/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.

  0 : [] --> o 
  div : [o * o] --> o 
  false : [] --> o 
  geq : [o * o] --> o 
  if : [o * o * o] --> o 
  minus : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  minus(0, X) => 0 
  minus(s(X), s(Y)) => minus(X, Y) 
  geq(X, 0) => true 
  geq(0, s(X)) => false 
  geq(s(X), s(Y)) => geq(X, Y) 
  div(0, s(X)) => 0 
  div(s(X), s(Y)) => if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) 
  if(true, X, Y) => X 
  if(false, X, Y) => 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:

  0 : [] --> ic 
  div : [ic * ic] --> ic 
  false : [] --> ob 
  geq : [ic * ic] --> ob 
  if : [ob * ic * ic] --> ic 
  minus : [ic * ic] --> ic 
  s : [ic] --> ic 
  true : [] --> ob 

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] minus#(s(X), s(Y)) =#> minus#(X, Y)   
    1] geq#(s(X), s(Y)) =#> geq#(X, Y)   
    2] div#(s(X), s(Y)) =#> if#(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)   
    3] div#(s(X), s(Y)) =#> geq#(X, Y)   
    4] div#(s(X), s(Y)) =#> div#(minus(X, Y), s(Y))   
    5] div#(s(X), s(Y)) =#> minus#(X, Y)   

  Rules R_0:

    minus(0, X) => 0 
    minus(s(X), s(Y)) => minus(X, Y) 
    geq(X, 0) => true 
    geq(0, s(X)) => false 
    geq(s(X), s(Y)) => geq(X, Y) 
    div(0, s(X)) => 0 
    div(s(X), s(Y)) => if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) 
    if(true, X, Y) => X 
    if(false, X, Y) => 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 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 : 1 
    * 2 :  
    * 3 : 1 
    * 4 :  
    * 5 : 0 

This graph has the following strongly connected components:

  P_1:

    minus#(s(X), s(Y)) =#> minus#(X, Y)   

  P_2:

    geq#(s(X), s(Y)) =#> geq#(X, 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) 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(geq#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(geq#(s(X), s(Y))) = s(X) |> X = nu(geq#(X, Y)) 

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(minus#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(minus#(s(X), s(Y))) = s(X) |> X = nu(minus#(X, Y)) 

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.