We consider the system h13.

  Alphabet:

    0 : [] --> a 
    rec : [a -> b -> b * b * a] --> b 
    s : [a] --> a 
    xap : [a -> b -> b * a] --> b -> b 
    yap : [b -> b * b] --> b 

  Rules:

    rec(/\x./\y.yap(xap(f, x), y), z, 0) => z 
    rec(/\x./\y.yap(xap(f, x), y), z, s(u)) => yap(xap(f, s(u)), rec(/\v./\w.yap(xap(f, v), w), z, u)) 
    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 : [] --> a 
    rec : [a -> b -> b * b * a] --> b 
    s : [a] --> a 
    yap : [b -> b * b] --> b 

  Rules:

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

  Rules R_0:

    rec(/\x./\y.yap(F(x), y), X, 0) => X 
    rec(/\x./\y.yap(F(x), y), X, s(Y)) => yap(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y)) 
    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:

    rec#(/\x./\y.yap(F(x), y), X, s(Y)) =#> yap#(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y))   
    rec#(/\x./\y.yap(F(x), y), X, s(Y)) =#> F(s(Y))   
    rec#(/\x./\y.yap(F(x), y), X, s(Y)) =#> rec#(/\z./\u.yap(F(z), u), X, Y)   
    rec#(/\x./\y.yap(F(x), y), X, s(Y)) =#> 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:

  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >? yap#(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y)) 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >? F(s(Y)) ~c0 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >? rec#(/\z./\u.yap(F(z), u), X, Y) 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >? F(~c2) ~c1 
  yap#(F, X) >? F(X) 
  rec(/\x./\y.yap(F(x), y), X, 0) >= X 
  rec(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y)) 
  yap(F, X) >= F X 
  rec(F, X, Y) >= rec#(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 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 
  [[~c2]] = _|_ 

We choose Lex = {} and Mul = {0, @_{o -> o}, rec, rec#, s, yap, yap#}, and the following precedence: 0 > yap > rec = rec# > @_{o -> o} > s > yap#

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

  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >= @_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y)) 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >= @_{o -> o}(F(s(Y)), _|_) 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >= rec#(/\x./\y.yap(F(x), y), X, Y) 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) > @_{o -> o}(F(_|_), _|_) 
  @_{o -> o}(F, X) >= @_{o -> o}(F, X) 
  rec(/\x./\y.yap(F(x), y), X, 0) >= X 
  rec(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y)) 
  yap(F, X) >= @_{o -> o}(F, X) 
  rec(F, X, Y) >= rec#(F, X, Y) 
  yap(F, X) >= @_{o -> o}(F, X) 

With these choices, we have:

  1] rec#(/\x./\y.yap(F(x), y), X, s(Y)) >= @_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because [2], by (Star) 
  2] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= @_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because [3], by (Select) 
  3] yap(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= @_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because [4], by (Star) 
  4] yap*(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= @_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because yap > @_{o -> o}, [5] and [12], by (Copy) 
  5] yap*(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= F(s(Y))  because [6], by (Select) 
  6] F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(s(Y))  because [7], by (Meta) 
  7] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= s(Y)  because rec# > s and [8], by (Copy) 
  8] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= Y  because [9], by (Select) 
  9] s(Y) >= Y  because [10], by (Star) 
  10] s*(Y) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 
  12] yap*(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= rec(/\x./\y.yap(F(x), y), X, Y)  because [13], by (Select) 
  13] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y)  because rec# = rec, rec# 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] s(Y) > Y  because [22], by definition 
  22] s*(Y) >= Y  because [11], by (Select) 

  23] rec#(/\x./\y.yap(F(x), y), X, s(Y)) >= @_{o -> o}(F(s(Y)), _|_)  because [24], by (Star) 
  24] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= @_{o -> o}(F(s(Y)), _|_)  because rec# > @_{o -> o}, [25] and [27], by (Copy) 
  25] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= F(s(Y))  because [26], by (Select) 
  26] /\x.yap(F(rec#*(/\y./\z.yap(F(y), z), X, s(Y))), x) >= F(s(Y))  because [6], by (Eta)[Kop13:2] 
  27] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= _|_  by (Bot) 

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

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

  37] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [38] and [39], by (Fun) 
  38] F >= F  by (Meta) 
  39] X >= X  by (Meta) 

  40] rec(/\x./\y.yap(F(x), y), X, 0) >= X  because [41], by (Star) 
  41] rec*(/\x./\y.yap(F(x), y), X, 0) >= X  because [42], by (Select) 
  42] X >= X  by (Meta) 

  43] rec(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because [44], by (Star) 
  44] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because [45], by (Select) 
  45] yap(F(rec*(/\x./\y.yap(F(x), y), X, s(Y))), rec*(/\z./\u.yap(F(z), u), X, s(Y))) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because yap in Mul, [46] and [50], by (Fun) 
  46] F(rec*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(s(Y))  because [47], by (Meta) 
  47] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= s(Y)  because [48], by (Select) 
  48] s(Y) >= s(Y)  because s in Mul and [49], by (Fun) 
  49] Y >= Y  by (Meta) 
  50] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y)  because rec in Mul, [14], [20] and [21], by (Stat) 

  51] yap(F, X) >= @_{o -> o}(F, X)  because [52], by (Star) 
  52] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [53] and [54], by (Copy) 
  53] yap*(F, X) >= F  because [38], by (Select) 
  54] yap*(F, X) >= X  because [39], by (Select) 

  55] rec(F, X, Y) >= rec#(F, X, Y)  because rec = rec#, rec in Mul, [56], [57] and [58], by (Fun) 
  56] F >= F  by (Meta) 
  57] X >= X  by (Meta) 
  58] Y >= Y  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 [63], by (Copy) 
  61] yap*(F, X) >= F  because [62], by (Select) 
  62] F >= F  by (Meta) 
  63] yap*(F, X) >= X  because [64], by (Select) 
  64] 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:

  rec#(/\x./\y.yap(F(x), y), X, s(Y)) =#> yap#(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y))   
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) =#> F(s(Y))   
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) =#> rec#(/\z./\u.yap(F(z), u), 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:

  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >? yap#(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y)) 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >? F(s(Y)) ~c0 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >? rec#(/\z./\u.yap(F(z), u), X, Y) 
  yap#(F, X) >? F(X) 
  rec(/\x./\y.yap(F(x), y), X, 0) >= X 
  rec(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y)) 
  yap(F, X) >= F X 
  rec(F, X, Y) >= rec#(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]] = _|_ 

We choose Lex = {} and Mul = {#argfun-yap##, 0, @_{o -> o}, rec, rec#, s, yap, yap#}, and the following precedence: 0 > rec = rec# > s > yap > @_{o -> o} > yap# > #argfun-yap##

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

  rec#(/\x./\y.yap(F(x), y), X, s(Y)) > #argfun-yap##(@_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))) 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) > @_{o -> o}(F(s(Y)), _|_) 
  rec#(/\x./\y.yap(F(x), y), X, s(Y)) >= rec#(/\x./\y.yap(F(x), y), X, Y) 
  #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X) 
  rec(/\x./\y.yap(F(x), y), X, 0) >= X 
  rec(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y)) 
  yap(F, X) >= @_{o -> o}(F, X) 
  rec(F, X, Y) >= rec#(F, X, Y) 
  yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X)) 

With these choices, we have:

  1] rec#(/\x./\y.yap(F(x), y), X, s(Y)) > #argfun-yap##(@_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y)))  because [2], by definition 
  2] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= #argfun-yap##(@_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y)))  because [3], by (Select) 
  3] yap(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= #argfun-yap##(@_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y)))  because [4], by (Star) 
  4] yap*(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= #argfun-yap##(@_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y)))  because yap > #argfun-yap## and [5], by (Copy) 
  5] yap*(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= @_{o -> o}(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because yap > @_{o -> o}, [6] and [11], by (Copy) 
  6] yap*(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= F(s(Y))  because [7], by (Select) 
  7] F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(s(Y))  because [8], by (Meta) 
  8] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= s(Y)  because [9], by (Select) 
  9] s(Y) >= s(Y)  because s in Mul and [10], by (Fun) 
  10] Y >= Y  by (Meta) 
  11] yap*(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= rec(/\x./\y.yap(F(x), y), X, Y)  because [12], by (Select) 
  12] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y)  because rec# = rec, rec# in Mul, [13], [19] and [20], by (Stat) 
  13] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [14], by (Abs) 
  14] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [15], by (Abs) 
  15] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [16] and [18], by (Fun) 
  16] F(y) >= F(y)  because [17], by (Meta) 
  17] y >= y  by (Var) 
  18] x >= x  by (Var) 
  19] X >= X  by (Meta) 
  20] s(Y) > Y  because [21], by definition 
  21] s*(Y) >= Y  because [10], by (Select) 

  22] rec#(/\x./\y.yap(F(x), y), X, s(Y)) > @_{o -> o}(F(s(Y)), _|_)  because [23], by definition 
  23] rec#*(/\x./\y.yap(F(x), y), X, s(Y)) >= @_{o -> o}(F(s(Y)), _|_)  because [24], by (Select) 
  24] yap(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= @_{o -> o}(F(s(Y)), _|_)  because [25], by (Star) 
  25] yap*(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= @_{o -> o}(F(s(Y)), _|_)  because yap > @_{o -> o}, [6] and [26], by (Copy) 
  26] yap*(F(rec#*(/\x./\y.yap(F(x), y), X, s(Y))), rec#*(/\z./\u.yap(F(z), u), X, s(Y))) >= _|_  by (Bot) 

  27] rec#(/\x./\y.yap(F(x), y), X, s(Y)) >= rec#(/\x./\y.yap(F(x), y), X, Y)  because rec# in Mul, [13], [19] and [28], by (Fun) 
  28] s(Y) >= Y  because [21], by (Star) 

  29] #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [30], by (Star) 
  30] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [31], by (Select) 
  31] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [32] and [33], by (Fun) 
  32] F >= F  by (Meta) 
  33] X >= X  by (Meta) 

  34] rec(/\x./\y.yap(F(x), y), X, 0) >= X  because [35], by (Star) 
  35] rec*(/\x./\y.yap(F(x), y), X, 0) >= X  because [36], by (Select) 
  36] X >= X  by (Meta) 

  37] rec(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because [38], by (Star) 
  38] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\x./\y.yap(F(x), y), X, Y))  because rec > yap, [39] and [43], by (Copy) 
  39] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= F(s(Y))  because [40], by (Select) 
  40] /\x.yap(F(rec*(/\y./\z.yap(F(y), z), X, s(Y))), x) >= F(s(Y))  because [41], by (Eta)[Kop13:2] 
  41] F(rec*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(s(Y))  because [42], by (Meta) 
  42] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= s(Y)  because [9], by (Select) 
  43] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y)  because rec in Mul, [13], [19] and [20], by (Stat) 

  44] yap(F, X) >= @_{o -> o}(F, X)  because [45], by (Star) 
  45] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [46] and [47], by (Copy) 
  46] yap*(F, X) >= F  because [32], by (Select) 
  47] yap*(F, X) >= X  because [33], by (Select) 

  48] rec(F, X, Y) >= rec#(F, X, Y)  because rec = rec#, rec in Mul, [49], [50] and [51], by (Fun) 
  49] F >= F  by (Meta) 
  50] X >= X  by (Meta) 
  51] Y >= Y  by (Meta) 

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

  rec#(/\x./\y.yap(F(x), y), X, s(Y)) =#> rec#(/\z./\u.yap(F(z), u), 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 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 : 0, 1 

This graph has the following strongly connected components:

  P_4:

    rec#(/\x./\y.yap(F(x), y), X, s(Y)) =#> rec#(/\z./\u.yap(F(z), u), X, Y)   

  P_5:

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

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_3, R_0, m, f) by (P_4, R_0, m, f) and (P_5, R_0, m, f).

Thus, the original system is terminating if each of (P_4, R_0, minimal, all) and (P_5, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_5, 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) 
  rec(/\x./\y.yap(F(x), y), X, 0) >= X 
  rec(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(s(Y)), rec(/\z./\u.yap(F(z), u), X, Y)) 
  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##, 0, @_{o -> o}, rec, s, yap, yap#}, and the following precedence: 0 > rec > s > 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] rec(/\x./\y.yap(F(x), y), X, 0) >= X  because [7], by (Star) 
  7] rec*(/\x./\y.yap(F(x), y), X, 0) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

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

  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_5, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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 apply the subterm criterion with the following projection function:

  nu(rec#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(rec#(/\x./\y.yap(F(x), y), X, s(Y))) = s(Y) |> Y = nu(rec#(/\z./\u.yap(F(z), u), X, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_4, R_0, minimal, f) by ({}, R_0, minimal, f).  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 +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[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.