/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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.

  !870 : [o * o] --> o 
  !dot : [o * o] --> o 
  and : [o * o] --> o 
  del : [o] --> o 
  f : [o * o * o * o] --> o 
  false : [] --> o 
  nil : [] --> o 
  true : [] --> o 
  u : [] --> o 
  v : [] --> o 

  del(!dot(X, !dot(Y, Z))) => f(!870(X, Y), X, Y, Z) 
  f(true, X, Y, Z) => del(!dot(Y, Z)) 
  f(false, X, Y, Z) => !dot(X, del(!dot(Y, Z))) 
  !870(nil, nil) => true 
  !870(!dot(X, Y), nil) => false 
  !870(nil, !dot(X, Y)) => false 
  !870(!dot(X, Y), !dot(u, v)) => and(!870(X, u), !870(Y, v)) 

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:

  !870 : [bc * bc] --> kc 
  !dot : [bc * bc] --> bc 
  and : [kc * kc] --> kc 
  del : [bc] --> bc 
  f : [kc * bc * bc * bc] --> bc 
  false : [] --> kc 
  nil : [] --> bc 
  true : [] --> kc 
  u : [] --> bc 
  v : [] --> bc 

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] del#(!dot(X, !dot(Y, Z))) =#> f#(!870(X, Y), X, Y, Z)   
    1] del#(!dot(X, !dot(Y, Z))) =#> !870#(X, Y)   
    2] f#(true, X, Y, Z) =#> del#(!dot(Y, Z))   
    3] f#(false, X, Y, Z) =#> del#(!dot(Y, Z))   
    4] !870#(!dot(X, Y), !dot(u, v)) =#> !870#(X, u)   
    5] !870#(!dot(X, Y), !dot(u, v)) =#> !870#(Y, v)   

  Rules R_0:

    del(!dot(X, !dot(Y, Z))) => f(!870(X, Y), X, Y, Z) 
    f(true, X, Y, Z) => del(!dot(Y, Z)) 
    f(false, X, Y, Z) => !dot(X, del(!dot(Y, Z))) 
    !870(nil, nil) => true 
    !870(!dot(X, Y), nil) => false 
    !870(nil, !dot(X, Y)) => false 
    !870(!dot(X, Y), !dot(u, v)) => and(!870(X, u), !870(Y, v)) 

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 : 2, 3 
    * 1 : 4, 5 
    * 2 : 0, 1 
    * 3 : 0, 1 
    * 4 :  
    * 5 :  

This graph has the following strongly connected components:

  P_1:

    del#(!dot(X, !dot(Y, Z))) =#> f#(!870(X, Y), X, Y, Z)   
    f#(true, X, Y, Z) =#> del#(!dot(Y, Z))   
    f#(false, X, Y, Z) =#> del#(!dot(Y, Z))   

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).

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).

The formative rules of (P_1, R_0) are R_1 ::=

  del(!dot(X, !dot(Y, Z))) => f(!870(X, Y), X, Y, Z) 
  f(true, X, Y, Z) => del(!dot(Y, Z)) 
  f(false, X, Y, Z) => !dot(X, del(!dot(Y, Z))) 
  !870(nil, nil) => true 
  !870(!dot(X, Y), nil) => false 
  !870(nil, !dot(X, Y)) => false 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_1, R_1, minimal, formative).

Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_1, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_1, R_1) are:

  !870(nil, nil) => true 
  !870(!dot(X, Y), nil) => false 
  !870(nil, !dot(X, Y)) => false 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  del#(!dot(X, !dot(Y, Z))) >? f#(!870(X, Y), X, Y, Z) 
  f#(true, X, Y, Z) >? del#(!dot(Y, Z)) 
  f#(false, X, Y, Z) >? del#(!dot(Y, Z)) 
  !870(nil, nil) >= true 
  !870(!dot(X, Y), nil) >= false 
  !870(nil, !dot(X, Y)) >= false 

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

We consider usable_rules with respect to the following argument filtering:

  f#(x_1,x_2,x_3,x_4) = f#(x_2x_3,x_4) 

This leaves the following ordering requirements:

  del#(!dot(X, !dot(Y, Z))) >= f#(!870(X, Y), X, Y, Z) 
  f#(true, X, Y, Z) > del#(!dot(Y, Z)) 
  f#(false, X, Y, Z) >= del#(!dot(Y, Z)) 

The following interpretation satisfies the requirements:

  !870 = \y0y1.0 
  !dot = \y0y1.1 + 3y0 + 3y1 
  del# = \y0.y0 
  f# = \y0y1y2y3.2 + 3y2 + 3y3 
  false = 0 
  nil = 3 
  true = 0 

Using this interpretation, the requirements translate to:

  [[del#(!dot(_x0, !dot(_x1, _x2)))]] = 4 + 3x0 + 9x1 + 9x2 > 2 + 3x1 + 3x2 = [[f#(!870(_x0, _x1), _x0, _x1, _x2)]] 
  [[f#(true, _x0, _x1, _x2)]] = 2 + 3x1 + 3x2 > 1 + 3x1 + 3x2 = [[del#(!dot(_x1, _x2))]] 
  [[f#(false, _x0, _x1, _x2)]] = 2 + 3x1 + 3x2 > 1 + 3x1 + 3x2 = [[del#(!dot(_x1, _x2))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_1) by ({}, R_1).  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.