We consider the system h62.

  Alphabet:

    app : [list * list] --> list 
    cons : [nat * list] --> list 
    foldl : [list -> nat -> list * list * list] --> list 
    iconsc : [] --> list -> nat -> list 
    nil : [] --> list 
    reverse : [list] --> list 
    reverse1 : [list] --> list 
    xap : [list -> nat -> list * list] --> nat -> list 
    yap : [nat -> list * nat] --> list 

  Rules:

    app(nil, x) => x 
    app(cons(x, y), z) => cons(x, app(y, z)) 
    foldl(/\x./\y.yap(xap(f, x), y), z, nil) => z 
    foldl(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => foldl(/\w./\x'.yap(xap(f, w), x'), yap(xap(f, z), u), v) 
    iconsc x y => cons(y, x) 
    reverse(x) => foldl(/\y./\z.yap(xap(iconsc, y), z), nil, x) 
    reverse1(x) => foldl(/\y./\z.app(cons(z, nil), y), nil, x) 
    xap(f, x) => f x 
    yap(f, x) => f x 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

Symbol xap is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:

  Alphabet:

    app : [list * list] --> list 
    cons : [nat * list] --> list 
    foldl : [list -> nat -> list * list * list] --> list 
    iconsc : [list] --> nat -> list 
    nil : [] --> list 
    reverse : [list] --> list 
    reverse1 : [list] --> list 
    yap : [nat -> list * nat] --> list 

  Rules:

    app(nil, X) => X 
    app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
    foldl(/\x./\y.yap(F(x), y), X, nil) => X 
    foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
    iconsc(X) Y => cons(Y, X) 
    reverse(X) => foldl(/\x./\y.yap(iconsc(x), y), nil, X) 
    reverse1(X) => foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
    yap(F, X) => F X 

We use rule removal, following [Kop12, Theorem 2.23].

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

  app(nil, X) >? X 
  app(cons(X, Y), Z) >? cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, nil) >? X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  iconsc(X) Y >? cons(Y, X) 
  reverse(X) >? foldl(/\x./\y.yap(iconsc(x), y), nil, X) 
  reverse1(X) >? foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
  yap(F, X) >? F X 

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

Argument functions:

  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
  [[nil]] = _|_ 

We choose Lex = {foldl} and Mul = {@_{o -> o}, app, cons, iconsc, reverse, reverse1, yap}, and the following precedence: reverse1 > reverse > foldl > @_{o -> o} = yap > app > cons > iconsc

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

  app(_|_, X) > X 
  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, _|_) > X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
  @_{o -> o}(iconsc(X), Y) >= cons(Y, X) 
  reverse(X) > foldl(/\x./\y.yap(iconsc(x), y), _|_, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X) 
  yap(F, X) >= @_{o -> o}(F, X) 

With these choices, we have:

  1] app(_|_, X) > X  because [2], by definition 
  2] app*(_|_, X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] app(cons(X, Y), Z) >= cons(X, app(Y, Z))  because [5], by (Star) 
  5] app*(cons(X, Y), Z) >= cons(X, app(Y, Z))  because app > cons, [6] and [10], by (Copy) 
  6] app*(cons(X, Y), Z) >= X  because [7], by (Select) 
  7] cons(X, Y) >= X  because [8], by (Star) 
  8] cons*(X, Y) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] app*(cons(X, Y), Z) >= app(Y, Z)  because app in Mul, [11] and [14], by (Stat) 
  11] cons(X, Y) > Y  because [12], by definition 
  12] cons*(X, Y) >= Y  because [13], by (Select) 
  13] Y >= Y  by (Meta) 
  14] Z >= Z  by (Meta) 

  15] foldl(/\x./\y.yap(F(x), y), X, _|_) > X  because [16], by definition 
  16] foldl*(/\x./\y.yap(F(x), y), X, _|_) >= X  because [17], by (Select) 
  17] X >= X  by (Meta) 

  18] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [19], by (Star) 
  19] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [20], by (Select) 
  20] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [21], by (Star) 
  21] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [22], by (Select) 
  22] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [23], [26], [34] and [47], by (Stat) 
  23] cons(Y, Z) > Z  because [24], by definition 
  24] cons*(Y, Z) >= Z  because [25], by (Select) 
  25] Z >= Z  by (Meta) 
  26] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [27], by (F-Abs) 
  27] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= /\x.yap(F(z), x)  because [28], by (Select) 
  28] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z), z)), x) >= /\x.yap(F(z), x)  because [29], by (Abs) 
  29] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z)), u) >= yap(F(z), u)  because yap in Mul, [30] and [33], by (Fun) 
  30] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z)) >= F(z)  because [31], by (Meta) 
  31] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= z  because [32], by (Select) 
  32] z >= z  by (Var) 
  33] u >= u  by (Var) 
  34] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because [35], by (Select) 
  35] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= yap(F(X), Y)  because [36], by (Star) 
  36] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= yap(F(X), Y)  because [37], by (Select) 
  37] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [38] and [43], by (Copy) 
  38] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [39], by (Select) 
  39] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z))), x) >= F(X)  because [40], by (Eta)[Kop13:2] 
  40] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [41], by (Meta) 
  41] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [42], by (Select) 
  42] X >= X  by (Meta) 
  43] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [44], by (Select) 
  44] cons(Y, Z) >= Y  because [45], by (Star) 
  45] cons*(Y, Z) >= Y  because [46], by (Select) 
  46] Y >= Y  by (Meta) 
  47] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [48], by (Select) 
  48] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= Z  because [49], by (Star) 
  49] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= Z  because [50], by (Select) 
  50] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [51], by (Select) 
  51] cons(Y, Z) >= Z  because [24], by (Star) 

  52] @_{o -> o}(iconsc(X), Y) >= cons(Y, X)  because [53], by (Star) 
  53] @_{o -> o}*(iconsc(X), Y) >= cons(Y, X)  because @_{o -> o} > cons, [54] and [56], by (Copy) 
  54] @_{o -> o}*(iconsc(X), Y) >= Y  because [55], by (Select) 
  55] Y >= Y  by (Meta) 
  56] @_{o -> o}*(iconsc(X), Y) >= X  because [57], by (Select) 
  57] iconsc(X) @_{o -> o}*(iconsc(X), Y) >= X  because [58] 
  58] iconsc*(X, @_{o -> o}*(iconsc(X), Y)) >= X  because [59], by (Select) 
  59] X >= X  by (Meta) 

  60] reverse(X) > foldl(/\x./\y.yap(iconsc(x), y), _|_, X)  because [61], by definition 
  61] reverse*(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X)  because reverse > foldl, [62], [70] and [71], by (Copy) 
  62] reverse*(X) >= /\y./\z.yap(iconsc(y), z)  because [63], by (F-Abs) 
  63] reverse*(X, x) >= /\z.yap(iconsc(x), z)  because [64], by (F-Abs) 
  64] reverse*(X, x, y) >= yap(iconsc(x), y)  because reverse > yap, [65] and [68], by (Copy) 
  65] reverse*(X, x, y) >= iconsc(x)  because reverse > iconsc and [66], by (Copy) 
  66] reverse*(X, x, y) >= x  because [67], by (Select) 
  67] x >= x  by (Var) 
  68] reverse*(X, x, y) >= y  because [69], by (Select) 
  69] y >= y  by (Var) 
  70] reverse*(X) >= _|_  by (Bot) 
  71] reverse*(X) >= X  because [72], by (Select) 
  72] X >= X  by (Meta) 

  73] reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [74], by (Star) 
  74] reverse1*(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because reverse1 > foldl, [75], [84] and [85], by (Copy) 
  75] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [76], by (F-Abs) 
  76] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [77], by (F-Abs) 
  77] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [78] and [82], by (Copy) 
  78] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [79] and [81], by (Copy) 
  79] reverse1*(X, x, y) >= y  because [80], by (Select) 
  80] y >= y  by (Var) 
  81] reverse1*(X, x, y) >= _|_  by (Bot) 
  82] reverse1*(X, x, y) >= x  because [83], by (Select) 
  83] x >= x  by (Var) 
  84] reverse1*(X) >= _|_  by (Bot) 
  85] reverse1*(X) >= X  because [86], by (Select) 
  86] X >= X  by (Meta) 

  87] yap(F, X) >= @_{o -> o}(F, X)  because yap = @_{o -> o}, yap in Mul, [88] and [89], by (Fun) 
  88] F >= F  by (Meta) 
  89] X >= X  by (Meta) 

We can thus remove the following rules:

  app(nil, X) => X 
  foldl(/\x./\y.yap(F(x), y), X, nil) => X 
  reverse(X) => foldl(/\x./\y.yap(iconsc(x), y), nil, X) 

We use rule removal, following [Kop12, Theorem 2.23].

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

  app(cons(X, Y), Z) >? cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  iconsc(X, Y) >? cons(Y, X) 
  reverse1(X) >? foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
  yap(F, X) >? F X 

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

Argument functions:

  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
  [[nil]] = _|_ 

We choose Lex = {foldl} and Mul = {@_{o -> o}, app, cons, iconsc, reverse1, yap}, and the following precedence: reverse1 > foldl > app > cons = iconsc > @_{o -> o} = yap

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

  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
  iconsc(X, Y) >= cons(Y, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X) 
  yap(F, X) >= @_{o -> o}(F, X) 

With these choices, we have:

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

  12] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [13], by definition 
  13] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [14], [17], [33] and [43], by (Stat) 
  14] cons(Y, Z) > Z  because [15], by definition 
  15] cons*(Y, Z) >= Z  because [16], by (Select) 
  16] Z >= Z  by (Meta) 
  17] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [18], by (F-Abs) 
  18] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= /\x.yap(F(z), x)  because [19], by (F-Abs) 
  19] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= yap(F(z), u)  because foldl > yap, [20] and [31], by (Copy) 
  20] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= F(z)  because [21], by (Select) 
  21] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z), z, u)), x) >= F(z)  because [22], by (Eta)[Kop13:2] 
  22] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)) >= F(z)  because [23], by (Meta) 
  23] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= z  because [24], by (Select) 
  24] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z), z, u)) >= z  because [25], by (Star) 
  25] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z), z, u)) >= z  because [26], by (Select) 
  26] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= z  because [27], by (Select) 
  27] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z), z, u)) >= z  because [28], by (Star) 
  28] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z), z, u)) >= z  because [29], by (Select) 
  29] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= z  because [30], by (Select) 
  30] z >= z  by (Var) 
  31] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= u  because [32], by (Select) 
  32] u >= u  by (Var) 
  33] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [34] and [39], by (Copy) 
  34] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [35], by (Select) 
  35] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z))), x) >= F(X)  because [36], by (Eta)[Kop13:2] 
  36] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [37], by (Meta) 
  37] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [38], by (Select) 
  38] X >= X  by (Meta) 
  39] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [40], by (Select) 
  40] cons(Y, Z) >= Y  because [41], by (Star) 
  41] cons*(Y, Z) >= Y  because [42], by (Select) 
  42] Y >= Y  by (Meta) 
  43] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [44], by (Select) 
  44] cons(Y, Z) >= Z  because [15], by (Star) 

  45] iconsc(X, Y) >= cons(Y, X)  because iconsc = cons, iconsc in Mul, [46] and [47], by (Fun) 
  46] X >= X  by (Meta) 
  47] Y >= Y  by (Meta) 

  48] reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [49], by (Star) 
  49] reverse1*(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because reverse1 > foldl, [50], [59] and [60], by (Copy) 
  50] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [51], by (F-Abs) 
  51] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [52], by (F-Abs) 
  52] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [53] and [57], by (Copy) 
  53] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [54] and [56], by (Copy) 
  54] reverse1*(X, x, y) >= y  because [55], by (Select) 
  55] y >= y  by (Var) 
  56] reverse1*(X, x, y) >= _|_  by (Bot) 
  57] reverse1*(X, x, y) >= x  because [58], by (Select) 
  58] x >= x  by (Var) 
  59] reverse1*(X) >= _|_  by (Bot) 
  60] reverse1*(X) >= X  because [61], by (Select) 
  61] X >= X  by (Meta) 

  62] yap(F, X) >= @_{o -> o}(F, X)  because yap = @_{o -> o}, yap in Mul, [63] and [64], by (Fun) 
  63] F >= F  by (Meta) 
  64] X >= X  by (Meta) 

We can thus remove the following rules:

  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 

We use rule removal, following [Kop12, Theorem 2.23].

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

  app(cons(X, Y), Z) >? cons(X, app(Y, Z)) 
  iconsc(X, Y) >? cons(Y, X) 
  reverse1(X) >? foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
  yap(F, X) >? F X 

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

Argument functions:

  [[nil]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, app, cons, foldl, iconsc, reverse1, yap}, and the following precedence: yap > reverse1 > @_{o -> o} > foldl > app > cons = iconsc

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

  app(cons(X, Y), Z) > cons(X, app(Y, Z)) 
  iconsc(X, Y) >= cons(Y, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X) 
  yap(F, X) > @_{o -> o}(F, X) 

With these choices, we have:

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

  12] iconsc(X, Y) >= cons(Y, X)  because iconsc = cons, iconsc in Mul, [13] and [14], by (Fun) 
  13] X >= X  by (Meta) 
  14] Y >= Y  by (Meta) 

  15] reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [16], by (Star) 
  16] reverse1*(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because reverse1 > foldl, [17], [26] and [27], by (Copy) 
  17] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [18], by (F-Abs) 
  18] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [19], by (F-Abs) 
  19] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [20] and [24], by (Copy) 
  20] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [21] and [23], by (Copy) 
  21] reverse1*(X, x, y) >= y  because [22], by (Select) 
  22] y >= y  by (Var) 
  23] reverse1*(X, x, y) >= _|_  by (Bot) 
  24] reverse1*(X, x, y) >= x  because [25], by (Select) 
  25] x >= x  by (Var) 
  26] reverse1*(X) >= _|_  by (Bot) 
  27] reverse1*(X) >= X  because [28], by (Select) 
  28] X >= X  by (Meta) 

  29] yap(F, X) > @_{o -> o}(F, X)  because [30], by definition 
  30] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [31] and [33], by (Copy) 
  31] yap*(F, X) >= F  because [32], by (Select) 
  32] F >= F  by (Meta) 
  33] yap*(F, X) >= X  because [34], by (Select) 
  34] X >= X  by (Meta) 

We can thus remove the following rules:

  app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
  yap(F, X) => F X 

We use rule removal, following [Kop12, Theorem 2.23].

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

  iconsc(X, Y) >? cons(Y, X) 
  reverse1(X) >? foldl(/\x./\y.app(cons(y, nil), x), nil, X) 

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

Argument functions:

  [[nil]] = _|_ 

We choose Lex = {} and Mul = {app, cons, foldl, iconsc, reverse1}, and the following precedence: reverse1 > app > iconsc > cons > foldl

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

  iconsc(X, Y) > cons(Y, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X) 

With these choices, we have:

  1] iconsc(X, Y) > cons(Y, X)  because [2], by definition 
  2] iconsc*(X, Y) >= cons(Y, X)  because iconsc > cons, [3] and [5], by (Copy) 
  3] iconsc*(X, Y) >= Y  because [4], by (Select) 
  4] Y >= Y  by (Meta) 
  5] iconsc*(X, Y) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 

  7] reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [8], by (Star) 
  8] reverse1*(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because reverse1 > foldl, [9], [18] and [19], by (Copy) 
  9] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [10], by (F-Abs) 
  10] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [11], by (F-Abs) 
  11] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [12] and [16], by (Copy) 
  12] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [13] and [15], by (Copy) 
  13] reverse1*(X, x, y) >= y  because [14], by (Select) 
  14] y >= y  by (Var) 
  15] reverse1*(X, x, y) >= _|_  by (Bot) 
  16] reverse1*(X, x, y) >= x  because [17], by (Select) 
  17] x >= x  by (Var) 
  18] reverse1*(X) >= _|_  by (Bot) 
  19] reverse1*(X) >= X  because [20], by (Select) 
  20] X >= X  by (Meta) 

We can thus remove the following rules:

  iconsc(X, Y) => cons(Y, X) 

We use rule removal, following [Kop12, Theorem 2.23].

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

  reverse1(X) >? foldl(/\x./\y.app(cons(y, nil), x), nil, X) 

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

Argument functions:

  [[nil]] = _|_ 

We choose Lex = {} and Mul = {app, cons, foldl, reverse1}, and the following precedence: reverse1 > cons > app > foldl

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

  reverse1(X) > foldl(/\x./\y.app(cons(y, _|_), x), _|_, X) 

With these choices, we have:

  1] reverse1(X) > foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [2], by definition 
  2] reverse1*(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because reverse1 > foldl, [3], [12] and [13], by (Copy) 
  3] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [4], by (F-Abs) 
  4] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [5], by (F-Abs) 
  5] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [6] and [10], by (Copy) 
  6] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [7] and [9], by (Copy) 
  7] reverse1*(X, x, y) >= y  because [8], by (Select) 
  8] y >= y  by (Var) 
  9] reverse1*(X, x, y) >= _|_  by (Bot) 
  10] reverse1*(X, x, y) >= x  because [11], by (Select) 
  11] x >= x  by (Var) 
  12] reverse1*(X) >= _|_  by (Bot) 
  13] reverse1*(X) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 

We can thus remove the following rules:

  reverse1(X) => foldl(/\x./\y.app(cons(y, nil), x), nil, 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.
[Kop13:2]  C. Kop.  StarHorpo with an Eta-Rule.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/etahorpo.pdf, 2013.