We consider the system h54.

  Alphabet:

    0 : [] --> nat 
    cons : [nat * list] --> list 
    foldl : [nat -> nat -> nat * nat * list] --> nat 
    nil : [] --> list 
    plusc : [] --> nat -> nat -> nat 
    s : [nat] --> nat 
    sum : [list] --> nat 
    xap : [nat -> nat -> nat * nat] --> nat -> nat 
    yap : [nat -> nat * nat] --> nat 

  Rules:

    foldl(/\x./\y.yap(xap(f, x), y), z, nil) => z 
    foldl(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => foldl(/\w./\x'.yap(xap(f, w), x'), yap(xap(f, z), u), v) 
    plusc x 0 => x 
    plusc x s(y) => s(plusc x y) 
    sum(x) => foldl(/\y./\z.yap(xap(plusc, y), z), 0, 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 : [] --> nat 
    cons : [nat * list] --> list 
    foldl : [nat -> nat -> nat * nat * list] --> nat 
    nil : [] --> list 
    plusc : [nat] --> nat -> nat 
    s : [nat] --> nat 
    sum : [list] --> nat 
    yap : [nat -> nat * nat] --> nat 

  Rules:

    foldl(/\x./\y.yap(F(x), y), X, nil) => X 
    foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
    plusc(X) 0 => X 
    plusc(X) s(Y) => s(plusc(X) Y) 
    sum(X) => foldl(/\x./\y.yap(plusc(x), y), 0, X) 
    yap(F, X) => F X 

We observe that the rules contain a first-order subset:

  plusc(X) 0 => X 
  plusc(X) s(Y) => s(plusc(X) Y) 

Moreover, the system is finitely branching.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS.

According to the external first-order termination prover, this system is indeed Ce-terminating:

 || proof of resources/system.trs
 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
 || 
 || 
 || Termination w.r.t. Q of the given QTRS could be proven:
 || 
 || (0) QTRS
 || (1) QTRSRRRProof [EQUIVALENT]
 || (2) QTRS
 || (3) RisEmptyProof [EQUIVALENT]
 || (4) YES
 || 
 || 
 || ----------------------------------------
 || 
 || (0)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    plusc(%X, 0) -> %X
 ||    plusc(%X, s(%Y)) -> s(plusc(%X, %Y))
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Polynomial interpretation [POLO]:
 || 
 ||    POL(0) = 2
 ||    POL(plusc(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
 ||    POL(s(x_1)) = 1 + x_1
 ||    POL(~PAIR(x_1, x_2)) = 2 + x_1 + x_2
 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
 || 
 ||    plusc(%X, 0) -> %X
 ||    plusc(%X, s(%Y)) -> s(plusc(%X, %Y))
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (2)
 || Obligation:
 || Q restricted rewrite system:
 || R is empty.
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (3) RisEmptyProof (EQUIVALENT)
 || The TRS R is empty. Hence, termination is trivially proven.
 || ----------------------------------------
 || 
 || (4)
 || YES
 || 
We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, all):

  Dependency Pairs P_0:

    0] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z)   
    1] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> yap#(F(z), u)   
    2] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(z)   
    3] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> yap#(F(X), Y)   
    4] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(X)   
    5] sum#(X) =#> foldl#(/\x./\y.yap(plusc(x), y), 0, X)   
    6] sum#(X) =#> yap#(plusc(x), y)   
    7] sum#(X) =#> plusc#(x)   
    8] yap#(F, X) =#> F(X)   

  Rules R_0:

    foldl(/\x./\y.yap(F(x), y), X, nil) => X 
    foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
    plusc(X) 0 => X 
    plusc(X) s(Y) => s(plusc(X) Y) 
    sum(X) => foldl(/\x./\y.yap(plusc(x), y), 0, 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, 8 
    * 3 : 8 
    * 4 : 0, 1, 2, 3, 4, 5, 6, 7, 8 
    * 5 : 0, 1, 2, 3, 4 
    * 6 :  
    * 7 :  
    * 8 : 0, 1, 2, 3, 4, 5, 6, 7, 8 

This graph has the following strongly connected components:

  P_1:

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

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).

Thus, the original system is terminating if (P_1, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, all).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(~c1) ~c0 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap#(F(X), Y) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(X) ~c2 
  sum#(X) >? foldl#(/\x./\y.yap(plusc(x), y), 0, X) 
  yap#(F, X) >? F(X) 
  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  plusc(X) 0 >= X 
  plusc(X) s(Y) >= s(plusc(X) Y) 
  sum(X) >= foldl(/\x./\y.yap(plusc(x), y), 0, X) 
  yap(F, X) >= F X 
  foldl(F, X, Y) >= foldl#(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##(foldl#(/\x./\y.yap(plusc(x), y), 0, X)) 
  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-sum##(x_1)]] = x_1 
  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
  [[foldl#(x_1, x_2, x_3)]] = foldl#(x_3, x_1, x_2) 
  [[s(x_1)]] = x_1 
  [[sum#(x_1)]] = x_1 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 
  [[~c2]] = _|_ 

We choose Lex = {foldl, foldl#} and Mul = {#argfun-yap##, 0, @_{o -> o}, cons, nil, plusc, sum, yap, yap#}, and the following precedence: nil > sum > foldl > foldl# > yap > #argfun-yap## > cons > plusc > @_{o -> o} > 0 > yap#

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

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > foldl#(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(_|_), _|_) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= #argfun-yap##(@_{o -> o}(F(X), Y)) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), _|_) 
  foldl#(/\x./\y.yap(plusc(x), y), 0, X) >= foldl#(/\x./\y.yap(plusc(x), y), 0, X) 
  #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X) 
  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
  @_{o -> o}(plusc(X), 0) >= X 
  @_{o -> o}(plusc(X), Y) >= @_{o -> o}(plusc(X), Y) 
  sum(X) >= foldl(/\x./\y.yap(plusc(x), y), 0, X) 
  yap(F, X) >= @_{o -> o}(F, X) 
  foldl(F, X, Y) >= foldl#(F, X, Y) 
  sum(X) >= foldl#(/\x./\y.yap(plusc(x), y), 0, X) 
  yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X)) 

With these choices, we have:

  1] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > foldl#(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [2], by definition 
  2] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl#(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [3], [6], [13] and [23], by (Stat) 
  3] cons(Y, Z) > Z  because [4], by definition 
  4] cons*(Y, Z) >= Z  because [5], by (Select) 
  5] Z >= Z  by (Meta) 
  6] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [7], by (Select) 
  7] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [8], by (Abs) 
  8] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [9], by (Abs) 
  9] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [10] and [12], by (Fun) 
  10] F(y) >= F(y)  because [11], by (Meta) 
  11] y >= y  by (Var) 
  12] x >= x  by (Var) 
  13] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl# > yap, [14] and [19], by (Copy) 
  14] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= F(X)  because [15], by (Select) 
  15] /\z.yap(F(foldl#*(/\u./\v.yap(F(u), v), X, cons(Y, Z))), z) >= F(X)  because [16], by (Eta)[Kop13:2] 
  16] F(foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [17], by (Meta) 
  17] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [18], by (Select) 
  18] X >= X  by (Meta) 
  19] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y  because [20], by (Select) 
  20] cons(Y, Z) >= Y  because [21], by (Star) 
  21] cons*(Y, Z) >= Y  because [22], by (Select) 
  22] Y >= Y  by (Meta) 
  23] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z  because [24], by (Select) 
  24] cons(Y, Z) >= Z  because [4], by (Star) 

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

  33] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because [34], by (Star) 
  34] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because [35], by (Select) 
  35] yap(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because [36], by (Star) 
  36] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because yap > #argfun-yap## and [37], by (Copy) 
  37] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= @_{o -> o}(F(X), Y)  because yap > @_{o -> o}, [38] and [39], by (Copy) 
  38] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [16], by (Select) 
  39] yap*(F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Y  because [19], by (Select) 

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

  45] foldl#(/\x./\y.yap(plusc(x), y), 0, X) >= foldl#(/\x./\y.yap(plusc(x), y), 0, X)  because [46], [52] and [53], by (Fun) 
  46] /\x./\z.yap(plusc(x), z) >= /\x./\z.yap(plusc(x), z)  because [47], by (Abs) 
  47] /\z.yap(plusc(y), z) >= /\z.yap(plusc(y), z)  because [48], by (Abs) 
  48] yap(plusc(y), x) >= yap(plusc(y), x)  because yap in Mul, [49] and [51], by (Fun) 
  49] plusc(y) >= plusc(y)  because plusc in Mul and [50], by (Fun) 
  50] y >= y  by (Var) 
  51] x >= x  by (Var) 
  52] 0 >= 0  by (Fun) 
  53] X >= X  by (Meta) 

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

  59] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [60], by (Star) 
  60] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [61], by (Select) 
  61] X >= X  by (Meta) 

  62] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [63], by (Star) 
  63] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [3], [64], [65] and [71], by (Stat) 
  64] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [7], by (Select) 
  65] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [66] and [70], by (Copy) 
  66] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [67], by (Select) 
  67] /\x.yap(F(foldl*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(X)  because [68], by (Eta)[Kop13:2] 
  68] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [69], by (Meta) 
  69] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [18], by (Select) 
  70] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [20], by (Select) 
  71] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [72], by (Select) 
  72] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [73], by (Star) 
  73] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [74], by (Select) 
  74] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [75], by (Select) 
  75] cons(Y, Z) >= Z  because [4], by (Star) 

  76] @_{o -> o}(plusc(X), 0) >= X  because [77], by (Star) 
  77] @_{o -> o}*(plusc(X), 0) >= X  because [78], by (Select) 
  78] plusc(X) @_{o -> o}*(plusc(X), 0) >= X  because [79] 
  79] plusc*(X, @_{o -> o}*(plusc(X), 0)) >= X  because [80], by (Select) 
  80] X >= X  by (Meta) 

  81] @_{o -> o}(plusc(X), Y) >= @_{o -> o}(plusc(X), Y)  because @_{o -> o} in Mul, [82] and [84], by (Fun) 
  82] plusc(X) >= plusc(X)  because plusc in Mul and [83], by (Fun) 
  83] X >= X  by (Meta) 
  84] Y >= Y  by (Meta) 

  85] sum(X) >= foldl(/\x./\y.yap(plusc(x), y), 0, X)  because [86], by (Star) 
  86] sum*(X) >= foldl(/\x./\y.yap(plusc(x), y), 0, X)  because sum > foldl, [87], [95] and [96], by (Copy) 
  87] sum*(X) >= /\y./\z.yap(plusc(y), z)  because [88], by (F-Abs) 
  88] sum*(X, x) >= /\z.yap(plusc(x), z)  because [89], by (F-Abs) 
  89] sum*(X, x, y) >= yap(plusc(x), y)  because sum > yap, [90] and [93], by (Copy) 
  90] sum*(X, x, y) >= plusc(x)  because sum > plusc and [91], by (Copy) 
  91] sum*(X, x, y) >= x  because [92], by (Select) 
  92] x >= x  by (Var) 
  93] sum*(X, x, y) >= y  because [94], by (Select) 
  94] y >= y  by (Var) 
  95] sum*(X) >= 0  because sum > 0, by (Copy) 
  96] sum*(X) >= X  because [53], by (Select) 

  97] yap(F, X) >= @_{o -> o}(F, X)  because [98], by (Star) 
  98] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [99] and [100], by (Copy) 
  99] yap*(F, X) >= F  because [57], by (Select) 
  100] yap*(F, X) >= X  because [58], by (Select) 

  101] foldl(F, X, Y) >= foldl#(F, X, Y)  because [102], by (Star) 
  102] foldl*(F, X, Y) >= foldl#(F, X, Y)  because foldl > foldl#, [103], [105] and [107], by (Copy) 
  103] foldl*(F, X, Y) >= F  because [104], by (Select) 
  104] F >= F  by (Meta) 
  105] foldl*(F, X, Y) >= X  because [106], by (Select) 
  106] X >= X  by (Meta) 
  107] foldl*(F, X, Y) >= Y  because [108], by (Select) 
  108] Y >= Y  by (Meta) 

  109] sum(X) >= foldl#(/\x./\y.yap(plusc(x), y), 0, X)  because [110], by (Star) 
  110] sum*(X) >= foldl#(/\x./\y.yap(plusc(x), y), 0, X)  because sum > foldl#, [111], [119] and [120], by (Copy) 
  111] sum*(X) >= /\y./\z.yap(plusc(y), z)  because [112], by (F-Abs) 
  112] sum*(X, x) >= /\z.yap(plusc(x), z)  because [113], by (F-Abs) 
  113] sum*(X, x, y) >= yap(plusc(x), y)  because sum > yap, [114] and [117], by (Copy) 
  114] sum*(X, x, y) >= plusc(x)  because sum > plusc and [115], by (Copy) 
  115] sum*(X, x, y) >= x  because [116], by (Select) 
  116] x >= x  by (Var) 
  117] sum*(X, x, y) >= y  because [118], by (Select) 
  118] y >= y  by (Var) 
  119] sum*(X) >= 0  because sum > 0, by (Copy) 
  120] sum*(X) >= X  because [121], by (Select) 
  121] X >= X  by (Meta) 

  122] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [123], by (Star) 
  123] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [124], by (Copy) 
  124] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [125] and [127], by (Copy) 
  125] yap*(F, X) >= F  because [126], by (Select) 
  126] F >= F  by (Meta) 
  127] yap*(F, X) >= X  because [128], by (Select) 
  128] X >= X  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_0, minimal, all) by (P_2, R_0, minimal, all), where P_2 consists of:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(z)   
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> yap#(F(X), Y)   
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(X)   
  sum#(X) =#> foldl#(/\x./\y.yap(plusc(x), y), 0, X)   
  yap#(F, X) =#> F(X)   

Thus, the original system is terminating if (P_2, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, all).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(~c1) ~c0 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap#(F(X), Y) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(X) ~c2 
  sum#(X) >? foldl#(/\x./\y.yap(plusc(x), y), 0, X) 
  yap#(F, X) >? F(X) 
  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  plusc(X) 0 >= X 
  plusc(X) s(Y) >= s(plusc(X) Y) 
  sum(X) >= foldl(/\x./\y.yap(plusc(x), y), 0, X) 
  yap(F, X) >= F X 
  foldl(F, X, Y) >= foldl#(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##(foldl#(/\x./\y.yap(plusc(x), y), 0, X)) 
  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]] = _|_ 
  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
  [[sum#(x_1)]] = x_1 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 
  [[~c2]] = _|_ 

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

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

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

With these choices, we have:

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

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

  20] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > @_{o -> o}(F(X), _|_)  because [21], by definition 
  21] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), _|_)  because foldl# > @_{o -> o}, [11] and [22], by (Copy) 
  22] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 

  23] #argfun-sum##(foldl#(/\x./\y.yap(plusc(x), y), _|_, X)) > foldl#(/\x./\y.yap(plusc(x), y), _|_, X)  because [24], by definition 
  24] #argfun-sum##*(foldl#(/\x./\y.yap(plusc(x), y), _|_, X)) >= foldl#(/\x./\y.yap(plusc(x), y), _|_, X)  because #argfun-sum## > foldl#, [25], [33] and [34], by (Copy) 
  25] #argfun-sum##*(foldl#(/\x./\y.yap(plusc(x), y), _|_, X)) >= /\x./\y.yap(plusc(x), y)  because [26], by (F-Abs) 
  26] #argfun-sum##*(foldl#(/\x./\y.yap(plusc(x), y), _|_, X), z) >= /\x.yap(plusc(z), x)  because [27], by (F-Abs) 
  27] #argfun-sum##*(foldl#(/\x./\y.yap(plusc(x), y), _|_, X), z, u) >= yap(plusc(z), u)  because #argfun-sum## > yap, [28] and [31], by (Copy) 
  28] #argfun-sum##*(foldl#(/\x./\y.yap(plusc(x), y), _|_, X), z, u) >= plusc(z)  because #argfun-sum## > plusc and [29], by (Copy) 
  29] #argfun-sum##*(foldl#(/\x./\y.yap(plusc(x), y), _|_, X), z, u) >= z  because [30], by (Select) 
  30] z >= z  by (Var) 
  31] #argfun-sum##*(foldl#(/\x./\y.yap(plusc(x), y), _|_, X), z, u) >= u  because [32], by (Select) 
  32] u >= u  by (Var) 
  33] #argfun-sum##*(foldl#(/\x./\y.yap(plusc(x), y), _|_, X)) >= _|_  by (Bot) 
  34] #argfun-sum##*(foldl#(/\x./\y.yap(plusc(x), y), _|_, X)) >= X  because [35], by (Select) 
  35] foldl#(/\x./\y.yap(plusc(x), y), _|_, X) >= X  because [36], by (Star) 
  36] foldl#*(/\x./\y.yap(plusc(x), y), _|_, X) >= X  because [37], by (Select) 
  37] X >= X  by (Meta) 

  38] #argfun-yap##(@_{o -> o}(F, X)) > @_{o -> o}(F, X)  because [39], by definition 
  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] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [44], by (Star) 
  44] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [45], by (Select) 
  45] X >= X  by (Meta) 

  46] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [47], by (Star) 
  47] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [48], [51], [58] and [68], by (Stat) 
  48] cons(Y, Z) > Z  because [49], by definition 
  49] cons*(Y, Z) >= Z  because [50], by (Select) 
  50] Z >= Z  by (Meta) 
  51] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [52], by (Select) 
  52] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [53], by (Abs) 
  53] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [54], by (Abs) 
  54] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [55] and [57], by (Fun) 
  55] F(y) >= F(y)  because [56], by (Meta) 
  56] y >= y  by (Var) 
  57] x >= x  by (Var) 
  58] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [59] and [63], by (Copy) 
  59] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= F(X)  because [60], by (Select) 
  60] /\z.yap(F(foldl*(/\u./\v.yap(F(u), v), X, cons(Y, Z))), z) >= F(X)  because [61], by (Eta)[Kop13:2] 
  61] F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [62], by (Meta) 
  62] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [15], by (Select) 
  63] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y  because [64], by (Select) 
  64] yap(F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= Y  because [65], by (Star) 
  65] yap*(F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= Y  because [66], by (Select) 
  66] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y  because [67], by (Select) 
  67] cons(Y, Z) >= Y  because [18], by (Star) 
  68] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z  because [69], by (Select) 
  69] cons(Y, Z) >= Z  because [49], by (Star) 

  70] @_{o -> o}(plusc(X), _|_) >= X  because [71], by (Star) 
  71] @_{o -> o}*(plusc(X), _|_) >= X  because [72], by (Select) 
  72] plusc(X) @_{o -> o}*(plusc(X), _|_) >= X  because [73] 
  73] plusc*(X, @_{o -> o}*(plusc(X), _|_)) >= X  because [74], by (Select) 
  74] X >= X  by (Meta) 

  75] @_{o -> o}(plusc(X), s(Y)) >= s(@_{o -> o}(plusc(X), Y))  because [76], by (Star) 
  76] @_{o -> o}*(plusc(X), s(Y)) >= s(@_{o -> o}(plusc(X), Y))  because [77], by (Select) 
  77] plusc(X) @_{o -> o}*(plusc(X), s(Y)) >= s(@_{o -> o}(plusc(X), Y))  because [78] 
  78] plusc*(X, @_{o -> o}*(plusc(X), s(Y))) >= s(@_{o -> o}(plusc(X), Y))  because plusc > s and [79], by (Copy) 
  79] plusc*(X, @_{o -> o}*(plusc(X), s(Y))) >= @_{o -> o}(plusc(X), Y)  because [80], by (Select) 
  80] @_{o -> o}*(plusc(X), s(Y)) >= @_{o -> o}(plusc(X), Y)  because @_{o -> o} in Mul, [81] and [83], by (Stat) 
  81] plusc(X) >= plusc(X)  because plusc in Mul and [82], by (Fun) 
  82] X >= X  by (Meta) 
  83] s(Y) > Y  because [84], by definition 
  84] s*(Y) >= Y  because [85], by (Select) 
  85] Y >= Y  by (Meta) 

  86] sum(X) >= foldl(/\x./\y.yap(plusc(x), y), _|_, X)  because [87], by (Star) 
  87] sum*(X) >= foldl(/\x./\y.yap(plusc(x), y), _|_, X)  because sum > foldl, [88], [96] and [97], by (Copy) 
  88] sum*(X) >= /\y./\z.yap(plusc(y), z)  because [89], by (F-Abs) 
  89] sum*(X, x) >= /\z.yap(plusc(x), z)  because [90], by (F-Abs) 
  90] sum*(X, x, y) >= yap(plusc(x), y)  because sum > yap, [91] and [94], by (Copy) 
  91] sum*(X, x, y) >= plusc(x)  because sum > plusc and [92], by (Copy) 
  92] sum*(X, x, y) >= x  because [93], by (Select) 
  93] x >= x  by (Var) 
  94] sum*(X, x, y) >= y  because [95], by (Select) 
  95] y >= y  by (Var) 
  96] sum*(X) >= _|_  by (Bot) 
  97] sum*(X) >= X  because [37], by (Select) 

  98] yap(F, X) >= @_{o -> o}(F, X)  because [99], by (Star) 
  99] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [100] and [101], by (Copy) 
  100] yap*(F, X) >= F  because [41], by (Select) 
  101] yap*(F, X) >= X  because [42], by (Select) 

  102] foldl(F, X, Y) >= foldl#(F, X, Y)  because [103], by (Star) 
  103] foldl*(F, X, Y) >= foldl#(F, X, Y)  because foldl > foldl#, [104], [106] and [108], by (Copy) 
  104] foldl*(F, X, Y) >= F  because [105], by (Select) 
  105] F >= F  by (Meta) 
  106] foldl*(F, X, Y) >= X  because [107], by (Select) 
  107] X >= X  by (Meta) 
  108] foldl*(F, X, Y) >= Y  because [109], by (Select) 
  109] Y >= Y  by (Meta) 

  110] sum(X) >= #argfun-sum##(foldl#(/\x./\y.yap(plusc(x), y), _|_, X))  because [111], by (Star) 
  111] sum*(X) >= #argfun-sum##(foldl#(/\x./\y.yap(plusc(x), y), _|_, X))  because sum > #argfun-sum## and [112], by (Copy) 
  112] sum*(X) >= foldl#(/\x./\y.yap(plusc(x), y), _|_, X)  because sum > foldl#, [113], [121] and [122], by (Copy) 
  113] sum*(X) >= /\y./\z.yap(plusc(y), z)  because [114], by (F-Abs) 
  114] sum*(X, x) >= /\z.yap(plusc(x), z)  because [115], by (F-Abs) 
  115] sum*(X, x, y) >= yap(plusc(x), y)  because sum > yap, [116] and [119], by (Copy) 
  116] sum*(X, x, y) >= plusc(x)  because sum > plusc and [117], by (Copy) 
  117] sum*(X, x, y) >= x  because [118], by (Select) 
  118] x >= x  by (Var) 
  119] sum*(X, x, y) >= y  because [120], by (Select) 
  120] y >= y  by (Var) 
  121] sum*(X) >= _|_  by (Bot) 
  122] sum*(X) >= X  because [123], by (Select) 
  123] X >= X  by (Meta) 

  124] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [125], by (Star) 
  125] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [126], by (Copy) 
  126] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [127] and [129], by (Copy) 
  127] yap*(F, X) >= F  because [128], by (Select) 
  128] F >= F  by (Meta) 
  129] yap*(F, X) >= X  because [130], by (Select) 
  130] X >= X  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_2, R_0, minimal, all) by (P_3, R_0, minimal, all), where P_3 consists of:

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

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

We consider the dependency pair problem (P_3, R_0, minimal, all).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(~c1) ~c0 
  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  plusc(X) 0 >= X 
  plusc(X) s(Y) >= s(plusc(X) Y) 
  sum(X) >= foldl(/\x./\y.yap(plusc(x), y), 0, X) 
  yap(F, X) >= F X 
  foldl(F, X, Y) >= foldl#(F, X, Y) 

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

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

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

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

With these choices, we have:

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

  8] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [9], by (Star) 
  9] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 

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

  35] @_{o -> o}(plusc(X), _|_) >= X  because [36], by (Star) 
  36] @_{o -> o}*(plusc(X), _|_) >= X  because [37], by (Select) 
  37] plusc(X) @_{o -> o}*(plusc(X), _|_) >= X  because [38] 
  38] plusc*(X, @_{o -> o}*(plusc(X), _|_)) >= X  because [39], by (Select) 
  39] X >= X  by (Meta) 

  40] @_{o -> o}(plusc(X), Y) >= @_{o -> o}(plusc(X), Y)  because @_{o -> o} in Mul, [41] and [43], by (Fun) 
  41] plusc(X) >= plusc(X)  because plusc in Mul and [42], by (Fun) 
  42] X >= X  by (Meta) 
  43] Y >= Y  by (Meta) 

  44] sum(X) >= foldl(/\x./\y.yap(plusc(x), y), _|_, X)  because [45], by (Star) 
  45] sum*(X) >= foldl(/\x./\y.yap(plusc(x), y), _|_, X)  because sum > foldl, [46], [54] and [55], by (Copy) 
  46] sum*(X) >= /\y./\z.yap(plusc(y), z)  because [47], by (F-Abs) 
  47] sum*(X, x) >= /\z.yap(plusc(x), z)  because [48], by (F-Abs) 
  48] sum*(X, x, y) >= yap(plusc(x), y)  because sum > yap, [49] and [52], by (Copy) 
  49] sum*(X, x, y) >= plusc(x)  because sum > plusc and [50], by (Copy) 
  50] sum*(X, x, y) >= x  because [51], by (Select) 
  51] x >= x  by (Var) 
  52] sum*(X, x, y) >= y  because [53], by (Select) 
  53] y >= y  by (Var) 
  54] sum*(X) >= _|_  by (Bot) 
  55] sum*(X) >= X  because [56], by (Select) 
  56] X >= X  by (Meta) 

  57] yap(F, X) >= @_{o -> o}(F, X)  because [58], by (Star) 
  58] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [59] and [61], by (Copy) 
  59] yap*(F, X) >= F  because [60], by (Select) 
  60] F >= F  by (Meta) 
  61] yap*(F, X) >= X  because [62], by (Select) 
  62] X >= X  by (Meta) 

  63] foldl(F, X, Y) >= foldl#(F, X, Y)  because foldl = foldl#, [64], [65] and [66], by (Fun) 
  64] F >= F  by (Meta) 
  65] X >= X  by (Meta) 
  66] Y >= Y  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.