/export/starexec/sandbox2/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
We consider the system theBenchmark.

  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) 

about to try horpo

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}, asort, cons, dsort, max, s, sort}, and the following precedence: asort > dsort > max > 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}(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) 
  asort(X) >= sort(_|_, max, X) 
  dsort(X) >= sort(max, _|_, 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 (Star) 
  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] and [43], 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] @_{o -> o}(@_{o -> o -> o}(max, _|_), X) >= X  because [46], by (Star) 
  46] @_{o -> o}*(@_{o -> o -> o}(max, _|_), X) >= X  because [47], by (Select) 
  47] X >= X  by (Meta) 

  48] @_{o -> o}(@_{o -> o -> o}(max, X), _|_) >= X  because [49], by (Star) 
  49] @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [50], by (Select) 
  50] @_{o -> o -> o}(max, X) @_{o -> o}*(@_{o -> o -> o}(max, X), _|_) >= X  because [51] 
  51] @_{o -> o -> o}*(max, X, @_{o -> o}*(@_{o -> o -> o}(max, X), _|_)) >= X  because [52], by (Select) 
  52] X >= X  by (Meta) 

  53] @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) > @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because [54], by definition 
  54] @_{o -> o}*(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y)  because @_{o -> o} in Mul, [55] and [60], by (Stat) 
  55] @_{o -> o -> o}(max, s(X)) >= @_{o -> o -> o}(max, X)  because @_{o -> o -> o} in Mul, [56] and [57], by (Fun) 
  56] max >= max  by (Fun) 
  57] s(X) >= X  because [58], by (Star) 
  58] s*(X) >= X  because [59], by (Select) 
  59] X >= X  by (Meta) 
  60] s(Y) > Y  because [61], by definition 
  61] s*(Y) >= Y  because [62], by (Select) 
  62] Y >= Y  by (Meta) 

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

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

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

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

  80] dsort(X) >= sort(max, _|_, X)  because [81], by (Star) 
  81] dsort*(X) >= sort(max, _|_, X)  because dsort > sort, [82], [83] and [84], by (Copy) 
  82] dsort*(X) >= max  because dsort > max, by (Copy) 
  83] dsort*(X) >= _|_  by (Bot) 
  84] dsort*(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) 
  insert(F, G, cons(X, Y), Z) => cons(F X Z, insert(F, G, Y, G X Z)) 
  max s(X) s(Y) => max 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]):

  sort(F, G, nil) >? nil 
  sort(F, G, cons(X, Y)) >? insert(F, G, sort(F, G, Y), X) 
  max 0 X >? X 
  max X 0 >? X 
  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 = \y0.3 + 3y0 
  cons = \y0y1.3 + 3y0 + 3y1 
  dsort = \y0.3 + 3y0 
  insert = \G0G1y2y3.y2 + y3 + G0(y3,y3) + G1(0,0) 
  max = \y0y1.0 
  min = \y0y1.0 
  nil = 0 
  s = \y0.3 + 3y0 
  sort = \G0G1y2.2y2 + G0(0,0) + G1(0,0) + 2y2y2G1(y2,y2) + 2G0(y2,y2) + y2y2G0(y2,y2) 

Using this interpretation, the requirements translate to:

  [[sort(_F0, _F1, nil)]] = F1(0,0) + 3F0(0,0) >= 0 = [[nil]] 
  [[sort(_F0, _F1, cons(_x2, _x3))]] = 6 + 6x2 + 6x3 + F0(0,0) + F1(0,0) + 9x2x2F0(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 9x3x3F0(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 11F0(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 18x2x2F1(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 18x2x3F0(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 18x2F0(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 18x3x3F1(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 18x3F0(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 18F1(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 36x2x3F1(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 36x2F1(3 + 3x2 + 3x3,3 + 3x2 + 3x3) + 36x3F1(3 + 3x2 + 3x3,3 + 3x2 + 3x3) > x2 + 2x3 + F0(0,0) + F0(x2,x2) + 2x3x3F1(x3,x3) + 2F0(x3,x3) + 2F1(0,0) + x3x3F0(x3,x3) = [[insert(_F0, _F1, sort(_F0, _F1, _x3), _x2)]] 
  [[max 0 _x0]] = 3 + x0 > x0 = [[_x0]] 
  [[max _x0 0]] = 3 + x0 > x0 = [[_x0]] 
  [[min 0 _x0]] = 3 + x0 >= 3 = [[0]] 
  [[min _x0 0]] = 3 + x0 >= 3 = [[0]] 
  [[min s(_x0) s(_x1)]] = 6 + 3x0 + 3x1 > x0 + x1 = [[min _x0 _x1]] 
  [[asort(_x0)]] = 3 + 3x0 > 2x0 = [[sort(min, max, _x0)]] 
  [[dsort(_x0)]] = 3 + 3x0 > 2x0 = [[sort(max, min, _x0)]] 

We can thus remove the following rules:

  sort(F, G, cons(X, Y)) => insert(F, G, sort(F, G, Y), X) 
  max 0 X => X 
  max X 0 => X 
  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 = 0 
  min = \y0y1.3 + 3y0 + 3y1 
  nil = 0 
  sort = \G0G1y2.3 + 3y2 + G0(0,0) + G1(0,0) 

Using this interpretation, the requirements translate to:

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