We consider the system h02. Alphabet: 0 : [] --> c 1 : [] --> c add : [] --> c -> a -> c cons : [] --> a -> b -> b fold : [] --> (c -> a -> c) -> b -> c -> c mul : [] --> c -> a -> c nil : [] --> b prod : [] --> b -> c sum : [] --> b -> c Rules: fold (/\x./\y.f x y) nil z => z fold (/\x./\y.f x y) (cons z u) v => fold (/\w./\x'.f w x') u (f v z) sum x => fold (/\y./\z.add y z) x 0 fold (/\x./\y.mul x y) z 1 => prod z 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. 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: Alphabet: 0 : [] --> c 1 : [] --> c add : [] --> c -> a -> c cons : [a * b] --> b fold : [c -> a -> c * b * c] --> c mul : [c * a] --> c nil : [] --> b prod : [b] --> c sum : [b] --> c ~AP1 : [c -> a -> c * c] --> a -> c Rules: fold(/\x./\y.~AP1(F, x) y, nil, X) => X fold(/\x./\y.~AP1(F, x) y, cons(X, Y), Z) => fold(/\z./\u.~AP1(F, z) u, Y, ~AP1(F, Z) X) sum(X) => fold(/\x./\y.~AP1(add, x) y, X, 0) fold(/\x./\y.mul(x, y), X, 1) => prod(X) fold(/\x./\y.mul(x, y), nil, X) => X fold(/\x./\y.mul(x, y), cons(X, Y), Z) => fold(/\z./\u.mul(z, u), Y, mul(Z, X)) ~AP1(F, X) => F X 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: Alphabet: 0 : [] --> c 1 : [] --> c add : [c * a] --> c cons : [a * b] --> b fold : [c -> a -> c * b * c] --> c mul : [c * a] --> c nil : [] --> b prod : [b] --> c sum : [b] --> c Rules: fold(/\x./\y.X(x, y), nil, Y) => Y fold(/\x./\y.X(x, y), cons(Y, Z), U) => fold(/\z./\u.X(z, u), Z, X(U, Y)) sum(X) => fold(/\x./\y.add(x, y), X, 0) fold(/\x./\y.mul(x, y), X, 1) => prod(X) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): fold(/\x./\y.X(x, y), nil, Y) >? Y fold(/\x./\y.X(x, y), cons(Y, Z), U) >? fold(/\z./\u.X(z, u), Z, X(U, Y)) sum(X) >? fold(/\x./\y.add(x, y), X, 0) fold(/\x./\y.mul(x, y), X, 1) >? prod(X) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ [[fold(x_1, x_2, x_3)]] = fold(x_2, x_3, x_1) We choose Lex = {fold} and Mul = {1, add, cons, mul, nil, prod, sum}, and the following precedence: nil > sum > cons > 1 > fold > mul > add > prod Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: fold(/\x./\y.X(x, y), nil, Y) > Y fold(/\x./\y.X(x, y), cons(Y, Z), U) >= fold(/\x./\y.X(x, y), Z, X(U, Y)) sum(X) >= fold(/\x./\y.add(x, y), X, _|_) fold(/\x./\y.mul(x, y), X, 1) > prod(X) With these choices, we have: 1] fold(/\x./\y.X(x, y), nil, Y) > Y because [2], by definition 2] fold*(/\x./\y.X(x, y), nil, Y) >= Y because [3], by (Select) 3] Y >= Y by (Meta) 4] fold(/\x./\y.X(x, y), cons(Y, Z), U) >= fold(/\x./\y.X(x, y), Z, X(U, Y)) because [5], by (Star) 5] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= fold(/\x./\y.X(x, y), Z, X(U, Y)) because [6], [9], [15] and [17], by (Stat) 6] cons(Y, Z) > Z because [7], by definition 7] cons*(Y, Z) >= Z because [8], by (Select) 8] Z >= Z by (Meta) 9] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= /\x./\y.X(x, y) because [10], by (Select) 10] /\x./\z.X(x, z) >= /\x./\z.X(x, z) because [11], by (Abs) 11] /\z.X(y, z) >= /\z.X(y, z) because [12], by (Abs) 12] X(y, x) >= X(y, x) because [13] and [14], by (Meta) 13] y >= y by (Var) 14] x >= x by (Var) 15] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= Z because [16], by (Select) 16] cons(Y, Z) >= Z because [7], by (Star) 17] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= X(U, Y) because [18], by (Select) 18] X(fold*(/\z./\u.X(z, u), cons(Y, Z), U), fold*(/\v./\w.X(v, w), cons(Y, Z), U)) >= X(U, Y) because [19] and [21], by (Meta) 19] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= U because [20], by (Select) 20] U >= U by (Meta) 21] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= Y because [22], by (Select) 22] cons(Y, Z) >= Y because [23], by (Star) 23] cons*(Y, Z) >= Y because [24], by (Select) 24] Y >= Y by (Meta) 25] sum(X) >= fold(/\x./\y.add(x, y), X, _|_) because [26], by (Star) 26] sum*(X) >= fold(/\x./\y.add(x, y), X, _|_) because sum > fold, [27], [34] and [36], by (Copy) 27] sum*(X) >= /\y./\z.add(y, z) because [28], by (F-Abs) 28] sum*(X, x) >= /\z.add(x, z) because [29], by (F-Abs) 29] sum*(X, x, y) >= add(x, y) because sum > add, [30] and [32], by (Copy) 30] sum*(X, x, y) >= x because [31], by (Select) 31] x >= x by (Var) 32] sum*(X, x, y) >= y because [33], by (Select) 33] y >= y by (Var) 34] sum*(X) >= X because [35], by (Select) 35] X >= X by (Meta) 36] sum*(X) >= _|_ by (Bot) 37] fold(/\x./\y.mul(x, y), X, 1) > prod(X) because [38], by definition 38] fold*(/\x./\y.mul(x, y), X, 1) >= prod(X) because fold > prod and [39], by (Copy) 39] fold*(/\x./\y.mul(x, y), X, 1) >= X because [40], by (Select) 40] X >= X by (Meta) We can thus remove the following rules: fold(/\x./\y.X(x, y), nil, Y) => Y fold(/\x./\y.mul(x, y), X, 1) => prod(X) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): fold(/\x./\y.X(x, y), cons(Y, Z), U) >? fold(/\z./\u.X(z, u), Z, X(U, Y)) sum(X) >? fold(/\x./\y.add(x, y), X, 0) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ [[fold(x_1, x_2, x_3)]] = fold(x_2, x_1, x_3) We choose Lex = {fold} and Mul = {add, cons, sum}, and the following precedence: cons > sum > add > fold Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: fold(/\x./\y.X(x, y), cons(Y, Z), U) >= fold(/\x./\y.X(x, y), Z, X(U, Y)) sum(X) > fold(/\x./\y.add(x, y), X, _|_) With these choices, we have: 1] fold(/\x./\y.X(x, y), cons(Y, Z), U) >= fold(/\x./\y.X(x, y), Z, X(U, Y)) because [2], by (Star) 2] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= fold(/\x./\y.X(x, y), Z, X(U, Y)) because [3], [6], [14] and [16], 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] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= /\x./\y.X(x, y) because [7], by (F-Abs) 7] fold*(/\x./\y.X(x, y), cons(Y, Z), U, z) >= /\x.X(z, x) because [8], by (F-Abs) 8] fold*(/\x./\y.X(x, y), cons(Y, Z), U, z, u) >= X(z, u) because [9], by (Select) 9] X(fold*(/\x./\y.X(x, y), cons(Y, Z), U, z, u), fold*(/\v./\w.X(v, w), cons(Y, Z), U, z, u)) >= X(z, u) because [10] and [12], by (Meta) 10] fold*(/\x./\y.X(x, y), cons(Y, Z), U, z, u) >= z because [11], by (Select) 11] z >= z by (Var) 12] fold*(/\x./\y.X(x, y), cons(Y, Z), U, z, u) >= u because [13], by (Select) 13] u >= u by (Var) 14] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= Z because [15], by (Select) 15] cons(Y, Z) >= Z because [4], by (Star) 16] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= X(U, Y) because [17], by (Select) 17] X(fold*(/\x./\y.X(x, y), cons(Y, Z), U), fold*(/\v./\w.X(v, w), cons(Y, Z), U)) >= X(U, Y) because [18] and [20], by (Meta) 18] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= U because [19], by (Select) 19] U >= U by (Meta) 20] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= Y because [21], by (Select) 21] cons(Y, Z) >= Y because [22], by (Star) 22] cons*(Y, Z) >= Y because [23], by (Select) 23] Y >= Y by (Meta) 24] sum(X) > fold(/\x./\y.add(x, y), X, _|_) because [25], by definition 25] sum*(X) >= fold(/\x./\y.add(x, y), X, _|_) because sum > fold, [26], [33] and [35], by (Copy) 26] sum*(X) >= /\y./\z.add(y, z) because [27], by (F-Abs) 27] sum*(X, x) >= /\z.add(x, z) because [28], by (F-Abs) 28] sum*(X, x, y) >= add(x, y) because sum > add, [29] and [31], by (Copy) 29] sum*(X, x, y) >= x because [30], by (Select) 30] x >= x by (Var) 31] sum*(X, x, y) >= y because [32], by (Select) 32] y >= y by (Var) 33] sum*(X) >= X because [34], by (Select) 34] X >= X by (Meta) 35] sum*(X) >= _|_ by (Bot) We can thus remove the following rules: sum(X) => fold(/\x./\y.add(x, y), X, 0) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): fold(/\x./\y.X(x, y), cons(Y, Z), U) >? fold(/\z./\u.X(z, u), Z, X(U, Y)) We use a recursive path ordering as defined in [Kop12, Chapter 5]. We choose Lex = {fold} and Mul = {cons}, and the following precedence: fold > cons With these choices, we have: 1] fold(/\x./\y.X(x, y), cons(Y, Z), U) > fold(/\x./\y.X(x, y), Z, X(U, Y)) because [2], by definition 2] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= fold(/\x./\y.X(x, y), Z, X(U, Y)) because [3], [8], [11], [12] and [14], by (Stat) 3] /\x./\z.X(x, z) >= /\x./\z.X(x, z) because [4], by (Abs) 4] /\z.X(y, z) >= /\z.X(y, z) because [5], by (Abs) 5] X(y, x) >= X(y, x) because [6] and [7], by (Meta) 6] y >= y by (Var) 7] x >= x by (Var) 8] cons(Y, Z) > Z because [9], by definition 9] cons*(Y, Z) >= Z because [10], by (Select) 10] Z >= Z by (Meta) 11] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= /\z./\u.X(z, u) because [3], by (Select) 12] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= Z because [13], by (Select) 13] cons(Y, Z) >= Z because [9], by (Star) 14] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= X(U, Y) because [15], by (Select) 15] X(fold*(/\z./\u.X(z, u), cons(Y, Z), U), fold*(/\v./\w.X(v, w), cons(Y, Z), U)) >= X(U, Y) because [16] and [18], by (Meta) 16] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= U because [17], by (Select) 17] U >= U by (Meta) 18] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= Y because [19], by (Select) 19] cons(Y, Z) >= Y because [20], by (Star) 20] cons*(Y, Z) >= Y because [21], by (Select) 21] Y >= Y by (Meta) We can thus remove the following rules: fold(/\x./\y.X(x, y), cons(Y, Z), U) => fold(/\z./\u.X(z, u), Z, X(U, Y)) 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. +++ Citations +++ [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012.