/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.

  !770!870 : [o * o] --> o 
  !870 : [o * o] --> o 
  !dot : [o * o] --> o 
  del : [o * o] --> o 
  if : [o * o * o] --> o 
  min : [o * o] --> o 
  msort : [o] --> o 
  nil : [] --> o 

  msort(nil) => nil 
  msort(!dot(X, Y)) => !dot(min(X, Y), msort(del(min(X, Y), !dot(X, Y)))) 
  min(X, nil) => X 
  min(X, !dot(Y, Z)) => if(!770!870(X, Y), min(X, Z), min(Y, Z)) 
  del(X, nil) => nil 
  del(X, !dot(Y, Z)) => if(!870(X, Y), Z, !dot(Y, del(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 : [ac * ac] --> tb 
  !870 : [ac * ac] --> tb 
  !dot : [ac * ac] --> ac 
  del : [ac * ac] --> ac 
  if : [tb * ac * ac] --> ac 
  min : [ac * ac] --> ac 
  msort : [ac] --> ac 
  nil : [] --> ac 

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] msort#(!dot(X, Y)) =#> min#(X, Y)   
    1] msort#(!dot(X, Y)) =#> msort#(del(min(X, Y), !dot(X, Y)))   
    2] msort#(!dot(X, Y)) =#> del#(min(X, Y), !dot(X, Y))   
    3] msort#(!dot(X, Y)) =#> min#(X, Y)   
    4] min#(X, !dot(Y, Z)) =#> min#(X, Z)   
    5] min#(X, !dot(Y, Z)) =#> min#(Y, Z)   
    6] del#(X, !dot(Y, Z)) =#> del#(X, Z)   

  Rules R_0:

    msort(nil) => nil 
    msort(!dot(X, Y)) => !dot(min(X, Y), msort(del(min(X, Y), !dot(X, Y)))) 
    min(X, nil) => X 
    min(X, !dot(Y, Z)) => if(!770!870(X, Y), min(X, Z), min(Y, Z)) 
    del(X, nil) => nil 
    del(X, !dot(Y, Z)) => if(!870(X, Y), Z, !dot(Y, del(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 : 4, 5 
    * 1 :  
    * 2 : 6 
    * 3 : 4, 5 
    * 4 : 4, 5 
    * 5 : 4, 5 
    * 6 : 6 

This graph has the following strongly connected components:

  P_1:

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

  P_2:

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

Thus, we can orient the dependency pairs as follows:

  nu(del#(X, !dot(Y, Z))) = !dot(Y, Z) |> Z = nu(del#(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(min#) = 2 

Thus, we can orient the dependency pairs as follows:

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

[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.