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


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YES
We consider the system theBenchmark.

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

  a : [] --> o 
  b : [] --> o 
  c : [] --> o 
  d : [] --> o 
  e : [] --> o 
  f : [o] --> o 
  g : [o * o] --> o 
  h : [o * o] --> o 

  f(a) => b 
  f(c) => d 
  f(g(X, Y)) => g(f(X), f(Y)) 
  f(h(X, Y)) => g(h(Y, f(X)), h(X, f(Y))) 
  g(X, X) => h(e, X) 

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(a) >? b 
  f(c) >? d 
  f(g(X, Y)) >? g(f(X), f(Y)) 
  f(h(X, Y)) >? g(h(Y, f(X)), h(X, f(Y))) 
  g(X, X) >? h(e, X) 

about to try horpo

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

Argument functions:

  [[b]] = _|_ 
  [[d]] = _|_ 
  [[e]] = _|_ 

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

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

  f(a) > _|_ 
  f(c) >= _|_ 
  f(g(X, Y)) >= g(f(X), f(Y)) 
  f(h(X, Y)) > g(h(Y, f(X)), h(X, f(Y))) 
  g(X, X) >= h(_|_, X) 

With these choices, we have:

  1] f(a) > _|_  because [2], by definition 
  2] f*(a) >= _|_  by (Bot) 

  3] f(c) >= _|_  by (Bot) 

  4] f(g(X, Y)) >= g(f(X), f(Y))  because [5], by (Star) 
  5] f*(g(X, Y)) >= g(f(X), f(Y))  because f > g, [6] and [10], by (Copy) 
  6] f*(g(X, Y)) >= f(X)  because f in Mul and [7], by (Stat) 
  7] g(X, Y) > X  because [8], by definition 
  8] g*(X, Y) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] f*(g(X, Y)) >= f(Y)  because f in Mul and [11], by (Stat) 
  11] g(X, Y) > Y  because [12], by definition 
  12] g*(X, Y) >= Y  because [13], by (Select) 
  13] Y >= Y  by (Meta) 

  14] f(h(X, Y)) > g(h(Y, f(X)), h(X, f(Y)))  because [15], by definition 
  15] f*(h(X, Y)) >= g(h(Y, f(X)), h(X, f(Y)))  because f > g, [16] and [23], by (Copy) 
  16] f*(h(X, Y)) >= h(Y, f(X))  because f > h, [17] and [20], by (Copy) 
  17] f*(h(X, Y)) >= Y  because [18], by (Select) 
  18] h(X, Y) >= Y  because [19], by (Star) 
  19] h*(X, Y) >= Y  because [13], by (Select) 
  20] f*(h(X, Y)) >= f(X)  because f in Mul and [21], by (Stat) 
  21] h(X, Y) > X  because [22], by definition 
  22] h*(X, Y) >= X  because [9], by (Select) 
  23] f*(h(X, Y)) >= h(X, f(Y))  because f > h, [24] and [26], by (Copy) 
  24] f*(h(X, Y)) >= X  because [25], by (Select) 
  25] h(X, Y) >= X  because [22], by (Star) 
  26] f*(h(X, Y)) >= f(Y)  because f in Mul and [27], by (Stat) 
  27] h(X, Y) > Y  because [28], by definition 
  28] h*(X, Y) >= Y  because [13], by (Select) 

  29] g(X, X) >= h(_|_, X)  because [30], by (Star) 
  30] g*(X, X) >= h(_|_, X)  because g > h, [31] and [32], by (Copy) 
  31] g*(X, X) >= _|_  by (Bot) 
  32] g*(X, X) >= X  because [9], by (Select) 

We can thus remove the following rules:

  f(a) => b 
  f(h(X, Y)) => g(h(Y, f(X)), h(X, f(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(c) >? d 
  f(g(X, Y)) >? g(f(X), f(Y)) 
  g(X, X) >? h(e, X) 

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

The following interpretation satisfies the requirements:

  c = 3 
  d = 0 
  e = 0 
  f = \y0.3y0 
  g = \y0y1.3 + y0 + y1 
  h = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[f(c)]] = 9 > 0 = [[d]] 
  [[f(g(_x0, _x1))]] = 9 + 3x0 + 3x1 > 3 + 3x0 + 3x1 = [[g(f(_x0), f(_x1))]] 
  [[g(_x0, _x0)]] = 3 + 2x0 > x0 = [[h(e, _x0)]] 

We can thus remove the following rules:

  f(c) => d 
  f(g(X, Y)) => g(f(X), f(Y)) 
  g(X, X) => h(e, 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.