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


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YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  a : [] --> o 
  active : [o] --> o 
  c : [o] --> o 
  f : [o] --> o 
  g : [o] --> o 
  mark : [o] --> o 
  ok : [o] --> o 
  proper : [o] --> o 
  top : [o] --> o 

  active(f(f(a))) => mark(c(f(g(f(a))))) 
  active(f(X)) => f(active(X)) 
  active(g(X)) => g(active(X)) 
  f(mark(X)) => mark(f(X)) 
  g(mark(X)) => mark(g(X)) 
  proper(f(X)) => f(proper(X)) 
  proper(a) => ok(a) 
  proper(c(X)) => c(proper(X)) 
  proper(g(X)) => g(proper(X)) 
  f(ok(X)) => ok(f(X)) 
  c(ok(X)) => ok(c(X)) 
  g(ok(X)) => ok(g(X)) 
  top(mark(X)) => top(proper(X)) 
  top(ok(X)) => top(active(X)) 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]).

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] active#(f(f(a))) =#> c#(f(g(f(a))))   
    1] active#(f(f(a))) =#> f#(g(f(a)))   
    2] active#(f(f(a))) =#> g#(f(a))   
    3] active#(f(f(a))) =#> f#(a)   
    4] active#(f(X)) =#> f#(active(X))   
    5] active#(f(X)) =#> active#(X)   
    6] active#(g(X)) =#> g#(active(X))   
    7] active#(g(X)) =#> active#(X)   
    8] f#(mark(X)) =#> f#(X)   
    9] g#(mark(X)) =#> g#(X)   
    10] proper#(f(X)) =#> f#(proper(X))   
    11] proper#(f(X)) =#> proper#(X)   
    12] proper#(c(X)) =#> c#(proper(X))   
    13] proper#(c(X)) =#> proper#(X)   
    14] proper#(g(X)) =#> g#(proper(X))   
    15] proper#(g(X)) =#> proper#(X)   
    16] f#(ok(X)) =#> f#(X)   
    17] c#(ok(X)) =#> c#(X)   
    18] g#(ok(X)) =#> g#(X)   
    19] top#(mark(X)) =#> top#(proper(X))   
    20] top#(mark(X)) =#> proper#(X)   
    21] top#(ok(X)) =#> top#(active(X))   
    22] top#(ok(X)) =#> active#(X)   

  Rules R_0:

    active(f(f(a))) => mark(c(f(g(f(a))))) 
    active(f(X)) => f(active(X)) 
    active(g(X)) => g(active(X)) 
    f(mark(X)) => mark(f(X)) 
    g(mark(X)) => mark(g(X)) 
    proper(f(X)) => f(proper(X)) 
    proper(a) => ok(a) 
    proper(c(X)) => c(proper(X)) 
    proper(g(X)) => g(proper(X)) 
    f(ok(X)) => ok(f(X)) 
    c(ok(X)) => ok(c(X)) 
    g(ok(X)) => ok(g(X)) 
    top(mark(X)) => top(proper(X)) 
    top(ok(X)) => top(active(X)) 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 17 
    * 1 : 8, 16 
    * 2 : 9, 18 
    * 3 :  
    * 4 : 8, 16 
    * 5 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 6 : 9, 18 
    * 7 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 8 : 8, 16 
    * 9 : 9, 18 
    * 10 : 8, 16 
    * 11 : 10, 11, 12, 13, 14, 15 
    * 12 : 17 
    * 13 : 10, 11, 12, 13, 14, 15 
    * 14 : 9, 18 
    * 15 : 10, 11, 12, 13, 14, 15 
    * 16 : 8, 16 
    * 17 : 17 
    * 18 : 9, 18 
    * 19 : 19, 20, 21, 22 
    * 20 : 10, 11, 12, 13, 14, 15 
    * 21 : 19, 20, 21, 22 
    * 22 : 0, 1, 2, 3, 4, 5, 6, 7 

This graph has the following strongly connected components:

  P_1:

    active#(f(X)) =#> active#(X)   
    active#(g(X)) =#> active#(X)   

  P_2:

    f#(mark(X)) =#> f#(X)   
    f#(ok(X)) =#> f#(X)   

  P_3:

    g#(mark(X)) =#> g#(X)   
    g#(ok(X)) =#> g#(X)   

  P_4:

    proper#(f(X)) =#> proper#(X)   
    proper#(c(X)) =#> proper#(X)   
    proper#(g(X)) =#> proper#(X)   

  P_5:

    c#(ok(X)) =#> c#(X)   

  P_6:

    top#(mark(X)) =#> top#(proper(X))   
    top#(ok(X)) =#> top#(active(X))   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f), (P_2, R_0, m, f), (P_3, R_0, m, f), (P_4, R_0, m, f), (P_5, R_0, m, f) and (P_6, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative), (P_5, R_0, minimal, formative) and (P_6, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_6, R_0, minimal, formative).

The formative rules of (P_6, R_0) are R_1 ::=

  active(f(f(a))) => mark(c(f(g(f(a))))) 
  active(f(X)) => f(active(X)) 
  active(g(X)) => g(active(X)) 
  f(mark(X)) => mark(f(X)) 
  g(mark(X)) => mark(g(X)) 
  proper(f(X)) => f(proper(X)) 
  proper(a) => ok(a) 
  proper(c(X)) => c(proper(X)) 
  proper(g(X)) => g(proper(X)) 
  f(ok(X)) => ok(f(X)) 
  c(ok(X)) => ok(c(X)) 
  g(ok(X)) => ok(g(X)) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_6, R_0, minimal, formative) by (P_6, R_1, minimal, formative).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative), (P_5, R_0, minimal, formative) and (P_6, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_6, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  top#(mark(X)) >? top#(proper(X)) 
  top#(ok(X)) >? top#(active(X)) 
  active(f(f(a))) >= mark(c(f(g(f(a))))) 
  active(f(X)) >= f(active(X)) 
  active(g(X)) >= g(active(X)) 
  f(mark(X)) >= mark(f(X)) 
  g(mark(X)) >= mark(g(X)) 
  proper(f(X)) >= f(proper(X)) 
  proper(a) >= ok(a) 
  proper(c(X)) >= c(proper(X)) 
  proper(g(X)) >= g(proper(X)) 
  f(ok(X)) >= ok(f(X)) 
  c(ok(X)) >= ok(c(X)) 
  g(ok(X)) >= ok(g(X)) 

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

The following interpretation satisfies the requirements:

  a = 1 
  active = \y0.y0 
  c = \y0.0 
  f = \y0.2y0 
  g = \y0.y0 
  mark = \y0.1 + 2y0 
  ok = \y0.y0 
  proper = \y0.y0 
  top# = \y0.2y0 

Using this interpretation, the requirements translate to:

  [[top#(mark(_x0))]] = 2 + 4x0 > 2x0 = [[top#(proper(_x0))]] 
  [[top#(ok(_x0))]] = 2x0 >= 2x0 = [[top#(active(_x0))]] 
  [[active(f(f(a)))]] = 4 >= 1 = [[mark(c(f(g(f(a)))))]] 
  [[active(f(_x0))]] = 2x0 >= 2x0 = [[f(active(_x0))]] 
  [[active(g(_x0))]] = x0 >= x0 = [[g(active(_x0))]] 
  [[f(mark(_x0))]] = 2 + 4x0 >= 1 + 4x0 = [[mark(f(_x0))]] 
  [[g(mark(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[mark(g(_x0))]] 
  [[proper(f(_x0))]] = 2x0 >= 2x0 = [[f(proper(_x0))]] 
  [[proper(a)]] = 1 >= 1 = [[ok(a)]] 
  [[proper(c(_x0))]] = 0 >= 0 = [[c(proper(_x0))]] 
  [[proper(g(_x0))]] = x0 >= x0 = [[g(proper(_x0))]] 
  [[f(ok(_x0))]] = 2x0 >= 2x0 = [[ok(f(_x0))]] 
  [[c(ok(_x0))]] = 0 >= 0 = [[ok(c(_x0))]] 
  [[g(ok(_x0))]] = x0 >= x0 = [[ok(g(_x0))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_6, R_1, minimal, formative) by (P_7, R_1, minimal, formative), where P_7 consists of:

  top#(ok(X)) =#> top#(active(X))   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative), (P_5, R_0, minimal, formative) and (P_7, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_7, R_1, minimal, formative).

The formative rules of (P_7, R_1) are R_2 ::=

  active(f(X)) => f(active(X)) 
  active(g(X)) => g(active(X)) 
  proper(f(X)) => f(proper(X)) 
  proper(a) => ok(a) 
  proper(c(X)) => c(proper(X)) 
  proper(g(X)) => g(proper(X)) 
  f(ok(X)) => ok(f(X)) 
  c(ok(X)) => ok(c(X)) 
  g(ok(X)) => ok(g(X)) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_7, R_1, minimal, formative) by (P_7, R_2, minimal, formative).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative), (P_5, R_0, minimal, formative) and (P_7, R_2, minimal, formative) is finite.

We consider the dependency pair problem (P_7, R_2, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_7, R_2) are:

  active(f(X)) => f(active(X)) 
  active(g(X)) => g(active(X)) 
  f(ok(X)) => ok(f(X)) 
  g(ok(X)) => ok(g(X)) 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  top#(ok(X)) >? top#(active(X)) 
  active(f(X)) >= f(active(X)) 
  active(g(X)) >= g(active(X)) 
  f(ok(X)) >= ok(f(X)) 
  g(ok(X)) >= ok(g(X)) 

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

The following interpretation satisfies the requirements:

  active = \y0.0 
  f = \y0.y0 
  g = \y0.y0 
  ok = \y0.3 + 3y0 
  top# = \y0.3y0 

Using this interpretation, the requirements translate to:

  [[top#(ok(_x0))]] = 9 + 9x0 > 0 = [[top#(active(_x0))]] 
  [[active(f(_x0))]] = 0 >= 0 = [[f(active(_x0))]] 
  [[active(g(_x0))]] = 0 >= 0 = [[g(active(_x0))]] 
  [[f(ok(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[ok(f(_x0))]] 
  [[g(ok(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[ok(g(_x0))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_7, R_2) by ({}, R_2).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative), (P_4, R_0, minimal, formative) and (P_5, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_5, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(c#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(c#(ok(X))) = ok(X) |> X = nu(c#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_5, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative), (P_3, R_0, minimal, formative) and (P_4, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_4, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(proper#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(proper#(f(X))) = f(X) |> X = nu(proper#(X)) 
  nu(proper#(c(X))) = c(X) |> X = nu(proper#(X)) 
  nu(proper#(g(X))) = g(X) |> X = nu(proper#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_4, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_3, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(g#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(g#(mark(X))) = mark(X) |> X = nu(g#(X)) 
  nu(g#(ok(X))) = ok(X) |> X = nu(g#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_3, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(f#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(f#(mark(X))) = mark(X) |> X = nu(f#(X)) 
  nu(f#(ok(X))) = ok(X) |> X = nu(f#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(active#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(active#(f(X))) = f(X) |> X = nu(active#(X)) 
  nu(active#(g(X))) = g(X) |> X = nu(active#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.