We consider the system Applicative_first_order_05__11.

  Alphabet:

    !facminus : [a * a] --> a 
    !facplus : [a * a] --> a 
    !factimes : [a * a] --> a 
    0 : [] --> a 
    1 : [] --> a 
    2 : [] --> a 
    D : [a] --> a 
    cons : [c * d] --> d 
    constant : [] --> a 
    div : [a * a] --> a 
    false : [] --> b 
    filter : [c -> b * d] --> d 
    filter2 : [b * c -> b * c * d] --> d 
    ln : [a] --> a 
    map : [c -> c * d] --> d 
    minus : [a] --> a 
    nil : [] --> d 
    pow : [a * a] --> a 
    t : [] --> a 
    true : [] --> b 

  Rules:

    D(t) => 1 
    D(constant) => 0 
    D(!facplus(x, y)) => !facplus(D(x), D(y)) 
    D(!factimes(x, y)) => !facplus(!factimes(y, D(x)), !factimes(x, D(y))) 
    D(!facminus(x, y)) => !facminus(D(x), D(y)) 
    D(minus(x)) => minus(D(x)) 
    D(div(x, y)) => !facminus(div(D(x), y), div(!factimes(x, D(y)), pow(y, 2))) 
    D(ln(x)) => div(D(x), x) 
    D(pow(x, y)) => !facplus(!factimes(!factimes(y, pow(x, !facminus(y, 1))), D(x)), !factimes(!factimes(pow(x, y), ln(x)), D(y))) 
    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 rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  D(t) >? 1 
  D(constant) >? 0 
  D(!facplus(X, Y)) >? !facplus(D(X), D(Y)) 
  D(!factimes(X, Y)) >? !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) 
  D(!facminus(X, Y)) >? !facminus(D(X), D(Y)) 
  D(minus(X)) >? minus(D(X)) 
  D(div(X, Y)) >? !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, 2))) 
  D(ln(X)) >? div(D(X), X) 
  D(pow(X, Y)) >? !facplus(!factimes(!factimes(Y, pow(X, !facminus(Y, 1))), D(X)), !factimes(!factimes(pow(X, Y), ln(X)), D(Y))) 
  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 use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[1]] = _|_ 
  [[2]] = _|_ 
  [[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 = {filter, filter2} and Mul = {!facminus, !facplus, !factimes, @_{o -> o}, D, cons, constant, div, false, ln, map, minus, pow, t, true}, and the following precedence: true > false > constant > t > D = ln = minus > !facminus > filter = filter2 > div > !facplus > pow > map > @_{o -> o} > cons > !factimes

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

  D(t) > _|_ 
  D(constant) > _|_ 
  D(!facplus(X, Y)) >= !facplus(D(X), D(Y)) 
  D(!factimes(X, Y)) > !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) 
  D(!facminus(X, Y)) > !facminus(D(X), D(Y)) 
  D(minus(X)) >= minus(D(X)) 
  D(div(X, Y)) >= !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, _|_))) 
  D(ln(X)) >= div(D(X), X) 
  D(pow(X, Y)) > !facplus(!factimes(!factimes(Y, pow(X, !facminus(Y, _|_))), D(X)), !factimes(!factimes(pow(X, Y), ln(X)), D(Y))) 
  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] D(t) > _|_  because [2], by definition 
  2] D*(t) >= _|_  by (Bot) 

  3] D(constant) > _|_  because [4], by definition 
  4] D*(constant) >= _|_  by (Bot) 

  5] D(!facplus(X, Y)) >= !facplus(D(X), D(Y))  because [6], by (Star) 
  6] D*(!facplus(X, Y)) >= !facplus(D(X), D(Y))  because D > !facplus, [7] and [11], by (Copy) 
  7] D*(!facplus(X, Y)) >= D(X)  because D in Mul and [8], by (Stat) 
  8] !facplus(X, Y) > X  because [9], by definition 
  9] !facplus*(X, Y) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] D*(!facplus(X, Y)) >= D(Y)  because D in Mul and [12], by (Stat) 
  12] !facplus(X, Y) > Y  because [13], by definition 
  13] !facplus*(X, Y) >= Y  because [14], by (Select) 
  14] Y >= Y  by (Meta) 

  15] D(!factimes(X, Y)) > !facplus(!factimes(Y, D(X)), !factimes(X, D(Y)))  because [16], by definition 
  16] D*(!factimes(X, Y)) >= !facplus(!factimes(Y, D(X)), !factimes(X, D(Y)))  because D > !facplus, [17] and [26], by (Copy) 
  17] D*(!factimes(X, Y)) >= !factimes(Y, D(X))  because D > !factimes, [18] and [22], by (Copy) 
  18] D*(!factimes(X, Y)) >= Y  because [19], by (Select) 
  19] !factimes(X, Y) >= Y  because [20], by (Star) 
  20] !factimes*(X, Y) >= Y  because [21], by (Select) 
  21] Y >= Y  by (Meta) 
  22] D*(!factimes(X, Y)) >= D(X)  because D in Mul and [23], by (Stat) 
  23] !factimes(X, Y) > X  because [24], by definition 
  24] !factimes*(X, Y) >= X  because [25], by (Select) 
  25] X >= X  by (Meta) 
  26] D*(!factimes(X, Y)) >= !factimes(X, D(Y))  because D > !factimes, [27] and [29], by (Copy) 
  27] D*(!factimes(X, Y)) >= X  because [28], by (Select) 
  28] !factimes(X, Y) >= X  because [24], by (Star) 
  29] D*(!factimes(X, Y)) >= D(Y)  because D in Mul and [30], by (Stat) 
  30] !factimes(X, Y) > Y  because [31], by definition 
  31] !factimes*(X, Y) >= Y  because [21], by (Select) 

  32] D(!facminus(X, Y)) > !facminus(D(X), D(Y))  because [33], by definition 
  33] D*(!facminus(X, Y)) >= !facminus(D(X), D(Y))  because D > !facminus, [34] and [38], by (Copy) 
  34] D*(!facminus(X, Y)) >= D(X)  because D in Mul and [35], by (Stat) 
  35] !facminus(X, Y) > X  because [36], by definition 
  36] !facminus*(X, Y) >= X  because [37], by (Select) 
  37] X >= X  by (Meta) 
  38] D*(!facminus(X, Y)) >= D(Y)  because D in Mul and [39], by (Stat) 
  39] !facminus(X, Y) > Y  because [40], by definition 
  40] !facminus*(X, Y) >= Y  because [41], by (Select) 
  41] Y >= Y  by (Meta) 

  42] D(minus(X)) >= minus(D(X))  because D = minus, D in Mul and [43], by (Fun) 
  43] minus(X) >= D(X)  because minus = D, minus in Mul and [44], by (Fun) 
  44] X >= X  by (Meta) 

  45] D(div(X, Y)) >= !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, _|_)))  because [46], by (Star) 
  46] D*(div(X, Y)) >= !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, _|_)))  because D > !facminus, [47] and [56], by (Copy) 
  47] D*(div(X, Y)) >= div(D(X), Y)  because D > div, [48] and [52], by (Copy) 
  48] D*(div(X, Y)) >= D(X)  because D in Mul and [49], by (Stat) 
  49] div(X, Y) > X  because [50], by definition 
  50] div*(X, Y) >= X  because [51], by (Select) 
  51] X >= X  by (Meta) 
  52] D*(div(X, Y)) >= Y  because [53], by (Select) 
  53] div(X, Y) >= Y  because [54], by (Star) 
  54] div*(X, Y) >= Y  because [55], by (Select) 
  55] Y >= Y  by (Meta) 
  56] D*(div(X, Y)) >= div(!factimes(X, D(Y)), pow(Y, _|_))  because D > div, [57] and [63], by (Copy) 
  57] D*(div(X, Y)) >= !factimes(X, D(Y))  because D > !factimes, [58] and [60], by (Copy) 
  58] D*(div(X, Y)) >= X  because [59], by (Select) 
  59] div(X, Y) >= X  because [50], by (Star) 
  60] D*(div(X, Y)) >= D(Y)  because D in Mul and [61], by (Stat) 
  61] div(X, Y) > Y  because [62], by definition 
  62] div*(X, Y) >= Y  because [55], by (Select) 
  63] D*(div(X, Y)) >= pow(Y, _|_)  because D > pow, [52] and [64], by (Copy) 
  64] D*(div(X, Y)) >= _|_  by (Bot) 

  65] D(ln(X)) >= div(D(X), X)  because [66], by (Star) 
  66] D*(ln(X)) >= div(D(X), X)  because D > div, [67] and [71], by (Copy) 
  67] D*(ln(X)) >= D(X)  because D in Mul and [68], by (Stat) 
  68] ln(X) > X  because [69], by definition 
  69] ln*(X) >= X  because [70], by (Select) 
  70] X >= X  by (Meta) 
  71] D*(ln(X)) >= X  because [72], by (Select) 
  72] ln(X) >= X  because [69], by (Star) 

  73] D(pow(X, Y)) > !facplus(!factimes(!factimes(Y, pow(X, !facminus(Y, _|_))), D(X)), !factimes(!factimes(pow(X, Y), ln(X)), D(Y)))  because [74], by definition 
  74] D*(pow(X, Y)) >= !facplus(!factimes(!factimes(Y, pow(X, !facminus(Y, _|_))), D(X)), !factimes(!factimes(pow(X, Y), ln(X)), D(Y)))  because D > !facplus, [75] and [91], by (Copy) 
  75] D*(pow(X, Y)) >= !factimes(!factimes(Y, pow(X, !facminus(Y, _|_))), D(X))  because D > !factimes, [76] and [88], by (Copy) 
  76] D*(pow(X, Y)) >= !factimes(Y, pow(X, !facminus(Y, _|_)))  because D > !factimes, [77] and [81], by (Copy) 
  77] D*(pow(X, Y)) >= Y  because [78], by (Select) 
  78] pow(X, Y) >= Y  because [79], by (Star) 
  79] pow*(X, Y) >= Y  because [80], by (Select) 
  80] Y >= Y  by (Meta) 
  81] D*(pow(X, Y)) >= pow(X, !facminus(Y, _|_))  because D > pow, [82] and [86], by (Copy) 
  82] D*(pow(X, Y)) >= X  because [83], by (Select) 
  83] pow(X, Y) >= X  because [84], by (Star) 
  84] pow*(X, Y) >= X  because [85], by (Select) 
  85] X >= X  by (Meta) 
  86] D*(pow(X, Y)) >= !facminus(Y, _|_)  because D > !facminus, [77] and [87], by (Copy) 
  87] D*(pow(X, Y)) >= _|_  by (Bot) 
  88] D*(pow(X, Y)) >= D(X)  because D in Mul and [89], by (Stat) 
  89] pow(X, Y) > X  because [90], by definition 
  90] pow*(X, Y) >= X  because [85], by (Select) 
  91] D*(pow(X, Y)) >= !factimes(!factimes(pow(X, Y), ln(X)), D(Y))  because D > !factimes, [92] and [95], by (Copy) 
  92] D*(pow(X, Y)) >= !factimes(pow(X, Y), ln(X))  because D > !factimes, [93] and [94], by (Copy) 
  93] D*(pow(X, Y)) >= pow(X, Y)  because D > pow, [82] and [77], by (Copy) 
  94] D*(pow(X, Y)) >= ln(X)  because D = ln, D in Mul and [89], by (Stat) 
  95] D*(pow(X, Y)) >= D(Y)  because D in Mul and [96], by (Stat) 
  96] pow(X, Y) > Y  because [97], by definition 
  97] pow*(X, Y) >= Y  because [80], by (Select) 

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

  99] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [100], by (Star) 
  100] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [101] and [108], by (Copy) 
  101] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [102] and [104], by (Copy) 
  102] map*(F, cons(X, Y)) >= F  because [103], by (Select) 
  103] F >= F  by (Meta) 
  104] map*(F, cons(X, Y)) >= X  because [105], by (Select) 
  105] cons(X, Y) >= X  because [106], by (Star) 
  106] cons*(X, Y) >= X  because [107], by (Select) 
  107] X >= X  by (Meta) 
  108] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [109] and [110], by (Stat) 
  109] F >= F  by (Meta) 
  110] cons(X, Y) > Y  because [111], by definition 
  111] cons*(X, Y) >= Y  because [112], by (Select) 
  112] Y >= Y  by (Meta) 

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

  114] filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because [115], by (Star) 
  115] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [116], [119], [120], [122] and [126], by (Stat) 
  116] cons(X, Y) > Y  because [117], by definition 
  117] cons*(X, Y) >= Y  because [118], by (Select) 
  118] Y >= Y  by (Meta) 
  119] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [120] and [122], by (Copy) 
  120] filter*(F, cons(X, Y)) >= F  because [121], by (Select) 
  121] F >= F  by (Meta) 
  122] filter*(F, cons(X, Y)) >= X  because [123], by (Select) 
  123] cons(X, Y) >= X  because [124], by (Star) 
  124] cons*(X, Y) >= X  because [125], by (Select) 
  125] X >= X  by (Meta) 
  126] filter*(F, cons(X, Y)) >= Y  because [127], by (Select) 
  127] cons(X, Y) >= Y  because [117], by (Star) 

  128] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [129], by (Star) 
  129] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [130] and [132], by (Copy) 
  130] filter2*(true, F, X, Y) >= X  because [131], by (Select) 
  131] X >= X  by (Meta) 
  132] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [133], [134], [135] and [136], by (Stat) 
  133] F >= F  by (Meta) 
  134] Y >= Y  by (Meta) 
  135] filter2*(true, F, X, Y) >= F  because [133], by (Select) 
  136] filter2*(true, F, X, Y) >= Y  because [134], by (Select) 

  137] filter2(false, F, X, Y) >= filter(F, Y)  because [138], by (Star) 
  138] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [139], [140], [141] and [142], by (Stat) 
  139] F >= F  by (Meta) 
  140] Y >= Y  by (Meta) 
  141] filter2*(false, F, X, Y) >= F  because [139], by (Select) 
  142] filter2*(false, F, X, Y) >= Y  because [140], by (Select) 

We can thus remove the following rules:

  D(t) => 1 
  D(constant) => 0 
  D(!factimes(X, Y)) => !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) 
  D(!facminus(X, Y)) => !facminus(D(X), D(Y)) 
  D(pow(X, Y)) => !facplus(!factimes(!factimes(Y, pow(X, !facminus(Y, 1))), D(X)), !factimes(!factimes(pow(X, Y), ln(X)), D(Y))) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  D(!facplus(X, Y)) >? !facplus(D(X), D(Y)) 
  D(minus(X)) >? minus(D(X)) 
  D(div(X, Y)) >? !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, 2))) 
  D(ln(X)) >? div(D(X), X) 
  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 use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[2]] = _|_ 
  [[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 = {filter, filter2} and Mul = {!facminus, !facplus, !factimes, @_{o -> o}, D, cons, div, false, ln, map, minus, pow, true}, and the following precedence: ln > D > !facplus > false > pow > minus > !factimes > div > filter = filter2 > true > map > @_{o -> o} > cons > !facminus

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

  D(!facplus(X, Y)) > !facplus(D(X), D(Y)) 
  D(minus(X)) > minus(D(X)) 
  D(div(X, Y)) >= !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, _|_))) 
  D(ln(X)) > div(D(X), X) 
  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] D(!facplus(X, Y)) > !facplus(D(X), D(Y))  because [2], by definition 
  2] D*(!facplus(X, Y)) >= !facplus(D(X), D(Y))  because D > !facplus, [3] and [7], by (Copy) 
  3] D*(!facplus(X, Y)) >= D(X)  because D in Mul and [4], by (Stat) 
  4] !facplus(X, Y) > X  because [5], by definition 
  5] !facplus*(X, Y) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] D*(!facplus(X, Y)) >= D(Y)  because D in Mul and [8], by (Stat) 
  8] !facplus(X, Y) > Y  because [9], by definition 
  9] !facplus*(X, Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 

  11] D(minus(X)) > minus(D(X))  because [12], by definition 
  12] D*(minus(X)) >= minus(D(X))  because D > minus and [13], by (Copy) 
  13] D*(minus(X)) >= D(X)  because D in Mul and [14], by (Stat) 
  14] minus(X) > X  because [15], by definition 
  15] minus*(X) >= X  because [16], by (Select) 
  16] X >= X  by (Meta) 

  17] D(div(X, Y)) >= !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, _|_)))  because [18], by (Star) 
  18] D*(div(X, Y)) >= !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, _|_)))  because D > !facminus, [19] and [28], by (Copy) 
  19] D*(div(X, Y)) >= div(D(X), Y)  because D > div, [20] and [24], by (Copy) 
  20] D*(div(X, Y)) >= D(X)  because D in Mul and [21], by (Stat) 
  21] div(X, Y) > X  because [22], by definition 
  22] div*(X, Y) >= X  because [23], by (Select) 
  23] X >= X  by (Meta) 
  24] D*(div(X, Y)) >= Y  because [25], by (Select) 
  25] div(X, Y) >= Y  because [26], by (Star) 
  26] div*(X, Y) >= Y  because [27], by (Select) 
  27] Y >= Y  by (Meta) 
  28] D*(div(X, Y)) >= div(!factimes(X, D(Y)), pow(Y, _|_))  because D > div, [29] and [35], by (Copy) 
  29] D*(div(X, Y)) >= !factimes(X, D(Y))  because D > !factimes, [30] and [32], by (Copy) 
  30] D*(div(X, Y)) >= X  because [31], by (Select) 
  31] div(X, Y) >= X  because [22], by (Star) 
  32] D*(div(X, Y)) >= D(Y)  because D in Mul and [33], by (Stat) 
  33] div(X, Y) > Y  because [34], by definition 
  34] div*(X, Y) >= Y  because [27], by (Select) 
  35] D*(div(X, Y)) >= pow(Y, _|_)  because D > pow, [24] and [36], by (Copy) 
  36] D*(div(X, Y)) >= _|_  by (Bot) 

  37] D(ln(X)) > div(D(X), X)  because [38], by definition 
  38] D*(ln(X)) >= div(D(X), X)  because D > div, [39] and [44], by (Copy) 
  39] D*(ln(X)) >= D(X)  because [40], by (Select) 
  40] ln(X) >= D(X)  because [41], by (Star) 
  41] ln*(X) >= D(X)  because ln > D and [42], by (Copy) 
  42] ln*(X) >= X  because [43], by (Select) 
  43] X >= X  by (Meta) 
  44] D*(ln(X)) >= X  because [45], by (Select) 
  45] ln(X) >= X  because [42], by (Star) 

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

  47] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [48], by (Star) 
  48] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [49] and [56], by (Copy) 
  49] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [50] and [52], by (Copy) 
  50] map*(F, cons(X, Y)) >= F  because [51], by (Select) 
  51] F >= F  by (Meta) 
  52] map*(F, cons(X, Y)) >= X  because [53], by (Select) 
  53] cons(X, Y) >= X  because [54], by (Star) 
  54] cons*(X, Y) >= X  because [55], by (Select) 
  55] X >= X  by (Meta) 
  56] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [57] and [58], by (Stat) 
  57] F >= F  by (Meta) 
  58] cons(X, Y) > Y  because [59], by definition 
  59] cons*(X, Y) >= Y  because [60], by (Select) 
  60] Y >= Y  by (Meta) 

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

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

We can thus remove the following rules:

  D(!facplus(X, Y)) => !facplus(D(X), D(Y)) 
  D(minus(X)) => minus(D(X)) 
  D(ln(X)) => div(D(X), X) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  D(div(X, Y)) >? !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, 2))) 
  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 use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[2]] = _|_ 
  [[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 = {filter, filter2} and Mul = {!facminus, !factimes, @_{o -> o}, D, cons, div, false, map, pow, true}, and the following precedence: false > map > D > pow > div > filter = filter2 > cons > !factimes > @_{o -> o} > !facminus > true

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

  D(div(X, Y)) > !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, _|_))) 
  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] D(div(X, Y)) > !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, _|_)))  because [2], by definition 
  2] D*(div(X, Y)) >= !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, _|_)))  because D > !facminus, [3] and [12], by (Copy) 
  3] D*(div(X, Y)) >= div(D(X), Y)  because D > div, [4] and [8], by (Copy) 
  4] D*(div(X, Y)) >= D(X)  because D in Mul and [5], by (Stat) 
  5] div(X, Y) > X  because [6], by definition 
  6] div*(X, Y) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] D*(div(X, Y)) >= Y  because [9], by (Select) 
  9] div(X, Y) >= Y  because [10], by (Star) 
  10] div*(X, Y) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 
  12] D*(div(X, Y)) >= div(!factimes(X, D(Y)), pow(Y, _|_))  because D > div, [13] and [19], by (Copy) 
  13] D*(div(X, Y)) >= !factimes(X, D(Y))  because D > !factimes, [14] and [16], by (Copy) 
  14] D*(div(X, Y)) >= X  because [15], by (Select) 
  15] div(X, Y) >= X  because [6], by (Star) 
  16] D*(div(X, Y)) >= D(Y)  because D in Mul and [17], by (Stat) 
  17] div(X, Y) > Y  because [18], by definition 
  18] div*(X, Y) >= Y  because [11], by (Select) 
  19] D*(div(X, Y)) >= pow(Y, _|_)  because D > pow, [8] and [20], by (Copy) 
  20] D*(div(X, Y)) >= _|_  by (Bot) 

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

  22] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [23], by (Star) 
  23] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [24] and [31], by (Copy) 
  24] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [25] and [27], by (Copy) 
  25] map*(F, cons(X, Y)) >= F  because [26], by (Select) 
  26] F >= F  by (Meta) 
  27] map*(F, cons(X, Y)) >= X  because [28], by (Select) 
  28] cons(X, Y) >= X  because [29], by (Star) 
  29] cons*(X, Y) >= X  because [30], by (Select) 
  30] X >= X  by (Meta) 
  31] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [32] and [33], by (Stat) 
  32] F >= F  by (Meta) 
  33] cons(X, Y) > Y  because [34], by definition 
  34] cons*(X, Y) >= Y  because [35], by (Select) 
  35] Y >= Y  by (Meta) 

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

  37] filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because [38], by (Star) 
  38] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [39], [42], [43], [45] and [49], by (Stat) 
  39] cons(X, Y) > Y  because [40], by definition 
  40] cons*(X, Y) >= Y  because [41], by (Select) 
  41] Y >= Y  by (Meta) 
  42] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [43] and [45], by (Copy) 
  43] filter*(F, cons(X, Y)) >= F  because [44], by (Select) 
  44] F >= F  by (Meta) 
  45] filter*(F, cons(X, Y)) >= X  because [46], by (Select) 
  46] cons(X, Y) >= X  because [47], by (Star) 
  47] cons*(X, Y) >= X  because [48], by (Select) 
  48] X >= X  by (Meta) 
  49] filter*(F, cons(X, Y)) >= Y  because [50], by (Select) 
  50] cons(X, Y) >= Y  because [40], by (Star) 

  51] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [52], by (Star) 
  52] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [53] and [55], by (Copy) 
  53] filter2*(true, F, X, Y) >= X  because [54], by (Select) 
  54] X >= X  by (Meta) 
  55] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [56], [57], [58] and [59], by (Stat) 
  56] F >= F  by (Meta) 
  57] Y >= Y  by (Meta) 
  58] filter2*(true, F, X, Y) >= F  because [56], by (Select) 
  59] filter2*(true, F, X, Y) >= Y  because [57], by (Select) 

  60] filter2(false, F, X, Y) > filter(F, Y)  because [61], by definition 
  61] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [62], [63], [64] and [65], by (Stat) 
  62] F >= F  by (Meta) 
  63] Y >= Y  by (Meta) 
  64] filter2*(false, F, X, Y) >= F  because [62], by (Select) 
  65] filter2*(false, F, X, Y) >= Y  because [63], by (Select) 

We can thus remove the following rules:

  D(div(X, Y)) => !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, 2))) 
  filter2(false, F, X, Y) => filter(F, Y) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  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)) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[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 = {filter, filter2} and Mul = {@_{o -> o}, cons, map, true}, and the following precedence: filter = filter2 > true > map > @_{o -> o} > cons

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

  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)) 

With these choices, we have:

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

  2] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [3], by (Star) 
  3] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [4] and [11], by (Copy) 
  4] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [5] and [7], by (Copy) 
  5] map*(F, cons(X, Y)) >= F  because [6], by (Select) 
  6] F >= F  by (Meta) 
  7] map*(F, cons(X, Y)) >= X  because [8], by (Select) 
  8] cons(X, Y) >= X  because [9], by (Star) 
  9] cons*(X, Y) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [12] and [13], by (Stat) 
  12] F >= F  by (Meta) 
  13] cons(X, Y) > Y  because [14], by definition 
  14] cons*(X, Y) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 

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

  17] filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because [18], by (Star) 
  18] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [19], [22], [23], [25] and [29], by (Stat) 
  19] cons(X, Y) > Y  because [20], by definition 
  20] cons*(X, Y) >= Y  because [21], by (Select) 
  21] Y >= Y  by (Meta) 
  22] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [23] and [25], by (Copy) 
  23] filter*(F, cons(X, Y)) >= F  because [24], by (Select) 
  24] F >= F  by (Meta) 
  25] filter*(F, cons(X, Y)) >= X  because [26], by (Select) 
  26] cons(X, Y) >= X  because [27], by (Star) 
  27] cons*(X, Y) >= X  because [28], by (Select) 
  28] X >= X  by (Meta) 
  29] filter*(F, cons(X, Y)) >= Y  because [30], by (Select) 
  30] cons(X, Y) >= Y  because [20], by (Star) 

  31] filter2(true, F, X, Y) > cons(X, filter(F, Y))  because [32], by definition 
  32] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [33] and [35], by (Copy) 
  33] filter2*(true, F, X, Y) >= X  because [34], by (Select) 
  34] X >= X  by (Meta) 
  35] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [36], [37], [38] and [39], by (Stat) 
  36] F >= F  by (Meta) 
  37] Y >= Y  by (Meta) 
  38] filter2*(true, F, X, Y) >= F  because [36], by (Select) 
  39] filter2*(true, F, X, Y) >= Y  because [37], by (Select) 

We can thus remove the following rules:

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

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  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) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

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

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

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

  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) 

With these choices, we have:

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

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

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

  14] filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because [15], by (Star) 
  15] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [16], [19], [20], [22] and [24], by (Stat) 
  16] cons(X, Y) > X  because [17], by definition 
  17] cons*(X, Y) >= X  because [18], by (Select) 
  18] X >= X  by (Meta) 
  19] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [20] and [22], by (Copy) 
  20] filter*(F, cons(X, Y)) >= F  because [21], by (Select) 
  21] F >= F  by (Meta) 
  22] filter*(F, cons(X, Y)) >= X  because [23], by (Select) 
  23] cons(X, Y) >= X  because [17], by (Star) 
  24] filter*(F, cons(X, Y)) >= Y  because [25], by (Select) 
  25] cons(X, Y) >= Y  because [26], by (Star) 
  26] cons*(X, Y) >= Y  because [27], by (Select) 
  27] Y >= Y  by (Meta) 

We can thus remove the following rules:

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

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  map(F, nil) >? nil 
  filter(F, nil) >? nil 
  filter(F, cons(X, Y)) >? filter2(F X, F, X, Y) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[nil]] = _|_ 

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

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

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

With these choices, we have:

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

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

  3] filter(F, cons(X, Y)) > filter2(@_{o -> o}(F, X), F, X, Y)  because [4], by definition 
  4] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter > filter2, [5], [6], [8] and [12], by (Copy) 
  5] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [6] and [8], by (Copy) 
  6] filter*(F, cons(X, Y)) >= F  because [7], by (Select) 
  7] F >= F  by (Meta) 
  8] filter*(F, cons(X, Y)) >= X  because [9], by (Select) 
  9] cons(X, Y) >= X  because [10], by (Star) 
  10] cons*(X, Y) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] filter*(F, cons(X, Y)) >= Y  because [13], by (Select) 
  13] cons(X, Y) >= Y  because [14], by (Star) 
  14] cons*(X, Y) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 

We can thus remove the following rules:

  filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  map(F, nil) >? nil 
  filter(F, nil) >? nil 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[nil]] = _|_ 

We choose Lex = {} and Mul = {filter, map}, and the following precedence: filter > map

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

  map(F, _|_) > _|_ 
  filter(F, _|_) >= _|_ 

With these choices, we have:

  1] map(F, _|_) > _|_  because [2], by definition 
  2] map*(F, _|_) >= _|_  by (Bot) 

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

We can thus remove the following rules:

  map(F, nil) => nil 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  filter(F, nil) >? nil 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[nil]] = _|_ 

We choose Lex = {} and Mul = {filter}, and the following precedence: filter

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

  filter(F, _|_) > _|_ 

With these choices, we have:

  1] filter(F, _|_) > _|_  because [2], by definition 
  2] filter*(F, _|_) >= _|_  by (Bot) 

We can thus remove the following rules:

  filter(F, nil) => nil 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.