/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.

  a : [] --> o 
  b : [] --> o 
  c : [] --> o 
  f : [o * o] --> o 
  s : [o] --> o 

  f(a, a) => f(a, b) 
  f(a, b) => f(s(a), c) 
  f(s(X), c) => f(X, c) 
  f(c, c) => f(a, a) 

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:

  a : [] --> aa 
  b : [] --> aa 
  c : [] --> aa 
  f : [aa * aa] --> ma 
  s : [aa] --> aa 

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] f#(a, a) =#> f#(a, b)   
    1] f#(a, b) =#> f#(s(a), c)   
    2] f#(s(X), c) =#> f#(X, c)   
    3] f#(c, c) =#> f#(a, a)   

  Rules R_0:

    f(a, a) => f(a, b) 
    f(a, b) => f(s(a), c) 
    f(s(X), c) => f(X, c) 
    f(c, c) => f(a, a) 

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).

This combination (P_0, R_0) has no formative rules!  We will name the empty set of rules:R_1.

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_0, R_1, minimal, formative).

Thus, the original system is terminating if (P_0, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  f#(a, a) >? f#(a, b) 
  f#(a, b) >? f#(s(a), c) 
  f#(s(X), c) >? f#(X, c) 
  f#(c, c) >? f#(a, a) 

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

The following interpretation satisfies the requirements:

  a = 0 
  b = 0 
  c = 1 
  f# = \y0y1.y0 
  s = \y0.2y0 

Using this interpretation, the requirements translate to:

  [[f#(a, a)]] = 0 >= 0 = [[f#(a, b)]] 
  [[f#(a, b)]] = 0 >= 0 = [[f#(s(a), c)]] 
  [[f#(s(_x0), c)]] = 2x0 >= x0 = [[f#(_x0, c)]] 
  [[f#(c, c)]] = 1 > 0 = [[f#(a, a)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_1, minimal, formative) by (P_1, R_1, minimal, formative), where P_1 consists of:

  f#(a, a) =#> f#(a, b)   
  f#(a, b) =#> f#(s(a), c)   
  f#(s(X), c) =#> f#(X, c)   

Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_1, 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 
    * 1 : 2 
    * 2 : 2 

This graph has the following strongly connected components:

  P_2:

    f#(s(X), c) =#> f#(X, c)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_1, R_1, m, f) by (P_2, R_1, m, f).

Thus, the original system is terminating if (P_2, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_1, 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), c)) = s(X) |> X = nu(f#(X, c)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_1, minimal, f) by ({}, R_1, 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.