/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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 
  false : [] --> o 
  function : [o * o * o * o] --> o 
  if : [] --> o 
  iszero : [] --> o 
  p : [] --> o 
  plus : [] --> o 
  s : [o] --> o 
  third : [] --> o 
  true : [] --> o 

  function(iszero, 0, X, Y) => true 
  function(iszero, s(X), Y, Z) => false 
  function(p, 0, X, Y) => 0 
  function(p, s(0), X, Y) => 0 
  function(p, s(s(X)), Y, Z) => s(function(p, s(X), X, X)) 
  function(plus, X, Y, Z) => function(if, function(iszero, Y, Y, Y), Y, Z) 
  function(if, true, X, Y) => Y 
  function(if, false, X, Y) => function(plus, function(third, X, Y, Y), function(p, X, X, Y), s(Y)) 
  function(third, X, Y, Z) => 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 : [] --> ld 
  false : [] --> ld 
  function : [sc * ld * ld * ld] --> ld 
  if : [] --> sc 
  iszero : [] --> sc 
  p : [] --> sc 
  plus : [] --> sc 
  s : [ld] --> ld 
  third : [] --> sc 
  true : [] --> ld 


+++ 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.