/export/starexec/sandbox2/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
We consider the system theBenchmark.

  Alphabet:

    pair : [nat -> nat * nat] --> nat 
    split : [nat] --> nat 

  Rules:

    split(f x) => pair(f, x) 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    pair : [nat -> nat * nat] --> nat 
    split : [nat] --> nat 
    ~AP1 : [nat -> nat * nat] --> nat 

  Rules:

    split(~AP1(F, X)) => pair(F, X) 
    ~AP1(F, X) => F X 

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

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

  split(~AP1(F, X)) >? pair(F, X) 
  ~AP1(F, X) >? F X 

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

The following interpretation satisfies the requirements:

  pair = \G0y1.y1 + G0(0) 
  split = \y0.3 + 3y0 
  ~AP1 = \G0y1.3 + 3y1 + 3G0(0) + 3G0(y1) 

Using this interpretation, the requirements translate to:

  [[split(~AP1(_F0, _x1))]] = 12 + 9x1 + 9F0(0) + 9F0(x1) > x1 + F0(0) = [[pair(_F0, _x1)]] 
  [[~AP1(_F0, _x1)]] = 3 + 3x1 + 3F0(0) + 3F0(x1) > x1 + F0(x1) = [[_F0 _x1]] 

We can thus remove the following rules:

  split(~AP1(F, X)) => pair(F, X) 
  ~AP1(F, X) => F X 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.