We consider the system h57.

  Alphabet:

    0 : [] --> nat 
    cons : [] --> nat -> list -> list 
    foldr : [] --> (nat -> nat -> nat) -> nat -> list -> nat 
    length : [] --> list -> nat 
    nil : [] --> list 
    s : [] --> nat -> nat 

  Rules:

    foldr (/\x./\y.f x y) z nil => z 
    foldr (/\x./\y.f x y) z (cons u v) => f u (foldr (/\w./\x'.f w x') z v) 
    length x => foldr (/\y./\z.s z) 0 x 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    0 : [] --> nat 
    cons : [nat * list] --> list 
    foldr : [nat -> nat -> nat * nat * list] --> nat 
    length : [list] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 
    ~AP1 : [nat -> nat -> nat * nat] --> nat -> nat 

  Rules:

    foldr(/\x./\y.~AP1(F, x) y, X, nil) => X 
    foldr(/\x./\y.~AP1(F, x) y, X, cons(Y, Z)) => ~AP1(F, Y) foldr(/\z./\u.~AP1(F, z) u, X, Z) 
    length(X) => foldr(/\x./\y.s(y), 0, X) 
    ~AP1(F, X) => F X 

Symbol ~AP1 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 : [] --> nat 
    cons : [nat * list] --> list 
    foldr : [nat -> nat -> nat * nat * list] --> nat 
    length : [list] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 

  Rules:

    foldr(/\x./\y.X(x, y), Y, nil) => Y 
    foldr(/\x./\y.X(x, y), Y, cons(Z, U)) => X(Z, foldr(/\z./\u.X(z, u), Y, U)) 
    length(X) => foldr(/\x./\y.s(y), 0, 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.X(x, y), Y, nil) >? Y 
  foldr(/\x./\y.X(x, y), Y, cons(Z, U)) >? X(Z, foldr(/\z./\u.X(z, u), Y, U)) 
  length(X) >? foldr(/\x./\y.s(y), 0, X) 

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

Argument functions:

  [[0]] = _|_ 
  [[s(x_1)]] = x_1 

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

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

  foldr(/\x./\y.X(x, y), Y, nil) >= Y 
  foldr(/\x./\y.X(x, y), Y, cons(Z, U)) > X(Z, foldr(/\x./\y.X(x, y), Y, U)) 
  length(X) > foldr(/\x./\y.y, _|_, X) 

With these choices, we have:

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

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

  21] length(X) > foldr(/\x./\y.y, _|_, X)  because [22], by definition 
  22] length*(X) >= foldr(/\x./\y.y, _|_, X)  because length > foldr, [23], [27] and [28], by (Copy) 
  23] length*(X) >= /\y./\z.z  because [24], by (F-Abs) 
  24] length*(X, x) >= /\z.z  because [25], by (F-Abs) 
  25] length*(X, x, y) >= y  because [26], by (Select) 
  26] y >= y  by (Var) 
  27] length*(X) >= _|_  by (Bot) 
  28] length*(X) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 

We can thus remove the following rules:

  foldr(/\x./\y.X(x, y), Y, cons(Z, U)) => X(Z, foldr(/\z./\u.X(z, u), Y, U)) 
  length(X) => foldr(/\x./\y.s(y), 0, 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.X(x, y), Y, nil) >? Y 

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

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

With these choices, we have:

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

We can thus remove the following rules:

  foldr(/\x./\y.X(x, y), Y, nil) => 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 +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.