We consider the system apply.

  Alphabet:

    cons : [a -> a * listf] --> listf 
    dapply : [a * a -> a * a -> a] --> a 
    lapply : [a * listf] --> a 
    nil : [] --> listf 

  Rules:

    dapply(x, f, g) => f (g x) 
    lapply(x, nil) => x 
    lapply(x, cons(f, y)) => f lapply(x, y) 

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

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]).

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

  Dependency Pairs P_0:

    0] lapply#(X, cons(F, Y)) =#> lapply#(X, Y)   

  Rules R_0:

    dapply(X, F, G) => F (G X) 
    lapply(X, nil) => X 
    lapply(X, cons(F, Y)) => F lapply(X, Y) 

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

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

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_0, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  lapply#(X, cons(F, Y)) >? lapply#(X, Y) 

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

The following interpretation satisfies the requirements:

  cons = \G0y1.1 + 2y1 
  lapply# = \y0y1.y1 

Using this interpretation, the requirements translate to:

  [[lapply#(_x0, cons(_F1, _x2))]] = 1 + 2x2 > x2 = [[lapply#(_x0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_0, R_0) by ({}, R_0).  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.
[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.