/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.

  0 : [] --> o 
  f : [o * o * o * o] --> o 
  false : [] --> o 
  if : [o * o * o] --> o 
  le : [o * o] --> o 
  minus : [o * o] --> o 
  perfectp : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  perfectp(0) => false 
  perfectp(s(X)) => f(X, s(0), s(X), s(X)) 
  f(0, X, 0, Y) => true 
  f(0, X, s(Y), Z) => false 
  f(s(X), 0, Y, Z) => f(X, Z, minus(Y, s(X)), Z) 
  f(s(X), s(Y), Z, U) => if(le(X, Y), f(s(X), minus(Y, X), Z, U), f(X, U, Z, U)) 

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 : [] --> bc 
  f : [bc * bc * bc * bc] --> kc 
  false : [] --> kc 
  if : [wb * kc * kc] --> kc 
  le : [bc * bc] --> wb 
  minus : [bc * bc] --> bc 
  perfectp : [bc] --> kc 
  s : [bc] --> bc 
  true : [] --> kc 

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] perfectp#(s(X)) =#> f#(X, s(0), s(X), s(X))   
    1] f#(s(X), 0, Y, Z) =#> f#(X, Z, minus(Y, s(X)), Z)   
    2] f#(s(X), s(Y), Z, U) =#> f#(s(X), minus(Y, X), Z, U)   
    3] f#(s(X), s(Y), Z, U) =#> f#(X, U, Z, U)   

  Rules R_0:

    perfectp(0) => false 
    perfectp(s(X)) => f(X, s(0), s(X), s(X)) 
    f(0, X, 0, Y) => true 
    f(0, X, s(Y), Z) => false 
    f(s(X), 0, Y, Z) => f(X, Z, minus(Y, s(X)), Z) 
    f(s(X), s(Y), Z, U) => if(le(X, Y), f(s(X), minus(Y, X), Z, U), f(X, U, Z, U)) 

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 : 2, 3 
    * 1 : 1, 2, 3 
    * 2 :  
    * 3 : 1, 2, 3 

This graph has the following strongly connected components:

  P_1:

    f#(s(X), 0, Y, Z) =#> f#(X, Z, minus(Y, s(X)), Z)   
    f#(s(X), s(Y), Z, U) =#> f#(X, U, Z, U)   

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).

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(f#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(f#(s(X), 0, Y, Z)) = s(X) |> X = nu(f#(X, Z, minus(Y, s(X)), Z)) 
  nu(f#(s(X), s(Y), Z, U)) = s(X) |> X = nu(f#(X, U, Z, U)) 

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.