We consider the system h48. Alphabet: 0 : [] --> nat add : [nat * nat] --> nat rec : [nat -> nat -> nat * nat * nat] --> nat s : [nat] --> nat succ : [] --> nat -> nat -> nat xap : [nat -> nat -> nat * nat] --> nat -> nat yap : [nat -> nat * nat] --> nat Rules: rec(/\x./\y.yap(xap(f, x), y), z, 0) => z rec(/\x./\y.yap(xap(f, x), y), z, s(u)) => yap(xap(f, u), rec(/\v./\w.yap(xap(f, v), w), z, u)) succ x y => s(y) add(x, y) => rec(/\z./\u.yap(xap(succ, z), u), x, y) add(x, 0) => x add(x, s(y)) => s(add(x, y)) 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 add : [nat * nat] --> nat rec : [nat -> nat -> nat * nat * nat] --> nat s : [nat] --> nat succ : [nat] --> nat -> nat yap : [nat -> nat * nat] --> nat Rules: rec(/\x./\y.yap(F(x), y), X, 0) => X rec(/\x./\y.yap(F(x), y), X, s(Y)) => yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) succ(X) Y => s(Y) add(X, Y) => rec(/\x./\y.yap(succ(x), y), X, Y) add(X, 0) => X add(X, s(Y)) => s(add(X, 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]): rec(/\x./\y.yap(F(x), y), X, 0) >? X rec(/\x./\y.yap(F(x), y), X, s(Y)) >? yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) succ(X) Y >? s(Y) add(X, Y) >? rec(/\x./\y.yap(succ(x), y), X, Y) add(X, 0) >? X add(X, s(Y)) >? s(add(X, Y)) yap(F, X) >? F X We use a recursive path ordering as defined in [Kop12, Chapter 5]. We choose Lex = {} and Mul = {0, @_{o -> o}, add, rec, s, succ, yap}, and the following precedence: 0 > add > succ > s > rec > yap > @_{o -> o} With these choices, we have: 1] rec(/\x./\y.yap(F(x), y), X, 0) > X because [2], by definition 2] rec*(/\x./\y.yap(F(x), y), X, 0) >= X because [3], by (Select) 3] X >= X by (Meta) 4] rec(/\x./\y.yap(F(x), y), X, s(Y)) > yap(F(Y), rec(/\x./\y.yap(F(x), y), X, Y)) because [5], by definition 5] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(Y), rec(/\x./\y.yap(F(x), y), X, Y)) because rec > yap, [6] and [13], by (Copy) 6] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= F(Y) because [7], by (Select) 7] /\x.yap(F(rec*(/\y./\z.yap(F(y), z), X, s(Y))), x) >= F(Y) because [8], by (Eta)[Kop13:2] 8] F(rec*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(Y) because [9], by (Meta) 9] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= Y because [10], by (Select) 10] s(Y) >= Y because [11], by (Star) 11] s*(Y) >= Y because [12], by (Select) 12] Y >= Y by (Meta) 13] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y) because rec 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] s(Y) > Y because [22], by definition 22] s*(Y) >= Y because [12], by (Select) 23] @_{o -> o}(succ(X), Y) > s(Y) because [24], by definition 24] @_{o -> o}*(succ(X), Y) >= s(Y) because [25], by (Select) 25] succ(X) @_{o -> o}*(succ(X), Y) >= s(Y) because [26] 26] succ*(X, @_{o -> o}*(succ(X), Y)) >= s(Y) because succ > s and [27], by (Copy) 27] succ*(X, @_{o -> o}*(succ(X), Y)) >= Y because [28], by (Select) 28] @_{o -> o}*(succ(X), Y) >= Y because [29], by (Select) 29] Y >= Y by (Meta) 30] add(X, Y) >= rec(/\x./\y.yap(succ(x), y), X, Y) because [31], by (Star) 31] add*(X, Y) >= rec(/\x./\y.yap(succ(x), y), X, Y) because add > rec, [32], [40] and [42], by (Copy) 32] add*(X, Y) >= /\y./\z.yap(succ(y), z) because [33], by (F-Abs) 33] add*(X, Y, x) >= /\z.yap(succ(x), z) because [34], by (F-Abs) 34] add*(X, Y, x, y) >= yap(succ(x), y) because add > yap, [35] and [38], by (Copy) 35] add*(X, Y, x, y) >= succ(x) because add > succ and [36], by (Copy) 36] add*(X, Y, x, y) >= x because [37], by (Select) 37] x >= x by (Var) 38] add*(X, Y, x, y) >= y because [39], by (Select) 39] y >= y by (Var) 40] add*(X, Y) >= X because [41], by (Select) 41] X >= X by (Meta) 42] add*(X, Y) >= Y because [43], by (Select) 43] Y >= Y by (Meta) 44] add(X, 0) >= X because [45], by (Star) 45] add*(X, 0) >= X because [46], by (Select) 46] X >= X by (Meta) 47] add(X, s(Y)) >= s(add(X, Y)) because [48], by (Star) 48] add*(X, s(Y)) >= s(add(X, Y)) because add > s and [49], by (Copy) 49] add*(X, s(Y)) >= add(X, Y) because add in Mul, [50] and [51], by (Stat) 50] X >= X by (Meta) 51] s(Y) > Y because [52], by definition 52] s*(Y) >= Y because [53], by (Select) 53] Y >= Y by (Meta) 54] yap(F, X) >= @_{o -> o}(F, X) because [55], by (Star) 55] yap*(F, X) >= @_{o -> o}(F, X) because yap > @_{o -> o}, [56] and [58], by (Copy) 56] yap*(F, X) >= F because [57], by (Select) 57] F >= F by (Meta) 58] yap*(F, X) >= X because [59], by (Select) 59] X >= X by (Meta) We can thus remove the following rules: rec(/\x./\y.yap(F(x), y), X, 0) => X rec(/\x./\y.yap(F(x), y), X, s(Y)) => yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) 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]): add(X, Y) >? rec(/\x./\y.yap(succ(x), y), X, Y) add(X, 0) >? X add(X, s(Y)) >? s(add(X, Y)) yap(F, X) >? F X We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 3 add = \y0y1.3 + 3y0 + 3y1 rec = \G0y1y2.y1 + y2 + G0(0,0) s = \y0.3 + y0 succ = \y0y1.y0 yap = \G0y1.y1 + G0(y1) Using this interpretation, the requirements translate to: [[add(_x0, _x1)]] = 3 + 3x0 + 3x1 > x0 + x1 = [[rec(/\x./\y.yap(succ(x), y), _x0, _x1)]] [[add(_x0, 0)]] = 12 + 3x0 > x0 = [[_x0]] [[add(_x0, s(_x1))]] = 12 + 3x0 + 3x1 > 6 + 3x0 + 3x1 = [[s(add(_x0, _x1))]] [[yap(_F0, _x1)]] = x1 + F0(x1) >= x1 + F0(x1) = [[_F0 _x1]] We can thus remove the following rules: add(X, Y) => rec(/\x./\y.yap(succ(x), y), X, Y) add(X, 0) => X add(X, s(Y)) => s(add(X, 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]): yap(F, X) >? F X We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: yap = \G0y1.1 + y1 + G0(y1) Using this interpretation, the requirements translate to: [[yap(_F0, _x1)]] = 1 + x1 + F0(x1) > x1 + F0(x1) = [[_F0 _x1]] We can thus remove the following rules: yap(F, X) => F 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.