/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 
  and : [o * o] --> o 
  cons : [o * o] --> o 
  from : [o] --> o 
  fromCond : [o * o] --> o 
  is!6220nat : [o] --> o 
  is!6220natlist : [o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  tt : [] --> o 

  and(tt, tt) => tt 
  is!6220nat(0) => tt 
  is!6220nat(s(X)) => is!6220nat(X) 
  is!6220natlist(nil) => tt 
  is!6220natlist(cons(X, Y)) => and(is!6220nat(X), is!6220natlist(Y)) 
  from(X) => fromCond(is!6220natlist(X), X) 
  fromCond(tt, cons(X, Y)) => from(cons(s(X), cons(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 : [] --> nb 
  and : [za * za] --> za 
  cons : [nb * nb] --> nb 
  from : [nb] --> ob 
  fromCond : [za * nb] --> ob 
  is!6220nat : [nb] --> za 
  is!6220natlist : [nb] --> za 
  nil : [] --> nb 
  s : [nb] --> nb 
  tt : [] --> za 


+++ 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.