/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. 0 : [] --> o U11 : [o * o * o] --> o U12 : [o * o] --> o U13 : [o] --> o U21 : [o * o] --> o U22 : [o] --> o U31 : [o * o] --> o U41 : [o * o * o] --> o a!6220!6220U11 : [o * o * o] --> o a!6220!6220U12 : [o * o] --> o a!6220!6220U13 : [o] --> o a!6220!6220U21 : [o * o] --> o a!6220!6220U22 : [o] --> o a!6220!6220U31 : [o * o] --> o a!6220!6220U41 : [o * o * o] --> o a!6220!6220and : [o * o] --> o a!6220!6220isNat : [o] --> o a!6220!6220isNatKind : [o] --> o a!6220!6220plus : [o * o] --> o and : [o * o] --> o isNat : [o] --> o isNatKind : [o] --> o mark : [o] --> o plus : [o * o] --> o s : [o] --> o tt : [] --> o a!6220!6220U11(tt, X, Y) => a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) => a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U13(tt) => tt a!6220!6220U21(tt, X) => a!6220!6220U22(a!6220!6220isNat(X)) a!6220!6220U22(tt) => tt a!6220!6220U31(tt, X) => mark(X) a!6220!6220U41(tt, X, Y) => s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220and(tt, X) => mark(X) a!6220!6220isNat(0) => tt a!6220!6220isNat(plus(X, Y)) => a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) a!6220!6220isNat(s(X)) => a!6220!6220U21(a!6220!6220isNatKind(X), X) a!6220!6220isNatKind(0) => tt a!6220!6220isNatKind(plus(X, Y)) => a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) a!6220!6220isNatKind(s(X)) => a!6220!6220isNatKind(X) a!6220!6220plus(X, 0) => a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) => a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) mark(U11(X, Y, Z)) => a!6220!6220U11(mark(X), Y, Z) mark(U12(X, Y)) => a!6220!6220U12(mark(X), Y) mark(isNat(X)) => a!6220!6220isNat(X) mark(U13(X)) => a!6220!6220U13(mark(X)) mark(U21(X, Y)) => a!6220!6220U21(mark(X), Y) mark(U22(X)) => a!6220!6220U22(mark(X)) mark(U31(X, Y)) => a!6220!6220U31(mark(X), Y) mark(U41(X, Y, Z)) => a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) => a!6220!6220plus(mark(X), mark(Y)) mark(and(X, Y)) => a!6220!6220and(mark(X), Y) mark(isNatKind(X)) => a!6220!6220isNatKind(X) mark(tt) => tt mark(s(X)) => s(mark(X)) mark(0) => 0 a!6220!6220U11(X, Y, Z) => U11(X, Y, Z) a!6220!6220U12(X, Y) => U12(X, Y) a!6220!6220isNat(X) => isNat(X) a!6220!6220U13(X) => U13(X) a!6220!6220U21(X, Y) => U21(X, Y) a!6220!6220U22(X) => U22(X) a!6220!6220U31(X, Y) => U31(X, Y) a!6220!6220U41(X, Y, Z) => U41(X, Y, Z) a!6220!6220plus(X, Y) => plus(X, Y) a!6220!6220and(X, Y) => and(X, Y) a!6220!6220isNatKind(X) => isNatKind(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]): a!6220!6220U11(tt, X, Y) >? a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) >? a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U13(tt) >? tt a!6220!6220U21(tt, X) >? a!6220!6220U22(a!6220!6220isNat(X)) a!6220!6220U22(tt) >? tt a!6220!6220U31(tt, X) >? mark(X) a!6220!6220U41(tt, X, Y) >? s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220and(tt, X) >? mark(X) a!6220!6220isNat(0) >? tt a!6220!6220isNat(plus(X, Y)) >? a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) a!6220!6220isNat(s(X)) >? a!6220!6220U21(a!6220!6220isNatKind(X), X) a!6220!6220isNatKind(0) >? tt a!6220!6220isNatKind(plus(X, Y)) >? a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) a!6220!6220plus(X, 0) >? a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) >? a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) mark(U11(X, Y, Z)) >? a!6220!6220U11(mark(X), Y, Z) mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(isNat(X)) >? a!6220!6220isNat(X) mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X, Y)) >? a!6220!6220U31(mark(X), Y) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt mark(s(X)) >? s(mark(X)) mark(0) >? 0 a!6220!6220U11(X, Y, Z) >? U11(X, Y, Z) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220isNat(X) >? isNat(X) a!6220!6220U13(X) >? U13(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X, Y) >? U31(X, Y) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isNatKind(X) >? isNatKind(X) about to try horpo We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ [[U13(x_1)]] = x_1 [[U22(x_1)]] = x_1 [[U41(x_1, x_2, x_3)]] = U41(x_3, x_2, x_1) [[a!6220!6220U13(x_1)]] = x_1 [[a!6220!6220U22(x_1)]] = x_1 [[a!6220!6220U41(x_1, x_2, x_3)]] = a!6220!6220U41(x_3, x_2, x_1) [[mark(x_1)]] = x_1 We choose Lex = {U41, a!6220!6220U41, a!6220!6220plus, plus} and Mul = {U11, U12, U21, U31, a!6220!6220U11, a!6220!6220U12, a!6220!6220U21, a!6220!6220U31, a!6220!6220and, a!6220!6220isNat, a!6220!6220isNatKind, and, isNat, isNatKind, s, tt}, and the following precedence: U41 = a!6220!6220U41 = a!6220!6220plus = plus > s > U21 = a!6220!6220U21 > a!6220!6220and = and > a!6220!6220isNatKind = isNatKind > U11 = a!6220!6220U11 > U12 = a!6220!6220U12 > U31 = a!6220!6220U31 > a!6220!6220isNat = isNat > tt Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) >= a!6220!6220isNat(X) tt >= tt a!6220!6220U21(tt, X) >= a!6220!6220isNat(X) tt >= tt a!6220!6220U31(tt, X) > X a!6220!6220U41(tt, X, Y) >= s(a!6220!6220plus(Y, X)) a!6220!6220and(tt, X) > X a!6220!6220isNat(_|_) >= tt a!6220!6220isNat(plus(X, Y)) >= a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) a!6220!6220isNat(s(X)) >= a!6220!6220U21(a!6220!6220isNatKind(X), X) a!6220!6220isNatKind(_|_) >= tt a!6220!6220isNatKind(plus(X, Y)) > a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) a!6220!6220plus(X, _|_) >= a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) U12(X, Y) >= a!6220!6220U12(X, Y) isNat(X) >= a!6220!6220isNat(X) X >= X U21(X, Y) >= a!6220!6220U21(X, Y) X >= X U31(X, Y) >= a!6220!6220U31(X, Y) U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) plus(X, Y) >= a!6220!6220plus(X, Y) and(X, Y) >= a!6220!6220and(X, Y) isNatKind(X) >= a!6220!6220isNatKind(X) tt >= tt s(X) >= s(X) _|_ >= _|_ a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) a!6220!6220U12(X, Y) >= U12(X, Y) a!6220!6220isNat(X) >= isNat(X) X >= X a!6220!6220U21(X, Y) >= U21(X, Y) X >= X a!6220!6220U31(X, Y) >= U31(X, Y) a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) a!6220!6220plus(X, Y) >= plus(X, Y) a!6220!6220and(X, Y) >= and(X, Y) a!6220!6220isNatKind(X) >= isNatKind(X) With these choices, we have: 1] a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) because [2], by (Star) 2] a!6220!6220U11*(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) because a!6220!6220U11 > a!6220!6220U12, [3] and [6], by (Copy) 3] a!6220!6220U11*(tt, X, Y) >= a!6220!6220isNat(X) because a!6220!6220U11 > a!6220!6220isNat and [4], by (Copy) 4] a!6220!6220U11*(tt, X, Y) >= X because [5], by (Select) 5] X >= X by (Meta) 6] a!6220!6220U11*(tt, X, Y) >= Y because [7], by (Select) 7] Y >= Y by (Meta) 8] a!6220!6220U12(tt, X) >= a!6220!6220isNat(X) because [9], by (Star) 9] a!6220!6220U12*(tt, X) >= a!6220!6220isNat(X) because a!6220!6220U12 > a!6220!6220isNat and [10], by (Copy) 10] a!6220!6220U12*(tt, X) >= X because [7], by (Select) 11] tt >= tt by (Fun) 12] a!6220!6220U21(tt, X) >= a!6220!6220isNat(X) because [13], by (Star) 13] a!6220!6220U21*(tt, X) >= a!6220!6220isNat(X) because a!6220!6220U21 > a!6220!6220isNat and [14], by (Copy) 14] a!6220!6220U21*(tt, X) >= X because [5], by (Select) 15] tt >= tt by (Fun) 16] a!6220!6220U31(tt, X) > X because [17], by definition 17] a!6220!6220U31*(tt, X) >= X because [18], by (Select) 18] X >= X by (Meta) 19] a!6220!6220U41(tt, X, Y) >= s(a!6220!6220plus(Y, X)) because [20], by (Star) 20] a!6220!6220U41*(tt, X, Y) >= s(a!6220!6220plus(Y, X)) because a!6220!6220U41 > s and [21], by (Copy) 21] a!6220!6220U41*(tt, X, Y) >= a!6220!6220plus(Y, X) because a!6220!6220U41 = a!6220!6220plus, [22], [23], [24] and [25], by (Stat) 22] X >= X by (Meta) 23] Y >= Y by (Meta) 24] a!6220!6220U41*(tt, X, Y) >= Y because [23], by (Select) 25] a!6220!6220U41*(tt, X, Y) >= X because [22], by (Select) 26] a!6220!6220and(tt, X) > X because [27], by definition 27] a!6220!6220and*(tt, X) >= X because [28], by (Select) 28] X >= X by (Meta) 29] a!6220!6220isNat(_|_) >= tt because [30], by (Star) 30] a!6220!6220isNat*(_|_) >= tt because a!6220!6220isNat > tt, by (Copy) 31] a!6220!6220isNat(plus(X, Y)) >= a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) because [32], by (Star) 32] a!6220!6220isNat*(plus(X, Y)) >= a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) because [33], by (Select) 33] plus(X, Y) >= a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) because [34], by (Star) 34] plus*(X, Y) >= a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) because plus > a!6220!6220U11, [35], [37] and [39], by (Copy) 35] plus*(X, Y) >= a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) because plus > a!6220!6220and, [36] and [38], by (Copy) 36] plus*(X, Y) >= a!6220!6220isNatKind(X) because plus > a!6220!6220isNatKind and [37], by (Copy) 37] plus*(X, Y) >= X because [5], by (Select) 38] plus*(X, Y) >= isNatKind(Y) because plus > isNatKind and [39], by (Copy) 39] plus*(X, Y) >= Y because [7], by (Select) 40] a!6220!6220isNat(s(X)) >= a!6220!6220U21(a!6220!6220isNatKind(X), X) because [41], by (Star) 41] a!6220!6220isNat*(s(X)) >= a!6220!6220U21(a!6220!6220isNatKind(X), X) because [42], by (Select) 42] s(X) >= a!6220!6220U21(a!6220!6220isNatKind(X), X) because [43], by (Star) 43] s*(X) >= a!6220!6220U21(a!6220!6220isNatKind(X), X) because s > a!6220!6220U21, [44] and [45], by (Copy) 44] s*(X) >= a!6220!6220isNatKind(X) because s > a!6220!6220isNatKind and [45], by (Copy) 45] s*(X) >= X because [5], by (Select) 46] a!6220!6220isNatKind(_|_) >= tt because [47], by (Star) 47] a!6220!6220isNatKind*(_|_) >= tt because a!6220!6220isNatKind > tt, by (Copy) 48] a!6220!6220isNatKind(plus(X, Y)) > a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) because [49], by definition 49] a!6220!6220isNatKind*(plus(X, Y)) >= a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) because [50], by (Select) 50] plus(X, Y) >= a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) because [35], by (Star) 51] a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) because [52], by (Star) 52] a!6220!6220isNatKind*(s(X)) >= a!6220!6220isNatKind(X) because [53], by (Select) 53] s(X) >= a!6220!6220isNatKind(X) because [44], by (Star) 54] a!6220!6220plus(X, _|_) >= a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) because [55], by (Star) 55] a!6220!6220plus*(X, _|_) >= a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) because a!6220!6220plus > a!6220!6220U31, [56] and [58], by (Copy) 56] a!6220!6220plus*(X, _|_) >= a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)) because a!6220!6220plus > a!6220!6220and, [57] and [59], by (Copy) 57] a!6220!6220plus*(X, _|_) >= a!6220!6220isNat(X) because a!6220!6220plus > a!6220!6220isNat and [58], by (Copy) 58] a!6220!6220plus*(X, _|_) >= X because [23], by (Select) 59] a!6220!6220plus*(X, _|_) >= isNatKind(X) because a!6220!6220plus > isNatKind and [58], by (Copy) 60] a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) because [61], by (Star) 61] a!6220!6220plus*(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) because a!6220!6220plus = a!6220!6220U41, [23], [62], [64], [67] and [72], by (Stat) 62] s(Y) > Y because [63], by definition 63] s*(Y) >= Y because [22], by (Select) 64] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))) because a!6220!6220plus > a!6220!6220and, [65] and [70], by (Copy) 65] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because a!6220!6220plus > a!6220!6220and, [66] and [69], by (Copy) 66] a!6220!6220plus*(X, s(Y)) >= a!6220!6220isNat(Y) because a!6220!6220plus > a!6220!6220isNat and [67], by (Copy) 67] a!6220!6220plus*(X, s(Y)) >= Y because [68], by (Select) 68] s(Y) >= Y because [63], by (Star) 69] a!6220!6220plus*(X, s(Y)) >= isNatKind(Y) because a!6220!6220plus > isNatKind and [67], by (Copy) 70] a!6220!6220plus*(X, s(Y)) >= and(isNat(X), isNatKind(X)) because a!6220!6220plus > and, [71] and [73], by (Copy) 71] a!6220!6220plus*(X, s(Y)) >= isNat(X) because a!6220!6220plus > isNat and [72], by (Copy) 72] a!6220!6220plus*(X, s(Y)) >= X because [23], by (Select) 73] a!6220!6220plus*(X, s(Y)) >= isNatKind(X) because a!6220!6220plus > isNatKind and [72], by (Copy) 74] U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) because U11 = a!6220!6220U11, U11 in Mul, [75], [76] and [77], by (Fun) 75] X >= X by (Meta) 76] Y >= Y by (Meta) 77] Z >= Z by (Meta) 78] U12(X, Y) >= a!6220!6220U12(X, Y) because U12 = a!6220!6220U12, U12 in Mul, [75] and [76], by (Fun) 79] isNat(X) >= a!6220!6220isNat(X) because isNat = a!6220!6220isNat, isNat in Mul and [80], by (Fun) 80] X >= X by (Meta) 81] X >= X by (Meta) 82] U21(X, Y) >= a!6220!6220U21(X, Y) because U21 = a!6220!6220U21, U21 in Mul, [75] and [76], by (Fun) 83] X >= X by (Meta) 84] U31(X, Y) >= a!6220!6220U31(X, Y) because U31 = a!6220!6220U31, U31 in Mul, [75] and [76], by (Fun) 85] U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) because U41 = a!6220!6220U41, [75], [76] and [77], by (Fun) 86] plus(X, Y) >= a!6220!6220plus(X, Y) because plus = a!6220!6220plus, [75] and [87], by (Fun) 87] Y >= Y by (Meta) 88] and(X, Y) >= a!6220!6220and(X, Y) because and = a!6220!6220and, and in Mul, [75] and [87], by (Fun) 89] isNatKind(X) >= a!6220!6220isNatKind(X) because isNatKind = a!6220!6220isNatKind, isNatKind in Mul and [80], by (Fun) 90] tt >= tt by (Fun) 91] s(X) >= s(X) because s in Mul and [92], by (Fun) 92] X >= X by (Meta) 93] _|_ >= _|_ by (Bot) 94] a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) because a!6220!6220U11 = U11, a!6220!6220U11 in Mul, [75], [87] and [77], by (Fun) 95] a!6220!6220U12(X, Y) >= U12(X, Y) because a!6220!6220U12 = U12, a!6220!6220U12 in Mul, [75] and [87], by (Fun) 96] a!6220!6220isNat(X) >= isNat(X) because a!6220!6220isNat = isNat, a!6220!6220isNat in Mul and [92], by (Fun) 97] X >= X by (Meta) 98] a!6220!6220U21(X, Y) >= U21(X, Y) because a!6220!6220U21 = U21, a!6220!6220U21 in Mul, [75] and [87], by (Fun) 99] X >= X by (Meta) 100] a!6220!6220U31(X, Y) >= U31(X, Y) because a!6220!6220U31 = U31, a!6220!6220U31 in Mul, [75] and [87], by (Fun) 101] a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) because a!6220!6220U41 = U41, [75], [87] and [77], by (Fun) 102] a!6220!6220plus(X, Y) >= plus(X, Y) because a!6220!6220plus = plus, [75] and [87], by (Fun) 103] a!6220!6220and(X, Y) >= and(X, Y) because a!6220!6220and = and, a!6220!6220and in Mul, [75] and [87], by (Fun) 104] a!6220!6220isNatKind(X) >= isNatKind(X) because a!6220!6220isNatKind = isNatKind, a!6220!6220isNatKind in Mul and [92], by (Fun) We can thus remove the following rules: a!6220!6220U31(tt, X) => mark(X) a!6220!6220and(tt, X) => mark(X) a!6220!6220isNatKind(plus(X, Y)) => a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(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]): a!6220!6220U11(tt, X, Y) >? a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) >? a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U13(tt) >? tt a!6220!6220U21(tt, X) >? a!6220!6220U22(a!6220!6220isNat(X)) a!6220!6220U22(tt) >? tt a!6220!6220U41(tt, X, Y) >? s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220isNat(0) >? tt a!6220!6220isNat(plus(X, Y)) >? a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) a!6220!6220isNat(s(X)) >? a!6220!6220U21(a!6220!6220isNatKind(X), X) a!6220!6220isNatKind(0) >? tt a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) a!6220!6220plus(X, 0) >? a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) >? a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) mark(U11(X, Y, Z)) >? a!6220!6220U11(mark(X), Y, Z) mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(isNat(X)) >? a!6220!6220isNat(X) mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X, Y)) >? a!6220!6220U31(mark(X), Y) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt mark(s(X)) >? s(mark(X)) mark(0) >? 0 a!6220!6220U11(X, Y, Z) >? U11(X, Y, Z) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220isNat(X) >? isNat(X) a!6220!6220U13(X) >? U13(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X, Y) >? U31(X, Y) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isNatKind(X) >? isNatKind(X) about to try horpo We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[U13(x_1)]] = x_1 [[U22(x_1)]] = x_1 [[U41(x_1, x_2, x_3)]] = U41(x_3, x_2, x_1) [[a!6220!6220U13(x_1)]] = x_1 [[a!6220!6220U22(x_1)]] = x_1 [[a!6220!6220U41(x_1, x_2, x_3)]] = a!6220!6220U41(x_3, x_2, x_1) [[a!6220!6220isNatKind(x_1)]] = x_1 [[isNatKind(x_1)]] = x_1 [[mark(x_1)]] = x_1 [[tt]] = _|_ We choose Lex = {U41, a!6220!6220U41, a!6220!6220plus, plus} and Mul = {0, U11, U12, U21, U31, a!6220!6220U11, a!6220!6220U12, a!6220!6220U21, a!6220!6220U31, a!6220!6220and, a!6220!6220isNat, and, isNat, s}, and the following precedence: U41 = a!6220!6220U41 = a!6220!6220plus = plus > U31 = a!6220!6220U31 > s > a!6220!6220and = and > U11 = a!6220!6220U11 > 0 > U12 = a!6220!6220U12 > U21 = a!6220!6220U21 > a!6220!6220isNat = isNat Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: a!6220!6220U11(_|_, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(_|_, X) >= a!6220!6220isNat(X) _|_ >= _|_ a!6220!6220U21(_|_, X) >= a!6220!6220isNat(X) _|_ >= _|_ a!6220!6220U41(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) a!6220!6220isNat(0) >= _|_ a!6220!6220isNat(plus(X, Y)) > a!6220!6220U11(a!6220!6220and(X, Y), X, Y) a!6220!6220isNat(s(X)) > a!6220!6220U21(X, X) 0 >= _|_ s(X) >= X a!6220!6220plus(X, 0) >= a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), X), X) a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)), Y, X) U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) U12(X, Y) >= a!6220!6220U12(X, Y) isNat(X) >= a!6220!6220isNat(X) X >= X U21(X, Y) >= a!6220!6220U21(X, Y) X >= X U31(X, Y) >= a!6220!6220U31(X, Y) U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) plus(X, Y) >= a!6220!6220plus(X, Y) and(X, Y) >= a!6220!6220and(X, Y) X >= X _|_ >= _|_ s(X) >= s(X) 0 >= 0 a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) a!6220!6220U12(X, Y) >= U12(X, Y) a!6220!6220isNat(X) >= isNat(X) X >= X a!6220!6220U21(X, Y) >= U21(X, Y) X >= X a!6220!6220U31(X, Y) >= U31(X, Y) a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) a!6220!6220plus(X, Y) >= plus(X, Y) a!6220!6220and(X, Y) >= and(X, Y) X >= X With these choices, we have: 1] a!6220!6220U11(_|_, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) because [2], by (Star) 2] a!6220!6220U11*(_|_, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) because a!6220!6220U11 > a!6220!6220U12, [3] and [6], by (Copy) 3] a!6220!6220U11*(_|_, X, Y) >= a!6220!6220isNat(X) because a!6220!6220U11 > a!6220!6220isNat and [4], by (Copy) 4] a!6220!6220U11*(_|_, X, Y) >= X because [5], by (Select) 5] X >= X by (Meta) 6] a!6220!6220U11*(_|_, X, Y) >= Y because [7], by (Select) 7] Y >= Y by (Meta) 8] a!6220!6220U12(_|_, X) >= a!6220!6220isNat(X) because [9], by (Star) 9] a!6220!6220U12*(_|_, X) >= a!6220!6220isNat(X) because a!6220!6220U12 > a!6220!6220isNat and [10], by (Copy) 10] a!6220!6220U12*(_|_, X) >= X because [7], by (Select) 11] _|_ >= _|_ by (Bot) 12] a!6220!6220U21(_|_, X) >= a!6220!6220isNat(X) because [13], by (Star) 13] a!6220!6220U21*(_|_, X) >= a!6220!6220isNat(X) because a!6220!6220U21 > a!6220!6220isNat and [14], by (Copy) 14] a!6220!6220U21*(_|_, X) >= X because [5], by (Select) 15] _|_ >= _|_ by (Bot) 16] a!6220!6220U41(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) because [17], by (Star) 17] a!6220!6220U41*(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) because a!6220!6220U41 > s and [18], by (Copy) 18] a!6220!6220U41*(_|_, X, Y) >= a!6220!6220plus(Y, X) because a!6220!6220U41 = a!6220!6220plus, [19], [20], [21] and [22], by (Stat) 19] X >= X by (Meta) 20] Y >= Y by (Meta) 21] a!6220!6220U41*(_|_, X, Y) >= Y because [20], by (Select) 22] a!6220!6220U41*(_|_, X, Y) >= X because [19], by (Select) 23] a!6220!6220isNat(0) >= _|_ by (Bot) 24] a!6220!6220isNat(plus(X, Y)) > a!6220!6220U11(a!6220!6220and(X, Y), X, Y) because [25], by definition 25] a!6220!6220isNat*(plus(X, Y)) >= a!6220!6220U11(a!6220!6220and(X, Y), X, Y) because [26], by (Select) 26] plus(X, Y) >= a!6220!6220U11(a!6220!6220and(X, Y), X, Y) because [27], by (Star) 27] plus*(X, Y) >= a!6220!6220U11(a!6220!6220and(X, Y), X, Y) because plus > a!6220!6220U11, [28], [29] and [30], by (Copy) 28] plus*(X, Y) >= a!6220!6220and(X, Y) because plus > a!6220!6220and, [29] and [30], by (Copy) 29] plus*(X, Y) >= X because [5], by (Select) 30] plus*(X, Y) >= Y because [7], by (Select) 31] a!6220!6220isNat(s(X)) > a!6220!6220U21(X, X) because [32], by definition 32] a!6220!6220isNat*(s(X)) >= a!6220!6220U21(X, X) because [33], by (Select) 33] s(X) >= a!6220!6220U21(X, X) because [34], by (Star) 34] s*(X) >= a!6220!6220U21(X, X) because s > a!6220!6220U21, [35] and [35], by (Copy) 35] s*(X) >= X because [5], by (Select) 36] 0 >= _|_ by (Bot) 37] s(X) >= X because [35], by (Star) 38] a!6220!6220plus(X, 0) >= a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), X), X) because [39], by (Star) 39] a!6220!6220plus*(X, 0) >= a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), X), X) because a!6220!6220plus > a!6220!6220U31, [40] and [43], by (Copy) 40] a!6220!6220plus*(X, 0) >= a!6220!6220and(a!6220!6220isNat(X), X) because a!6220!6220plus > a!6220!6220and, [41] and [43], by (Copy) 41] a!6220!6220plus*(X, 0) >= a!6220!6220isNat(X) because a!6220!6220plus > a!6220!6220isNat and [42], by (Copy) 42] a!6220!6220plus*(X, 0) >= X because [20], by (Select) 43] a!6220!6220plus*(X, 0) >= X because [20], by (Select) 44] a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)), Y, X) because [45], by (Star) 45] a!6220!6220plus*(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)), Y, X) because a!6220!6220plus = a!6220!6220U41, [20], [46], [48], [53] and [57], by (Stat) 46] s(Y) > Y because [47], by definition 47] s*(Y) >= Y because [19], by (Select) 48] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)) because a!6220!6220plus > a!6220!6220and, [49] and [54], by (Copy) 49] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220isNat(Y), Y) because a!6220!6220plus > a!6220!6220and, [50] and [53], by (Copy) 50] a!6220!6220plus*(X, s(Y)) >= a!6220!6220isNat(Y) because a!6220!6220plus > a!6220!6220isNat and [51], by (Copy) 51] a!6220!6220plus*(X, s(Y)) >= Y because [52], by (Select) 52] s(Y) >= Y because [47], by (Star) 53] a!6220!6220plus*(X, s(Y)) >= Y because [52], by (Select) 54] a!6220!6220plus*(X, s(Y)) >= and(isNat(X), X) because a!6220!6220plus > and, [55] and [57], by (Copy) 55] a!6220!6220plus*(X, s(Y)) >= isNat(X) because a!6220!6220plus > isNat and [56], by (Copy) 56] a!6220!6220plus*(X, s(Y)) >= X because [20], by (Select) 57] a!6220!6220plus*(X, s(Y)) >= X because [20], by (Select) 58] U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) because U11 = a!6220!6220U11, U11 in Mul, [59], [60] and [61], by (Fun) 59] X >= X by (Meta) 60] Y >= Y by (Meta) 61] Z >= Z by (Meta) 62] U12(X, Y) >= a!6220!6220U12(X, Y) because U12 = a!6220!6220U12, U12 in Mul, [59] and [60], by (Fun) 63] isNat(X) >= a!6220!6220isNat(X) because isNat = a!6220!6220isNat, isNat in Mul and [64], by (Fun) 64] X >= X by (Meta) 65] X >= X by (Meta) 66] U21(X, Y) >= a!6220!6220U21(X, Y) because U21 = a!6220!6220U21, U21 in Mul, [59] and [60], by (Fun) 67] X >= X by (Meta) 68] U31(X, Y) >= a!6220!6220U31(X, Y) because U31 = a!6220!6220U31, U31 in Mul, [59] and [60], by (Fun) 69] U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) because U41 = a!6220!6220U41, [59], [60] and [61], by (Fun) 70] plus(X, Y) >= a!6220!6220plus(X, Y) because plus = a!6220!6220plus, [59] and [71], by (Fun) 71] Y >= Y by (Meta) 72] and(X, Y) >= a!6220!6220and(X, Y) because and = a!6220!6220and, and in Mul, [59] and [71], by (Fun) 73] X >= X by (Meta) 74] _|_ >= _|_ by (Bot) 75] s(X) >= s(X) because s in Mul and [76], by (Fun) 76] X >= X by (Meta) 77] 0 >= 0 by (Fun) 78] a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) because a!6220!6220U11 = U11, a!6220!6220U11 in Mul, [59], [71] and [61], by (Fun) 79] a!6220!6220U12(X, Y) >= U12(X, Y) because a!6220!6220U12 = U12, a!6220!6220U12 in Mul, [59] and [71], by (Fun) 80] a!6220!6220isNat(X) >= isNat(X) because a!6220!6220isNat = isNat, a!6220!6220isNat in Mul and [76], by (Fun) 81] X >= X by (Meta) 82] a!6220!6220U21(X, Y) >= U21(X, Y) because a!6220!6220U21 = U21, a!6220!6220U21 in Mul, [59] and [71], by (Fun) 83] X >= X by (Meta) 84] a!6220!6220U31(X, Y) >= U31(X, Y) because a!6220!6220U31 = U31, a!6220!6220U31 in Mul, [59] and [71], by (Fun) 85] a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) because a!6220!6220U41 = U41, [59], [71] and [61], by (Fun) 86] a!6220!6220plus(X, Y) >= plus(X, Y) because a!6220!6220plus = plus, [59] and [71], by (Fun) 87] a!6220!6220and(X, Y) >= and(X, Y) because a!6220!6220and = and, a!6220!6220and in Mul, [59] and [71], by (Fun) 88] X >= X by (Meta) We can thus remove the following rules: a!6220!6220isNat(plus(X, Y)) => a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) a!6220!6220isNat(s(X)) => a!6220!6220U21(a!6220!6220isNatKind(X), 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]): a!6220!6220U11(tt, X, Y) >? a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) >? a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U13(tt) >? tt a!6220!6220U21(tt, X) >? a!6220!6220U22(a!6220!6220isNat(X)) a!6220!6220U22(tt) >? tt a!6220!6220U41(tt, X, Y) >? s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220isNat(0) >? tt a!6220!6220isNatKind(0) >? tt a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) a!6220!6220plus(X, 0) >? a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) >? a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) mark(U11(X, Y, Z)) >? a!6220!6220U11(mark(X), Y, Z) mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(isNat(X)) >? a!6220!6220isNat(X) mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X, Y)) >? a!6220!6220U31(mark(X), Y) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt mark(s(X)) >? s(mark(X)) mark(0) >? 0 a!6220!6220U11(X, Y, Z) >? U11(X, Y, Z) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220isNat(X) >? isNat(X) a!6220!6220U13(X) >? U13(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X, Y) >? U31(X, Y) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isNatKind(X) >? isNatKind(X) about to try horpo We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[U41(x_1, x_2, x_3)]] = U41(x_2, x_3, x_1) [[a!6220!6220U41(x_1, x_2, x_3)]] = a!6220!6220U41(x_2, x_3, x_1) [[a!6220!6220plus(x_1, x_2)]] = a!6220!6220plus(x_2, x_1) [[mark(x_1)]] = x_1 [[plus(x_1, x_2)]] = plus(x_2, x_1) We choose Lex = {U41, a!6220!6220U41, a!6220!6220plus, plus} and Mul = {0, U11, U12, U13, U21, U22, U31, a!6220!6220U11, a!6220!6220U12, a!6220!6220U13, a!6220!6220U21, a!6220!6220U22, a!6220!6220U31, a!6220!6220and, a!6220!6220isNat, a!6220!6220isNatKind, and, isNat, isNatKind, s, tt}, and the following precedence: U41 = a!6220!6220U41 = a!6220!6220plus = plus > U31 = a!6220!6220U31 > s > a!6220!6220and = and > a!6220!6220isNatKind = isNatKind > 0 > U11 = a!6220!6220U11 > U21 = a!6220!6220U21 > U12 = a!6220!6220U12 > a!6220!6220isNat = isNat > tt > U22 = a!6220!6220U22 > U13 = a!6220!6220U13 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) >= a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U13(tt) > tt a!6220!6220U21(tt, X) > a!6220!6220U22(a!6220!6220isNat(X)) a!6220!6220U22(tt) > tt a!6220!6220U41(tt, X, Y) >= s(a!6220!6220plus(Y, X)) a!6220!6220isNat(0) >= tt a!6220!6220isNatKind(0) > tt a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) a!6220!6220plus(X, 0) > a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) U12(X, Y) >= a!6220!6220U12(X, Y) isNat(X) >= a!6220!6220isNat(X) U13(X) >= a!6220!6220U13(X) U21(X, Y) >= a!6220!6220U21(X, Y) U22(X) >= a!6220!6220U22(X) U31(X, Y) >= a!6220!6220U31(X, Y) U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) plus(X, Y) >= a!6220!6220plus(X, Y) and(X, Y) >= a!6220!6220and(X, Y) isNatKind(X) >= a!6220!6220isNatKind(X) tt >= tt s(X) >= s(X) 0 >= 0 a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) a!6220!6220U12(X, Y) >= U12(X, Y) a!6220!6220isNat(X) >= isNat(X) a!6220!6220U13(X) >= U13(X) a!6220!6220U21(X, Y) >= U21(X, Y) a!6220!6220U22(X) >= U22(X) a!6220!6220U31(X, Y) >= U31(X, Y) a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) a!6220!6220plus(X, Y) >= plus(X, Y) a!6220!6220and(X, Y) >= and(X, Y) a!6220!6220isNatKind(X) >= isNatKind(X) With these choices, we have: 1] a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) because [2], by (Star) 2] a!6220!6220U11*(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) because a!6220!6220U11 > a!6220!6220U12, [3] and [6], by (Copy) 3] a!6220!6220U11*(tt, X, Y) >= a!6220!6220isNat(X) because a!6220!6220U11 > a!6220!6220isNat and [4], by (Copy) 4] a!6220!6220U11*(tt, X, Y) >= X because [5], by (Select) 5] X >= X by (Meta) 6] a!6220!6220U11*(tt, X, Y) >= Y because [7], by (Select) 7] Y >= Y by (Meta) 8] a!6220!6220U12(tt, X) >= a!6220!6220U13(a!6220!6220isNat(X)) because [9], by (Star) 9] a!6220!6220U12*(tt, X) >= a!6220!6220U13(a!6220!6220isNat(X)) because a!6220!6220U12 > a!6220!6220U13 and [10], by (Copy) 10] a!6220!6220U12*(tt, X) >= a!6220!6220isNat(X) because a!6220!6220U12 > a!6220!6220isNat and [11], by (Copy) 11] a!6220!6220U12*(tt, X) >= X because [7], by (Select) 12] a!6220!6220U13(tt) > tt because [13], by definition 13] a!6220!6220U13*(tt) >= tt because [14], by (Select) 14] tt >= tt by (Fun) 15] a!6220!6220U21(tt, X) > a!6220!6220U22(a!6220!6220isNat(X)) because [16], by definition 16] a!6220!6220U21*(tt, X) >= a!6220!6220U22(a!6220!6220isNat(X)) because a!6220!6220U21 > a!6220!6220U22 and [17], by (Copy) 17] a!6220!6220U21*(tt, X) >= a!6220!6220isNat(X) because a!6220!6220U21 > a!6220!6220isNat and [18], by (Copy) 18] a!6220!6220U21*(tt, X) >= X because [5], by (Select) 19] a!6220!6220U22(tt) > tt because [20], by definition 20] a!6220!6220U22*(tt) >= tt because [14], by (Select) 21] a!6220!6220U41(tt, X, Y) >= s(a!6220!6220plus(Y, X)) because [22], by (Star) 22] a!6220!6220U41*(tt, X, Y) >= s(a!6220!6220plus(Y, X)) because a!6220!6220U41 > s and [23], by (Copy) 23] a!6220!6220U41*(tt, X, Y) >= a!6220!6220plus(Y, X) because a!6220!6220U41 = a!6220!6220plus, [24], [25], [26] and [27], by (Stat) 24] X >= X by (Meta) 25] Y >= Y by (Meta) 26] a!6220!6220U41*(tt, X, Y) >= Y because [25], by (Select) 27] a!6220!6220U41*(tt, X, Y) >= X because [24], by (Select) 28] a!6220!6220isNat(0) >= tt because [29], by (Star) 29] a!6220!6220isNat*(0) >= tt because [30], by (Select) 30] 0 >= tt because [31], by (Star) 31] 0* >= tt because 0 > tt, by (Copy) 32] a!6220!6220isNatKind(0) > tt because [33], by definition 33] a!6220!6220isNatKind*(0) >= tt because [30], by (Select) 34] a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) because a!6220!6220isNatKind in Mul and [35], by (Fun) 35] s(X) >= X because [36], by (Star) 36] s*(X) >= X because [5], by (Select) 37] a!6220!6220plus(X, 0) > a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) because [38], by definition 38] a!6220!6220plus*(X, 0) >= a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) because a!6220!6220plus > a!6220!6220U31, [39] and [41], by (Copy) 39] a!6220!6220plus*(X, 0) >= a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)) because a!6220!6220plus > a!6220!6220and, [40] and [42], by (Copy) 40] a!6220!6220plus*(X, 0) >= a!6220!6220isNat(X) because a!6220!6220plus > a!6220!6220isNat and [41], by (Copy) 41] a!6220!6220plus*(X, 0) >= X because [25], by (Select) 42] a!6220!6220plus*(X, 0) >= isNatKind(X) because a!6220!6220plus > isNatKind and [41], by (Copy) 43] a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) because [44], by (Star) 44] a!6220!6220plus*(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) because a!6220!6220plus = a!6220!6220U41, [45], [47], [57] and [55], by (Stat) 45] s(Y) > Y because [46], by definition 46] s*(Y) >= Y because [24], by (Select) 47] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))) because a!6220!6220plus > a!6220!6220and, [48] and [53], by (Copy) 48] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because [49], by (Select) 49] s(Y) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because [50], by (Star) 50] s*(Y) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because s > a!6220!6220and, [51] and [52], by (Copy) 51] s*(Y) >= a!6220!6220isNat(Y) because s > a!6220!6220isNat and [46], by (Copy) 52] s*(Y) >= isNatKind(Y) because s > isNatKind and [46], by (Copy) 53] a!6220!6220plus*(X, s(Y)) >= and(isNat(X), isNatKind(X)) because a!6220!6220plus > and, [54] and [56], by (Copy) 54] a!6220!6220plus*(X, s(Y)) >= isNat(X) because a!6220!6220plus > isNat and [55], by (Copy) 55] a!6220!6220plus*(X, s(Y)) >= X because [25], by (Select) 56] a!6220!6220plus*(X, s(Y)) >= isNatKind(X) because a!6220!6220plus > isNatKind and [55], by (Copy) 57] a!6220!6220plus*(X, s(Y)) >= Y because [58], by (Select) 58] s(Y) >= Y because [46], by (Star) 59] U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) because U11 = a!6220!6220U11, U11 in Mul, [60], [61] and [62], by (Fun) 60] X >= X by (Meta) 61] Y >= Y by (Meta) 62] Z >= Z by (Meta) 63] U12(X, Y) >= a!6220!6220U12(X, Y) because U12 = a!6220!6220U12, U12 in Mul, [60] and [61], by (Fun) 64] isNat(X) >= a!6220!6220isNat(X) because isNat = a!6220!6220isNat, isNat in Mul and [65], by (Fun) 65] X >= X by (Meta) 66] U13(X) >= a!6220!6220U13(X) because U13 = a!6220!6220U13, U13 in Mul and [67], by (Fun) 67] X >= X by (Meta) 68] U21(X, Y) >= a!6220!6220U21(X, Y) because U21 = a!6220!6220U21, U21 in Mul, [60] and [61], by (Fun) 69] U22(X) >= a!6220!6220U22(X) because U22 = a!6220!6220U22, U22 in Mul and [67], by (Fun) 70] U31(X, Y) >= a!6220!6220U31(X, Y) because U31 = a!6220!6220U31, U31 in Mul, [60] and [61], by (Fun) 71] U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) because U41 = a!6220!6220U41, [60], [61] and [62], by (Fun) 72] plus(X, Y) >= a!6220!6220plus(X, Y) because plus = a!6220!6220plus, [60] and [73], by (Fun) 73] Y >= Y by (Meta) 74] and(X, Y) >= a!6220!6220and(X, Y) because and = a!6220!6220and, and in Mul, [60] and [73], by (Fun) 75] isNatKind(X) >= a!6220!6220isNatKind(X) because isNatKind = a!6220!6220isNatKind, isNatKind in Mul and [67], by (Fun) 76] tt >= tt by (Fun) 77] s(X) >= s(X) because s in Mul and [67], by (Fun) 78] 0 >= 0 by (Fun) 79] a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) because a!6220!6220U11 = U11, a!6220!6220U11 in Mul, [60], [73] and [62], by (Fun) 80] a!6220!6220U12(X, Y) >= U12(X, Y) because a!6220!6220U12 = U12, a!6220!6220U12 in Mul, [60] and [73], by (Fun) 81] a!6220!6220isNat(X) >= isNat(X) because a!6220!6220isNat = isNat, a!6220!6220isNat in Mul and [67], by (Fun) 82] a!6220!6220U13(X) >= U13(X) because a!6220!6220U13 = U13, a!6220!6220U13 in Mul and [67], by (Fun) 83] a!6220!6220U21(X, Y) >= U21(X, Y) because a!6220!6220U21 = U21, a!6220!6220U21 in Mul, [60] and [73], by (Fun) 84] a!6220!6220U22(X) >= U22(X) because a!6220!6220U22 = U22, a!6220!6220U22 in Mul and [67], by (Fun) 85] a!6220!6220U31(X, Y) >= U31(X, Y) because a!6220!6220U31 = U31, a!6220!6220U31 in Mul, [60] and [73], by (Fun) 86] a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) because a!6220!6220U41 = U41, [60], [73] and [62], by (Fun) 87] a!6220!6220plus(X, Y) >= plus(X, Y) because a!6220!6220plus = plus, [60] and [73], by (Fun) 88] a!6220!6220and(X, Y) >= and(X, Y) because a!6220!6220and = and, a!6220!6220and in Mul, [60] and [73], by (Fun) 89] a!6220!6220isNatKind(X) >= isNatKind(X) because a!6220!6220isNatKind = isNatKind, a!6220!6220isNatKind in Mul and [67], by (Fun) We can thus remove the following rules: a!6220!6220U13(tt) => tt a!6220!6220U21(tt, X) => a!6220!6220U22(a!6220!6220isNat(X)) a!6220!6220U22(tt) => tt a!6220!6220isNatKind(0) => tt a!6220!6220plus(X, 0) => a!6220!6220U31(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), 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]): a!6220!6220U11(tt, X, Y) >? a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) >? a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U41(tt, X, Y) >? s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220isNat(0) >? tt a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) a!6220!6220plus(X, s(Y)) >? a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) mark(U11(X, Y, Z)) >? a!6220!6220U11(mark(X), Y, Z) mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(isNat(X)) >? a!6220!6220isNat(X) mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X, Y)) >? a!6220!6220U31(mark(X), Y) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt mark(s(X)) >? s(mark(X)) mark(0) >? 0 a!6220!6220U11(X, Y, Z) >? U11(X, Y, Z) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220isNat(X) >? isNat(X) a!6220!6220U13(X) >? U13(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X, Y) >? U31(X, Y) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isNatKind(X) >? isNatKind(X) about to try horpo We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ [[U13(x_1)]] = x_1 [[U41(x_1, x_2, x_3)]] = U41(x_3, x_2, x_1) [[a!6220!6220U13(x_1)]] = x_1 [[a!6220!6220U41(x_1, x_2, x_3)]] = a!6220!6220U41(x_3, x_2, x_1) [[mark(x_1)]] = x_1 We choose Lex = {U41, a!6220!6220U41, a!6220!6220plus, plus} and Mul = {U11, U12, U21, U22, U31, a!6220!6220U11, a!6220!6220U12, a!6220!6220U21, a!6220!6220U22, a!6220!6220U31, a!6220!6220and, a!6220!6220isNat, a!6220!6220isNatKind, and, isNat, isNatKind, s, tt}, and the following precedence: U11 = a!6220!6220U11 > U12 = a!6220!6220U12 > U31 = a!6220!6220U31 > U41 = a!6220!6220U41 = a!6220!6220plus = plus > s > a!6220!6220and = and > U22 = a!6220!6220U22 > U21 = a!6220!6220U21 > a!6220!6220isNat = isNat > a!6220!6220isNatKind = isNatKind > tt Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) >= a!6220!6220isNat(X) a!6220!6220U41(tt, X, Y) >= s(a!6220!6220plus(Y, X)) a!6220!6220isNat(_|_) > tt a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) U12(X, Y) >= a!6220!6220U12(X, Y) isNat(X) >= a!6220!6220isNat(X) X >= X U21(X, Y) >= a!6220!6220U21(X, Y) U22(X) >= a!6220!6220U22(X) U31(X, Y) >= a!6220!6220U31(X, Y) U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) plus(X, Y) >= a!6220!6220plus(X, Y) and(X, Y) >= a!6220!6220and(X, Y) isNatKind(X) >= a!6220!6220isNatKind(X) tt >= tt s(X) >= s(X) _|_ >= _|_ a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) a!6220!6220U12(X, Y) >= U12(X, Y) a!6220!6220isNat(X) >= isNat(X) X >= X a!6220!6220U21(X, Y) >= U21(X, Y) a!6220!6220U22(X) >= U22(X) a!6220!6220U31(X, Y) >= U31(X, Y) a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) a!6220!6220plus(X, Y) >= plus(X, Y) a!6220!6220and(X, Y) >= and(X, Y) a!6220!6220isNatKind(X) >= isNatKind(X) With these choices, we have: 1] a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) because [2], by (Star) 2] a!6220!6220U11*(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) because a!6220!6220U11 > a!6220!6220U12, [3] and [6], by (Copy) 3] a!6220!6220U11*(tt, X, Y) >= a!6220!6220isNat(X) because a!6220!6220U11 > a!6220!6220isNat and [4], by (Copy) 4] a!6220!6220U11*(tt, X, Y) >= X because [5], by (Select) 5] X >= X by (Meta) 6] a!6220!6220U11*(tt, X, Y) >= Y because [7], by (Select) 7] Y >= Y by (Meta) 8] a!6220!6220U12(tt, X) >= a!6220!6220isNat(X) because [9], by (Star) 9] a!6220!6220U12*(tt, X) >= a!6220!6220isNat(X) because a!6220!6220U12 > a!6220!6220isNat and [10], by (Copy) 10] a!6220!6220U12*(tt, X) >= X because [7], by (Select) 11] a!6220!6220U41(tt, X, Y) >= s(a!6220!6220plus(Y, X)) because [12], by (Star) 12] a!6220!6220U41*(tt, X, Y) >= s(a!6220!6220plus(Y, X)) because a!6220!6220U41 > s and [13], by (Copy) 13] a!6220!6220U41*(tt, X, Y) >= a!6220!6220plus(Y, X) because a!6220!6220U41 = a!6220!6220plus, [14], [15], [16] and [17], by (Stat) 14] X >= X by (Meta) 15] Y >= Y by (Meta) 16] a!6220!6220U41*(tt, X, Y) >= Y because [15], by (Select) 17] a!6220!6220U41*(tt, X, Y) >= X because [14], by (Select) 18] a!6220!6220isNat(_|_) > tt because [19], by definition 19] a!6220!6220isNat*(_|_) >= tt because a!6220!6220isNat > tt, by (Copy) 20] a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) because [21], by (Star) 21] a!6220!6220isNatKind*(s(X)) >= a!6220!6220isNatKind(X) because [22], by (Select) 22] s(X) >= a!6220!6220isNatKind(X) because [23], by (Star) 23] s*(X) >= a!6220!6220isNatKind(X) because s > a!6220!6220isNatKind and [24], by (Copy) 24] s*(X) >= X because [5], by (Select) 25] a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) because [26], by (Star) 26] a!6220!6220plus*(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) because a!6220!6220plus = a!6220!6220U41, [15], [27], [29], [39] and [37], by (Stat) 27] s(Y) > Y because [28], by definition 28] s*(Y) >= Y because [14], by (Select) 29] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))) because a!6220!6220plus > a!6220!6220and, [30] and [35], by (Copy) 30] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because [31], by (Select) 31] s(Y) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because [32], by (Star) 32] s*(Y) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because s > a!6220!6220and, [33] and [34], by (Copy) 33] s*(Y) >= a!6220!6220isNat(Y) because s > a!6220!6220isNat and [28], by (Copy) 34] s*(Y) >= isNatKind(Y) because s > isNatKind and [28], by (Copy) 35] a!6220!6220plus*(X, s(Y)) >= and(isNat(X), isNatKind(X)) because a!6220!6220plus > and, [36] and [38], by (Copy) 36] a!6220!6220plus*(X, s(Y)) >= isNat(X) because a!6220!6220plus > isNat and [37], by (Copy) 37] a!6220!6220plus*(X, s(Y)) >= X because [15], by (Select) 38] a!6220!6220plus*(X, s(Y)) >= isNatKind(X) because a!6220!6220plus > isNatKind and [37], by (Copy) 39] a!6220!6220plus*(X, s(Y)) >= Y because [40], by (Select) 40] s(Y) >= Y because [28], by (Star) 41] U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) because U11 = a!6220!6220U11, U11 in Mul, [42], [43] and [44], by (Fun) 42] X >= X by (Meta) 43] Y >= Y by (Meta) 44] Z >= Z by (Meta) 45] U12(X, Y) >= a!6220!6220U12(X, Y) because U12 = a!6220!6220U12, U12 in Mul, [42] and [43], by (Fun) 46] isNat(X) >= a!6220!6220isNat(X) because isNat = a!6220!6220isNat, isNat in Mul and [47], by (Fun) 47] X >= X by (Meta) 48] X >= X by (Meta) 49] U21(X, Y) >= a!6220!6220U21(X, Y) because U21 = a!6220!6220U21, U21 in Mul, [42] and [43], by (Fun) 50] U22(X) >= a!6220!6220U22(X) because U22 = a!6220!6220U22, U22 in Mul and [51], by (Fun) 51] X >= X by (Meta) 52] U31(X, Y) >= a!6220!6220U31(X, Y) because U31 = a!6220!6220U31, U31 in Mul, [42] and [43], by (Fun) 53] U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) because U41 = a!6220!6220U41, [42], [43] and [44], by (Fun) 54] plus(X, Y) >= a!6220!6220plus(X, Y) because plus = a!6220!6220plus, [42] and [55], by (Fun) 55] Y >= Y by (Meta) 56] and(X, Y) >= a!6220!6220and(X, Y) because and = a!6220!6220and, and in Mul, [42] and [55], by (Fun) 57] isNatKind(X) >= a!6220!6220isNatKind(X) because isNatKind = a!6220!6220isNatKind, isNatKind in Mul and [51], by (Fun) 58] tt >= tt by (Fun) 59] s(X) >= s(X) because s in Mul and [51], by (Fun) 60] _|_ >= _|_ by (Bot) 61] a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) because a!6220!6220U11 = U11, a!6220!6220U11 in Mul, [42], [55] and [44], by (Fun) 62] a!6220!6220U12(X, Y) >= U12(X, Y) because a!6220!6220U12 = U12, a!6220!6220U12 in Mul, [42] and [55], by (Fun) 63] a!6220!6220isNat(X) >= isNat(X) because a!6220!6220isNat = isNat, a!6220!6220isNat in Mul and [51], by (Fun) 64] X >= X by (Meta) 65] a!6220!6220U21(X, Y) >= U21(X, Y) because a!6220!6220U21 = U21, a!6220!6220U21 in Mul, [42] and [55], by (Fun) 66] a!6220!6220U22(X) >= U22(X) because a!6220!6220U22 = U22, a!6220!6220U22 in Mul and [51], by (Fun) 67] a!6220!6220U31(X, Y) >= U31(X, Y) because a!6220!6220U31 = U31, a!6220!6220U31 in Mul, [42] and [55], by (Fun) 68] a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) because a!6220!6220U41 = U41, [42], [55] and [44], by (Fun) 69] a!6220!6220plus(X, Y) >= plus(X, Y) because a!6220!6220plus = plus, [42] and [55], by (Fun) 70] a!6220!6220and(X, Y) >= and(X, Y) because a!6220!6220and = and, a!6220!6220and in Mul, [42] and [55], by (Fun) 71] a!6220!6220isNatKind(X) >= isNatKind(X) because a!6220!6220isNatKind = isNatKind, a!6220!6220isNatKind in Mul and [51], by (Fun) We can thus remove the following rules: a!6220!6220isNat(0) => tt We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220U11(tt, X, Y) >? a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) >? a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U41(tt, X, Y) >? s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) a!6220!6220plus(X, s(Y)) >? a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) mark(U11(X, Y, Z)) >? a!6220!6220U11(mark(X), Y, Z) mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(isNat(X)) >? a!6220!6220isNat(X) mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X, Y)) >? a!6220!6220U31(mark(X), Y) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt mark(s(X)) >? s(mark(X)) mark(0) >? 0 a!6220!6220U11(X, Y, Z) >? U11(X, Y, Z) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220isNat(X) >? isNat(X) a!6220!6220U13(X) >? U13(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X, Y) >? U31(X, Y) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isNatKind(X) >? isNatKind(X) about to try horpo We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[U13(x_1)]] = x_1 [[U41(x_1, x_2, x_3)]] = U41(x_2, x_3, x_1) [[a!6220!6220U13(x_1)]] = x_1 [[a!6220!6220U41(x_1, x_2, x_3)]] = a!6220!6220U41(x_2, x_3, x_1) [[a!6220!6220plus(x_1, x_2)]] = a!6220!6220plus(x_2, x_1) [[mark(x_1)]] = x_1 [[plus(x_1, x_2)]] = plus(x_2, x_1) We choose Lex = {U41, a!6220!6220U41, a!6220!6220plus, plus} and Mul = {0, U11, U12, U21, U22, U31, a!6220!6220U11, a!6220!6220U12, a!6220!6220U21, a!6220!6220U22, a!6220!6220U31, a!6220!6220and, a!6220!6220isNat, a!6220!6220isNatKind, and, isNat, isNatKind, s, tt}, and the following precedence: U41 = a!6220!6220U41 = a!6220!6220plus = plus > s > U31 = a!6220!6220U31 > a!6220!6220and = and > U11 = a!6220!6220U11 > a!6220!6220isNatKind = isNatKind > 0 > U22 = a!6220!6220U22 > U12 = a!6220!6220U12 > tt > a!6220!6220isNat = isNat > U21 = a!6220!6220U21 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: a!6220!6220U11(tt, X, Y) > a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U12(tt, X) >= a!6220!6220isNat(X) a!6220!6220U41(tt, X, Y) >= s(a!6220!6220plus(Y, X)) a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) a!6220!6220plus(X, s(Y)) > a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) U12(X, Y) >= a!6220!6220U12(X, Y) isNat(X) >= a!6220!6220isNat(X) X >= X U21(X, Y) >= a!6220!6220U21(X, Y) U22(X) >= a!6220!6220U22(X) U31(X, Y) >= a!6220!6220U31(X, Y) U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) plus(X, Y) >= a!6220!6220plus(X, Y) and(X, Y) >= a!6220!6220and(X, Y) isNatKind(X) >= a!6220!6220isNatKind(X) tt >= tt s(X) >= s(X) 0 >= 0 a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) a!6220!6220U12(X, Y) >= U12(X, Y) a!6220!6220isNat(X) >= isNat(X) X >= X a!6220!6220U21(X, Y) >= U21(X, Y) a!6220!6220U22(X) >= U22(X) a!6220!6220U31(X, Y) >= U31(X, Y) a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) a!6220!6220plus(X, Y) >= plus(X, Y) a!6220!6220and(X, Y) >= and(X, Y) a!6220!6220isNatKind(X) >= isNatKind(X) With these choices, we have: 1] a!6220!6220U11(tt, X, Y) > a!6220!6220U12(a!6220!6220isNat(X), Y) because [2], by definition 2] a!6220!6220U11*(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNat(X), Y) because a!6220!6220U11 > a!6220!6220U12, [3] and [6], by (Copy) 3] a!6220!6220U11*(tt, X, Y) >= a!6220!6220isNat(X) because a!6220!6220U11 > a!6220!6220isNat and [4], by (Copy) 4] a!6220!6220U11*(tt, X, Y) >= X because [5], by (Select) 5] X >= X by (Meta) 6] a!6220!6220U11*(tt, X, Y) >= Y because [7], by (Select) 7] Y >= Y by (Meta) 8] a!6220!6220U12(tt, X) >= a!6220!6220isNat(X) because [9], by (Star) 9] a!6220!6220U12*(tt, X) >= a!6220!6220isNat(X) because a!6220!6220U12 > a!6220!6220isNat and [10], by (Copy) 10] a!6220!6220U12*(tt, X) >= X because [7], by (Select) 11] a!6220!6220U41(tt, X, Y) >= s(a!6220!6220plus(Y, X)) because [12], by (Star) 12] a!6220!6220U41*(tt, X, Y) >= s(a!6220!6220plus(Y, X)) because a!6220!6220U41 > s and [13], by (Copy) 13] a!6220!6220U41*(tt, X, Y) >= a!6220!6220plus(Y, X) because a!6220!6220U41 = a!6220!6220plus, [14], [15], [16] and [17], by (Stat) 14] X >= X by (Meta) 15] Y >= Y by (Meta) 16] a!6220!6220U41*(tt, X, Y) >= Y because [15], by (Select) 17] a!6220!6220U41*(tt, X, Y) >= X because [14], by (Select) 18] a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) because [19], by (Star) 19] a!6220!6220isNatKind*(s(X)) >= a!6220!6220isNatKind(X) because a!6220!6220isNatKind in Mul and [20], by (Stat) 20] s(X) > X because [21], by definition 21] s*(X) >= X because [5], by (Select) 22] a!6220!6220plus(X, s(Y)) > a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) because [23], by definition 23] a!6220!6220plus*(X, s(Y)) >= a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) because a!6220!6220plus = a!6220!6220U41, [24], [26], [36] and [34], by (Stat) 24] s(Y) > Y because [25], by definition 25] s*(Y) >= Y because [14], by (Select) 26] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))) because a!6220!6220plus > a!6220!6220and, [27] and [32], by (Copy) 27] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because [28], by (Select) 28] s(Y) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because [29], by (Star) 29] s*(Y) >= a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)) because s > a!6220!6220and, [30] and [31], by (Copy) 30] s*(Y) >= a!6220!6220isNat(Y) because s > a!6220!6220isNat and [25], by (Copy) 31] s*(Y) >= isNatKind(Y) because s > isNatKind and [25], by (Copy) 32] a!6220!6220plus*(X, s(Y)) >= and(isNat(X), isNatKind(X)) because a!6220!6220plus > and, [33] and [35], by (Copy) 33] a!6220!6220plus*(X, s(Y)) >= isNat(X) because a!6220!6220plus > isNat and [34], by (Copy) 34] a!6220!6220plus*(X, s(Y)) >= X because [15], by (Select) 35] a!6220!6220plus*(X, s(Y)) >= isNatKind(X) because a!6220!6220plus > isNatKind and [34], by (Copy) 36] a!6220!6220plus*(X, s(Y)) >= Y because [37], by (Select) 37] s(Y) >= Y because [25], by (Star) 38] U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) because U11 = a!6220!6220U11, U11 in Mul, [39], [40] and [41], by (Fun) 39] X >= X by (Meta) 40] Y >= Y by (Meta) 41] Z >= Z by (Meta) 42] U12(X, Y) >= a!6220!6220U12(X, Y) because U12 = a!6220!6220U12, U12 in Mul, [39] and [40], by (Fun) 43] isNat(X) >= a!6220!6220isNat(X) because isNat = a!6220!6220isNat, isNat in Mul and [44], by (Fun) 44] X >= X by (Meta) 45] X >= X by (Meta) 46] U21(X, Y) >= a!6220!6220U21(X, Y) because U21 = a!6220!6220U21, U21 in Mul, [39] and [40], by (Fun) 47] U22(X) >= a!6220!6220U22(X) because U22 = a!6220!6220U22, U22 in Mul and [48], by (Fun) 48] X >= X by (Meta) 49] U31(X, Y) >= a!6220!6220U31(X, Y) because U31 = a!6220!6220U31, U31 in Mul, [39] and [40], by (Fun) 50] U41(X, Y, Z) >= a!6220!6220U41(X, Y, Z) because U41 = a!6220!6220U41, [39], [40] and [41], by (Fun) 51] plus(X, Y) >= a!6220!6220plus(X, Y) because plus = a!6220!6220plus, [39] and [52], by (Fun) 52] Y >= Y by (Meta) 53] and(X, Y) >= a!6220!6220and(X, Y) because and = a!6220!6220and, and in Mul, [39] and [52], by (Fun) 54] isNatKind(X) >= a!6220!6220isNatKind(X) because isNatKind = a!6220!6220isNatKind, isNatKind in Mul and [48], by (Fun) 55] tt >= tt by (Fun) 56] s(X) >= s(X) because s in Mul and [48], by (Fun) 57] 0 >= 0 by (Fun) 58] a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) because a!6220!6220U11 = U11, a!6220!6220U11 in Mul, [39], [52] and [41], by (Fun) 59] a!6220!6220U12(X, Y) >= U12(X, Y) because a!6220!6220U12 = U12, a!6220!6220U12 in Mul, [39] and [52], by (Fun) 60] a!6220!6220isNat(X) >= isNat(X) because a!6220!6220isNat = isNat, a!6220!6220isNat in Mul and [48], by (Fun) 61] X >= X by (Meta) 62] a!6220!6220U21(X, Y) >= U21(X, Y) because a!6220!6220U21 = U21, a!6220!6220U21 in Mul, [39] and [52], by (Fun) 63] a!6220!6220U22(X) >= U22(X) because a!6220!6220U22 = U22, a!6220!6220U22 in Mul and [48], by (Fun) 64] a!6220!6220U31(X, Y) >= U31(X, Y) because a!6220!6220U31 = U31, a!6220!6220U31 in Mul, [39] and [52], by (Fun) 65] a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) because a!6220!6220U41 = U41, [39], [52] and [41], by (Fun) 66] a!6220!6220plus(X, Y) >= plus(X, Y) because a!6220!6220plus = plus, [39] and [52], by (Fun) 67] a!6220!6220and(X, Y) >= and(X, Y) because a!6220!6220and = and, a!6220!6220and in Mul, [39] and [52], by (Fun) 68] a!6220!6220isNatKind(X) >= isNatKind(X) because a!6220!6220isNatKind = isNatKind, a!6220!6220isNatKind in Mul and [48], by (Fun) We can thus remove the following rules: a!6220!6220U11(tt, X, Y) => a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220plus(X, s(Y)) => a!6220!6220U41(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, 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]): a!6220!6220U12(tt, X) >? a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U41(tt, X, Y) >? s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) mark(U11(X, Y, Z)) >? a!6220!6220U11(mark(X), Y, Z) mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(isNat(X)) >? a!6220!6220isNat(X) mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X, Y)) >? a!6220!6220U31(mark(X), Y) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt mark(s(X)) >? s(mark(X)) mark(0) >? 0 a!6220!6220U11(X, Y, Z) >? U11(X, Y, Z) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220isNat(X) >? isNat(X) a!6220!6220U13(X) >? U13(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X, Y) >? U31(X, Y) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isNatKind(X) >? isNatKind(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 0 U11 = \y0y1y2.y0 + y1 + y2 U12 = \y0y1.y0 + 2y1 U13 = \y0.y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.y0 U31 = \y0y1.y1 + 2y0 U41 = \y0y1y2.2y0 + 2y1 + 2y2 a!6220!6220U11 = \y0y1y2.y0 + y1 + y2 a!6220!6220U12 = \y0y1.y0 + 2y1 a!6220!6220U13 = \y0.y0 a!6220!6220U21 = \y0y1.y1 + 2y0 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0y1.y1 + 2y0 a!6220!6220U41 = \y0y1y2.2y0 + 2y1 + 2y2 a!6220!6220and = \y0y1.y0 + y1 a!6220!6220isNat = \y0.y0 a!6220!6220isNatKind = \y0.y0 a!6220!6220plus = \y0y1.y0 + y1 and = \y0y1.y0 + y1 isNat = \y0.y0 isNatKind = \y0.y0 mark = \y0.y0 plus = \y0y1.y0 + y1 s = \y0.2y0 tt = 2 Using this interpretation, the requirements translate to: [[a!6220!6220U12(tt, _x0)]] = 2 + 2x0 > x0 = [[a!6220!6220U13(a!6220!6220isNat(_x0))]] [[a!6220!6220U41(tt, _x0, _x1)]] = 4 + 2x0 + 2x1 > 2x0 + 2x1 = [[s(a!6220!6220plus(mark(_x1), mark(_x0)))]] [[a!6220!6220isNatKind(s(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(U11(_x0, _x1, _x2))]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[a!6220!6220U11(mark(_x0), _x1, _x2)]] [[mark(U12(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(isNat(_x0))]] = x0 >= x0 = [[a!6220!6220isNat(_x0)]] [[mark(U13(_x0))]] = x0 >= x0 = [[a!6220!6220U13(mark(_x0))]] [[mark(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = x0 >= x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U31(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U31(mark(_x0), _x1)]] [[mark(U41(_x0, _x1, _x2))]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[a!6220!6220U41(mark(_x0), _x1, _x2)]] [[mark(plus(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isNatKind(_x0))]] = x0 >= x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(tt)]] = 2 >= 2 = [[tt]] [[mark(s(_x0))]] = 2x0 >= 2x0 = [[s(mark(_x0))]] [[mark(0)]] = 0 >= 0 = [[0]] [[a!6220!6220U11(_x0, _x1, _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[a!6220!6220U12(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[U12(_x0, _x1)]] [[a!6220!6220isNat(_x0)]] = x0 >= x0 = [[isNat(_x0)]] [[a!6220!6220U13(_x0)]] = x0 >= x0 = [[U13(_x0)]] [[a!6220!6220U21(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U31(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[U31(_x0, _x1)]] [[a!6220!6220U41(_x0, _x1, _x2)]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U41(_x0, _x1, _x2)]] [[a!6220!6220plus(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isNatKind(_x0)]] = x0 >= x0 = [[isNatKind(_x0)]] We can thus remove the following rules: a!6220!6220U12(tt, X) => a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U41(tt, X, Y) => s(a!6220!6220plus(mark(Y), mark(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]): a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) mark(U11(X, Y, Z)) >? a!6220!6220U11(mark(X), Y, Z) mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(isNat(X)) >? a!6220!6220isNat(X) mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X, Y)) >? a!6220!6220U31(mark(X), Y) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt mark(s(X)) >? s(mark(X)) mark(0) >? 0 a!6220!6220U11(X, Y, Z) >? U11(X, Y, Z) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220isNat(X) >? isNat(X) a!6220!6220U13(X) >? U13(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X, Y) >? U31(X, Y) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isNatKind(X) >? isNatKind(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 2 U11 = \y0y1y2.1 + y0 + y2 + 2y1 U12 = \y0y1.y0 + y1 U13 = \y0.y0 U21 = \y0y1.2 + y0 + y1 U22 = \y0.y0 U31 = \y0y1.1 + y0 + y1 U41 = \y0y1y2.y0 + y1 + y2 a!6220!6220U11 = \y0y1y2.1 + y0 + y2 + 2y1 a!6220!6220U12 = \y0y1.y0 + 2y1 a!6220!6220U13 = \y0.y0 a!6220!6220U21 = \y0y1.3 + y0 + y1 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0y1.1 + y0 + y1 a!6220!6220U41 = \y0y1y2.y0 + y1 + y2 a!6220!6220and = \y0y1.2 + y0 + y1 a!6220!6220isNat = \y0.1 + 2y0 a!6220!6220isNatKind = \y0.2y0 a!6220!6220plus = \y0y1.y0 + y1 and = \y0y1.2 + y0 + y1 isNat = \y0.1 + 2y0 isNatKind = \y0.2y0 mark = \y0.2y0 plus = \y0y1.y0 + y1 s = \y0.2y0 tt = 0 Using this interpretation, the requirements translate to: [[a!6220!6220isNatKind(s(_x0))]] = 4x0 >= 2x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(U11(_x0, _x1, _x2))]] = 2 + 2x0 + 2x2 + 4x1 > 1 + x2 + 2x0 + 2x1 = [[a!6220!6220U11(mark(_x0), _x1, _x2)]] [[mark(U12(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(isNat(_x0))]] = 2 + 4x0 > 1 + 2x0 = [[a!6220!6220isNat(_x0)]] [[mark(U13(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U13(mark(_x0))]] [[mark(U21(_x0, _x1))]] = 4 + 2x0 + 2x1 > 3 + x1 + 2x0 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U31(_x0, _x1))]] = 2 + 2x0 + 2x1 > 1 + x1 + 2x0 = [[a!6220!6220U31(mark(_x0), _x1)]] [[mark(U41(_x0, _x1, _x2))]] = 2x0 + 2x1 + 2x2 >= x1 + x2 + 2x0 = [[a!6220!6220U41(mark(_x0), _x1, _x2)]] [[mark(plus(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = 4 + 2x0 + 2x1 > 2 + x1 + 2x0 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isNatKind(_x0))]] = 4x0 >= 2x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(s(_x0))]] = 4x0 >= 4x0 = [[s(mark(_x0))]] [[mark(0)]] = 4 > 2 = [[0]] [[a!6220!6220U11(_x0, _x1, _x2)]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[U11(_x0, _x1, _x2)]] [[a!6220!6220U12(_x0, _x1)]] = x0 + 2x1 >= x0 + x1 = [[U12(_x0, _x1)]] [[a!6220!6220isNat(_x0)]] = 1 + 2x0 >= 1 + 2x0 = [[isNat(_x0)]] [[a!6220!6220U13(_x0)]] = x0 >= x0 = [[U13(_x0)]] [[a!6220!6220U21(_x0, _x1)]] = 3 + x0 + x1 > 2 + x0 + x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U31(_x0, _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[U31(_x0, _x1)]] [[a!6220!6220U41(_x0, _x1, _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U41(_x0, _x1, _x2)]] [[a!6220!6220plus(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isNatKind(_x0)]] = 2x0 >= 2x0 = [[isNatKind(_x0)]] We can thus remove the following rules: mark(U11(X, Y, Z)) => a!6220!6220U11(mark(X), Y, Z) mark(isNat(X)) => a!6220!6220isNat(X) mark(U21(X, Y)) => a!6220!6220U21(mark(X), Y) mark(U31(X, Y)) => a!6220!6220U31(mark(X), Y) mark(and(X, Y)) => a!6220!6220and(mark(X), Y) mark(0) => 0 a!6220!6220U21(X, Y) => U21(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]): a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt mark(s(X)) >? s(mark(X)) a!6220!6220U11(X, Y, Z) >? U11(X, Y, Z) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220isNat(X) >? isNat(X) a!6220!6220U13(X) >? U13(X) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X, Y) >? U31(X, Y) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isNatKind(X) >? isNatKind(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U11 = \y0y1y2.y0 + y1 + y2 U12 = \y0y1.y0 + y1 U13 = \y0.y0 U22 = \y0.y0 U31 = \y0y1.y0 + y1 U41 = \y0y1y2.y0 + y1 + 2y2 a!6220!6220U11 = \y0y1y2.3 + 3y0 + 3y1 + 3y2 a!6220!6220U12 = \y0y1.y0 + 2y1 a!6220!6220U13 = \y0.y0 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0y1.3 + 3y0 + 3y1 a!6220!6220U41 = \y0y1y2.y0 + y1 + 2y2 a!6220!6220and = \y0y1.3 + 3y0 + 3y1 a!6220!6220isNat = \y0.3 + 3y0 a!6220!6220isNatKind = \y0.2y0 a!6220!6220plus = \y0y1.y0 + y1 and = \y0y1.y0 + y1 isNat = \y0.y0 isNatKind = \y0.y0 mark = \y0.2y0 plus = \y0y1.y0 + y1 s = \y0.1 + 2y0 tt = 1 Using this interpretation, the requirements translate to: [[a!6220!6220isNatKind(s(_x0))]] = 2 + 4x0 > 2x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(U12(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(U13(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U13(mark(_x0))]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U41(_x0, _x1, _x2))]] = 2x0 + 2x1 + 4x2 >= x1 + 2x0 + 2x2 = [[a!6220!6220U41(mark(_x0), _x1, _x2)]] [[mark(plus(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] [[mark(isNatKind(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(tt)]] = 2 > 1 = [[tt]] [[mark(s(_x0))]] = 2 + 4x0 > 1 + 4x0 = [[s(mark(_x0))]] [[a!6220!6220U11(_x0, _x1, _x2)]] = 3 + 3x0 + 3x1 + 3x2 > x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[a!6220!6220U12(_x0, _x1)]] = x0 + 2x1 >= x0 + x1 = [[U12(_x0, _x1)]] [[a!6220!6220isNat(_x0)]] = 3 + 3x0 > x0 = [[isNat(_x0)]] [[a!6220!6220U13(_x0)]] = x0 >= x0 = [[U13(_x0)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U31(_x0, _x1)]] = 3 + 3x0 + 3x1 > x0 + x1 = [[U31(_x0, _x1)]] [[a!6220!6220U41(_x0, _x1, _x2)]] = x0 + x1 + 2x2 >= x0 + x1 + 2x2 = [[U41(_x0, _x1, _x2)]] [[a!6220!6220plus(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = 3 + 3x0 + 3x1 > x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isNatKind(_x0)]] = 2x0 >= x0 = [[isNatKind(_x0)]] We can thus remove the following rules: a!6220!6220isNatKind(s(X)) => a!6220!6220isNatKind(X) mark(tt) => tt mark(s(X)) => s(mark(X)) a!6220!6220U11(X, Y, Z) => U11(X, Y, Z) a!6220!6220isNat(X) => isNat(X) a!6220!6220U31(X, Y) => U31(X, Y) a!6220!6220and(X, Y) => and(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]): mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220U13(X) >? U13(X) a!6220!6220U22(X) >? U22(X) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220isNatKind(X) >? isNatKind(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U12 = \y0y1.2 + y0 + y1 U13 = \y0.y0 U22 = \y0.1 + y0 U41 = \y0y1y2.y0 + y1 + y2 a!6220!6220U12 = \y0y1.2 + y0 + 2y1 a!6220!6220U13 = \y0.y0 a!6220!6220U22 = \y0.1 + y0 a!6220!6220U41 = \y0y1y2.y0 + y1 + y2 a!6220!6220isNatKind = \y0.2 + 2y0 a!6220!6220plus = \y0y1.3 + y0 + y1 isNatKind = \y0.2 + y0 mark = \y0.2y0 plus = \y0y1.2 + y0 + y1 Using this interpretation, the requirements translate to: [[mark(U12(_x0, _x1))]] = 4 + 2x0 + 2x1 > 2 + 2x0 + 2x1 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(U13(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U13(mark(_x0))]] [[mark(U22(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U41(_x0, _x1, _x2))]] = 2x0 + 2x1 + 2x2 >= x1 + x2 + 2x0 = [[a!6220!6220U41(mark(_x0), _x1, _x2)]] [[mark(plus(_x0, _x1))]] = 4 + 2x0 + 2x1 > 3 + 2x0 + 2x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] [[mark(isNatKind(_x0))]] = 4 + 2x0 > 2 + 2x0 = [[a!6220!6220isNatKind(_x0)]] [[a!6220!6220U12(_x0, _x1)]] = 2 + x0 + 2x1 >= 2 + x0 + x1 = [[U12(_x0, _x1)]] [[a!6220!6220U13(_x0)]] = x0 >= x0 = [[U13(_x0)]] [[a!6220!6220U22(_x0)]] = 1 + x0 >= 1 + x0 = [[U22(_x0)]] [[a!6220!6220U41(_x0, _x1, _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U41(_x0, _x1, _x2)]] [[a!6220!6220plus(_x0, _x1)]] = 3 + x0 + x1 > 2 + x0 + x1 = [[plus(_x0, _x1)]] [[a!6220!6220isNatKind(_x0)]] = 2 + 2x0 >= 2 + x0 = [[isNatKind(_x0)]] We can thus remove the following rules: mark(U12(X, Y)) => a!6220!6220U12(mark(X), Y) mark(U22(X)) => a!6220!6220U22(mark(X)) mark(plus(X, Y)) => a!6220!6220plus(mark(X), mark(Y)) mark(isNatKind(X)) => a!6220!6220isNatKind(X) a!6220!6220plus(X, Y) => plus(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]): mark(U13(X)) >? a!6220!6220U13(mark(X)) mark(U41(X, Y, Z)) >? a!6220!6220U41(mark(X), Y, Z) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220U13(X) >? U13(X) a!6220!6220U22(X) >? U22(X) a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) a!6220!6220isNatKind(X) >? isNatKind(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U12 = \y0y1.y0 + y1 U13 = \y0.2 + y0 U22 = \y0.y0 U41 = \y0y1y2.1 + y0 + y1 + y2 a!6220!6220U12 = \y0y1.3 + y0 + y1 a!6220!6220U13 = \y0.3 + y0 a!6220!6220U22 = \y0.3 + y0 a!6220!6220U41 = \y0y1y2.1 + y0 + y1 + y2 a!6220!6220isNatKind = \y0.3 + 2y0 isNatKind = \y0.y0 mark = \y0.3 + 2y0 Using this interpretation, the requirements translate to: [[mark(U13(_x0))]] = 7 + 2x0 > 6 + 2x0 = [[a!6220!6220U13(mark(_x0))]] [[mark(U41(_x0, _x1, _x2))]] = 5 + 2x0 + 2x1 + 2x2 > 4 + x1 + x2 + 2x0 = [[a!6220!6220U41(mark(_x0), _x1, _x2)]] [[a!6220!6220U12(_x0, _x1)]] = 3 + x0 + x1 > x0 + x1 = [[U12(_x0, _x1)]] [[a!6220!6220U13(_x0)]] = 3 + x0 > 2 + x0 = [[U13(_x0)]] [[a!6220!6220U22(_x0)]] = 3 + x0 > x0 = [[U22(_x0)]] [[a!6220!6220U41(_x0, _x1, _x2)]] = 1 + x0 + x1 + x2 >= 1 + x0 + x1 + x2 = [[U41(_x0, _x1, _x2)]] [[a!6220!6220isNatKind(_x0)]] = 3 + 2x0 > x0 = [[isNatKind(_x0)]] We can thus remove the following rules: mark(U13(X)) => a!6220!6220U13(mark(X)) mark(U41(X, Y, Z)) => a!6220!6220U41(mark(X), Y, Z) a!6220!6220U12(X, Y) => U12(X, Y) a!6220!6220U13(X) => U13(X) a!6220!6220U22(X) => U22(X) a!6220!6220isNatKind(X) => isNatKind(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]): a!6220!6220U41(X, Y, Z) >? U41(X, Y, Z) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U41 = \y0y1y2.y0 + y1 + y2 a!6220!6220U41 = \y0y1y2.3 + 3y0 + 3y1 + 3y2 Using this interpretation, the requirements translate to: [[a!6220!6220U41(_x0, _x1, _x2)]] = 3 + 3x0 + 3x1 + 3x2 > x0 + x1 + x2 = [[U41(_x0, _x1, _x2)]] We can thus remove the following rules: a!6220!6220U41(X, Y, Z) => U41(X, Y, Z) 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.