We consider the system h52. Alphabet: 0 : [] --> nat cons : [] --> nat -> list -> list foldl : [] --> (nat -> nat -> nat) -> nat -> list -> nat nil : [] --> list plusc : [] --> nat -> nat -> nat s : [] --> nat -> nat sum : [] --> list -> nat Rules: foldl (/\x./\y.f x y) z nil => z foldl (/\x./\y.f x y) z (cons u v) => foldl (/\w./\x'.f w x') (f z u) v plusc x 0 => x plusc x (s y) => s (plusc x y) sum x => foldl (/\y./\z.plusc y z) 0 x 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 : [] --> nat cons : [nat * list] --> list foldl : [nat -> nat -> nat * nat * list] --> nat nil : [] --> list plusc : [nat * nat] --> nat s : [nat] --> nat sum : [list] --> nat ~AP1 : [nat -> nat -> nat * nat] --> nat -> nat Rules: foldl(/\x./\y.~AP1(F, x) y, X, nil) => X foldl(/\x./\y.~AP1(F, x) y, X, cons(Y, Z)) => foldl(/\z./\u.~AP1(F, z) u, ~AP1(F, X) Y, Z) plusc(X, 0) => X plusc(X, s(Y)) => s(plusc(X, Y)) sum(X) => foldl(/\x./\y.plusc(x, y), 0, X) foldl(/\x./\y.plusc(x, y), X, nil) => X foldl(/\x./\y.plusc(x, y), X, cons(Y, Z)) => foldl(/\z./\u.plusc(z, u), plusc(X, Y), Z) ~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 : [] --> nat cons : [nat * list] --> list foldl : [nat -> nat -> nat * nat * list] --> nat nil : [] --> list plusc : [nat * nat] --> nat s : [nat] --> nat sum : [list] --> nat Rules: foldl(/\x./\y.X(x, y), Y, nil) => Y foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) plusc(X, 0) => X plusc(X, s(Y)) => s(plusc(X, Y)) sum(X) => foldl(/\x./\y.plusc(x, y), 0, 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]): foldl(/\x./\y.X(x, y), Y, nil) >? Y foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >? foldl(/\z./\u.X(z, u), X(Y, Z), U) plusc(X, 0) >? X plusc(X, s(Y)) >? s(plusc(X, Y)) sum(X) >? foldl(/\x./\y.plusc(x, y), 0, X) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_2, x_1) We choose Lex = {foldl} and Mul = {0, cons, nil, plusc, s, sum}, and the following precedence: cons > sum > foldl > nil > plusc > s > 0 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: foldl(/\x./\y.X(x, y), Y, nil) >= Y foldl(/\x./\y.X(x, y), Y, cons(Z, U)) > foldl(/\x./\y.X(x, y), X(Y, Z), U) plusc(X, 0) >= X plusc(X, s(Y)) > s(plusc(X, Y)) sum(X) > foldl(/\x./\y.plusc(x, y), 0, X) With these choices, we have: 1] foldl(/\x./\y.X(x, y), Y, nil) >= Y because [2], by (Star) 2] foldl*(/\x./\y.X(x, y), Y, nil) >= Y because [3], by (Select) 3] Y >= Y by (Meta) 4] foldl(/\x./\y.X(x, y), Y, cons(Z, U)) > foldl(/\x./\y.X(x, y), X(Y, Z), U) because [5], by definition 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) 6] cons(Z, U) > U because [7], by definition 7] cons*(Z, U) >= U because [8], by (Select) 8] U >= U by (Meta) 9] foldl*(/\x./\y.X(x, y), Y, cons(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] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= X(Y, Z) because [16], by (Select) 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) 17] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Y because [18], by (Select) 18] Y >= Y by (Meta) 19] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Z because [20], by (Select) 20] cons(Z, U) >= Z because [21], by (Star) 21] cons*(Z, U) >= Z because [22], by (Select) 22] Z >= Z by (Meta) 23] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= U because [24], by (Select) 24] cons(Z, U) >= U because [7], by (Star) 25] plusc(X, 0) >= X because [26], by (Star) 26] plusc*(X, 0) >= X because [27], by (Select) 27] X >= X by (Meta) 28] plusc(X, s(Y)) > s(plusc(X, Y)) because [29], by definition 29] plusc*(X, s(Y)) >= s(plusc(X, Y)) because plusc > s and [30], by (Copy) 30] plusc*(X, s(Y)) >= plusc(X, Y) because plusc in Mul, [31] and [32], by (Stat) 31] X >= X by (Meta) 32] s(Y) > Y because [33], by definition 33] s*(Y) >= Y because [34], by (Select) 34] Y >= Y by (Meta) 35] sum(X) > foldl(/\x./\y.plusc(x, y), 0, X) because [36], by definition 36] sum*(X) >= foldl(/\x./\y.plusc(x, y), 0, X) because sum > foldl, [37], [44] and [45], by (Copy) 37] sum*(X) >= /\y./\z.plusc(y, z) because [38], by (F-Abs) 38] sum*(X, x) >= /\z.plusc(x, z) because [39], by (F-Abs) 39] sum*(X, x, y) >= plusc(x, y) because sum > plusc, [40] and [42], by (Copy) 40] sum*(X, x, y) >= x because [41], by (Select) 41] x >= x by (Var) 42] sum*(X, x, y) >= y because [43], by (Select) 43] y >= y by (Var) 44] sum*(X) >= 0 because sum > 0, by (Copy) 45] sum*(X) >= X because [46], by (Select) 46] X >= X by (Meta) We can thus remove the following rules: foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) plusc(X, s(Y)) => s(plusc(X, Y)) sum(X) => foldl(/\x./\y.plusc(x, y), 0, 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]): foldl(/\x./\y.X(x, y), Y, nil) >? Y plusc(X, 0) >? X We use a recursive path ordering as defined in [Kop12, Chapter 5]. We choose Lex = {} and Mul = {0, foldl, nil, plusc}, and the following precedence: foldl > 0 > plusc > nil With these choices, we have: 1] foldl(/\x./\y.X(x, y), Y, nil) > Y because [2], by definition 2] foldl*(/\x./\y.X(x, y), Y, nil) >= Y because [3], by (Select) 3] Y >= Y by (Meta) 4] plusc(X, 0) >= X because [5], by (Star) 5] plusc*(X, 0) >= X because [6], by (Select) 6] X >= X by (Meta) We can thus remove the following rules: foldl(/\x./\y.X(x, y), Y, nil) => Y We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): plusc(X, 0) >? X We use a recursive path ordering as defined in [Kop12, Chapter 5]. We choose Lex = {} and Mul = {0, plusc}, and the following precedence: 0 > plusc With these choices, we have: 1] plusc(X, 0) > X because [2], by definition 2] plusc*(X, 0) >= X because [3], by (Select) 3] X >= X by (Meta) We can thus remove the following rules: plusc(X, 0) => X 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.