] @_{o -> o -> o}(min, s(X)) >= @_{o -> o -> o}(min, X)  because [22], by (Star) 
  22] @_{o -> o -> o}*(min, s(X)) >= @_{o -> o -> o}(min, X)  because @_{o -> o -> o} in Mul, [23] and [24], by (Stat) 
  23] min >= min  by (Fun) 
  24] s(X) > X  because [25], by definition 
  25] s*(X) >= X  because [26], by (Select) 
  26] X >= X  by (Meta) 
  27] s(Y) >= Y  because [28], by (Star) 
  28] s*(Y) >= Y  because [29], by (Select) 
  29] Y >= Y  by (Meta) 

  30] insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because [31], by (Star) 
  31] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because insert > cons, [32] and [42], by (Copy) 
  32] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [33] and [38], by (Copy) 
  33] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [34] and [36], by (Copy) 
  34] insert*(F, G, cons(X, Y), Z) >= F  because [35], by (Select) 
  35] F >= F  by (Meta) 
  36] insert*(F, G, cons(X, Y), Z) >= Z  because [37], by (Select) 
  37] Z >= Z  by (Meta) 
  38] insert*(F, G, cons(X, Y), Z) >= X  because [39], by (Select) 
  39] cons(X, Y) >= X  because [40], by (Star) 
  40] cons*(X, Y) >= X  because [41], by (Select) 
  41] X >= X  by (Meta) 
  42] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [43], [34], [46], [48] and [50], by (Stat) 
  43] cons(X, Y) > Y  because [44], by definition 
  44] cons*(X, Y) >= Y  because [45], by (Select) 
  45] Y >= Y  by (Meta) 
  46] insert*(F, G, cons(X, Y), Z) >= G  because [47], by (Select) 
  47] G >= G  by (Meta) 
  48] insert*(F, G, cons(X, Y), Z) >= Y  because [49], by (Select) 
  49] cons(X, Y) >= Y  because [44], by (Star) 
  50] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [51] and [38], by (Copy) 
  51] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [46] and [36], by (Copy) 

  52] sort(F, G, _|_) >= _|_  by (Bot) 

  53] ascending!fac6220sort(X) >= sort(min, max, X)  because [54], by (Star) 
  54] ascending!fac6220sort*(X) >= sort(min, max, X)  because ascending!fac6220sort > sort, [55], [56] and [57], by (Copy) 
  55] ascending!fac6220sort*(X) >= min  because ascending!fac6220sort = min and ascending!fac6220sort in Mul, by (Stat) 
  56] ascending!fac6220sort*(X) >= max  because ascending!fac6220sort > max, by (Copy) 
  57] ascending!fac6220sort*(X) >= X  because [58], by (Select) 
  58] X >= X  by (Meta) 

  59] descending!fac6220sort(X) >= sort(max, min, X)  because [60], by (Star) 
  60] descending!fac6220sort*(X) >= sort(max, min, X)  because descending!fac6220sort > sort, [61], [62] and [63], by (Copy) 
  61] descending!fac6220sort*(X) >= max  because descending!fac6220sort > max, by (Copy) 
  62] descending!fac6220sort*(X) >= min  because descending!fac6220sort > min, by (Copy) 
  63] descending!fac6220sort*(X) >= X  because [64], by (Select) 
  64] X >= X  by (Meta) 

We can thus remove the following rules:

  max 0 X => X 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  max X 0 >? X 
  max s(X) s(Y) >? max X Y 
  min 0 X >? 0 
  min X 0 >? 0 
  min s(X) s(Y) >? min X Y 
  insert(F, G, cons(X, Y), Z) >? cons(F Z X, insert(F, G, Y, G Z X)) 
  sort(F, G, nil) >? nil 
  ascending!fac6220sort(X) >? sort(min, max, X) 
  descending!fac6220sort(X) >? sort(max, min, X) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[@_{o -> o}(x_1, x_2)]] = @_{o -> o}(x_2, x_1) 
  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_3, x_1, x_4, x_2) 
  [[nil]] = _|_ 

We choose Lex = {@_{o -> o}, insert} and Mul = {@_{o -> o -> o}, ascending!fac6220sort, cons, descending!fac6220sort, max, min, s, sort}, and the following precedence: descending!fac6220sort > insert > @_{o -> o} > @_{o -> o -> o} > ascending!fac6220sort > min > cons > max > s > sort

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  @_{o -> o}(@_{o -> o -> o}(max, X), _|_) > X 
  @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y) 
  @_{o -> o}(@_{o -> o -> o}(min, _|_), X) >= _|_ 
  @_{o -> o}(@_{o -> o -> o}(min, X), _|_) >= _|_ 
  @_{o -> o}(@_{o -> o -> o}(min, s(X)), s(Y)) > @_{o -> o}(@_{o -> o -> o}(min, X), Y) 
  insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}, insert(F, G, Y, @_{o -> o} @_{o -> o -> o}(G, Z) X)) 
  sort(F, G, _|_) >= _|_ 
  ascending!fac6220sort(X) >= sort(min, max, X) 
  descending!fac6220sort(X) >= sort(max, min, X) 

With these choices, we have:

  1] @_{o -> o}(@_{o -> o -> o}(max, X), _|_) > X  because [2], by definition 
  2] @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [3], by (Select) 
  3] @_{o -> o -> o}(max, X) @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [4] 
  4] @_{o -> o -> o}*(max, X, @_{o -> o}*(@_{o -> o -> o}(max, X), _|_)) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 

  6] @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because [7] and [12], by (Fun) 
  7] @_{o -> o -> o}(max, s(X)) >= @_{o -> o -> o}(max, X)  because @_{o -> o -> o} in Mul, [8] and [9], by (Fun) 
  8] max >= max  by (Fun) 
  9] s(X) >= X  because [10], by (Star) 
  10] s*(X) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] s(Y) >= Y  because [13], by (Star) 
  13] s*(Y) >= Y  because [14], by (Select) 
  14] Y >= Y  by (Meta) 

  15] @_{o -> o}(@_{o -> o -> o}(min, _|_), X) >= _|_  by (Bot) 

  16] @_{o -> o}(@_{o -> o -> o}(min, X), _|_) >= _|_  by (Bot) 

  17] @_{o -> o}(@_{o -> o -> o}(min, s(X)), s(Y)) > @_{o -> o}(@_{o -> o -> o}(min, X), Y)  because [18], by definition 
  18] @_{o -> o}*(@_{o -> o -> o}(min, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(min, X), Y)  because [19], [22] and [30], by (Stat) 
  19] s(Y) > Y  because [20], by definition 
  20] s*(Y) >= Y  because [21], by (Select) 
  21] Y >= Y  by (Meta) 
  22] @_{o -> o}*(@_{o -> o -> o}(min, s(X)), s(Y)) >= @_{o -> o -> o}(min, X)  because @_{o -> o} > @_{o -> o -> o}, [23] and [24], by (Copy) 
  23] @_{o -> o}*(@_{o -> o -> o}(min, s(X)), s(Y)) >= min  because @_{o -> o} > min, by (Copy) 
  24] @_{o -> o}*(@_{o -> o -> o}(min, s(X)), s(Y)) >= X  because [25], by (Select) 
  25] @_{o -> o -> o}(min, s(X)) @_{o -> o}*(@_{o -> o -> o}(min, s(X)), s(Y)) >= X  because [26] 
  26] @_{o -> o -> o}*(min, s(X), @_{o -> o}*(@_{o -> o -> o}(min, s(X)), s(Y))) >= X  because [27], by (Select) 
  27] s(X) >= X  because [28], by (Star) 
  28] s*(X) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 
  30] @_{o -> o}*(@_{o -> o -> o}(min, s(X)), s(Y)) >= Y  because [31], by (Select) 
  31] s(Y) >= Y  because [20], by (Star) 

  32] insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}, insert(F, G, Y, @_{o -> o} @_{o -> o -> o}(G, Z) X))  because [33], by (Star) 
  33] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}, insert(F, G, Y, @_{o -> o} @_{o -> o -> o}(G, Z) X))  because insert > cons, [34] and [44], by (Copy) 
  34] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [35] and [40], by (Copy) 
  35] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [36] and [38], by (Copy) 
  36] insert*(F, G, cons(X, Y), Z) >= F  because [37], by (Select) 
  37] F >= F  by (Meta) 
  38] insert*(F, G, cons(X, Y), Z) >= Z  because [39], by (Select) 
  39] Z >= Z  by (Meta) 
  40] insert*(F, G, cons(X, Y), Z) >= X  because [41], by (Select) 
  41] cons(X, Y) >= X  because [42], by (Star) 
  42] cons*(X, Y) >= X  because [43], by (Select) 
  43] X >= X  by (Meta) 
  44] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [45], [36], [48], [50] and [52], by (Stat) 
  45] cons(X, Y) > Y  because [46], by definition 
  46] cons*(X, Y) >= Y  because [47], by (Select) 
  47] Y >= Y  by (Meta) 
  48] insert*(F, G, cons(X, Y), Z) >= G  because [49], by (Select) 
  49] G >= G  by (Meta) 
  50] insert*(F, G, cons(X, Y), Z) >= Y  because [51], by (Select) 
  51] cons(X, Y) >= Y  because [46], by (Star) 
  52] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [53] and [40], by (Copy) 
  53] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [48] and [38], by (Copy) 

  54] sort(F, G, _|_) >= _|_  by (Bot) 

  55] ascending!fac6220sort(X) >= sort(min, max, X)  because [56], by (Star) 
  56] ascending!fac6220sort*(X) >= sort(min, max, X)  because ascending!fac6220sort > sort, [57], [58] and [59], by (Copy) 
  57] ascending!fac6220sort*(X) >= min  because ascending!fac6220sort > min, by (Copy) 
  58] ascending!fac6220sort*(X) >= max  because ascending!fac6220sort > max, by (Copy) 
  59] ascending!fac6220sort*(X) >= X  because [60], by (Select) 
  60] X >= X  by (Meta) 

  61] descending!fac6220sort(X) >= sort(max, min, X)  because [62], by (Star) 
  62] descending!fac6220sort*(X) >= sort(max, min, X)  because descending!fac6220sort > sort, [63], [64] and [65], by (Copy) 
  63] descending!fac6220sort*(X) >= max  because descending!fac6220sort > max, by (Copy) 
  64] descending!fac6220sort*(X) >= min  because descending!fac6220sort > min, by (Copy) 
  65] descending!fac6220sort*(X) >= X  because [66], by (Select) 
  66] X >= X  by (Meta) 

We can thus remove the following rules:

  max X 0 => X 
  min s(X) s(Y) => min X Y 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  max s(X) s(Y) >? max X Y 
  min 0 X >? 0 
  min X 0 >? 0 
  insert(F, G, cons(X, Y), Z) >? cons(F Z X, insert(F, G, Y, G Z X)) 
  sort(F, G, nil) >? nil 
  ascending!fac6220sort(X) >? sort(min, max, X) 
  descending!fac6220sort(X) >? sort(max, min, X) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_3, x_1, x_4, x_2) 
  [[min]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, ascending!fac6220sort, cons, descending!fac6220sort, max, s, sort}, and the following precedence: ascending!fac6220sort > descending!fac6220sort > insert > @_{o -> o -> o} > @_{o -> o} > cons > max > s > sort

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y) 
  @_{o -> o}(@_{o -> o -> o}(_|_, _|_), X) > _|_ 
  @_{o -> o}(@_{o -> o -> o}(_|_, X), _|_) >= _|_ 
  insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))) 
  sort(F, G, _|_) >= _|_ 
  ascending!fac6220sort(X) >= sort(_|_, max, X) 
  descending!fac6220sort(X) >= sort(max, _|_, X) 

With these choices, we have:

  1] @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because @_{o -> o} in Mul, [2] and [7], by (Fun) 
  2] @_{o -> o -> o}(max, s(X)) >= @_{o -> o -> o}(max, X)  because @_{o -> o -> o} in Mul, [3] and [4], by (Fun) 
  3] max >= max  by (Fun) 
  4] s(X) >= X  because [5], by (Star) 
  5] s*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] s(Y) >= Y  because [8], by (Star) 
  8] s*(Y) >= Y  because [9], by (Select) 
  9] Y >= Y  by (Meta) 

  10] @_{o -> o}(@_{o -> o -> o}(_|_, _|_), X) > _|_  because [11], by definition 
  11] @_{o -> o}*(@_{o -> o -> o}(_|_, _|_), X) >= _|_  by (Bot) 

  12] @_{o -> o}(@_{o -> o -> o}(_|_, X), _|_) >= _|_  by (Bot) 

  13] insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because [14], by (Star) 
  14] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because insert > cons, [15] and [25], by (Copy) 
  15] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [16] and [21], by (Copy) 
  16] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [17] and [19], by (Copy) 
  17] insert*(F, G, cons(X, Y), Z) >= F  because [18], by (Select) 
  18] F >= F  by (Meta) 
  19] insert*(F, G, cons(X, Y), Z) >= Z  because [20], by (Select) 
  20] Z >= Z  by (Meta) 
  21] insert*(F, G, cons(X, Y), Z) >= X  because [22], by (Select) 
  22] cons(X, Y) >= X  because [23], by (Star) 
  23] cons*(X, Y) >= X  because [24], by (Select) 
  24] X >= X  by (Meta) 
  25] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [26], [17], [29], [31] and [33], by (Stat) 
  26] cons(X, Y) > Y  because [27], by definition 
  27] cons*(X, Y) >= Y  because [28], by (Select) 
  28] Y >= Y  by (Meta) 
  29] insert*(F, G, cons(X, Y), Z) >= G  because [30], by (Select) 
  30] G >= G  by (Meta) 
  31] insert*(F, G, cons(X, Y), Z) >= Y  because [32], by (Select) 
  32] cons(X, Y) >= Y  because [27], by (Star) 
  33] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [34] and [21], by (Copy) 
  34] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [29] and [19], by (Copy) 

  35] sort(F, G, _|_) >= _|_  by (Bot) 

  36] ascending!fac6220sort(X) >= sort(_|_, max, X)  because [37], by (Star) 
  37] ascending!fac6220sort*(X) >= sort(_|_, max, X)  because ascending!fac6220sort > sort, [38], [39] and [40], by (Copy) 
  38] ascending!fac6220sort*(X) >= _|_  by (Bot) 
  39] ascending!fac6220sort*(X) >= max  because ascending!fac6220sort > max, by (Copy) 
  40] ascending!fac6220sort*(X) >= X  because [41], by (Select) 
  41] X >= X  by (Meta) 

  42] descending!fac6220sort(X) >= sort(max, _|_, X)  because [43], by (Star) 
  43] descending!fac6220sort*(X) >= sort(max, _|_, X)  because descending!fac6220sort > sort, [44], [45] and [46], by (Copy) 
  44] descending!fac6220sort*(X) >= max  because descending!fac6220sort > max, by (Copy) 
  45] descending!fac6220sort*(X) >= _|_  by (Bot) 
  46] descending!fac6220sort*(X) >= X  because [47], by (Select) 
  47] X >= X  by (Meta) 

We can thus remove the following rules:

  min 0 X => 0 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  max s(X) s(Y) >? max X Y 
  min X 0 >? 0 
  insert(F, G, cons(X, Y), Z) >? cons(F Z X, insert(F, G, Y, G Z X)) 
  sort(F, G, nil) >? nil 
  ascending!fac6220sort(X) >? sort(min, max, X) 
  descending!fac6220sort(X) >? sort(max, min, X) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_3, x_1, x_2, x_4) 
  [[min]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, ascending!fac6220sort, cons, descending!fac6220sort, max, s, sort}, and the following precedence: ascending!fac6220sort > insert > @_{o -> o -> o} > @_{o -> o} > cons > descending!fac6220sort > max > s > sort

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y) 
  @_{o -> o}(@_{o -> o -> o}(_|_, X), _|_) > _|_ 
  insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))) 
  sort(F, G, _|_) >= _|_ 
  ascending!fac6220sort(X) >= sort(_|_, max, X) 
  descending!fac6220sort(X) >= sort(max, _|_, X) 

With these choices, we have:

  1] @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because @_{o -> o} in Mul, [2] and [7], by (Fun) 
  2] @_{o -> o -> o}(max, s(X)) >= @_{o -> o -> o}(max, X)  because @_{o -> o -> o} in Mul, [3] and [4], by (Fun) 
  3] max >= max  by (Fun) 
  4] s(X) >= X  because [5], by (Star) 
  5] s*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] s(Y) >= Y  because [8], by (Star) 
  8] s*(Y) >= Y  because [9], by (Select) 
  9] Y >= Y  by (Meta) 

  10] @_{o -> o}(@_{o -> o -> o}(_|_, X), _|_) > _|_  because [11], by definition 
  11] @_{o -> o}*(@_{o -> o -> o}(_|_, X), _|_) >= _|_  by (Bot) 

  12] insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because [13], by (Star) 
  13] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because insert > cons, [14] and [24], by (Copy) 
  14] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [15] and [20], by (Copy) 
  15] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [16] and [18], by (Copy) 
  16] insert*(F, G, cons(X, Y), Z) >= F  because [17], by (Select) 
  17] F >= F  by (Meta) 
  18] insert*(F, G, cons(X, Y), Z) >= Z  because [19], by (Select) 
  19] Z >= Z  by (Meta) 
  20] insert*(F, G, cons(X, Y), Z) >= X  because [21], by (Select) 
  21] cons(X, Y) >= X  because [22], by (Star) 
  22] cons*(X, Y) >= X  because [23], by (Select) 
  23] X >= X  by (Meta) 
  24] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [25], [16], [28], [30] and [32], by (Stat) 
  25] cons(X, Y) > Y  because [26], by definition 
  26] cons*(X, Y) >= Y  because [27], by (Select) 
  27] Y >= Y  by (Meta) 
  28] insert*(F, G, cons(X, Y), Z) >= G  because [29], by (Select) 
  29] G >= G  by (Meta) 
  30] insert*(F, G, cons(X, Y), Z) >= Y  because [31], by (Select) 
  31] cons(X, Y) >= Y  because [26], by (Star) 
  32] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [33] and [20], by (Copy) 
  33] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [28] and [18], by (Copy) 

  34] sort(F, G, _|_) >= _|_  by (Bot) 

  35] ascending!fac6220sort(X) >= sort(_|_, max, X)  because [36], by (Star) 
  36] ascending!fac6220sort*(X) >= sort(_|_, max, X)  because ascending!fac6220sort > sort, [37], [38] and [39], by (Copy) 
  37] ascending!fac6220sort*(X) >= _|_  by (Bot) 
  38] ascending!fac6220sort*(X) >= max  because ascending!fac6220sort > max, by (Copy) 
  39] ascending!fac6220sort*(X) >= X  because [40], by (Select) 
  40] X >= X  by (Meta) 

  41] descending!fac6220sort(X) >= sort(max, _|_, X)  because [42], by (Star) 
  42] descending!fac6220sort*(X) >= sort(max, _|_, X)  because descending!fac6220sort > sort, [43], [44] and [45], by (Copy) 
  43] descending!fac6220sort*(X) >= max  because descending!fac6220sort > max, by (Copy) 
  44] descending!fac6220sort*(X) >= _|_  by (Bot) 
  45] descending!fac6220sort*(X) >= X  because [46], by (Select) 
  46] X >= X  by (Meta) 

We can thus remove the following rules:

  min X 0 => 0 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  max s(X) s(Y) >? max X Y 
  insert(F, G, cons(X, Y), Z) >? cons(F Z X, insert(F, G, Y, G Z X)) 
  sort(F, G, nil) >? nil 
  ascending!fac6220sort(X) >? sort(min, max, X) 
  descending!fac6220sort(X) >? sort(max, min, X) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_1, x_3, x_2, x_4) 
  [[min]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, ascending!fac6220sort, cons, descending!fac6220sort, max, s, sort}, and the following precedence: ascending!fac6220sort > descending!fac6220sort > insert > @_{o -> o -> o} > cons > max > s > sort > @_{o -> o}

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y) 
  insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))) 
  sort(F, G, _|_) > _|_ 
  ascending!fac6220sort(X) >= sort(_|_, max, X) 
  descending!fac6220sort(X) >= sort(max, _|_, X) 

With these choices, we have:

  1] @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because @_{o -> o} in Mul, [2] and [7], by (Fun) 
  2] @_{o -> o -> o}(max, s(X)) >= @_{o -> o -> o}(max, X)  because @_{o -> o -> o} in Mul, [3] and [4], by (Fun) 
  3] max >= max  by (Fun) 
  4] s(X) >= X  because [5], by (Star) 
  5] s*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] s(Y) >= Y  because [8], by (Star) 
  8] s*(Y) >= Y  because [9], by (Select) 
  9] Y >= Y  by (Meta) 

  10] insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because [11], by (Star) 
  11] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because insert > cons, [12] and [22], by (Copy) 
  12] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [13] and [18], by (Copy) 
  13] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [14] and [16], by (Copy) 
  14] insert*(F, G, cons(X, Y), Z) >= F  because [15], by (Select) 
  15] F >= F  by (Meta) 
  16] insert*(F, G, cons(X, Y), Z) >= Z  because [17], by (Select) 
  17] Z >= Z  by (Meta) 
  18] insert*(F, G, cons(X, Y), Z) >= X  because [19], by (Select) 
  19] cons(X, Y) >= X  because [20], by (Star) 
  20] cons*(X, Y) >= X  because [21], by (Select) 
  21] X >= X  by (Meta) 
  22] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [23], [24], [14], [27], [29] and [31], by (Stat) 
  23] F >= F  by (Meta) 
  24] cons(X, Y) > Y  because [25], by definition 
  25] cons*(X, Y) >= Y  because [26], by (Select) 
  26] Y >= Y  by (Meta) 
  27] insert*(F, G, cons(X, Y), Z) >= G  because [28], by (Select) 
  28] G >= G  by (Meta) 
  29] insert*(F, G, cons(X, Y), Z) >= Y  because [30], by (Select) 
  30] cons(X, Y) >= Y  because [25], by (Star) 
  31] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [32] and [18], by (Copy) 
  32] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [27] and [16], by (Copy) 

  33] sort(F, G, _|_) > _|_  because [34], by definition 
  34] sort*(F, G, _|_) >= _|_  by (Bot) 

  35] ascending!fac6220sort(X) >= sort(_|_, max, X)  because [36], by (Star) 
  36] ascending!fac6220sort*(X) >= sort(_|_, max, X)  because ascending!fac6220sort > sort, [37], [38] and [39], by (Copy) 
  37] ascending!fac6220sort*(X) >= _|_  by (Bot) 
  38] ascending!fac6220sort*(X) >= max  because ascending!fac6220sort > max, by (Copy) 
  39] ascending!fac6220sort*(X) >= X  because [40], by (Select) 
  40] X >= X  by (Meta) 

  41] descending!fac6220sort(X) >= sort(max, _|_, X)  because [42], by (Star) 
  42] descending!fac6220sort*(X) >= sort(max, _|_, X)  because descending!fac6220sort > sort, [43], [44] and [45], by (Copy) 
  43] descending!fac6220sort*(X) >= max  because descending!fac6220sort > max, by (Copy) 
  44] descending!fac6220sort*(X) >= _|_  by (Bot) 
  45] descending!fac6220sort*(X) >= X  because [46], by (Select) 
  46] X >= X  by (Meta) 

We can thus remove the following rules:

  sort(F, G, nil) => nil 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  max s(X) s(Y) >? max X Y 
  insert(F, G, cons(X, Y), Z) >? cons(F Z X, insert(F, G, Y, G Z X)) 
  ascending!fac6220sort(X) >? sort(min, max, X) 
  descending!fac6220sort(X) >? sort(max, min, X) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_1, x_3, x_2, x_4) 
  [[min]] = _|_ 

We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, ascending!fac6220sort, cons, descending!fac6220sort, max, s, sort}, and the following precedence: ascending!fac6220sort > descending!fac6220sort > insert > @_{o -> o -> o} > @_{o -> o} > cons > max > s > sort

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y) 
  insert(F, G, cons(X, Y), Z) > cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))) 
  ascending!fac6220sort(X) >= sort(_|_, max, X) 
  descending!fac6220sort(X) >= sort(max, _|_, X) 

With these choices, we have:

  1] @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because @_{o -> o} in Mul, [2] and [7], by (Fun) 
  2] @_{o -> o -> o}(max, s(X)) >= @_{o -> o -> o}(max, X)  because @_{o -> o -> o} in Mul, [3] and [4], by (Fun) 
  3] max >= max  by (Fun) 
  4] s(X) >= X  because [5], by (Star) 
  5] s*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] s(Y) >= Y  because [8], by (Star) 
  8] s*(Y) >= Y  because [9], by (Select) 
  9] Y >= Y  by (Meta) 

  10] insert(F, G, cons(X, Y), Z) > cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because [11], by definition 
  11] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because insert > cons, [12] and [22], by (Copy) 
  12] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [13] and [18], by (Copy) 
  13] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [14] and [16], by (Copy) 
  14] insert*(F, G, cons(X, Y), Z) >= F  because [15], by (Select) 
  15] F >= F  by (Meta) 
  16] insert*(F, G, cons(X, Y), Z) >= Z  because [17], by (Select) 
  17] Z >= Z  by (Meta) 
  18] insert*(F, G, cons(X, Y), Z) >= X  because [19], by (Select) 
  19] cons(X, Y) >= X  because [20], by (Star) 
  20] cons*(X, Y) >= X  because [21], by (Select) 
  21] X >= X  by (Meta) 
  22] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [23], [24], [14], [27], [29] and [31], by (Stat) 
  23] F >= F  by (Meta) 
  24] cons(X, Y) > Y  because [25], by definition 
  25] cons*(X, Y) >= Y  because [26], by (Select) 
  26] Y >= Y  by (Meta) 
  27] insert*(F, G, cons(X, Y), Z) >= G  because [28], by (Select) 
  28] G >= G  by (Meta) 
  29] insert*(F, G, cons(X, Y), Z) >= Y  because [30], by (Select) 
  30] cons(X, Y) >= Y  because [25], by (Star) 
  31] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [32] and [18], by (Copy) 
  32] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [27] and [16], by (Copy) 

  33] ascending!fac6220sort(X) >= sort(_|_, max, X)  because [34], by (Star) 
  34] ascending!fac6220sort*(X) >= sort(_|_, max, X)  because ascending!fac6220sort > sort, [35], [36] and [37], by (Copy) 
  35] ascending!fac6220sort*(X) >= _|_  by (Bot) 
  36] ascending!fac6220sort*(X) >= max  because ascending!fac6220sort > max, by (Copy) 
  37] ascending!fac6220sort*(X) >= X  because [38], by (Select) 
  38] X >= X  by (Meta) 

  39] descending!fac6220sort(X) >= sort(max, _|_, X)  because [40], by (Star) 
  40] descending!fac6220sort*(X) >= sort(max, _|_, X)  because descending!fac6220sort > sort, [41], [42] and [43], by (Copy) 
  41] descending!fac6220sort*(X) >= max  because descending!fac6220sort > max, by (Copy) 
  42] descending!fac6220sort*(X) >= _|_  by (Bot) 
  43] descending!fac6220sort*(X) >= X  because [44], by (Select) 
  44] X >= X  by (Meta) 

We can thus remove the following rules:

  insert(F, G, cons(X, Y), Z) => cons(F Z X, insert(F, G, Y, G Z X)) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  max s(X) s(Y) >? max X Y 
  ascending!fac6220sort(X) >? sort(min, max, X) 
  descending!fac6220sort(X) >? sort(max, min, X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  ascending!fac6220sort = \y0.3 + 3y0 
  descending!fac6220sort = \y0.3 + 3y0 
  max = \y0y1.0 
  min = \y0y1.0 
  s = \y0.3 + 3y0 
  sort = \G0G1y2.y2 + G0(0,0) + G1(0,0) 

Using this interpretation, the requirements translate to:

  [[max s(_x0) s(_x1)]] = 6 + 3x0 + 3x1 > x0 + x1 = [[max _x0 _x1]] 
  [[ascending!fac6220sort(_x0)]] = 3 + 3x0 > x0 = [[sort(min, max, _x0)]] 
  [[descending!fac6220sort(_x0)]] = 3 + 3x0 > x0 = [[sort(max, min, _x0)]] 

We can thus remove the following rules:

  max s(X) s(Y) => max X Y 
  ascending!fac6220sort(X) => sort(min, max, X) 
  descending!fac6220sort(X) => sort(max, min, X) 

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.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.