We consider the system h56.

  Alphabet:

    cons : [a * b] --> b 
    foldr : [a -> b -> b * b * b] --> b 
    nil : [] --> b 
    xap : [a -> b -> b * a] --> b -> b 
    yap : [b -> b * b] --> b 

  Rules:

    foldr(/\x./\y.yap(xap(f, x), y), z, nil) => z 
    foldr(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => yap(xap(f, u), foldr(/\w./\x'.yap(xap(f, w), x'), z, v)) 
    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:

    cons : [a * b] --> b 
    foldr : [a -> b -> b * b * b] --> b 
    nil : [] --> b 
    yap : [b -> b * b] --> b 

  Rules:

    foldr(/\x./\y.yap(F(x), y), X, nil) => X 
    foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 
    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]):

  foldr(/\x./\y.yap(F(x), y), X, nil) >? X 
  foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 
  yap(F, X) >? F X 

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

We choose Lex = {} and Mul = {@_{o -> o}, cons, foldr, nil, yap}, and the following precedence: cons > nil > foldr > yap > @_{o -> o}

With these choices, we have:

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

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

  24] yap(F, X) >= @_{o -> o}(F, X)  because [25], by (Star) 
  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:

  foldr(/\x./\y.yap(F(x), y), X, nil) => 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]):

  foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 
  yap(F, X) >? F X 

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

We choose Lex = {} and Mul = {@_{o -> o}, cons, foldr, yap}, and the following precedence: cons > foldr > yap > @_{o -> o}

With these choices, we have:

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

  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:

  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]):

  foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 

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

We choose Lex = {} and Mul = {cons, foldr, yap}, and the following precedence: yap > foldr > cons

With these choices, we have:

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

We can thus remove the following rules:

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