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


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We consider the system theBenchmark.

  Alphabet:

    minus : [N * N] --> N 
    s : [N] --> N 
    z : [] --> N 

  Rules:

    minus(z, x) => z 
    minus(x, z) => x 
    minus(s(x), s(y)) => minus(x, y) 
    minus(x, x) => z 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

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

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

  minus(z, X) >? z 
  minus(X, z) >? X 
  minus(s(X), s(Y)) >? minus(X, Y) 
  minus(X, X) >? z 

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

The following interpretation satisfies the requirements:

  minus = \y0y1.3 + y1 + 3y0 
  s = \y0.3 + 3y0 
  z = 0 

Using this interpretation, the requirements translate to:

  [[minus(z, _x0)]] = 3 + x0 > 0 = [[z]] 
  [[minus(_x0, z)]] = 3 + 3x0 > x0 = [[_x0]] 
  [[minus(s(_x0), s(_x1))]] = 15 + 3x1 + 9x0 > 3 + x1 + 3x0 = [[minus(_x0, _x1)]] 
  [[minus(_x0, _x0)]] = 3 + 4x0 > 0 = [[z]] 

We can thus remove the following rules:

  minus(z, X) => z 
  minus(X, z) => X 
  minus(s(X), s(Y)) => minus(X, Y) 
  minus(X, X) => z 

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 +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.