/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.

  !minus : [o * o] --> o 
  !plus : [o * o] --> o 
  !times : [o * o] --> o 
  0 : [] --> o 
  fact : [o] --> o 
  false : [] --> o 
  ge : [o * o] --> o 
  iffact : [o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  !plus(X, 0) => X 
  !plus(X, s(Y)) => s(!plus(X, Y)) 
  !times(X, 0) => 0 
  !times(X, s(Y)) => !plus(!times(X, Y), X) 
  ge(X, 0) => true 
  ge(0, s(X)) => false 
  ge(s(X), s(Y)) => ge(X, Y) 
  !minus(X, 0) => X 
  !minus(s(X), s(Y)) => !minus(X, Y) 
  fact(X) => iffact(X, ge(X, s(s(0)))) 
  iffact(X, true) => !times(X, fact(!minus(X, s(0)))) 
  iffact(X, false) => 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:

  !minus : [id * id] --> id 
  !plus : [id * id] --> id 
  !times : [id * id] --> id 
  0 : [] --> id 
  fact : [id] --> id 
  false : [] --> qc 
  ge : [id * id] --> qc 
  iffact : [id * qc] --> id 
  s : [id] --> id 
  true : [] --> qc 


+++ 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.