We consider the system Applicative_first_order_05__12.

  Alphabet:

    and : [a * a] --> 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 
    not : [a] --> a 
    or : [a * a] --> a 
    true : [] --> b 

  Rules:

    not(not(x)) => x 
    not(or(x, y)) => and(not(x), not(y)) 
    not(and(x, y)) => or(not(x), not(y)) 
    and(x, or(y, z)) => or(and(x, y), and(x, z)) 
    and(or(x, y), z) => or(and(z, x), and(z, 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]):

  not(not(X)) >? X 
  not(or(X, Y)) >? and(not(X), not(Y)) 
  not(and(X, Y)) >? or(not(X), not(Y)) 
  and(X, or(Y, Z)) >? or(and(X, Y), and(X, Z)) 
  and(or(X, Y), Z) >? or(and(Z, X), and(Z, 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:

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

We choose Lex = {cons, filter, filter2, map, not, or} and Mul = {@_{o -> o}, and, false, true}, and the following precedence: not > and > map > or > true > filter = filter2 > @_{o -> o} > cons > false

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

  not(not(X)) >= X 
  not(or(X, Y)) >= and(not(X), not(Y)) 
  not(and(X, Y)) >= or(not(X), not(Y)) 
  and(X, or(Y, Z)) >= or(and(X, Y), and(X, Z)) 
  and(or(X, Y), Z) >= or(and(Z, X), and(Z, 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] not(not(X)) >= X  because [2], by (Star) 
  2] not*(not(X)) >= X  because [3], by (Select) 
  3] not(X) >= X  because [4], by (Star) 
  4] not*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 

  6] not(or(X, Y)) >= and(not(X), not(Y))  because [7], by (Star) 
  7] not*(or(X, Y)) >= and(not(X), not(Y))  because not > and, [8] and [14], by (Copy) 
  8] not*(or(X, Y)) >= not(X)  because [9], [12] and [9], by (Stat) 
  9] or(X, Y) > X  because [10], by definition 
  10] or*(X, Y) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] not*(or(X, Y)) >= X  because [13], by (Select) 
  13] or(X, Y) >= X  because [10], by (Star) 
  14] not*(or(X, Y)) >= not(Y)  because [15], [18] and [15], by (Stat) 
  15] or(X, Y) > Y  because [16], by definition 
  16] or*(X, Y) >= Y  because [17], by (Select) 
  17] Y >= Y  by (Meta) 
  18] not*(or(X, Y)) >= Y  because [19], by (Select) 
  19] or(X, Y) >= Y  because [16], by (Star) 

  20] not(and(X, Y)) >= or(not(X), not(Y))  because [21], by (Star) 
  21] not*(and(X, Y)) >= or(not(X), not(Y))  because not > or, [22] and [28], by (Copy) 
  22] not*(and(X, Y)) >= not(X)  because [23], [26] and [23], by (Stat) 
  23] and(X, Y) > X  because [24], by definition 
  24] and*(X, Y) >= X  because [25], by (Select) 
  25] X >= X  by (Meta) 
  26] not*(and(X, Y)) >= X  because [27], by (Select) 
  27] and(X, Y) >= X  because [24], by (Star) 
  28] not*(and(X, Y)) >= not(Y)  because [29], [32] and [29], by (Stat) 
  29] and(X, Y) > Y  because [30], by definition 
  30] and*(X, Y) >= Y  because [31], by (Select) 
  31] Y >= Y  by (Meta) 
  32] not*(and(X, Y)) >= Y  because [33], by (Select) 
  33] and(X, Y) >= Y  because [30], by (Star) 

  34] and(X, or(Y, Z)) >= or(and(X, Y), and(X, Z))  because [35], by (Star) 
  35] and*(X, or(Y, Z)) >= or(and(X, Y), and(X, Z))  because and > or, [36] and [41], by (Copy) 
  36] and*(X, or(Y, Z)) >= and(X, Y)  because and in Mul, [37] and [38], by (Stat) 
  37] X >= X  by (Meta) 
  38] or(Y, Z) > Y  because [39], by definition 
  39] or*(Y, Z) >= Y  because [40], by (Select) 
  40] Y >= Y  by (Meta) 
  41] and*(X, or(Y, Z)) >= and(X, Z)  because and in Mul, [37] and [42], by (Stat) 
  42] or(Y, Z) > Z  because [43], by definition 
  43] or*(Y, Z) >= Z  because [44], by (Select) 
  44] Z >= Z  by (Meta) 

  45] and(or(X, Y), Z) >= or(and(Z, X), and(Z, Y))  because [46], by (Star) 
  46] and*(or(X, Y), Z) >= or(and(Z, X), and(Z, Y))  because and > or, [47] and [52], by (Copy) 
  47] and*(or(X, Y), Z) >= and(Z, X)  because and in Mul, [48] and [51], by (Stat) 
  48] or(X, Y) > X  because [49], by definition 
  49] or*(X, Y) >= X  because [50], by (Select) 
  50] X >= X  by (Meta) 
  51] Z >= Z  by (Meta) 
  52] and*(or(X, Y), Z) >= and(Z, Y)  because and in Mul, [53] and [51], by (Stat) 
  53] or(X, Y) > Y  because [54], by definition 
  54] or*(X, Y) >= Y  because [55], by (Select) 
  55] Y >= Y  by (Meta) 

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

  57] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [58], by (Star) 
  58] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [59] and [66], by (Copy) 
  59] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [60] and [62], by (Copy) 
  60] map*(F, cons(X, Y)) >= F  because [61], by (Select) 
  61] F >= F  by (Meta) 
  62] map*(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] map*(F, cons(X, Y)) >= map(F, Y)  because [67], [60], [70], [72] and [67], by (Stat) 
  67] cons(X, Y) > Y  because [68], by definition 
  68] cons*(X, Y) >= Y  because [69], by (Select) 
  69] Y >= Y  by (Meta) 
  70] map*(F, cons(X, Y)) >= Y  because [71], by (Select) 
  71] cons(X, Y) >= Y  because [68], by (Star) 
  72] F >= F  by (Meta) 

  73] filter(F, _|_) > _|_  because [74], by definition 
  74] filter*(F, _|_) >= _|_  by (Bot) 

  75] filter(F, cons(X, Y)) > filter2(@_{o -> o}(F, X), F, X, Y)  because [76], by definition 
  76] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [77], [78], [81], [82], [83] and [87], by (Stat) 
  77] F >= F  by (Meta) 
  78] cons(X, Y) > Y  because [79], by definition 
  79] cons*(X, Y) >= Y  because [80], by (Select) 
  80] Y >= Y  by (Meta) 
  81] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [82] and [83], by (Copy) 
  82] filter*(F, cons(X, Y)) >= F  because [77], by (Select) 
  83] filter*(F, cons(X, Y)) >= X  because [84], by (Select) 
  84] cons(X, Y) >= X  because [85], by (Star) 
  85] cons*(X, Y) >= X  because [86], by (Select) 
  86] X >= X  by (Meta) 
  87] filter*(F, cons(X, Y)) >= Y  because [88], by (Select) 
  88] cons(X, Y) >= Y  because [79], by (Star) 

  89] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [90], by (Star) 
  90] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [91] and [93], by (Copy) 
  91] filter2*(true, F, X, Y) >= X  because [92], by (Select) 
  92] X >= X  by (Meta) 
  93] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [94], [95], [96], [97], [94] and [95], by (Stat) 
  94] F >= F  by (Meta) 
  95] Y >= Y  by (Meta) 
  96] filter2*(true, F, X, Y) >= F  because [94], by (Select) 
  97] filter2*(true, F, X, Y) >= Y  because [95], by (Select) 

  98] filter2(false, F, X, Y) >= filter(F, Y)  because [99], by (Star) 
  99] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [100], [101], [102], [103], [100] and [101], by (Stat) 
  100] F >= F  by (Meta) 
  101] Y >= Y  by (Meta) 
  102] filter2*(false, F, X, Y) >= F  because [100], by (Select) 
  103] filter2*(false, F, X, Y) >= Y  because [101], by (Select) 

We can thus remove the following rules:

  filter(F, nil) => nil 
  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]):

  not(not(X)) >? X 
  not(or(X, Y)) >? and(not(X), not(Y)) 
  not(and(X, Y)) >? or(not(X), not(Y)) 
  and(X, or(Y, Z)) >? or(and(X, Y), and(X, Z)) 
  and(or(X, Y), Z) >? or(and(Z, X), and(Z, Y)) 
  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(F, 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:

  [[@_{o -> o}(x_1, x_2)]] = @_{o -> o}(x_2, x_1) 
  [[cons(x_1, x_2)]] = cons(x_2, x_1) 
  [[filter2(x_1, x_2, x_3, x_4)]] = filter2(x_1, x_3, x_4, x_2) 
  [[map(x_1, x_2)]] = map(x_2, x_1) 
  [[or(x_1, x_2)]] = or(x_2, x_1) 

We choose Lex = {@_{o -> o}, cons, filter2, map, or} and Mul = {and, false, filter, nil, not, true}, and the following precedence: @_{o -> o} = filter2 = map > false > filter > cons > nil > not > and > or > true

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

  not(not(X)) >= X 
  not(or(X, Y)) > and(not(X), not(Y)) 
  not(and(X, Y)) >= or(not(X), not(Y)) 
  and(X, or(Y, Z)) > or(and(X, Y), and(X, Z)) 
  and(or(X, Y), Z) > or(and(Z, X), and(Z, Y)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, 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] not(not(X)) >= X  because [2], by (Star) 
  2] not*(not(X)) >= X  because [3], by (Select) 
  3] not(X) >= X  because [4], by (Star) 
  4] not*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 

  6] not(or(X, Y)) > and(not(X), not(Y))  because [7], by definition 
  7] not*(or(X, Y)) >= and(not(X), not(Y))  because not > and, [8] and [12], by (Copy) 
  8] not*(or(X, Y)) >= not(X)  because not in Mul and [9], by (Stat) 
  9] or(X, Y) > X  because [10], by definition 
  10] or*(X, Y) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] not*(or(X, Y)) >= not(Y)  because not in Mul and [13], by (Stat) 
  13] or(X, Y) > Y  because [14], by definition 
  14] or*(X, Y) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 

  16] not(and(X, Y)) >= or(not(X), not(Y))  because [17], by (Star) 
  17] not*(and(X, Y)) >= or(not(X), not(Y))  because not > or, [18] and [22], by (Copy) 
  18] not*(and(X, Y)) >= not(X)  because not in Mul and [19], by (Stat) 
  19] and(X, Y) > X  because [20], by definition 
  20] and*(X, Y) >= X  because [21], by (Select) 
  21] X >= X  by (Meta) 
  22] not*(and(X, Y)) >= not(Y)  because not in Mul and [23], by (Stat) 
  23] and(X, Y) > Y  because [24], by definition 
  24] and*(X, Y) >= Y  because [25], by (Select) 
  25] Y >= Y  by (Meta) 

  26] and(X, or(Y, Z)) > or(and(X, Y), and(X, Z))  because [27], by definition 
  27] and*(X, or(Y, Z)) >= or(and(X, Y), and(X, Z))  because and > or, [28] and [33], by (Copy) 
  28] and*(X, or(Y, Z)) >= and(X, Y)  because and in Mul, [29] and [30], by (Stat) 
  29] X >= X  by (Meta) 
  30] or(Y, Z) > Y  because [31], by definition 
  31] or*(Y, Z) >= Y  because [32], by (Select) 
  32] Y >= Y  by (Meta) 
  33] and*(X, or(Y, Z)) >= and(X, Z)  because and in Mul, [29] and [34], by (Stat) 
  34] or(Y, Z) > Z  because [35], by definition 
  35] or*(Y, Z) >= Z  because [36], by (Select) 
  36] Z >= Z  by (Meta) 

  37] and(or(X, Y), Z) > or(and(Z, X), and(Z, Y))  because [38], by definition 
  38] and*(or(X, Y), Z) >= or(and(Z, X), and(Z, Y))  because and > or, [39] and [44], by (Copy) 
  39] and*(or(X, Y), Z) >= and(Z, X)  because and in Mul, [40] and [43], by (Stat) 
  40] or(X, Y) > X  because [41], by definition 
  41] or*(X, Y) >= X  because [42], by (Select) 
  42] X >= X  by (Meta) 
  43] Z >= Z  by (Meta) 
  44] and*(or(X, Y), Z) >= and(Z, Y)  because and in Mul, [45] and [43], by (Stat) 
  45] or(X, Y) > Y  because [46], by definition 
  46] or*(X, Y) >= Y  because [47], by (Select) 
  47] Y >= Y  by (Meta) 

  48] map(F, nil) >= nil  because [49], by (Star) 
  49] map*(F, nil) >= nil  because map > nil, by (Copy) 

  50] map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, Y))  because [51], by definition 
  51] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [52] and [61], by (Copy) 
  52] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map = @_{o -> o}, [53], [56], [58], [60] and [53], by (Stat) 
  53] cons(X, Y) > X  because [54], by definition 
  54] cons*(X, Y) >= X  because [55], by (Select) 
  55] X >= X  by (Meta) 
  56] map*(F, cons(X, Y)) >= F  because [57], by (Select) 
  57] F >= F  by (Meta) 
  58] map*(F, cons(X, Y)) >= X  because [59], by (Select) 
  59] cons(X, Y) >= X  because [54], by (Star) 
  60] F >= F  by (Meta) 
  61] map*(F, cons(X, Y)) >= map(F, Y)  because [62], [56], [65], [60] and [62], by (Stat) 
  62] cons(X, Y) > Y  because [63], by definition 
  63] cons*(X, Y) >= Y  because [64], by (Select) 
  64] Y >= Y  by (Meta) 
  65] map*(F, cons(X, Y)) >= Y  because [66], by (Select) 
  66] cons(X, Y) >= Y  because [63], by (Star) 

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

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

We can thus remove the following rules:

  not(or(X, Y)) => and(not(X), not(Y)) 
  and(X, or(Y, Z)) => or(and(X, Y), and(X, Z)) 
  and(or(X, Y), Z) => or(and(Z, X), and(Z, Y)) 
  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
  filter2(true, F, X, Y) => cons(X, filter(F, 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]):

  not(not(X)) >? X 
  not(and(X, Y)) >? or(not(X), not(Y)) 
  map(F, nil) >? nil 

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

The following interpretation satisfies the requirements:

  and = Lam[y0;y1].3 + 3*y0 + 3*y1 
  map = Lam[G0;y1].3 + 3*y1 + G0(0) 
  nil = 2 
  not = Lam[y0].3*y0 
  or = Lam[y0;y1].y0 + y1 

Using this interpretation, the requirements translate to:

  [[not(not(_x0))]] = 9*x0 >= x0 = [[_x0]] 
  [[not(and(_x0, _x1))]] = 9 + 9*x0 + 9*x1 > 3*x0 + 3*x1 = [[or(not(_x0), not(_x1))]] 
  [[map(_F0, nil)]] = 9 + F0(0) > 2 = [[nil]] 

We can thus remove the following rules:

  not(and(X, Y)) => or(not(X), not(Y)) 
  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]):

  not(not(X)) >? X 

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

The following interpretation satisfies the requirements:

  not = Lam[y0].3 + 2*y0 

Using this interpretation, the requirements translate to:

  [[not(not(_x0))]] = 9 + 4*x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  not(not(X)) => X 

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.