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:

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

We choose Lex = {cons, insert} and Mul = {0, @_{o -> o -> o}, @_{o -> o}, ascending!fac6220sort, descending!fac6220sort, s, sort}, and the following precedence: ascending!fac6220sort > descending!fac6220sort > s > sort > insert > 0 > @_{o -> o -> o} > @_{o -> o} > cons

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}(_|_, 0), X) >= X 
  @_{o -> o}(@_{o -> o -> o}(_|_, X), 0) > X 
  @_{o -> o}(@_{o -> o -> o}(_|_, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(_|_, X), Y) 
  @_{o -> o}(@_{o -> o -> o}(_|_, 0), X) > 0 
  @_{o -> o}(@_{o -> o -> o}(_|_, X), 0) > 0 
  @_{o -> o}(@_{o -> o -> o}(_|_, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(_|_, 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(_|_, _|_, X) 
  descending!fac6220sort(X) >= sort(_|_, _|_, X) 

With these choices, we have:

  1] @_{o -> o}(@_{o -> o -> o}(_|_, 0), X) >= X  because [2], by (Star) 
  2] @_{o -> o}*(@_{o -> o -> o}(_|_, 0), X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] @_{o -> o}(@_{o -> o -> o}(_|_, X), 0) > X  because [5], by definition 
  5] @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0) >= X  because [6], by (Select) 
  6] @_{o -> o -> o}(_|_, X) @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0) >= X  because [7] 
  7] @_{o -> o -> o}*(_|_, X, @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0)) >= X  because [8], by (Select) 
  8] @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0) >= X  because [9], by (Select) 
  9] @_{o -> o -> o}(_|_, X) @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0) >= X  because [10] 
  10] @_{o -> o -> o}*(_|_, X, @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0)) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 

  12] @_{o -> o}(@_{o -> o -> o}(_|_, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(_|_, X), Y)  because @_{o -> o} in Mul, [13] and [18], by (Fun) 
  13] @_{o -> o -> o}(_|_, s(X)) >= @_{o -> o -> o}(_|_, X)  because @_{o -> o -> o} in Mul, [14] and [15], by (Fun) 
  14] _|_ >= _|_  by (Bot) 
  15] s(X) >= X  because [16], by (Star) 
  16] s*(X) >= X  because [17], by (Select) 
  17] X >= X  by (Meta) 
  18] s(Y) >= Y  because [19], by (Star) 
  19] s*(Y) >= Y  because [20], by (Select) 
  20] Y >= Y  by (Meta) 

  21] @_{o -> o}(@_{o -> o -> o}(_|_, 0), X) > 0  because [22], by definition 
  22] @_{o -> o}*(@_{o -> o -> o}(_|_, 0), X) >= 0  because [23], by (Select) 
  23] @_{o -> o -> o}(_|_, 0) @_{o -> o}*(@_{o -> o -> o}(_|_, 0), X) >= 0  because [24] 
  24] @_{o -> o -> o}*(_|_, 0, @_{o -> o}*(@_{o -> o -> o}(_|_, 0), X)) >= 0  because [25], by (Select) 
  25] 0 >= 0  by (Fun) 

  26] @_{o -> o}(@_{o -> o -> o}(_|_, X), 0) > 0  because [27], by definition 
  27] @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0) >= 0  because [28], by (Select) 
  28] @_{o -> o -> o}(_|_, X) @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0) >= 0  because [29] 
  29] @_{o -> o -> o}*(_|_, X, @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0)) >= 0  because [30], by (Select) 
  30] @_{o -> o}*(@_{o -> o -> o}(_|_, X), 0) >= 0  because [31], by (Select) 
  31] 0 >= 0  by (Fun) 

  32] @_{o -> o}(@_{o -> o -> o}(_|_, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(_|_, X), Y)  because @_{o -> o} in Mul, [33] and [38], by (Fun) 
  33] @_{o -> o -> o}(_|_, s(X)) >= @_{o -> o -> o}(_|_, X)  because @_{o -> o -> o} in Mul, [34] and [35], by (Fun) 
  34] _|_ >= _|_  by (Bot) 
  35] s(X) >= X  because [36], by (Star) 
  36] s*(X) >= X  because [37], by (Select) 
  37] X >= X  by (Meta) 
  38] s(Y) >= Y  because [39], by (Star) 
  39] s*(Y) >= Y  because [40], by (Select) 
  40] Y >= Y  by (Meta) 

  41] insert(F, G, _|_, X) > cons(X, _|_)  because [42], by definition 
  42] insert*(F, G, _|_, X) >= cons(X, _|_)  because insert > cons, [43] and [45], by (Copy) 
  43] insert*(F, G, _|_, X) >= X  because [44], by (Select) 
  44] X >= X  by (Meta) 
  45] insert*(F, G, _|_, X) >= _|_  by (Bot) 

  46] 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 [47], by definition 
  47] 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, [48] and [58], by (Copy) 
  48] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [49] and [54], by (Copy) 
  49] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [50] and [52], by (Copy) 
  50] insert*(F, G, cons(X, Y), Z) >= F  because [51], by (Select) 
  51] F >= F  by (Meta) 
  52] insert*(F, G, cons(X, Y), Z) >= Z  because [53], by (Select) 
  53] Z >= Z  by (Meta) 
  54] insert*(F, G, cons(X, Y), Z) >= X  because [55], by (Select) 
  55] cons(X, Y) >= X  because [56], by (Star) 
  56] cons*(X, Y) >= X  because [57], by (Select) 
  57] X >= X  by (Meta) 
  58] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [59], [50], [62], [64], [66], [68] and [69], by (Stat) 
  59] cons(X, Y) > Y  because [60], by definition 
  60] cons*(X, Y) >= Y  because [61], by (Select) 
  61] Y >= Y  by (Meta) 
  62] insert*(F, G, cons(X, Y), Z) >= G  because [63], by (Select) 
  63] G >= G  by (Meta) 
  64] insert*(F, G, cons(X, Y), Z) >= Y  because [65], by (Select) 
  65] cons(X, Y) >= Y  because [60], by (Star) 
  66] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [67] and [54], by (Copy) 
  67] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [62] and [52], by (Copy) 
  68] F >= F  by (Meta) 
  69] G >= G  by (Meta) 

  70] sort(F, G, _|_) > _|_  because [71], by definition 
  71] sort*(F, G, _|_) >= _|_  by (Bot) 

  72] sort(F, G, cons(X, Y)) > insert(F, G, sort(F, G, Y), X)  because [73], by definition 
  73] sort*(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because sort > insert, [74], [76], [78] and [84], by (Copy) 
  74] sort*(F, G, cons(X, Y)) >= F  because [75], by (Select) 
  75] F >= F  by (Meta) 
  76] sort*(F, G, cons(X, Y)) >= G  because [77], by (Select) 
  77] G >= G  by (Meta) 
  78] sort*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort in Mul, [79], [80] and [81], by (Stat) 
  79] F >= F  by (Meta) 
  80] G >= G  by (Meta) 
  81] cons(X, Y) > Y  because [82], by definition 
  82] cons*(X, Y) >= Y  because [83], by (Select) 
  83] Y >= Y  by (Meta) 
  84] sort*(F, G, cons(X, Y)) >= X  because [85], by (Select) 
  85] cons(X, Y) >= X  because [86], by (Star) 
  86] cons*(X, Y) >= X  because [87], by (Select) 
  87] X >= X  by (Meta) 

  88] ascending!fac6220sort(X) >= sort(_|_, _|_, X)  because [89], by (Star) 
  89] ascending!fac6220sort*(X) >= sort(_|_, _|_, X)  because ascending!fac6220sort > sort, [90], [91] and [92], by (Copy) 
  90] ascending!fac6220sort*(X) >= _|_  by (Bot) 
  91] ascending!fac6220sort*(X) >= _|_  by (Bot) 
  92] ascending!fac6220sort*(X) >= X  because [93], by (Select) 
  93] X >= X  by (Meta) 

  94] descending!fac6220sort(X) >= sort(_|_, _|_, X)  because [95], by (Star) 
  95] descending!fac6220sort*(X) >= sort(_|_, _|_, X)  because descending!fac6220sort > sort, [96], [97] and [98], by (Copy) 
  96] descending!fac6220sort*(X) >= _|_  by (Bot) 
  97] descending!fac6220sort*(X) >= _|_  by (Bot) 
  98] descending!fac6220sort*(X) >= X  because [99], by (Select) 
  99] X >= X  by (Meta) 

We can thus remove the following rules:

  max X 0 => X 
  min 0 X => 0 
  min X 0 => 0 
  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) 

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 s(X) s(Y) >? max X Y 
  min s(X) s(Y) >? min 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:

  0 = 3 
  ascending!fac6220sort = Lam[y0].3 + 3*y0 
  descending!fac6220sort = Lam[y0].3 + 3*y0 
  max = Lam[y0;y1].0 
  min = Lam[y0;y1].0 
  s = Lam[y0].3 + 3*y0 
  sort = Lam[G0;G1;y2].y2 + 3*G0(0,0) + 3*G1(0,0) 

Using this interpretation, the requirements translate to:

  [[max 0 _x0]] = 3 + x0 > x0 = [[_x0]] 
  [[max s(_x0) s(_x1)]] = 6 + 3*x0 + 3*x1 > x0 + x1 = [[max _x0 _x1]] 
  [[min s(_x0) s(_x1)]] = 6 + 3*x0 + 3*x1 > x0 + x1 = [[min _x0 _x1]] 
  [[ascending!fac6220sort(_x0)]] = 3 + 3*x0 > x0 = [[sort(min, max, _x0)]] 
  [[descending!fac6220sort(_x0)]] = 3 + 3*x0 > x0 = [[sort(max, min, _x0)]] 

We can thus remove the following rules:

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