/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.

  a : [o] --> o 
  b : [o] --> o 
  p : [o * o] --> o 

  p(a(X), p(a(b(Y)), Z)) => p(a(b(a(Z))), p(a(a(Y)), Z)) 

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

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

  p(a(X), p(a(b(Y)), Z)) >? p(a(b(a(Z))), p(a(a(Y)), Z)) 

about to try horpo

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

We choose Lex = {} and Mul = {a, b, p}, and the following precedence: p > b > a

With these choices, we have:

  1] p(a(X), p(a(b(Y)), Z)) > p(a(b(a(Z))), p(a(a(Y)), Z))  because [2], by definition 
  2] p*(a(X), p(a(b(Y)), Z)) >= p(a(b(a(Z))), p(a(a(Y)), Z))  because p in Mul, [3] and [9], by (Stat) 
  3] p(a(b(Y)), Z) > a(b(a(Z)))  because [4], by definition 
  4] p*(a(b(Y)), Z) >= a(b(a(Z)))  because p > a and [5], by (Copy) 
  5] p*(a(b(Y)), Z) >= b(a(Z))  because p > b and [6], by (Copy) 
  6] p*(a(b(Y)), Z) >= a(Z)  because p > a and [7], by (Copy) 
  7] p*(a(b(Y)), Z) >= Z  because [8], by (Select) 
  8] Z >= Z  by (Meta) 
  9] p(a(b(Y)), Z) > p(a(a(Y)), Z)  because [10], by definition 
  10] p*(a(b(Y)), Z) >= p(a(a(Y)), Z)  because p in Mul, [11] and [17], by (Stat) 
  11] a(b(Y)) > a(a(Y))  because [12], by definition 
  12] a*(b(Y)) >= a(a(Y))  because a in Mul and [13], by (Stat) 
  13] b(Y) > a(Y)  because [14], by definition 
  14] b*(Y) >= a(Y)  because b > a and [15], by (Copy) 
  15] b*(Y) >= Y  because [16], by (Select) 
  16] Y >= Y  by (Meta) 
  17] Z >= Z  by (Meta) 

We can thus remove the following rules:

  p(a(X), p(a(b(Y)), Z)) => p(a(b(a(Z))), p(a(a(Y)), 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.