/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 
  cadr : [o] --> o 
  car : [o] --> o 
  cddr : [o] --> o 
  cons : [o * o] --> o 
  false : [] --> o 
  if : [o * o * o] --> o 
  if2 : [o * o] --> o 
  ifplus : [o * o * o] --> o 
  iftimes : [o * o * o] --> o 
  isZero : [o] --> o 
  nil : [] --> o 
  p : [o] --> o 
  plus : [o * o] --> o 
  prod : [o] --> o 
  s : [o] --> o 
  shorter : [o * o] --> o 
  times : [o * o] --> o 
  true : [] --> o 

  car(cons(X, Y)) => X 
  cddr(nil) => nil 
  cddr(cons(X, nil)) => nil 
  cddr(cons(X, cons(Y, Z))) => Z 
  cadr(cons(X, cons(Y, Z))) => Y 
  isZero(0) => true 
  isZero(s(X)) => false 
  plus(X, Y) => ifplus(isZero(X), X, Y) 
  ifplus(true, X, Y) => Y 
  ifplus(false, X, Y) => s(plus(p(X), Y)) 
  times(X, Y) => iftimes(isZero(X), X, Y) 
  iftimes(true, X, Y) => 0 
  iftimes(false, X, Y) => plus(Y, times(p(X), Y)) 
  p(s(X)) => X 
  p(0) => 0 
  shorter(nil, X) => true 
  shorter(cons(X, Y), 0) => false 
  shorter(cons(X, Y), s(Z)) => shorter(Y, Z) 
  prod(X) => if(shorter(X, 0), shorter(X, s(0)), X) 
  if(true, X, Y) => s(0) 
  if(false, X, Y) => if2(X, Y) 
  if2(true, X) => car(X) 
  if2(false, X) => prod(cons(times(car(X), cadr(X)), cddr(X))) 

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 : [] --> fh 
  cadr : [eh] --> fh 
  car : [eh] --> fh 
  cddr : [eh] --> eh 
  cons : [fh * eh] --> eh 
  false : [] --> zf 
  if : [zf * zf * eh] --> fh 
  if2 : [zf * eh] --> fh 
  ifplus : [zf * fh * fh] --> fh 
  iftimes : [zf * fh * fh] --> fh 
  isZero : [fh] --> zf 
  nil : [] --> eh 
  p : [fh] --> fh 
  plus : [fh * fh] --> fh 
  prod : [eh] --> fh 
  s : [fh] --> fh 
  shorter : [eh * fh] --> zf 
  times : [fh * fh] --> fh 
  true : [] --> zf 


+++ 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.