We consider the system h58. Alphabet: 0 : [] --> nat cons : [nat * list] --> list foldr : [nat -> nat -> nat * nat * list] --> nat length : [list] --> nat nil : [] --> list s : [nat] --> nat succ : [] --> nat -> nat -> nat xap : [nat -> nat -> nat * nat] --> nat -> nat yap : [nat -> nat * nat] --> nat Rules: foldr(/\x./\y.yap(xap(f, x), y), z, nil) => z foldr(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => yap(xap(f, u), foldr(/\w./\x'.yap(xap(f, w), x'), z, v)) succ x y => s(y) length(x) => foldr(/\y./\z.yap(xap(succ, y), z), 0, x) xap(f, x) => f x yap(f, x) => f x This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). Symbol xap 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: Alphabet: 0 : [] --> nat cons : [nat * list] --> list foldr : [nat -> nat -> nat * nat * list] --> nat length : [list] --> nat nil : [] --> list s : [nat] --> nat succ : [nat] --> nat -> nat yap : [nat -> nat * nat] --> nat Rules: foldr(/\x./\y.yap(F(x), y), X, nil) => X foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) succ(X) Y => s(Y) length(X) => foldr(/\x./\y.yap(succ(x), y), 0, X) yap(F, X) => F 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]): foldr(/\x./\y.yap(F(x), y), X, nil) >? X foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) succ(X) Y >? s(Y) length(X) >? foldr(/\x./\y.yap(succ(x), y), 0, X) yap(F, X) >? F X We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ [[s(x_1)]] = x_1 We choose Lex = {} and Mul = {@_{o -> o}, cons, foldr, length, nil, succ, yap}, and the following precedence: cons > length > succ > foldr > yap > @_{o -> o} > nil Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: foldr(/\x./\y.yap(F(x), y), X, nil) >= X foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)) @_{o -> o}(succ(X), Y) > Y length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) yap(F, X) >= @_{o -> o}(F, X) With these choices, we have: 1] foldr(/\x./\y.yap(F(x), y), X, nil) >= X because [2], by (Star) 2] foldr*(/\x./\y.yap(F(x), y), X, nil) >= X because [3], by (Select) 3] X >= X by (Meta) 4] foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)) because [5], by definition 5] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)) because foldr > yap, [6] and [16], by (Copy) 6] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(Y) because [7], by (Select) 7] /\x.yap(F(foldr*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(Y) because [8], by (Eta)[Kop13:2] 8] F(foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(Y) because [9], by (Meta) 9] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y because [10], by (Select) 10] yap(F(foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Y because [11], by (Star) 11] yap*(F(foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldr*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Y because [12], by (Select) 12] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y because [13], by (Select) 13] cons(Y, Z) >= Y because [14], by (Star) 14] cons*(Y, Z) >= Y because [15], by (Select) 15] Y >= Y by (Meta) 16] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z) because foldr in Mul, [17], [23] and [24], by (Stat) 17] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z) because [18], by (Abs) 18] /\z.yap(F(y), z) >= /\z.yap(F(y), z) because [19], by (Abs) 19] yap(F(y), x) >= yap(F(y), x) because yap in Mul, [20] and [22], by (Fun) 20] F(y) >= F(y) because [21], by (Meta) 21] y >= y by (Var) 22] x >= x by (Var) 23] X >= X by (Meta) 24] cons(Y, Z) > Z because [25], by definition 25] cons*(Y, Z) >= Z because [26], by (Select) 26] Z >= Z by (Meta) 27] @_{o -> o}(succ(X), Y) > Y because [28], by definition 28] @_{o -> o}*(succ(X), Y) >= Y because [29], by (Select) 29] Y >= Y by (Meta) 30] length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because [31], by (Star) 31] length*(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because length > foldr, [32], [40] and [41], by (Copy) 32] length*(X) >= /\y./\z.yap(succ(y), z) because [33], by (F-Abs) 33] length*(X, x) >= /\z.yap(succ(x), z) because [34], by (F-Abs) 34] length*(X, x, y) >= yap(succ(x), y) because length > yap, [35] and [38], by (Copy) 35] length*(X, x, y) >= succ(x) because length > succ and [36], by (Copy) 36] length*(X, x, y) >= x because [37], by (Select) 37] x >= x by (Var) 38] length*(X, x, y) >= y because [39], by (Select) 39] y >= y by (Var) 40] length*(X) >= _|_ by (Bot) 41] length*(X) >= X because [42], by (Select) 42] X >= X by (Meta) 43] yap(F, X) >= @_{o -> o}(F, X) because [44], by (Star) 44] yap*(F, X) >= @_{o -> o}(F, X) because yap > @_{o -> o}, [45] and [47], by (Copy) 45] yap*(F, X) >= F because [46], by (Select) 46] F >= F by (Meta) 47] yap*(F, X) >= X because [48], by (Select) 48] X >= X by (Meta) We can thus remove the following rules: foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) succ(X) Y => s(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]): foldr(/\x./\y.yap(F(x), y), X, nil) >? X length(X) >? foldr(/\x./\y.yap(succ(x), y), 0, X) yap(F, X) >? F X We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ We choose Lex = {} and Mul = {@_{o -> o}, foldr, length, nil, succ, yap}, and the following precedence: length > @_{o -> o} = yap > foldr > nil > succ Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: foldr(/\x./\y.yap(F(x), y), X, nil) > X length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) yap(F, X) >= @_{o -> o}(F, X) With these choices, we have: 1] foldr(/\x./\y.yap(F(x), y), X, nil) > X because [2], by definition 2] foldr*(/\x./\y.yap(F(x), y), X, nil) >= X because [3], by (Select) 3] X >= X by (Meta) 4] length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because [5], by (Star) 5] length*(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because length > foldr, [6], [14] and [15], by (Copy) 6] length*(X) >= /\y./\z.yap(succ(y), z) because [7], by (F-Abs) 7] length*(X, x) >= /\z.yap(succ(x), z) because [8], by (F-Abs) 8] length*(X, x, y) >= yap(succ(x), y) because length > yap, [9] and [12], by (Copy) 9] length*(X, x, y) >= succ(x) because length > succ and [10], by (Copy) 10] length*(X, x, y) >= x because [11], by (Select) 11] x >= x by (Var) 12] length*(X, x, y) >= y because [13], by (Select) 13] y >= y by (Var) 14] length*(X) >= _|_ by (Bot) 15] length*(X) >= X because [16], by (Select) 16] X >= X by (Meta) 17] yap(F, X) >= @_{o -> o}(F, X) because yap = @_{o -> o}, yap in Mul, [18] and [19], by (Fun) 18] F >= F by (Meta) 19] X >= X by (Meta) We can thus remove the following rules: foldr(/\x./\y.yap(F(x), y), X, nil) => 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]): length(X) >? foldr(/\x./\y.yap(succ(x), y), 0, X) yap(F, X) >? F X We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ We choose Lex = {} and Mul = {@_{o -> o}, foldr, length, succ, yap}, and the following precedence: length > succ > yap > @_{o -> o} > foldr Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) yap(F, X) > @_{o -> o}(F, X) With these choices, we have: 1] length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because [2], by (Star) 2] length*(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because length > foldr, [3], [11] and [12], by (Copy) 3] length*(X) >= /\y./\z.yap(succ(y), z) because [4], by (F-Abs) 4] length*(X, x) >= /\z.yap(succ(x), z) because [5], by (F-Abs) 5] length*(X, x, y) >= yap(succ(x), y) because length > yap, [6] and [9], by (Copy) 6] length*(X, x, y) >= succ(x) because length > succ and [7], by (Copy) 7] length*(X, x, y) >= x because [8], by (Select) 8] x >= x by (Var) 9] length*(X, x, y) >= y because [10], by (Select) 10] y >= y by (Var) 11] length*(X) >= _|_ by (Bot) 12] length*(X) >= X because [13], by (Select) 13] X >= X by (Meta) 14] yap(F, X) > @_{o -> o}(F, X) because [15], by definition 15] yap*(F, X) >= @_{o -> o}(F, X) because yap > @_{o -> o}, [16] and [18], by (Copy) 16] yap*(F, X) >= F because [17], by (Select) 17] F >= F by (Meta) 18] yap*(F, X) >= X because [19], by (Select) 19] X >= X by (Meta) We can thus remove the following rules: yap(F, X) => F 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]): length(X) >? foldr(/\x./\y.yap(succ(x), y), 0, X) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ We choose Lex = {} and Mul = {foldr, length, succ, yap}, and the following precedence: length > foldr > succ > yap Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: length(X) > foldr(/\x./\y.yap(succ(x), y), _|_, X) With these choices, we have: 1] length(X) > foldr(/\x./\y.yap(succ(x), y), _|_, X) because [2], by definition 2] length*(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because length > foldr, [3], [11] and [12], by (Copy) 3] length*(X) >= /\y./\z.yap(succ(y), z) because [4], by (F-Abs) 4] length*(X, x) >= /\z.yap(succ(x), z) because [5], by (F-Abs) 5] length*(X, x, y) >= yap(succ(x), y) because length > yap, [6] and [9], by (Copy) 6] length*(X, x, y) >= succ(x) because length > succ and [7], by (Copy) 7] length*(X, x, y) >= x because [8], by (Select) 8] x >= x by (Var) 9] length*(X, x, y) >= y because [10], by (Select) 10] y >= y by (Var) 11] length*(X) >= _|_ by (Bot) 12] length*(X) >= X because [13], by (Select) 13] X >= X by (Meta) We can thus remove the following rules: length(X) => foldr(/\x./\y.yap(succ(x), y), 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 +++ [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. [Kop13:2] C. Kop. StarHorpo with an Eta-Rule. Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/etahorpo.pdf, 2013.