/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.

  !dot : [o * o] --> o 
  a : [] --> o 
  b : [] --> o 
  b!450 : [] --> o 
  c : [] --> o 
  d : [] --> o 
  d!450 : [] --> o 
  e : [] --> o 
  f : [o * o] --> o 
  g : [o * o] --> o 
  h : [o * o] --> o 
  i : [o * o * o] --> o 
  if : [o * o * o] --> o 

  f(g(i(a, b, b!450), c), d) => if(e, f(!dot(b, c), d!450), f(!dot(b!450, c), d!450)) 
  f(g(h(a, b), c), d) => if(e, f(!dot(b, g(h(a, b), c)), d), f(c, d!450)) 

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:

  !dot : [k * ya] --> ya 
  a : [] --> g 
  b : [] --> k 
  b!450 : [] --> k 
  c : [] --> ya 
  d : [] --> aa 
  d!450 : [] --> aa 
  e : [] --> u 
  f : [ya * aa] --> eb 
  g : [va * ya] --> ya 
  h : [g * k] --> va 
  i : [g * k * k] --> va 
  if : [u * eb * eb] --> eb 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  f(g(i(a, b, b!450), c), d) >? if(e, f(!dot(b, c), d!450), f(!dot(b!450, c), d!450)) 
  f(g(h(a, b), c), d) >? if(e, f(!dot(b, g(h(a, b), c)), d), f(c, d!450)) 

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

The following interpretation satisfies the requirements:

  !dot = \y0y1.y0 + y1 
  a = 3 
  b = 0 
  b!450 = 0 
  c = 0 
  d = 3 
  d!450 = 0 
  e = 0 
  f = \y0y1.y0 + 3y1 
  g = \y0y1.3 + 3y0 + 3y1 
  h = \y0y1.y0 + y1 
  i = \y0y1y2.3 + 3y0 + 3y1 + 3y2 
  if = \y0y1y2.y0 + y1 + y2 

Using this interpretation, the requirements translate to:

  [[f(g(i(a, b, b!450), c), d)]] = 48 > 0 = [[if(e, f(!dot(b, c), d!450), f(!dot(b!450, c), d!450))]] 
  [[f(g(h(a, b), c), d)]] = 21 >= 21 = [[if(e, f(!dot(b, g(h(a, b), c)), d), f(c, d!450))]] 

We can thus remove the following rules:

  f(g(i(a, b, b!450), c), d) => if(e, f(!dot(b, c), d!450), f(!dot(b!450, c), d!450)) 

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#(g(h(a, b), c), d) =#> f#(!dot(b, g(h(a, b), c)), d)   
    1] f#(g(h(a, b), c), d) =#> f#(c, d!450)   

  Rules R_0:

    f(g(h(a, b), c), d) => if(e, f(!dot(b, g(h(a, b), c)), d), f(c, d!450)) 

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 :  
    * 1 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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.