0.00/0.22 YES 0.00/0.23 We consider the system theBenchmark. 0.00/0.23 0.00/0.23 Alphabet: 0.00/0.23 0.00/0.23 cons : [] --> a -> alist -> alist 0.00/0.23 foldl : [] --> (a -> a -> a) -> a -> alist -> a 0.00/0.23 nil : [] --> alist 0.00/0.23 0.00/0.23 Rules: 0.00/0.23 0.00/0.23 foldl (/\x./\y.f x y) z nil => z 0.00/0.23 foldl (/\x./\y.f x y) z (cons u v) => foldl (/\w./\x'.f w x') (f z u) v 0.00/0.23 0.00/0.23 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. 0.00/0.23 0.00/0.23 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: 0.00/0.23 0.00/0.23 Alphabet: 0.00/0.23 0.00/0.23 cons : [a * alist] --> alist 0.00/0.23 foldl : [a -> a -> a * a * alist] --> a 0.00/0.23 nil : [] --> alist 0.00/0.23 ~AP1 : [a -> a -> a * a] --> a -> a 0.00/0.23 0.00/0.23 Rules: 0.00/0.23 0.00/0.23 foldl(/\x./\y.~AP1(F, x) y, X, nil) => X 0.00/0.23 foldl(/\x./\y.~AP1(F, x) y, X, cons(Y, Z)) => foldl(/\z./\u.~AP1(F, z) u, ~AP1(F, X) Y, Z) 0.00/0.23 ~AP1(F, X) => F X 0.00/0.23 0.00/0.23 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. This gives: 0.00/0.23 0.00/0.23 Alphabet: 0.00/0.23 0.00/0.23 cons : [a * alist] --> alist 0.00/0.23 foldl : [a -> a -> a * a * alist] --> a 0.00/0.23 nil : [] --> alist 0.00/0.23 0.00/0.23 Rules: 0.00/0.23 0.00/0.23 foldl(/\x./\y.X(x, y), Y, nil) => Y 0.00/0.23 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) 0.00/0.23 0.00/0.23 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.23 0.00/0.23 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.23 0.00/0.23 foldl(/\x./\y.X(x, y), Y, nil) >? Y 0.00/0.23 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >? foldl(/\z./\u.X(z, u), X(Y, Z), U) 0.00/0.23 0.00/0.23 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 0.00/0.23 0.00/0.23 Argument functions: 0.00/0.23 0.00/0.23 [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_2, x_1) 0.00/0.23 0.00/0.23 We choose Lex = {foldl} and Mul = {cons, nil}, and the following precedence: cons > foldl > nil 0.00/0.23 0.00/0.23 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 0.00/0.23 0.00/0.23 foldl(/\x./\y.X(x, y), Y, nil) > Y 0.00/0.23 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) 0.00/0.23 0.00/0.23 With these choices, we have: 0.00/0.23 0.00/0.23 1] foldl(/\x./\y.X(x, y), Y, nil) > Y because [2], by definition 0.00/0.23 2] foldl*(/\x./\y.X(x, y), Y, nil) >= Y because [3], by (Select) 0.00/0.23 3] Y >= Y by (Meta) 0.00/0.23 0.00/0.23 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) 0.00/0.23 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) 0.00/0.23 6] cons(Z, U) > U because [7], by definition 0.00/0.23 7] cons*(Z, U) >= U because [8], by (Select) 0.00/0.23 8] U >= U by (Meta) 0.00/0.23 9] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= /\x./\y.X(x, y) because [10], by (F-Abs) 0.00/0.23 10] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z) >= /\x.X(z, x) because [11], by (Select) 0.00/0.23 11] /\x.X(foldl*(/\y./\v.X(y, v), Y, cons(Z, U), z), x) >= /\x.X(z, x) because [12], by (Abs) 0.00/0.23 12] X(foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z), u) >= X(z, u) because [13] and [15], by (Meta) 0.00/0.23 13] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z) >= z because [14], by (Select) 0.00/0.23 14] z >= z by (Var) 0.00/0.23 15] u >= u by (Var) 0.00/0.23 16] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(Y, Z) because [17], by (Select) 0.00/0.23 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) 0.00/0.23 18] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= Y because [19], by (Select) 0.00/0.23 19] Y >= Y by (Meta) 0.00/0.23 20] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= Z because [21], by (Select) 0.00/0.23 21] cons(Z, U) >= Z because [22], by (Star) 0.00/0.23 22] cons*(Z, U) >= Z because [23], by (Select) 0.00/0.23 23] Z >= Z by (Meta) 0.00/0.23 24] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= U because [25], by (Select) 0.00/0.23 25] cons(Z, U) >= U because [7], by (Star) 0.00/0.23 0.00/0.23 We can thus remove the following rules: 0.00/0.23 0.00/0.23 foldl(/\x./\y.X(x, y), Y, nil) => Y 0.00/0.23 0.00/0.23 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.23 0.00/0.23 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.23 0.00/0.23 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >? foldl(/\z./\u.X(z, u), X(Y, Z), U) 0.00/0.23 0.00/0.23 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 0.00/0.23 0.00/0.23 Argument functions: 0.00/0.23 0.00/0.23 [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_2, x_1) 0.00/0.23 0.00/0.23 We choose Lex = {foldl} and Mul = {cons}, and the following precedence: cons > foldl 0.00/0.23 0.00/0.23 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 0.00/0.23 0.00/0.23 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) > foldl(/\x./\y.X(x, y), X(Y, Z), U) 0.00/0.23 0.00/0.23 With these choices, we have: 0.00/0.23 0.00/0.23 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 0.00/0.23 2] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) because [3], [6], [14] and [22], by (Stat) 0.00/0.23 3] cons(Z, U) > U because [4], by definition 0.00/0.23 4] cons*(Z, U) >= U because [5], by (Select) 0.00/0.23 5] U >= U by (Meta) 0.00/0.23 6] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= /\x./\y.X(x, y) because [7], by (F-Abs) 0.00/0.23 7] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z) >= /\x.X(z, x) because [8], by (F-Abs) 0.00/0.23 8] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z, u) >= X(z, u) because [9], by (Select) 0.00/0.23 9] X(foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z, u), foldl*(/\v./\w.X(v, w), Y, cons(Z, U), z, u)) >= X(z, u) because [10] and [12], by (Meta) 0.00/0.23 10] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z, u) >= z because [11], by (Select) 0.00/0.23 11] z >= z by (Var) 0.00/0.23 12] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z, u) >= u because [13], by (Select) 0.00/0.23 13] u >= u by (Var) 0.00/0.23 14] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(Y, Z) because [15], by (Select) 0.00/0.23 15] 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 [16] and [18], by (Meta) 0.00/0.23 16] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= Y because [17], by (Select) 0.00/0.23 17] Y >= Y by (Meta) 0.00/0.23 18] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= Z because [19], by (Select) 0.00/0.23 19] cons(Z, U) >= Z because [20], by (Star) 0.00/0.23 20] cons*(Z, U) >= Z because [21], by (Select) 0.00/0.23 21] Z >= Z by (Meta) 0.00/0.23 22] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= U because [23], by (Select) 0.00/0.23 23] cons(Z, U) >= U because [4], by (Star) 0.00/0.23 0.00/0.23 We can thus remove the following rules: 0.00/0.23 0.00/0.23 foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) 0.00/0.23 0.00/0.23 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. 0.00/0.23 0.00/0.23 0.00/0.23 +++ Citations +++ 0.00/0.23 0.00/0.23 [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. 0.00/0.23 [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. 0.00/0.23 EOF