0.00/0.28 YES 0.00/0.29 We consider the system theBenchmark. 0.00/0.29 0.00/0.29 Alphabet: 0.00/0.29 0.00/0.29 0 : [] --> c 0.00/0.29 1 : [] --> c 0.00/0.29 add : [] --> c -> a -> c 0.00/0.29 cons : [] --> a -> b -> b 0.00/0.29 fold : [] --> (c -> a -> c) -> b -> c -> c 0.00/0.29 mul : [] --> c -> a -> c 0.00/0.29 nil : [] --> b 0.00/0.29 prod : [] --> b -> c 0.00/0.29 sum : [] --> b -> c 0.00/0.29 0.00/0.29 Rules: 0.00/0.29 0.00/0.29 fold (/\x./\y.f x y) nil z => z 0.00/0.29 fold (/\x./\y.f x y) (cons z u) v => fold (/\w./\x'.f w x') u (f v z) 0.00/0.29 sum x => fold (/\y./\z.add y z) x 0 0.00/0.29 fold (/\x./\y.mul x y) z 1 => prod z 0.00/0.29 0.00/0.29 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. 0.00/0.29 0.00/0.29 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: 0.00/0.29 0.00/0.29 Alphabet: 0.00/0.29 0.00/0.29 0 : [] --> c 0.00/0.29 1 : [] --> c 0.00/0.29 add : [] --> c -> a -> c 0.00/0.29 cons : [a * b] --> b 0.00/0.29 fold : [c -> a -> c * b * c] --> c 0.00/0.29 mul : [c * a] --> c 0.00/0.29 nil : [] --> b 0.00/0.29 prod : [b] --> c 0.00/0.29 sum : [b] --> c 0.00/0.29 ~AP1 : [c -> a -> c * c] --> a -> c 0.00/0.29 0.00/0.29 Rules: 0.00/0.29 0.00/0.29 fold(/\x./\y.~AP1(F, x) y, nil, X) => X 0.00/0.29 fold(/\x./\y.~AP1(F, x) y, cons(X, Y), Z) => fold(/\z./\u.~AP1(F, z) u, Y, ~AP1(F, Z) X) 0.00/0.29 sum(X) => fold(/\x./\y.~AP1(add, x) y, X, 0) 0.00/0.29 fold(/\x./\y.mul(x, y), X, 1) => prod(X) 0.00/0.29 fold(/\x./\y.mul(x, y), nil, X) => X 0.00/0.29 fold(/\x./\y.mul(x, y), cons(X, Y), Z) => fold(/\z./\u.mul(z, u), Y, mul(Z, X)) 0.00/0.29 ~AP1(F, X) => F X 0.00/0.29 0.00/0.29 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: 0.00/0.29 0.00/0.29 Alphabet: 0.00/0.29 0.00/0.29 0 : [] --> c 0.00/0.29 1 : [] --> c 0.00/0.29 add : [c * a] --> c 0.00/0.29 cons : [a * b] --> b 0.00/0.29 fold : [c -> a -> c * b * c] --> c 0.00/0.29 mul : [c * a] --> c 0.00/0.29 nil : [] --> b 0.00/0.29 prod : [b] --> c 0.00/0.29 sum : [b] --> c 0.00/0.29 0.00/0.29 Rules: 0.00/0.29 0.00/0.29 fold(/\x./\y.X(x, y), nil, Y) => Y 0.00/0.29 fold(/\x./\y.X(x, y), cons(Y, Z), U) => fold(/\z./\u.X(z, u), Z, X(U, Y)) 0.00/0.29 sum(X) => fold(/\x./\y.add(x, y), X, 0) 0.00/0.29 fold(/\x./\y.mul(x, y), X, 1) => prod(X) 0.00/0.29 0.00/0.29 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.29 0.00/0.29 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.29 0.00/0.29 fold(/\x./\y.X(x, y), nil, Y) >? Y 0.00/0.29 fold(/\x./\y.X(x, y), cons(Y, Z), U) >? fold(/\z./\u.X(z, u), Z, X(U, Y)) 0.00/0.29 sum(X) >? fold(/\x./\y.add(x, y), X, 0) 0.00/0.29 fold(/\x./\y.mul(x, y), X, 1) >? prod(X) 0.00/0.29 0.00/0.29 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 0.00/0.29 0.00/0.29 Argument functions: 0.00/0.29 0.00/0.29 [[0]] = _|_ 0.00/0.29 [[prod(x_1)]] = x_1 0.00/0.29 0.00/0.29 We choose Lex = {fold} and Mul = {1, add, cons, mul, nil, sum}, and the following precedence: 1 > cons > mul > nil > sum > add > fold 0.00/0.29 0.00/0.29 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 0.00/0.29 0.00/0.29 fold(/\x./\y.X(x, y), nil, Y) >= Y 0.00/0.29 fold(/\x./\y.X(x, y), cons(Y, Z), U) > fold(/\x./\y.X(x, y), Z, X(U, Y)) 0.00/0.29 sum(X) >= fold(/\x./\y.add(x, y), X, _|_) 0.00/0.29 fold(/\x./\y.mul(x, y), X, 1) >= X 0.00/0.29 0.00/0.29 With these choices, we have: 0.00/0.29 0.00/0.29 1] fold(/\x./\y.X(x, y), nil, Y) >= Y because [2], by (Star) 0.00/0.29 2] fold*(/\x./\y.X(x, y), nil, Y) >= Y because [3], by (Select) 0.00/0.29 3] Y >= Y by (Meta) 0.00/0.29 0.00/0.29 4] fold(/\x./\y.X(x, y), cons(Y, Z), U) > fold(/\x./\y.X(x, y), Z, X(U, Y)) because [5], by definition 0.00/0.29 5] fold*(/\x./\y.X(x, y), cons(Y, Z), U) >= fold(/\x./\y.X(x, y), Z, X(U, Y)) because [6], [11], [14], [15] and [17], by (Stat) 0.00/0.29 6] /\x./\z.X(x, z) >= /\x./\z.X(x, z) because [7], by (Abs) 0.00/0.29 7] /\z.X(y, z) >= /\z.X(y, z) because [8], by (Abs) 0.00/0.29 8] X(y, x) >= X(y, x) because [9] and [10], by (Meta) 0.00/0.29 9] y >= y by (Var) 0.00/0.29 10] x >= x by (Var) 0.00/0.29 11] cons(Y, Z) > Z because [12], by definition 0.00/0.29 12] cons*(Y, Z) >= Z because [13], by (Select) 0.00/0.29 13] Z >= Z by (Meta) 0.00/0.29 14] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= /\z./\u.X(z, u) because [6], by (Select) 0.00/0.29 15] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= Z because [16], by (Select) 0.00/0.29 16] cons(Y, Z) >= Z because [12], by (Star) 0.00/0.29 17] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= X(U, Y) because [18], by (Select) 0.00/0.29 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) 0.00/0.29 19] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= U because [20], by (Select) 0.00/0.29 20] U >= U by (Meta) 0.00/0.29 21] fold*(/\z./\u.X(z, u), cons(Y, Z), U) >= Y because [22], by (Select) 0.00/0.29 22] cons(Y, Z) >= Y because [23], by (Star) 0.00/0.29 23] cons*(Y, Z) >= Y because [24], by (Select) 0.00/0.29 24] Y >= Y by (Meta) 0.00/0.29 0.00/0.29 25] sum(X) >= fold(/\x./\y.add(x, y), X, _|_) because [26], by (Star) 0.00/0.29 26] sum*(X) >= fold(/\x./\y.add(x, y), X, _|_) because sum > fold, [27], [34] and [36], by (Copy) 0.00/0.29 27] sum*(X) >= /\y./\z.add(y, z) because [28], by (F-Abs) 0.00/0.29 28] sum*(X, x) >= /\z.add(x, z) because [29], by (F-Abs) 0.00/0.29 29] sum*(X, x, y) >= add(x, y) because sum > add, [30] and [32], by (Copy) 0.00/0.29 30] sum*(X, x, y) >= x because [31], by (Select) 0.00/0.29 31] x >= x by (Var) 0.00/0.29 32] sum*(X, x, y) >= y because [33], by (Select) 0.00/0.29 33] y >= y by (Var) 0.00/0.29 34] sum*(X) >= X because [35], by (Select) 0.00/0.29 35] X >= X by (Meta) 0.00/0.29 36] sum*(X) >= _|_ by (Bot) 0.00/0.29 0.00/0.29 37] fold(/\x./\y.mul(x, y), X, 1) >= X because [38], by (Star) 0.00/0.29 38] fold*(/\x./\y.mul(x, y), X, 1) >= X because [39], by (Select) 0.00/0.29 39] X >= X by (Meta) 0.00/0.29 0.00/0.29 We can thus remove the following rules: 0.00/0.29 0.00/0.29 fold(/\x./\y.X(x, y), cons(Y, Z), U) => fold(/\z./\u.X(z, u), Z, X(U, Y)) 0.00/0.29 0.00/0.29 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.29 0.00/0.29 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.29 0.00/0.29 fold(/\x./\y.X(x, y), nil, Y) >? Y 0.00/0.29 sum(X) >? fold(/\x./\y.add(x, y), X, 0) 0.00/0.29 fold(/\x./\y.mul(x, y), X, 1) >? prod(X) 0.00/0.29 0.00/0.29 We orient these requirements with a polynomial interpretation in the natural numbers. 0.00/0.29 0.00/0.29 The following interpretation satisfies the requirements: 0.00/0.29 0.00/0.29 0 = 0 0.00/0.29 1 = 3 0.00/0.29 add = \y0y1.y0 + y1 0.00/0.29 fold = \G0y1y2.y1 + y2 + G0(0,0) 0.00/0.29 mul = \y0y1.3 + 3y0 + 3y1 0.00/0.30 nil = 3 0.00/0.30 prod = \y0.y0 0.00/0.30 sum = \y0.3 + 3y0 0.00/0.30 0.00/0.30 Using this interpretation, the requirements translate to: 0.00/0.30 0.00/0.30 [[fold(/\x./\y._x0(x, y), nil, _x1)]] = 3 + x1 + F0(0,0) > x1 = [[_x1]] 0.00/0.30 [[sum(_x0)]] = 3 + 3x0 > x0 = [[fold(/\x./\y.add(x, y), _x0, 0)]] 0.00/0.30 [[fold(/\x./\y.mul(x, y), _x0, 1)]] = 6 + x0 > x0 = [[prod(_x0)]] 0.00/0.30 0.00/0.30 We can thus remove the following rules: 0.00/0.30 0.00/0.30 fold(/\x./\y.X(x, y), nil, Y) => Y 0.00/0.30 sum(X) => fold(/\x./\y.add(x, y), X, 0) 0.00/0.30 fold(/\x./\y.mul(x, y), X, 1) => prod(X) 0.00/0.30 0.00/0.30 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. 0.00/0.30 0.00/0.30 0.00/0.30 +++ Citations +++ 0.00/0.30 0.00/0.30 [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. 0.00/0.30 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 0.00/0.30 EOF