/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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.

  !dot : [o * o] --> o 
  0 : [] --> o 
  int : [o * o] --> o 
  int!6220list : [o] --> o 
  nil : [] --> o 
  s : [o] --> o 

  int(0, 0) => !dot(0, nil) 
  int(0, s(X)) => !dot(0, int(s(0), s(X))) 
  int(s(X), 0) => nil 
  int(s(X), s(Y)) => int!6220list(int(X, Y)) 
  int!6220list(nil) => nil 
  int!6220list(!dot(X, Y)) => !dot(s(X), int!6220list(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]):

  int(0, 0) >? !dot(0, nil) 
  int(0, s(X)) >? !dot(0, int(s(0), s(X))) 
  int(s(X), 0) >? nil 
  int(s(X), s(Y)) >? int!6220list(int(X, Y)) 
  int!6220list(nil) >? nil 
  int!6220list(!dot(X, Y)) >? !dot(s(X), int!6220list(Y)) 

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

The following interpretation satisfies the requirements:

  !dot = \y0y1.y0 + y1 
  0 = 0 
  int = \y0y1.2 + 2y0 + 2y1 
  int!6220list = \y0.y0 
  nil = 0 
  s = \y0.y0 

Using this interpretation, the requirements translate to:

  [[int(0, 0)]] = 2 > 0 = [[!dot(0, nil)]] 
  [[int(0, s(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[!dot(0, int(s(0), s(_x0)))]] 
  [[int(s(_x0), 0)]] = 2 + 2x0 > 0 = [[nil]] 
  [[int(s(_x0), s(_x1))]] = 2 + 2x0 + 2x1 >= 2 + 2x0 + 2x1 = [[int!6220list(int(_x0, _x1))]] 
  [[int!6220list(nil)]] = 0 >= 0 = [[nil]] 
  [[int!6220list(!dot(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!dot(s(_x0), int!6220list(_x1))]] 

We can thus remove the following rules:

  int(0, 0) => !dot(0, nil) 
  int(s(X), 0) => nil 

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

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

  int(0, s(X)) >? !dot(0, int(s(0), s(X))) 
  int(s(X), s(Y)) >? int!6220list(int(X, Y)) 
  int!6220list(nil) >? nil 
  int!6220list(!dot(X, Y)) >? !dot(s(X), int!6220list(Y)) 

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

The following interpretation satisfies the requirements:

  !dot = \y0y1.y1 + 2y0 
  0 = 0 
  int = \y0y1.y0 + y1 
  int!6220list = \y0.2y0 
  nil = 2 
  s = \y0.2y0 

Using this interpretation, the requirements translate to:

  [[int(0, s(_x0))]] = 2x0 >= 2x0 = [[!dot(0, int(s(0), s(_x0)))]] 
  [[int(s(_x0), s(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[int!6220list(int(_x0, _x1))]] 
  [[int!6220list(nil)]] = 4 > 2 = [[nil]] 
  [[int!6220list(!dot(_x0, _x1))]] = 2x1 + 4x0 >= 2x1 + 4x0 = [[!dot(s(_x0), int!6220list(_x1))]] 

We can thus remove the following rules:

  int!6220list(nil) => nil 

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] int#(0, s(X)) =#> int#(s(0), s(X))   
    1] int#(s(X), s(Y)) =#> int!6220list#(int(X, Y))   
    2] int#(s(X), s(Y)) =#> int#(X, Y)   
    3] int!6220list#(!dot(X, Y)) =#> int!6220list#(Y)   

  Rules R_0:

    int(0, s(X)) => !dot(0, int(s(0), s(X))) 
    int(s(X), s(Y)) => int!6220list(int(X, Y)) 
    int!6220list(!dot(X, Y)) => !dot(s(X), int!6220list(Y)) 

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 : 1, 2 
    * 1 : 3 
    * 2 : 0, 1, 2 
    * 3 : 3 

This graph has the following strongly connected components:

  P_1:

    int#(0, s(X)) =#> int#(s(0), s(X))   
    int#(s(X), s(Y)) =#> int#(X, Y)   

  P_2:

    int!6220list#(!dot(X, Y)) =#> int!6220list#(Y)   

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) and (P_2, R_0, m, f).

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(int!6220list#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(int!6220list#(!dot(X, Y))) = !dot(X, Y) |> Y = nu(int!6220list#(Y)) 

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(int#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(int#(0, s(X))) = s(X) = s(X) = nu(int#(s(0), s(X))) 
  nu(int#(s(X), s(Y))) = s(Y) |> Y = nu(int#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by (P_3, R_0, minimal, f), where P_3 contains:

  int#(0, s(X)) =#> int#(s(0), s(X))   

Thus, the original system is terminating if (P_3, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_3, 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 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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.