/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 
  ge : [o * o] --> o 
  gen : [o * o * o] --> o 
  generate : [o * o] --> o 
  if : [o * o * o * o] --> o 
  nil : [] --> o 
  s : [o] --> o 
  sum : [o] --> o 
  times : [o * o] --> o 
  true : [] --> o 

  times(X, Y) => sum(generate(X, Y)) 
  generate(X, Y) => gen(X, Y, 0) 
  gen(X, Y, Z) => if(ge(Z, X), X, Y, Z) 
  if(true, X, Y, Z) => nil 
  if(false, X, Y, Z) => cons(Y, gen(X, Y, s(Z))) 
  sum(nil) => 0 
  sum(cons(0, X)) => sum(X) 
  sum(cons(s(X), Y)) => s(sum(cons(X, Y))) 
  ge(X, 0) => true 
  ge(0, s(X)) => false 
  ge(s(X), s(Y)) => ge(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 : [] --> hd 
  cons : [hd * pc] --> pc 
  false : [] --> ld 
  ge : [hd * hd] --> ld 
  gen : [hd * hd * hd] --> pc 
  generate : [hd * hd] --> pc 
  if : [ld * hd * hd * hd] --> pc 
  nil : [] --> pc 
  s : [hd] --> hd 
  sum : [pc] --> hd 
  times : [hd * hd] --> hd 
  true : [] --> ld 


+++ 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.