We consider the system h53.

  Alphabet:

    cons : [a * alist] --> alist 
    foldl : [a -> a -> a * a * alist] --> a 
    nil : [] --> alist 
    xap : [a -> a -> a * a] --> a -> a 
    yap : [a -> a * a] --> a 

  Rules:

    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) 
    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:

    cons : [a * alist] --> alist 
    foldl : [a -> a -> a * a * alist] --> a 
    nil : [] --> alist 
    yap : [a -> a * a] --> a 

  Rules:

    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) 
    yap(F, X) => F X 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, all):

  Dependency Pairs P_0:

    0] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z)   
    1] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> yap#(F(z), u)   
    2] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(z)   
    3] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> yap#(F(X), Y)   
    4] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(X)   
    5] yap#(F, X) =#> F(X)   

  Rules R_0:

    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) 
    yap(F, X) => F X 

Thus, the original system is terminating if (P_0, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, all).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0, 1, 2, 3, 4 
    * 1 :  
    * 2 : 0, 1, 2, 3, 4, 5 
    * 3 : 5 
    * 4 : 0, 1, 2, 3, 4, 5 
    * 5 : 0, 1, 2, 3, 4, 5 

This graph has the following strongly connected components:

  P_1:

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

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).

Thus, the original system is terminating if (P_1, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, all).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(~c1) ~c0 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap#(F(X), Y) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(X) ~c2 
  yap#(F, X) >? F(X) 
  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) 
  yap(F, X) >= F X 
  foldl(F, X, Y) >= foldl#(F, X, Y) 
  yap(F, X) >= yap#(F, X) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( yap#(F, X) ) = #argfun-yap##(F X) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

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_2, x_1) 
  [[foldl#(x_1, x_2, x_3)]] = foldl#(x_3, x_2, x_1) 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 
  [[~c2]] = _|_ 

We choose Lex = {foldl, foldl#} and Mul = {#argfun-yap##, @_{o -> o}, cons, nil, yap, yap#}, and the following precedence: cons > foldl > nil > foldl# > yap > #argfun-yap## > @_{o -> o} > yap#

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

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

With these choices, we have:

  1] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > foldl#(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [2], by definition 
  2] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl#(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [3], [6], [13] and [29], by (Stat) 
  3] cons(Y, Z) > Z  because [4], by definition 
  4] cons*(Y, Z) >= Z  because [5], by (Select) 
  5] Z >= Z  by (Meta) 
  6] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [7], by (Select) 
  7] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [8], by (Abs) 
  8] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [9], by (Abs) 
  9] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [10] and [12], by (Fun) 
  10] F(y) >= F(y)  because [11], by (Meta) 
  11] y >= y  by (Var) 
  12] x >= x  by (Var) 
  13] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl# > yap, [14] and [25], by (Copy) 
  14] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= F(X)  because [15], by (Select) 
  15] /\z.yap(F(foldl#*(/\u./\v.yap(F(u), v), X, cons(Y, Z))), z) >= F(X)  because [16], by (Eta)[Kop13:2] 
  16] F(foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [17], by (Meta) 
  17] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [18], by (Select) 
  18] yap(F(foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl#*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= X  because [19], by (Star) 
  19] yap*(F(foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl#*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= X  because [20], by (Select) 
  20] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [21], by (Select) 
  21] yap(F(foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl#*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= X  because [22], by (Star) 
  22] yap*(F(foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl#*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= X  because [23], by (Select) 
  23] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [24], by (Select) 
  24] X >= X  by (Meta) 
  25] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y  because [26], by (Select) 
  26] cons(Y, Z) >= Y  because [27], by (Star) 
  27] cons*(Y, Z) >= Y  because [28], by (Select) 
  28] Y >= Y  by (Meta) 
  29] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z  because [30], by (Select) 
  30] cons(Y, Z) >= Z  because [4], by (Star) 

  31] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(_|_), _|_)  because [32], by (Star) 
  32] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(_|_), _|_)  because foldl# > @_{o -> o}, [33] and [37], by (Copy) 
  33] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(_|_)  because [34], by (Select) 
  34] /\x.yap(F(foldl#*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(_|_)  because [35], by (Eta)[Kop13:2] 
  35] F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(_|_)  because [36], by (Meta) 
  36] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 
  37] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 

  38] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > #argfun-yap##(@_{o -> o}(F(X), Y))  because [39], by definition 
  39] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because [40], by (Select) 
  40] yap(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because [41], by (Star) 
  41] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because yap > #argfun-yap## and [42], by (Copy) 
  42] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= @_{o -> o}(F(X), Y)  because [43], by (Select) 
  43] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), Y)  because foldl# > @_{o -> o}, [14] and [25], by (Copy) 

  44] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > @_{o -> o}(F(X), _|_)  because [45], by definition 
  45] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), _|_)  because foldl# > @_{o -> o}, [14] and [46], by (Copy) 
  46] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 

  47] #argfun-yap##(@_{o -> o}(F, X)) > @_{o -> o}(F, X)  because [48], by definition 
  48] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [49], by (Select) 
  49] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [50] and [51], by (Fun) 
  50] F >= F  by (Meta) 
  51] X >= X  by (Meta) 

  52] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [53], by (Star) 
  53] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [54], by (Select) 
  54] X >= X  by (Meta) 

  55] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [56], by (Star) 
  56] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [3], [57], [58] and [64], by (Stat) 
  57] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [7], by (Select) 
  58] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [59] and [63], by (Copy) 
  59] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [60], by (Select) 
  60] /\x.yap(F(foldl*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(X)  because [61], by (Eta)[Kop13:2] 
  61] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [62], by (Meta) 
  62] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [24], by (Select) 
  63] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [26], by (Select) 
  64] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [30], by (Select) 

  65] yap(F, X) >= @_{o -> o}(F, X)  because [66], by (Star) 
  66] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [67] and [68], by (Copy) 
  67] yap*(F, X) >= F  because [50], by (Select) 
  68] yap*(F, X) >= X  because [51], by (Select) 

  69] foldl(F, X, Y) >= foldl#(F, X, Y)  because [70], by (Star) 
  70] foldl*(F, X, Y) >= foldl#(F, X, Y)  because foldl > foldl#, [71], [73] and [75], by (Copy) 
  71] foldl*(F, X, Y) >= F  because [72], by (Select) 
  72] F >= F  by (Meta) 
  73] foldl*(F, X, Y) >= X  because [74], by (Select) 
  74] X >= X  by (Meta) 
  75] foldl*(F, X, Y) >= Y  because [76], by (Select) 
  76] Y >= Y  by (Meta) 

  77] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [78], by (Star) 
  78] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [79], by (Copy) 
  79] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [80] and [82], by (Copy) 
  80] yap*(F, X) >= F  because [81], by (Select) 
  81] F >= F  by (Meta) 
  82] yap*(F, X) >= X  because [83], by (Select) 
  83] X >= X  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_0, minimal, all) by (P_2, R_0, minimal, all), where P_2 consists of:

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

Thus, the original system is terminating if (P_2, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, all).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(~c1) ~c0 
  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) 
  yap(F, X) >= F X 
  foldl(F, X, Y) >= foldl#(F, X, Y) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

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) 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 

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

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

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

With these choices, we have:

  1] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > @_{o -> o}(F(_|_), _|_)  because [2], by definition 
  2] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(_|_), _|_)  because foldl# > @_{o -> o}, [3] and [7], by (Copy) 
  3] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(_|_)  because [4], by (Select) 
  4] /\x.yap(F(foldl#*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(_|_)  because [5], by (Eta)[Kop13:2] 
  5] F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(_|_)  because [6], by (Meta) 
  6] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 
  7] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 

  8] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [9], by (Star) 
  9] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 

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

  41] yap(F, X) >= @_{o -> o}(F, X)  because [42], by (Star) 
  42] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [43] and [45], by (Copy) 
  43] yap*(F, X) >= F  because [44], by (Select) 
  44] F >= F  by (Meta) 
  45] yap*(F, X) >= X  because [46], by (Select) 
  46] X >= X  by (Meta) 

  47] foldl(F, X, Y) >= foldl#(F, X, Y)  because [48], by (Star) 
  48] foldl*(F, X, Y) >= foldl#(F, X, Y)  because foldl > foldl#, [49], [51] and [53], by (Copy) 
  49] foldl*(F, X, Y) >= F  because [50], by (Select) 
  50] F >= F  by (Meta) 
  51] foldl*(F, X, Y) >= X  because [52], by (Select) 
  52] X >= X  by (Meta) 
  53] foldl*(F, X, Y) >= Y  because [54], by (Select) 
  54] Y >= Y  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_2, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ 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.