0.52/0.81 YES 0.62/0.87 We consider the system theBenchmark. 0.62/0.87 0.62/0.87 Alphabet: 0.62/0.87 0.62/0.87 0 : [] --> nat 0.62/0.87 cons : [] --> nat -> list -> list 0.62/0.87 foldl : [] --> (nat -> nat -> nat) -> nat -> list -> nat 0.62/0.87 nil : [] --> list 0.62/0.87 plusc : [] --> nat -> nat -> nat 0.62/0.87 s : [] --> nat -> nat 0.62/0.87 sum : [] --> list -> nat 0.62/0.87 0.62/0.87 Rules: 0.62/0.87 0.62/0.87 foldl (/\x./\y.f x y) z nil => z 0.62/0.87 foldl (/\x./\y.f x y) z (cons u v) => foldl (/\w./\x'.f w x') (f z u) v 0.62/0.87 plusc x 0 => x 0.62/0.87 plusc x (s y) => s (plusc x y) 0.62/0.87 sum x => foldl (/\y./\z.plusc y z) 0 x 0.62/0.87 0.62/0.87 Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. 0.62/0.87 0.62/0.87 We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: 0.62/0.87 0.62/0.87 Alphabet: 0.62/0.87 0.62/0.87 0 : [] --> nat 0.62/0.87 cons : [nat * list] --> list 0.62/0.87 foldl : [nat -> nat -> nat * nat * list] --> nat 0.62/0.87 nil : [] --> list 0.62/0.87 plusc : [nat * nat] --> nat 0.62/0.87 s : [nat] --> nat 0.62/0.87 sum : [list] --> nat 0.62/0.87 ~AP1 : [nat -> nat -> nat * nat] --> nat -> nat 0.62/0.87 0.62/0.87 Rules: 0.62/0.87 0.62/0.87 foldl(/\x./\y.~AP1(F, x) y, X, nil) => X 0.62/0.87 foldl(/\x./\y.~AP1(F, x) y, X, cons(Y, Z)) => foldl(/\z./\u.~AP1(F, z) u, ~AP1(F, X) Y, Z) 0.62/0.87 plusc(X, 0) => X 0.62/0.87 plusc(X, s(Y)) => s(plusc(X, Y)) 0.62/0.87 sum(X) => foldl(/\x./\y.plusc(x, y), 0, X) 0.62/0.87 foldl(/\x./\y.plusc(x, y), X, nil) => X 0.62/0.87 foldl(/\x./\y.plusc(x, y), X, cons(Y, Z)) => foldl(/\z./\u.plusc(z, u), plusc(X, Y), Z) 0.62/0.87 ~AP1(F, X) => F X 0.62/0.87 0.62/0.87 Symbol ~AP1 is an encoding for application that is only used in innocuous ways. We can simplify the program (without losing non-termination) by removing it. Additionally, we can remove some (now-)redundant rules. This gives: 0.62/0.87 0.62/0.87 Alphabet: 0.62/0.87 0.62/0.87 0 : [] --> nat 0.62/0.87 cons : [nat * list] --> list 0.62/0.87 foldl : [nat -> nat -> nat * nat * list] --> nat 0.62/0.87 nil : [] --> list 0.62/0.87 plusc : [nat * nat] --> nat 0.62/0.87 s : [nat] --> nat 0.62/0.87 sum : [list] --> nat 0.62/0.87 0.62/0.87 Rules: 0.62/0.87 0.62/0.87 foldl(/\x./\y.X(x, y), Y, nil) => Y 0.62/0.87 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) 0.62/0.87 plusc(X, 0) => X 0.62/0.87 plusc(X, s(Y)) => s(plusc(X, Y)) 0.62/0.87 sum(X) => foldl(/\x./\y.plusc(x, y), 0, X) 0.62/0.87 0.62/0.87 We use rule removal, following [Kop12, Theorem 2.23]. 0.62/0.87 0.62/0.87 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.62/0.87 0.62/0.87 foldl(/\x./\y.X(x, y), Y, nil) >? Y 0.62/0.87 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >? foldl(/\z./\u.X(z, u), X(Y, Z), U) 0.62/0.87 plusc(X, 0) >? X 0.62/0.87 plusc(X, s(Y)) >? s(plusc(X, Y)) 0.62/0.87 sum(X) >? foldl(/\x./\y.plusc(x, y), 0, X) 0.62/0.87 0.62/0.87 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 0.62/0.87 0.62/0.87 Argument functions: 0.62/0.87 0.62/0.87 [[0]] = _|_ 0.62/0.87 [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 0.62/0.87 0.62/0.87 We choose Lex = {foldl} and Mul = {cons, nil, plusc, s, sum}, and the following precedence: cons > nil > sum > foldl > plusc > s 0.62/0.87 0.62/0.87 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 0.62/0.87 0.62/0.87 foldl(/\x./\y.X(x, y), Y, nil) >= Y 0.62/0.87 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) 0.62/0.87 plusc(X, _|_) > X 0.62/0.87 plusc(X, s(Y)) >= s(plusc(X, Y)) 0.62/0.87 sum(X) >= foldl(/\x./\y.plusc(x, y), _|_, X) 0.62/0.87 0.62/0.87 With these choices, we have: 0.62/0.87 0.62/0.87 1] foldl(/\x./\y.X(x, y), Y, nil) >= Y because [2], by (Star) 0.62/0.87 2] foldl*(/\x./\y.X(x, y), Y, nil) >= Y because [3], by (Select) 0.62/0.87 3] Y >= Y by (Meta) 0.62/0.87 0.62/0.87 4] foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) because [5], by (Star) 0.62/0.87 5] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) because [6], [9], [15] and [23], by (Stat) 0.62/0.87 6] cons(Z, U) > U because [7], by definition 0.62/0.87 7] cons*(Z, U) >= U because [8], by (Select) 0.62/0.87 8] U >= U by (Meta) 0.62/0.87 9] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= /\x./\y.X(x, y) because [10], by (Select) 0.62/0.87 10] /\x./\z.X(x, z) >= /\x./\z.X(x, z) because [11], by (Abs) 0.62/0.87 11] /\z.X(y, z) >= /\z.X(y, z) because [12], by (Abs) 0.62/0.87 12] X(y, x) >= X(y, x) because [13] and [14], by (Meta) 0.62/0.87 13] y >= y by (Var) 0.62/0.87 14] x >= x by (Var) 0.62/0.87 15] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= X(Y, Z) because [16], by (Select) 0.62/0.87 16] X(foldl*(/\z./\u.X(z, u), Y, cons(Z, U)), foldl*(/\v./\w.X(v, w), Y, cons(Z, U))) >= X(Y, Z) because [17] and [19], by (Meta) 0.62/0.87 17] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Y because [18], by (Select) 0.62/0.87 18] Y >= Y by (Meta) 0.62/0.87 19] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Z because [20], by (Select) 0.62/0.87 20] cons(Z, U) >= Z because [21], by (Star) 0.62/0.87 21] cons*(Z, U) >= Z because [22], by (Select) 0.62/0.87 22] Z >= Z by (Meta) 0.62/0.87 23] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= U because [24], by (Select) 0.62/0.87 24] cons(Z, U) >= U because [7], by (Star) 0.62/0.87 0.62/0.87 25] plusc(X, _|_) > X because [26], by definition 0.62/0.87 26] plusc*(X, _|_) >= X because [27], by (Select) 0.62/0.87 27] X >= X by (Meta) 0.62/0.87 0.62/0.87 28] plusc(X, s(Y)) >= s(plusc(X, Y)) because [29], by (Star) 0.62/0.87 29] plusc*(X, s(Y)) >= s(plusc(X, Y)) because plusc > s and [30], by (Copy) 0.62/0.87 30] plusc*(X, s(Y)) >= plusc(X, Y) because plusc in Mul, [31] and [32], by (Stat) 0.62/0.87 31] X >= X by (Meta) 0.62/0.87 32] s(Y) > Y because [33], by definition 0.62/0.87 33] s*(Y) >= Y because [34], by (Select) 0.62/0.87 34] Y >= Y by (Meta) 0.62/0.87 0.62/0.87 35] sum(X) >= foldl(/\x./\y.plusc(x, y), _|_, X) because [36], by (Star) 0.62/0.87 36] sum*(X) >= foldl(/\x./\y.plusc(x, y), _|_, X) because sum > foldl, [37], [44] and [45], by (Copy) 0.62/0.87 37] sum*(X) >= /\y./\z.plusc(y, z) because [38], by (F-Abs) 0.62/0.87 38] sum*(X, x) >= /\z.plusc(x, z) because [39], by (F-Abs) 0.62/0.87 39] sum*(X, x, y) >= plusc(x, y) because sum > plusc, [40] and [42], by (Copy) 0.62/0.87 40] sum*(X, x, y) >= x because [41], by (Select) 0.62/0.87 41] x >= x by (Var) 0.62/0.87 42] sum*(X, x, y) >= y because [43], by (Select) 0.62/0.87 43] y >= y by (Var) 0.62/0.87 44] sum*(X) >= _|_ by (Bot) 0.62/0.87 45] sum*(X) >= X because [46], by (Select) 0.62/0.87 46] X >= X by (Meta) 0.62/0.87 0.62/0.87 We can thus remove the following rules: 0.62/0.87 0.62/0.87 plusc(X, 0) => X 0.62/0.87 0.62/0.87 We use rule removal, following [Kop12, Theorem 2.23]. 0.62/0.87 0.62/0.87 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.62/0.87 0.62/0.87 foldl(/\x./\y.X(x, y), Y, nil) >? Y 0.62/0.87 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >? foldl(/\z./\u.X(z, u), X(Y, Z), U) 0.62/0.87 plusc(X, s(Y)) >? s(plusc(X, Y)) 0.62/0.87 sum(X) >? foldl(/\x./\y.plusc(x, y), 0, X) 0.62/0.87 0.62/0.87 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 0.62/0.87 0.62/0.87 Argument functions: 0.62/0.87 0.62/0.87 [[0]] = _|_ 0.62/0.87 [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_2, x_1) 0.62/0.87 0.62/0.87 We choose Lex = {foldl} and Mul = {cons, nil, plusc, s, sum}, and the following precedence: cons > nil > sum > foldl > plusc > s 0.62/0.87 0.62/0.87 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 0.62/0.87 0.62/0.87 foldl(/\x./\y.X(x, y), Y, nil) > Y 0.62/0.87 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) 0.62/0.87 plusc(X, s(Y)) >= s(plusc(X, Y)) 0.62/0.87 sum(X) >= foldl(/\x./\y.plusc(x, y), _|_, X) 0.62/0.87 0.62/0.87 With these choices, we have: 0.62/0.87 0.62/0.87 1] foldl(/\x./\y.X(x, y), Y, nil) > Y because [2], by definition 0.62/0.87 2] foldl*(/\x./\y.X(x, y), Y, nil) >= Y because [3], by (Select) 0.62/0.87 3] Y >= Y by (Meta) 0.62/0.87 0.62/0.87 4] foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) because [5], by (Star) 0.62/0.87 5] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) because [6], [9], [16] and [24], by (Stat) 0.62/0.87 6] cons(Z, U) > U because [7], by definition 0.62/0.87 7] cons*(Z, U) >= U because [8], by (Select) 0.62/0.87 8] U >= U by (Meta) 0.62/0.87 9] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= /\x./\y.X(x, y) because [10], by (F-Abs) 0.62/0.87 10] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z) >= /\x.X(z, x) because [11], by (Select) 0.62/0.87 11] /\x.X(foldl*(/\y./\v.X(y, v), Y, cons(Z, U), z), x) >= /\x.X(z, x) because [12], by (Abs) 0.62/0.87 12] X(foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z), u) >= X(z, u) because [13] and [15], by (Meta) 0.62/0.87 13] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z) >= z because [14], by (Select) 0.62/0.87 14] z >= z by (Var) 0.62/0.87 15] u >= u by (Var) 0.62/0.87 16] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(Y, Z) because [17], by (Select) 0.62/0.87 17] X(foldl*(/\x./\y.X(x, y), Y, cons(Z, U)), foldl*(/\v./\w.X(v, w), Y, cons(Z, U))) >= X(Y, Z) because [18] and [20], by (Meta) 0.62/0.87 18] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= Y because [19], by (Select) 0.62/0.87 19] Y >= Y by (Meta) 0.62/0.87 20] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= Z because [21], by (Select) 0.62/0.87 21] cons(Z, U) >= Z because [22], by (Star) 0.62/0.87 22] cons*(Z, U) >= Z because [23], by (Select) 0.62/0.87 23] Z >= Z by (Meta) 0.62/0.87 24] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= U because [25], by (Select) 0.62/0.87 25] cons(Z, U) >= U because [7], by (Star) 0.62/0.87 0.62/0.87 26] plusc(X, s(Y)) >= s(plusc(X, Y)) because [27], by (Star) 0.62/0.87 27] plusc*(X, s(Y)) >= s(plusc(X, Y)) because plusc > s and [28], by (Copy) 0.62/0.87 28] plusc*(X, s(Y)) >= plusc(X, Y) because plusc in Mul, [29] and [30], by (Stat) 0.62/0.87 29] X >= X by (Meta) 0.62/0.87 30] s(Y) > Y because [31], by definition 0.62/0.87 31] s*(Y) >= Y because [32], by (Select) 0.62/0.87 32] Y >= Y by (Meta) 0.62/0.87 0.62/0.87 33] sum(X) >= foldl(/\x./\y.plusc(x, y), _|_, X) because [34], by (Star) 0.62/0.87 34] sum*(X) >= foldl(/\x./\y.plusc(x, y), _|_, X) because sum > foldl, [35], [42] and [43], by (Copy) 0.62/0.87 35] sum*(X) >= /\y./\z.plusc(y, z) because [36], by (F-Abs) 0.62/0.87 36] sum*(X, x) >= /\z.plusc(x, z) because [37], by (F-Abs) 0.62/0.87 37] sum*(X, x, y) >= plusc(x, y) because sum > plusc, [38] and [40], by (Copy) 0.62/0.87 38] sum*(X, x, y) >= x because [39], by (Select) 0.62/0.87 39] x >= x by (Var) 0.62/0.87 40] sum*(X, x, y) >= y because [41], by (Select) 0.62/0.87 41] y >= y by (Var) 0.62/0.87 42] sum*(X) >= _|_ by (Bot) 0.62/0.87 43] sum*(X) >= X because [44], by (Select) 0.62/0.87 44] X >= X by (Meta) 0.62/0.87 0.62/0.87 We can thus remove the following rules: 0.62/0.87 0.62/0.87 foldl(/\x./\y.X(x, y), Y, nil) => Y 0.62/0.87 0.62/0.87 We use rule removal, following [Kop12, Theorem 2.23]. 0.62/0.87 0.62/0.87 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.62/0.87 0.62/0.87 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >? foldl(/\z./\u.X(z, u), X(Y, Z), U) 0.62/0.87 plusc(X, s(Y)) >? s(plusc(X, Y)) 0.62/0.87 sum(X) >? foldl(/\x./\y.plusc(x, y), 0, X) 0.62/0.87 0.62/0.87 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 0.62/0.87 0.62/0.87 Argument functions: 0.62/0.87 0.62/0.87 [[0]] = _|_ 0.62/0.87 [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_2, x_1) 0.62/0.87 [[s(x_1)]] = x_1 0.62/0.87 0.62/0.87 We choose Lex = {foldl} and Mul = {cons, plusc, sum}, and the following precedence: cons > sum > foldl > plusc 0.62/0.87 0.62/0.87 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 0.62/0.87 0.62/0.87 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) > foldl(/\x./\y.X(x, y), X(Y, Z), U) 0.62/0.87 plusc(X, Y) >= plusc(X, Y) 0.62/0.87 sum(X) >= foldl(/\x./\y.plusc(x, y), _|_, X) 0.62/0.87 0.62/0.87 With these choices, we have: 0.62/0.87 0.62/0.87 1] foldl(/\x./\y.X(x, y), Y, cons(Z, U)) > foldl(/\x./\y.X(x, y), X(Y, Z), U) because [2], by definition 0.62/0.87 2] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) because [3], [6], [12] and [20], by (Stat) 0.62/0.87 3] cons(Z, U) > U because [4], by definition 0.62/0.87 4] cons*(Z, U) >= U because [5], by (Select) 0.62/0.87 5] U >= U by (Meta) 0.62/0.87 6] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= /\x./\y.X(x, y) because [7], by (Select) 0.62/0.87 7] /\x./\z.X(x, z) >= /\x./\z.X(x, z) because [8], by (Abs) 0.62/0.87 8] /\z.X(y, z) >= /\z.X(y, z) because [9], by (Abs) 0.62/0.87 9] X(y, x) >= X(y, x) because [10] and [11], by (Meta) 0.62/0.87 10] y >= y by (Var) 0.62/0.87 11] x >= x by (Var) 0.62/0.87 12] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= X(Y, Z) because [13], by (Select) 0.62/0.87 13] X(foldl*(/\z./\u.X(z, u), Y, cons(Z, U)), foldl*(/\v./\w.X(v, w), Y, cons(Z, U))) >= X(Y, Z) because [14] and [16], by (Meta) 0.62/0.87 14] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Y because [15], by (Select) 0.62/0.87 15] Y >= Y by (Meta) 0.62/0.87 16] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Z because [17], by (Select) 0.62/0.87 17] cons(Z, U) >= Z because [18], by (Star) 0.62/0.87 18] cons*(Z, U) >= Z because [19], by (Select) 0.62/0.87 19] Z >= Z by (Meta) 0.62/0.87 20] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= U because [21], by (Select) 0.62/0.87 21] cons(Z, U) >= U because [4], by (Star) 0.62/0.87 0.62/0.87 22] plusc(X, Y) >= plusc(X, Y) because plusc in Mul, [23] and [24], by (Fun) 0.62/0.87 23] X >= X by (Meta) 0.62/0.87 24] Y >= Y by (Meta) 0.62/0.87 0.62/0.87 25] sum(X) >= foldl(/\x./\y.plusc(x, y), _|_, X) because [26], by (Star) 0.62/0.87 26] sum*(X) >= foldl(/\x./\y.plusc(x, y), _|_, X) because sum > foldl, [27], [34] and [35], by (Copy) 0.62/0.87 27] sum*(X) >= /\y./\z.plusc(y, z) because [28], by (F-Abs) 0.62/0.87 28] sum*(X, x) >= /\z.plusc(x, z) because [29], by (F-Abs) 0.62/0.87 29] sum*(X, x, y) >= plusc(x, y) because sum > plusc, [30] and [32], by (Copy) 0.62/0.87 30] sum*(X, x, y) >= x because [31], by (Select) 0.62/0.87 31] x >= x by (Var) 0.62/0.87 32] sum*(X, x, y) >= y because [33], by (Select) 0.62/0.87 33] y >= y by (Var) 0.62/0.87 34] sum*(X) >= _|_ by (Bot) 0.62/0.87 35] sum*(X) >= X because [36], by (Select) 0.62/0.87 36] X >= X by (Meta) 0.62/0.87 0.62/0.87 We can thus remove the following rules: 0.62/0.87 0.62/0.87 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) 0.62/0.87 0.62/0.87 We use rule removal, following [Kop12, Theorem 2.23]. 0.62/0.87 0.62/0.87 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.62/0.87 0.62/0.87 plusc(X, s(Y)) >? s(plusc(X, Y)) 0.62/0.87 sum(X) >? foldl(/\x./\y.plusc(x, y), 0, X) 0.62/0.87 0.62/0.87 We orient these requirements with a polynomial interpretation in the natural numbers. 0.62/0.87 0.62/0.87 The following interpretation satisfies the requirements: 0.62/0.87 0.62/0.87 0 = 0 0.62/0.87 foldl = \G0y1y2.y1 + y2 + G0(0,0) 0.62/0.87 plusc = \y0y1.y0 + 2y1 0.62/0.87 s = \y0.1 + y0 0.62/0.87 sum = \y0.3 + 3y0 0.62/0.87 0.62/0.87 Using this interpretation, the requirements translate to: 0.62/0.87 0.62/0.87 [[plusc(_x0, s(_x1))]] = 2 + x0 + 2x1 > 1 + x0 + 2x1 = [[s(plusc(_x0, _x1))]] 0.62/0.87 [[sum(_x0)]] = 3 + 3x0 > x0 = [[foldl(/\x./\y.plusc(x, y), 0, _x0)]] 0.62/0.87 0.62/0.87 We can thus remove the following rules: 0.62/0.87 0.62/0.87 plusc(X, s(Y)) => s(plusc(X, Y)) 0.62/0.87 sum(X) => foldl(/\x./\y.plusc(x, y), 0, X) 0.62/0.87 0.62/0.87 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. 0.62/0.87 0.62/0.87 0.62/0.87 +++ Citations +++ 0.62/0.87 0.62/0.87 [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. 0.62/0.87 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 0.62/0.87 EOF