/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 
  empty : [o] --> o 
  f : [o * o] --> o 
  false : [] --> o 
  fstsplit : [o * o] --> o 
  if1 : [o * o * o] --> o 
  if2 : [o * o * o] --> o 
  if3 : [o * o * o] --> o 
  length : [o] --> o 
  leq : [o * o] --> o 
  map!6220f : [o * o] --> o 
  nil : [] --> o 
  process : [o * o] --> o 
  s : [o] --> o 
  self : [] --> o 
  sndsplit : [o * o] --> o 
  true : [] --> o 

  fstsplit(0, X) => nil 
  fstsplit(s(X), nil) => nil 
  fstsplit(s(X), cons(Y, Z)) => cons(Y, fstsplit(X, Z)) 
  sndsplit(0, X) => X 
  sndsplit(s(X), nil) => nil 
  sndsplit(s(X), cons(Y, Z)) => sndsplit(X, Z) 
  empty(nil) => true 
  empty(cons(X, Y)) => false 
  leq(0, X) => true 
  leq(s(X), 0) => false 
  leq(s(X), s(Y)) => leq(X, Y) 
  length(nil) => 0 
  length(cons(X, Y)) => s(length(Y)) 
  app(nil, X) => X 
  app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
  map!6220f(X, nil) => nil 
  map!6220f(X, cons(Y, Z)) => app(f(X, Y), map!6220f(X, Z)) 
  process(X, Y) => if1(X, Y, leq(Y, length(X))) 
  if1(X, Y, true) => if2(X, Y, empty(fstsplit(Y, X))) 
  if1(X, Y, false) => if3(X, Y, empty(fstsplit(Y, app(map!6220f(self, nil), X)))) 
  if2(X, Y, false) => process(app(map!6220f(self, nil), sndsplit(Y, X)), Y) 
  if3(X, Y, false) => process(sndsplit(Y, app(map!6220f(self, nil), X)), Y) 

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 : [] --> qf 
  app : [xh * xh] --> xh 
  cons : [ra * xh] --> xh 
  empty : [xh] --> wg 
  f : [qg * ra] --> xh 
  false : [] --> wg 
  fstsplit : [qf * xh] --> xh 
  if1 : [xh * qf * wg] --> zh 
  if2 : [xh * qf * wg] --> zh 
  if3 : [xh * qf * wg] --> zh 
  length : [xh] --> qf 
  leq : [qf * qf] --> wg 
  map!6220f : [qg * xh] --> xh 
  nil : [] --> xh 
  process : [xh * qf] --> zh 
  s : [qf] --> qf 
  self : [] --> qg 
  sndsplit : [qf * xh] --> xh 
  true : [] --> wg 


+++ 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.