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


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MAYBE
We consider the system theBenchmark.

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

  !minus : [o * o] --> o 
  !plus : [o * o] --> o 
  !times : [o * o] --> o 
  0 : [] --> o 
  f : [o] --> o 
  s : [o] --> o 
  twice : [o] --> o 

  !plus(0, X) => X 
  !plus(s(X), Y) => s(!plus(X, Y)) 
  !times(X, 0) => 0 
  !times(X, s(Y)) => !plus(X, !times(X, Y)) 
  twice(0) => 0 
  twice(s(X)) => s(s(twice(X))) 
  !minus(X, 0) => X 
  !minus(s(X), s(Y)) => !minus(X, Y) 
  f(s(X)) => f(!minus(!times(s(s(X)), s(s(X))), !plus(!times(s(X), s(s(X))), s(s(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:

  !minus : [pc * pc] --> pc 
  !plus : [pc * pc] --> pc 
  !times : [pc * pc] --> pc 
  0 : [] --> pc 
  f : [pc] --> qc 
  s : [pc] --> pc 
  twice : [pc] --> pc 


+++ 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.