/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 
  evenodd : [o * o] --> o 
  false : [] --> o 
  not : [o] --> o 
  s : [o] --> o 
  true : [] --> o 

  not(true) => false 
  not(false) => true 
  evenodd(X, 0) => not(evenodd(X, s(0))) 
  evenodd(0, s(0)) => false 
  evenodd(s(X), s(0)) => evenodd(X, 0) 

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 : [] --> ma 
  evenodd : [ma * ma] --> qa 
  false : [] --> qa 
  not : [qa] --> qa 
  s : [ma] --> ma 
  true : [] --> qa 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  not(true) >? false 
  not(false) >? true 
  evenodd(X, 0) >? not(evenodd(X, s(0))) 
  evenodd(0, s(0)) >? false 
  evenodd(s(X), s(0)) >? evenodd(X, 0) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  evenodd = \y0y1.1 + y0 + 2y1 
  false = 0 
  not = \y0.y0 
  s = \y0.2y0 
  true = 0 

Using this interpretation, the requirements translate to:

  [[not(true)]] = 0 >= 0 = [[false]] 
  [[not(false)]] = 0 >= 0 = [[true]] 
  [[evenodd(_x0, 0)]] = 1 + x0 >= 1 + x0 = [[not(evenodd(_x0, s(0)))]] 
  [[evenodd(0, s(0))]] = 1 > 0 = [[false]] 
  [[evenodd(s(_x0), s(0))]] = 1 + 2x0 >= 1 + x0 = [[evenodd(_x0, 0)]] 

We can thus remove the following rules:

  evenodd(0, s(0)) => false 

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] evenodd#(X, 0) =#> not#(evenodd(X, s(0)))   
    1] evenodd#(X, 0) =#> evenodd#(X, s(0))   
    2] evenodd#(s(X), s(0)) =#> evenodd#(X, 0)   

  Rules R_0:

    not(true) => false 
    not(false) => true 
    evenodd(X, 0) => not(evenodd(X, s(0))) 
    evenodd(s(X), s(0)) => evenodd(X, 0) 

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 :  
    * 1 : 2 
    * 2 : 0, 1 

This graph has the following strongly connected components:

  P_1:

    evenodd#(X, 0) =#> evenodd#(X, s(0))   
    evenodd#(s(X), s(0)) =#> evenodd#(X, 0)   

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(evenodd#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(evenodd#(X, 0)) = X = X = nu(evenodd#(X, s(0))) 
  nu(evenodd#(s(X), s(0))) = s(X) |> X = nu(evenodd#(X, 0)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by (P_2, R_0, minimal, f), where P_2 contains:

  evenodd#(X, 0) =#> evenodd#(X, s(0))   

Thus, the original system is terminating if (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, 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.