/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.

  0 : [] --> o 
  10 : [] --> o 
  cons : [o * o] --> o 
  entry : [o * o * o] --> o 
  false : [] --> o 
  gen : [o] --> o 
  if1 : [o * o] --> o 
  if2 : [o * o] --> o 
  if3 : [o * o * o] --> o 
  le : [o * o] --> o 
  nil : [] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  table : [] --> o 
  times : [o * o] --> o 
  true : [] --> o 

  table => gen(s(0)) 
  gen(X) => if1(le(X, 10), X) 
  if1(false, X) => nil 
  if1(true, X) => if2(X, X) 
  if2(X, Y) => if3(le(Y, 10), X, Y) 
  if3(true, X, Y) => cons(entry(X, Y, times(X, Y)), if2(X, s(Y))) 
  if3(false, X, Y) => gen(s(X)) 
  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  plus(0, X) => X 
  plus(s(X), Y) => s(plus(X, Y)) 
  times(0, X) => 0 
  times(s(X), Y) => plus(Y, times(X, Y)) 
  10 => s(s(s(s(s(s(s(s(s(s(0)))))))))) 

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 : [] --> we 
  10 : [] --> we 
  cons : [yb * lc] --> lc 
  entry : [we * we * we] --> yb 
  false : [] --> fd 
  gen : [we] --> lc 
  if1 : [fd * we] --> lc 
  if2 : [we * we] --> lc 
  if3 : [fd * we * we] --> lc 
  le : [we * we] --> fd 
  nil : [] --> lc 
  plus : [we * we] --> we 
  s : [we] --> we 
  table : [] --> lc 
  times : [we * we] --> we 
  true : [] --> fd 


+++ 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.