We consider the system 07ordinal.

  Alphabet:

    lim : [] --> (N -> O) -> O 
    plus : [] --> O -> O -> O 
    s : [] --> O -> O 
    z : [] --> O 

  Rules:

    plus z x => x 
    plus (s x) y => s (plus x y) 
    plus (lim (/\x.f x)) y => lim (/\u.plus (f u) y) 

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:

    lim : [N -> O] --> O 
    plus : [O * O] --> O 
    s : [O] --> O 
    z : [] --> O 
    ~AP1 : [N -> O * N] --> O 

  Rules:

    plus(z, X) => X 
    plus(s(X), Y) => s(plus(X, Y)) 
    plus(lim(/\x.~AP1(F, x)), X) => lim(/\y.plus(~AP1(F, y), 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:

    lim : [N -> O] --> O 
    plus : [O * O] --> O 
    s : [O] --> O 
    z : [] --> O 

  Rules:

    plus(z, X) => X 
    plus(s(X), Y) => s(plus(X, Y)) 
    plus(lim(/\x.X(x)), Y) => lim(/\y.plus(X(y), Y)) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  plus(z, X) >? X 
  plus(s(X), Y) >? s(plus(X, Y)) 
  plus(lim(/\x.X(x)), Y) >? lim(/\y.plus(X(y), Y)) 

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

The following interpretation satisfies the requirements:

  lim = \G0.3 + G0(0) 
  plus = \y0y1.3 + y1 + 3y0 
  s = \y0.3 + y0 
  z = 3 

Using this interpretation, the requirements translate to:

  [[plus(z, _x0)]] = 12 + x0 > x0 = [[_x0]] 
  [[plus(s(_x0), _x1)]] = 12 + x1 + 3x0 > 6 + x1 + 3x0 = [[s(plus(_x0, _x1))]] 
  [[plus(lim(/\x._x0(x)), _x1)]] = 12 + x1 + 3F0(0) > 6 + x1 + 3F0(0) = [[lim(/\x.plus(_x0(x), _x1))]] 

We can thus remove the following rules:

  plus(z, X) => X 
  plus(s(X), Y) => s(plus(X, Y)) 
  plus(lim(/\x.X(x)), Y) => lim(/\y.plus(X(y), 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.