We consider the system h60.

  Alphabet:

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

  Rules:

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

  Rules:

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

  Rules R_0:

    rec(0, X, /\x./\y.yap(F(x), y)) => X 
    rec(s(X), Y, /\x./\y.yap(F(x), y)) => yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 
    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#(s(X), Y, /\x./\y.yap(F(x), y)) =#> yap#(F(X), rec(X, Y, /\z./\u.yap(F(z), u)))   
    rec#(s(X), Y, /\x./\y.yap(F(x), y)) =#> F(X)   
    rec#(s(X), Y, /\x./\y.yap(F(x), y)) =#> rec#(X, Y, /\z./\u.yap(F(z), u))   
    rec#(s(X), Y, /\x./\y.yap(F(x), 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#(s(X), Y, /\x./\y.yap(F(x), y)) >? yap#(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 
  rec#(s(X), Y, /\x./\y.yap(F(x), y)) >? F(X) ~c0 
  rec#(s(X), Y, /\x./\y.yap(F(x), y)) >? rec#(X, Y, /\z./\u.yap(F(z), u)) 
  rec#(s(X), Y, /\x./\y.yap(F(x), y)) >? F(~c2) ~c1 
  yap#(F, X) >? F(X) 
  rec(0, X, /\x./\y.yap(F(x), y)) >= X 
  rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 
  yap(F, X) >= F X 
  rec(X, Y, F) >= rec#(X, Y, F) 
  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##, 0, @_{o -> o}, rec, rec#, s, yap, yap#}, and the following precedence: 0 > s > yap > rec = rec# > #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:

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

With these choices, we have:

  1] rec#(s(X), Y, /\x./\y.yap(F(x), y)) >= #argfun-yap##(@_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y))))  because [2], by (Star) 
  2] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= #argfun-yap##(@_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y))))  because [3], by (Select) 
  3] yap(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= #argfun-yap##(@_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y))))  because [4], by (Star) 
  4] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= #argfun-yap##(@_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y))))  because yap > #argfun-yap## and [5], by (Copy) 
  5] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= @_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because yap > @_{o -> o}, [6] and [12], by (Copy) 
  6] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= F(X)  because [7], by (Select) 
  7] F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(X)  because [8], by (Meta) 
  8] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= X  because [9], by (Select) 
  9] s(X) >= X  because [10], by (Star) 
  10] s*(X) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= rec(X, Y, /\x./\y.yap(F(x), y))  because [13], by (Select) 
  13] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= rec(X, Y, /\x./\y.yap(F(x), y))  because rec# = rec, rec# in Mul, [14], [16] and [17], by (Stat) 
  14] s(X) > X  because [15], by definition 
  15] s*(X) >= X  because [11], by (Select) 
  16] Y >= Y  by (Meta) 
  17] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [18], by (Abs) 
  18] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [19], by (Abs) 
  19] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [20] and [22], by (Fun) 
  20] F(y) >= F(y)  because [21], by (Meta) 
  21] y >= y  by (Var) 
  22] x >= x  by (Var) 

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

  28] rec#(s(X), Y, /\x./\y.yap(F(x), y)) >= rec#(X, Y, /\x./\y.yap(F(x), y))  because rec# in Mul, [29], [16] and [17], by (Fun) 
  29] s(X) >= X  because [15], by (Star) 

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

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

  42] rec(0, X, /\x./\y.yap(F(x), y)) >= X  because [43], by (Star) 
  43] rec*(0, X, /\x./\y.yap(F(x), y)) >= X  because [44], by (Select) 
  44] X >= X  by (Meta) 

  45] rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [46], by (Star) 
  46] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [47], by (Select) 
  47] yap(F(rec*(s(X), Y, /\x./\y.yap(F(x), y))), rec*(s(X), Y, /\z./\u.yap(F(z), u))) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because yap in Mul, [48] and [50], by (Fun) 
  48] F(rec*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(X)  because [49], by (Meta) 
  49] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= X  because [29], by (Select) 
  50] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= rec(X, Y, /\x./\y.yap(F(x), y))  because rec in Mul, [14], [16] and [17], 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 [40], by (Select) 
  54] yap*(F, X) >= X  because [41], by (Select) 

  55] rec(X, Y, F) >= rec#(X, Y, F)  because rec = rec#, rec in Mul, [56], [57] and [58], by (Fun) 
  56] X >= X  by (Meta) 
  57] Y >= Y  by (Meta) 
  58] F >= F  by (Meta) 

  59] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [60], by (Star) 
  60] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [61], by (Copy) 
  61] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [62] and [64], by (Copy) 
  62] yap*(F, X) >= F  because [63], by (Select) 
  63] F >= F  by (Meta) 
  64] yap*(F, X) >= X  because [65], by (Select) 
  65] 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#(s(X), Y, /\x./\y.yap(F(x), y)) =#> yap#(F(X), rec(X, Y, /\z./\u.yap(F(z), u)))   
  rec#(s(X), Y, /\x./\y.yap(F(x), y)) =#> rec#(X, Y, /\z./\u.yap(F(z), u))   
  rec#(s(X), Y, /\x./\y.yap(F(x), y)) =#> 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:

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

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

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

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

With these choices, we have:

  1] rec#(s(X), Y, /\x./\y.yap(F(x), y)) >= @_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [2], by (Star) 
  2] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= @_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [3], by (Select) 
  3] yap(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= @_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [4], by (Star) 
  4] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= @_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because yap > @_{o -> o}, [5] and [11], by (Copy) 
  5] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= F(X)  because [6], by (Select) 
  6] F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(X)  because [7], by (Meta) 
  7] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= X  because [8], by (Select) 
  8] s(X) >= X  because [9], by (Star) 
  9] s*(X) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= rec(X, Y, /\x./\y.yap(F(x), y))  because [12], by (Select) 
  12] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= rec(X, Y, /\x./\y.yap(F(x), y))  because rec# = rec, rec# in Mul, [13], [15] and [16], by (Stat) 
  13] s(X) > X  because [14], by definition 
  14] s*(X) >= X  because [10], by (Select) 
  15] Y >= Y  by (Meta) 
  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] rec#(s(X), Y, /\x./\y.yap(F(x), y)) >= rec#(X, Y, /\x./\y.yap(F(x), y))  because rec# in Mul, [23], [15] and [16], by (Fun) 
  23] s(X) >= X  because [14], by (Star) 

  24] rec#(s(X), Y, /\x./\y.yap(F(x), y)) > @_{o -> o}(F(_|_), _|_)  because [25], by definition 
  25] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= @_{o -> o}(F(_|_), _|_)  because [26], by (Select) 
  26] yap(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= @_{o -> o}(F(_|_), _|_)  because [27], by (Star) 
  27] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= @_{o -> o}(F(_|_), _|_)  because yap > @_{o -> o}, [28] and [31], by (Copy) 
  28] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= F(_|_)  because [29], by (Select) 
  29] F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(_|_)  because [30], by (Meta) 
  30] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= _|_  by (Bot) 
  31] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= _|_  by (Bot) 

  32] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [33] and [34], by (Fun) 
  33] F >= F  by (Meta) 
  34] X >= X  by (Meta) 

  35] rec(0, X, /\x./\y.yap(F(x), y)) >= X  because [36], by (Star) 
  36] rec*(0, X, /\x./\y.yap(F(x), y)) >= X  because [37], by (Select) 
  37] X >= X  by (Meta) 

  38] rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [39], by (Star) 
  39] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [40], by (Select) 
  40] yap(F(rec*(s(X), Y, /\x./\y.yap(F(x), y))), rec*(s(X), Y, /\z./\u.yap(F(z), u))) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because yap in Mul, [41] and [43], by (Fun) 
  41] F(rec*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(X)  because [42], by (Meta) 
  42] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= X  because [23], by (Select) 
  43] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= rec(X, Y, /\x./\y.yap(F(x), y))  because rec in Mul, [13], [15] and [16], 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 [33], by (Select) 
  47] yap*(F, X) >= X  because [34], by (Select) 

  48] rec(X, Y, F) >= rec#(X, Y, F)  because rec = rec#, rec in Mul, [49], [50] and [51], by (Fun) 
  49] X >= X  by (Meta) 
  50] Y >= Y  by (Meta) 
  51] F >= F  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_2, R_0, minimal, all) by (P_3, R_0, minimal, all), where P_3 consists of:

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

  rec#(s(X), Y, /\x./\y.yap(F(x), y)) >? yap#(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 
  rec#(s(X), Y, /\x./\y.yap(F(x), y)) >? rec#(X, Y, /\z./\u.yap(F(z), u)) 
  yap#(F, X) >? F(X) 
  rec(0, X, /\x./\y.yap(F(x), y)) >= X 
  rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 
  yap(F, X) >= F X 
  rec(X, Y, F) >= rec#(X, Y, F) 
  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, rec#, s, yap, yap#}, and the following precedence: 0 > rec = rec# > s > yap > #argfun-yap## > @_{o -> o} > yap#

With these choices, we have:

  1] rec#(s(X), Y, /\x./\y.yap(F(x), y)) > #argfun-yap##(@_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y))))  because [2], by definition 
  2] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= #argfun-yap##(@_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y))))  because [3], by (Select) 
  3] yap(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= #argfun-yap##(@_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y))))  because [4], by (Star) 
  4] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= #argfun-yap##(@_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y))))  because yap > #argfun-yap## and [5], by (Copy) 
  5] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= @_{o -> o}(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because yap > @_{o -> o}, [6] and [12], by (Copy) 
  6] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= F(X)  because [7], by (Select) 
  7] F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(X)  because [8], by (Meta) 
  8] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= X  because [9], by (Select) 
  9] s(X) >= X  because [10], by (Star) 
  10] s*(X) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] yap*(F(rec#*(s(X), Y, /\x./\y.yap(F(x), y))), rec#*(s(X), Y, /\z./\u.yap(F(z), u))) >= rec(X, Y, /\x./\y.yap(F(x), y))  because [13], by (Select) 
  13] rec#*(s(X), Y, /\x./\y.yap(F(x), y)) >= rec(X, Y, /\x./\y.yap(F(x), y))  because rec# = rec, rec# in Mul, [14], [16] and [17], by (Stat) 
  14] s(X) > X  because [15], by definition 
  15] s*(X) >= X  because [11], by (Select) 
  16] Y >= Y  by (Meta) 
  17] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [18], by (Abs) 
  18] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [19], by (Abs) 
  19] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [20] and [22], by (Fun) 
  20] F(y) >= F(y)  because [21], by (Meta) 
  21] y >= y  by (Var) 
  22] x >= x  by (Var) 

  23] rec#(s(X), Y, /\x./\y.yap(F(x), y)) >= rec#(X, Y, /\x./\y.yap(F(x), y))  because rec# in Mul, [24], [16] and [17], by (Fun) 
  24] s(X) >= X  because [15], by (Star) 

  25] #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [26], by (Star) 
  26] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [27], by (Select) 
  27] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [28] and [29], by (Fun) 
  28] F >= F  by (Meta) 
  29] X >= X  by (Meta) 

  30] rec(0, X, /\x./\y.yap(F(x), y)) >= X  because [31], by (Star) 
  31] rec*(0, X, /\x./\y.yap(F(x), y)) >= X  because [32], by (Select) 
  32] X >= X  by (Meta) 

  33] rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [34], by (Star) 
  34] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because rec > yap, [35] and [39], by (Copy) 
  35] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= F(X)  because [36], by (Select) 
  36] /\x.yap(F(rec*(s(X), Y, /\y./\z.yap(F(y), z))), x) >= F(X)  because [37], by (Eta)[Kop13:2] 
  37] F(rec*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(X)  because [38], by (Meta) 
  38] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= X  because [24], by (Select) 
  39] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= rec(X, Y, /\x./\y.yap(F(x), y))  because rec in Mul, [14], [16] and [17], by (Stat) 

  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 [28], by (Select) 
  43] yap*(F, X) >= X  because [29], by (Select) 

  44] rec(X, Y, F) >= rec#(X, Y, F)  because rec = rec#, rec in Mul, [45], [46] and [47], by (Fun) 
  45] X >= X  by (Meta) 
  46] Y >= Y  by (Meta) 
  47] F >= F  by (Meta) 

  48] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [49], by (Star) 
  49] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [50], by (Copy) 
  50] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [51] and [53], by (Copy) 
  51] yap*(F, X) >= F  because [52], by (Select) 
  52] F >= F  by (Meta) 
  53] yap*(F, X) >= X  because [54], by (Select) 
  54] 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:

  rec#(s(X), Y, /\x./\y.yap(F(x), y)) =#> rec#(X, Y, /\z./\u.yap(F(z), u))   
  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 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_5:

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

  P_6:

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

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_4, R_0, m, f) by (P_5, R_0, m, f) and (P_6, R_0, m, f).

Thus, the original system is terminating if each of (P_5, R_0, minimal, all) and (P_6, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_6, 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(0, X, /\x./\y.yap(F(x), y)) >= X 
  rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 
  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 > s > yap > #argfun-yap## > rec > @_{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(0, X, /\x./\y.yap(F(x), y)) >= X  because [7], by (Star) 
  7] rec*(0, X, /\x./\y.yap(F(x), y)) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

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

  27] yap(F, X) >= @_{o -> o}(F, X)  because [28], by (Star) 
  28] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [29] and [30], by (Copy) 
  29] yap*(F, X) >= F  because [4], by (Select) 
  30] yap*(F, X) >= X  because [5], by (Select) 

  31] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [32], by (Star) 
  32] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [33], by (Copy) 
  33] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [34] and [36], by (Copy) 
  34] yap*(F, X) >= F  because [35], by (Select) 
  35] F >= F  by (Meta) 
  36] yap*(F, X) >= X  because [37], by (Select) 
  37] 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_6, 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_5, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_5, R_0, minimal, all).

We apply the subterm criterion with the following projection function:

  nu(rec#) = 1 

Thus, we can orient the dependency pairs as follows:

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

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_5, 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.