/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


MAYBE
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  1 : [] --> o 
  bin2s : [o] --> o 
  bin2ss : [o * o] --> o 
  bug!6220list!6220not : [] --> o 
  cons : [o * o] --> o 
  double : [o] --> o 
  eq : [o * o] --> o 
  false : [] --> o 
  half : [o] --> o 
  if1 : [o * o * o * o] --> o 
  if2 : [o * o * o * o] --> o 
  log : [o] --> o 
  lt : [o * o] --> o 
  more : [o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  s2bin : [o] --> o 
  s2bin1 : [o * o * o] --> o 
  s2bin2 : [o * o] --> o 
  true : [] --> o 

  eq(0, 0) => true 
  eq(0, s(X)) => false 
  eq(s(X), 0) => false 
  eq(s(X), s(Y)) => eq(X, Y) 
  lt(0, s(X)) => true 
  lt(X, 0) => false 
  lt(s(X), s(Y)) => lt(X, Y) 
  bin2s(nil) => 0 
  bin2s(cons(X, Y)) => bin2ss(X, Y) 
  bin2ss(X, nil) => X 
  bin2ss(X, cons(0, Y)) => bin2ss(double(X), Y) 
  bin2ss(X, cons(1, Y)) => bin2ss(s(double(X)), Y) 
  half(0) => 0 
  half(s(0)) => 0 
  half(s(s(X))) => s(half(X)) 
  log(0) => 0 
  log(s(0)) => 0 
  log(s(s(X))) => s(log(half(s(s(X))))) 
  more(nil) => nil 
  more(cons(X, Y)) => cons(cons(0, X), cons(cons(1, X), cons(X, Y))) 
  s2bin(X) => s2bin1(X, 0, cons(nil, nil)) 
  s2bin1(X, Y, Z) => if1(lt(Y, log(X)), X, Y, Z) 
  if1(true, X, Y, Z) => s2bin1(X, s(Y), more(Z)) 
  if1(false, X, Y, Z) => s2bin2(X, Z) 
  s2bin2(X, nil) => bug!6220list!6220not 
  s2bin2(X, cons(Y, Z)) => if2(eq(X, bin2s(Y)), X, Y, Z) 
  if2(true, X, Y, Z) => Y 
  if2(false, X, Y, Z) => s2bin2(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:

  0 : [] --> lj 
  1 : [] --> lj 
  bin2s : [lj] --> lj 
  bin2ss : [lj * lj] --> lj 
  bug!6220list!6220not : [] --> lj 
  cons : [lj * lj] --> lj 
  double : [lj] --> lj 
  eq : [lj * lj] --> ri 
  false : [] --> ri 
  half : [lj] --> lj 
  if1 : [ri * lj * lj * lj] --> lj 
  if2 : [ri * lj * lj * lj] --> lj 
  log : [lj] --> lj 
  lt : [lj * lj] --> ri 
  more : [lj] --> lj 
  nil : [] --> lj 
  s : [lj] --> lj 
  s2bin : [lj] --> lj 
  s2bin1 : [lj * lj * lj] --> lj 
  s2bin2 : [lj * lj] --> lj 
  true : [] --> ri 


+++ 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.