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:

  [[@_{o -> o}(x_1, x_2)]] = @_{o -> o}(x_2, x_1) 
  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
  [[foldl#(x_1, x_2, x_3)]] = foldl#(x_3, x_2, x_1) 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 
  [[~c2]] = _|_ 

We choose Lex = {@_{o -> o}, foldl, foldl#} and Mul = {#argfun-yap##, cons, nil, yap, yap#}, and the following precedence: nil > yap > foldl > cons > yap# > @_{o -> o} = foldl# > #argfun-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 [22], 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 [14], by (Select) 
  14] yap(F(foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl#*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= yap(F(X), Y)  because yap in Mul, [15] and [18], by (Fun) 
  15] F(foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [16], by (Meta) 
  16] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [17], by (Select) 
  17] X >= X  by (Meta) 
  18] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y  because [19], by (Select) 
  19] cons(Y, Z) >= Y  because [20], by (Star) 
  20] cons*(Y, Z) >= Y  because [21], by (Select) 
  21] Y >= Y  by (Meta) 
  22] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z  because [23], by (Select) 
  23] cons(Y, Z) >= Z  because [4], by (Star) 

  24] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(_|_), _|_)  because [25], by (Star) 
  25] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(_|_), _|_)  because foldl# = @_{o -> o}, [26], [28] and [32], by (Stat) 
  26] cons(Y, Z) > _|_  because [27], by definition 
  27] cons*(Y, Z) >= _|_  by (Bot) 
  28] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(_|_)  because [29], by (Select) 
  29] /\x.yap(F(foldl#*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(_|_)  because [30], by (Eta)[Kop13:2] 
  30] F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(_|_)  because [31], by (Meta) 
  31] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 
  32] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 

  33] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because [34], by (Star) 
  34] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because [35], by (Select) 
  35] 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 [36], by (Star) 
  36] 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 [37], by (Copy) 
  37] 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 yap > @_{o -> o}, [38] and [39], by (Copy) 
  38] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [15], by (Select) 
  39] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Y  because [18], by (Select) 

  40] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > @_{o -> o}(F(X), _|_)  because [41], by definition 
  41] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), _|_)  because [42], by (Select) 
  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), _|_)  because [43], by (Star) 
  43] 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), _|_)  because yap > @_{o -> o}, [38] and [44], by (Copy) 
  44] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= _|_  by (Bot) 

  45] #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [46], by (Star) 
  46] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [47], by (Select) 
  47] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because [48] and [49], by (Fun) 
  48] F >= F  by (Meta) 
  49] X >= X  by (Meta) 

  50] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [51], by (Star) 
  51] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [52], by (Select) 
  52] X >= X  by (Meta) 

  53] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [54], by (Star) 
  54] 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], [55], [56] and [61], by (Stat) 
  55] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [7], by (Select) 
  56] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because [57], by (Select) 
  57] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= yap(F(X), Y)  because yap in Mul, [58] and [60], by (Fun) 
  58] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [59], by (Meta) 
  59] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [17], by (Select) 
  60] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [19], by (Select) 
  61] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [23], by (Select) 

  62] yap(F, X) >= @_{o -> o}(F, X)  because [63], by (Star) 
  63] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [64] and [65], by (Copy) 
  64] yap*(F, X) >= F  because [48], by (Select) 
  65] yap*(F, X) >= X  because [49], by (Select) 

  66] foldl(F, X, Y) >= foldl#(F, X, Y)  because [67], by (Star) 
  67] foldl*(F, X, Y) >= foldl#(F, X, Y)  because foldl > foldl#, [68], [70] and [72], by (Copy) 
  68] foldl*(F, X, Y) >= F  because [69], by (Select) 
  69] F >= F  by (Meta) 
  70] foldl*(F, X, Y) >= X  because [71], by (Select) 
  71] X >= X  by (Meta) 
  72] foldl*(F, X, Y) >= Y  because [73], by (Select) 
  73] Y >= Y  by (Meta) 

  74] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [75], by (Star) 
  75] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [76], by (Copy) 
  76] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [77] and [79], by (Copy) 
  77] yap*(F, X) >= F  because [78], by (Select) 
  78] F >= F  by (Meta) 
  79] yap*(F, X) >= X  because [80], by (Select) 
  80] 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)   
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> yap#(F(X), Y)   
  yap#(F, X) =#> F(X)   

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, cons(Y, Z)) >? yap#(F(X), Y) 
  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:

  [[#argfun-yap##(x_1)]] = x_1 
  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_2, x_1) 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 

We choose Lex = {foldl} and Mul = {@_{o -> o}, cons, foldl#, nil, yap, yap#}, and the following precedence: nil > foldl > yap > cons > foldl# > @_{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)) > @_{o -> o}(F(_|_), _|_) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), Y) 
  @_{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) >= @_{o -> o}(F, X) 

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, cons(Y, Z)) >= @_{o -> o}(F(X), Y)  because [9], by (Star) 
  9] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), Y)  because foldl# > @_{o -> o}, [10] and [15], by (Copy) 
  10] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [11], by (Select) 
  11] /\x.yap(F(foldl#*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(X)  because [12], by (Eta)[Kop13:2] 
  12] F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [13], by (Meta) 
  13] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 
  15] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [16], by (Select) 
  16] yap(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Y  because [17], by (Star) 
  17] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Y  because [18], by (Select) 
  18] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [19], by (Select) 
  19] cons(Y, Z) >= Y  because [20], by (Star) 
  20] cons*(Y, Z) >= Y  because [21], by (Select) 
  21] Y >= Y  by (Meta) 

  22] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [23] and [24], by (Fun) 
  23] F >= F  by (Meta) 
  24] X >= X  by (Meta) 

  25] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [26], by (Star) 
  26] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [27], by (Select) 
  27] X >= X  by (Meta) 

  28] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [29], by (Star) 
  29] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [30], [33], [46] and [52], by (Stat) 
  30] cons(Y, Z) > Z  because [31], by definition 
  31] cons*(Y, Z) >= Z  because [32], by (Select) 
  32] Z >= Z  by (Meta) 
  33] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [34], by (F-Abs) 
  34] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= /\x.yap(F(z), x)  because [35], by (F-Abs) 
  35] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= yap(F(z), u)  because foldl > yap, [36] and [44], by (Copy) 
  36] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= F(z)  because [37], by (Select) 
  37] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z), z, u)), x) >= F(z)  because [38], by (Eta)[Kop13:2] 
  38] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)) >= F(z)  because [39], by (Meta) 
  39] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= z  because [40], by (Select) 
  40] 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 [41], by (Star) 
  41] 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 [42], by (Select) 
  42] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= z  because [43], by (Select) 
  43] z >= z  by (Var) 
  44] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= u  because [45], by (Select) 
  45] u >= u  by (Var) 
  46] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [47] and [51], by (Copy) 
  47] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [48], by (Select) 
  48] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z))), x) >= F(X)  because [49], by (Eta)[Kop13:2] 
  49] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [50], by (Meta) 
  50] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [14], by (Select) 
  51] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [19], by (Select) 
  52] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [53], by (Select) 
  53] cons(Y, Z) >= Z  because [31], by (Star) 

  54] yap(F, X) >= @_{o -> o}(F, X)  because [55], by (Star) 
  55] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [56] and [57], by (Copy) 
  56] yap*(F, X) >= F  because [23], by (Select) 
  57] yap*(F, X) >= X  because [24], by (Select) 

  58] foldl(F, X, Y) >= foldl#(F, X, Y)  because [59], by (Star) 
  59] foldl*(F, X, Y) >= foldl#(F, X, Y)  because foldl > foldl#, [60], [62] and [64], by (Copy) 
  60] foldl*(F, X, Y) >= F  because [61], by (Select) 
  61] F >= F  by (Meta) 
  62] foldl*(F, X, Y) >= X  because [63], by (Select) 
  63] X >= X  by (Meta) 
  64] foldl*(F, X, Y) >= Y  because [65], by (Select) 
  65] Y >= Y  by (Meta) 

  66] yap(F, X) >= @_{o -> o}(F, X)  because [67], by (Star) 
  67] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [68] and [70], by (Copy) 
  68] yap*(F, X) >= F  because [69], by (Select) 
  69] F >= F  by (Meta) 
  70] yap*(F, X) >= X  because [71], by (Select) 
  71] 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_2, R_0, minimal, all) by (P_3, R_0, minimal, all), where P_3 consists of:

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

Thus, the original system is terminating if (P_3, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_3, 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)) >? yap#(F(X), Y) 
  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:

  [[#argfun-yap##(x_1)]] = x_1 
  [[foldl(x_1, x_2, x_3)]] = foldl(x_1, x_3, x_2) 

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

With these choices, we have:

  1] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > @_{o -> o}(F(X), Y)  because [2], by definition 
  2] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), Y)  because foldl# > @_{o -> o}, [3] and [8], by (Copy) 
  3] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [4], by (Select) 
  4] /\x.yap(F(foldl#*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(X)  because [5], by (Eta)[Kop13:2] 
  5] F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [6], by (Meta) 
  6] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [9], by (Select) 
  9] cons(Y, Z) >= Y  because [10], by (Star) 
  10] cons*(Y, Z) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 

  12] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [13] and [14], by (Fun) 
  13] F >= F  by (Meta) 
  14] X >= X  by (Meta) 

  15] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [16], by (Star) 
  16] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [17], by (Select) 
  17] yap(F(foldl*(/\x./\y.yap(F(x), y), X, nil)), foldl*(/\z./\u.yap(F(z), u), X, nil)) >= X  because [18], by (Star) 
  18] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, nil)), foldl*(/\z./\u.yap(F(z), u), X, nil)) >= X  because [19], by (Select) 
  19] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [20], by (Select) 
  20] X >= X  by (Meta) 

  21] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [22], by (Star) 
  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], [29], [32], [33] and [38], by (Stat) 
  23] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [24], by (Abs) 
  24] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [25], by (Abs) 
  25] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [26] and [28], by (Fun) 
  26] F(y) >= F(y)  because [27], by (Meta) 
  27] y >= y  by (Var) 
  28] x >= x  by (Var) 
  29] cons(Y, Z) > Z  because [30], by definition 
  30] cons*(Y, Z) >= Z  because [31], by (Select) 
  31] Z >= Z  by (Meta) 
  32] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= /\z./\u.yap(F(z), u)  because [23], by (Select) 
  33] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y)  because [34], by (Select) 
  34] yap(F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= yap(F(X), Y)  because yap in Mul, [35] and [37], by (Fun) 
  35] F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [36], by (Meta) 
  36] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [7], by (Select) 
  37] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y  because [9], by (Select) 
  38] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z  because [39], by (Select) 
  39] cons(Y, Z) >= Z  because [30], by (Star) 

  40] yap(F, X) >= @_{o -> o}(F, X)  because [41], by (Star) 
  41] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [42] and [43], by (Copy) 
  42] yap*(F, X) >= F  because [13], by (Select) 
  43] yap*(F, X) >= X  because [14], by (Select) 

  44] foldl(F, X, Y) >= foldl#(F, X, Y)  because [45], by (Star) 
  45] foldl*(F, X, Y) >= foldl#(F, X, Y)  because foldl > foldl#, [46], [48] and [50], by (Copy) 
  46] foldl*(F, X, Y) >= F  because [47], by (Select) 
  47] F >= F  by (Meta) 
  48] foldl*(F, X, Y) >= X  because [49], by (Select) 
  49] X >= X  by (Meta) 
  50] foldl*(F, X, Y) >= Y  because [51], by (Select) 
  51] Y >= Y  by (Meta) 

  52] yap(F, X) >= @_{o -> o}(F, X)  because [53], by (Star) 
  53] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [54] and [56], by (Copy) 
  54] yap*(F, X) >= F  because [55], by (Select) 
  55] F >= F  by (Meta) 
  56] yap*(F, X) >= X  because [57], by (Select) 
  57] 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_3, R_0, minimal, all) by (P_4, R_0, minimal, all), where P_4 consists of:

  yap#(F, X) =#> F(X)   

Thus, the original system is terminating if (P_4, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_4, 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:

  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 
  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) 

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

  #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) 
  yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X)) 

With these choices, we have:

  1] #argfun-yap##(@_{o -> o}(F, X)) > @_{o -> o}(F, X)  because [2], by definition 
  2] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [3], by (Select) 
  3] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [4] and [5], by (Fun) 
  4] F >= F  by (Meta) 
  5] X >= X  by (Meta) 

  6] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [7], by (Star) 
  7] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

  9] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [10], by (Star) 
  10] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [11], [14], [24] and [34], by (Stat) 
  11] cons(Y, Z) > Z  because [12], by definition 
  12] cons*(Y, Z) >= Z  because [13], by (Select) 
  13] Z >= Z  by (Meta) 
  14] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [15], by (F-Abs) 
  15] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= /\x.yap(F(z), x)  because [16], by (F-Abs) 
  16] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= yap(F(z), u)  because foldl > yap, [17] and [22], by (Copy) 
  17] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= F(z)  because [18], by (Select) 
  18] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z), z, u)), x) >= F(z)  because [19], by (Eta)[Kop13:2] 
  19] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)) >= F(z)  because [20], by (Meta) 
  20] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= z  because [21], by (Select) 
  21] z >= z  by (Var) 
  22] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= u  because [23], by (Select) 
  23] u >= u  by (Var) 
  24] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [25] and [30], by (Copy) 
  25] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [26], by (Select) 
  26] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z))), x) >= F(X)  because [27], by (Eta)[Kop13:2] 
  27] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [28], by (Meta) 
  28] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 
  30] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [31], by (Select) 
  31] cons(Y, Z) >= Y  because [32], by (Star) 
  32] cons*(Y, Z) >= Y  because [33], by (Select) 
  33] Y >= Y  by (Meta) 
  34] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [35], by (Select) 
  35] cons(Y, Z) >= Z  because [12], by (Star) 

  36] yap(F, X) >= @_{o -> o}(F, X)  because [37], by (Star) 
  37] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [38] and [39], by (Copy) 
  38] yap*(F, X) >= F  because [4], by (Select) 
  39] yap*(F, X) >= X  because [5], by (Select) 

  40] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [41], by (Star) 
  41] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [42], by (Copy) 
  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) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_4, 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.