/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 
  app : [o * o] --> o 
  cons : [o * o] --> o 
  false : [] --> o 
  ge : [o * o] --> o 
  helpa : [o * o * o * o] --> o 
  helpb : [o * o * o * o] --> o 
  if : [o * o * o * o * o] --> o 
  length : [o] --> o 
  nil : [] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  take : [o * o * o] --> o 
  true : [] --> o 
  xs : [] --> o 

  app(X, Y) => helpa(0, plus(length(X), length(Y)), X, Y) 
  plus(X, 0) => X 
  plus(X, s(Y)) => s(plus(X, Y)) 
  length(nil) => 0 
  length(cons(X, Y)) => s(length(Y)) 
  helpa(X, Y, Z, U) => if(ge(X, Y), X, Y, Z, U) 
  ge(X, 0) => true 
  ge(0, s(X)) => false 
  ge(s(X), s(Y)) => ge(X, Y) 
  if(true, X, Y, Z, U) => nil 
  if(false, X, Y, Z, U) => helpb(X, Y, Z, U) 
  take(0, cons(X, xs), Y) => X 
  take(0, nil, cons(X, Y)) => X 
  take(s(X), cons(Y, xs), Z) => take(X, xs, Z) 
  take(s(X), nil, cons(Y, Z)) => take(X, nil, Z) 
  helpb(X, Y, Z, U) => cons(take(X, Z, U), helpa(s(X), Y, Z, U)) 

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 : [] --> dg 
  app : [dg * dg] --> dg 
  cons : [dg * dg] --> dg 
  false : [] --> bd 
  ge : [dg * dg] --> bd 
  helpa : [dg * dg * dg * dg] --> dg 
  helpb : [dg * dg * dg * dg] --> dg 
  if : [bd * dg * dg * dg * dg] --> dg 
  length : [dg] --> dg 
  nil : [] --> dg 
  plus : [dg * dg] --> dg 
  s : [dg] --> dg 
  take : [dg * dg * dg] --> dg 
  true : [] --> bd 
  xs : [] --> dg 


+++ 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.