/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.

  cons : [o * o] --> o 
  nil : [] --> o 
  r : [o * o * o * o] --> o 
  succ : [o] --> o 
  zero : [] --> o 

  r(X, Y, Z, nil) => X 
  r(X, nil, Y, cons(Z, U)) => r(X, X, cons(succ(zero), Y), U) 
  r(X, cons(Y, Z), nil, cons(U, V)) => r(X, X, cons(succ(zero), nil), V) 
  r(X, cons(Y, Z), cons(U, V), cons(W, Q)) => r(Z, cons(Y, Z), V, cons(succ(zero), cons(W, Q))) 

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:

  cons : [wb * bc] --> bc 
  nil : [] --> bc 
  r : [bc * bc * bc * bc] --> bc 
  succ : [ea] --> wb 
  zero : [] --> ea 


+++ 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.