/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.

  !770!870 : [o * o] --> o 
  !dot : [o * o] --> o 
  0 : [] --> o 
  bsort : [o] --> o 
  bubble : [o] --> o 
  butlast : [o] --> o 
  if : [o * o * o] --> o 
  last : [o] --> o 
  nil : [] --> o 

  bsort(nil) => nil 
  bsort(!dot(X, Y)) => last(!dot(bubble(!dot(X, Y)), bsort(butlast(bubble(!dot(X, Y)))))) 
  bubble(nil) => nil 
  bubble(!dot(X, nil)) => !dot(X, nil) 
  bubble(!dot(X, !dot(Y, Z))) => if(!770!870(X, Y), !dot(Y, bubble(!dot(X, Z))), !dot(X, bubble(!dot(Y, Z)))) 
  last(nil) => 0 
  last(!dot(X, nil)) => X 
  last(!dot(X, !dot(Y, Z))) => last(!dot(Y, Z)) 
  butlast(nil) => nil 
  butlast(!dot(X, nil)) => nil 
  butlast(!dot(X, !dot(Y, Z))) => !dot(X, butlast(!dot(Y, 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:

  !770!870 : [rd * rd] --> lb 
  !dot : [rd * rd] --> rd 
  0 : [] --> rd 
  bsort : [rd] --> rd 
  bubble : [rd] --> rd 
  butlast : [rd] --> rd 
  if : [lb * rd * rd] --> rd 
  last : [rd] --> rd 
  nil : [] --> rd 


+++ 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.