/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] --> o 
  inc : [o] --> o 
  isEmpty : [o] --> o 
  isZero : [o] --> o 
  nil : [] --> o 
  p : [o] --> o 
  s : [o] --> o 
  sum : [o] --> o 
  sumList : [o * o] --> o 
  tail : [o] --> o 
  true : [] --> o 

  isEmpty(cons(X, Y)) => false 
  isEmpty(nil) => true 
  isZero(0) => true 
  isZero(s(X)) => false 
  head(cons(X, Y)) => X 
  tail(cons(X, Y)) => Y 
  tail(nil) => nil 
  p(s(s(X))) => s(p(s(X))) 
  p(s(0)) => 0 
  p(0) => 0 
  inc(s(X)) => s(inc(X)) 
  inc(0) => s(0) 
  sumList(X, Y) => if(isEmpty(X), isZero(head(X)), Y, tail(X), cons(p(head(X)), tail(X)), inc(Y)) 
  if(true, X, Y, Z, U, V) => Y 
  if(false, true, X, Y, Z, U) => sumList(Y, X) 
  if(false, false, X, Y, Z, U) => sumList(Z, U) 
  sum(X) => sumList(X, 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:

  0 : [] --> cf 
  cons : [cf * wd] --> wd 
  false : [] --> sd 
  head : [wd] --> cf 
  if : [sd * sd * cf * wd * wd * cf] --> cf 
  inc : [cf] --> cf 
  isEmpty : [wd] --> sd 
  isZero : [cf] --> sd 
  nil : [] --> wd 
  p : [cf] --> cf 
  s : [cf] --> cf 
  sum : [wd] --> cf 
  sumList : [wd * cf] --> cf 
  tail : [wd] --> wd 
  true : [] --> sd 


+++ 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.