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 > foldr > nil > succ > yap > @_{o -> o} 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 (Star) 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 [13], 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] cons(Y, Z) >= Y because [11], by (Star) 11] cons*(Y, Z) >= Y because [12], by (Select) 12] Y >= Y by (Meta) 13] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z) because foldr in Mul, [14], [20] and [21], by (Stat) 14] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z) because [15], by (Abs) 15] /\z.yap(F(y), z) >= /\z.yap(F(y), z) because [16], by (Abs) 16] yap(F(y), x) >= yap(F(y), x) because yap in Mul, [17] and [19], by (Fun) 17] F(y) >= F(y) because [18], by (Meta) 18] y >= y by (Var) 19] x >= x by (Var) 20] X >= X by (Meta) 21] cons(Y, Z) > Z because [22], by definition 22] cons*(Y, Z) >= Z because [23], by (Select) 23] Z >= Z by (Meta) 24] @_{o -> o}(succ(X), Y) > Y because [25], by definition 25] @_{o -> o}*(succ(X), Y) >= Y because [26], by (Select) 26] Y >= Y by (Meta) 27] length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because [28], by (Star) 28] length*(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because length > foldr, [29], [37] and [38], by (Copy) 29] length*(X) >= /\y./\z.yap(succ(y), z) because [30], by (F-Abs) 30] length*(X, x) >= /\z.yap(succ(x), z) because [31], by (F-Abs) 31] length*(X, x, y) >= yap(succ(x), y) because length > yap, [32] and [35], by (Copy) 32] length*(X, x, y) >= succ(x) because length > succ and [33], by (Copy) 33] length*(X, x, y) >= x because [34], by (Select) 34] x >= x by (Var) 35] length*(X, x, y) >= y because [36], by (Select) 36] y >= y by (Var) 37] length*(X) >= _|_ by (Bot) 38] length*(X) >= X because [39], by (Select) 39] X >= X by (Meta) 40] yap(F, X) > @_{o -> o}(F, X) because [41], by definition 41] yap*(F, X) >= @_{o -> o}(F, X) because yap > @_{o -> o}, [42] and [44], by (Copy) 42] yap*(F, X) >= F because [43], by (Select) 43] F >= F by (Meta) 44] yap*(F, X) >= X because [45], by (Select) 45] X >= X by (Meta) We can thus remove the following rules: succ(X) Y => s(Y) 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)) 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 = {cons, foldr, length, nil, succ, yap}, and the following precedence: length > foldr > nil > succ > yap > cons 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)) length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, 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] 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 (Star) 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 [13], 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] cons(Y, Z) >= Y because [11], by (Star) 11] cons*(Y, Z) >= Y because [12], by (Select) 12] Y >= Y by (Meta) 13] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z) because foldr in Mul, [14], [20] and [21], by (Stat) 14] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z) because [15], by (Abs) 15] /\z.yap(F(y), z) >= /\z.yap(F(y), z) because [16], by (Abs) 16] yap(F(y), x) >= yap(F(y), x) because yap in Mul, [17] and [19], by (Fun) 17] F(y) >= F(y) because [18], by (Meta) 18] y >= y by (Var) 19] x >= x by (Var) 20] X >= X by (Meta) 21] cons(Y, Z) > Z because [22], by definition 22] cons*(Y, Z) >= Z because [23], by (Select) 23] Z >= Z by (Meta) 24] length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because [25], by (Star) 25] length*(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because length > foldr, [26], [34] and [35], by (Copy) 26] length*(X) >= /\y./\z.yap(succ(y), z) because [27], by (F-Abs) 27] length*(X, x) >= /\z.yap(succ(x), z) because [28], by (F-Abs) 28] length*(X, x, y) >= yap(succ(x), y) because length > yap, [29] and [32], by (Copy) 29] length*(X, x, y) >= succ(x) because length > succ and [30], by (Copy) 30] length*(X, x, y) >= x because [31], by (Select) 31] x >= x by (Var) 32] length*(X, x, y) >= y because [33], by (Select) 33] y >= y by (Var) 34] length*(X) >= _|_ by (Bot) 35] length*(X) >= X because [36], by (Select) 36] 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]): foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 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 = {cons, foldr, length, succ, yap}, and the following precedence: length > foldr > succ > yap > cons 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, cons(Y, Z)) > yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)) length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) With these choices, we have: 1] foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)) because [2], by definition 2] 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, [3] and [10], by (Copy) 3] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(Y) because [4], by (Select) 4] /\x.yap(F(foldr*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(Y) because [5], by (Eta)[Kop13:2] 5] F(foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(Y) because [6], by (Meta) 6] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y because [7], by (Select) 7] cons(Y, Z) >= Y because [8], by (Star) 8] cons*(Y, Z) >= Y because [9], by (Select) 9] Y >= Y by (Meta) 10] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z) because foldr in Mul, [11], [17] and [18], by (Stat) 11] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z) because [12], by (Abs) 12] /\z.yap(F(y), z) >= /\z.yap(F(y), z) because [13], by (Abs) 13] yap(F(y), x) >= yap(F(y), x) because yap in Mul, [14] and [16], by (Fun) 14] F(y) >= F(y) because [15], by (Meta) 15] y >= y by (Var) 16] x >= x by (Var) 17] X >= X by (Meta) 18] cons(Y, Z) > Z because [19], by definition 19] cons*(Y, Z) >= Z because [20], by (Select) 20] Z >= Z by (Meta) 21] length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because [22], by (Star) 22] length*(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) because length > foldr, [23], [31] and [32], by (Copy) 23] length*(X) >= /\y./\z.yap(succ(y), z) because [24], by (F-Abs) 24] length*(X, x) >= /\z.yap(succ(x), z) because [25], by (F-Abs) 25] length*(X, x, y) >= yap(succ(x), y) because length > yap, [26] and [29], by (Copy) 26] length*(X, x, y) >= succ(x) because length > succ and [27], by (Copy) 27] length*(X, x, y) >= x because [28], by (Select) 28] x >= x by (Var) 29] length*(X, x, y) >= y because [30], by (Select) 30] y >= y by (Var) 31] length*(X) >= _|_ by (Bot) 32] length*(X) >= X because [33], by (Select) 33] 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)) 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 orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 0 foldr = \G0y1y2.y1 + y2 + G0(0,0) length = \y0.3 + 3y0 succ = \y0y1.y0 yap = \G0y1.y1 + G0(0) Using this interpretation, the requirements translate to: [[length(_x0)]] = 3 + 3x0 > x0 = [[foldr(/\x./\y.yap(succ(x), y), 0, _x0)]] 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.