We consider the system uncurry.

  Alphabet:

    f : [] --> a -> b -> c 
    f1 : [a] --> b -> c 
    f2 : [a * b] --> c 

  Rules:

    f x => f1(x) 
    f1(x) y => f2(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 in [Kop12, Ch. 7.8]).

In order to do so, we start by eta-expanding the system, which gives:

  f(X, Y) => f1(X, Y) 
  f1(X, Y) => f2(X, Y) 

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

  Dependency Pairs P_0:

    0] f#(X, Y) =#> f1#(X, Y)   

  Rules R_0:

    f(X, Y) => f1(X, Y) 
    f1(X, Y) => f2(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].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  f#(X, Y) >? f1#(X, Y) 

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

  pi( f#(X, Y) ) = #argfun-f##(f1#(X, Y)) 

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

The following interpretation satisfies the requirements:

  #argfun-f## = \y0.1 + y0 
  f1# = \y0y1.0 
  f# = \y0y1.0 

Using this interpretation, the requirements translate to:

  [[#argfun-f##(f1#(_x0, _x1))]] = 1 > 0 = [[f1#(_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.
[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.