We consider the system Applicative_first_order_05__08.

  Alphabet:

    !facminus : [a * a] --> a 
    !facplus : [a * a] --> a 
    !factimes : [a * a] --> a 
    0 : [] --> a 
    1 : [] --> a 
    D : [a] --> a 
    cons : [c * d] --> d 
    constant : [] --> a 
    false : [] --> b 
    filter : [c -> b * d] --> d 
    filter2 : [b * c -> b * c * d] --> d 
    map : [c -> c * d] --> d 
    nil : [] --> d 
    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)) 
    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)) 
  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]] = _|_ 
  [[@_{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 = {@_{o -> o}, filter, filter2} and Mul = {!facminus, !facplus, !factimes, D, cons, constant, false, map, t, true}, and the following precedence: constant > map > @_{o -> o} = filter = filter2 > cons > t > D > !factimes > true > false > !facminus > !facplus

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

  2] D(constant) >= _|_  by (Bot) 

  3] D(!facplus(X, Y)) >= !facplus(D(X), D(Y))  because [4], by (Star) 
  4] D*(!facplus(X, Y)) >= !facplus(D(X), D(Y))  because D > !facplus, [5] and [9], by (Copy) 
  5] D*(!facplus(X, Y)) >= D(X)  because D in Mul and [6], by (Stat) 
  6] !facplus(X, Y) > X  because [7], by definition 
  7] !facplus*(X, Y) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 
  9] D*(!facplus(X, Y)) >= D(Y)  because D in Mul and [10], by (Stat) 
  10] !facplus(X, Y) > Y  because [11], by definition 
  11] !facplus*(X, Y) >= Y  because [12], by (Select) 
  12] Y >= Y  by (Meta) 

  13] D(!factimes(X, Y)) > !facplus(!factimes(Y, D(X)), !factimes(X, D(Y)))  because [14], by definition 
  14] D*(!factimes(X, Y)) >= !facplus(!factimes(Y, D(X)), !factimes(X, D(Y)))  because D > !facplus, [15] and [24], by (Copy) 
  15] D*(!factimes(X, Y)) >= !factimes(Y, D(X))  because D > !factimes, [16] and [20], by (Copy) 
  16] D*(!factimes(X, Y)) >= Y  because [17], by (Select) 
  17] !factimes(X, Y) >= Y  because [18], by (Star) 
  18] !factimes*(X, Y) >= Y  because [19], by (Select) 
  19] Y >= Y  by (Meta) 
  20] D*(!factimes(X, Y)) >= D(X)  because D in Mul and [21], by (Stat) 
  21] !factimes(X, Y) > X  because [22], by definition 
  22] !factimes*(X, Y) >= X  because [23], by (Select) 
  23] X >= X  by (Meta) 
  24] D*(!factimes(X, Y)) >= !factimes(X, D(Y))  because D > !factimes, [25] and [27], by (Copy) 
  25] D*(!factimes(X, Y)) >= X  because [26], by (Select) 
  26] !factimes(X, Y) >= X  because [22], by (Star) 
  27] D*(!factimes(X, Y)) >= D(Y)  because D in Mul and [28], by (Stat) 
  28] !factimes(X, Y) > Y  because [29], by definition 
  29] !factimes*(X, Y) >= Y  because [19], by (Select) 

  30] D(!facminus(X, Y)) >= !facminus(D(X), D(Y))  because [31], by (Star) 
  31] D*(!facminus(X, Y)) >= !facminus(D(X), D(Y))  because D > !facminus, [32] and [36], by (Copy) 
  32] D*(!facminus(X, Y)) >= D(X)  because D in Mul and [33], by (Stat) 
  33] !facminus(X, Y) > X  because [34], by definition 
  34] !facminus*(X, Y) >= X  because [35], by (Select) 
  35] X >= X  by (Meta) 
  36] D*(!facminus(X, Y)) >= D(Y)  because D in Mul and [37], by (Stat) 
  37] !facminus(X, Y) > Y  because [38], by definition 
  38] !facminus*(X, Y) >= Y  because [39], by (Select) 
  39] Y >= Y  by (Meta) 

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

  41] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [42], by (Star) 
  42] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [43] and [50], by (Copy) 
  43] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [44] and [46], by (Copy) 
  44] map*(F, cons(X, Y)) >= F  because [45], by (Select) 
  45] F >= F  by (Meta) 
  46] map*(F, cons(X, Y)) >= X  because [47], by (Select) 
  47] cons(X, Y) >= X  because [48], by (Star) 
  48] cons*(X, Y) >= X  because [49], by (Select) 
  49] X >= X  by (Meta) 
  50] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [51] and [52], by (Stat) 
  51] F >= F  by (Meta) 
  52] cons(X, Y) > Y  because [53], by definition 
  53] cons*(X, Y) >= Y  because [54], by (Select) 
  54] Y >= Y  by (Meta) 

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

  56] filter(F, cons(X, Y)) > filter2(@_{o -> o}(F, X), F, X, Y)  because [57], by definition 
  57] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [58], [61], [65], [67] and [69], by (Stat) 
  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, cons(X, Y)) >= @_{o -> o}(F, X)  because filter = @_{o -> o}, [62], [65] and [67], by (Stat) 
  62] cons(X, Y) > X  because [63], by definition 
  63] cons*(X, Y) >= X  because [64], by (Select) 
  64] X >= X  by (Meta) 
  65] filter*(F, cons(X, Y)) >= F  because [66], by (Select) 
  66] F >= F  by (Meta) 
  67] filter*(F, cons(X, Y)) >= X  because [68], by (Select) 
  68] cons(X, Y) >= X  because [63], by (Star) 
  69] filter*(F, cons(X, Y)) >= Y  because [70], by (Select) 
  70] cons(X, Y) >= Y  because [59], by (Star) 

  71] filter2(true, F, X, Y) > cons(X, filter(F, Y))  because [72], by definition 
  72] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [73] and [75], by (Copy) 
  73] filter2*(true, F, X, Y) >= X  because [74], by (Select) 
  74] X >= X  by (Meta) 
  75] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [76], [77], [78] and [79], by (Stat) 
  76] F >= F  by (Meta) 
  77] Y >= Y  by (Meta) 
  78] filter2*(true, F, X, Y) >= F  because [76], by (Select) 
  79] filter2*(true, F, X, Y) >= Y  because [77], by (Select) 

  80] filter2(false, F, X, Y) >= filter(F, Y)  because [81], by (Star) 
  81] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [82], [83], [84] and [85], by (Stat) 
  82] F >= F  by (Meta) 
  83] Y >= Y  by (Meta) 
  84] filter2*(false, F, X, Y) >= F  because [82], by (Select) 
  85] filter2*(false, F, X, Y) >= Y  because [83], by (Select) 

We can thus remove the following rules:

  D(!factimes(X, Y)) => !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) 
  filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
  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]):

  D(t) >? 1 
  D(constant) >? 0 
  D(!facplus(X, Y)) >? !facplus(D(X), D(Y)) 
  D(!facminus(X, Y)) >? !facminus(D(X), D(Y)) 
  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 
  filter(F, nil) >? nil 
  filter2(false, F, X, Y) >? filter(F, Y) 

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

Argument functions:

  [[0]] = _|_ 
  [[1]] = _|_ 

We choose Lex = {} and Mul = {!facminus, !facplus, @_{o -> o}, D, cons, constant, false, filter, filter2, map, nil, t}, and the following precedence: t > D > !facminus > constant > !facplus > false > map > cons > nil > @_{o -> o} > filter2 > filter

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(!facminus(X, Y)) > !facminus(D(X), D(Y)) 
  map(F, nil) >= nil 
  map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y)) 
  filter(F, nil) > nil 
  filter2(false, F, X, Y) >= filter(F, Y) 

With these choices, we have:

  1] D(t) >= _|_  by (Bot) 

  2] D(constant) >= _|_  by (Bot) 

  3] D(!facplus(X, Y)) >= !facplus(D(X), D(Y))  because [4], by (Star) 
  4] D*(!facplus(X, Y)) >= !facplus(D(X), D(Y))  because D > !facplus, [5] and [9], by (Copy) 
  5] D*(!facplus(X, Y)) >= D(X)  because D in Mul and [6], by (Stat) 
  6] !facplus(X, Y) > X  because [7], by definition 
  7] !facplus*(X, Y) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 
  9] D*(!facplus(X, Y)) >= D(Y)  because D in Mul and [10], by (Stat) 
  10] !facplus(X, Y) > Y  because [11], by definition 
  11] !facplus*(X, Y) >= Y  because [12], by (Select) 
  12] Y >= Y  by (Meta) 

  13] D(!facminus(X, Y)) > !facminus(D(X), D(Y))  because [14], by definition 
  14] D*(!facminus(X, Y)) >= !facminus(D(X), D(Y))  because D > !facminus, [15] and [19], by (Copy) 
  15] D*(!facminus(X, Y)) >= D(X)  because D in Mul and [16], by (Stat) 
  16] !facminus(X, Y) > X  because [17], by definition 
  17] !facminus*(X, Y) >= X  because [18], by (Select) 
  18] X >= X  by (Meta) 
  19] D*(!facminus(X, Y)) >= D(Y)  because D in Mul and [20], by (Stat) 
  20] !facminus(X, Y) > Y  because [21], by definition 
  21] !facminus*(X, Y) >= Y  because [22], by (Select) 
  22] Y >= Y  by (Meta) 

  23] map(F, nil) >= nil  because [24], by (Star) 
  24] map*(F, nil) >= nil  because [25], by (Select) 
  25] nil >= nil  by (Fun) 

  26] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [27], by (Star) 
  27] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [28] and [35], by (Copy) 
  28] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [29] and [31], by (Copy) 
  29] map*(F, cons(X, Y)) >= F  because [30], by (Select) 
  30] F >= F  by (Meta) 
  31] map*(F, cons(X, Y)) >= X  because [32], by (Select) 
  32] cons(X, Y) >= X  because [33], by (Star) 
  33] cons*(X, Y) >= X  because [34], by (Select) 
  34] X >= X  by (Meta) 
  35] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [36] and [37], by (Stat) 
  36] F >= F  by (Meta) 
  37] cons(X, Y) > Y  because [38], by definition 
  38] cons*(X, Y) >= Y  because [39], by (Select) 
  39] Y >= Y  by (Meta) 

  40] filter(F, nil) > nil  because [41], by definition 
  41] filter*(F, nil) >= nil  because [25], by (Select) 

  42] filter2(false, F, X, Y) >= filter(F, Y)  because [43], by (Star) 
  43] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 > filter, [44] and [46], by (Copy) 
  44] filter2*(false, F, X, Y) >= F  because [45], by (Select) 
  45] F >= F  by (Meta) 
  46] filter2*(false, F, X, Y) >= Y  because [47], by (Select) 
  47] Y >= Y  by (Meta) 

We can thus remove the following rules:

  D(!facminus(X, Y)) => !facminus(D(X), D(Y)) 
  filter(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]):

  D(t) >? 1 
  D(constant) >? 0 
  D(!facplus(X, Y)) >? !facplus(D(X), D(Y)) 
  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(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]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {} and Mul = {!facplus, @_{o -> o}, D, cons, constant, false, filter, filter2, map, t}, and the following precedence: filter2 > filter > D > !facplus > constant > map > cons > false > @_{o -> o} > t

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

With these choices, we have:

  1] D(t) >= _|_  by (Bot) 

  2] D(constant) >= _|_  by (Bot) 

  3] D(!facplus(X, Y)) > !facplus(D(X), D(Y))  because [4], by definition 
  4] D*(!facplus(X, Y)) >= !facplus(D(X), D(Y))  because D > !facplus, [5] and [9], by (Copy) 
  5] D*(!facplus(X, Y)) >= D(X)  because D in Mul and [6], by (Stat) 
  6] !facplus(X, Y) > X  because [7], by definition 
  7] !facplus*(X, Y) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 
  9] D*(!facplus(X, Y)) >= D(Y)  because D in Mul and [10], by (Stat) 
  10] !facplus(X, Y) > Y  because [11], by definition 
  11] !facplus*(X, Y) >= Y  because [12], by (Select) 
  12] Y >= Y  by (Meta) 

  13] map(F, _|_) > _|_  because [14], by definition 
  14] map*(F, _|_) >= _|_  by (Bot) 

  15] map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, Y))  because [16], by definition 
  16] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [17] and [24], by (Copy) 
  17] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [18] and [20], by (Copy) 
  18] map*(F, cons(X, Y)) >= F  because [19], by (Select) 
  19] F >= F  by (Meta) 
  20] map*(F, cons(X, Y)) >= X  because [21], by (Select) 
  21] cons(X, Y) >= X  because [22], by (Star) 
  22] cons*(X, Y) >= X  because [23], by (Select) 
  23] X >= X  by (Meta) 
  24] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [25] and [26], by (Stat) 
  25] F >= F  by (Meta) 
  26] cons(X, Y) > Y  because [27], by definition 
  27] cons*(X, Y) >= Y  because [28], by (Select) 
  28] Y >= Y  by (Meta) 

  29] filter2(false, F, X, Y) >= filter(F, Y)  because [30], by (Star) 
  30] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 > filter, [31] and [33], by (Copy) 
  31] filter2*(false, F, X, Y) >= F  because [32], by (Select) 
  32] F >= F  by (Meta) 
  33] filter2*(false, F, X, Y) >= Y  because [34], by (Select) 
  34] Y >= Y  by (Meta) 

We can thus remove the following rules:

  D(!facplus(X, Y)) => !facplus(D(X), D(Y)) 
  map(F, nil) => nil 
  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]):

  D(t) >? 1 
  D(constant) >? 0 
  filter2(false, F, X, Y) >? filter(F, Y) 

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

Argument functions:

  [[0]] = _|_ 
  [[1]] = _|_ 
  [[D(x_1)]] = x_1 

We choose Lex = {} and Mul = {constant, false, filter, filter2, t}, and the following precedence: constant > false > filter2 > filter > t

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

  t >= _|_ 
  constant >= _|_ 
  filter2(false, F, X, Y) > filter(F, Y) 

With these choices, we have:

  1] t >= _|_  by (Bot) 

  2] constant >= _|_  by (Bot) 

  3] filter2(false, F, X, Y) > filter(F, Y)  because [4], by definition 
  4] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 > filter, [5] and [7], by (Copy) 
  5] filter2*(false, F, X, Y) >= F  because [6], by (Select) 
  6] F >= F  by (Meta) 
  7] filter2*(false, F, X, Y) >= Y  because [8], by (Select) 
  8] Y >= Y  by (Meta) 

We can thus remove the following rules:

  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]):

  D(t) >? 1 
  D(constant) >? 0 

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

Argument functions:

  [[0]] = _|_ 
  [[1]] = _|_ 

We choose Lex = {} and Mul = {D, constant, t}, and the following precedence: D > constant > t

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

  D(t) > _|_ 
  D(constant) >= _|_ 

With these choices, we have:

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

  3] D(constant) >= _|_  by (Bot) 

We can thus remove the following rules:

  D(t) => 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]):

  D(constant) >? 0 

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

Argument functions:

  [[0]] = _|_ 

We choose Lex = {} and Mul = {D, constant}, and the following precedence: D > constant

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

  D(constant) > _|_ 

With these choices, we have:

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

We can thus remove the following rules:

  D(constant) => 0 

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.