/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.

  add : [o * o] --> o 
  app : [o * o] --> o 
  false : [] --> o 
  head : [o] --> o 
  if : [o * o * o * o] --> o 
  nil : [] --> o 
  null : [o] --> o 
  reverse : [o] --> o 
  shuff : [o * o] --> o 
  shuffle : [o] --> o 
  tail : [o] --> o 
  true : [] --> o 

  null(nil) => true 
  null(add(X, Y)) => false 
  tail(add(X, Y)) => Y 
  tail(nil) => nil 
  head(add(X, Y)) => X 
  app(nil, X) => X 
  app(add(X, Y), Z) => add(X, app(Y, Z)) 
  reverse(nil) => nil 
  reverse(add(X, Y)) => app(reverse(Y), add(X, nil)) 
  shuffle(X) => shuff(X, nil) 
  shuff(X, Y) => if(null(X), X, Y, app(Y, add(head(X), nil))) 
  if(true, X, Y, Z) => Y 
  if(false, X, Y, Z) => shuff(reverse(tail(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:

  add : [wc * td] --> td 
  app : [td * td] --> td 
  false : [] --> rc 
  head : [td] --> wc 
  if : [rc * td * td * td] --> td 
  nil : [] --> td 
  null : [td] --> rc 
  reverse : [td] --> td 
  shuff : [td * td] --> td 
  shuffle : [td] --> td 
  tail : [td] --> td 
  true : [] --> rc 


+++ 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.