9.22/9.29	YES
9.29/9.33	We consider the system theBenchmark.
9.29/9.33	
9.29/9.33	  Alphabet:
9.29/9.33	
9.29/9.33	    0 : [] --> nat 
9.29/9.33	    cons : [nat * list] --> list 
9.29/9.33	    foldl : [nat -> nat -> nat * nat * list] --> nat 
9.29/9.33	    nil : [] --> list 
9.29/9.33	    plusc : [] --> nat -> nat -> nat 
9.29/9.33	    s : [nat] --> nat 
9.29/9.33	    sum : [list] --> nat 
9.29/9.33	    xap : [nat -> nat -> nat * nat] --> nat -> nat 
9.29/9.33	    yap : [nat -> nat * nat] --> nat 
9.29/9.33	
9.29/9.33	  Rules:
9.29/9.33	
9.29/9.33	    foldl(/\x./\y.yap(xap(f, x), y), z, nil) => z 
9.29/9.33	    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) 
9.29/9.33	    plusc x 0 => x 
9.29/9.33	    plusc x s(y) => s(plusc x y) 
9.29/9.33	    sum(x) => foldl(/\y./\z.yap(xap(plusc, y), z), 0, x) 
9.29/9.33	    xap(f, x) => f x 
9.29/9.33	    yap(f, x) => f x 
9.29/9.33	
9.29/9.33	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
9.29/9.33	
9.29/9.33	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:
9.29/9.33	
9.29/9.33	  Alphabet:
9.29/9.33	
9.29/9.33	    0 : [] --> nat 
9.29/9.33	    cons : [nat * list] --> list 
9.29/9.33	    foldl : [nat -> nat -> nat * nat * list] --> nat 
9.29/9.33	    nil : [] --> list 
9.29/9.33	    plusc : [nat] --> nat -> nat 
9.29/9.33	    s : [nat] --> nat 
9.29/9.33	    sum : [list] --> nat 
9.29/9.33	    yap : [nat -> nat * nat] --> nat 
9.29/9.33	
9.29/9.33	  Rules:
9.29/9.33	
9.29/9.33	    foldl(/\x./\y.yap(F(x), y), X, nil) => X 
9.29/9.33	    foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
9.29/9.33	    plusc(X) 0 => X 
9.29/9.33	    plusc(X) s(Y) => s(plusc(X) Y) 
9.29/9.33	    sum(X) => foldl(/\x./\y.yap(plusc(x), y), 0, X) 
9.29/9.33	    yap(F, X) => F X 
9.29/9.33	
9.29/9.33	We use rule removal, following [Kop12, Theorem 2.23].
9.29/9.33	
9.29/9.33	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
9.29/9.33	
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, nil) >? X 
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
9.29/9.33	  plusc(X) 0 >? X 
9.29/9.33	  plusc(X) s(Y) >? s(plusc(X) Y) 
9.29/9.33	  sum(X) >? foldl(/\x./\y.yap(plusc(x), y), 0, X) 
9.29/9.33	  yap(F, X) >? F X 
9.29/9.33	
9.29/9.33	We use a recursive path ordering as defined in [Kop12, Chapter 5].
9.29/9.33	
9.29/9.33	Argument functions:
9.29/9.33	
9.29/9.33	  [[0]] = _|_ 
9.29/9.33	  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
9.29/9.33	  [[s(x_1)]] = x_1 
9.29/9.33	
9.29/9.33	We choose Lex = {foldl} and Mul = {@_{o -> o}, cons, nil, plusc, sum, yap}, and the following precedence: nil > sum > plusc > cons > yap > @_{o -> o} > foldl
9.29/9.33	
9.29/9.33	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
9.29/9.33	
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
9.29/9.33	  @_{o -> o}(plusc(X), _|_) >= X 
9.29/9.33	  @_{o -> o}(plusc(X), Y) >= @_{o -> o}(plusc(X), Y) 
9.29/9.33	  sum(X) >= foldl(/\x./\y.yap(plusc(x), y), _|_, X) 
9.29/9.33	  yap(F, X) > @_{o -> o}(F, X) 
9.29/9.33	
9.29/9.33	With these choices, we have:
9.29/9.33	
9.29/9.33	  1] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [2], by (Star) 
9.29/9.33	  2] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [3], by (Select) 
9.29/9.33	  3] X >= X  by (Meta) 
9.29/9.33	
9.29/9.33	  4] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [5], by (Star) 
9.29/9.33	  5] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [6], [9], [16] and [25], by (Stat) 
9.29/9.33	  6] cons(Y, Z) > Z  because [7], by definition 
9.29/9.33	  7] cons*(Y, Z) >= Z  because [8], by (Select) 
9.29/9.33	  8] Z >= Z  by (Meta) 
9.29/9.33	  9] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [10], by (Select) 
9.29/9.33	  10] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [11], by (Abs) 
9.29/9.33	  11] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [12], by (Abs) 
9.29/9.33	  12] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [13] and [15], by (Fun) 
9.29/9.33	  13] F(y) >= F(y)  because [14], by (Meta) 
9.29/9.33	  14] y >= y  by (Var) 
9.29/9.33	  15] x >= x  by (Var) 
9.29/9.33	  16] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y)  because [17], by (Select) 
9.29/9.33	  17] yap(F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= yap(F(X), Y)  because yap in Mul, [18] and [21], by (Fun) 
9.29/9.33	  18] F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [19], by (Meta) 
9.29/9.33	  19] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [20], by (Select) 
9.29/9.33	  20] X >= X  by (Meta) 
9.29/9.33	  21] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y  because [22], by (Select) 
9.29/9.33	  22] cons(Y, Z) >= Y  because [23], by (Star) 
9.29/9.33	  23] cons*(Y, Z) >= Y  because [24], by (Select) 
9.29/9.33	  24] Y >= Y  by (Meta) 
9.29/9.33	  25] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z  because [26], by (Select) 
9.29/9.33	  26] cons(Y, Z) >= Z  because [7], by (Star) 
9.29/9.33	
9.29/9.33	  27] @_{o -> o}(plusc(X), _|_) >= X  because [28], by (Star) 
9.29/9.33	  28] @_{o -> o}*(plusc(X), _|_) >= X  because [29], by (Select) 
9.29/9.33	  29] plusc(X) @_{o -> o}*(plusc(X), _|_) >= X  because [30] 
9.29/9.33	  30] plusc*(X, @_{o -> o}*(plusc(X), _|_)) >= X  because [31], by (Select) 
9.29/9.33	  31] X >= X  by (Meta) 
9.29/9.33	
9.29/9.33	  32] @_{o -> o}(plusc(X), Y) >= @_{o -> o}(plusc(X), Y)  because @_{o -> o} in Mul, [33] and [35], by (Fun) 
9.29/9.33	  33] plusc(X) >= plusc(X)  because plusc in Mul and [34], by (Fun) 
9.29/9.33	  34] X >= X  by (Meta) 
9.29/9.33	  35] Y >= Y  by (Meta) 
9.29/9.33	
9.29/9.33	  36] sum(X) >= foldl(/\x./\y.yap(plusc(x), y), _|_, X)  because [37], by (Star) 
9.29/9.33	  37] sum*(X) >= foldl(/\x./\y.yap(plusc(x), y), _|_, X)  because sum > foldl, [38], [46] and [47], by (Copy) 
9.29/9.33	  38] sum*(X) >= /\y./\z.yap(plusc(y), z)  because [39], by (F-Abs) 
9.29/9.33	  39] sum*(X, x) >= /\z.yap(plusc(x), z)  because [40], by (F-Abs) 
9.29/9.33	  40] sum*(X, x, y) >= yap(plusc(x), y)  because sum > yap, [41] and [44], by (Copy) 
9.29/9.33	  41] sum*(X, x, y) >= plusc(x)  because sum > plusc and [42], by (Copy) 
9.29/9.33	  42] sum*(X, x, y) >= x  because [43], by (Select) 
9.29/9.33	  43] x >= x  by (Var) 
9.29/9.33	  44] sum*(X, x, y) >= y  because [45], by (Select) 
9.29/9.33	  45] y >= y  by (Var) 
9.29/9.33	  46] sum*(X) >= _|_  by (Bot) 
9.29/9.33	  47] sum*(X) >= X  because [48], by (Select) 
9.29/9.33	  48] X >= X  by (Meta) 
9.29/9.33	
9.29/9.33	  49] yap(F, X) > @_{o -> o}(F, X)  because [50], by definition 
9.29/9.33	  50] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [51] and [53], by (Copy) 
9.29/9.33	  51] yap*(F, X) >= F  because [52], by (Select) 
9.29/9.33	  52] F >= F  by (Meta) 
9.29/9.33	  53] yap*(F, X) >= X  because [54], by (Select) 
9.29/9.33	  54] X >= X  by (Meta) 
9.29/9.33	
9.29/9.33	We can thus remove the following rules:
9.29/9.33	
9.29/9.33	  yap(F, X) => F X 
9.29/9.33	
9.29/9.33	We use rule removal, following [Kop12, Theorem 2.23].
9.29/9.33	
9.29/9.33	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
9.29/9.33	
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, nil) >? X 
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
9.29/9.33	  plusc(X) 0 >? X 
9.29/9.33	  plusc(X) s(Y) >? s(plusc(X) Y) 
9.29/9.33	  sum(X) >? foldl(/\x./\y.yap(plusc(x), y), 0, X) 
9.29/9.33	
9.29/9.33	We use a recursive path ordering as defined in [Kop12, Chapter 5].
9.29/9.33	
9.29/9.33	Argument functions:
9.29/9.33	
9.29/9.33	  [[0]] = _|_ 
9.29/9.33	  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
9.29/9.33	
9.29/9.33	We choose Lex = {foldl} and Mul = {@_{o -> o}, cons, nil, plusc, s, sum, yap}, and the following precedence: nil > sum > plusc > @_{o -> o} > s > foldl > cons > yap
9.29/9.33	
9.29/9.33	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
9.29/9.33	
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
9.29/9.33	  @_{o -> o}(plusc(X), _|_) > X 
9.29/9.33	  @_{o -> o}(plusc(X), s(Y)) > s(@_{o -> o}(plusc(X), Y)) 
9.29/9.33	  sum(X) > foldl(/\x./\y.yap(plusc(x), y), _|_, X) 
9.29/9.33	
9.29/9.33	With these choices, we have:
9.29/9.33	
9.29/9.33	  1] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [2], by (Star) 
9.29/9.33	  2] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [3], by (Select) 
9.29/9.33	  3] X >= X  by (Meta) 
9.29/9.33	
9.29/9.33	  4] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [5], by definition 
9.29/9.33	  5] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [6], [9], [19] and [29], by (Stat) 
9.29/9.33	  6] cons(Y, Z) > Z  because [7], by definition 
9.29/9.33	  7] cons*(Y, Z) >= Z  because [8], by (Select) 
9.29/9.33	  8] Z >= Z  by (Meta) 
9.29/9.33	  9] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [10], by (F-Abs) 
9.29/9.33	  10] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= /\x.yap(F(z), x)  because [11], by (F-Abs) 
9.29/9.33	  11] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= yap(F(z), u)  because foldl > yap, [12] and [17], by (Copy) 
9.29/9.33	  12] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= F(z)  because [13], by (Select) 
9.29/9.33	  13] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z), z, u)), x) >= F(z)  because [14], by (Eta)[Kop13:2] 
9.29/9.33	  14] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)) >= F(z)  because [15], by (Meta) 
9.29/9.33	  15] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= z  because [16], by (Select) 
9.29/9.33	  16] z >= z  by (Var) 
9.29/9.33	  17] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= u  because [18], by (Select) 
9.29/9.33	  18] u >= u  by (Var) 
9.29/9.33	  19] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [20] and [25], by (Copy) 
9.29/9.33	  20] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [21], by (Select) 
9.29/9.33	  21] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z))), x) >= F(X)  because [22], by (Eta)[Kop13:2] 
9.29/9.33	  22] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [23], by (Meta) 
9.29/9.33	  23] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [24], by (Select) 
9.29/9.33	  24] X >= X  by (Meta) 
9.29/9.33	  25] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [26], by (Select) 
9.29/9.33	  26] cons(Y, Z) >= Y  because [27], by (Star) 
9.29/9.33	  27] cons*(Y, Z) >= Y  because [28], by (Select) 
9.29/9.33	  28] Y >= Y  by (Meta) 
9.29/9.33	  29] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [30], by (Select) 
9.29/9.33	  30] cons(Y, Z) >= Z  because [7], by (Star) 
9.29/9.33	
9.29/9.33	  31] @_{o -> o}(plusc(X), _|_) > X  because [32], by definition 
9.29/9.33	  32] @_{o -> o}*(plusc(X), _|_) >= X  because [33], by (Select) 
9.29/9.33	  33] plusc(X) @_{o -> o}*(plusc(X), _|_) >= X  because [34] 
9.29/9.33	  34] plusc*(X, @_{o -> o}*(plusc(X), _|_)) >= X  because [35], by (Select) 
9.29/9.33	  35] X >= X  by (Meta) 
9.29/9.33	
9.29/9.33	  36] @_{o -> o}(plusc(X), s(Y)) > s(@_{o -> o}(plusc(X), Y))  because [37], by definition 
9.29/9.33	  37] @_{o -> o}*(plusc(X), s(Y)) >= s(@_{o -> o}(plusc(X), Y))  because [38], by (Select) 
9.29/9.33	  38] plusc(X) @_{o -> o}*(plusc(X), s(Y)) >= s(@_{o -> o}(plusc(X), Y))  because [39] 
9.29/9.33	  39] plusc*(X, @_{o -> o}*(plusc(X), s(Y))) >= s(@_{o -> o}(plusc(X), Y))  because plusc > s and [40], by (Copy) 
9.29/9.33	  40] plusc*(X, @_{o -> o}*(plusc(X), s(Y))) >= @_{o -> o}(plusc(X), Y)  because [41], by (Select) 
9.29/9.33	  41] @_{o -> o}*(plusc(X), s(Y)) >= @_{o -> o}(plusc(X), Y)  because @_{o -> o} in Mul, [42] and [44], by (Stat) 
9.29/9.33	  42] plusc(X) >= plusc(X)  because plusc in Mul and [43], by (Fun) 
9.29/9.33	  43] X >= X  by (Meta) 
9.29/9.33	  44] s(Y) > Y  because [45], by definition 
9.29/9.33	  45] s*(Y) >= Y  because [46], by (Select) 
9.29/9.33	  46] Y >= Y  by (Meta) 
9.29/9.33	
9.29/9.33	  47] sum(X) > foldl(/\x./\y.yap(plusc(x), y), _|_, X)  because [48], by definition 
9.29/9.33	  48] sum*(X) >= foldl(/\x./\y.yap(plusc(x), y), _|_, X)  because sum > foldl, [49], [57] and [58], by (Copy) 
9.29/9.33	  49] sum*(X) >= /\y./\z.yap(plusc(y), z)  because [50], by (F-Abs) 
9.29/9.33	  50] sum*(X, x) >= /\z.yap(plusc(x), z)  because [51], by (F-Abs) 
9.29/9.33	  51] sum*(X, x, y) >= yap(plusc(x), y)  because sum > yap, [52] and [55], by (Copy) 
9.29/9.33	  52] sum*(X, x, y) >= plusc(x)  because sum > plusc and [53], by (Copy) 
9.29/9.33	  53] sum*(X, x, y) >= x  because [54], by (Select) 
9.29/9.33	  54] x >= x  by (Var) 
9.29/9.33	  55] sum*(X, x, y) >= y  because [56], by (Select) 
9.29/9.33	  56] y >= y  by (Var) 
9.29/9.33	  57] sum*(X) >= _|_  by (Bot) 
9.29/9.33	  58] sum*(X) >= X  because [59], by (Select) 
9.29/9.33	  59] X >= X  by (Meta) 
9.29/9.33	
9.29/9.33	We can thus remove the following rules:
9.29/9.33	
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
9.29/9.33	  plusc(X) 0 => X 
9.29/9.33	  plusc(X) s(Y) => s(plusc(X) Y) 
9.29/9.33	  sum(X) => foldl(/\x./\y.yap(plusc(x), y), 0, X) 
9.29/9.33	
9.29/9.33	We use rule removal, following [Kop12, Theorem 2.23].
9.29/9.33	
9.29/9.33	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
9.29/9.33	
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, nil) >? X 
9.29/9.33	
9.29/9.33	We orient these requirements with a polynomial interpretation in the natural numbers.
9.29/9.33	
9.29/9.33	The following interpretation satisfies the requirements:
9.29/9.33	
9.29/9.33	  foldl = \G0y1y2.3 + y1 + y2 + G0(0,0) 
9.29/9.33	  nil = 3 
9.29/9.33	  yap = \G0y1.3 + y1 + G0(0) 
9.29/9.33	
9.29/9.33	Using this interpretation, the requirements translate to:
9.29/9.33	
9.29/9.33	  [[foldl(/\x./\y.yap(_F0(x), y), _x1, nil)]] = 9 + x1 + F0(0,0) > x1 = [[_x1]] 
9.29/9.33	
9.29/9.33	We can thus remove the following rules:
9.29/9.33	
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, nil) => X 
9.29/9.33	
9.29/9.33	All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.
9.29/9.33	
9.29/9.33	
9.29/9.33	+++ Citations +++
9.29/9.33	
9.29/9.33	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
9.29/9.33	[Kop13:2]  C. Kop.  StarHorpo with an Eta-Rule.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/etahorpo.pdf, 2013.
9.29/9.34	EOF