We consider the system 473.

  Alphabet:

    dots : [] --> o 
    f : [o -> o] --> o 
    g : [(o -> o) -> o] --> o 

  Rules:

    g(/\h.X(h)) => X(/\x.X(/\y.x)) 
    f(/\x.X(x)) => dots 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] g#(/\h.X(h)) =#> X(/\x.X(/\y.x))   
    1] g#(/\h.X(h)) =#> X(/\x.y) {X : 1}  

  Rules R_0:

    g(/\h.X(h)) => X(/\x.X(/\y.x)) 
    f(/\x.X(x)) => dots 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

This combination (P_0, R_0) has no formative rules!  We will name the empty set of rules:R_1.

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_0, R_1, minimal, formative).

Thus, the original system is terminating if (P_0, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

  g#(/\h.X(h)) >? X(/\x.X(/\y.x)) 
  g#(/\h.X(h)) >? X(/\x.~c0) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( g#(F) ) = #argfun-g##(F (/\x.F (/\y.x)), F (/\z.~c0)) 

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

The following interpretation satisfies the requirements:

  #argfun-g## = \y0y1.3 + max(y0, y1) 
  g# = \G0.0 
  ~c0 = 0 

Using this interpretation, the requirements translate to:

  [[#argfun-g##((/\h._x0(h)) (/\x.(/\i._x0(i)) (/\y.x)), (/\j._x0(j)) (/\z.~c0))]] = 3 + max(F0(\y0.0), F0(\y0.max(y0, F0(\y1.y1)))) > F0(\y0.F0(\y1.y1)) = [[_x0(/\x._x0(/\y.x))]] 
  [[#argfun-g##((/\h._x0(h)) (/\x.(/\i._x0(i)) (/\y.x)), (/\j._x0(j)) (/\z.~c0))]] = 3 + max(F0(\y0.0), F0(\y0.max(y0, F0(\y1.y1)))) > F0(\y0.0) = [[_x0(/\x.~c0)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_0, R_1) by ({}, R_1).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.