We consider the system Applicative_first_order_05__30.

  Alphabet:

    !faccolon : [a * a] --> a 
    C : [] --> a 
    cons : [c * d] --> d 
    false : [] --> b 
    filter : [c -> b * d] --> d 
    filter2 : [b * c -> b * c * d] --> d 
    map : [c -> c * d] --> d 
    nil : [] --> d 
    true : [] --> b 

  Rules:

    !faccolon(!faccolon(!faccolon(!faccolon(C, x), y), z), u) => !faccolon(!faccolon(x, z), !faccolon(!faccolon(!faccolon(x, y), z), u)) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    filter(f, nil) => nil 
    filter(f, cons(x, y)) => filter2(f x, f, x, y) 
    filter2(true, f, x, y) => cons(x, filter(f, y)) 
    filter2(false, f, x, y) => filter(f, y) 

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]).

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

  Dependency Pairs P_0:

    0] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))   
    1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Z)   
    2] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(!faccolon(X, Y), Z), U)   
    3] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Y), Z)   
    4] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   
    5] map#(F, cons(X, Y)) =#> map#(F, Y)   
    6] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    7] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    8] filter2#(false, F, X, Y) =#> filter#(F, Y)   

  Rules R_0:

    !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) => !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(F, Y) 

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, 2, 3, 4 
    * 1 : 0, 1, 2, 3, 4 
    * 2 : 0, 1, 2, 3, 4 
    * 3 : 0, 1, 2, 3, 4 
    * 4 : 0, 1, 2, 3, 4 
    * 5 : 5 
    * 6 : 7, 8 
    * 7 : 6 
    * 8 : 6 

This graph has the following strongly connected components:

  P_1:

    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))   
    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Z)   
    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(!faccolon(X, Y), Z), U)   
    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Y), Z)   
    !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   

  P_2:

    map#(F, cons(X, Y)) =#> map#(F, Y)   

  P_3:

    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    filter2#(true, F, X, Y) =#> filter#(F, Y)   
    filter2#(false, F, X, Y) =#> filter#(F, Y)   

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), (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, static, formative), (P_2, R_0, static, formative) and (P_3, R_0, static, formative) is finite.

We consider the dependency pair problem (P_3, 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:

  filter#(F, cons(X, Y)) >? filter2#(F X, F, X, Y) 
  filter2#(true, F, X, Y) >? filter#(F, Y) 
  filter2#(false, F, X, Y) >? filter#(F, Y) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) >= cons(F X, map(F, Y)) 
  filter(F, nil) >= nil 
  filter(F, cons(X, Y)) >= filter2(F X, F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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

The following interpretation satisfies the requirements:

  !faccolon = \y0y1.0 
  C = 0 
  cons = \y0y1.2 + y1 
  false = 3 
  filter = \G0y1.2y1 
  filter2 = \y0G1y2y3.2 + 2y3 
  filter2# = \y0G1y2y3.1 + 2y3 + 2y3G1(y3) 
  filter# = \G0y1.2y1 + 2y1G0(y1) 
  map = \G0y1.y1 
  nil = 0 
  true = 3 

Using this interpretation, the requirements translate to:

  [[filter#(_F0, cons(_x1, _x2))]] = 4 + 2x2 + 2x2F0(2 + x2) + 4F0(2 + x2) > 1 + 2x2 + 2x2F0(x2) = [[filter2#(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2#(true, _F0, _x1, _x2)]] = 1 + 2x2 + 2x2F0(x2) > 2x2 + 2x2F0(x2) = [[filter#(_F0, _x2)]] 
  [[filter2#(false, _F0, _x1, _x2)]] = 1 + 2x2 + 2x2F0(x2) > 2x2 + 2x2F0(x2) = [[filter#(_F0, _x2)]] 
  [[!faccolon(!faccolon(!faccolon(!faccolon(C, _x0), _x1), _x2), _x3)]] = 0 >= 0 = [[!faccolon(!faccolon(_x0, _x2), !faccolon(!faccolon(!faccolon(_x0, _x1), _x2), _x3))]] 
  [[map(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 2 + x2 >= 2 + x2 = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 4 + 2x2 >= 2 + 2x2 = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 2 + 2x2 >= 2 + 2x2 = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 2 + 2x2 >= 2x2 = [[filter(_F0, _x2)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_3, 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 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 with usable rules [Kop12, Thm. 7.44].  (P_2, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  map#(F, cons(X, Y)) >? map#(F, Y) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.3 + 2y1 
  map# = \G0y1.y1 

Using this interpretation, the requirements translate to:

  [[map#(_F0, cons(_x1, _x2))]] = 3 + 2x2 > x2 = [[map#(_F0, _x2)]] 

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 with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_1, R_0) are:

  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) => !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(!faccolon(X, Y), Z), U) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Y), Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Y) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

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:

  [[!faccolon#(x_1, x_2)]] = !faccolon#(x_1) 

We choose Lex = {!faccolon} and Mul = {!faccolon#, C}, and the following precedence: !faccolon# > C > !faccolon

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

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(!faccolon(!faccolon(X, Y), Z)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Y)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z))  because !faccolon# in Mul and [2], by (Fun) 
  2] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [3] and [8], by (Fun) 
  3] !faccolon(!faccolon(C, X), Y) >= X  because [4], by (Star) 
  4] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 
  5] !faccolon(C, X) >= X  because [6], by (Star) 
  6] !faccolon*(C, X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] Z >= Z  by (Meta) 

  9] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because [10], by (Star) 
  10] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because !faccolon# in Mul and [11], by (Stat) 
  11] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > X  because [12], by definition 
  12] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [3], by (Select) 

  13] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(!faccolon(!faccolon(X, Y), Z))  because [14], by definition 
  14] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(!faccolon(X, Y), Z))  because !faccolon# in Mul and [15], by (Stat) 
  15] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [16], by definition 
  16] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [17], [24] and [28], by (Stat) 
  17] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [18], by definition 
  18] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [19], [21] and [22], by (Stat) 
  19] !faccolon(C, X) > X  because [20], by definition 
  20] !faccolon*(C, X) >= X  because [7], by (Select) 
  21] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 
  22] !faccolon*(!faccolon(C, X), Y) >= Y  because [23], by (Select) 
  23] Y >= Y  by (Meta) 
  24] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [25], by (Select) 
  25] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [26] and [27], by (Fun) 
  26] !faccolon(C, X) >= X  because [20], by (Star) 
  27] Y >= Y  by (Meta) 
  28] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [8], by (Select) 

  29] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Y))  because !faccolon# in Mul and [30], by (Fun) 
  30] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [24], by (Star) 

  31] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because [32], by (Star) 
  32] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because !faccolon# in Mul and [11], by (Stat) 

  33] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [34], by (Star) 
  34] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [35], [38] and [39], by (Stat) 
  35] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [36], by definition 
  36] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [37], [12] and [28], by (Stat) 
  37] !faccolon(!faccolon(C, X), Y) > X  because [21], by definition 
  38] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [2], by (Select) 
  39] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [15], [40] and [43], by (Stat) 
  40] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [41], by (Select) 
  41] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [42] and [8], by (Fun) 
  42] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [26] and [27], by (Fun) 
  43] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [44], by (Select) 
  44] U >= U  by (Meta) 

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_4, R_0, static, formative), where P_4 consists of:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Z)   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Y), Z)   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   

Thus, the original system is terminating if (P_4, R_0, static, formative) is finite.

We consider the dependency pair problem (P_4, R_0, static, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_4, R_0) are:

  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) => !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Y), Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Y) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

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:

  [[!faccolon#(x_1, x_2)]] = !faccolon#(x_1) 

We choose Lex = {!faccolon} and Mul = {!faccolon#, C}, and the following precedence: C > !faccolon > !faccolon#

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

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(X) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Y)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z))  because !faccolon# in Mul and [2], by (Fun) 
  2] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [3] and [8], by (Fun) 
  3] !faccolon(!faccolon(C, X), Y) >= X  because [4], by (Star) 
  4] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 
  5] !faccolon(C, X) >= X  because [6], by (Star) 
  6] !faccolon*(C, X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] Z >= Z  by (Meta) 

  9] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(X)  because [10], by definition 
  10] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because [11], by (Select) 
  11] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X)  because [12], by (Star) 
  12] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X)  because [13], by (Select) 
  13] !faccolon(!faccolon(C, X), Y) >= !faccolon#(X)  because [14], by (Star) 
  14] !faccolon*(!faccolon(C, X), Y) >= !faccolon#(X)  because !faccolon > !faccolon# and [15], by (Copy) 
  15] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 

  16] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Y))  because !faccolon# in Mul and [17], by (Fun) 
  17] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [18], by (Star) 
  18] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [19], by (Select) 
  19] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [20] and [21], by (Fun) 
  20] !faccolon(C, X) >= X  because [6], by (Star) 
  21] Y >= Y  by (Meta) 

  22] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because [23], by (Star) 
  23] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because [24], by (Select) 
  24] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X)  because [25], by (Star) 
  25] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X)  because !faccolon > !faccolon# and [26], by (Copy) 
  26] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [3], by (Select) 

  27] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [28], by (Star) 
  28] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [29], [33] and [34], by (Stat) 
  29] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [30], by definition 
  30] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [31], [26] and [32], by (Stat) 
  31] !faccolon(!faccolon(C, X), Y) > X  because [15], by definition 
  32] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [8], by (Select) 
  33] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [2], by (Select) 
  34] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [35], [43] and [46], by (Stat) 
  35] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [36], by definition 
  36] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [37], [42] and [32], by (Stat) 
  37] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [38], by definition 
  38] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [39], [15] and [41], by (Stat) 
  39] !faccolon(C, X) > X  because [40], by definition 
  40] !faccolon*(C, X) >= X  because [7], by (Select) 
  41] !faccolon*(!faccolon(C, X), Y) >= Y  because [21], by (Select) 
  42] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [19], by (Select) 
  43] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [44], by (Select) 
  44] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [45] and [8], by (Fun) 
  45] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [20] and [21], by (Fun) 
  46] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [47], by (Select) 
  47] U >= U  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_4, R_0, static, formative) by (P_5, R_0, static, formative), where P_5 consists of:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Y), Z)   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   

Thus, the original system is terminating if (P_5, R_0, static, formative) is finite.

We consider the dependency pair problem (P_5, R_0, static, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_5, R_0) are:

  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) => !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Y), Z) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Y) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

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:

  [[!faccolon#(x_1, x_2)]] = !faccolon#(x_1) 

We choose Lex = {!faccolon} and Mul = {!faccolon#, C}, and the following precedence: !faccolon# > !faccolon > C

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

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(!faccolon(X, Y)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z))  because !faccolon# in Mul and [2], by (Fun) 
  2] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [3] and [8], by (Fun) 
  3] !faccolon(!faccolon(C, X), Y) >= X  because [4], by (Star) 
  4] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 
  5] !faccolon(C, X) >= X  because [6], by (Star) 
  6] !faccolon*(C, X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] Z >= Z  by (Meta) 

  9] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(!faccolon(X, Y))  because [10], by definition 
  10] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Y))  because !faccolon# in Mul and [11], by (Stat) 
  11] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Y)  because [12], by definition 
  12] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [13], by (Select) 
  13] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [14] and [15], by (Fun) 
  14] !faccolon(C, X) >= X  because [6], by (Star) 
  15] Y >= Y  by (Meta) 

  16] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because [17], by (Star) 
  17] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because !faccolon# in Mul and [18], by (Stat) 
  18] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > X  because [19], by definition 
  19] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [3], by (Select) 

  20] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [21], by (Star) 
  21] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [22], [27] and [28], by (Stat) 
  22] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [23], by definition 
  23] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [24], [19] and [26], by (Stat) 
  24] !faccolon(!faccolon(C, X), Y) > X  because [25], by definition 
  25] !faccolon*(!faccolon(C, X), Y) >= X  because [14], by (Select) 
  26] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [8], by (Select) 
  27] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [2], by (Select) 
  28] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [29], [36] and [39], by (Stat) 
  29] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [30], by definition 
  30] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [31], [12] and [26], by (Stat) 
  31] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [32], by definition 
  32] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [33], [25] and [35], by (Stat) 
  33] !faccolon(C, X) > X  because [34], by definition 
  34] !faccolon*(C, X) >= X  because [7], by (Select) 
  35] !faccolon*(!faccolon(C, X), Y) >= Y  because [15], by (Select) 
  36] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [37], by (Select) 
  37] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [38] and [8], by (Fun) 
  38] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [14] and [15], by (Fun) 
  39] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [40], by (Select) 
  40] U >= U  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_5, R_0, static, formative) by (P_6, R_0, static, formative), where P_6 consists of:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))   
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(X, Y)   

Thus, the original system is terminating if (P_6, R_0, static, formative) is finite.

We consider the dependency pair problem (P_6, R_0, static, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_6, R_0) are:

  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) => !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(X, Y) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

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:

  [[!faccolon#(x_1, x_2)]] = !faccolon#(x_1) 

We choose Lex = {!faccolon} and Mul = {!faccolon#, C}, and the following precedence: !faccolon > !faccolon# > C

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

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z)) 
  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(X) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z))  because !faccolon# in Mul and [2], by (Fun) 
  2] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [3] and [8], by (Fun) 
  3] !faccolon(!faccolon(C, X), Y) >= X  because [4], by (Star) 
  4] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 
  5] !faccolon(C, X) >= X  because [6], by (Star) 
  6] !faccolon*(C, X) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] Z >= Z  by (Meta) 

  9] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(X)  because [10], by definition 
  10] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because !faccolon# in Mul and [11], by (Stat) 
  11] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > X  because [12], by definition 
  12] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [3], by (Select) 

  13] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [14], by (Star) 
  14] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [15], [20] and [21], by (Stat) 
  15] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [16], by definition 
  16] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [17], [12] and [19], by (Stat) 
  17] !faccolon(!faccolon(C, X), Y) > X  because [18], by definition 
  18] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 
  19] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [8], by (Select) 
  20] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [2], by (Select) 
  21] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [22], [34] and [37], by (Stat) 
  22] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [23], by definition 
  23] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [24], [30] and [19], by (Stat) 
  24] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [25], by definition 
  25] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [26], [18] and [28], by (Stat) 
  26] !faccolon(C, X) > X  because [27], by definition 
  27] !faccolon*(C, X) >= X  because [7], by (Select) 
  28] !faccolon*(!faccolon(C, X), Y) >= Y  because [29], by (Select) 
  29] Y >= Y  by (Meta) 
  30] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [31], by (Select) 
  31] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [32] and [33], by (Fun) 
  32] !faccolon(C, X) >= X  because [27], by (Star) 
  33] Y >= Y  by (Meta) 
  34] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [35], by (Select) 
  35] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [36] and [8], by (Fun) 
  36] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [32] and [33], by (Fun) 
  37] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [38], by (Select) 
  38] U >= U  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_6, R_0, static, formative) by (P_7, R_0, static, formative), where P_7 consists of:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) =#> !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))   

Thus, the original system is terminating if (P_7, R_0, static, formative) is finite.

We consider the dependency pair problem (P_7, R_0, static, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_7, R_0) are:

  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) => !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

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:

  [[!faccolon#(x_1, x_2)]] = !faccolon#(x_1) 

We choose Lex = {!faccolon} and Mul = {!faccolon#, C}, and the following precedence: !faccolon > !faccolon# > C

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

  !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(!faccolon(X, Z)) 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) > !faccolon#(!faccolon(X, Z))  because [2], by definition 
  2] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(!faccolon(X, Z))  because [3], by (Select) 
  3] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(!faccolon(X, Z))  because [4], by (Star) 
  4] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(!faccolon(X, Z))  because !faccolon > !faccolon# and [5], by (Copy) 
  5] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [6], [11] and [13], by (Stat) 
  6] !faccolon(!faccolon(C, X), Y) > X  because [7], by definition 
  7] !faccolon*(!faccolon(C, X), Y) >= X  because [8], by (Select) 
  8] !faccolon(C, X) >= X  because [9], by (Star) 
  9] !faccolon*(C, X) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [12], by (Select) 
  12] !faccolon(!faccolon(C, X), Y) >= X  because [7], by (Star) 
  13] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [14], by (Select) 
  14] Z >= Z  by (Meta) 

  15] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [16], by (Star) 
  16] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [17], [18] and [24], by (Stat) 
  17] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [5], by definition 
  18] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [19], [20] and [22], by (Stat) 
  19] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > X  because [11], by definition 
  20] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= X  because [21], by (Select) 
  21] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= X  because [11], by (Star) 
  22] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= Z  because [23], by (Select) 
  23] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [13], by (Star) 
  24] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [25], [37] and [41], by (Stat) 
  25] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [26], by definition 
  26] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [27], [33] and [13], by (Stat) 
  27] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [28], by definition 
  28] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [29], [7] and [31], by (Stat) 
  29] !faccolon(C, X) > X  because [30], by definition 
  30] !faccolon*(C, X) >= X  because [10], by (Select) 
  31] !faccolon*(!faccolon(C, X), Y) >= Y  because [32], by (Select) 
  32] Y >= Y  by (Meta) 
  33] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [34], by (Select) 
  34] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [35] and [36], by (Fun) 
  35] !faccolon(C, X) >= X  because [30], by (Star) 
  36] Y >= Y  by (Meta) 
  37] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [38], [39] and [22], by (Stat) 
  38] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Y)  because [33], by definition 
  39] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Y)  because [40], by (Select) 
  40] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [33], by (Star) 
  41] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [42], by (Select) 
  42] U >= U  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_7, 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.