We consider the system Applicative_05__ReverseLastInit.

  Alphabet:

    compose : [a -> a * a -> a] --> a -> a 
    cons : [a * a] --> a 
    hd : [] --> a -> a 
    init : [] --> a -> a 
    last : [] --> a -> a 
    nil : [] --> a 
    reverse : [] --> a -> a 
    reverse2 : [a * a] --> a 
    tl : [] --> a -> a 

  Rules:

    compose(f, g) x => g (f x) 
    reverse x => reverse2(x, nil) 
    reverse2(nil, x) => x 
    reverse2(cons(x, y), z) => reverse2(y, cons(x, z)) 
    hd cons(x, y) => x 
    tl cons(x, y) => y 
    last => compose(hd, reverse) 
    init => compose(reverse, compose(tl, reverse)) 

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:

  compose(F, G, X) => G (F X) 
  reverse(X) => reverse2(X, nil) 
  reverse2(nil, X) => X 
  reverse2(cons(X, Y), Z) => reverse2(Y, cons(X, Z)) 
  hd(cons(X, Y)) => X 
  tl(cons(X, Y)) => Y 
  last(X) => compose(/\x.hd(x), /\y.reverse(y), X) 
  init(X) => compose(/\x.reverse(x), /\y.compose(/\z.tl(z), /\u.reverse(u), y), X) 

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

  Dependency Pairs P_0:

    0] reverse#(X) =#> reverse2#(X, nil)   
    1] reverse2#(cons(X, Y), Z) =#> reverse2#(Y, cons(X, Z))   
    2] last#(X) =#> compose#(/\x.hd(x), /\y.reverse(y), X)   
    3] last#(X) =#> hd#(Y)   
    4] last#(X) =#> reverse#(Y)   
    5] init#(X) =#> compose#(/\x.reverse(x), /\y.compose(/\z.tl(z), /\u.reverse(u), y), X)   
    6] init#(X) =#> reverse#(Y)   
    7] init#(X) =#> compose#(/\x.tl(x), /\y.reverse(y), Y)   
    8] init#(X) =#> tl#(Y)   
    9] init#(X) =#> reverse#(Y)   

  Rules R_0:

    compose(F, G, X) => G (F X) 
    reverse(X) => reverse2(X, nil) 
    reverse2(nil, X) => X 
    reverse2(cons(X, Y), Z) => reverse2(Y, cons(X, Z)) 
    hd(cons(X, Y)) => X 
    tl(cons(X, Y)) => Y 
    last(X) => compose(/\x.hd(x), /\y.reverse(y), X) 
    init(X) => compose(/\x.reverse(x), /\y.compose(/\z.tl(z), /\u.reverse(u), y), 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 : 1 
    * 1 : 1 
    * 2 :  
    * 3 :  
    * 4 : 0 
    * 5 :  
    * 6 : 0 
    * 7 :  
    * 8 :  
    * 9 : 0 

This graph has the following strongly connected components:

  P_1:

    reverse2#(cons(X, Y), Z) =#> reverse2#(Y, cons(X, Z))   

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:

  reverse2#(cons(X, Y), Z) >? reverse2#(Y, cons(X, Z)) 
  compose(F, G, X) >= G (F X) 
  reverse(X) >= reverse2(X, nil) 
  reverse2(nil, X) >= X 
  reverse2(cons(X, Y), Z) >= reverse2(Y, cons(X, Z)) 
  hd(cons(X, Y)) >= X 
  tl(cons(X, Y)) >= Y 
  last(X) >= compose(/\x.hd(x), /\y.reverse(y), X) 
  init(X) >= compose(/\x.reverse(x), /\y.compose(/\z.tl(z), /\u.reverse(u), y), X) 

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

  pi( compose(F, G, X) ) = #argfun-compose#(G (F X)) 
  pi( reverse(X) ) = #argfun-reverse#(reverse2(X, nil)) 
  pi( init(X) ) = #argfun-init#(#argfun-compose#((/\x.#argfun-compose#((/\y.#argfun-reverse#(reverse2(y, nil))) ((/\z.tl(z)) x))) ((/\u.#argfun-reverse#(reverse2(u, nil))) X))) 
  pi( last(X) ) = #argfun-last#(#argfun-compose#((/\x.#argfun-reverse#(reverse2(x, nil))) ((/\y.hd(y)) X))) 

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

The following interpretation satisfies the requirements:

  #argfun-compose# = \y0.3 + y0 
  #argfun-init# = \y0.3 + y0 
  #argfun-last# = \y0.3 + y0 
  #argfun-reverse# = \y0.3 + y0 
  compose = \G0G1y2.0 
  cons = \y0y1.3 + y1 + 3y0 
  hd = \y0.y0 
  init = \y0.0 
  last = \y0.0 
  nil = 0 
  reverse = \y0.0 
  reverse2 = \y0y1.2y1 + 3y0 
  reverse2# = \y0y1.y0 
  tl = \y0.y0 

Using this interpretation, the requirements translate to:

  [[reverse2#(cons(_x0, _x1), _x2)]] = 3 + x1 + 3x0 > x1 = [[reverse2#(_x1, cons(_x0, _x2))]] 
  [[#argfun-compose#(_F0 (_F1 _x2))]] = 3 + F0(F1(x2)) >= F0(F1(x2)) = [[_F0 (_F1 _x2)]] 
  [[#argfun-reverse#(reverse2(_x0, nil))]] = 3 + 3x0 >= 3x0 = [[reverse2(_x0, nil)]] 
  [[reverse2(nil, _x0)]] = 2x0 >= x0 = [[_x0]] 
  [[reverse2(cons(_x0, _x1), _x2)]] = 9 + 2x2 + 3x1 + 9x0 >= 6 + 2x2 + 3x1 + 6x0 = [[reverse2(_x1, cons(_x0, _x2))]] 
  [[hd(cons(_x0, _x1))]] = 3 + x1 + 3x0 >= x0 = [[_x0]] 
  [[tl(cons(_x0, _x1))]] = 3 + x1 + 3x0 >= x1 = [[_x1]] 
  [[#argfun-last#(#argfun-compose#((/\x.#argfun-reverse#(reverse2(x, nil))) ((/\y.hd(y)) _x0)))]] = 9 + 3x0 >= 6 + 3x0 = [[#argfun-compose#((/\x.#argfun-reverse#(reverse2(x, nil))) ((/\y.hd(y)) _x0))]] 
  [[#argfun-init#(#argfun-compose#((/\x.#argfun-compose#((/\y.#argfun-reverse#(reverse2(y, nil))) ((/\z.tl(z)) x))) ((/\u.#argfun-reverse#(reverse2(u, nil))) _x0)))]] = 21 + 9x0 >= 18 + 9x0 = [[#argfun-compose#((/\x.#argfun-compose#((/\y.#argfun-reverse#(reverse2(y, nil))) ((/\z.tl(z)) x))) ((/\u.#argfun-reverse#(reverse2(u, nil))) _x0))]] 

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.