We consider the system Applicative_first_order_05__30.

  Alphabet:

    !faccolon : [a * a] --> a 
    C : [] --> 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 
    true : [] --> b 

  Rules:

    !faccolon(!faccolon(!faccolon(!faccolon(C, x), y), z), u) => !faccolon(!faccolon(x, z), !faccolon(!faccolon(!faccolon(x, y), z), u)) 
    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]):

  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >? !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  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 = {!faccolon, cons, filter, filter2, true} and Mul = {@_{o -> o}, C, false, map, nil}, and the following precedence: C > false > !faccolon > map > nil > filter = filter2 = true > cons > @_{o -> o}

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

  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) > !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  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] !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) > !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [2], by definition 
  2] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U))  because [3], [15] and [18], by (Stat) 
  3] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(X, Z)  because [4], by definition 
  4] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [5], [10], [12], [5] and [14], by (Stat) 
  5] !faccolon(!faccolon(C, X), Y) > X  because [6], by definition 
  6] !faccolon*(!faccolon(C, X), Y) >= X  because [7], by (Select) 
  7] !faccolon(C, X) >= X  because [8], by (Star) 
  8] !faccolon*(C, X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= X  because [11], by (Select) 
  11] !faccolon(!faccolon(C, X), Y) >= X  because [6], by (Star) 
  12] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= Z  because [13], by (Select) 
  13] Z >= Z  by (Meta) 
  14] Z >= Z  by (Meta) 
  15] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(X, Z)  because [16], by (Select) 
  16] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Z)  because [17] and [14], by (Fun) 
  17] !faccolon(!faccolon(C, X), Y) >= X  because [6], by (Star) 
  18] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(!faccolon(X, Y), Z), U)  because [19], [31] and [34], by (Stat) 
  19] !faccolon(!faccolon(!faccolon(C, X), Y), Z) > !faccolon(!faccolon(X, Y), Z)  because [20], by definition 
  20] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [21], [28] and [12], by (Stat) 
  21] !faccolon(!faccolon(C, X), Y) > !faccolon(X, Y)  because [22], by definition 
  22] !faccolon*(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [23], [6], [25], [23] and [27], by (Stat) 
  23] !faccolon(C, X) > X  because [24], by definition 
  24] !faccolon*(C, X) >= X  because [9], by (Select) 
  25] !faccolon*(!faccolon(C, X), Y) >= Y  because [26], by (Select) 
  26] Y >= Y  by (Meta) 
  27] Y >= Y  by (Meta) 
  28] !faccolon*(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(X, Y)  because [29], by (Select) 
  29] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [30] and [27], by (Fun) 
  30] !faccolon(C, X) >= X  because [24], by (Star) 
  31] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= !faccolon(!faccolon(X, Y), Z)  because [32], by (Select) 
  32] !faccolon(!faccolon(!faccolon(C, X), Y), Z) >= !faccolon(!faccolon(X, Y), Z)  because [33] and [14], by (Fun) 
  33] !faccolon(!faccolon(C, X), Y) >= !faccolon(X, Y)  because [30] and [27], by (Fun) 
  34] !faccolon*(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) >= U  because [35], by (Select) 
  35] U >= U  by (Meta) 

  36] map(F, nil) > nil  because [37], by definition 
  37] map*(F, nil) >= nil  because [38], by (Select) 
  38] nil >= nil  by (Fun) 

  39] map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, Y))  because [40], by definition 
  40] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [41] and [48], by (Copy) 
  41] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [42] and [44], by (Copy) 
  42] map*(F, cons(X, Y)) >= F  because [43], by (Select) 
  43] F >= F  by (Meta) 
  44] map*(F, cons(X, Y)) >= X  because [45], by (Select) 
  45] cons(X, Y) >= X  because [46], by (Star) 
  46] cons*(X, Y) >= X  because [47], by (Select) 
  47] X >= X  by (Meta) 
  48] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [49] and [50], by (Stat) 
  49] F >= F  by (Meta) 
  50] cons(X, Y) > Y  because [51], by definition 
  51] cons*(X, Y) >= Y  because [52], by (Select) 
  52] Y >= Y  by (Meta) 

  53] filter(F, nil) > nil  because [54], by definition 
  54] filter*(F, nil) >= nil  because [38], by (Select) 

  55] filter(F, cons(X, Y)) > filter2(@_{o -> o}(F, X), F, X, Y)  because [56], by definition 
  56] filter*(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y)  because filter = filter2, [57], [60], [61], [63] and [67], by (Stat) 
  57] cons(X, Y) > Y  because [58], by definition 
  58] cons*(X, Y) >= Y  because [59], by (Select) 
  59] Y >= Y  by (Meta) 
  60] filter*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because filter > @_{o -> o}, [61] and [63], by (Copy) 
  61] filter*(F, cons(X, Y)) >= F  because [62], by (Select) 
  62] F >= F  by (Meta) 
  63] filter*(F, cons(X, Y)) >= X  because [64], by (Select) 
  64] cons(X, Y) >= X  because [65], by (Star) 
  65] cons*(X, Y) >= X  because [66], by (Select) 
  66] X >= X  by (Meta) 
  67] filter*(F, cons(X, Y)) >= Y  because [68], by (Select) 
  68] cons(X, Y) >= Y  because [58], by (Star) 

  69] filter2(true, F, X, Y) > cons(X, filter(F, Y))  because [70], by definition 
  70] filter2*(true, F, X, Y) >= cons(X, filter(F, Y))  because filter2 > cons, [71] and [73], by (Copy) 
  71] filter2*(true, F, X, Y) >= X  because [72], by (Select) 
  72] X >= X  by (Meta) 
  73] filter2*(true, F, X, Y) >= filter(F, Y)  because filter2 = filter, [74], [75], [76], [77], [74] and [75], by (Stat) 
  74] F >= F  by (Meta) 
  75] Y >= Y  by (Meta) 
  76] filter2*(true, F, X, Y) >= F  because [74], by (Select) 
  77] filter2*(true, F, X, Y) >= Y  because [75], by (Select) 

  78] filter2(false, F, X, Y) > filter(F, Y)  because [79], by definition 
  79] filter2*(false, F, X, Y) >= filter(F, Y)  because filter2 = filter, [80], [81], [82], [83], [80] and [81], by (Stat) 
  80] F >= F  by (Meta) 
  81] Y >= Y  by (Meta) 
  82] filter2*(false, F, X, Y) >= F  because [80], by (Select) 
  83] filter2*(false, F, X, Y) >= Y  because [81], by (Select) 

We can thus remove the following rules:

  !faccolon(!faccolon(!faccolon(!faccolon(C, X), Y), Z), U) => !faccolon(!faccolon(X, Z), !faccolon(!faccolon(!faccolon(X, Y), Z), U)) 
  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) 

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.