/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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.

  0 : [] --> o 
  cons : [o * o] --> o 
  first : [o * o] --> o 
  from : [o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  sel : [o * o] --> o 

  from(X) => cons(X, from(s(X))) 
  first(0, X) => nil 
  first(s(X), cons(Y, Z)) => cons(Y, first(X, Z)) 
  sel(0, cons(X, Y)) => X 
  sel(s(X), cons(Y, Z)) => sel(X, Z) 

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 : [] --> db 
  cons : [db * za] --> za 
  first : [db * za] --> za 
  from : [db] --> za 
  nil : [] --> za 
  s : [db] --> db 
  sel : [db * za] --> db 


+++ 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.