/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  1 : [] --> o 
  c : [o] --> o 
  f : [o] --> o 
  false : [] --> o 
  g : [o * o] --> o 
  if : [o * o * o] --> o 
  s : [o] --> o 
  true : [] --> o 

  f(0) => true 
  f(1) => false 
  f(s(X)) => f(X) 
  if(true, X, Y) => X 
  if(false, X, Y) => Y 
  g(s(X), s(Y)) => if(f(X), s(X), s(Y)) 
  g(X, c(Y)) => g(X, g(s(c(Y)), 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 : [] --> pb 
  1 : [] --> pb 
  c : [pb] --> pb 
  f : [pb] --> xa 
  false : [] --> xa 
  g : [pb * pb] --> pb 
  if : [xa * pb * pb] --> pb 
  s : [pb] --> pb 
  true : [] --> xa 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]).

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] f#(s(X)) =#> f#(X)   
    1] g#(s(X), s(Y)) =#> if#(f(X), s(X), s(Y))   
    2] g#(s(X), s(Y)) =#> f#(X)   
    3] g#(X, c(Y)) =#> g#(X, g(s(c(Y)), Y))   
    4] g#(X, c(Y)) =#> g#(s(c(Y)), Y)   

  Rules R_0:

    f(0) => true 
    f(1) => false 
    f(s(X)) => f(X) 
    if(true, X, Y) => X 
    if(false, X, Y) => Y 
    g(s(X), s(Y)) => if(f(X), s(X), s(Y)) 
    g(X, c(Y)) => g(X, g(s(c(Y)), Y)) 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0 
    * 1 :  
    * 2 : 0 
    * 3 : 1, 2, 3, 4 
    * 4 : 1, 2, 3, 4 

This graph has the following strongly connected components:

  P_1:

    f#(s(X)) =#> f#(X)   

  P_2:

    g#(X, c(Y)) =#> g#(X, g(s(c(Y)), Y))   
    g#(X, c(Y)) =#> g#(s(c(Y)), Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  g#(X, c(Y)) >? g#(X, g(s(c(Y)), Y)) 
  g#(X, c(Y)) >? g#(s(c(Y)), Y) 
  f(0) >= true 
  f(1) >= false 
  f(s(X)) >= f(X) 
  if(true, X, Y) >= X 
  if(false, X, Y) >= Y 
  g(s(X), s(Y)) >= if(f(X), s(X), s(Y)) 
  g(X, c(Y)) >= g(X, g(s(c(Y)), Y)) 

We orient these requirements with a polynomial interpretation in the natural numbers.

We consider usable_rules with respect to the following argument filtering:

  if(x_1,x_2,x_3) = if(x_2x_3) 

This leaves the following ordering requirements:

  g#(X, c(Y)) > g#(X, g(s(c(Y)), Y)) 
  g#(X, c(Y)) >= g#(s(c(Y)), Y) 
  if(true, X, Y) >= X 
  if(false, X, Y) >= Y 
  g(s(X), s(Y)) >= if(f(X), s(X), s(Y)) 
  g(X, c(Y)) >= g(X, g(s(c(Y)), Y)) 

The following interpretation satisfies the requirements:

  0 = 3 
  1 = 3 
  c = \y0.2 + 2y0 
  f = \y0.0 
  false = 0 
  g = \y0y1.1 
  g# = \y0y1.y1 
  if = \y0y1y2.2y1 + 2y2 
  s = \y0.0 
  true = 0 

Using this interpretation, the requirements translate to:

  [[g#(_x0, c(_x1))]] = 2 + 2x1 > 1 = [[g#(_x0, g(s(c(_x1)), _x1))]] 
  [[g#(_x0, c(_x1))]] = 2 + 2x1 > x1 = [[g#(s(c(_x1)), _x1)]] 
  [[if(true, _x0, _x1)]] = 2x0 + 2x1 >= x0 = [[_x0]] 
  [[if(false, _x0, _x1)]] = 2x0 + 2x1 >= x1 = [[_x1]] 
  [[g(s(_x0), s(_x1))]] = 1 >= 0 = [[if(f(_x0), s(_x0), s(_x1))]] 
  [[g(_x0, c(_x1))]] = 1 >= 1 = [[g(_x0, g(s(c(_x1)), _x1))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_2, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(f#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(f#(s(X))) = s(X) |> X = nu(f#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ 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.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.