We consider the system Applicative_first_order_05__18.

  Alphabet:

    !facplus : [b * b] --> b 
    !factimes : [a * b] --> b 
    cons : [d * e] --> e 
    false : [] --> c 
    filter : [d -> c * e] --> e 
    filter2 : [c * d -> c * d * e] --> e 
    map : [d -> d * e] --> e 
    nil : [] --> e 
    true : [] --> c 

  Rules:

    !factimes(x, !facplus(y, z)) => !facplus(!factimes(x, y), !factimes(x, z)) 
    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]):

  !factimes(X, !facplus(Y, Z)) >? !facplus(!factimes(X, Y), !factimes(X, Z)) 
  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:

  [[cons(x_1, x_2)]] = cons(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_1, x_3) 

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

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

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

  12] map(F, nil) >= nil  because [13], by (Star) 
  13] map*(F, nil) >= nil  because [14], by (Select) 
  14] nil >= nil  by (Fun) 

  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] filter(F, nil) > nil  because [30], by definition 
  30] filter*(F, nil) >= nil  because [14], by (Select) 

  31] filter(F, cons(X, Y)) > filter2(@_{o -> o}(F, X), F, X, Y)  because [32], by definition 
  32] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [33], [36], [37], [39] and [43], by (Stat) 
  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, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [37] and [39], by (Copy) 
  37] filter*(F, cons(X, Y)) >= F  because [38], by (Select) 
  38] F >= F  by (Meta) 
  39] filter*(F, cons(X, Y)) >= X  because [40], by (Select) 
  40] cons(X, Y) >= X  because [41], by (Star) 
  41] cons*(X, Y) >= X  because [42], by (Select) 
  42] X >= X  by (Meta) 
  43] filter*(F, cons(X, Y)) >= Y  because [44], by (Select) 
  44] cons(X, Y) >= Y  because [34], by (Star) 

  45] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [46], by (Star) 
  46] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [47] and [49], by (Copy) 
  47] filter2*(true, F, X, Y) >= X  because [48], by (Select) 
  48] X >= X  by (Meta) 
  49] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [50], [51], [52], [53], [50] and [51], by (Stat) 
  50] F >= F  by (Meta) 
  51] Y >= Y  by (Meta) 
  52] filter2*(true, F, X, Y) >= F  because [50], by (Select) 
  53] filter2*(true, F, X, Y) >= Y  because [51], by (Select) 

  54] filter2(false, F, X, Y) >= filter(F, Y)  because [55], by (Star) 
  55] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [56], [57], [58], [59], [56] and [57], by (Stat) 
  56] F >= F  by (Meta) 
  57] Y >= Y  by (Meta) 
  58] filter2*(false, F, X, Y) >= F  because [56], by (Select) 
  59] filter2*(false, F, X, Y) >= Y  because [57], by (Select) 

We can thus remove the following rules:

  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 rule removal, following [Kop12, Theorem 2.23].

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

  !factimes(X, !facplus(Y, Z)) >? !facplus(!factimes(X, Y), !factimes(X, Z)) 
  map(F, nil) >? nil 
  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].

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

With these choices, we have:

  1] !factimes(X, !facplus(Y, Z)) >= !facplus(!factimes(X, Y), !factimes(X, Z))  because [2], by (Star) 
  2] !factimes*(X, !facplus(Y, Z)) >= !facplus(!factimes(X, Y), !factimes(X, Z))  because !factimes > !facplus, [3] and [8], by (Copy) 
  3] !factimes*(X, !facplus(Y, Z)) >= !factimes(X, Y)  because !factimes in Mul, [4] and [5], by (Stat) 
  4] X >= X  by (Meta) 
  5] !facplus(Y, Z) > Y  because [6], by definition 
  6] !facplus*(Y, Z) >= Y  because [7], by (Select) 
  7] Y >= Y  by (Meta) 
  8] !factimes*(X, !facplus(Y, Z)) >= !factimes(X, Z)  because !factimes in Mul, [4] and [9], by (Stat) 
  9] !facplus(Y, Z) > Z  because [10], by definition 
  10] !facplus*(Y, Z) >= Z  because [11], by (Select) 
  11] Z >= Z  by (Meta) 

  12] map(F, nil) >= nil  because [13], by (Star) 
  13] map*(F, nil) >= nil  because map > nil, by (Copy) 

  14] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [15], by (Star) 
  15] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [16] and [18], by (Copy) 
  16] filter2*(true, F, X, Y) >= X  because [17], by (Select) 
  17] X >= X  by (Meta) 
  18] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 > filter, [19] and [21], by (Copy) 
  19] filter2*(true, F, X, Y) >= F  because [20], by (Select) 
  20] F >= F  by (Meta) 
  21] filter2*(true, F, X, Y) >= Y  because [22], by (Select) 
  22] Y >= Y  by (Meta) 

  23] filter2(false, F, X, Y) > filter(F, Y)  because [24], by definition 
  24] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 > filter, [25] and [27], by (Copy) 
  25] filter2*(false, F, X, Y) >= F  because [26], by (Select) 
  26] F >= F  by (Meta) 
  27] filter2*(false, F, X, Y) >= Y  because [28], by (Select) 
  28] 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]):

  !factimes(X, !facplus(Y, Z)) >? !facplus(!factimes(X, Y), !factimes(X, Z)) 
  map(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].

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

With these choices, we have:

  1] !factimes(X, !facplus(Y, Z)) >= !facplus(!factimes(X, Y), !factimes(X, Z))  because [2], by (Star) 
  2] !factimes*(X, !facplus(Y, Z)) >= !facplus(!factimes(X, Y), !factimes(X, Z))  because !factimes > !facplus, [3] and [8], by (Copy) 
  3] !factimes*(X, !facplus(Y, Z)) >= !factimes(X, Y)  because !factimes in Mul, [4] and [5], by (Stat) 
  4] X >= X  by (Meta) 
  5] !facplus(Y, Z) > Y  because [6], by definition 
  6] !facplus*(Y, Z) >= Y  because [7], by (Select) 
  7] Y >= Y  by (Meta) 
  8] !factimes*(X, !facplus(Y, Z)) >= !factimes(X, Z)  because !factimes in Mul, [4] and [9], by (Stat) 
  9] !facplus(Y, Z) > Z  because [10], by definition 
  10] !facplus*(Y, Z) >= Z  because [11], by (Select) 
  11] Z >= Z  by (Meta) 

  12] map(F, nil) > nil  because [13], by definition 
  13] map*(F, nil) >= nil  because [14], by (Select) 
  14] nil >= nil  by (Fun) 

  15] filter2(true, F, X, Y) >= cons(X, filter(F, Y))  because [16], by (Star) 
  16] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [17] and [19], by (Copy) 
  17] filter2*(true, F, X, Y) >= X  because [18], by (Select) 
  18] X >= X  by (Meta) 
  19] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 > filter, [20] and [22], by (Copy) 
  20] filter2*(true, F, X, Y) >= F  because [21], by (Select) 
  21] F >= F  by (Meta) 
  22] filter2*(true, F, X, Y) >= Y  because [23], by (Select) 
  23] Y >= Y  by (Meta) 

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]):

  !factimes(X, !facplus(Y, Z)) >? !facplus(!factimes(X, Y), !factimes(X, Z)) 
  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) 

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

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

  !factimes(X, !facplus(Y, Z)) >= !facplus(!factimes(X, Y), !factimes(X, Z)) 
  filter2(true, F, X, Y) > cons(X, filter(F, Y)) 

With these choices, we have:

  1] !factimes(X, !facplus(Y, Z)) >= !facplus(!factimes(X, Y), !factimes(X, Z))  because [2], by (Star) 
  2] !factimes*(X, !facplus(Y, Z)) >= !facplus(!factimes(X, Y), !factimes(X, Z))  because !factimes > !facplus, [3] and [8], by (Copy) 
  3] !factimes*(X, !facplus(Y, Z)) >= !factimes(X, Y)  because !factimes in Mul, [4] and [5], by (Stat) 
  4] X >= X  by (Meta) 
  5] !facplus(Y, Z) > Y  because [6], by definition 
  6] !facplus*(Y, Z) >= Y  because [7], by (Select) 
  7] Y >= Y  by (Meta) 
  8] !factimes*(X, !facplus(Y, Z)) >= !factimes(X, Z)  because !factimes in Mul, [4] and [9], by (Stat) 
  9] !facplus(Y, Z) > Z  because [10], by definition 
  10] !facplus*(Y, Z) >= Z  because [11], by (Select) 
  11] Z >= Z  by (Meta) 

  12] filter2(true, F, X, Y) > cons(X, filter(F, Y))  because [13], by definition 
  13] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [14] and [16], by (Copy) 
  14] filter2*(true, F, X, Y) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 
  16] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 > filter, [17] and [19], by (Copy) 
  17] filter2*(true, F, X, Y) >= F  because [18], by (Select) 
  18] F >= F  by (Meta) 
  19] filter2*(true, F, X, Y) >= Y  because [20], by (Select) 
  20] 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]):

  !factimes(X, !facplus(Y, Z)) >? !facplus(!factimes(X, Y), !factimes(X, Z)) 

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

Argument functions:

  [[!facplus(x_1, x_2)]] = !facplus(x_2, x_1) 

We choose Lex = {!facplus} and Mul = {!factimes}, and the following precedence: !factimes > !facplus

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

  !factimes(X, !facplus(Y, Z)) > !facplus(!factimes(X, Y), !factimes(X, Z)) 

With these choices, we have:

  1] !factimes(X, !facplus(Y, Z)) > !facplus(!factimes(X, Y), !factimes(X, Z))  because [2], by definition 
  2] !factimes*(X, !facplus(Y, Z)) >= !facplus(!factimes(X, Y), !factimes(X, Z))  because !factimes > !facplus, [3] and [8], by (Copy) 
  3] !factimes*(X, !facplus(Y, Z)) >= !factimes(X, Y)  because !factimes in Mul, [4] and [5], by (Stat) 
  4] X >= X  by (Meta) 
  5] !facplus(Y, Z) > Y  because [6], by definition 
  6] !facplus*(Y, Z) >= Y  because [7], by (Select) 
  7] Y >= Y  by (Meta) 
  8] !factimes*(X, !facplus(Y, Z)) >= !factimes(X, Z)  because !factimes in Mul, [4] and [9], by (Stat) 
  9] !facplus(Y, Z) > Z  because [10], by definition 
  10] !facplus*(Y, Z) >= Z  because [11], by (Select) 
  11] Z >= Z  by (Meta) 

We can thus remove the following rules:

  !factimes(X, !facplus(Y, Z)) => !facplus(!factimes(X, Y), !factimes(X, Z)) 

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.