/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 
  2ndsneg : [o * o] --> o 
  2ndspos : [o * o] --> o 
  cons : [o * o] --> o 
  cons2 : [o * o] --> o 
  from : [o] --> o 
  negrecip : [o] --> o 
  pi : [o] --> o 
  plus : [o * o] --> o 
  posrecip : [o] --> o 
  rcons : [o * o] --> o 
  rnil : [] --> o 
  s : [o] --> o 
  square : [o] --> o 
  times : [o * o] --> o 

  from(X) => cons(X, from(s(X))) 
  2ndspos(0, X) => rnil 
  2ndspos(s(X), cons(Y, Z)) => 2ndspos(s(X), cons2(Y, Z)) 
  2ndspos(s(X), cons2(Y, cons(Z, U))) => rcons(posrecip(Z), 2ndsneg(X, U)) 
  2ndsneg(0, X) => rnil 
  2ndsneg(s(X), cons(Y, Z)) => 2ndsneg(s(X), cons2(Y, Z)) 
  2ndsneg(s(X), cons2(Y, cons(Z, U))) => rcons(negrecip(Z), 2ndspos(X, U)) 
  pi(X) => 2ndspos(X, from(0)) 
  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)) 
  square(X) => times(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 : [] --> qe 
  2ndsneg : [qe * dd] --> ed 
  2ndspos : [qe * dd] --> ed 
  cons : [qe * dd] --> dd 
  cons2 : [qe * dd] --> dd 
  from : [qe] --> dd 
  negrecip : [qe] --> sc 
  pi : [qe] --> ed 
  plus : [qe * qe] --> qe 
  posrecip : [qe] --> sc 
  rcons : [sc * ed] --> ed 
  rnil : [] --> ed 
  s : [qe] --> qe 
  square : [qe] --> qe 
  times : [qe * qe] --> qe 


+++ 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.