We consider the system Applicative_05__Ex7Sorting.

  Alphabet:

    0 : [] --> a 
    asort : [b] --> b 
    cons : [a * b] --> b 
    dsort : [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:

    sort(f, g, nil) => nil 
    sort(f, g, cons(x, y)) => insert(f, g, sort(f, g, y), x) 
    insert(f, g, nil, x) => cons(x, nil) 
    insert(f, g, cons(x, y), z) => cons(f x z, insert(f, g, y, g x z)) 
    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 
    asort(x) => sort(min, max, x) 
    dsort(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]):

  sort(F, G, nil) >? nil 
  sort(F, G, cons(X, Y)) >? insert(F, G, sort(F, G, Y), X) 
  insert(F, G, nil, X) >? cons(X, nil) 
  insert(F, G, cons(X, Y), Z) >? cons(F X Z, insert(F, G, Y, G X Z)) 
  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 
  asort(X) >? sort(min, max, X) 
  dsort(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) 
  [[max]] = _|_ 
  [[min]] = _|_ 
  [[nil]] = _|_ 

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

  sort(F, G, _|_) >= _|_ 
  sort(F, G, cons(X, Y)) > insert(F, G, sort(F, G, Y), X) 
  insert(F, G, _|_, X) > cons(X, _|_) 
  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))) 
  @_{o -> o}(@_{o -> o -> o}(_|_, _|_), X) >= X 
  @_{o -> o}(@_{o -> o -> o}(_|_, X), _|_) > X 
  @_{o -> o}(@_{o -> o -> o}(_|_, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(_|_, 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) 
  asort(X) >= sort(_|_, _|_, X) 
  dsort(X) >= sort(_|_, _|_, X) 

With these choices, we have:

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

  2] sort(F, G, cons(X, Y)) > insert(F, G, sort(F, G, Y), X)  because [3], by definition 
  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) 
  4] sort*(F, G, cons(X, Y)) >= F  because [5], by (Select) 
  5] F >= F  by (Meta) 
  6] sort*(F, G, cons(X, Y)) >= G  because [7], by (Select) 
  7] G >= G  by (Meta) 
  8] sort*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort in Mul, [9], [10] and [11], by (Stat) 
  9] F >= F  by (Meta) 
  10] G >= G  by (Meta) 
  11] cons(X, Y) > Y  because [12], by definition 
  12] cons*(X, Y) >= Y  because [13], by (Select) 
  13] Y >= Y  by (Meta) 
  14] sort*(F, G, cons(X, Y)) >= X  because [15], by (Select) 
  15] cons(X, Y) >= X  because [16], by (Star) 
  16] cons*(X, Y) >= X  because [17], by (Select) 
  17] X >= X  by (Meta) 

  18] insert(F, G, _|_, X) > cons(X, _|_)  because [19], by definition 
  19] insert*(F, G, _|_, X) >= cons(X, _|_)  because insert > cons, [20] and [22], by (Copy) 
  20] insert*(F, G, _|_, X) >= X  because [21], by (Select) 
  21] X >= X  by (Meta) 
  22] insert*(F, G, _|_, X) >= _|_  by (Bot) 

  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 
  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) 
  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) 
  26] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, X)  because insert > @_{o -> o -> o}, [27] and [29], by (Copy) 
  27] insert*(F, G, cons(X, Y), Z) >= F  because [28], by (Select) 
  28] F >= F  by (Meta) 
  29] insert*(F, G, cons(X, Y), Z) >= X  because [30], by (Select) 
  30] cons(X, Y) >= X  because [31], by (Star) 
  31] cons*(X, Y) >= X  because [32], by (Select) 
  32] X >= X  by (Meta) 
  33] insert*(F, G, cons(X, Y), Z) >= Z  because [34], by (Select) 
  34] Z >= Z  by (Meta) 
  35] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z))  because [36], [27], [39], [41], [43], [45] and [46], by (Stat) 
  36] cons(X, Y) > Y  because [37], by definition 
  37] cons*(X, Y) >= Y  because [38], by (Select) 
  38] Y >= Y  by (Meta) 
  39] insert*(F, G, cons(X, Y), Z) >= G  because [40], by (Select) 
  40] G >= G  by (Meta) 
  41] insert*(F, G, cons(X, Y), Z) >= Y  because [42], by (Select) 
  42] cons(X, Y) >= Y  because [37], by (Star) 
  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) 
  44] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, X)  because insert > @_{o -> o -> o}, [39] and [29], by (Copy) 
  45] F >= F  by (Meta) 
  46] G >= G  by (Meta) 

  47] @_{o -> o}(@_{o -> o -> o}(_|_, _|_), X) >= X  because [48], by (Star) 
  48] @_{o -> o}*(@_{o -> o -> o}(_|_, _|_), X) >= X  because [49], by (Select) 
  49] X >= X  by (Meta) 

  50] @_{o -> o}(@_{o -> o -> o}(_|_, X), _|_) > X  because [51], by definition 
  51] @_{o -> o}*(@_{o -> o -> o}(_|_, X), _|_) >= X  because [52], by (Select) 
  52] @_{o -> o -> o}(_|_, X) @_{o -> o}*(@_{o -> o -> o}(_|_, X), _|_) >= X  because [53] 
  53] @_{o -> o -> o}*(_|_, X, @_{o -> o}*(@_{o -> o -> o}(_|_, X), _|_)) >= X  because [54], by (Select) 
  54] X >= X  by (Meta) 

  55] @_{o -> o}(@_{o -> o -> o}(_|_, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(_|_, X), Y)  because @_{o -> o} in Mul, [56] and [61], by (Fun) 
  56] @_{o -> o -> o}(_|_, s(X)) >= @_{o -> o -> o}(_|_, X)  because @_{o -> o -> o} in Mul, [57] and [58], by (Fun) 
  57] _|_ >= _|_  by (Bot) 
  58] s(X) >= X  because [59], by (Star) 
  59] s*(X) >= X  because [60], by (Select) 
  60] X >= X  by (Meta) 
  61] s(Y) >= Y  because [62], by (Star) 
  62] s*(Y) >= Y  because [63], by (Select) 
  63] Y >= Y  by (Meta) 

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

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

  66] @_{o -> o}(@_{o -> o -> o}(_|_, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(_|_, X), Y)  because @_{o -> o} in Mul, [67] and [72], by (Fun) 
  67] @_{o -> o -> o}(_|_, s(X)) >= @_{o -> o -> o}(_|_, X)  because @_{o -> o -> o} in Mul, [68] and [69], by (Fun) 
  68] _|_ >= _|_  by (Bot) 
  69] s(X) >= X  because [70], by (Star) 
  70] s*(X) >= X  because [71], by (Select) 
  71] X >= X  by (Meta) 
  72] s(Y) >= Y  because [73], by (Star) 
  73] s*(Y) >= Y  because [74], by (Select) 
  74] Y >= Y  by (Meta) 

  75] asort(X) >= sort(_|_, _|_, X)  because [76], by (Star) 
  76] asort*(X) >= sort(_|_, _|_, X)  because asort > sort, [77], [78] and [79], by (Copy) 
  77] asort*(X) >= _|_  by (Bot) 
  78] asort*(X) >= _|_  by (Bot) 
  79] asort*(X) >= X  because [80], by (Select) 
  80] X >= X  by (Meta) 

  81] dsort(X) >= sort(_|_, _|_, X)  because [82], by (Star) 
  82] dsort*(X) >= sort(_|_, _|_, X)  because dsort > sort, [83], [84] and [85], by (Copy) 
  83] dsort*(X) >= _|_  by (Bot) 
  84] dsort*(X) >= _|_  by (Bot) 
  85] dsort*(X) >= X  because [86], by (Select) 
  86] X >= X  by (Meta) 

We can thus remove the following rules:

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

  sort(F, G, nil) >? nil 
  max 0 X >? 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 
  asort(X) >? sort(min, max, X) 
  dsort(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 
  asort = Lam[y0].3 + 3*y0 
  dsort = Lam[y0].3 + 3*y0 
  max = Lam[y0;y1].0 
  min = Lam[y0;y1].0 
  nil = 0 
  s = Lam[y0].3 + 3*y0 
  sort = Lam[G0;G1;y2].3*y2 + G0(0,0) + G1(0,0) 

Using this interpretation, the requirements translate to:

  [[sort(_F0, _F1, nil)]] = F0(0,0) + F1(0,0) >= 0 = [[nil]] 
  [[max 0 _x0]] = 3 + x0 > x0 = [[_x0]] 
  [[max s(_x0) s(_x1)]] = 6 + 3*x0 + 3*x1 > x0 + x1 = [[max _x0 _x1]] 
  [[min 0 _x0]] = 3 + x0 >= 3 = [[0]] 
  [[min _x0 0]] = 3 + x0 >= 3 = [[0]] 
  [[min s(_x0) s(_x1)]] = 6 + 3*x0 + 3*x1 > x0 + x1 = [[min _x0 _x1]] 
  [[asort(_x0)]] = 3 + 3*x0 > 3*x0 = [[sort(min, max, _x0)]] 
  [[dsort(_x0)]] = 3 + 3*x0 > 3*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 
  asort(X) => sort(min, max, X) 
  dsort(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]):

  sort(F, G, nil) >? nil 
  min(0, X) >? 0 
  min(X, 0) >? 0 

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

The following interpretation satisfies the requirements:

  0 = 2 
  min = Lam[y0;y1].3 + 3*y0 + 3*y1 
  nil = 2 
  sort = Lam[G0;G1;y2].3 + 3*y2 + 2*G0(0,0) + 2*G0(y2,y2) + 2*G1(y2,y2) 

Using this interpretation, the requirements translate to:

  [[sort(_F0, _F1, nil)]] = 9 + 2*F0(0,0) + 2*F0(2,2) + 2*F1(2,2) > 2 = [[nil]] 
  [[min(0, _x0)]] = 9 + 3*x0 > 2 = [[0]] 
  [[min(_x0, 0)]] = 9 + 3*x0 > 2 = [[0]] 

We can thus remove the following rules:

  sort(F, G, nil) => nil 
  min(0, X) => 0 
  min(X, 0) => 0 

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.