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 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) 
  explode#(cons(X, Y), F, Z) >? explode#(Y, /\x.op(F, F, x), F Z) 
  explode#(cons(X, Y), F, Z) >? op#(F, F, U) 
  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)]] = _|_ 
  [[implode#(x_1, x_2, x_3)]] = implode#(x_1) 
  [[op#(x_1, x_2, x_3)]] = _|_ 

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

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

  implode#(cons(X, Y)) > implode#(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] implode#(cons(X, Y)) > implode#(Y)  because [2], by definition 
  2] implode#*(cons(X, Y)) >= implode#(Y)  because [3], by (Select) 
  3] cons(X, Y) >= implode#(Y)  because [4], by (Star) 
  4] cons*(X, Y) >= implode#(Y)  because cons > implode# and [5], by (Copy) 
  5] cons*(X, Y) >= Y  because [6], by (Select) 
  6] Y >= Y  by (Meta) 

  7] _|_ >= _|_  by (Bot) 

  8] _|_ >= _|_  by (Bot) 

  9] #argfun-op#(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [10], by (Star) 
  10] #argfun-op#*(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [11], by (Select) 
  11] @_{o -> o}(F, @_{o -> o}(G, X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because @_{o -> o} in Mul, [12] and [13], by (Fun) 
  12] F >= F  by (Meta) 
  13] @_{o -> o}(G, X) >= @_{o -> o}(G, X)  because @_{o -> o} in Mul, [14] and [15], by (Fun) 
  14] G >= G  by (Meta) 
  15] X >= X  by (Meta) 

  16] implode(nil, F, X) >= X  because [17], by (Star) 
  17] implode*(nil, F, X) >= X  because [18], by (Select) 
  18] X >= X  by (Meta) 

  19] implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [20], by (Star) 
  20] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [21], [22], [24] and [26], by (Stat) 
  21] cons(X, Y) > Y  because [5], by definition 
  22] implode*(cons(X, Y), F, Z) >= Y  because [23], by (Select) 
  23] cons(X, Y) >= Y  because [5], by (Star) 
  24] implode*(cons(X, Y), F, Z) >= F  because [25], by (Select) 
  25] F >= F  by (Meta) 
  26] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode > @_{o -> o}, [24] and [27], by (Copy) 
  27] implode*(cons(X, Y), F, Z) >= Z  because [28], by (Select) 
  28] Z >= Z  by (Meta) 

  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 the dependency pair problem (P_0, R_0, static, formative) by (P_1, R_0, static, formative), where P_1 consists of:

  explode#(cons(X, Y), F, Z) =#> explode#(Y, /\x.op(F, F, x), F Z)   
  explode#(cons(X, Y), F, Z) =#> op#(F, F, U)   

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:

  explode#(cons(X, Y), F, Z) >? explode#(Y, /\x.op(F, F, x), F Z) 
  explode#(cons(X, Y), F, Z) >? op#(F, F, U) 
  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)]] = explode#(x_1, x_3, x_2) 
  [[implode(x_1, x_2, x_3)]] = implode(x_2, x_1, x_3) 
  [[op#(x_1, x_2, x_3)]] = _|_ 

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

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

  explode#(cons(X, Y), F, Z) > explode#(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z)) 
  explode#(cons(X, Y), 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] explode#(cons(X, Y), F, Z) > explode#(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z))  because [2], by definition 
  2] explode#*(cons(X, Y), F, Z) >= explode#(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z))  because [3], [6], [8] and [16], 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] explode#*(cons(X, Y), F, Z) >= Y  because [7], by (Select) 
  7] cons(X, Y) >= Y  because [4], by (Star) 
  8] explode#*(cons(X, Y), F, Z) >= /\y.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, y)))  because [9], by (F-Abs) 
  9] explode#*(cons(X, Y), F, Z, x) >= #argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x)))  because explode# > #argfun-op# and [10], by (Copy) 
  10] explode#*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, @_{o -> o}(F, x))  because explode# > @_{o -> o}, [11] and [13], by (Copy) 
  11] explode#*(cons(X, Y), F, Z, x) >= F  because [12], by (Select) 
  12] F >= F  by (Meta) 
  13] explode#*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, x)  because explode# > @_{o -> o}, [11] and [14], by (Copy) 
  14] explode#*(cons(X, Y), F, Z, x) >= x  because [15], by (Select) 
  15] x >= x  by (Var) 
  16] explode#*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because explode# > @_{o -> o}, [17] and [18], by (Copy) 
  17] explode#*(cons(X, Y), F, Z) >= F  because [12], by (Select) 
  18] explode#*(cons(X, Y), F, Z) >= Z  because [19], by (Select) 
  19] Z >= Z  by (Meta) 

  20] explode#(cons(X, Y), F, Z) >= _|_  by (Bot) 

  21] #argfun-op#(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [22], by (Star) 
  22] #argfun-op#*(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [23], by (Select) 
  23] @_{o -> o}(F, @_{o -> o}(G, X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because @_{o -> o} in Mul, [24] and [25], by (Fun) 
  24] F >= F  by (Meta) 
  25] @_{o -> o}(G, X) >= @_{o -> o}(G, X)  because @_{o -> o} in Mul, [26] and [27], by (Fun) 
  26] G >= G  by (Meta) 
  27] X >= X  by (Meta) 

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

  31] implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [32], by (Star) 
  32] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [33], [36], [37], [39] and [40], by (Stat) 
  33] cons(X, Y) > Y  because [34], by definition 
  34] cons*(X, Y) >= Y  because [35], by (Select) 
  35] Y >= Y  by (Meta) 
  36] F >= F  by (Meta) 
  37] implode*(cons(X, Y), F, Z) >= Y  because [38], by (Select) 
  38] cons(X, Y) >= Y  because [34], by (Star) 
  39] implode*(cons(X, Y), F, Z) >= F  because [36], by (Select) 
  40] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode > @_{o -> o}, [39] and [41], by (Copy) 
  41] implode*(cons(X, Y), F, Z) >= Z  because [42], by (Select) 
  42] Z >= Z  by (Meta) 

  43] explode(nil, F, X) >= X  because [44], by (Star) 
  44] explode*(nil, F, X) >= X  because [45], by (Select) 
  45] X >= X  by (Meta) 

  46] explode(cons(X, Y), F, Z) >= explode(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z))  because [47], by (Star) 
  47] explode*(cons(X, Y), F, Z) >= explode(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z))  because [3], [48], [49] and [56], by (Stat) 
  48] explode*(cons(X, Y), F, Z) >= Y  because [7], by (Select) 
  49] explode*(cons(X, Y), F, Z) >= /\y.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, y)))  because [50], by (F-Abs) 
  50] explode*(cons(X, Y), F, Z, x) >= #argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x)))  because explode > #argfun-op# and [51], by (Copy) 
  51] explode*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, @_{o -> o}(F, x))  because explode > @_{o -> o}, [52] and [53], by (Copy) 
  52] explode*(cons(X, Y), F, Z, x) >= F  because [12], by (Select) 
  53] explode*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, x)  because explode > @_{o -> o}, [52] and [54], by (Copy) 
  54] explode*(cons(X, Y), F, Z, x) >= x  because [55], by (Select) 
  55] x >= x  by (Var) 
  56] explode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because explode > @_{o -> o}, [57] and [58], by (Copy) 
  57] explode*(cons(X, Y), F, Z) >= F  because [12], by (Select) 
  58] explode*(cons(X, Y), F, Z) >= Z  because [19], by (Select) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_0, static, formative) by (P_2, R_0, static, formative), where P_2 consists of:

  explode#(cons(X, Y), F, Z) =#> op#(F, F, U)   

Thus, the original system is terminating if (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) >? op#(F, F, U) 
  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:

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

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

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

  explode#(cons(X, Y), F, Z) > op#(F, F) 
  #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] explode#(cons(X, Y), F, Z) > op#(F, F)  because [2], by definition 
  2] explode#*(cons(X, Y), F, Z) >= op#(F, F)  because explode# > op#, [3] and [3], by (Copy) 
  3] explode#*(cons(X, Y), F, Z) >= F  because [4], by (Select) 
  4] F >= F  by (Meta) 

  5] #argfun-op#(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [6], by (Star) 
  6] #argfun-op#*(@_{o -> o}(F, @_{o -> o}(G, X))) >= @_{o -> o}(F, @_{o -> o}(G, X))  because [7], by (Select) 
  7] @_{o -> o}(F, @_{o -> o}(G, X)) >= @_{o -> o}(F, @_{o -> o}(G, X))  because @_{o -> o} in Mul, [8] and [9], by (Fun) 
  8] F >= F  by (Meta) 
  9] @_{o -> o}(G, X) >= @_{o -> o}(G, X)  because @_{o -> o} in Mul, [10] and [11], by (Fun) 
  10] G >= G  by (Meta) 
  11] X >= X  by (Meta) 

  12] implode(nil, F, X) >= X  because [13], by (Star) 
  13] implode*(nil, F, X) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 

  15] implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [16], by (Star) 
  16] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z))  because [17], [20], [22] and [24], by (Stat) 
  17] cons(X, Y) > Y  because [18], by definition 
  18] cons*(X, Y) >= Y  because [19], by (Select) 
  19] Y >= Y  by (Meta) 
  20] implode*(cons(X, Y), F, Z) >= Y  because [21], by (Select) 
  21] cons(X, Y) >= Y  because [18], by (Star) 
  22] implode*(cons(X, Y), F, Z) >= F  because [23], by (Select) 
  23] F >= F  by (Meta) 
  24] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because implode > @_{o -> o}, [22] and [25], by (Copy) 
  25] implode*(cons(X, Y), F, Z) >= Z  because [26], by (Select) 
  26] Z >= Z  by (Meta) 

  27] explode(nil, F, X) >= X  because [28], by (Star) 
  28] explode*(nil, F, X) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 

  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], by (Star) 
  31] explode*(cons(X, Y), F, Z) >= explode(Y, /\x.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x))), @_{o -> o}(F, Z))  because [32], [35], [37] and [44], by (Stat) 
  32] cons(X, Y) > Y  because [33], by definition 
  33] cons*(X, Y) >= Y  because [34], by (Select) 
  34] Y >= Y  by (Meta) 
  35] explode*(cons(X, Y), F, Z) >= Y  because [36], by (Select) 
  36] cons(X, Y) >= Y  because [33], by (Star) 
  37] explode*(cons(X, Y), F, Z) >= /\y.#argfun-op#(@_{o -> o}(F, @_{o -> o}(F, y)))  because [38], by (F-Abs) 
  38] explode*(cons(X, Y), F, Z, x) >= #argfun-op#(@_{o -> o}(F, @_{o -> o}(F, x)))  because explode > #argfun-op# and [39], by (Copy) 
  39] explode*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, @_{o -> o}(F, x))  because explode > @_{o -> o}, [40] and [41], by (Copy) 
  40] explode*(cons(X, Y), F, Z, x) >= F  because [4], by (Select) 
  41] explode*(cons(X, Y), F, Z, x) >= @_{o -> o}(F, x)  because explode > @_{o -> o}, [40] and [42], by (Copy) 
  42] explode*(cons(X, Y), F, Z, x) >= x  because [43], by (Select) 
  43] x >= x  by (Var) 
  44] explode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z)  because explode > @_{o -> o}, [45] and [46], by (Copy) 
  45] explode*(cons(X, Y), F, Z) >= F  because [4], by (Select) 
  46] explode*(cons(X, Y), F, Z) >= Z  because [47], by (Select) 
  47] 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.

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.