We consider the system Applicative_first_order_05__17.

  Alphabet:

    !facdot : [a * a] --> a 
    1 : [] --> a 
    cons : [c * d] --> d 
    false : [] --> b 
    filter : [c -> b * d] --> d 
    filter2 : [b * c -> b * c * d] --> d 
    i : [a] --> a 
    map : [c -> c * d] --> d 
    nil : [] --> d 
    true : [] --> b 

  Rules:

    !facdot(1, x) => x 
    !facdot(x, 1) => x 
    !facdot(i(x), x) => 1 
    !facdot(x, i(x)) => 1 
    !facdot(i(x), !facdot(x, y)) => y 
    !facdot(x, !facdot(i(x), y)) => y 
    !facdot(!facdot(x, y), z) => !facdot(x, !facdot(y, z)) 
    i(1) => 1 
    i(i(x)) => x 
    i(!facdot(x, y)) => !facdot(i(y), i(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) 

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]):

  !facdot(1, X) >? X 
  !facdot(X, 1) >? X 
  !facdot(i(X), X) >? 1 
  !facdot(X, i(X)) >? 1 
  !facdot(i(X), !facdot(X, Y)) >? Y 
  !facdot(X, !facdot(i(X), Y)) >? Y 
  !facdot(!facdot(X, Y), Z) >? !facdot(X, !facdot(Y, Z)) 
  i(1) >? 1 
  i(i(X)) >? X 
  i(!facdot(X, Y)) >? !facdot(i(Y), i(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:

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

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

  !facdot(_|_, X) >= X 
  !facdot(X, _|_) > X 
  !facdot(i(X), X) >= _|_ 
  !facdot(X, i(X)) >= _|_ 
  !facdot(i(X), !facdot(X, Y)) > Y 
  !facdot(X, !facdot(i(X), Y)) >= Y 
  !facdot(!facdot(X, Y), Z) >= !facdot(X, !facdot(Y, Z)) 
  i(_|_) >= _|_ 
  i(i(X)) >= X 
  i(!facdot(X, Y)) > !facdot(i(Y), i(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] !facdot(_|_, X) >= X  because [2], by (Star) 
  2] !facdot*(_|_, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] !facdot(X, _|_) > X  because [5], by definition 
  5] !facdot*(X, _|_) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 

  7] !facdot(i(X), X) >= _|_  by (Bot) 

  8] !facdot(X, i(X)) >= _|_  by (Bot) 

  9] !facdot(i(X), !facdot(X, Y)) > Y  because [10], by definition 
  10] !facdot*(i(X), !facdot(X, Y)) >= Y  because [11], by (Select) 
  11] !facdot(X, Y) >= Y  because [12], by (Star) 
  12] !facdot*(X, Y) >= Y  because [13], by (Select) 
  13] Y >= Y  by (Meta) 

  14] !facdot(X, !facdot(i(X), Y)) >= Y  because [15], by (Star) 
  15] !facdot*(X, !facdot(i(X), Y)) >= Y  because [16], by (Select) 
  16] !facdot(i(X), Y) >= Y  because [17], by (Star) 
  17] !facdot*(i(X), Y) >= Y  because [18], by (Select) 
  18] Y >= Y  by (Meta) 

  19] !facdot(!facdot(X, Y), Z) >= !facdot(X, !facdot(Y, Z))  because [20], by (Star) 
  20] !facdot*(!facdot(X, Y), Z) >= !facdot(X, !facdot(Y, Z))  because [21], [24] and [26], by (Stat) 
  21] !facdot(X, Y) > X  because [22], by definition 
  22] !facdot*(X, Y) >= X  because [23], by (Select) 
  23] X >= X  by (Meta) 
  24] !facdot*(!facdot(X, Y), Z) >= X  because [25], by (Select) 
  25] !facdot(X, Y) >= X  because [22], by (Star) 
  26] !facdot*(!facdot(X, Y), Z) >= !facdot(Y, Z)  because [27], [30] and [32], by (Stat) 
  27] !facdot(X, Y) > Y  because [28], by definition 
  28] !facdot*(X, Y) >= Y  because [29], by (Select) 
  29] Y >= Y  by (Meta) 
  30] !facdot*(!facdot(X, Y), Z) >= Y  because [31], by (Select) 
  31] !facdot(X, Y) >= Y  because [28], by (Star) 
  32] !facdot*(!facdot(X, Y), Z) >= Z  because [33], by (Select) 
  33] Z >= Z  by (Meta) 

  34] i(_|_) >= _|_  by (Bot) 

  35] i(i(X)) >= X  because [36], by (Star) 
  36] i*(i(X)) >= X  because [37], by (Select) 
  37] i(X) >= X  because [38], by (Star) 
  38] i*(X) >= X  because [39], by (Select) 
  39] X >= X  by (Meta) 

  40] i(!facdot(X, Y)) > !facdot(i(Y), i(X))  because [41], by definition 
  41] i*(!facdot(X, Y)) >= !facdot(i(Y), i(X))  because i > !facdot, [42] and [46], by (Copy) 
  42] i*(!facdot(X, Y)) >= i(Y)  because i in Mul and [43], by (Stat) 
  43] !facdot(X, Y) > Y  because [44], by definition 
  44] !facdot*(X, Y) >= Y  because [45], by (Select) 
  45] Y >= Y  by (Meta) 
  46] i*(!facdot(X, Y)) >= i(X)  because i in Mul and [47], by (Stat) 
  47] !facdot(X, Y) > X  because [48], by definition 
  48] !facdot*(X, Y) >= X  because [49], by (Select) 
  49] X >= X  by (Meta) 

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

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

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

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

  80] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [81], by (Star) 
  81] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [82] and [84], by (Copy) 
  82] filter2*(true, F, X, Y) >= X  because [83], by (Select) 
  83] X >= X  by (Meta) 
  84] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [85], [86], [87] and [88], by (Stat) 
  85] F >= F  by (Meta) 
  86] Y >= Y  by (Meta) 
  87] filter2*(true, F, X, Y) >= F  because [85], by (Select) 
  88] filter2*(true, F, X, Y) >= Y  because [86], by (Select) 

  89] filter2(false, F, X, Y) > filter(F, Y)  because [90], by definition 
  90] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [91], [92], [93] and [94], by (Stat) 
  91] F >= F  by (Meta) 
  92] Y >= Y  by (Meta) 
  93] filter2*(false, F, X, Y) >= F  because [91], by (Select) 
  94] filter2*(false, F, X, Y) >= Y  because [92], by (Select) 

We can thus remove the following rules:

  !facdot(X, 1) => X 
  !facdot(i(X), !facdot(X, Y)) => Y 
  i(!facdot(X, Y)) => !facdot(i(Y), i(X)) 
  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
  filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
  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]):

  !facdot(1, X) >? X 
  !facdot(i(X), X) >? 1 
  !facdot(X, i(X)) >? 1 
  !facdot(X, !facdot(i(X), Y)) >? Y 
  !facdot(!facdot(X, Y), Z) >? !facdot(X, !facdot(Y, Z)) 
  i(1) >? 1 
  i(i(X)) >? X 
  map(F, nil) >? nil 
  filter(F, nil) >? nil 
  filter2(true, F, X, Y) >? cons(X, filter(F, Y)) 

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

Argument functions:

  [[1]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {!facdot} and Mul = {cons, filter, filter2, i, map, true}, and the following precedence: map > filter2 > filter > i > !facdot > cons > true

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

  !facdot(_|_, X) >= X 
  !facdot(i(X), X) >= _|_ 
  !facdot(X, i(X)) >= _|_ 
  !facdot(X, !facdot(i(X), Y)) > Y 
  !facdot(!facdot(X, Y), Z) > !facdot(X, !facdot(Y, Z)) 
  i(_|_) > _|_ 
  i(i(X)) > X 
  map(F, _|_) >= _|_ 
  filter(F, _|_) >= _|_ 
  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 

With these choices, we have:

  1] !facdot(_|_, X) >= X  because [2], by (Star) 
  2] !facdot*(_|_, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] !facdot(i(X), X) >= _|_  by (Bot) 

  5] !facdot(X, i(X)) >= _|_  by (Bot) 

  6] !facdot(X, !facdot(i(X), Y)) > Y  because [7], by definition 
  7] !facdot*(X, !facdot(i(X), Y)) >= Y  because [8], by (Select) 
  8] !facdot(i(X), Y) >= Y  because [9], by (Star) 
  9] !facdot*(i(X), Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 

  11] !facdot(!facdot(X, Y), Z) > !facdot(X, !facdot(Y, Z))  because [12], by definition 
  12] !facdot*(!facdot(X, Y), Z) >= !facdot(X, !facdot(Y, Z))  because [13], [16] and [18], by (Stat) 
  13] !facdot(X, Y) > X  because [14], by definition 
  14] !facdot*(X, Y) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 
  16] !facdot*(!facdot(X, Y), Z) >= X  because [17], by (Select) 
  17] !facdot(X, Y) >= X  because [14], by (Star) 
  18] !facdot*(!facdot(X, Y), Z) >= !facdot(Y, Z)  because [19], [22] and [24], by (Stat) 
  19] !facdot(X, Y) > Y  because [20], by definition 
  20] !facdot*(X, Y) >= Y  because [21], by (Select) 
  21] Y >= Y  by (Meta) 
  22] !facdot*(!facdot(X, Y), Z) >= Y  because [23], by (Select) 
  23] !facdot(X, Y) >= Y  because [20], by (Star) 
  24] !facdot*(!facdot(X, Y), Z) >= Z  because [25], by (Select) 
  25] Z >= Z  by (Meta) 

  26] i(_|_) > _|_  because [27], by definition 
  27] i*(_|_) >= _|_  by (Bot) 

  28] i(i(X)) > X  because [29], by definition 
  29] i*(i(X)) >= X  because [30], by (Select) 
  30] i(X) >= X  because [31], by (Star) 
  31] i*(X) >= X  because [32], by (Select) 
  32] X >= X  by (Meta) 

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

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

  35] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [36], by (Star) 
  36] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [37] and [39], by (Copy) 
  37] filter2*(true, F, X, Y) >= X  because [38], by (Select) 
  38] X >= X  by (Meta) 
  39] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 > filter, [40] and [42], by (Copy) 
  40] filter2*(true, F, X, Y) >= F  because [41], by (Select) 
  41] F >= F  by (Meta) 
  42] filter2*(true, F, X, Y) >= Y  because [43], by (Select) 
  43] Y >= Y  by (Meta) 

We can thus remove the following rules:

  !facdot(X, !facdot(i(X), Y)) => Y 
  !facdot(!facdot(X, Y), Z) => !facdot(X, !facdot(Y, Z)) 
  i(1) => 1 
  i(i(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]):

  !facdot(1, X) >? X 
  !facdot(i(X), X) >? 1 
  !facdot(X, i(X)) >? 1 
  map(F, nil) >? nil 
  filter(F, nil) >? nil 
  filter2(true, F, X, Y) >? cons(X, filter(F, Y)) 

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

Argument functions:

  [[1]] = _|_ 
  [[i(x_1)]] = x_1 
  [[nil]] = _|_ 

We choose Lex = {} and Mul = {!facdot, cons, filter, filter2, map, true}, and the following precedence: true > !facdot > filter2 > filter > cons > map

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

  !facdot(_|_, X) >= X 
  !facdot(X, X) >= _|_ 
  !facdot(X, X) >= _|_ 
  map(F, _|_) >= _|_ 
  filter(F, _|_) >= _|_ 
  filter2(true, F, X, Y) > cons(X, filter(F, Y)) 

With these choices, we have:

  1] !facdot(_|_, X) >= X  because [2], by (Star) 
  2] !facdot*(_|_, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] !facdot(X, X) >= _|_  by (Bot) 

  5] !facdot(X, X) >= _|_  by (Bot) 

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

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

  8] filter2(true, F, X, Y) > cons(X, filter(F, Y))  because [9], by definition 
  9] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [10] and [12], by (Copy) 
  10] filter2*(true, F, X, Y) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 > filter, [13] and [15], by (Copy) 
  13] filter2*(true, F, X, Y) >= F  because [14], by (Select) 
  14] F >= F  by (Meta) 
  15] filter2*(true, F, X, Y) >= Y  because [16], by (Select) 
  16] Y >= Y  by (Meta) 

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]):

  !facdot(1, X) >? X 
  !facdot(i(X), X) >? 1 
  !facdot(X, i(X)) >? 1 
  map(F, nil) >? nil 
  filter(F, nil) >? nil 

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

Argument functions:

  [[1]] = _|_ 
  [[i(x_1)]] = x_1 
  [[nil]] = _|_ 

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

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

  !facdot(_|_, X) >= X 
  !facdot(X, X) > _|_ 
  !facdot(X, X) >= _|_ 
  map(F, _|_) >= _|_ 
  filter(F, _|_) >= _|_ 

With these choices, we have:

  1] !facdot(_|_, X) >= X  because [2], by (Star) 
  2] !facdot*(_|_, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] !facdot(X, X) > _|_  because [5], by definition 
  5] !facdot*(X, X) >= _|_  by (Bot) 

  6] !facdot(X, X) >= _|_  by (Bot) 

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

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

We can thus remove the following rules:

  !facdot(i(X), X) => 1 

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

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

  !facdot(1, X) >? X 
  !facdot(X, i(X)) >? 1 
  map(F, nil) >? nil 
  filter(F, nil) >? nil 

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

Argument functions:

  [[1]] = _|_ 
  [[i(x_1)]] = x_1 
  [[nil]] = _|_ 

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

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

  !facdot(_|_, X) >= X 
  !facdot(X, X) > _|_ 
  map(F, _|_) >= _|_ 
  filter(F, _|_) >= _|_ 

With these choices, we have:

  1] !facdot(_|_, X) >= X  because [2], by (Star) 
  2] !facdot*(_|_, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] !facdot(X, X) > _|_  because [5], by definition 
  5] !facdot*(X, X) >= _|_  by (Bot) 

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

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

We can thus remove the following rules:

  !facdot(X, i(X)) => 1 

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

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

  !facdot(1, X) >? X 
  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 = {!facdot, 1, filter, map}, and the following precedence: !facdot > filter > map > 1

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

  !facdot(1, X) > X 
  map(F, _|_) >= _|_ 
  filter(F, _|_) >= _|_ 

With these choices, we have:

  1] !facdot(1, X) > X  because [2], by definition 
  2] !facdot*(1, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

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

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

We can thus remove the following rules:

  !facdot(1, 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]):

  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.