/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 
  cons : [o * o] --> o 
  false : [] --> o 
  head : [o] --> o 
  if : [o * o * o * o * o] --> o 
  length : [o] --> o 
  lt : [o * o] --> o 
  nil : [] --> o 
  rev : [o * o * o * o] --> o 
  reverse : [o] --> o 
  s : [o] --> o 
  tail : [o] --> o 
  true : [] --> o 
  undefined : [] --> o 

  length(nil) => 0 
  length(cons(X, Y)) => s(length(Y)) 
  lt(X, 0) => false 
  lt(0, s(X)) => true 
  lt(s(X), s(Y)) => lt(X, Y) 
  head(cons(X, Y)) => X 
  head(nil) => undefined 
  tail(nil) => nil 
  tail(cons(X, Y)) => Y 
  reverse(X) => rev(0, X, nil, X) 
  rev(X, Y, Z, U) => if(lt(X, length(U)), X, Y, Z, U) 
  if(true, X, Y, Z, U) => rev(s(X), tail(Y), cons(head(Y), Z), U) 
  if(false, X, Y, Z, U) => 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 : [] --> od 
  cons : [od * qd] --> qd 
  false : [] --> wc 
  head : [qd] --> od 
  if : [wc * od * qd * qd * qd] --> qd 
  length : [qd] --> od 
  lt : [od * od] --> wc 
  nil : [] --> qd 
  rev : [od * qd * qd * qd] --> qd 
  reverse : [qd] --> qd 
  s : [od] --> od 
  tail : [qd] --> qd 
  true : [] --> wc 
  undefined : [] --> od 


+++ 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.