We consider the system Applicative_05__Ex6Folding.

  Alphabet:

    0 : [] --> c 
    1 : [] --> c 
    add : [] --> c -> a -> c 
    cons : [a * b] --> b 
    fold : [c -> a -> c * b * c] --> c 
    mul : [] --> c -> a -> c 
    nil : [] --> b 
    prod : [b] --> c 
    sum : [b] --> c 

  Rules:

    fold(f, nil, x) => x 
    fold(f, cons(x, y), z) => fold(f, y, f z x) 
    sum(x) => fold(add, x, 0) 
    fold(mul, x, 1) => prod(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:

  fold(F, nil, X) => X 
  fold(F, cons(X, Y), Z) => fold(F, Y, F Z X) 
  sum(X) => fold(/\x./\y.add(x, y), X, 0) 
  fold(/\x./\y.mul(x, y), X, 1) => prod(X) 

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

  Dependency Pairs P_0:

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

  Rules R_0:

    fold(F, nil, X) => X 
    fold(F, cons(X, Y), Z) => fold(F, Y, F Z X) 
    sum(X) => fold(/\x./\y.add(x, y), X, 0) 
    fold(/\x./\y.mul(x, y), X, 1) => prod(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).

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

    * 0 : 0 
    * 1 : 0 

This graph has the following strongly connected components:

  P_1:

    fold#(F, cons(X, Y), Z) =#> fold#(F, Y, F Z X)   

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, formative) is finite.

We consider the dependency pair problem (P_1, R_0, 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:

  fold#(F, cons(X, Y), Z) >? fold#(F, Y, F Z X) 
  fold(F, nil, X) >= X 
  fold(F, cons(X, Y), Z) >= fold(F, Y, F Z X) 
  sum(X) >= fold(/\x./\y.add(x, y), X, 0) 
  fold(/\x./\y.mul(x, y), X, 1) >= prod(X) 

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

  pi( sum(X) ) = #argfun-sum#(fold(/\x./\y.add(x, y), X, 0)) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[add(x_1, x_2)]] = _|_ 
  [[fold(x_1, x_2, x_3)]] = fold(x_2, x_1, x_3) 
  [[fold#(x_1, x_2, x_3)]] = x_2 
  [[mul(x_1, x_2)]] = mul 
  [[prod(x_1)]] = _|_ 
  [[sum(x_1)]] = x_1 

We choose Lex = {fold} and Mul = {#argfun-sum#, 1, @_{o -> o -> o}, @_{o -> o}, cons, mul, nil}, and the following precedence: mul > 1 > nil > #argfun-sum# > fold > @_{o -> o -> o} > cons > @_{o -> o}

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  cons(X, Y) > Y 
  fold(F, nil, X) >= X 
  fold(F, cons(X, Y), Z) >= fold(F, Y, @_{o -> o}(@_{o -> o -> o}(F, Z), X)) 
  #argfun-sum#(fold(/\x./\y._|_, X, _|_)) >= fold(/\x./\y._|_, X, _|_) 
  fold(/\x./\y.mul, X, 1) >= _|_ 

With these choices, we have:

  1] cons(X, Y) > Y  because [2], by definition 
  2] cons*(X, Y) >= Y  because [3], by (Select) 
  3] Y >= Y  by (Meta) 

  4] fold(F, nil, X) >= X  because [5], by (Star) 
  5] fold*(F, nil, X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 

  7] fold(F, cons(X, Y), Z) >= fold(F, Y, @_{o -> o}(@_{o -> o -> o}(F, Z), X))  because [8], by (Star) 
  8] fold*(F, cons(X, Y), Z) >= fold(F, Y, @_{o -> o}(@_{o -> o -> o}(F, Z), X))  because [9], [10], [12] and [14], by (Stat) 
  9] cons(X, Y) > Y  because [2], by definition 
  10] fold*(F, cons(X, Y), Z) >= F  because [11], by (Select) 
  11] F >= F  by (Meta) 
  12] fold*(F, cons(X, Y), Z) >= Y  because [13], by (Select) 
  13] cons(X, Y) >= Y  because [2], by (Star) 
  14] fold*(F, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because fold > @_{o -> o}, [15] and [18], by (Copy) 
  15] fold*(F, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because fold > @_{o -> o -> o}, [10] and [16], by (Copy) 
  16] fold*(F, cons(X, Y), Z) >= Z  because [17], by (Select) 
  17] Z >= Z  by (Meta) 
  18] fold*(F, cons(X, Y), Z) >= X  because [19], by (Select) 
  19] cons(X, Y) >= X  because [20], by (Star) 
  20] cons*(X, Y) >= X  because [21], by (Select) 
  21] X >= X  by (Meta) 

  22] #argfun-sum#(fold(/\x./\y._|_, X, _|_)) >= fold(/\x./\y._|_, X, _|_)  because [23], by (Star) 
  23] #argfun-sum#*(fold(/\x./\y._|_, X, _|_)) >= fold(/\x./\y._|_, X, _|_)  because #argfun-sum# > fold, [24], [27] and [31], by (Copy) 
  24] #argfun-sum#*(fold(/\x./\y._|_, X, _|_)) >= /\x./\y._|_  because [25], by (F-Abs) 
  25] #argfun-sum#*(fold(/\x./\y._|_, X, _|_), z) >= /\x._|_  because [26], by (F-Abs) 
  26] #argfun-sum#*(fold(/\x./\y._|_, X, _|_), z, u) >= _|_  by (Bot) 
  27] #argfun-sum#*(fold(/\x./\y._|_, X, _|_)) >= X  because [28], by (Select) 
  28] fold(/\x./\y._|_, X, _|_) >= X  because [29], by (Star) 
  29] fold*(/\x./\y._|_, X, _|_) >= X  because [30], by (Select) 
  30] X >= X  by (Meta) 
  31] #argfun-sum#*(fold(/\x./\y._|_, X, _|_)) >= _|_  by (Bot) 

  32] fold(/\x./\y.mul, X, 1) >= _|_  by (Bot) 

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 +++

[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.