/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:

    0 : [] --> a 
    id : [] --> a -> a 
    plus : [a] --> a -> a 
    s : [a] --> a 

  Rules:

    id x => x 
    plus(0) => id 
    plus(s(x)) y => s(plus(x) y) 

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]):

  id X >? X 
  plus(0) >? id 
  plus(s(X)) Y >? s(plus(X) Y) 

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

The following interpretation satisfies the requirements:

  0 = 3 
  id = \y0.3y0 
  plus = \y0y1.3 + 3y0 + 3y1 
  s = \y0.3 + y0 

Using this interpretation, the requirements translate to:

  [[id _x0]] = 4x0 >= x0 = [[_x0]] 
  [[plus(0)]] = \y0.12 + 3y0 > \y0.3y0 = [[id]] 
  [[plus(s(_x0)) _x1]] = 12 + 3x0 + 4x1 > 6 + 3x0 + 4x1 = [[s(plus(_x0) _x1)]] 

We can thus remove the following rules:

  plus(0) => id 
  plus(s(X)) Y => s(plus(X) Y) 

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

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

  id(X) >? X 

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

The following interpretation satisfies the requirements:

  id = \y0.1 + y0 

Using this interpretation, the requirements translate to:

  [[id(_x0)]] = 1 + x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  id(X) => 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 +++

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