We consider the system plode.

  Alphabet:

    cons : [nat * list] --> list 
    explode : [list * nat -> nat * nat] --> nat 
    implode : [list * nat -> nat * nat] --> nat 
    nil : [] --> list 
    op : [nat -> nat * nat -> nat] --> nat -> nat 

  Rules:

    op(f, g) x => f (g x) 
    implode(nil, f, x) => x 
    implode(cons(x, y), f, z) => implode(y, f, f z) 
    explode(nil, f, x) => x 
    explode(cons(x, y), f, z) => explode(y, op(f, f), f z) 

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:

  op(F, G, X) => F (G X) 
  implode(nil, F, X) => X 
  implode(cons(X, Y), F, Z) => implode(Y, F, F Z) 
  explode(nil, F, X) => X 
  explode(cons(X, Y), F, Z) => explode(Y, /\x.op(F, F, x), F Z) 

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

  Dependency Pairs P_0:

    0] implode#(cons(X, Y), F, Z) =#> implode#(Y, F, F Z)   
    1] explode#(cons(X, Y), F, Z) =#> explode#(Y, /\x.op(F, F, x), F Z)   
    2] explode#(cons(X, Y), F, Z) =#> op#(F, F, U)   

  Rules R_0:

    op(F, G, X) => F (G X) 
    implode(nil, F, X) => X 
    implode(cons(X, Y), F, Z) => implode(Y, F, F Z) 
    explode(nil, F, X) => X 
    explode(cons(X, Y), F, Z) => explode(Y, /\x.op(F, F, x), F Z) 

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 : 1, 2 
    * 2 :  

This graph has the following strongly connected components:

  P_1:

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

  P_2:

    explode#(cons(X, Y), F, Z) =#> explode#(Y, /\x.op(F, F, x), F 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) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite.

We consider the dependency pair problem (P_2, 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:

  explode#(cons(X, Y), F, Z) >? explode#(Y, /\x.op(F, F, x), F Z) 
  op(F, G, X) >= F (G X) 
  implode(nil, F, X) >= X 
  implode(cons(X, Y), F, Z) >= implode(Y, F, F Z) 
  explode(nil, F, X) >= X 
  explode(cons(X, Y), F, Z) >= explode(Y, /\x.op(F, F, x), F Z) 

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

  pi( op(F, G, X) ) = #argfun-op#(F (G X)) 

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:

  [[explode(x_1, x_2, x_3)]] = explode(x_1, x_3, x_2) 
  [[explode#(x_1, x_2, x_3)]] = x_1 

We choose Lex = {explode, implode} and Mul = {#argfun-op#, @_{o -> o}, cons, nil, op}, and the following precedence: implode > explode > @_{o -> o} > nil > op > cons > #argfun-op#

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

  cons(X, Y) > Y 
  #argfun-op#(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X)) 
  implode(nil, F, X) >= X 
  implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z)) 
  explode(nil, F, X) >= X 
  explode(cons(X, Y), F, Z) >= explode(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z)) 

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] #argfun-op#(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [5], by (Star) 
  5] #argfun-op#*(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [6], by (Select) 
  6] @_{o -> o}(F, @_{o -> o}(G, X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because @_{o -> o} in Mul, [7] and [8], by (Fun) 
  7] F >= F  by (Meta) 
  8] @_{o -> o}(G, X) >= @_{o -> o}(G, X)  because @_{o -> o} in Mul, [9] and [10], by (Fun) 
  9] G >= G  by (Meta) 
  10] X >= X  by (Meta) 

  11] implode(nil, F, X) >= X  because [12], by (Star) 
  12] implode*(nil, F, X) >= X  because [13], by (Select) 
  13] X >= X  by (Meta) 

  14] implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [15], by (Star) 
  15] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [16], [19], [21] and [23], by (Stat) 
  16] cons(X, Y) > Y  because [17], by definition 
  17] cons*(X, Y) >= Y  because [18], by (Select) 
  18] Y >= Y  by (Meta) 
  19] implode*(cons(X, Y), F, Z) >= Y  because [20], by (Select) 
  20] cons(X, Y) >= Y  because [17], by (Star) 
  21] implode*(cons(X, Y), F, Z) >= F  because [22], by (Select) 
  22] F >= F  by (Meta) 
  23] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode > @_{o -> o}, [21] and [24], by (Copy) 
  24] implode*(cons(X, Y), F, Z) >= Z  because [25], by (Select) 
  25] Z >= Z  by (Meta) 

  26] explode(nil, F, X) >= X  because [27], by (Star) 
  27] explode*(nil, F, X) >= X  because [28], by (Select) 
  28] X >= X  by (Meta) 

  29] explode(cons(X, Y), F, Z) >= explode(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z))  because [30], by (Star) 
  30] explode*(cons(X, Y), F, Z) >= explode(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z))  because [31], [32], [34] and [42], by (Stat) 
  31] cons(X, Y) > Y  because [2], by definition 
  32] explode*(cons(X, Y), F, Z) >= Y  because [33], by (Select) 
  33] cons(X, Y) >= Y  because [2], by (Star) 
  34] explode*(cons(X, Y), F, Z) >= /\y.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, y)))  because [35], by (F-Abs) 
  35] explode*(cons(X, Y), F, Z, x) >= #argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x)))  because explode > #argfun-op# and [36], by (Copy) 
  36] explode*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, @_{o -> o}(F, x))  because explode > @_{o -> o}, [37] and [39], by (Copy) 
  37] explode*(cons(X, Y), F, Z, x) >= F  because [38], by (Select) 
  38] F >= F  by (Meta) 
  39] explode*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, x)  because explode > @_{o -> o}, [37] and [40], by (Copy) 
  40] explode*(cons(X, Y), F, Z, x) >= x  because [41], by (Select) 
  41] x >= x  by (Var) 
  42] explode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because explode > @_{o -> o}, [43] and [44], by (Copy) 
  43] explode*(cons(X, Y), F, Z) >= F  because [38], by (Select) 
  44] explode*(cons(X, Y), F, Z) >= Z  because [45], by (Select) 
  45] Z >= Z  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_2, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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:

  implode#(cons(X, Y), F, Z) >? implode#(Y, F, F Z) 
  op(F, G, X) >= F (G X) 
  implode(nil, F, X) >= X 
  implode(cons(X, Y), F, Z) >= implode(Y, F, F Z) 
  explode(nil, F, X) >= X 
  explode(cons(X, Y), F, Z) >= explode(Y, /\x.op(F, F, x), F Z) 

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

  pi( op(F, G, X) ) = #argfun-op#(F (G X)) 

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:

  [[explode(x_1, x_2, x_3)]] = explode(x_1, x_3, x_2) 
  [[implode#(x_1, x_2, x_3)]] = implode#(x_2, x_1, x_3) 

We choose Lex = {explode, implode, implode#} and Mul = {#argfun-op#, @_{o -> o}, cons, nil, op}, and the following precedence: cons > explode > implode > implode# > @_{o -> o} > #argfun-op# > nil > op

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

  implode#(cons(X, Y), F, Z) > implode#(Y, F, @_{o -> o}(F, Z)) 
  #argfun-op#(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X)) 
  implode(nil, F, X) >= X 
  implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z)) 
  explode(nil, F, X) >= X 
  explode(cons(X, Y), F, Z) >= explode(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z)) 

With these choices, we have:

  1] implode#(cons(X, Y), F, Z) > implode#(Y, F, @_{o -> o}(F, Z))  because [2], by definition 
  2] implode#*(cons(X, Y), F, Z) >= implode#(Y, F, @_{o -> o}(F, Z))  because [3], [6], [7], [9] and [10], by (Stat) 
  3] cons(X, Y) > Y  because [4], by definition 
  4] cons*(X, Y) >= Y  because [5], by (Select) 
  5] Y >= Y  by (Meta) 
  6] F >= F  by (Meta) 
  7] implode#*(cons(X, Y), F, Z) >= Y  because [8], by (Select) 
  8] cons(X, Y) >= Y  because [4], by (Star) 
  9] implode#*(cons(X, Y), F, Z) >= F  because [6], by (Select) 
  10] implode#*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode# > @_{o -> o}, [9] and [11], by (Copy) 
  11] implode#*(cons(X, Y), F, Z) >= Z  because [12], by (Select) 
  12] Z >= Z  by (Meta) 

  13] #argfun-op#(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [14], by (Star) 
  14] #argfun-op#*(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [15], by (Select) 
  15] @_{o -> o}(F, @_{o -> o}(G, X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because @_{o -> o} in Mul, [16] and [17], by (Fun) 
  16] F >= F  by (Meta) 
  17] @_{o -> o}(G, X) >= @_{o -> o}(G, X)  because @_{o -> o} in Mul, [18] and [19], by (Fun) 
  18] G >= G  by (Meta) 
  19] X >= X  by (Meta) 

  20] implode(nil, F, X) >= X  because [21], by (Star) 
  21] implode*(nil, F, X) >= X  because [22], by (Select) 
  22] X >= X  by (Meta) 

  23] implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [24], by (Star) 
  24] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [3], [25], [26] and [27], by (Stat) 
  25] implode*(cons(X, Y), F, Z) >= Y  because [8], by (Select) 
  26] implode*(cons(X, Y), F, Z) >= F  because [6], by (Select) 
  27] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode > @_{o -> o}, [26] and [28], by (Copy) 
  28] implode*(cons(X, Y), F, Z) >= Z  because [12], by (Select) 

  29] explode(nil, F, X) >= X  because [30], by (Star) 
  30] explode*(nil, F, X) >= X  because [31], by (Select) 
  31] X >= X  by (Meta) 

  32] explode(cons(X, Y), F, Z) >= explode(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z))  because [33], by (Star) 
  33] explode*(cons(X, Y), F, Z) >= explode(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z))  because [34], [37], [39] and [47], by (Stat) 
  34] cons(X, Y) > Y  because [35], by definition 
  35] cons*(X, Y) >= Y  because [36], by (Select) 
  36] Y >= Y  by (Meta) 
  37] explode*(cons(X, Y), F, Z) >= Y  because [38], by (Select) 
  38] cons(X, Y) >= Y  because [35], by (Star) 
  39] explode*(cons(X, Y), F, Z) >= /\y.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, y)))  because [40], by (F-Abs) 
  40] explode*(cons(X, Y), F, Z, x) >= #argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x)))  because explode > #argfun-op# and [41], by (Copy) 
  41] explode*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, @_{o -> o}(F, x))  because explode > @_{o -> o}, [42] and [44], by (Copy) 
  42] explode*(cons(X, Y), F, Z, x) >= F  because [43], by (Select) 
  43] F >= F  by (Meta) 
  44] explode*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, x)  because explode > @_{o -> o}, [42] and [45], by (Copy) 
  45] explode*(cons(X, Y), F, Z, x) >= x  because [46], by (Select) 
  46] x >= x  by (Var) 
  47] explode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because explode > @_{o -> o}, [48] and [49], by (Copy) 
  48] explode*(cons(X, Y), F, Z) >= F  because [43], by (Select) 
  49] explode*(cons(X, Y), F, Z) >= Z  because [50], by (Select) 
  50] Z >= Z  by (Meta) 

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.