/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.

  !colon : [o * o] --> o 
  !plus : [o * o] --> o 
  a : [] --> o 
  f : [o] --> o 
  g : [o * o] --> o 

  !colon(!colon(X, Y), Z) => !colon(X, !colon(Y, Z)) 
  !colon(!plus(X, Y), Z) => !plus(!colon(X, Z), !colon(Y, Z)) 
  !colon(X, !plus(Y, f(Z))) => !colon(g(X, Z), !plus(Y, a)) 

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] !colon#(!colon(X, Y), Z) =#> !colon#(X, !colon(Y, Z))   
    1] !colon#(!colon(X, Y), Z) =#> !colon#(Y, Z)   
    2] !colon#(!plus(X, Y), Z) =#> !colon#(X, Z)   
    3] !colon#(!plus(X, Y), Z) =#> !colon#(Y, Z)   
    4] !colon#(X, !plus(Y, f(Z))) =#> !colon#(g(X, Z), !plus(Y, a))   

  Rules R_0:

    !colon(!colon(X, Y), Z) => !colon(X, !colon(Y, Z)) 
    !colon(!plus(X, Y), Z) => !plus(!colon(X, Z), !colon(Y, Z)) 
    !colon(X, !plus(Y, f(Z))) => !colon(g(X, Z), !plus(Y, a)) 

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 : 0, 1, 2, 3, 4 
    * 1 : 0, 1, 2, 3, 4 
    * 2 : 0, 1, 2, 3, 4 
    * 3 : 0, 1, 2, 3, 4 
    * 4 :  

This graph has the following strongly connected components:

  P_1:

    !colon#(!colon(X, Y), Z) =#> !colon#(X, !colon(Y, Z))   
    !colon#(!colon(X, Y), Z) =#> !colon#(Y, Z)   
    !colon#(!plus(X, Y), Z) =#> !colon#(X, Z)   
    !colon#(!plus(X, Y), Z) =#> !colon#(Y, Z)   

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).

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(!colon#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(!colon#(!colon(X, Y), Z)) = !colon(X, Y) |> X = nu(!colon#(X, !colon(Y, Z))) 
  nu(!colon#(!colon(X, Y), Z)) = !colon(X, Y) |> Y = nu(!colon#(Y, Z)) 
  nu(!colon#(!plus(X, Y), Z)) = !plus(X, Y) |> X = nu(!colon#(X, Z)) 
  nu(!colon#(!plus(X, Y), Z)) = !plus(X, Y) |> Y = nu(!colon#(Y, Z)) 

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.