/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 
  eq : [o * o] --> o 
  f : [o] --> o 
  false : [] --> o 
  gt : [o * o] --> o 
  if : [o * o] --> o 
  mod2 : [o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 
  while : [o * o] --> o 

  while(true, s(s(s(s(s(X)))))) => while(gt(s(s(s(s(s(X))))), 0), f(s(s(s(s(s(X))))))) 
  f(X) => if(eq(mod2(X), 0), X) 
  gt(s(X), s(Y)) => gt(X, Y) 
  gt(s(X), 0) => true 
  gt(0, X) => false 
  if(true, X) => plus(X, s(0)) 
  if(false, X) => plus(X, s(s(s(0)))) 
  mod2(s(s(X))) => mod2(X) 
  mod2(s(0)) => s(0) 
  mod2(0) => 0 
  eq(s(X), s(Y)) => eq(X, Y) 
  eq(0, 0) => true 
  eq(0, s(X)) => false 
  eq(s(X), 0) => false 
  plus(s(X), Y) => plus(X, s(Y)) 
  plus(0, X) => 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 : [] --> af 
  eq : [af * af] --> me 
  f : [af] --> af 
  false : [] --> me 
  gt : [af * af] --> me 
  if : [me * af] --> af 
  mod2 : [af] --> af 
  plus : [af * af] --> af 
  s : [af] --> af 
  true : [] --> me 
  while : [me * af] --> ua 


+++ 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.