/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:

    !faccolon : [] --> b -> b -> b 
    !facplus : [] --> b -> b -> b 
    a : [] --> b 
    cons : [] --> d -> e -> e 
    false : [] --> c 
    filter : [] --> d -> c -> e -> e 
    filter2 : [] --> c -> d -> c -> d -> e -> e 
    g : [] --> b -> a -> b 
    map : [] --> d -> d -> e -> e 
    nil : [] --> e 
    true : [] --> c 

  Rules:

    !faccolon (!faccolon x y) z => !faccolon x (!faccolon y z) 
    !faccolon (!facplus x y) z => !facplus (!faccolon x z) (!faccolon y z) 
    !faccolon x (!facplus y (f z)) => !faccolon (g x z) (!facplus y a) 
    map f nil => nil 
    map f (cons x y) => cons (f x) (map f y) 
    filter f nil => nil 
    filter f (cons x y) => filter2 (f x) f x y 
    filter2 true f x y => cons x (filter f y) 
    filter2 false f x y => filter f y 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    !faccolon : [b * b] --> b 
    !facplus : [b * b] --> b 
    a : [] --> b 
    cons : [d * e] --> e 
    false : [] --> c 
    filter : [d -> c * e] --> e 
    filter2 : [c * d -> c * d * e] --> e 
    g : [b * a] --> b 
    map : [d -> d * e] --> e 
    nil : [] --> e 
    true : [] --> c 
    ~AP1 : [a -> b * a] --> b 

  Rules:

    !faccolon(!faccolon(X, Y), Z) => !faccolon(X, !faccolon(Y, Z)) 
    !faccolon(!facplus(X, Y), Z) => !facplus(!faccolon(X, Z), !faccolon(Y, Z)) 
    !faccolon(X, !facplus(Y, ~AP1(F, Z))) => !faccolon(g(X, Z), !facplus(Y, a)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(F, Y) 
    !faccolon(X, !facplus(Y, g(Z, U))) => !faccolon(g(X, U), !facplus(Y, a)) 
    ~AP1(F, X) => F X 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]).

We thus obtain the following dependency pair problem (P_0, R_0, static, formative):

  Dependency Pairs P_0:

    0] !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(X, !faccolon(Y, Z))   
    1] !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(Y, Z)   
    2] !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(X, Z)   
    3] !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(Y, Z)   
    4] !faccolon#(X, !facplus(Y, ~AP1(F, Z))) =#> !faccolon#(g(X, Z), !facplus(Y, a))   
    5] map#(F, cons(X, Y)) =#> map#(F, Y)   
    6] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    7] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    8] filter2#(false, F, X, Y) =#> filter#(F, Y)   
    9] !faccolon#(X, !facplus(Y, g(Z, U))) =#> !faccolon#(g(X, U), !facplus(Y, a))   

  Rules R_0:

    !faccolon(!faccolon(X, Y), Z) => !faccolon(X, !faccolon(Y, Z)) 
    !faccolon(!facplus(X, Y), Z) => !facplus(!faccolon(X, Z), !faccolon(Y, Z)) 
    !faccolon(X, !facplus(Y, ~AP1(F, Z))) => !faccolon(g(X, Z), !facplus(Y, a)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(F, Y) 
    !faccolon(X, !facplus(Y, g(Z, U))) => !faccolon(g(X, U), !facplus(Y, a)) 
    ~AP1(F, X) => F X 

Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.

We consider the dependency pair problem (P_0, R_0, static, 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, 9 
    * 1 : 0, 1, 2, 3, 4, 9 
    * 2 : 0, 1, 2, 3, 4, 9 
    * 3 : 0, 1, 2, 3, 4, 9 
    * 4 :  
    * 5 : 5 
    * 6 : 7, 8 
    * 7 : 6 
    * 8 : 6 
    * 9 :  

This graph has the following strongly connected components:

  P_1:

    !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(X, !faccolon(Y, Z))   
    !faccolon#(!faccolon(X, Y), Z) =#> !faccolon#(Y, Z)   
    !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(X, Z)   
    !faccolon#(!facplus(X, Y), Z) =#> !faccolon#(Y, Z)   

  P_2:

    map#(F, cons(X, Y)) =#> map#(F, Y)   

  P_3:

    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    filter2#(true, F, X, Y) =#> filter#(F, Y)   
    filter2#(false, F, X, Y) =#> filter#(F, 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), (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, static, formative), (P_2, R_0, static, formative) and (P_3, R_0, static, formative) is finite.

We consider the dependency pair problem (P_3, R_0, static, formative).

We apply the subterm criterion with the following projection function:

  nu(filter2#) = 4 
  nu(filter#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 
  nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
  nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_3, R_0, static, f) by (P_4, R_0, static, f), where P_4 contains:

  filter2#(true, F, X, Y) =#> filter#(F, Y)   
  filter2#(false, F, X, Y) =#> filter#(F, Y)   

Thus, the original system is terminating if each of (P_1, R_0, static, formative), (P_2, R_0, static, formative) and (P_4, R_0, static, formative) is finite.

We consider the dependency pair problem (P_4, R_0, static, 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 :  

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

Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite.

We consider the dependency pair problem (P_2, R_0, static, formative).

We apply the subterm criterion with the following projection function:

  nu(map#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_2, R_0, static, f) by ({}, R_0, static, 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, static, formative) is finite.

We consider the dependency pair problem (P_1, R_0, static, formative).

We apply the subterm criterion with the following projection function:

  nu(!faccolon#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(!faccolon#(!faccolon(X, Y), Z)) = !faccolon(X, Y) |> X = nu(!faccolon#(X, !faccolon(Y, Z))) 
  nu(!faccolon#(!faccolon(X, Y), Z)) = !faccolon(X, Y) |> Y = nu(!faccolon#(Y, Z)) 
  nu(!faccolon#(!facplus(X, Y), Z)) = !facplus(X, Y) |> X = nu(!faccolon#(X, Z)) 
  nu(!faccolon#(!facplus(X, Y), Z)) = !facplus(X, Y) |> Y = nu(!faccolon#(Y, Z)) 

By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_1, R_0, static, f) by ({}, R_0, static, 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 +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
[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.