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


--------------------------------------------------------------------------------


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!6220int : [o * o * o * o] --> o 
  if!6220intlist : [o * o] --> o 
  if1 : [o * o * o] --> o 
  if2 : [o * o * o] --> o 
  int : [o * o] --> o 
  intlist : [o] --> o 
  nil : [] --> o 
  p : [o] --> o 
  s : [o] --> o 
  tail : [o] --> o 
  true : [] --> o 
  zero : [o] --> o 

  empty(nil) => true 
  empty(cons(X, Y)) => false 
  tail(nil) => nil 
  tail(cons(X, Y)) => Y 
  head(cons(X, Y)) => X 
  zero(0) => true 
  zero(s(X)) => false 
  p(0) => 0 
  p(s(0)) => 0 
  p(s(s(X))) => s(p(s(X))) 
  intlist(X) => if!6220intlist(empty(X), X) 
  if!6220intlist(true, X) => nil 
  if!6220intlist(false, X) => cons(s(head(X)), intlist(tail(X))) 
  int(X, Y) => if!6220int(zero(X), zero(Y), X, Y) 
  if!6220int(true, X, Y, Z) => if1(X, Y, Z) 
  if!6220int(false, X, Y, Z) => if2(X, Y, Z) 
  if1(true, X, Y) => cons(0, nil) 
  if1(false, X, Y) => cons(0, int(s(0), Y)) 
  if2(true, X, Y) => nil 
  if2(false, X, Y) => intlist(int(p(X), p(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:

  0 : [] --> zf 
  cons : [zf * zf] --> zf 
  empty : [zf] --> wd 
  false : [] --> wd 
  head : [zf] --> zf 
  if!6220int : [wd * wd * zf * zf] --> zf 
  if!6220intlist : [wd * zf] --> zf 
  if1 : [wd * zf * zf] --> zf 
  if2 : [wd * zf * zf] --> zf 
  int : [zf * zf] --> zf 
  intlist : [zf] --> zf 
  nil : [] --> zf 
  p : [zf] --> zf 
  s : [zf] --> zf 
  tail : [zf] --> zf 
  true : [] --> wd 
  zero : [zf] --> wd 


+++ 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.