We consider the system h56.

  Alphabet:

    cons : [a * b] --> b 
    foldr : [a -> b -> b * b * b] --> b 
    nil : [] --> b 
    xap : [a -> b -> b * a] --> b -> b 
    yap : [b -> b * b] --> b 

  Rules:

    foldr(/\x./\y.yap(xap(f, x), y), z, nil) => z 
    foldr(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => yap(xap(f, u), foldr(/\w./\x'.yap(xap(f, w), x'), z, 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 * b] --> b 
    foldr : [a -> b -> b * b * b] --> b 
    nil : [] --> b 
    yap : [b -> b * b] --> b 

  Rules:

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

  Rules R_0:

    foldr(/\x./\y.yap(F(x), y), X, nil) => X 
    foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, 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 : 5 
    * 1 : 0, 1, 2, 3, 4, 5 
    * 2 : 0, 1, 2, 3, 4 
    * 3 :  
    * 4 : 0, 1, 2, 3, 4, 5 
    * 5 : 0, 1, 2, 3, 4, 5 

This graph has the following strongly connected components:

  P_1:

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

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

  [[~c0]] = _|_ 
  [[~c1]] = _|_ 
  [[~c2]] = _|_ 

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

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

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

With these choices, we have:

  1] foldr#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > #argfun-yap##(@_{o -> o}(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)))  because [2], by definition 
  2] foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= #argfun-yap##(@_{o -> o}(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)))  because [3], by (Select) 
  3] yap(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= #argfun-yap##(@_{o -> o}(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)))  because [4], by (Star) 
  4] yap*(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= #argfun-yap##(@_{o -> o}(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)))  because yap > #argfun-yap## and [5], by (Copy) 
  5] yap*(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= @_{o -> o}(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z))  because yap > @_{o -> o}, [6] and [12], by (Copy) 
  6] yap*(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(Y)  because [7], by (Select) 
  7] F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(Y)  because [8], by (Meta) 
  8] foldr#*(/\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] yap*(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= foldr(/\x./\y.yap(F(x), y), X, Z)  because [13], by (Select) 
  13] foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z)  because foldr# = foldr, foldr# in Mul, [14], [20] and [21], by (Stat) 
  14] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [15], by (Abs) 
  15] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [16], by (Abs) 
  16] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [17] and [19], by (Fun) 
  17] F(y) >= F(y)  because [18], by (Meta) 
  18] y >= y  by (Var) 
  19] x >= x  by (Var) 
  20] X >= X  by (Meta) 
  21] cons(Y, Z) > Z  because [22], by definition 
  22] cons*(Y, Z) >= Z  because [23], by (Select) 
  23] Z >= Z  by (Meta) 

  24] foldr#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(Y), _|_)  because [25], by (Star) 
  25] foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(Y), _|_)  because [26], by (Select) 
  26] yap(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= @_{o -> o}(F(Y), _|_)  because [27], by (Star) 
  27] yap*(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= @_{o -> o}(F(Y), _|_)  because yap > @_{o -> o}, [6] and [28], by (Copy) 
  28] yap*(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= _|_  by (Bot) 

  29] foldr#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr#(/\x./\y.yap(F(x), y), X, Z)  because foldr# in Mul, [14], [20] and [30], by (Fun) 
  30] cons(Y, Z) >= Z  because [22], by (Star) 

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

  38] #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [39], by (Star) 
  39] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [40], by (Select) 
  40] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [41] and [42], by (Fun) 
  41] F >= F  by (Meta) 
  42] X >= X  by (Meta) 

  43] foldr(/\x./\y.yap(F(x), y), X, nil) >= X  because [44], by (Star) 
  44] foldr*(/\x./\y.yap(F(x), y), X, nil) >= X  because [45], by (Select) 
  45] X >= X  by (Meta) 

  46] foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z))  because [47], by (Star) 
  47] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z))  because foldr > yap, [48] and [52], by (Copy) 
  48] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(Y)  because [49], by (Select) 
  49] /\x.yap(F(foldr*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(Y)  because [50], by (Eta)[Kop13:2] 
  50] F(foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(Y)  because [51], by (Meta) 
  51] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [9], by (Select) 
  52] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z)  because foldr in Mul, [14], [20] and [21], by (Stat) 

  53] yap(F, X) >= @_{o -> o}(F, X)  because [54], by (Star) 
  54] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [55] and [56], by (Copy) 
  55] yap*(F, X) >= F  because [41], by (Select) 
  56] yap*(F, X) >= X  because [42], by (Select) 

  57] foldr(F, X, Y) >= foldr#(F, X, Y)  because foldr = foldr#, foldr in Mul, [58], [59] and [60], by (Fun) 
  58] F >= F  by (Meta) 
  59] X >= X  by (Meta) 
  60] Y >= Y  by (Meta) 

  61] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [62], by (Star) 
  62] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [63], by (Copy) 
  63] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [64] and [66], by (Copy) 
  64] yap*(F, X) >= F  because [65], by (Select) 
  65] F >= F  by (Meta) 
  66] yap*(F, X) >= X  because [67], by (Select) 
  67] 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:

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

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

  [[~c0]] = _|_ 
  [[~c1]] = _|_ 
  [[~c2]] = _|_ 

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

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

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

With these choices, we have:

  1] foldr#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > @_{o -> o}(F(Y), _|_)  because [2], by definition 
  2] foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(Y), _|_)  because foldr# > @_{o -> o}, [3] and [13], by (Copy) 
  3] foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(Y)  because [4], by (Select) 
  4] /\x.yap(F(foldr#*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(Y)  because [5], by (Eta)[Kop13:2] 
  5] F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(Y)  because [6], by (Meta) 
  6] foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [7], by (Select) 
  7] yap(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Y  because [8], by (Star) 
  8] yap*(F(foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Y  because [9], by (Select) 
  9] foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [10], by (Select) 
  10] cons(Y, Z) >= Y  because [11], by (Star) 
  11] cons*(Y, Z) >= Y  because [12], by (Select) 
  12] Y >= Y  by (Meta) 
  13] foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 

  14] foldr#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > foldr#(/\x./\y.yap(F(x), y), X, Z)  because [15], by definition 
  15] foldr#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr#(/\x./\y.yap(F(x), y), X, Z)  because foldr# in Mul, [16], [22] and [23], by (Stat) 
  16] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [17], by (Abs) 
  17] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [18], by (Abs) 
  18] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [19] and [21], by (Fun) 
  19] F(y) >= F(y)  because [20], by (Meta) 
  20] y >= y  by (Var) 
  21] x >= x  by (Var) 
  22] X >= X  by (Meta) 
  23] cons(Y, Z) > Z  because [24], by definition 
  24] cons*(Y, Z) >= Z  because [25], by (Select) 
  25] Z >= Z  by (Meta) 

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

  33] #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [34], by (Star) 
  34] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [35], by (Select) 
  35] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [36] and [37], by (Fun) 
  36] F >= F  by (Meta) 
  37] X >= X  by (Meta) 

  38] foldr(/\x./\y.yap(F(x), y), X, nil) >= X  because [39], by (Star) 
  39] foldr*(/\x./\y.yap(F(x), y), X, nil) >= X  because [40], by (Select) 
  40] X >= X  by (Meta) 

  41] foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z))  because [42], by (Star) 
  42] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z))  because foldr > yap, [43] and [47], by (Copy) 
  43] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(Y)  because [44], by (Select) 
  44] /\x.yap(F(foldr*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(Y)  because [45], by (Eta)[Kop13:2] 
  45] F(foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(Y)  because [46], by (Meta) 
  46] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [10], by (Select) 
  47] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z)  because foldr in Mul, [16], [22] and [23], by (Stat) 

  48] yap(F, X) >= @_{o -> o}(F, X)  because [49], by (Star) 
  49] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [50] and [51], by (Copy) 
  50] yap*(F, X) >= F  because [36], by (Select) 
  51] yap*(F, X) >= X  because [37], by (Select) 

  52] foldr(F, X, Y) >= foldr#(F, X, Y)  because [53], by (Star) 
  53] foldr*(F, X, Y) >= foldr#(F, X, Y)  because foldr > foldr#, [54], [56] and [58], by (Copy) 
  54] foldr*(F, X, Y) >= F  because [55], by (Select) 
  55] F >= F  by (Meta) 
  56] foldr*(F, X, Y) >= X  because [57], by (Select) 
  57] X >= X  by (Meta) 
  58] foldr*(F, X, Y) >= Y  because [59], by (Select) 
  59] Y >= Y  by (Meta) 

  60] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [61], by (Star) 
  61] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [62], by (Copy) 
  62] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [63] and [65], by (Copy) 
  63] yap*(F, X) >= F  because [64], by (Select) 
  64] F >= F  by (Meta) 
  65] yap*(F, X) >= X  because [66], by (Select) 
  66] 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:

  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:

  yap#(F, X) >? F(X) 
  foldr(/\x./\y.yap(F(x), y), X, nil) >= X 
  foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, 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].

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

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

  9] foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z))  because [10], by (Star) 
  10] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z))  because foldr > yap, [11] and [18], by (Copy) 
  11] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(Y)  because [12], by (Select) 
  12] /\x.yap(F(foldr*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(Y)  because [13], by (Eta)[Kop13:2] 
  13] F(foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(Y)  because [14], by (Meta) 
  14] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [15], by (Select) 
  15] cons(Y, Z) >= Y  because [16], by (Star) 
  16] cons*(Y, Z) >= Y  because [17], by (Select) 
  17] Y >= Y  by (Meta) 
  18] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z)  because foldr in Mul, [19], [25] and [26], by (Stat) 
  19] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [20], by (Abs) 
  20] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [21], by (Abs) 
  21] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [22] and [24], by (Fun) 
  22] F(y) >= F(y)  because [23], by (Meta) 
  23] y >= y  by (Var) 
  24] x >= x  by (Var) 
  25] X >= X  by (Meta) 
  26] cons(Y, Z) > Z  because [27], by definition 
  27] cons*(Y, Z) >= Z  because [28], by (Select) 
  28] Z >= Z  by (Meta) 

  29] yap(F, X) >= @_{o -> o}(F, X)  because [30], by (Star) 
  30] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [31] and [32], by (Copy) 
  31] yap*(F, X) >= F  because [4], by (Select) 
  32] yap*(F, X) >= X  because [5], by (Select) 

  33] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [34], by (Star) 
  34] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [35], by (Copy) 
  35] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [36] and [38], by (Copy) 
  36] yap*(F, X) >= F  because [37], by (Select) 
  37] F >= F  by (Meta) 
  38] yap*(F, X) >= X  because [39], by (Select) 
  39] 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_3, 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.