We consider the system h59.

  Alphabet:

    0 : [] --> nat 
    rec : [] --> nat -> a -> (nat -> a -> a) -> a 
    s : [] --> nat -> nat 

  Rules:

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

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 
    rec : [nat * a * nat -> a -> a] --> a 
    s : [nat] --> nat 
    ~AP1 : [nat -> a -> a * nat] --> a -> a 

  Rules:

    rec(0, X, /\x./\y.~AP1(F, x) y) => X 
    rec(s(X), Y, /\x./\y.~AP1(F, x) y) => ~AP1(F, X) rec(X, Y, /\z./\u.~AP1(F, z) u) 
    ~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 
    rec : [nat * a * nat -> a -> a] --> a 
    s : [nat] --> nat 

  Rules:

    rec(0, X, /\x./\y.Y(x, y)) => X 
    rec(s(X), Y, /\x./\y.Z(x, y)) => Z(X, rec(X, Y, /\z./\u.Z(z, u))) 

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

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

Argument functions:

  [[rec(x_1, x_2, x_3)]] = rec(x_1, x_3, x_2) 

We choose Lex = {rec} and Mul = {0, s}, and the following precedence: rec > 0 > s

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

  rec(0, X, /\x./\y.Y(x, y)) > X 
  rec(s(X), Y, /\x./\y.Z(x, y)) >= Z(X, rec(X, Y, /\x./\y.Z(x, y))) 

With these choices, we have:

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

  4] rec(s(X), Y, /\x./\y.Z(x, y)) >= Z(X, rec(X, Y, /\x./\y.Z(x, y)))  because [5], by (Star) 
  5] rec*(s(X), Y, /\x./\y.Z(x, y)) >= Z(X, rec(X, Y, /\x./\y.Z(x, y)))  because [6], by (Select) 
  6] Z(rec*(s(X), Y, /\x./\y.Z(x, y)), rec*(s(X), Y, /\z./\u.Z(z, u))) >= Z(X, rec(X, Y, /\x./\y.Z(x, y)))  because [7] and [11], by (Meta) 
  7] rec*(s(X), Y, /\x./\y.Z(x, y)) >= X  because [8], by (Select) 
  8] s(X) >= X  because [9], by (Star) 
  9] s*(X) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] rec*(s(X), Y, /\x./\y.Z(x, y)) >= rec(X, Y, /\x./\y.Z(x, y))  because [12], [7], [14] and [16], by (Stat) 
  12] s(X) > X  because [13], by definition 
  13] s*(X) >= X  because [10], by (Select) 
  14] rec*(s(X), Y, /\x./\y.Z(x, y)) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 
  16] rec*(s(X), Y, /\x./\y.Z(x, y)) >= /\x./\y.Z(x, y)  because [17], by (Select) 
  17] /\x./\z.Z(x, z) >= /\x./\z.Z(x, z)  because [18], by (Abs) 
  18] /\z.Z(y, z) >= /\z.Z(y, z)  because [19], by (Abs) 
  19] Z(y, x) >= Z(y, x)  because [20] and [21], by (Meta) 
  20] y >= y  by (Var) 
  21] x >= x  by (Var) 

We can thus remove the following rules:

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

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

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

With these choices, we have:

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

We can thus remove the following rules:

  rec(s(X), Y, /\x./\y.Z(x, y)) => Z(X, rec(X, Y, /\z./\u.Z(z, u))) 

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.