/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.

  0 : [] --> o 
  cons : [o * o] --> o 
  empty : [o] --> o 
  false : [] --> o 
  head : [o] --> o 
  if : [o * o * o] --> o 
  if2 : [o * o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  sum : [o * o] --> o 
  tail : [o] --> o 
  true : [] --> o 
  weight : [o] --> o 
  weight!6220undefined!6220error : [] --> o 

  sum(cons(s(X), Y), cons(Z, U)) => sum(cons(X, Y), cons(s(Z), U)) 
  sum(cons(0, X), Y) => sum(X, Y) 
  sum(nil, X) => X 
  empty(nil) => true 
  empty(cons(X, Y)) => false 
  tail(nil) => nil 
  tail(cons(X, Y)) => Y 
  head(cons(X, Y)) => X 
  weight(X) => if(empty(X), empty(tail(X)), X) 
  if(true, X, Y) => weight!6220undefined!6220error 
  if(false, X, Y) => if2(X, Y) 
  if2(true, X) => head(X) 
  if2(false, X) => weight(sum(X, cons(0, tail(tail(X))))) 

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 : [] --> zd 
  cons : [zd * yd] --> yd 
  empty : [yd] --> uc 
  false : [] --> uc 
  head : [yd] --> zd 
  if : [uc * uc * yd] --> zd 
  if2 : [uc * yd] --> zd 
  nil : [] --> yd 
  s : [zd] --> zd 
  sum : [yd * yd] --> yd 
  tail : [yd] --> yd 
  true : [] --> uc 
  weight : [yd] --> zd 
  weight!6220undefined!6220error : [] --> zd 


+++ 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.