We consider the system loopy.

  Alphabet:

    f : [N -> N * N] --> N 
    g : [] --> N -> N 
    h : [N] --> N 

  Rules:

    f(g, x) => h(x) 
    f(i, x) => i x 
    h(x) => f(/\y.y, x) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

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

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

  f(g, X) >? h(X) 
  f(F, X) >? F X 
  h(X) >? f(/\x.x, X) 

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

We choose Lex = {} and Mul = {@_{o -> o}, f, g, h}, and the following precedence: g > h > f > @_{o -> o}

With these choices, we have:

  1] f(g, X) >= h(X)  because [2], by (Star) 
  2] f*(g, X) >= h(X)  because [3], by (Select) 
  3] g f*(g, X) >= h(X)  because [4] 
  4] g*(f*(g, X)) >= h(X)  because g > h and [5], by (Copy) 
  5] g*(f*(g, X)) >= X  because [6], by (Select) 
  6] f*(g, X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 

  8] f(F, X) > @_{o -> o}(F, X)  because [9], by definition 
  9] f*(F, X) >= @_{o -> o}(F, X)  because f > @_{o -> o}, [10] and [12], by (Copy) 
  10] f*(F, X) >= F  because [11], by (Select) 
  11] F >= F  by (Meta) 
  12] f*(F, X) >= X  because [13], by (Select) 
  13] X >= X  by (Meta) 

  14] h(X) >= f(/\x.x, X)  because [15], by (Star) 
  15] h*(X) >= f(/\x.x, X)  because h > f, [16] and [19], by (Copy) 
  16] h*(X) >= /\y.y  because [17], by (F-Abs) 
  17] h*(X, x) >= x  because [18], by (Select) 
  18] x >= x  by (Var) 
  19] h*(X) >= X  because [20], by (Select) 
  20] X >= X  by (Meta) 

We can thus remove the following rules:

  f(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]):

  f(g, X) >? h(X) 
  h(X) >? f(/\x.x, X) 

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

We choose Lex = {} and Mul = {f, g, h}, and the following precedence: g > h > f

With these choices, we have:

  1] f(g, X) >= h(X)  because [2], by (Star) 
  2] f*(g, X) >= h(X)  because [3], by (Select) 
  3] g f*(g, X) >= h(X)  because [4] 
  4] g*(f*(g, X)) >= h(X)  because g > h and [5], by (Copy) 
  5] g*(f*(g, X)) >= X  because [6], by (Select) 
  6] f*(g, X) >= X  because [7], by (Select) 
  7] g f*(g, X) >= X  because [8] 
  8] g*(f*(g, X)) >= X  because [9], by (Select) 
  9] f*(g, X) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 

  11] h(X) > f(/\x.x, X)  because [12], by definition 
  12] h*(X) >= f(/\x.x, X)  because h > f, [13] and [16], by (Copy) 
  13] h*(X) >= /\y.y  because [14], by (F-Abs) 
  14] h*(X, x) >= x  because [15], by (Select) 
  15] x >= x  by (Var) 
  16] h*(X) >= X  because [17], by (Select) 
  17] X >= X  by (Meta) 

We can thus remove the following rules:

  h(X) => f(/\x.x, 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]):

  f(g, X) >? h(X) 

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

Argument functions:

  [[h(x_1)]] = x_1 

We choose Lex = {} and Mul = {f, g}, and the following precedence: f > g

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

  f(g, X) > X 

With these choices, we have:

  1] f(g, X) > X  because [2], by definition 
  2] f*(g, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

We can thus remove the following rules:

  f(g, X) => h(X) 

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.