/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.

  0 : [] --> o 
  and : [o * o] --> o 
  edge : [o * o * o] --> o 
  empty : [] --> o 
  eq : [o * o] --> o 
  false : [] --> o 
  if1 : [o * o * o * o * o * o] --> o 
  if2 : [o * o * o * o * o * o] --> o 
  le : [o * o] --> o 
  or : [o * o] --> o 
  reach : [o * o * o * o * o] --> o 
  reachable : [o * o * o] --> o 
  s : [o] --> o 
  size : [o] --> o 
  true : [] --> o 

  eq(0, 0) => true 
  eq(0, s(X)) => false 
  eq(s(X), 0) => false 
  eq(s(X), s(Y)) => eq(X, Y) 
  or(true, X) => true 
  or(false, X) => X 
  and(true, X) => X 
  and(false, X) => false 
  size(empty) => 0 
  size(edge(X, Y, Z)) => s(size(Z)) 
  le(0, X) => true 
  le(s(X), 0) => false 
  le(s(X), s(Y)) => le(X, Y) 
  reachable(X, Y, Z) => reach(X, Y, 0, Z, Z) 
  reach(X, Y, Z, U, V) => if1(eq(X, Y), X, Y, Z, U, V) 
  if1(true, X, Y, Z, U, V) => true 
  if1(false, X, Y, Z, U, V) => if2(le(Z, size(V)), X, Y, Z, U, V) 
  if2(false, X, Y, Z, U, V) => false 
  if2(true, X, Y, Z, empty, U) => false 
  if2(true, X, Y, Z, edge(U, V, W), Q) => or(if2(true, X, Y, Z, W, Q), and(eq(X, U), reach(V, Y, s(Z), Q, Q))) 

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 : [] --> nh 
  and : [nh * nh] --> nh 
  edge : [nh * nh * sg] --> sg 
  empty : [] --> sg 
  eq : [nh * nh] --> nh 
  false : [] --> nh 
  if1 : [nh * nh * nh * nh * sg * sg] --> nh 
  if2 : [nh * nh * nh * nh * sg * sg] --> nh 
  le : [nh * nh] --> nh 
  or : [nh * nh] --> nh 
  reach : [nh * nh * nh * sg * sg] --> nh 
  reachable : [nh * nh * sg] --> nh 
  s : [nh] --> nh 
  size : [sg] --> nh 
  true : [] --> nh 


+++ 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.