We consider the system 482.

  Alphabet:

    apply : [o -> o * o] --> o 
    compo : [o -> o * o -> o * o] --> o 

  Rules:

    compo(/\x.X(x), /\y.Y(y), Z) => X(Y(Z)) 
    apply(/\x.X(x), Y) => X(Y) 

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] compo#(/\x.X(x), /\y.Y(y), Z) =#> X(Y(Z))   
    1] compo#(/\x.X(x), /\y.Y(y), Z) =#> Y(Z) {X : 1}  
    2] apply#(/\x.X(x), Y) =#> X(Y)   

  Rules R_0:

    compo(/\x.X(x), /\y.Y(y), Z) => X(Y(Z)) 
    apply(/\x.X(x), Y) => X(Y) 

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:

  compo#(/\x.X(x), /\y.Y(y), Z) >? X(Y(Z)) 
  compo#(/\x.X(x), /\y.Y(y), Z) >? Y(Z) 
  apply#(/\x.X(x), Y) >? X(Y) 

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

  pi( apply#(F, X) ) = #argfun-apply##(F X) 
  pi( compo#(F, G, X) ) = #argfun-compo##(F (G X), G X) 

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

The following interpretation satisfies the requirements:

  #argfun-apply## = \y0.3 + y0 
  #argfun-compo## = \y0y1.3 + max(y0, y1) 
  apply# = \G0y1.0 
  compo# = \G0G1y2.0 

Using this interpretation, the requirements translate to:

  [[#argfun-compo##((/\x._x0(x)) ((/\y._x1(y)) _x2), (/\z._x1(z)) _x2)]] = 3 + max(x2, F0(max(x2, F1(x2))), F1(x2)) > F0(F1(x2)) = [[_x0(_x1(_x2))]] 
  [[#argfun-compo##((/\x._x0(x)) ((/\y._x1(y)) _x2), (/\z._x1(z)) _x2)]] = 3 + max(x2, F0(max(x2, F1(x2))), F1(x2)) > F1(x2) = [[_x1(_x2)]] 
  [[#argfun-apply##((/\x._x0(x)) _x1)]] = 3 + max(x1, F0(x1)) > F0(x1) = [[_x0(_x1)]] 

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.