We consider the system h03.

  Alphabet:

    0 : [] --> c 
    1 : [] --> c 
    add : [] --> c -> a -> c 
    cons : [a * b] --> b 
    fold : [c -> a -> c * b * c] --> c 
    mul : [] --> c -> a -> c 
    nil : [] --> b 
    prod : [b] --> c 
    sum : [b] --> c 
    xap : [c -> a -> c * c] --> a -> c 
    yap : [a -> c * a] --> c 

  Rules:

    fold(/\x./\y.yap(xap(f, x), y), nil, z) => z 
    fold(/\x./\y.yap(xap(f, x), y), cons(z, u), v) => fold(/\w./\x'.yap(xap(f, w), x'), u, yap(xap(f, v), z)) 
    sum(x) => fold(/\y./\z.yap(xap(add, y), z), x, 0) 
    fold(mul, x, 1) => prod(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:

    0 : [] --> c 
    1 : [] --> c 
    add : [c] --> a -> c 
    cons : [a * b] --> b 
    fold : [c -> a -> c * b * c] --> c 
    mul : [] --> c -> a -> c 
    nil : [] --> b 
    prod : [b] --> c 
    sum : [b] --> c 
    yap : [a -> c * a] --> c 

  Rules:

    fold(/\x./\y.yap(F(x), y), nil, X) => X 
    fold(/\x./\y.yap(F(x), y), cons(X, Y), Z) => fold(/\z./\u.yap(F(z), u), Y, yap(F(Z), X)) 
    sum(X) => fold(/\x./\y.yap(add(x), y), X, 0) 
    fold(mul, X, 1) => prod(X) 
    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] fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) =#> fold#(/\z./\u.yap(F(z), u), Y, yap(F(Z), X))   
    1] fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) =#> yap#(F(z), u)   
    2] fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) =#> F(z)   
    3] fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) =#> yap#(F(Z), X)   
    4] fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) =#> F(Z)   
    5] sum#(X) =#> fold#(/\x./\y.yap(add(x), y), X, 0)   
    6] sum#(X) =#> yap#(add(x), y)   
    7] yap#(F, X) =#> F(X)   

  Rules R_0:

    fold(/\x./\y.yap(F(x), y), nil, X) => X 
    fold(/\x./\y.yap(F(x), y), cons(X, Y), Z) => fold(/\z./\u.yap(F(z), u), Y, yap(F(Z), X)) 
    sum(X) => fold(/\x./\y.yap(add(x), y), X, 0) 
    fold(mul, X, 1) => prod(X) 
    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, 6, 7 
    * 3 : 7 
    * 4 : 0, 1, 2, 3, 4, 5, 6, 7 
    * 5 : 0, 1, 2, 3, 4 
    * 6 :  
    * 7 : 0, 1, 2, 3, 4, 5, 6, 7 

This graph has the following strongly connected components:

  P_1:

    fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) =#> fold#(/\z./\u.yap(F(z), u), Y, yap(F(Z), X))   
    fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) =#> F(z)   
    fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) =#> yap#(F(Z), X)   
    fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) =#> F(Z)   
    sum#(X) =#> fold#(/\x./\y.yap(add(x), y), X, 0)   
    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:

  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) >? fold#(/\z./\u.yap(F(z), u), Y, yap(F(Z), X)) 
  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) >? F(~c1) ~c0 
  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) >? yap#(F(Z), X) 
  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) >? F(Z) ~c2 
  sum#(X) >? fold#(/\x./\y.yap(add(x), y), X, 0) 
  yap#(F, X) >? F(X) 
  fold(/\x./\y.yap(F(x), y), nil, X) >= X 
  fold(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= fold(/\z./\u.yap(F(z), u), Y, yap(F(Z), X)) 
  sum(X) >= fold(/\x./\y.yap(add(x), y), X, 0) 
  fold(mul, X, 1) >= prod(X) 
  yap(F, X) >= F X 
  fold(F, X, Y) >= fold#(F, X, Y) 
  sum(X) >= sum#(X) 
  yap(F, X) >= yap#(F, X) 

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

  pi( sum#(X) ) = #argfun-sum##(fold#(/\x./\y.yap(add(x), y), X, 0)) 
  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:

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

We choose Lex = {@_{o -> o}, fold, fold#} and Mul = {#argfun-sum##, #argfun-yap##, 1, add, cons, mul, nil, sum, yap, yap#}, and the following precedence: 1 > cons > sum > #argfun-sum## > nil > add > fold > fold# > yap > mul > #argfun-yap## > yap# > @_{o -> o}

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

  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) > fold#(/\x./\y.yap(F(x), y), Y, yap(F(Z), X)) 
  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= @_{o -> o}(F(_|_), _|_) 
  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) > #argfun-yap##(@_{o -> o}(F(Z), X)) 
  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) > @_{o -> o}(F(Z), _|_) 
  #argfun-sum##(fold#(/\x./\y.yap(add(x), y), X, _|_)) > fold#(/\x./\y.yap(add(x), y), X, _|_) 
  #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X) 
  fold(/\x./\y.yap(F(x), y), nil, X) >= X 
  fold(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= fold(/\x./\y.yap(F(x), y), Y, yap(F(Z), X)) 
  sum(X) >= fold(/\x./\y.yap(add(x), y), X, _|_) 
  fold(mul, X, 1) >= X 
  yap(F, X) >= @_{o -> o}(F, X) 
  fold(F, X, Y) >= fold#(F, X, Y) 
  sum(X) >= #argfun-sum##(fold#(/\x./\y.yap(add(x), y), X, _|_)) 
  yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X)) 

With these choices, we have:

  1] fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) > fold#(/\x./\y.yap(F(x), y), Y, yap(F(Z), X))  because [2], by definition 
  2] fold#*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= fold#(/\x./\y.yap(F(x), y), Y, yap(F(Z), X))  because [3], [6], [13] and [15], by (Stat) 
  3] cons(X, Y) > Y  because [4], by definition 
  4] cons*(X, Y) >= Y  because [5], by (Select) 
  5] Y >= Y  by (Meta) 
  6] fold#*(/\x./\y.yap(F(x), y), cons(X, 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] fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z) >= Y  because [14], by (Select) 
  14] cons(X, Y) >= Y  because [4], by (Star) 
  15] fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z) >= yap(F(Z), X)  because fold# > yap, [16] and [21], by (Copy) 
  16] fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z) >= F(Z)  because [17], by (Select) 
  17] /\z.yap(F(fold#*(/\u./\v.yap(F(u), v), cons(X, Y), Z)), z) >= F(Z)  because [18], by (Eta)[Kop13:2] 
  18] F(fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z)) >= F(Z)  because [19], by (Meta) 
  19] fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z) >= Z  because [20], by (Select) 
  20] Z >= Z  by (Meta) 
  21] fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z) >= X  because [22], by (Select) 
  22] cons(X, Y) >= X  because [23], by (Star) 
  23] cons*(X, Y) >= X  because [24], by (Select) 
  24] X >= X  by (Meta) 

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

  32] fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) > #argfun-yap##(@_{o -> o}(F(Z), X))  because [33], by definition 
  33] fold#*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= #argfun-yap##(@_{o -> o}(F(Z), X))  because fold# > #argfun-yap## and [34], by (Copy) 
  34] fold#*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= @_{o -> o}(F(Z), X)  because fold# > @_{o -> o}, [16] and [21], by (Copy) 

  35] fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) > @_{o -> o}(F(Z), _|_)  because [36], by definition 
  36] fold#*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= @_{o -> o}(F(Z), _|_)  because [37], by (Select) 
  37] yap(F(fold#*(/\x./\y.yap(F(x), y), cons(X, Y), Z)), fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z)) >= @_{o -> o}(F(Z), _|_)  because [38], by (Star) 
  38] yap*(F(fold#*(/\x./\y.yap(F(x), y), cons(X, Y), Z)), fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z)) >= @_{o -> o}(F(Z), _|_)  because yap > @_{o -> o}, [39] and [40], by (Copy) 
  39] yap*(F(fold#*(/\x./\y.yap(F(x), y), cons(X, Y), Z)), fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z)) >= F(Z)  because [18], by (Select) 
  40] yap*(F(fold#*(/\x./\y.yap(F(x), y), cons(X, Y), Z)), fold#*(/\z./\u.yap(F(z), u), cons(X, Y), Z)) >= _|_  by (Bot) 

  41] #argfun-sum##(fold#(/\x./\y.yap(add(x), y), X, _|_)) > fold#(/\x./\y.yap(add(x), y), X, _|_)  because [42], by definition 
  42] #argfun-sum##*(fold#(/\x./\y.yap(add(x), y), X, _|_)) >= fold#(/\x./\y.yap(add(x), y), X, _|_)  because [43], by (Select) 
  43] fold#(/\x./\y.yap(add(x), y), X, _|_) >= fold#(/\x./\y.yap(add(x), y), X, _|_)  because [44], [50] and [51], by (Fun) 
  44] /\x./\z.yap(add(x), z) >= /\x./\z.yap(add(x), z)  because [45], by (Abs) 
  45] /\z.yap(add(y), z) >= /\z.yap(add(y), z)  because [46], by (Abs) 
  46] yap(add(y), x) >= yap(add(y), x)  because yap in Mul, [47] and [49], by (Fun) 
  47] add(y) >= add(y)  because add in Mul and [48], by (Fun) 
  48] y >= y  by (Var) 
  49] x >= x  by (Var) 
  50] X >= X  by (Meta) 
  51] _|_ >= _|_  by (Bot) 

  52] #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [53], by (Star) 
  53] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [54], by (Select) 
  54] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because [55] and [56], by (Fun) 
  55] F >= F  by (Meta) 
  56] X >= X  by (Meta) 

  57] fold(/\x./\y.yap(F(x), y), nil, X) >= X  because [58], by (Star) 
  58] fold*(/\x./\y.yap(F(x), y), nil, X) >= X  because [59], by (Select) 
  59] X >= X  by (Meta) 

  60] fold(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= fold(/\x./\y.yap(F(x), y), Y, yap(F(Z), X))  because [61], by (Star) 
  61] fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= fold(/\x./\y.yap(F(x), y), Y, yap(F(Z), X))  because [3], [62], [63] and [64], by (Stat) 
  62] fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= /\x./\y.yap(F(x), y)  because [7], by (Select) 
  63] fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= Y  because [14], by (Select) 
  64] fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= yap(F(Z), X)  because fold > yap, [65] and [69], by (Copy) 
  65] fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= F(Z)  because [66], by (Select) 
  66] /\x.yap(F(fold*(/\y./\z.yap(F(y), z), cons(X, Y), Z)), x) >= F(Z)  because [67], by (Eta)[Kop13:2] 
  67] F(fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z)) >= F(Z)  because [68], by (Meta) 
  68] fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= Z  because [20], by (Select) 
  69] fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= X  because [70], by (Select) 
  70] yap(F(fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z)), fold*(/\z./\u.yap(F(z), u), cons(X, Y), Z)) >= X  because [71], by (Star) 
  71] yap*(F(fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z)), fold*(/\z./\u.yap(F(z), u), cons(X, Y), Z)) >= X  because [72], by (Select) 
  72] fold*(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= X  because [73], by (Select) 
  73] cons(X, Y) >= X  because [23], by (Star) 

  74] sum(X) >= fold(/\x./\y.yap(add(x), y), X, _|_)  because [75], by (Star) 
  75] sum*(X) >= fold(/\x./\y.yap(add(x), y), X, _|_)  because sum > fold, [76], [84] and [85], by (Copy) 
  76] sum*(X) >= /\y./\z.yap(add(y), z)  because [77], by (F-Abs) 
  77] sum*(X, x) >= /\z.yap(add(x), z)  because [78], by (F-Abs) 
  78] sum*(X, x, y) >= yap(add(x), y)  because sum > yap, [79] and [82], by (Copy) 
  79] sum*(X, x, y) >= add(x)  because sum > add and [80], by (Copy) 
  80] sum*(X, x, y) >= x  because [81], by (Select) 
  81] x >= x  by (Var) 
  82] sum*(X, x, y) >= y  because [83], by (Select) 
  83] y >= y  by (Var) 
  84] sum*(X) >= X  because [50], by (Select) 
  85] sum*(X) >= _|_  by (Bot) 

  86] fold(mul, X, 1) >= X  because [87], by (Star) 
  87] fold*(mul, X, 1) >= X  because [88], by (Select) 
  88] X >= X  by (Meta) 

  89] yap(F, X) >= @_{o -> o}(F, X)  because [90], by (Star) 
  90] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [91] and [92], by (Copy) 
  91] yap*(F, X) >= F  because [55], by (Select) 
  92] yap*(F, X) >= X  because [56], by (Select) 

  93] fold(F, X, Y) >= fold#(F, X, Y)  because [94], by (Star) 
  94] fold*(F, X, Y) >= fold#(F, X, Y)  because fold > fold#, [95], [97] and [99], by (Copy) 
  95] fold*(F, X, Y) >= F  because [96], by (Select) 
  96] F >= F  by (Meta) 
  97] fold*(F, X, Y) >= X  because [98], by (Select) 
  98] X >= X  by (Meta) 
  99] fold*(F, X, Y) >= Y  because [100], by (Select) 
  100] Y >= Y  by (Meta) 

  101] sum(X) >= #argfun-sum##(fold#(/\x./\y.yap(add(x), y), X, _|_))  because [102], by (Star) 
  102] sum*(X) >= #argfun-sum##(fold#(/\x./\y.yap(add(x), y), X, _|_))  because sum > #argfun-sum## and [103], by (Copy) 
  103] sum*(X) >= fold#(/\x./\y.yap(add(x), y), X, _|_)  because sum > fold#, [104], [112] and [114], by (Copy) 
  104] sum*(X) >= /\y./\z.yap(add(y), z)  because [105], by (F-Abs) 
  105] sum*(X, x) >= /\z.yap(add(x), z)  because [106], by (F-Abs) 
  106] sum*(X, x, y) >= yap(add(x), y)  because sum > yap, [107] and [110], by (Copy) 
  107] sum*(X, x, y) >= add(x)  because sum > add and [108], by (Copy) 
  108] sum*(X, x, y) >= x  because [109], by (Select) 
  109] x >= x  by (Var) 
  110] sum*(X, x, y) >= y  because [111], by (Select) 
  111] y >= y  by (Var) 
  112] sum*(X) >= X  because [113], by (Select) 
  113] X >= X  by (Meta) 
  114] sum*(X) >= _|_  by (Bot) 

  115] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [116], by (Star) 
  116] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [117], by (Copy) 
  117] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [118] and [120], by (Copy) 
  118] yap*(F, X) >= F  because [119], by (Select) 
  119] F >= F  by (Meta) 
  120] yap*(F, X) >= X  because [121], by (Select) 
  121] 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:

  fold#(/\x./\y.yap(F(x), y), cons(X, 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:

  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) >? F(~c1) ~c0 
  yap#(F, X) >? F(X) 
  fold(/\x./\y.yap(F(x), y), nil, X) >= X 
  fold(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= fold(/\z./\u.yap(F(z), u), Y, yap(F(Z), X)) 
  sum(X) >= fold(/\x./\y.yap(add(x), y), X, 0) 
  fold(mul, X, 1) >= prod(X) 
  yap(F, X) >= F X 
  fold(F, X, Y) >= fold#(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:

  [[0]] = _|_ 
  [[fold(x_1, x_2, x_3)]] = fold(x_2, x_1, x_3) 
  [[prod(x_1)]] = x_1 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 

We choose Lex = {fold} and Mul = {#argfun-yap##, 1, @_{o -> o}, add, cons, fold#, mul, nil, sum, yap, yap#}, and the following precedence: 1 > mul > nil > sum > add > yap# > fold > yap > #argfun-yap## > fold# > @_{o -> o} > cons

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

  fold#(/\x./\y.yap(F(x), y), cons(X, Y), Z) > @_{o -> o}(F(_|_), _|_) 
  #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X) 
  fold(/\x./\y.yap(F(x), y), nil, X) >= X 
  fold(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= fold(/\x./\y.yap(F(x), y), Y, yap(F(Z), X)) 
  sum(X) >= fold(/\x./\y.yap(add(x), y), X, _|_) 
  fold(mul, X, 1) >= X 
  yap(F, X) >= @_{o -> o}(F, X) 
  fold(F, X, Y) >= fold#(F, X, Y) 
  yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X)) 

With these choices, we have:

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

  8] #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [9], by (Star) 
  9] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [10], by (Select) 
  10] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [11] and [12], by (Fun) 
  11] F >= F  by (Meta) 
  12] X >= X  by (Meta) 

  13] fold(/\x./\y.yap(F(x), y), nil, X) >= X  because [14], by (Star) 
  14] fold*(/\x./\y.yap(F(x), y), nil, X) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 

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

  43] sum(X) >= fold(/\x./\y.yap(add(x), y), X, _|_)  because [44], by (Star) 
  44] sum*(X) >= fold(/\x./\y.yap(add(x), y), X, _|_)  because sum > fold, [45], [53] and [55], by (Copy) 
  45] sum*(X) >= /\y./\z.yap(add(y), z)  because [46], by (F-Abs) 
  46] sum*(X, x) >= /\z.yap(add(x), z)  because [47], by (F-Abs) 
  47] sum*(X, x, y) >= yap(add(x), y)  because sum > yap, [48] and [51], by (Copy) 
  48] sum*(X, x, y) >= add(x)  because sum > add and [49], by (Copy) 
  49] sum*(X, x, y) >= x  because [50], by (Select) 
  50] x >= x  by (Var) 
  51] sum*(X, x, y) >= y  because [52], by (Select) 
  52] y >= y  by (Var) 
  53] sum*(X) >= X  because [54], by (Select) 
  54] X >= X  by (Meta) 
  55] sum*(X) >= _|_  by (Bot) 

  56] fold(mul, X, 1) >= X  because [57], by (Star) 
  57] fold*(mul, X, 1) >= X  because [58], by (Select) 
  58] X >= X  by (Meta) 

  59] yap(F, X) >= @_{o -> o}(F, X)  because [60], by (Star) 
  60] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [61] and [62], by (Copy) 
  61] yap*(F, X) >= F  because [11], by (Select) 
  62] yap*(F, X) >= X  because [12], by (Select) 

  63] fold(F, X, Y) >= fold#(F, X, Y)  because [64], by (Star) 
  64] fold*(F, X, Y) >= fold#(F, X, Y)  because fold > fold#, [65], [67] and [69], by (Copy) 
  65] fold*(F, X, Y) >= F  because [66], by (Select) 
  66] F >= F  by (Meta) 
  67] fold*(F, X, Y) >= X  because [68], by (Select) 
  68] X >= X  by (Meta) 
  69] fold*(F, X, Y) >= Y  because [70], by (Select) 
  70] Y >= Y  by (Meta) 

  71] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [72], by (Star) 
  72] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [73], by (Copy) 
  73] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [74] and [76], by (Copy) 
  74] yap*(F, X) >= F  because [75], by (Select) 
  75] F >= F  by (Meta) 
  76] yap*(F, X) >= X  because [77], by (Select) 
  77] 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) 
  fold(/\x./\y.yap(F(x), y), nil, X) >= X 
  fold(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= fold(/\z./\u.yap(F(z), u), Y, yap(F(Z), X)) 
  sum(X) >= fold(/\x./\y.yap(add(x), y), X, 0) 
  fold(mul, X, 1) >= prod(X) 
  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:

  [[0]] = _|_ 
  [[fold(x_1, x_2, x_3)]] = fold(x_2, x_1, x_3) 
  [[prod(x_1)]] = x_1 

We choose Lex = {fold} and Mul = {#argfun-yap##, 1, @_{o -> o}, add, cons, mul, nil, sum, yap, yap#}, and the following precedence: 1 > cons > mul > nil > sum > add > fold > 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) 
  fold(/\x./\y.yap(F(x), y), nil, X) >= X 
  fold(/\x./\y.yap(F(x), y), cons(X, Y), Z) >= fold(/\x./\y.yap(F(x), y), Y, yap(F(Z), X)) 
  sum(X) >= fold(/\x./\y.yap(add(x), y), X, _|_) 
  fold(mul, X, 1) >= X 
  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] fold(/\x./\y.yap(F(x), y), nil, X) >= X  because [7], by (Star) 
  7] fold*(/\x./\y.yap(F(x), y), nil, X) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

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

  33] sum(X) >= fold(/\x./\y.yap(add(x), y), X, _|_)  because [34], by (Star) 
  34] sum*(X) >= fold(/\x./\y.yap(add(x), y), X, _|_)  because sum > fold, [35], [43] and [45], by (Copy) 
  35] sum*(X) >= /\y./\z.yap(add(y), z)  because [36], by (F-Abs) 
  36] sum*(X, x) >= /\z.yap(add(x), z)  because [37], by (F-Abs) 
  37] sum*(X, x, y) >= yap(add(x), y)  because sum > yap, [38] and [41], by (Copy) 
  38] sum*(X, x, y) >= add(x)  because sum > add and [39], by (Copy) 
  39] sum*(X, x, y) >= x  because [40], by (Select) 
  40] x >= x  by (Var) 
  41] sum*(X, x, y) >= y  because [42], by (Select) 
  42] y >= y  by (Var) 
  43] sum*(X) >= X  because [44], by (Select) 
  44] X >= X  by (Meta) 
  45] sum*(X) >= _|_  by (Bot) 

  46] fold(mul, X, 1) >= X  because [47], by (Star) 
  47] fold*(mul, X, 1) >= X  because [48], by (Select) 
  48] X >= X  by (Meta) 

  49] yap(F, X) >= @_{o -> o}(F, X)  because [50], by (Star) 
  50] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [51] and [52], by (Copy) 
  51] yap*(F, X) >= F  because [4], by (Select) 
  52] yap*(F, X) >= X  because [5], by (Select) 

  53] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [54], by (Star) 
  54] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [55], by (Copy) 
  55] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [56] and [58], by (Copy) 
  56] yap*(F, X) >= F  because [57], by (Select) 
  57] F >= F  by (Meta) 
  58] yap*(F, X) >= X  because [59], by (Select) 
  59] 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.