/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  cons : [o * o] --> o 
  f : [o] --> o 
  g : [o] --> o 
  h : [o] --> o 
  s : [o] --> o 

  f(s(X)) => f(X) 
  g(cons(0, X)) => g(X) 
  g(cons(s(X), Y)) => s(X) 
  h(cons(X, Y)) => h(g(cons(X, 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 : [] --> oa 
  cons : [oa * oa] --> oa 
  f : [oa] --> l 
  g : [oa] --> oa 
  h : [oa] --> pa 
  s : [oa] --> oa 

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

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

  f(s(X)) >? f(X) 
  g(cons(0, X)) >? g(X) 
  g(cons(s(X), Y)) >? s(X) 
  h(cons(X, Y)) >? h(g(cons(X, Y))) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  cons = \y0y1.y0 + y1 
  f = \y0.y0 
  g = \y0.y0 
  h = \y0.y0 
  s = \y0.y0 

Using this interpretation, the requirements translate to:

  [[f(s(_x0))]] = x0 >= x0 = [[f(_x0)]] 
  [[g(cons(0, _x0))]] = 3 + x0 > x0 = [[g(_x0)]] 
  [[g(cons(s(_x0), _x1))]] = x0 + x1 >= x0 = [[s(_x0)]] 
  [[h(cons(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[h(g(cons(_x0, _x1)))]] 

We can thus remove the following rules:

  g(cons(0, X)) => g(X) 

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

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

  f(s(X)) >? f(X) 
  g(cons(s(X), Y)) >? s(X) 
  h(cons(X, Y)) >? h(g(cons(X, Y))) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.1 + y0 + y1 
  f = \y0.y0 
  g = \y0.y0 
  h = \y0.2y0 
  s = \y0.2y0 

Using this interpretation, the requirements translate to:

  [[f(s(_x0))]] = 2x0 >= x0 = [[f(_x0)]] 
  [[g(cons(s(_x0), _x1))]] = 1 + x1 + 2x0 > 2x0 = [[s(_x0)]] 
  [[h(cons(_x0, _x1))]] = 2 + 2x0 + 2x1 >= 2 + 2x0 + 2x1 = [[h(g(cons(_x0, _x1)))]] 

We can thus remove the following rules:

  g(cons(s(X), Y)) => s(X) 

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

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

  f(s(X)) >? f(X) 
  h(cons(X, Y)) >? h(g(cons(X, Y))) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.y0 + y1 
  f = \y0.y0 
  g = \y0.y0 
  h = \y0.y0 
  s = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[f(s(_x0))]] = 3 + 3x0 > x0 = [[f(_x0)]] 
  [[h(cons(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[h(g(cons(_x0, _x1)))]] 

We can thus remove the following rules:

  f(s(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 in [Kop12, Ch. 7.8]).

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

  Dependency Pairs P_0:

    0] h#(cons(X, Y)) =#> h#(g(cons(X, Y)))   

  Rules R_0:

    h(cons(X, Y)) => h(g(cons(X, 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 :  

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.