We consider the system h46. Alphabet: 0 : [] --> nat plus : [] --> nat -> nat -> nat rec : [] --> nat -> nat -> (nat -> nat -> nat) -> nat s : [] --> nat -> nat succ : [] --> nat -> nat -> nat Rules: rec 0 x (/\y./\z.f y z) => x rec (s x) y (/\z./\u.f z u) => f x (rec x y (/\v./\w.f v w)) succ x y => s y plus x y => rec x y (/\z./\u.succ z u) 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 plus : [nat * nat] --> nat rec : [nat * nat * nat -> nat -> nat] --> nat s : [nat] --> nat succ : [nat * nat] --> nat ~AP1 : [nat -> nat -> nat * nat] --> nat -> nat Rules: rec(0, X, /\x./\y.~AP1(F, x) y) => X rec(s(X), Y, /\x./\y.~AP1(F, x) y) => ~AP1(F, X) rec(X, Y, /\z./\u.~AP1(F, z) u) succ(X, Y) => s(Y) plus(X, Y) => rec(X, Y, /\x./\y.succ(x, y)) rec(0, X, /\x./\y.plus(x, y)) => X rec(0, X, /\x./\y.succ(x, y)) => X rec(s(X), Y, /\x./\y.plus(x, y)) => plus(X, rec(X, Y, /\z./\u.plus(z, u))) rec(s(X), Y, /\x./\y.succ(x, y)) => succ(X, rec(X, Y, /\z./\u.succ(z, u))) ~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 plus : [nat * nat] --> nat rec : [nat * nat * nat -> nat -> nat] --> nat s : [nat] --> nat succ : [nat * nat] --> nat Rules: rec(0, X, /\x./\y.Y(x, y)) => X rec(s(X), Y, /\x./\y.Z(x, y)) => Z(X, rec(X, Y, /\z./\u.Z(z, u))) succ(X, Y) => s(Y) plus(X, Y) => rec(X, Y, /\x./\y.succ(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]): rec(0, X, /\x./\y.Y(x, y)) >? X rec(s(X), Y, /\x./\y.Z(x, y)) >? Z(X, rec(X, Y, /\z./\u.Z(z, u))) succ(X, Y) >? s(Y) plus(X, Y) >? rec(X, Y, /\x./\y.succ(x, y)) We use a recursive path ordering as defined in [Kop12, Chapter 5]. We choose Lex = {} and Mul = {0, plus, rec, s, succ}, and the following precedence: plus > rec > succ > 0 > s With these choices, we have: 1] rec(0, X, /\x./\y.Y(x, y)) >= X because [2], by (Star) 2] rec*(0, X, /\x./\y.Y(x, y)) >= X because [3], by (Select) 3] X >= X by (Meta) 4] rec(s(X), Y, /\x./\y.Z(x, y)) >= Z(X, rec(X, Y, /\x./\y.Z(x, y))) because [5], by (Star) 5] rec*(s(X), Y, /\x./\y.Z(x, y)) >= Z(X, rec(X, Y, /\x./\y.Z(x, y))) because [6], by (Select) 6] Z(rec*(s(X), Y, /\x./\y.Z(x, y)), rec*(s(X), Y, /\z./\u.Z(z, u))) >= Z(X, rec(X, Y, /\x./\y.Z(x, y))) because [7] and [11], by (Meta) 7] rec*(s(X), Y, /\x./\y.Z(x, y)) >= X because [8], by (Select) 8] s(X) >= X because [9], by (Star) 9] s*(X) >= X because [10], by (Select) 10] X >= X by (Meta) 11] rec*(s(X), Y, /\x./\y.Z(x, y)) >= rec(X, Y, /\x./\y.Z(x, y)) because rec in Mul, [12], [14] and [15], by (Stat) 12] s(X) > X because [13], by definition 13] s*(X) >= X because [10], by (Select) 14] Y >= Y by (Meta) 15] /\x./\z.Z(x, z) >= /\x./\z.Z(x, z) because [16], by (Abs) 16] /\z.Z(y, z) >= /\z.Z(y, z) because [17], by (Abs) 17] Z(y, x) >= Z(y, x) because [18] and [19], by (Meta) 18] y >= y by (Var) 19] x >= x by (Var) 20] succ(X, Y) >= s(Y) because [21], by (Star) 21] succ*(X, Y) >= s(Y) because succ > s and [22], by (Copy) 22] succ*(X, Y) >= Y because [23], by (Select) 23] Y >= Y by (Meta) 24] plus(X, Y) > rec(X, Y, /\x./\y.succ(x, y)) because [25], by definition 25] plus*(X, Y) >= rec(X, Y, /\x./\y.succ(x, y)) because plus > rec, [26], [28] and [30], by (Copy) 26] plus*(X, Y) >= X because [27], by (Select) 27] X >= X by (Meta) 28] plus*(X, Y) >= Y because [29], by (Select) 29] Y >= Y by (Meta) 30] plus*(X, Y) >= /\y./\z.succ(y, z) because [31], by (F-Abs) 31] plus*(X, Y, x) >= /\z.succ(x, z) because [32], by (F-Abs) 32] plus*(X, Y, x, y) >= succ(x, y) because plus > succ, [33] and [35], by (Copy) 33] plus*(X, Y, x, y) >= x because [34], by (Select) 34] x >= x by (Var) 35] plus*(X, Y, x, y) >= y because [36], by (Select) 36] y >= y by (Var) We can thus remove the following rules: plus(X, Y) => rec(X, Y, /\x./\y.succ(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]): rec(0, X, /\x./\y.Y(x, y)) >? X rec(s(X), Y, /\x./\y.Z(x, y)) >? Z(X, rec(X, Y, /\z./\u.Z(z, u))) succ(X, Y) >? s(Y) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[rec(x_1, x_2, x_3)]] = rec(x_1, x_3, x_2) We choose Lex = {rec} and Mul = {0, s, succ}, and the following precedence: succ > s > 0 > rec Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: rec(0, X, /\x./\y.Y(x, y)) > X rec(s(X), Y, /\x./\y.Z(x, y)) > Z(X, rec(X, Y, /\x./\y.Z(x, y))) succ(X, Y) >= s(Y) With these choices, we have: 1] rec(0, X, /\x./\y.Y(x, y)) > X because [2], by definition 2] rec*(0, X, /\x./\y.Y(x, y)) >= X because [3], by (Select) 3] X >= X by (Meta) 4] rec(s(X), Y, /\x./\y.Z(x, y)) > Z(X, rec(X, Y, /\x./\y.Z(x, y))) because [5], by definition 5] rec*(s(X), Y, /\x./\y.Z(x, y)) >= Z(X, rec(X, Y, /\x./\y.Z(x, y))) because [6], by (Select) 6] Z(rec*(s(X), Y, /\x./\y.Z(x, y)), rec*(s(X), Y, /\z./\u.Z(z, u))) >= Z(X, rec(X, Y, /\x./\y.Z(x, y))) because [7] and [11], by (Meta) 7] rec*(s(X), Y, /\x./\y.Z(x, y)) >= X because [8], by (Select) 8] s(X) >= X because [9], by (Star) 9] s*(X) >= X because [10], by (Select) 10] X >= X by (Meta) 11] rec*(s(X), Y, /\x./\y.Z(x, y)) >= rec(X, Y, /\x./\y.Z(x, y)) because [12], [7], [14] and [16], by (Stat) 12] s(X) > X because [13], by definition 13] s*(X) >= X because [10], by (Select) 14] rec*(s(X), Y, /\x./\y.Z(x, y)) >= Y because [15], by (Select) 15] Y >= Y by (Meta) 16] rec*(s(X), Y, /\x./\y.Z(x, y)) >= /\x./\y.Z(x, y) because [17], by (Select) 17] /\x./\z.Z(x, z) >= /\x./\z.Z(x, z) because [18], by (Abs) 18] /\z.Z(y, z) >= /\z.Z(y, z) because [19], by (Abs) 19] Z(y, x) >= Z(y, x) because [20] and [21], by (Meta) 20] y >= y by (Var) 21] x >= x by (Var) 22] succ(X, Y) >= s(Y) because [23], by (Star) 23] succ*(X, Y) >= s(Y) because succ > s and [24], by (Copy) 24] succ*(X, Y) >= Y because [25], by (Select) 25] Y >= Y by (Meta) We can thus remove the following rules: rec(0, X, /\x./\y.Y(x, y)) => X rec(s(X), Y, /\x./\y.Z(x, y)) => Z(X, rec(X, Y, /\z./\u.Z(z, u))) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): succ(X, Y) >? s(Y) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[s(x_1)]] = x_1 We choose Lex = {} and Mul = {succ}, and the following precedence: succ Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: succ(X, Y) > Y With these choices, we have: 1] succ(X, Y) > Y because [2], by definition 2] succ*(X, Y) >= Y because [3], by (Select) 3] Y >= Y by (Meta) We can thus remove the following rules: succ(X, Y) => s(Y) 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.