We consider the system typed_lam.

  Alphabet:

    app : [] --> arrab -> a -> b 
    lam : [] --> (a -> b) -> arrab 

  Rules:

    app (lam (/\x.f x)) y => f y 
    lam (/\x.app y x) => y 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    app : [arrab * a] --> b 
    lam : [a -> b] --> arrab 
    ~AP1 : [a -> b * a] --> b 

  Rules:

    app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) 
    lam(/\x.app(X, x)) => X 
    app(lam(/\x.app(X, x)), Y) => app(X, Y) 
    ~AP1(F, X) => F X 

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, all):

  Dependency Pairs P_0:

    0] app#(lam(/\x.~AP1(F, x)), X) =#> ~AP1#(F, X)   
    1] app#(lam(/\x.app(X, x)), Y) =#> app#(X, Y)   

  Rules R_0:

    app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) 
    lam(/\x.app(X, x)) => X 
    app(lam(/\x.app(X, x)), Y) => app(X, Y) 
    ~AP1(F, X) => F X 

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

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

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 :  
    * 1 : 0, 1 

This graph has the following strongly connected components:

  P_1:

    app#(lam(/\x.app(X, x)), Y) =#> app#(X, Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).

Thus, the original system is terminating if (P_1, R_0, static, all) is finite.

We consider the dependency pair problem (P_1, R_0, static, all).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_1, R_0) has no usable rules.

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

  app#(lam(/\x.app(X, x)), Y) >? app#(X, Y) 

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

The following interpretation satisfies the requirements:

  app = \y0y1.3 + 2y0 
  app# = \y0y1.y0 
  lam = \G0.3 + G0(0) 

Using this interpretation, the requirements translate to:

  [[app#(lam(/\x.app(_x0, x)), _x1)]] = 6 + 2x0 > x0 = [[app#(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, 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 +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[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.