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: s > rec > 0 > 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 [6], by (Select) 
  6] yap(F(rec*(/\x./\y.yap(F(x), y), X, s(Y))), rec*(/\z./\u.yap(F(z), u), X, s(Y))) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because yap in Mul, [7] and [11], by (Fun) 
  7] F(rec*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(s(Y))  because [8], by (Meta) 
  8] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= s(Y)  because [9], by (Select) 
  9] s(Y) >= s(Y)  because s in Mul and [10], by (Fun) 
  10] Y >= Y  by (Meta) 
  11] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y)  because rec in Mul, [12], [18] and [19], by (Stat) 
  12] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [13], by (Abs) 
  13] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [14], by (Abs) 
  14] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [15] and [17], by (Fun) 
  15] F(y) >= F(y)  because [16], by (Meta) 
  16] y >= y  by (Var) 
  17] x >= x  by (Var) 
  18] X >= X  by (Meta) 
  19] s(Y) > Y  because [20], by definition 
  20] s*(Y) >= Y  because [10], by (Select) 

  21] yap(F, X) > @_{o -> o}(F, X)  because [22], by definition 
  22] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [23] and [25], by (Copy) 
  23] yap*(F, X) >= F  because [24], by (Select) 
  24] F >= F  by (Meta) 
  25] yap*(F, X) >= X  because [26], by (Select) 
  26] 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 use a recursive path ordering as defined in [Kop12, Chapter 5].

We choose Lex = {} and Mul = {rec, s, yap}, and the following precedence: rec > s > yap

With these choices, we have:

  1] rec(/\x./\y.yap(F(x), y), X, s(Y)) > yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because [2], by definition 
  2] 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, [3] and [11], by (Copy) 
  3] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= F(s(Y))  because [4], by (Select) 
  4] /\x.yap(F(rec*(/\y./\z.yap(F(y), z), X, s(Y))), x) >= F(s(Y))  because [5], by (Eta)[Kop13:2] 
  5] F(rec*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(s(Y))  because [6], by (Meta) 
  6] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= s(Y)  because rec > s and [7], by (Copy) 
  7] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= Y  because [8], by (Select) 
  8] s(Y) >= Y  because [9], by (Star) 
  9] s*(Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 
  11] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y)  because rec in Mul, [12], [18] and [19], by (Stat) 
  12] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [13], by (Abs) 
  13] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [14], by (Abs) 
  14] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [15] and [17], by (Fun) 
  15] F(y) >= F(y)  because [16], by (Meta) 
  16] y >= y  by (Var) 
  17] x >= x  by (Var) 
  18] X >= X  by (Meta) 
  19] s(Y) > Y  because [20], by definition 
  20] s*(Y) >= Y  because [10], by (Select) 

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.