/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 
  div : [o * o] --> o 
  divByZeroError : [] --> o 
  false : [] --> o 
  ge : [o * o] --> o 
  if : [o * o * o] --> o 
  ify : [o * o * o] --> o 
  minus : [o * o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  ge(0, 0) => true 
  ge(s(X), 0) => ge(X, 0) 
  ge(0, s(0)) => false 
  ge(0, s(s(X))) => ge(0, s(X)) 
  ge(s(X), s(Y)) => ge(X, Y) 
  minus(0, 0) => 0 
  minus(0, s(X)) => minus(0, X) 
  minus(s(X), 0) => s(minus(X, 0)) 
  minus(s(X), s(Y)) => minus(X, Y) 
  plus(0, 0) => 0 
  plus(0, s(X)) => s(plus(0, X)) 
  plus(s(X), Y) => s(plus(X, Y)) 
  div(X, Y) => ify(ge(Y, s(0)), X, Y) 
  ify(false, X, Y) => divByZeroError 
  ify(true, X, Y) => if(ge(X, Y), X, Y) 
  if(false, X, Y) => 0 
  if(true, X, Y) => s(div(minus(X, Y), 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 : [] --> jf 
  div : [jf * jf] --> jf 
  divByZeroError : [] --> jf 
  false : [] --> pe 
  ge : [jf * jf] --> pe 
  if : [pe * jf * jf] --> jf 
  ify : [pe * jf * jf] --> jf 
  minus : [jf * jf] --> jf 
  plus : [jf * jf] --> jf 
  s : [jf] --> jf 
  true : [] --> pe 


+++ 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.