We consider the system h13.

  Alphabet:

    0 : [] --> a 
    rec : [a -> b -> b * b * a] --> b 
    s : [a] --> a 
    xap : [a -> b -> b * a] --> b -> b 
    yap : [b -> b * b] --> b 

  Rules:

    rec(/\x./\y.yap(xap(f, x), y), z, 0) => z 
    rec(/\x./\y.yap(xap(f, x), y), z, s(u)) => yap(xap(f, s(u)), rec(/\v./\w.yap(xap(f, v), w), z, u)) 
    xap(f, x) => f x 
    yap(f, x) => f x 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

Symbol xap is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:

  Alphabet:

    0 : [] --> a 
    rec : [a -> b -> b * b * a] --> b 
    s : [a] --> a 
    yap : [b -> b * b] --> b 

  Rules:

    rec(/\x./\y.yap(F(x), y), X, 0) => X 
    rec(/\x./\y.yap(F(x), y), X, s(Y)) => yap(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y)) 
    yap(F, X) => F 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]):

  rec(/\x./\y.yap(F(x), y), X, 0) >? X 
  rec(/\x./\y.yap(F(x), y), X, s(Y)) >? yap(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y)) 
  yap(F, X) >? F X 

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

We choose Lex = {} and Mul = {0, @_{o -> o}, rec, s, yap}, and the following precedence: 0 > rec > s > yap > @_{o -> o}

With these choices, we have:

  1] rec(/\x./\y.yap(F(x), y), X, 0) > X  because [2], by definition 
  2] rec*(/\x./\y.yap(F(x), y), X, 0) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] rec(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because [5], by (Star) 
  5] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because rec > yap, [6] and [14], by (Copy) 
  6] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= F(s(Y))  because [7], by (Select) 
  7] /\x.yap(F(rec*(/\y./\z.yap(F(y), z), X, s(Y))), x) >= F(s(Y))  because [8], by (Eta)[Kop13:2] 
  8] F(rec*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(s(Y))  because [9], by (Meta) 
  9] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= s(Y)  because rec > s and [10], by (Copy) 
  10] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= Y  because [11], by (Select) 
  11] s(Y) >= Y  because [12], by (Star) 
  12] s*(Y) >= Y  because [13], by (Select) 
  13] Y >= Y  by (Meta) 
  14] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y)  because rec in Mul, [15], [21] and [22], by (Stat) 
  15] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [16], by (Abs) 
  16] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [17], by (Abs) 
  17] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [18] and [20], by (Fun) 
  18] F(y) >= F(y)  because [19], by (Meta) 
  19] y >= y  by (Var) 
  20] x >= x  by (Var) 
  21] X >= X  by (Meta) 
  22] s(Y) > Y  because [23], by definition 
  23] s*(Y) >= Y  because [13], by (Select) 

  24] yap(F, X) > @_{o -> o}(F, X)  because [25], by definition 
  25] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [26] and [28], by (Copy) 
  26] yap*(F, X) >= F  because [27], by (Select) 
  27] F >= F  by (Meta) 
  28] yap*(F, X) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 

We can thus remove the following rules:

  rec(/\x./\y.yap(F(x), y), X, 0) => X 
  yap(F, X) => F 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]):

  rec(/\x./\y.yap(F(x), y), X, s(Y)) >? yap(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y)) 

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

The following interpretation satisfies the requirements:

  rec = \G0y1y2.y1 + y2 + G0(0,0) + 2y2y2G0(y2,y2) + 2G0(y2,y1) 
  s = \y0.3 + 3y0 
  yap = \G0y1.y1 + 2G0(0) 

Using this interpretation, the requirements translate to:

  [[rec(/\x./\y.yap(_F0(x), y), _x1, s(_x2))]] = 57 + 3x1 + 54x2x2x2 + 162x2x2 + 165x2 + 2F0(0,0) + 36x2x2F0(3 + 3x2,0) + 40F0(3 + 3x2,0) + 72x2F0(3 + 3x2,0) > x2 + 2x2x2x2 + 3x1 + 2F0(0,0) + 2F0(3 + 3x2,0) + 4x2x2F0(x2,0) + 4F0(x2,0) = [[yap(_F0(s(_x2)), rec(/\x./\y.yap(_F0(x), y), _x1, _x2))]] 

We can thus remove the following rules:

  rec(/\x./\y.yap(F(x), y), X, s(Y)) => yap(F(s(Y)), rec(/\z./\u.yap(F(z), u), 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.
[Kop13:2]  C. Kop.  StarHorpo with an Eta-Rule.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/etahorpo.pdf, 2013.