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_1, x_3, x_4, x_2) 
  [[min]] = _|_ 

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

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}(_|_, _|_), X) >= _|_ 
  @_{o -> o}(@_{o -> o -> o}(_|_, X), _|_) >= _|_ 
  @_{o -> o}(@_{o -> o -> o}(_|_, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(_|_, X), Y) 
  insert(F, G, nil, X) > cons(X, nil) 
  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, nil) >= nil 
  sort(F, G, cons(X, Y)) > insert(F, G, sort(F, G, Y), 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, _|_), 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 [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}(_|_, _|_), X) >= _|_  by (Bot) 

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

  21] @_{o -> o}(@_{o -> o -> o}(_|_, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(_|_, X), Y)  because @_{o -> o} in Mul, [22] and [27], by (Fun) 
  22] @_{o -> o -> o}(_|_, s(X)) >= @_{o -> o -> o}(_|_, X)  because @_{o -> o -> o} in Mul, [23] and [24], by (Fun) 
  23] _|_ >= _|_  by (Bot) 
  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, nil, X) > cons(X, nil)  because [31], by definition 
  31] insert*(F, G, nil, X) >= cons(X, nil)  because insert > cons, [32] and [34], by (Copy) 
  32] insert*(F, G, nil, X) >= X  because [33], by (Select) 
  33] X >= X  by (Meta) 
  34] insert*(F, G, nil, X) >= nil  because insert > nil, by (Copy) 

  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], [49], [39], [52], [54] and [56], by (Stat) 
  48] F >= F  by (Meta) 
  49] cons(X, Y) > Y  because [50], by definition 
  50] cons*(X, Y) >= Y  because [51], by (Select) 
  51] Y >= Y  by (Meta) 
  52] insert*(F, G, cons(X, Y), Z) >= G  because [53], by (Select) 
  53] G >= G  by (Meta) 
  54] insert*(F, G, cons(X, Y), Z) >= Y  because [55], by (Select) 
  55] cons(X, Y) >= Y  because [50], by (Star) 
  56] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [57] and [43], by (Copy) 
  57] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [52] and [41], by (Copy) 

  58] sort(F, G, nil) >= nil  because [59], by (Star) 
  59] sort*(F, G, nil) >= nil  because sort > nil, by (Copy) 

  60] sort(F, G, cons(X, Y)) > insert(F, G, sort(F, G, Y), X)  because [61], by definition 
  61] sort*(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because sort > insert, [62], [64], [66] and [72], by (Copy) 
  62] sort*(F, G, cons(X, Y)) >= F  because [63], by (Select) 
  63] F >= F  by (Meta) 
  64] sort*(F, G, cons(X, Y)) >= G  because [65], by (Select) 
  65] G >= G  by (Meta) 
  66] sort*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort in Mul, [67], [68] and [69], by (Stat) 
  67] F >= F  by (Meta) 
  68] G >= G  by (Meta) 
  69] cons(X, Y) > Y  because [70], by definition 
  70] cons*(X, Y) >= Y  because [71], by (Select) 
  71] Y >= Y  by (Meta) 
  72] sort*(F, G, cons(X, Y)) >= X  because [73], by (Select) 
  73] cons(X, Y) >= X  because [74], by (Star) 
  74] cons*(X, Y) >= X  because [75], by (Select) 
  75] X >= X  by (Meta) 

  76] ascending!fac6220sort(X) > sort(_|_, max, X)  because [77], by definition 
  77] ascending!fac6220sort*(X) >= sort(_|_, max, X)  because ascending!fac6220sort > sort, [78], [79] and [80], by (Copy) 
  78] ascending!fac6220sort*(X) >= _|_  by (Bot) 
  79] ascending!fac6220sort*(X) >= max  because ascending!fac6220sort > max, by (Copy) 
  80] ascending!fac6220sort*(X) >= X  because [81], by (Select) 
  81] X >= X  by (Meta) 

  82] descending!fac6220sort(X) >= sort(max, _|_, X)  because [83], by (Star) 
  83] descending!fac6220sort*(X) >= sort(max, _|_, X)  because descending!fac6220sort > sort, [84], [85] and [86], by (Copy) 
  84] descending!fac6220sort*(X) >= max  because descending!fac6220sort > max, by (Copy) 
  85] descending!fac6220sort*(X) >= _|_  by (Bot) 
  86] descending!fac6220sort*(X) >= X  because [87], by (Select) 
  87] X >= X  by (Meta) 

We can thus remove the following rules:

  max 0 X => X 
  insert(F, G, nil, X) => cons(X, nil) 
  sort(F, G, cons(X, Y)) => insert(F, G, sort(F, G, Y), X) 
  ascending!fac6220sort(X) => sort(min, max, 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 
  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_1, x_4) 

We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, cons, descending!fac6220sort, max, min, nil, s, sort}, and the following precedence: descending!fac6220sort > sort > insert > @_{o -> o -> o} > cons > nil > s > max > min > @_{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, 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, nil) > nil 
  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] @_{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 @_{o -> o} in Mul, [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) > _|_  because [16], by definition 
  16] @_{o -> o}*(@_{o -> o -> o}(min, _|_), X) >= _|_  by (Bot) 

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

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

  32] 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 [33], by (Star) 
  33] 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, [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, nil) > nil  because [55], by definition 
  55] sort*(F, G, nil) >= nil  because [56], by (Select) 
  56] nil >= nil  by (Fun) 

  57] descending!fac6220sort(X) > sort(max, min, X)  because [58], by definition 
  58] descending!fac6220sort*(X) >= sort(max, min, X)  because descending!fac6220sort > sort, [59], [60] and [61], by (Copy) 
  59] descending!fac6220sort*(X) >= max  because descending!fac6220sort > max, by (Copy) 
  60] descending!fac6220sort*(X) >= min  because descending!fac6220sort > min, by (Copy) 
  61] descending!fac6220sort*(X) >= X  because [62], by (Select) 
  62] X >= X  by (Meta) 

We can thus remove the following rules:

  min 0 X => 0 
  sort(F, G, nil) => nil 
  descending!fac6220sort(X) => sort(max, min, 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(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)) 

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) 

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

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

  max(X, _|_) > X 
  max(s(X), s(Y)) >= max(X, Y) 
  min(X, _|_) >= _|_ 
  min(s(X), s(Y)) >= 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))) 

With these choices, we have:

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

  4] max(s(X), s(Y)) >= max(X, Y)  because max in Mul, [5] and [8], by (Fun) 
  5] s(X) >= X  because [6], by (Star) 
  6] s*(X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] s(Y) >= Y  because [9], by (Star) 
  9] s*(Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 

  11] min(X, _|_) >= _|_  by (Bot) 

  12] min(s(X), s(Y)) >= min(X, Y)  because [13], by (Star) 
  13] min*(s(X), s(Y)) >= min(X, Y)  because min in Mul, [14] and [17], by (Stat) 
  14] s(X) > X  because [15], by definition 
  15] s*(X) >= X  because [16], by (Select) 
  16] X >= X  by (Meta) 
  17] s(Y) >= Y  because [18], by (Star) 
  18] s*(Y) >= Y  because [19], by (Select) 
  19] Y >= Y  by (Meta) 

  20] 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 [21], by (Star) 
  21] 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, [22] and [32], by (Copy) 
  22] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [23] and [28], by (Copy) 
  23] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [24] and [26], by (Copy) 
  24] insert*(F, G, cons(X, Y), Z) >= F  because [25], by (Select) 
  25] F >= F  by (Meta) 
  26] insert*(F, G, cons(X, Y), Z) >= Z  because [27], by (Select) 
  27] Z >= Z  by (Meta) 
  28] insert*(F, G, cons(X, Y), Z) >= X  because [29], by (Select) 
  29] cons(X, Y) >= X  because [30], by (Star) 
  30] cons*(X, Y) >= X  because [31], by (Select) 
  31] X >= X  by (Meta) 
  32] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [33], [24], [36], [38] and [40], by (Stat) 
  33] cons(X, Y) > Y  because [34], by definition 
  34] cons*(X, Y) >= Y  because [35], by (Select) 
  35] Y >= Y  by (Meta) 
  36] insert*(F, G, cons(X, Y), Z) >= G  because [37], by (Select) 
  37] G >= G  by (Meta) 
  38] insert*(F, G, cons(X, Y), Z) >= Y  because [39], by (Select) 
  39] cons(X, Y) >= Y  because [34], by (Star) 
  40] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [41] and [28], by (Copy) 
  41] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [36] and [26], by (Copy) 

We can thus remove the following rules:

  max(X, 0) => 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) 
  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)) 

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

Argument functions:

  [[0]] = _|_ 

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

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

  max(s(X), s(Y)) > max(X, Y) 
  min(X, _|_) >= _|_ 
  min(s(X), s(Y)) >= 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))) 

With these choices, we have:

  1] max(s(X), s(Y)) > max(X, Y)  because [2], by definition 
  2] max*(s(X), s(Y)) >= max(X, Y)  because max in Mul, [3] and [6], by (Stat) 
  3] s(X) >= X  because [4], by (Star) 
  4] s*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 
  6] s(Y) > Y  because [7], by definition 
  7] s*(Y) >= Y  because [8], by (Select) 
  8] Y >= Y  by (Meta) 

  9] min(X, _|_) >= _|_  by (Bot) 

  10] min(s(X), s(Y)) >= min(X, Y)  because [11], by (Star) 
  11] min*(s(X), s(Y)) >= min(X, Y)  because min in Mul, [12] and [15], by (Stat) 
  12] s(X) > X  because [13], by definition 
  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] 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 [19], by definition 
  19] 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, [20] and [30], by (Copy) 
  20] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [21] and [26], by (Copy) 
  21] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [22] and [24], by (Copy) 
  22] insert*(F, G, cons(X, Y), Z) >= F  because [23], by (Select) 
  23] F >= F  by (Meta) 
  24] insert*(F, G, cons(X, Y), Z) >= Z  because [25], by (Select) 
  25] Z >= Z  by (Meta) 
  26] insert*(F, G, cons(X, Y), Z) >= X  because [27], by (Select) 
  27] cons(X, Y) >= X  because [28], by (Star) 
  28] cons*(X, Y) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 
  30] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [31], [32], [33], [22], [36], [37] and [39], by (Stat) 
  31] F >= F  by (Meta) 
  32] G >= G  by (Meta) 
  33] cons(X, Y) > Y  because [34], by definition 
  34] cons*(X, Y) >= Y  because [35], by (Select) 
  35] Y >= Y  by (Meta) 
  36] insert*(F, G, cons(X, Y), Z) >= G  because [32], by (Select) 
  37] insert*(F, G, cons(X, Y), Z) >= Y  because [38], by (Select) 
  38] cons(X, Y) >= Y  because [34], by (Star) 
  39] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [40] and [26], by (Copy) 
  40] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [36] and [24], by (Copy) 

We can thus remove the following rules:

  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)) 

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

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

  min(X, 0) >? 0 
  min(s(X), s(Y)) >? min(X, Y) 

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

Argument functions:

  [[0]] = _|_ 

We choose Lex = {} and Mul = {min, s}, and the following precedence: min > s

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

  min(X, _|_) > _|_ 
  min(s(X), s(Y)) >= min(X, Y) 

With these choices, we have:

  1] min(X, _|_) > _|_  because [2], by definition 
  2] min*(X, _|_) >= _|_  by (Bot) 

  3] min(s(X), s(Y)) >= min(X, Y)  because [4], by (Star) 
  4] min*(s(X), s(Y)) >= min(X, Y)  because min in Mul, [5] and [8], by (Stat) 
  5] s(X) > X  because [6], by definition 
  6] s*(X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] s(Y) >= Y  because [9], by (Star) 
  9] s*(Y) >= Y  because [10], by (Select) 
  10] Y >= Y  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]):

  min(s(X), s(Y)) >? min(X, Y) 

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

We choose Lex = {} and Mul = {min, s}, and the following precedence: min > s

With these choices, we have:

  1] min(s(X), s(Y)) > min(X, Y)  because [2], by definition 
  2] min*(s(X), s(Y)) >= min(X, Y)  because min in Mul, [3] and [6], by (Stat) 
  3] s(X) >= X  because [4], by (Star) 
  4] s*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 
  6] s(Y) > Y  because [7], by definition 
  7] s*(Y) >= Y  because [8], by (Select) 
  8] Y >= Y  by (Meta) 

We can thus remove the following rules:

  min(s(X), s(Y)) => min(X, Y) 

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.