/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.

  c : [o * o] --> o 
  f : [o] --> o 
  g : [o] --> o 
  s : [o] --> o 

  f(c(X, s(Y))) => f(c(s(X), Y)) 
  g(c(s(X), Y)) => f(c(X, s(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:

  c : [aa * aa] --> ba 
  f : [ba] --> ca 
  g : [ba] --> ca 
  s : [aa] --> aa 

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(c(X, s(Y))) >? f(c(s(X), Y)) 
  g(c(s(X), Y)) >? f(c(X, s(Y))) 

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

The following interpretation satisfies the requirements:

  c = \y0y1.2y0 + 3y1 
  f = \y0.2y0 
  g = \y0.3 + 3y0 
  s = \y0.3 + y0 

Using this interpretation, the requirements translate to:

  [[f(c(_x0, s(_x1)))]] = 18 + 4x0 + 6x1 > 12 + 4x0 + 6x1 = [[f(c(s(_x0), _x1))]] 
  [[g(c(s(_x0), _x1))]] = 21 + 6x0 + 9x1 > 18 + 4x0 + 6x1 = [[f(c(_x0, s(_x1)))]] 

We can thus remove the following rules:

  f(c(X, s(Y))) => f(c(s(X), Y)) 
  g(c(s(X), Y)) => f(c(X, s(Y))) 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ 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.