/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.

  !770!870 : [o * o] --> o 
  !dot : [o * o] --> o 
  !plus!plus : [o * o] --> o 
  greaters : [o * o] --> o 
  if : [o * o * o] --> o 
  lowers : [o * o] --> o 
  nil : [] --> o 
  qsort : [o] --> o 

  qsort(nil) => nil 
  qsort(!dot(X, Y)) => !plus!plus(qsort(lowers(X, Y)), !dot(X, qsort(greaters(X, Y)))) 
  lowers(X, nil) => nil 
  lowers(X, !dot(Y, Z)) => if(!770!870(Y, X), !dot(Y, lowers(X, Z)), lowers(X, Z)) 
  greaters(X, nil) => nil 
  greaters(X, !dot(Y, Z)) => if(!770!870(Y, X), greaters(X, Z), !dot(Y, greaters(X, Z))) 

As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95].  Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows:

  !770!870 : [dc * dc] --> ub 
  !dot : [dc * dc] --> dc 
  !plus!plus : [dc * dc] --> dc 
  greaters : [dc * dc] --> dc 
  if : [ub * dc * dc] --> dc 
  lowers : [dc * dc] --> dc 
  nil : [] --> dc 
  qsort : [dc] --> dc 

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] qsort#(!dot(X, Y)) =#> qsort#(lowers(X, Y))   
    1] qsort#(!dot(X, Y)) =#> lowers#(X, Y)   
    2] qsort#(!dot(X, Y)) =#> qsort#(greaters(X, Y))   
    3] qsort#(!dot(X, Y)) =#> greaters#(X, Y)   
    4] lowers#(X, !dot(Y, Z)) =#> lowers#(X, Z)   
    5] lowers#(X, !dot(Y, Z)) =#> lowers#(X, Z)   
    6] greaters#(X, !dot(Y, Z)) =#> greaters#(X, Z)   
    7] greaters#(X, !dot(Y, Z)) =#> greaters#(X, Z)   

  Rules R_0:

    qsort(nil) => nil 
    qsort(!dot(X, Y)) => !plus!plus(qsort(lowers(X, Y)), !dot(X, qsort(greaters(X, Y)))) 
    lowers(X, nil) => nil 
    lowers(X, !dot(Y, Z)) => if(!770!870(Y, X), !dot(Y, lowers(X, Z)), lowers(X, Z)) 
    greaters(X, nil) => nil 
    greaters(X, !dot(Y, Z)) => if(!770!870(Y, X), greaters(X, Z), !dot(Y, greaters(X, Z))) 

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 : 4, 5 
    * 2 :  
    * 3 : 6, 7 
    * 4 : 4, 5 
    * 5 : 4, 5 
    * 6 : 6, 7 
    * 7 : 6, 7 

This graph has the following strongly connected components:

  P_1:

    lowers#(X, !dot(Y, Z)) =#> lowers#(X, Z)   
    lowers#(X, !dot(Y, Z)) =#> lowers#(X, Z)   

  P_2:

    greaters#(X, !dot(Y, Z)) =#> greaters#(X, Z)   
    greaters#(X, !dot(Y, Z)) =#> greaters#(X, 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) 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(greaters#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(greaters#(X, !dot(Y, Z))) = !dot(Y, Z) |> Z = nu(greaters#(X, Z)) 
  nu(greaters#(X, !dot(Y, Z))) = !dot(Y, Z) |> Z = nu(greaters#(X, Z)) 

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(lowers#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(lowers#(X, !dot(Y, Z))) = !dot(Y, Z) |> Z = nu(lowers#(X, Z)) 
  nu(lowers#(X, !dot(Y, Z))) = !dot(Y, Z) |> Z = nu(lowers#(X, 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 +++

[FuhGieParSchSwi11]  C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski.  Proving Termination by Dependency Pairs and Inductive Theorem Proving.  In volume 47(2) of Journal of Automated Reasoning.  133--160, 2011.
[Gra95]  B. Gramlich.  Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems.  In volume 24(1-2) of Fundamentae Informaticae.  3--23, 1995.
[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.