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


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YES
We consider the system theBenchmark.

  Alphabet:

    cons : [b * c] --> c 
    false : [] --> a 
    filter : [b -> a * c] --> c 
    filtersub : [a * b -> a * c] --> c 
    nil : [] --> c 
    true : [] --> a 

  Rules:

    filter(f, nil) => nil 
    filter(f, cons(x, y)) => filtersub(f x, f, cons(x, y)) 
    filtersub(true, f, cons(x, y)) => cons(x, filter(f, y)) 
    filtersub(false, f, cons(x, y)) => filter(f, y) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

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] filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y))   
    1] filtersub#(true, F, cons(X, Y)) =#> filter#(F, Y)   
    2] filtersub#(false, F, cons(X, Y)) =#> filter#(F, Y)   

  Rules R_0:

    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filtersub(F X, F, cons(X, Y)) 
    filtersub(true, F, cons(X, Y)) => cons(X, filter(F, Y)) 
    filtersub(false, F, cons(X, Y)) => filter(F, 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 apply the subterm criterion with the following projection function:

  nu(filter#) = 2 
  nu(filtersub#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(filter#(F, cons(X, Y))) = cons(X, Y) = cons(X, Y) = nu(filtersub#(F X, F, cons(X, Y))) 
  nu(filtersub#(true, F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) 
  nu(filtersub#(false, F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_0, R_0, minimal, f) by (P_1, R_0, minimal, f), where P_1 contains:

  filter#(F, cons(X, Y)) =#> filtersub#(F X, F, cons(X, Y))   

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 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.