/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.

  append : [o * o] --> o 
  cons : [o * o] --> o 
  false : [] --> o 
  hd : [o] --> o 
  ifappend : [o * o * o] --> o 
  is!6220empty : [o] --> o 
  nil : [] --> o 
  tl : [o] --> o 
  true : [] --> o 

  is!6220empty(nil) => true 
  is!6220empty(cons(X, Y)) => false 
  hd(cons(X, Y)) => X 
  tl(cons(X, Y)) => Y 
  append(X, Y) => ifappend(X, Y, X) 
  ifappend(X, Y, nil) => Y 
  ifappend(X, Y, cons(Z, U)) => cons(Z, append(U, Y)) 

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:

  append : [ob * ob] --> ob 
  cons : [q * ob] --> ob 
  false : [] --> w 
  hd : [ob] --> q 
  ifappend : [ob * ob * ob] --> ob 
  is!6220empty : [ob] --> w 
  nil : [] --> ob 
  tl : [ob] --> ob 
  true : [] --> w 

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] append#(X, Y) =#> ifappend#(X, Y, X)   
    1] ifappend#(X, Y, cons(Z, U)) =#> append#(U, Y)   

  Rules R_0:

    is!6220empty(nil) => true 
    is!6220empty(cons(X, Y)) => false 
    hd(cons(X, Y)) => X 
    tl(cons(X, Y)) => Y 
    append(X, Y) => ifappend(X, Y, X) 
    ifappend(X, Y, nil) => Y 
    ifappend(X, Y, cons(Z, U)) => cons(Z, append(U, 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(append#) = 1 
  nu(ifappend#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(append#(X, Y)) = X = X = nu(ifappend#(X, Y, X)) 
  nu(ifappend#(X, Y, cons(Z, U))) = cons(Z, U) |> U = nu(append#(U, 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:

  append#(X, Y) =#> ifappend#(X, Y, X)   

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 +++

[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.