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_1, x_3) 
  [[nil]] = _|_ 

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

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 [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 (Star) 
  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, _|_) >= _|_  by (Bot) 

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

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

  64] filter(F, cons(X, Y)) > filter2(@_{o -> o}(F, X), F, X, Y)  because [65], by definition 
  65] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [66], [67], [70], [74], [75] and [77], by (Stat) 
  66] F >= F  by (Meta) 
  67] cons(X, Y) > Y  because [68], by definition 
  68] cons*(X, Y) >= Y  because [69], by (Select) 
  69] Y >= Y  by (Meta) 
  70] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter = @_{o -> o}, [66], [71], [74] and [75], by (Stat) 
  71] cons(X, Y) > X  because [72], by definition 
  72] cons*(X, Y) >= X  because [73], by (Select) 
  73] X >= X  by (Meta) 
  74] filter*(F, cons(X, Y)) >= F  because [66], by (Select) 
  75] filter*(F, cons(X, Y)) >= X  because [76], by (Select) 
  76] cons(X, Y) >= X  because [72], by (Star) 
  77] filter*(F, cons(X, Y)) >= Y  because [78], by (Select) 
  78] cons(X, Y) >= Y  because [68], by (Star) 

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

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

We can thus remove the following rules:

  and(or(X, Y), Z) => or(and(Z, X), and(Z, Y)) 
  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 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)) 
  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 
  filter(F, nil) >? nil 

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

Argument functions:

  [[nil]] = _|_ 

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

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

With these choices, we have:

  1] not(not(X)) > X  because [2], by definition 
  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 (Star) 
  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] map(F, _|_) >= _|_  by (Bot) 

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

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

We can thus remove the following rules:

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

  not(and(X, Y)) >? or(not(X), not(Y)) 
  and(X, or(Y, Z)) >? or(and(X, Y), and(X, Z)) 
  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 = {and, filter, map, not, or}, and the following precedence: filter > map > and > not > or

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

  not(and(X, Y)) > or(not(X), not(Y)) 
  and(X, or(Y, Z)) > or(and(X, Y), and(X, Z)) 
  map(F, _|_) > _|_ 
  filter(F, _|_) >= _|_ 

With these choices, we have:

  1] not(and(X, Y)) > or(not(X), not(Y))  because [2], by definition 
  2] not*(and(X, Y)) >= or(not(X), not(Y))  because not > or, [3] and [8], by (Copy) 
  3] not*(and(X, Y)) >= not(X)  because [4], by (Select) 
  4] and(X, Y) >= not(X)  because [5], by (Star) 
  5] and*(X, Y) >= not(X)  because and > not and [6], by (Copy) 
  6] and*(X, Y) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] not*(and(X, Y)) >= not(Y)  because not in Mul and [9], by (Stat) 
  9] and(X, Y) > Y  because [10], by definition 
  10] and*(X, Y) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 

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

  23] map(F, _|_) > _|_  because [24], by definition 
  24] map*(F, _|_) >= _|_  by (Bot) 

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

We can thus remove the following rules:

  not(and(X, Y)) => or(not(X), not(Y)) 
  and(X, or(Y, Z)) => or(and(X, Y), and(X, Z)) 
  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.