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 [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  map#(F, cons(X, Y)) >? map#(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 = \y0y1.3 + y1 
  false = 3 
  filter = \G0y1.y1 + 3y1G0(y1) 
  filter2 = \y0G1y2y3.3 + y3 + 2G1(y3) + 3y3G1(y3) 
  map = \G0y1.2y1 + 2G0(0) + 2G0(y1) + 3y1G0(y1) 
  map# = \G0y1.y1 
  nil = 0 
  true = 3 

Using this interpretation, the requirements translate to:

  [[map#(_F0, cons(_x1, _x2))]] = 3 + x2 > x2 = [[map#(_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)]] = 4F0(0) >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 6 + 2x2 + 2F0(0) + 3x2F0(3 + x2) + 11F0(3 + x2) >= 3 + 2x2 + 2F0(0) + 2F0(x2) + 3x2F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[filter(_F0, nil)]] = 0 >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 3 + x2 + 3x2F0(3 + x2) + 9F0(3 + x2) >= 3 + x2 + 2F0(x2) + 3x2F0(x2) = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
  [[filter2(true, _F0, _x1, _x2)]] = 3 + x2 + 2F0(x2) + 3x2F0(x2) >= 3 + x2 + 3x2F0(x2) = [[cons(_x1, filter(_F0, _x2))]] 
  [[filter2(false, _F0, _x1, _x2)]] = 3 + x2 + 2F0(x2) + 3x2F0(x2) >= x2 + 3x2F0(x2) = [[filter(_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 [Kop12, Thm. 7.16].  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)) 
  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) 

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) 
  [[@_{o -> o}(x_1, x_2)]] = @_{o -> o}(x_2, x_1) 
  [[filter(x_1, x_2)]] = filter(x_2, x_1) 
  [[filter2(x_1, x_2, x_3, x_4)]] = filter2(x_4, x_2, x_3, x_1) 
  [[nil]] = _|_ 

We choose Lex = {!faccolon, @_{o -> o}, filter, filter2} and Mul = {!faccolon#, C, cons, false, map, true}, and the following precedence: C > false > map > @_{o -> o} = filter = filter2 > cons > !faccolon# > true > !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)) 
  map(F, _|_) >= _|_ 
  map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y)) 
  filter(F, _|_) >= _|_ 
  filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

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) 

  45] map(F, _|_) >= _|_  by (Bot) 

  46] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [47], by (Star) 
  47] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [48] and [55], by (Copy) 
  48] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [49] and [51], by (Copy) 
  49] map*(F, cons(X, Y)) >= F  because [50], by (Select) 
  50] F >= F  by (Meta) 
  51] map*(F, cons(X, Y)) >= X  because [52], by (Select) 
  52] cons(X, Y) >= X  because [53], by (Star) 
  53] cons*(X, Y) >= X  because [54], by (Select) 
  54] X >= X  by (Meta) 
  55] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [56] and [57], by (Stat) 
  56] F >= F  by (Meta) 
  57] cons(X, Y) > Y  because [58], by definition 
  58] cons*(X, Y) >= Y  because [59], by (Select) 
  59] Y >= Y  by (Meta) 

  60] filter(F, _|_) >= _|_  by (Bot) 

  61] filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because [62], by (Star) 
  62] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [63], [66], [70], [72] and [74], by (Stat) 
  63] cons(X, Y) > Y  because [64], by definition 
  64] cons*(X, Y) >= Y  because [65], by (Select) 
  65] Y >= Y  by (Meta) 
  66] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter = @_{o -> o}, [67], [70] and [72], by (Stat) 
  67] cons(X, Y) > X  because [68], by definition 
  68] cons*(X, Y) >= X  because [69], by (Select) 
  69] X >= X  by (Meta) 
  70] filter*(F, cons(X, Y)) >= F  because [71], by (Select) 
  71] F >= F  by (Meta) 
  72] filter*(F, cons(X, Y)) >= X  because [73], by (Select) 
  73] cons(X, Y) >= X  because [68], by (Star) 
  74] filter*(F, cons(X, Y)) >= Y  because [75], by (Select) 
  75] cons(X, Y) >= Y  because [64], by (Star) 

  76] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [77], by (Star) 
  77] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [78] and [80], by (Copy) 
  78] filter2*(true, F, X, Y) >= X  because [79], by (Select) 
  79] X >= X  by (Meta) 
  80] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [81], [82], [83] and [84], by (Stat) 
  81] F >= F  by (Meta) 
  82] Y >= Y  by (Meta) 
  83] filter2*(true, F, X, Y) >= F  because [81], by (Select) 
  84] filter2*(true, F, X, Y) >= Y  because [82], by (Select) 

  85] filter2(false, F, X, Y) >= filter(F, Y)  because [86], by (Star) 
  86] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [87], [88], [89] and [90], by (Stat) 
  87] F >= F  by (Meta) 
  88] Y >= Y  by (Meta) 
  89] filter2*(false, F, X, Y) >= F  because [87], by (Select) 
  90] filter2*(false, F, X, Y) >= Y  because [88], 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_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 [Kop12, Thm. 7.16].  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)) 
  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) 

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:

  [[cons(x_1, x_2)]] = _|_ 
  [[filter(x_1, x_2)]] = _|_ 
  [[filter2(x_1, x_2, x_3, x_4)]] = _|_ 
  [[map(x_1, x_2)]] = map 
  [[nil]] = _|_ 

We choose Lex = {!faccolon, !faccolon#} and Mul = {@_{o -> o}, C, false, map, true}, and the following precedence: @_{o -> o} > !faccolon# > !faccolon > C > false > map > true

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), 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)) 
  map >= _|_ 
  map >= _|_ 
  _|_ >= _|_ 
  _|_ >= _|_ 
  _|_ >= _|_ 
  _|_ >= _|_ 

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) > !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [2], by definition 
  2] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [3], [14] and [19], by (Stat) 
  3] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [4], by definition 
  4] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [5], [10] and [12], by (Stat) 
  5] !faccolon(!faccolon(C, X), Y) > X  because [6], by definition 
  6] !faccolon*(!faccolon(C, X), Y) >= X  because [7], by (Select) 
  7] !faccolon(C, X) >= X  because [8], by (Star) 
  8] !faccolon*(C, X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [11], by (Select) 
  11] !faccolon(!faccolon(C, X), Y) >= X  because [6], by (Star) 
  12] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [13], by (Select) 
  13] Z >= Z  by (Meta) 
  14] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because !faccolon# > !faccolon, [15] and [17], by (Copy) 
  15] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= X  because [16], by (Select) 
  16] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= X  because [10], by (Star) 
  17] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= Z  because [18], by (Select) 
  18] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [12], by (Star) 
  19] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because !faccolon# > !faccolon, [20] and [26], by (Copy) 
  20] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [21], by (Select) 
  21] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [22] and [25], by (Fun) 
  22] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [23] and [24], by (Fun) 
  23] !faccolon(C, X) >= X  because [8], by (Star) 
  24] Y >= Y  by (Meta) 
  25] Z >= Z  by (Meta) 
  26] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [27], by (Select) 
  27] U >= U  by (Meta) 

  28] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Z)  because [29], by (Star) 
  29] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Z)  because [30], [15] and [17], by (Stat) 
  30] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > X  because [10], by definition 

  31] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(!faccolon(X, Y), Z)  because [32], by (Star) 
  32] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(!faccolon(X, Y), Z)  because [33], [35] and [17], by (Stat) 
  33] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Y)  because [34], by definition 
  34] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [22], by (Select) 
  35] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Y)  because !faccolon# > !faccolon, [15] and [36], by (Copy) 
  36] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= Y  because [37], by (Select) 
  37] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= Y  because [38], by (Star) 
  38] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Y  because [39], by (Select) 
  39] !faccolon(!faccolon(C, X), Y) >= Y  because [40], by (Star) 
  40] !faccolon*(!faccolon(C, X), Y) >= Y  because [24], by (Select) 

  41] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Y)  because [42], by (Star) 
  42] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Y)  because [30], [15] and [36], by (Stat) 

  43] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [44], by (Star) 
  44] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [3], [45] and [48], by (Stat) 
  45] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [46], by (Select) 
  46] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [47] and [25], by (Fun) 
  47] !faccolon(!faccolon(C, X), Y) >= X  because [6], by (Star) 
  48] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [49], [56] and [57], by (Stat) 
  49] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [50], by definition 
  50] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [51], [34] and [12], by (Stat) 
  51] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [52], by definition 
  52] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [53], [6] and [55], by (Stat) 
  53] !faccolon(C, X) > X  because [54], by definition 
  54] !faccolon*(C, X) >= X  because [9], by (Select) 
  55] !faccolon*(!faccolon(C, X), Y) >= Y  because [24], by (Select) 
  56] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [21], by (Select) 
  57] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [27], by (Select) 

  58] map >= _|_  by (Bot) 

  59] map >= _|_  by (Bot) 

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

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

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

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

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#(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_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 [Kop12, Thm. 7.16].  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#(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)) 
  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) 

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:

  [[@_{o -> o}(x_1, x_2)]] = @_{o -> o}(x_2, x_1) 
  [[filter(x_1, x_2)]] = filter(x_2, x_1) 
  [[filter2(x_1, x_2, x_3, x_4)]] = filter2(x_4, x_2, x_3, x_1) 
  [[nil]] = _|_ 

We choose Lex = {!faccolon, @_{o -> o}, filter, filter2} and Mul = {!faccolon#, C, cons, false, map, true}, and the following precedence: !faccolon > !faccolon# > C > false > map > @_{o -> o} = filter = filter2 > cons > true

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), 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)) 
  map(F, _|_) >= _|_ 
  map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y)) 
  filter(F, _|_) >= _|_ 
  filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

With these choices, we have:

  1] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Z)  because [2], by (Star) 
  2] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Z)  because [3], by (Select) 
  3] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X, Z)  because [4], by (Star) 
  4] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X, Z)  because !faccolon > !faccolon#, [5] and [11], by (Copy) 
  5] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [6], by (Select) 
  6] !faccolon(!faccolon(C, X), Y) >= X  because [7], by (Star) 
  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) >= Z  because [12], by (Select) 
  12] Z >= Z  by (Meta) 

  13] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) > !faccolon#(!faccolon(X, Y), Z)  because [14], by definition 
  14] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(!faccolon(X, Y), Z)  because !faccolon# in Mul, [15] and [20], by (Stat) 
  15] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Y)  because [16], by definition 
  16] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [17], by (Select) 
  17] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [18] and [19], by (Fun) 
  18] !faccolon(C, X) >= X  because [9], by (Star) 
  19] Y >= Y  by (Meta) 
  20] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > Z  because [11], by definition 

  21] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Y)  because [22], by (Star) 
  22] !faccolon#*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon#(X, Y)  because [23], by (Select) 
  23] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X, Y)  because [24], by (Star) 
  24] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon#(X, Y)  because !faccolon > !faccolon#, [5] and [25], by (Copy) 
  25] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Y  because [26], by (Select) 
  26] !faccolon(!faccolon(C, X), Y) >= Y  because [27], by (Star) 
  27] !faccolon*(!faccolon(C, X), Y) >= Y  because [19], by (Select) 

  28] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [29], by (Star) 
  29] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [30], [34] and [38], by (Stat) 
  30] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [31], by definition 
  31] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [32], [5] and [11], by (Stat) 
  32] !faccolon(!faccolon(C, X), Y) > X  because [33], by definition 
  33] !faccolon*(!faccolon(C, X), Y) >= X  because [18], by (Select) 
  34] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [35], by (Select) 
  35] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [36] and [37], by (Fun) 
  36] !faccolon(!faccolon(C, X), Y) >= X  because [33], by (Star) 
  37] Z >= Z  by (Meta) 
  38] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [39], [46] and [49], by (Stat) 
  39] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [40], by definition 
  40] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [41], [16] and [11], by (Stat) 
  41] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [42], by definition 
  42] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [43], [33] and [45], by (Stat) 
  43] !faccolon(C, X) > X  because [44], by definition 
  44] !faccolon*(C, X) >= X  because [10], by (Select) 
  45] !faccolon*(!faccolon(C, X), Y) >= Y  because [19], by (Select) 
  46] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [47], by (Select) 
  47] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [48] and [37], by (Fun) 
  48] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [18] and [19], by (Fun) 
  49] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [50], by (Select) 
  50] U >= U  by (Meta) 

  51] map(F, _|_) >= _|_  by (Bot) 

  52] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [53], by (Star) 
  53] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [54] and [61], by (Copy) 
  54] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [55] and [57], by (Copy) 
  55] map*(F, cons(X, Y)) >= F  because [56], by (Select) 
  56] F >= F  by (Meta) 
  57] map*(F, cons(X, Y)) >= X  because [58], by (Select) 
  58] cons(X, Y) >= X  because [59], by (Star) 
  59] cons*(X, Y) >= X  because [60], by (Select) 
  60] X >= X  by (Meta) 
  61] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [62] and [63], by (Stat) 
  62] F >= F  by (Meta) 
  63] cons(X, Y) > Y  because [64], by definition 
  64] cons*(X, Y) >= Y  because [65], by (Select) 
  65] Y >= Y  by (Meta) 

  66] filter(F, _|_) >= _|_  by (Bot) 

  67] filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because [68], by (Star) 
  68] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [69], [72], [76], [78] and [80], by (Stat) 
  69] cons(X, Y) > Y  because [70], by definition 
  70] cons*(X, Y) >= Y  because [71], by (Select) 
  71] Y >= Y  by (Meta) 
  72] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter = @_{o -> o}, [73], [76] and [78], by (Stat) 
  73] cons(X, Y) > X  because [74], by definition 
  74] cons*(X, Y) >= X  because [75], by (Select) 
  75] X >= X  by (Meta) 
  76] filter*(F, cons(X, Y)) >= F  because [77], by (Select) 
  77] F >= F  by (Meta) 
  78] filter*(F, cons(X, Y)) >= X  because [79], by (Select) 
  79] cons(X, Y) >= X  because [74], by (Star) 
  80] filter*(F, cons(X, Y)) >= Y  because [81], by (Select) 
  81] cons(X, Y) >= Y  because [70], by (Star) 

  82] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [83], by (Star) 
  83] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [84] and [86], by (Copy) 
  84] filter2*(true, F, X, Y) >= X  because [85], by (Select) 
  85] X >= X  by (Meta) 
  86] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [87], [88], [89] and [90], by (Stat) 
  87] F >= F  by (Meta) 
  88] Y >= Y  by (Meta) 
  89] filter2*(true, F, X, Y) >= F  because [87], by (Select) 
  90] filter2*(true, F, X, Y) >= Y  because [88], by (Select) 

  91] filter2(false, F, X, Y) >= filter(F, Y)  because [92], by (Star) 
  92] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [93], [94], [95] and [96], by (Stat) 
  93] F >= F  by (Meta) 
  94] Y >= Y  by (Meta) 
  95] filter2*(false, F, X, Y) >= F  because [93], by (Select) 
  96] filter2*(false, F, X, Y) >= Y  because [94], by (Select) 

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#(X, Z)   
  !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 [Kop12, Thm. 7.16].  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#(X, 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)) 
  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) 

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) 
  [[cons(x_1, x_2)]] = _|_ 
  [[filter(x_1, x_2)]] = _|_ 
  [[filter2(x_1, x_2, x_3, x_4)]] = _|_ 
  [[map(x_1, x_2)]] = map 
  [[nil]] = _|_ 

We choose Lex = {!faccolon} and Mul = {!faccolon#, @_{o -> o}, C, false, map, true}, and the following precedence: !faccolon > !faccolon# > @_{o -> o} > C > false > map > true

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#(X) 
  !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)) 
  map >= _|_ 
  map >= _|_ 
  _|_ >= _|_ 
  _|_ >= _|_ 
  _|_ >= _|_ 
  _|_ >= _|_ 

With these choices, we have:

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

  10] !faccolon#(!faccolon(!faccolon(!faccolon(C, X), Y), Z)) >= !faccolon#(X)  because !faccolon# in Mul and [11], by (Fun) 
  11] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= X  because [4], by (Star) 

  12] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [13], by (Star) 
  13] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [14], [20] and [24], by (Stat) 
  14] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [15], by definition 
  15] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [16], [4] and [18], by (Stat) 
  16] !faccolon(!faccolon(C, X), Y) > X  because [17], by definition 
  17] !faccolon*(!faccolon(C, X), Y) >= X  because [7], by (Select) 
  18] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [19], by (Select) 
  19] Z >= Z  by (Meta) 
  20] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [21], by (Select) 
  21] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [22] and [23], by (Fun) 
  22] !faccolon(!faccolon(C, X), Y) >= X  because [17], by (Star) 
  23] Z >= Z  by (Meta) 
  24] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [25], [37] and [40], 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 [18], 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], [17] and [31], by (Stat) 
  29] !faccolon(C, X) > X  because [30], by definition 
  30] !faccolon*(C, X) >= X  because [9], 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], by (Select) 
  38] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [39] and [23], by (Fun) 
  39] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [35] and [36], by (Fun) 
  40] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [41], by (Select) 
  41] U >= U  by (Meta) 

  42] map >= _|_  by (Bot) 

  43] map >= _|_  by (Bot) 

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

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

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

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

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#(X, Y)   

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 [Kop12, Thm. 7.16].  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#(X, 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) 

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)]] = x_1 
  [[filter(x_1, x_2)]] = filter(x_2, x_1) 
  [[filter2(x_1, x_2, x_3, x_4)]] = filter2(x_4, x_2, x_1, x_3) 
  [[nil]] = _|_ 

We choose Lex = {!faccolon, filter, filter2} and Mul = {@_{o -> o}, C, cons, false, map, true}, and the following precedence: !faccolon > C > false > filter = filter2 > map > @_{o -> o} > cons > true

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

  !faccolon(!faccolon(!faccolon(C, X), Y), Z) > X 
  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  map(F, _|_) >= _|_ 
  map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y)) 
  filter(F, _|_) >= _|_ 
  filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y) 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
  filter2(false, F, X, Y) >= filter(F, Y) 

With these choices, we have:

  1] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > X  because [2], by definition 
  2] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [3], by (Select) 
  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] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [9], by (Star) 
  9] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [10], [16] and [20], by (Stat) 
  10] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [11], by definition 
  11] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [12], [2] and [14], by (Stat) 
  12] !faccolon(!faccolon(C, X), Y) > X  because [13], by definition 
  13] !faccolon*(!faccolon(C, X), Y) >= X  because [5], by (Select) 
  14] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [15], by (Select) 
  15] Z >= Z  by (Meta) 
  16] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [17], by (Select) 
  17] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [18] and [19], by (Fun) 
  18] !faccolon(!faccolon(C, X), Y) >= X  because [13], by (Star) 
  19] Z >= Z  by (Meta) 
  20] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [21], [33] and [36], by (Stat) 
  21] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [22], by definition 
  22] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [23], [29] and [14], by (Stat) 
  23] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [24], by definition 
  24] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [25], [13] and [27], by (Stat) 
  25] !faccolon(C, X) > X  because [26], by definition 
  26] !faccolon*(C, X) >= X  because [7], by (Select) 
  27] !faccolon*(!faccolon(C, X), Y) >= Y  because [28], by (Select) 
  28] Y >= Y  by (Meta) 
  29] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [30], by (Select) 
  30] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [31] and [32], by (Fun) 
  31] !faccolon(C, X) >= X  because [26], by (Star) 
  32] Y >= Y  by (Meta) 
  33] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [34], by (Select) 
  34] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [35] and [19], by (Fun) 
  35] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [31] and [32], by (Fun) 
  36] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [37], by (Select) 
  37] U >= U  by (Meta) 

  38] map(F, _|_) >= _|_  by (Bot) 

  39] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [40], by (Star) 
  40] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [41] and [48], by (Copy) 
  41] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [42] and [44], by (Copy) 
  42] map*(F, cons(X, Y)) >= F  because [43], by (Select) 
  43] F >= F  by (Meta) 
  44] map*(F, cons(X, Y)) >= X  because [45], by (Select) 
  45] cons(X, Y) >= X  because [46], by (Star) 
  46] cons*(X, Y) >= X  because [47], by (Select) 
  47] X >= X  by (Meta) 
  48] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [49] and [50], by (Stat) 
  49] F >= F  by (Meta) 
  50] cons(X, Y) > Y  because [51], by definition 
  51] cons*(X, Y) >= Y  because [52], by (Select) 
  52] Y >= Y  by (Meta) 

  53] filter(F, _|_) >= _|_  by (Bot) 

  54] filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because [55], by (Star) 
  55] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [56], [59], [60], [62] and [66], by (Stat) 
  56] cons(X, Y) > Y  because [57], by definition 
  57] cons*(X, Y) >= Y  because [58], by (Select) 
  58] Y >= Y  by (Meta) 
  59] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [60] and [62], by (Copy) 
  60] filter*(F, cons(X, Y)) >= F  because [61], by (Select) 
  61] F >= F  by (Meta) 
  62] filter*(F, cons(X, Y)) >= X  because [63], by (Select) 
  63] cons(X, Y) >= X  because [64], by (Star) 
  64] cons*(X, Y) >= X  because [65], by (Select) 
  65] X >= X  by (Meta) 
  66] filter*(F, cons(X, Y)) >= Y  because [67], by (Select) 
  67] cons(X, Y) >= Y  because [57], by (Star) 

  68] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [69], by (Star) 
  69] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [70] and [72], by (Copy) 
  70] filter2*(true, F, X, Y) >= X  because [71], by (Select) 
  71] X >= X  by (Meta) 
  72] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [73], [74], [75] and [76], by (Stat) 
  73] F >= F  by (Meta) 
  74] Y >= Y  by (Meta) 
  75] filter2*(true, F, X, Y) >= F  because [73], by (Select) 
  76] filter2*(true, F, X, Y) >= Y  because [74], by (Select) 

  77] filter2(false, F, X, Y) >= filter(F, Y)  because [78], by (Star) 
  78] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [79], [80], [81] and [82], by (Stat) 
  79] F >= F  by (Meta) 
  80] Y >= Y  by (Meta) 
  81] filter2*(false, F, X, Y) >= F  because [79], by (Select) 
  82] filter2*(false, F, X, Y) >= Y  because [80], by (Select) 

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.