We consider the system foldl.

  Alphabet:

    0 : [] --> nat 
    cons : [nat * list] --> list 
    foldl : [nat -> nat -> nat * nat * list] --> nat 
    nil : [] --> list 
    plus : [nat * nat] --> nat 
    plusc : [] --> nat -> nat -> nat 
    sum : [list] --> nat 

  Rules:

    foldl(f, x, nil) => x 
    foldl(f, x, cons(y, z)) => foldl(f, f x y, z) 
    plusc => /\x./\y.plus(x, y) 
    sum(x) => foldl(plusc, 0, x) 

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]).

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

  foldl(F, X, nil) => X 
  foldl(F, X, cons(Y, Z)) => foldl(F, F X Y, Z) 
  plusc(X, Y) => (/\x./\y.plus x y) X Y 
  sum(X) => foldl(/\x./\y.plusc(x, y), 0, X) 

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

  Dependency Pairs P_0:

    0] foldl#(F, X, cons(Y, Z)) =#> foldl#(F, F X Y, Z)   
    1] sum#(X) =#> foldl#(/\x./\y.plusc(x, y), 0, X)   
    2] sum#(X) =#> plusc#(Y, Z)   

  Rules R_0:

    foldl(F, X, nil) => X 
    foldl(F, X, cons(Y, Z)) => foldl(F, F X Y, Z) 
    plusc(X, Y) => (/\x./\y.plus x y) X Y 
    sum(X) => foldl(/\x./\y.plusc(x, y), 0, X) 

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).

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, static, formative) by (P_0, R_1, static, formative).

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

We consider the dependency pair problem (P_0, R_1, static, 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:

  foldl#(F, X, cons(Y, Z)) >? foldl#(F, F X Y, Z) 
  sum#(X) >? foldl#(/\x./\y.plusc(x, y), 0, X) 
  sum#(X) >? plusc#(Y, Z) 

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

The following interpretation satisfies the requirements:

  0 = 0 
  cons = \y0y1.3 + 2y1 
  foldl# = \G0y1y2.y2 
  plusc = \y0y1.0 
  plusc# = \y0y1.0 
  sum# = \y0.3 + 2y0 

Using this interpretation, the requirements translate to:

  [[foldl#(_F0, _x1, cons(_x2, _x3))]] = 3 + 2x3 > x3 = [[foldl#(_F0, _F0 _x1 _x2, _x3)]] 
  [[sum#(_x0)]] = 3 + 2x0 > x0 = [[foldl#(/\x./\y.plusc(x, y), 0, _x0)]] 
  [[sum#(_x0)]] = 3 + 2x0 > 0 = [[plusc#(_x1, _x2)]] 

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.
[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.