12.23/12.32 YES 12.33/12.37 We consider the system theBenchmark. 12.33/12.37 12.33/12.37 Alphabet: 12.33/12.37 12.33/12.37 0 : [] --> nat 12.33/12.37 cons : [nat * natlist] --> natlist 12.33/12.37 foldl : [nat -> nat -> nat * nat * natlist] --> nat 12.33/12.37 nil : [] --> natlist 12.33/12.37 plus : [] --> nat -> nat -> nat 12.33/12.37 sum : [natlist] --> nat 12.33/12.37 xap : [nat -> nat -> nat * nat] --> nat -> nat 12.33/12.37 yap : [nat -> nat * nat] --> nat 12.33/12.37 12.33/12.37 Rules: 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(xap(f, x), y), z, nil) => z 12.33/12.37 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) 12.33/12.37 sum(x) => foldl(/\y./\z.yap(xap(plus, y), z), 0, x) 12.33/12.37 xap(f, x) => f x 12.33/12.37 yap(f, x) => f x 12.33/12.37 12.33/12.37 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 12.33/12.37 12.33/12.37 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: 12.33/12.37 12.33/12.37 Alphabet: 12.33/12.37 12.33/12.37 0 : [] --> nat 12.33/12.37 cons : [nat * natlist] --> natlist 12.33/12.37 foldl : [nat -> nat -> nat * nat * natlist] --> nat 12.33/12.37 nil : [] --> natlist 12.33/12.37 plus : [nat] --> nat -> nat 12.33/12.37 sum : [natlist] --> nat 12.33/12.37 yap : [nat -> nat * nat] --> nat 12.33/12.37 12.33/12.37 Rules: 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, nil) => X 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 12.33/12.37 sum(X) => foldl(/\x./\y.yap(plus(x), y), 0, X) 12.33/12.37 yap(F, X) => F X 12.33/12.37 12.33/12.37 We use rule removal, following [Kop12, Theorem 2.23]. 12.33/12.37 12.33/12.37 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, nil) >? X 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 12.33/12.37 sum(X) >? foldl(/\x./\y.yap(plus(x), y), 0, X) 12.33/12.37 yap(F, X) >? F X 12.33/12.37 12.33/12.37 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 12.33/12.37 12.33/12.37 Argument functions: 12.33/12.37 12.33/12.37 [[0]] = _|_ 12.33/12.37 [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 12.33/12.37 12.33/12.37 We choose Lex = {foldl} and Mul = {@_{o -> o}, cons, nil, plus, sum, yap}, and the following precedence: cons > nil > sum > plus > yap > @_{o -> o} > foldl 12.33/12.37 12.33/12.37 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, nil) >= X 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 12.33/12.37 sum(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) 12.33/12.37 yap(F, X) > @_{o -> o}(F, X) 12.33/12.37 12.33/12.37 With these choices, we have: 12.33/12.37 12.33/12.37 1] foldl(/\x./\y.yap(F(x), y), X, nil) >= X because [2], by (Star) 12.33/12.37 2] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X because [3], by (Select) 12.33/12.37 3] X >= X by (Meta) 12.33/12.37 12.33/12.37 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) 12.33/12.37 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], [17] and [26], by (Stat) 12.33/12.37 6] cons(Y, Z) > Z because [7], by definition 12.33/12.37 7] cons*(Y, Z) >= Z because [8], by (Select) 12.33/12.37 8] Z >= Z by (Meta) 12.33/12.37 9] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y) because [10], by (F-Abs) 12.33/12.37 10] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= /\x.yap(F(z), x) because [11], by (Select) 12.33/12.37 11] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z), z)), x) >= /\x.yap(F(z), x) because [12], by (Abs) 12.33/12.37 12] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z)), u) >= yap(F(z), u) because yap in Mul, [13] and [16], by (Fun) 12.33/12.37 13] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z)) >= F(z) because [14], by (Meta) 12.33/12.37 14] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= z because [15], by (Select) 12.33/12.37 15] z >= z by (Var) 12.33/12.37 16] u >= u by (Var) 12.33/12.37 17] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y) because [18], by (Select) 12.33/12.37 18] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= yap(F(X), Y) because yap in Mul, [19] and [22], by (Fun) 12.33/12.37 19] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X) because [20], by (Meta) 12.33/12.37 20] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X because [21], by (Select) 12.33/12.37 21] X >= X by (Meta) 12.33/12.37 22] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y because [23], by (Select) 12.33/12.37 23] cons(Y, Z) >= Y because [24], by (Star) 12.33/12.37 24] cons*(Y, Z) >= Y because [25], by (Select) 12.33/12.37 25] Y >= Y by (Meta) 12.33/12.37 26] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z because [27], by (Select) 12.33/12.37 27] cons(Y, Z) >= Z because [7], by (Star) 12.33/12.37 12.33/12.37 28] sum(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) because [29], by (Star) 12.33/12.37 29] sum*(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) because sum > foldl, [30], [38] and [39], by (Copy) 12.33/12.37 30] sum*(X) >= /\y./\z.yap(plus(y), z) because [31], by (F-Abs) 12.33/12.37 31] sum*(X, x) >= /\z.yap(plus(x), z) because [32], by (F-Abs) 12.33/12.37 32] sum*(X, x, y) >= yap(plus(x), y) because sum > yap, [33] and [36], by (Copy) 12.33/12.37 33] sum*(X, x, y) >= plus(x) because sum > plus and [34], by (Copy) 12.33/12.37 34] sum*(X, x, y) >= x because [35], by (Select) 12.33/12.37 35] x >= x by (Var) 12.33/12.37 36] sum*(X, x, y) >= y because [37], by (Select) 12.33/12.37 37] y >= y by (Var) 12.33/12.37 38] sum*(X) >= _|_ by (Bot) 12.33/12.37 39] sum*(X) >= X because [40], by (Select) 12.33/12.37 40] X >= X by (Meta) 12.33/12.37 12.33/12.37 41] yap(F, X) > @_{o -> o}(F, X) because [42], by definition 12.33/12.37 42] yap*(F, X) >= @_{o -> o}(F, X) because yap > @_{o -> o}, [43] and [45], by (Copy) 12.33/12.37 43] yap*(F, X) >= F because [44], by (Select) 12.33/12.37 44] F >= F by (Meta) 12.33/12.37 45] yap*(F, X) >= X because [46], by (Select) 12.33/12.37 46] X >= X by (Meta) 12.33/12.37 12.33/12.37 We can thus remove the following rules: 12.33/12.37 12.33/12.37 yap(F, X) => F X 12.33/12.37 12.33/12.37 We use rule removal, following [Kop12, Theorem 2.23]. 12.33/12.37 12.33/12.37 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, nil) >? X 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 12.33/12.37 sum(X) >? foldl(/\x./\y.yap(plus(x), y), 0, X) 12.33/12.37 12.33/12.37 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 12.33/12.37 12.33/12.37 Argument functions: 12.33/12.37 12.33/12.37 [[0]] = _|_ 12.33/12.37 [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 12.33/12.37 12.33/12.37 We choose Lex = {foldl} and Mul = {cons, nil, plus, sum, yap}, and the following precedence: cons > nil > sum > plus > foldl > yap 12.33/12.37 12.33/12.37 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, nil) > X 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 12.33/12.37 sum(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) 12.33/12.37 12.33/12.37 With these choices, we have: 12.33/12.37 12.33/12.37 1] foldl(/\x./\y.yap(F(x), y), X, nil) > X because [2], by definition 12.33/12.37 2] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X because [3], by (Select) 12.33/12.37 3] X >= X by (Meta) 12.33/12.37 12.33/12.37 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) 12.33/12.37 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 [26], by (Stat) 12.33/12.37 6] cons(Y, Z) > Z because [7], by definition 12.33/12.37 7] cons*(Y, Z) >= Z because [8], by (Select) 12.33/12.37 8] Z >= Z by (Meta) 12.33/12.37 9] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y) because [10], by (Select) 12.33/12.37 10] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z) because [11], by (Abs) 12.33/12.37 11] /\z.yap(F(y), z) >= /\z.yap(F(y), z) because [12], by (Abs) 12.33/12.37 12] yap(F(y), x) >= yap(F(y), x) because yap in Mul, [13] and [15], by (Fun) 12.33/12.37 13] F(y) >= F(y) because [14], by (Meta) 12.33/12.37 14] y >= y by (Var) 12.33/12.37 15] x >= x by (Var) 12.33/12.37 16] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y) because foldl > yap, [17] and [22], by (Copy) 12.33/12.37 17] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= F(X) because [18], by (Select) 12.33/12.37 18] /\z.yap(F(foldl*(/\u./\v.yap(F(u), v), X, cons(Y, Z))), z) >= F(X) because [19], by (Eta)[Kop13:2] 12.33/12.37 19] F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X) because [20], by (Meta) 12.33/12.37 20] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X because [21], by (Select) 12.33/12.37 21] X >= X by (Meta) 12.33/12.37 22] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y because [23], by (Select) 12.33/12.37 23] cons(Y, Z) >= Y because [24], by (Star) 12.33/12.37 24] cons*(Y, Z) >= Y because [25], by (Select) 12.33/12.37 25] Y >= Y by (Meta) 12.33/12.37 26] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z because [27], by (Select) 12.33/12.37 27] cons(Y, Z) >= Z because [7], by (Star) 12.33/12.37 12.33/12.37 28] sum(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) because [29], by (Star) 12.33/12.37 29] sum*(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) because sum > foldl, [30], [38] and [39], by (Copy) 12.33/12.37 30] sum*(X) >= /\y./\z.yap(plus(y), z) because [31], by (F-Abs) 12.33/12.37 31] sum*(X, x) >= /\z.yap(plus(x), z) because [32], by (F-Abs) 12.33/12.37 32] sum*(X, x, y) >= yap(plus(x), y) because sum > yap, [33] and [36], by (Copy) 12.33/12.37 33] sum*(X, x, y) >= plus(x) because sum > plus and [34], by (Copy) 12.33/12.37 34] sum*(X, x, y) >= x because [35], by (Select) 12.33/12.37 35] x >= x by (Var) 12.33/12.37 36] sum*(X, x, y) >= y because [37], by (Select) 12.33/12.37 37] y >= y by (Var) 12.33/12.37 38] sum*(X) >= _|_ by (Bot) 12.33/12.37 39] sum*(X) >= X because [40], by (Select) 12.33/12.37 40] X >= X by (Meta) 12.33/12.37 12.33/12.37 We can thus remove the following rules: 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, nil) => X 12.33/12.37 12.33/12.37 We use rule removal, following [Kop12, Theorem 2.23]. 12.33/12.37 12.33/12.37 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 12.33/12.37 sum(X) >? foldl(/\x./\y.yap(plus(x), y), 0, X) 12.33/12.37 12.33/12.37 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 12.33/12.37 12.33/12.37 Argument functions: 12.33/12.37 12.33/12.37 [[0]] = _|_ 12.33/12.37 [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 12.33/12.37 12.33/12.37 We choose Lex = {foldl} and Mul = {cons, plus, sum, yap}, and the following precedence: sum > plus > foldl > yap > cons 12.33/12.37 12.33/12.37 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 12.33/12.37 sum(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) 12.33/12.37 12.33/12.37 With these choices, we have: 12.33/12.37 12.33/12.37 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 12.33/12.37 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) 12.33/12.37 3] cons(Y, Z) > Z because [4], by definition 12.33/12.37 4] cons*(Y, Z) >= Z because [5], by (Select) 12.33/12.37 5] Z >= Z by (Meta) 12.33/12.37 6] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y) because [7], by (Select) 12.33/12.37 7] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z) because [8], by (Abs) 12.33/12.37 8] /\z.yap(F(y), z) >= /\z.yap(F(y), z) because [9], by (Abs) 12.33/12.37 9] yap(F(y), x) >= yap(F(y), x) because yap in Mul, [10] and [12], by (Fun) 12.33/12.37 10] F(y) >= F(y) because [11], by (Meta) 12.33/12.37 11] y >= y by (Var) 12.33/12.37 12] x >= x by (Var) 12.33/12.37 13] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y) because foldl > yap, [14] and [19], by (Copy) 12.33/12.37 14] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= F(X) because [15], by (Select) 12.33/12.37 15] /\z.yap(F(foldl*(/\u./\v.yap(F(u), v), X, cons(Y, Z))), z) >= F(X) because [16], by (Eta)[Kop13:2] 12.33/12.37 16] F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X) because [17], by (Meta) 12.33/12.37 17] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X because [18], by (Select) 12.33/12.37 18] X >= X by (Meta) 12.33/12.37 19] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y because [20], by (Select) 12.33/12.37 20] cons(Y, Z) >= Y because [21], by (Star) 12.33/12.37 21] cons*(Y, Z) >= Y because [22], by (Select) 12.33/12.37 22] Y >= Y by (Meta) 12.33/12.37 23] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z because [24], by (Select) 12.33/12.37 24] cons(Y, Z) >= Z because [4], by (Star) 12.33/12.37 12.33/12.37 25] sum(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) because [26], by (Star) 12.33/12.37 26] sum*(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) because sum > foldl, [27], [35] and [36], by (Copy) 12.33/12.37 27] sum*(X) >= /\y./\z.yap(plus(y), z) because [28], by (F-Abs) 12.33/12.37 28] sum*(X, x) >= /\z.yap(plus(x), z) because [29], by (F-Abs) 12.33/12.37 29] sum*(X, x, y) >= yap(plus(x), y) because sum > yap, [30] and [33], by (Copy) 12.33/12.37 30] sum*(X, x, y) >= plus(x) because sum > plus and [31], by (Copy) 12.33/12.37 31] sum*(X, x, y) >= x because [32], by (Select) 12.33/12.37 32] x >= x by (Var) 12.33/12.37 33] sum*(X, x, y) >= y because [34], by (Select) 12.33/12.37 34] y >= y by (Var) 12.33/12.37 35] sum*(X) >= _|_ by (Bot) 12.33/12.37 36] sum*(X) >= X because [37], by (Select) 12.33/12.37 37] X >= X by (Meta) 12.33/12.37 12.33/12.37 We can thus remove the following rules: 12.33/12.37 12.33/12.37 foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 12.33/12.37 12.33/12.37 We use rule removal, following [Kop12, Theorem 2.23]. 12.33/12.37 12.33/12.37 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 12.33/12.37 12.33/12.37 sum(X) >? foldl(/\x./\y.yap(plus(x), y), 0, X) 12.33/12.37 12.33/12.37 We orient these requirements with a polynomial interpretation in the natural numbers. 12.33/12.37 12.33/12.37 The following interpretation satisfies the requirements: 12.33/12.37 12.33/12.37 0 = 0 12.33/12.37 foldl = \G0y1y2.y1 + y2 + G0(0,0) 12.33/12.37 plus = \y0y1.y0 12.33/12.37 sum = \y0.3 + 3y0 12.33/12.37 yap = \G0y1.y1 + G0(0) 12.33/12.37 12.33/12.37 Using this interpretation, the requirements translate to: 12.33/12.37 12.33/12.37 [[sum(_x0)]] = 3 + 3x0 > x0 = [[foldl(/\x./\y.yap(plus(x), y), 0, _x0)]] 12.33/12.37 12.33/12.37 We can thus remove the following rules: 12.33/12.37 12.33/12.37 sum(X) => foldl(/\x./\y.yap(plus(x), y), 0, X) 12.33/12.37 12.33/12.37 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. 12.33/12.37 12.33/12.37 12.33/12.37 +++ Citations +++ 12.33/12.37 12.33/12.37 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 12.33/12.37 [Kop13:2] C. Kop. StarHorpo with an Eta-Rule. Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/etahorpo.pdf, 2013. 12.33/12.38 EOF