/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.

  if : [o * o * o] --> o 

  if(if(X, Y, Z), U, V) => if(X, if(Y, U, V), if(Z, U, V)) 

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

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

  if(if(X, Y, Z), U, V) >? if(X, if(Y, U, V), if(Z, U, V)) 

about to try horpo

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

Argument functions:

  [[if(x_1, x_2, x_3)]] = if(x_1, x_3, x_2) 

We choose Lex = {if} and Mul = {}, and the following precedence: if

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  if(if(X, Y, Z), U, V) > if(X) (if(Y) U V) if(Z) 

With these choices, we have:

  1] if(if(X, Y, Z), U, V) > if(X) (if(Y) U V) if(Z)  because [2], by definition 
  2] if*(if(X, Y, Z), U, V) >= if(X) (if(Y) U V) if(Z)  because [3], [6], [8] and [18], by (Stat) 
  3] if(X, Y, Z) > X  because [4], by definition 
  4] if*(X, Y, Z) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 
  6] if*(if(X, Y, Z), U, V) >= X  because [7], by (Select) 
  7] if(X, Y, Z) >= X  because [4], by (Star) 
  8] if*(if(X, Y, Z), U, V) >= if(Y, U, V)  because [9], [12], [14] and [16], by (Stat) 
  9] if(X, Y, Z) > Y  because [10], by definition 
  10] if*(X, Y, Z) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 
  12] if*(if(X, Y, Z), U, V) >= Y  because [13], by (Select) 
  13] if(X, Y, Z) >= Y  because [10], by (Star) 
  14] if*(if(X, Y, Z), U, V) >= U  because [15], by (Select) 
  15] U >= U  by (Meta) 
  16] if*(if(X, Y, Z), U, V) >= V  because [17], by (Select) 
  17] V >= V  by (Meta) 
  18] if*(if(X, Y, Z), U, V) >= if(Z, U, V)  because [19], [22], [14] and [16], by (Stat) 
  19] if(X, Y, Z) > Z  because [20], by definition 
  20] if*(X, Y, Z) >= Z  because [21], by (Select) 
  21] Z >= Z  by (Meta) 
  22] if*(if(X, Y, Z), U, V) >= Z  because [23], by (Select) 
  23] if(X, Y, Z) >= Z  because [20], by (Star) 

We can thus remove the following rules:

  if(if(X, Y, Z), U, V) => if(X, if(Y, U, V), if(Z, U, V)) 

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.