/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.

  append : [o * o] --> o 
  cons : [o * o] --> o 
  dropLast : [o] --> o 
  false : [] --> o 
  if : [o * o * o * o] --> o 
  isEmpty : [o] --> o 
  last : [o] --> o 
  nil : [] --> o 
  rev : [o * o] --> o 
  reverse : [o] --> o 
  true : [] --> o 

  isEmpty(nil) => true 
  isEmpty(cons(X, Y)) => false 
  last(cons(X, nil)) => X 
  last(cons(X, cons(Y, Z))) => last(cons(Y, Z)) 
  dropLast(nil) => nil 
  dropLast(cons(X, nil)) => nil 
  dropLast(cons(X, cons(Y, Z))) => cons(X, dropLast(cons(Y, Z))) 
  append(nil, X) => X 
  append(cons(X, Y), Z) => cons(X, append(Y, Z)) 
  reverse(X) => rev(X, nil) 
  rev(X, Y) => if(isEmpty(X), dropLast(X), append(Y, last(X)), Y) 
  if(true, X, Y, Z) => Z 
  if(false, X, Y, Z) => rev(X, Y) 

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:

  append : [zd * zd] --> zd 
  cons : [zd * zd] --> zd 
  dropLast : [zd] --> zd 
  false : [] --> ad 
  if : [ad * zd * zd * zd] --> zd 
  isEmpty : [zd] --> ad 
  last : [zd] --> zd 
  nil : [] --> zd 
  rev : [zd * zd] --> zd 
  reverse : [zd] --> zd 
  true : [] --> ad 


+++ 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.