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


--------------------------------------------------------------------------------


YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  f : [o * o * o] --> o 
  g : [o] --> o 

  f(g(X), Y, Y) => g(f(X, 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]):

  f(g(X), Y, Y) >? g(f(X, X, Y)) 

about to try horpo

We use a recursive path ordering as defined in [Kop12, Chapter 5].

We choose Lex = {} and Mul = {f, g}, and the following precedence: f > g

With these choices, we have:

  1] f(g(X), Y, Y) > g(f(X, X, Y))  because [2], by definition 
  2] f*(g(X), Y, Y) >= g(f(X, X, Y))  because f > g and [3], by (Copy) 
  3] f*(g(X), Y, Y) >= f(X, X, Y)  because f in Mul, [4], [4] and [7], by (Stat) 
  4] g(X) > X  because [5], by definition 
  5] g*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] Y >= Y  by (Meta) 

We can thus remove the following rules:

  f(g(X), Y, Y) => g(f(X, X, Y)) 

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.