/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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] --> o U32 : [o * o] --> o U33 : [o] --> o U41 : [o * o] --> o U51 : [o * o * o] --> o U61 : [o] --> o U71 : [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] --> o a!6220!6220U32 : [o * o] --> o a!6220!6220U33 : [o] --> o a!6220!6220U41 : [o * o] --> o a!6220!6220U51 : [o * o * o] --> o a!6220!6220U61 : [o] --> o a!6220!6220U71 : [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 a!6220!6220x : [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 x : [o * o] --> 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, Y) => a!6220!6220U32(a!6220!6220isNat(X), Y) a!6220!6220U32(tt, X) => a!6220!6220U33(a!6220!6220isNat(X)) a!6220!6220U33(tt) => tt a!6220!6220U41(tt, X) => mark(X) a!6220!6220U51(tt, X, Y) => s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220U61(tt) => 0 a!6220!6220U71(tt, X, Y) => a!6220!6220plus(a!6220!6220x(mark(Y), mark(X)), mark(Y)) 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!6220isNat(x(X, Y)) => a!6220!6220U31(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) 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!6220isNatKind(x(X, Y)) => a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) a!6220!6220plus(X, 0) => a!6220!6220U41(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) => a!6220!6220U51(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) a!6220!6220x(X, 0) => a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X))) a!6220!6220x(X, s(Y)) => a!6220!6220U71(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, Z)) => a!6220!6220U31(mark(X), Y, Z) mark(U32(X, Y)) => a!6220!6220U32(mark(X), Y) mark(U33(X)) => a!6220!6220U33(mark(X)) mark(U41(X, Y)) => a!6220!6220U41(mark(X), Y) mark(U51(X, Y, Z)) => a!6220!6220U51(mark(X), Y, Z) mark(plus(X, Y)) => a!6220!6220plus(mark(X), mark(Y)) mark(U61(X)) => a!6220!6220U61(mark(X)) mark(U71(X, Y, Z)) => a!6220!6220U71(mark(X), Y, Z) mark(x(X, Y)) => a!6220!6220x(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, Z) => U31(X, Y, Z) a!6220!6220U32(X, Y) => U32(X, Y) a!6220!6220U33(X) => U33(X) a!6220!6220U41(X, Y) => U41(X, Y) a!6220!6220U51(X, Y, Z) => U51(X, Y, Z) a!6220!6220plus(X, Y) => plus(X, Y) a!6220!6220U61(X) => U61(X) a!6220!6220U71(X, Y, Z) => U71(X, Y, Z) a!6220!6220x(X, Y) => x(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, Y) >? a!6220!6220U32(a!6220!6220isNat(X), Y) a!6220!6220U32(tt, X) >? a!6220!6220U33(a!6220!6220isNat(X)) a!6220!6220U33(tt) >? tt a!6220!6220U41(tt, X) >? mark(X) a!6220!6220U51(tt, X, Y) >? s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220U61(tt) >? 0 a!6220!6220U71(tt, X, Y) >? a!6220!6220plus(a!6220!6220x(mark(Y), mark(X)), mark(Y)) 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!6220isNat(x(X, Y)) >? a!6220!6220U31(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) 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!6220isNatKind(x(X, Y)) >? a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) a!6220!6220plus(X, 0) >? a!6220!6220U41(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) >? a!6220!6220U51(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) a!6220!6220x(X, 0) >? a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X))) a!6220!6220x(X, s(Y)) >? a!6220!6220U71(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, Z)) >? a!6220!6220U31(mark(X), Y, Z) mark(U32(X, Y)) >? a!6220!6220U32(mark(X), Y) mark(U33(X)) >? a!6220!6220U33(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U51(X, Y, Z)) >? a!6220!6220U51(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U71(X, Y, Z)) >? a!6220!6220U71(mark(X), Y, Z) mark(x(X, Y)) >? a!6220!6220x(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, Z) >? U31(X, Y, Z) a!6220!6220U32(X, Y) >? U32(X, Y) a!6220!6220U33(X) >? U33(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U51(X, Y, Z) >? U51(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y, Z) >? U71(X, Y, Z) a!6220!6220x(X, Y) >? x(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 [[U51(x_1, x_2, x_3)]] = U51(x_2, x_3, x_1) [[U71(x_1, x_2, x_3)]] = U71(x_2, x_3, x_1) [[a!6220!6220U13(x_1)]] = x_1 [[a!6220!6220U22(x_1)]] = x_1 [[a!6220!6220U51(x_1, x_2, x_3)]] = a!6220!6220U51(x_2, x_3, x_1) [[a!6220!6220U71(x_1, x_2, x_3)]] = a!6220!6220U71(x_2, x_3, x_1) [[a!6220!6220isNatKind(x_1)]] = x_1 [[a!6220!6220plus(x_1, x_2)]] = a!6220!6220plus(x_2, x_1) [[a!6220!6220x(x_1, x_2)]] = a!6220!6220x(x_2, x_1) [[isNatKind(x_1)]] = x_1 [[mark(x_1)]] = x_1 [[plus(x_1, x_2)]] = plus(x_2, x_1) [[tt]] = _|_ [[x(x_1, x_2)]] = x(x_2, x_1) We choose Lex = {U51, U71, a!6220!6220U51, a!6220!6220U71, a!6220!6220plus, a!6220!6220x, plus, x} and Mul = {U11, U12, U21, U31, U32, U33, U41, U61, a!6220!6220U11, a!6220!6220U12, a!6220!6220U21, a!6220!6220U31, a!6220!6220U32, a!6220!6220U33, a!6220!6220U41, a!6220!6220U61, a!6220!6220and, a!6220!6220isNat, and, isNat, s}, and the following precedence: U71 = a!6220!6220U71 = a!6220!6220x = x > U51 = a!6220!6220U51 = a!6220!6220plus = plus > U31 = a!6220!6220U31 > U61 = a!6220!6220U61 > U32 = a!6220!6220U32 > U33 = a!6220!6220U33 > U11 = a!6220!6220U11 > U41 = a!6220!6220U41 > s > a!6220!6220and = and > U21 = a!6220!6220U21 > U12 = a!6220!6220U12 > 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!6220U31(_|_, X, Y) >= a!6220!6220U32(a!6220!6220isNat(X), Y) a!6220!6220U32(_|_, X) > a!6220!6220U33(a!6220!6220isNat(X)) a!6220!6220U33(_|_) > _|_ a!6220!6220U41(_|_, X) >= X a!6220!6220U51(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) a!6220!6220U61(_|_) >= _|_ a!6220!6220U71(_|_, X, Y) > a!6220!6220plus(a!6220!6220x(Y, X), Y) a!6220!6220and(_|_, X) >= X a!6220!6220isNat(_|_) >= _|_ 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) a!6220!6220isNat(x(X, Y)) > a!6220!6220U31(a!6220!6220and(X, Y), X, Y) _|_ >= _|_ plus(X, Y) > a!6220!6220and(X, Y) s(X) >= X x(X, Y) > a!6220!6220and(X, Y) a!6220!6220plus(X, _|_) >= a!6220!6220U41(a!6220!6220and(a!6220!6220isNat(X), X), X) a!6220!6220plus(X, s(Y)) >= a!6220!6220U51(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)), Y, X) a!6220!6220x(X, _|_) >= a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(X), X)) a!6220!6220x(X, s(Y)) >= a!6220!6220U71(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, Z) >= a!6220!6220U31(X, Y, Z) U32(X, Y) >= a!6220!6220U32(X, Y) U33(X) >= a!6220!6220U33(X) U41(X, Y) >= a!6220!6220U41(X, Y) U51(X, Y, Z) >= a!6220!6220U51(X, Y, Z) plus(X, Y) >= a!6220!6220plus(X, Y) U61(X) >= a!6220!6220U61(X) U71(X, Y, Z) >= a!6220!6220U71(X, Y, Z) x(X, Y) >= a!6220!6220x(X, Y) and(X, Y) >= a!6220!6220and(X, Y) X >= X _|_ >= _|_ 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, Z) >= U31(X, Y, Z) a!6220!6220U32(X, Y) >= U32(X, Y) a!6220!6220U33(X) >= U33(X) a!6220!6220U41(X, Y) >= U41(X, Y) a!6220!6220U51(X, Y, Z) >= U51(X, Y, Z) a!6220!6220plus(X, Y) >= plus(X, Y) a!6220!6220U61(X) >= U61(X) a!6220!6220U71(X, Y, Z) >= U71(X, Y, Z) a!6220!6220x(X, Y) >= x(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 definition 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 definition 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!6220U31(_|_, X, Y) >= a!6220!6220U32(a!6220!6220isNat(X), Y) because [17], by (Star) 17] a!6220!6220U31*(_|_, X, Y) >= a!6220!6220U32(a!6220!6220isNat(X), Y) because a!6220!6220U31 > a!6220!6220U32, [18] and [20], by (Copy) 18] a!6220!6220U31*(_|_, X, Y) >= a!6220!6220isNat(X) because a!6220!6220U31 > a!6220!6220isNat and [19], by (Copy) 19] a!6220!6220U31*(_|_, X, Y) >= X because [5], by (Select) 20] a!6220!6220U31*(_|_, X, Y) >= Y because [7], by (Select) 21] a!6220!6220U32(_|_, X) > a!6220!6220U33(a!6220!6220isNat(X)) because [22], by definition 22] a!6220!6220U32*(_|_, X) >= a!6220!6220U33(a!6220!6220isNat(X)) because a!6220!6220U32 > a!6220!6220U33 and [23], by (Copy) 23] a!6220!6220U32*(_|_, X) >= a!6220!6220isNat(X) because a!6220!6220U32 > a!6220!6220isNat and [24], by (Copy) 24] a!6220!6220U32*(_|_, X) >= X because [7], by (Select) 25] a!6220!6220U33(_|_) > _|_ because [26], by definition 26] a!6220!6220U33*(_|_) >= _|_ by (Bot) 27] a!6220!6220U41(_|_, X) >= X because [28], by (Star) 28] a!6220!6220U41*(_|_, X) >= X because [29], by (Select) 29] X >= X by (Meta) 30] a!6220!6220U51(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) because [31], by (Star) 31] a!6220!6220U51*(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) because a!6220!6220U51 > s and [32], by (Copy) 32] a!6220!6220U51*(_|_, X, Y) >= a!6220!6220plus(Y, X) because a!6220!6220U51 = a!6220!6220plus, [33], [34], [35] and [36], by (Stat) 33] X >= X by (Meta) 34] Y >= Y by (Meta) 35] a!6220!6220U51*(_|_, X, Y) >= Y because [34], by (Select) 36] a!6220!6220U51*(_|_, X, Y) >= X because [33], by (Select) 37] a!6220!6220U61(_|_) >= _|_ by (Bot) 38] a!6220!6220U71(_|_, X, Y) > a!6220!6220plus(a!6220!6220x(Y, X), Y) because [39], by definition 39] a!6220!6220U71*(_|_, X, Y) >= a!6220!6220plus(a!6220!6220x(Y, X), Y) because a!6220!6220U71 > a!6220!6220plus, [40] and [41], by (Copy) 40] a!6220!6220U71*(_|_, X, Y) >= a!6220!6220x(Y, X) because a!6220!6220U71 = a!6220!6220x, [33], [34], [41] and [42], by (Stat) 41] a!6220!6220U71*(_|_, X, Y) >= Y because [34], by (Select) 42] a!6220!6220U71*(_|_, X, Y) >= X because [33], by (Select) 43] a!6220!6220and(_|_, X) >= X because [44], by (Star) 44] a!6220!6220and*(_|_, X) >= X because [45], by (Select) 45] X >= X by (Meta) 46] a!6220!6220isNat(_|_) >= _|_ by (Bot) 47] a!6220!6220isNat(plus(X, Y)) >= a!6220!6220U11(a!6220!6220and(X, Y), X, Y) because [48], by (Star) 48] a!6220!6220isNat*(plus(X, Y)) >= a!6220!6220U11(a!6220!6220and(X, Y), X, Y) because [49], by (Select) 49] plus(X, Y) >= a!6220!6220U11(a!6220!6220and(X, Y), X, Y) because [50], by (Star) 50] plus*(X, Y) >= a!6220!6220U11(a!6220!6220and(X, Y), X, Y) because plus > a!6220!6220U11, [51], [52] and [53], by (Copy) 51] plus*(X, Y) >= a!6220!6220and(X, Y) because plus > a!6220!6220and, [52] and [53], by (Copy) 52] plus*(X, Y) >= X because [5], by (Select) 53] plus*(X, Y) >= Y because [7], by (Select) 54] a!6220!6220isNat(s(X)) > a!6220!6220U21(X, X) because [55], by definition 55] a!6220!6220isNat*(s(X)) >= a!6220!6220U21(X, X) because [56], by (Select) 56] s(X) >= a!6220!6220U21(X, X) because [57], by (Star) 57] s*(X) >= a!6220!6220U21(X, X) because s > a!6220!6220U21, [58] and [58], by (Copy) 58] s*(X) >= X because [5], by (Select) 59] a!6220!6220isNat(x(X, Y)) > a!6220!6220U31(a!6220!6220and(X, Y), X, Y) because [60], by definition 60] a!6220!6220isNat*(x(X, Y)) >= a!6220!6220U31(a!6220!6220and(X, Y), X, Y) because [61], by (Select) 61] x(X, Y) >= a!6220!6220U31(a!6220!6220and(X, Y), X, Y) because [62], by (Star) 62] x*(X, Y) >= a!6220!6220U31(a!6220!6220and(X, Y), X, Y) because x > a!6220!6220U31, [63], [64] and [65], by (Copy) 63] x*(X, Y) >= a!6220!6220and(X, Y) because x > a!6220!6220and, [64] and [65], by (Copy) 64] x*(X, Y) >= X because [5], by (Select) 65] x*(X, Y) >= Y because [7], by (Select) 66] _|_ >= _|_ by (Bot) 67] plus(X, Y) > a!6220!6220and(X, Y) because [51], by definition 68] s(X) >= X because [58], by (Star) 69] x(X, Y) > a!6220!6220and(X, Y) because [63], by definition 70] a!6220!6220plus(X, _|_) >= a!6220!6220U41(a!6220!6220and(a!6220!6220isNat(X), X), X) because [71], by (Star) 71] a!6220!6220plus*(X, _|_) >= a!6220!6220U41(a!6220!6220and(a!6220!6220isNat(X), X), X) because a!6220!6220plus > a!6220!6220U41, [72] and [75], by (Copy) 72] a!6220!6220plus*(X, _|_) >= a!6220!6220and(a!6220!6220isNat(X), X) because a!6220!6220plus > a!6220!6220and, [73] and [75], by (Copy) 73] a!6220!6220plus*(X, _|_) >= a!6220!6220isNat(X) because a!6220!6220plus > a!6220!6220isNat and [74], by (Copy) 74] a!6220!6220plus*(X, _|_) >= X because [34], by (Select) 75] a!6220!6220plus*(X, _|_) >= X because [34], by (Select) 76] a!6220!6220plus(X, s(Y)) >= a!6220!6220U51(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)), Y, X) because [77], by (Star) 77] a!6220!6220plus*(X, s(Y)) >= a!6220!6220U51(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)), Y, X) because a!6220!6220plus = a!6220!6220U51, [78], [80], [85] and [89], by (Stat) 78] s(Y) > Y because [79], by definition 79] s*(Y) >= Y because [33], by (Select) 80] 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, [81] and [86], by (Copy) 81] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220isNat(Y), Y) because a!6220!6220plus > a!6220!6220and, [82] and [85], by (Copy) 82] a!6220!6220plus*(X, s(Y)) >= a!6220!6220isNat(Y) because a!6220!6220plus > a!6220!6220isNat and [83], by (Copy) 83] a!6220!6220plus*(X, s(Y)) >= Y because [84], by (Select) 84] s(Y) >= Y because [79], by (Star) 85] a!6220!6220plus*(X, s(Y)) >= Y because [84], by (Select) 86] a!6220!6220plus*(X, s(Y)) >= and(isNat(X), X) because a!6220!6220plus > and, [87] and [89], by (Copy) 87] a!6220!6220plus*(X, s(Y)) >= isNat(X) because a!6220!6220plus > isNat and [88], by (Copy) 88] a!6220!6220plus*(X, s(Y)) >= X because [34], by (Select) 89] a!6220!6220plus*(X, s(Y)) >= X because [34], by (Select) 90] a!6220!6220x(X, _|_) >= a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(X), X)) because [91], by (Star) 91] a!6220!6220x*(X, _|_) >= a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(X), X)) because a!6220!6220x > a!6220!6220U61 and [92], by (Copy) 92] a!6220!6220x*(X, _|_) >= a!6220!6220and(a!6220!6220isNat(X), X) because a!6220!6220x > a!6220!6220and, [93] and [95], by (Copy) 93] a!6220!6220x*(X, _|_) >= a!6220!6220isNat(X) because a!6220!6220x > a!6220!6220isNat and [94], by (Copy) 94] a!6220!6220x*(X, _|_) >= X because [34], by (Select) 95] a!6220!6220x*(X, _|_) >= X because [34], by (Select) 96] a!6220!6220x(X, s(Y)) >= a!6220!6220U71(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)), Y, X) because [97], by (Star) 97] a!6220!6220x*(X, s(Y)) >= a!6220!6220U71(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)), Y, X) because a!6220!6220x = a!6220!6220U71, [78], [98], [102] and [106], by (Stat) 98] a!6220!6220x*(X, s(Y)) >= a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), Y), and(isNat(X), X)) because a!6220!6220x > a!6220!6220and, [99] and [103], by (Copy) 99] a!6220!6220x*(X, s(Y)) >= a!6220!6220and(a!6220!6220isNat(Y), Y) because a!6220!6220x > a!6220!6220and, [100] and [102], by (Copy) 100] a!6220!6220x*(X, s(Y)) >= a!6220!6220isNat(Y) because a!6220!6220x > a!6220!6220isNat and [101], by (Copy) 101] a!6220!6220x*(X, s(Y)) >= Y because [84], by (Select) 102] a!6220!6220x*(X, s(Y)) >= Y because [84], by (Select) 103] a!6220!6220x*(X, s(Y)) >= and(isNat(X), X) because a!6220!6220x > and, [104] and [106], by (Copy) 104] a!6220!6220x*(X, s(Y)) >= isNat(X) because a!6220!6220x > isNat and [105], by (Copy) 105] a!6220!6220x*(X, s(Y)) >= X because [34], by (Select) 106] a!6220!6220x*(X, s(Y)) >= X because [34], by (Select) 107] U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) because U11 = a!6220!6220U11, U11 in Mul, [108], [109] and [110], by (Fun) 108] X >= X by (Meta) 109] Y >= Y by (Meta) 110] Z >= Z by (Meta) 111] U12(X, Y) >= a!6220!6220U12(X, Y) because U12 = a!6220!6220U12, U12 in Mul, [108] and [109], by (Fun) 112] isNat(X) >= a!6220!6220isNat(X) because isNat = a!6220!6220isNat, isNat in Mul and [113], by (Fun) 113] X >= X by (Meta) 114] X >= X by (Meta) 115] U21(X, Y) >= a!6220!6220U21(X, Y) because U21 = a!6220!6220U21, U21 in Mul, [108] and [109], by (Fun) 116] X >= X by (Meta) 117] U31(X, Y, Z) >= a!6220!6220U31(X, Y, Z) because U31 = a!6220!6220U31, U31 in Mul, [108], [109] and [110], by (Fun) 118] U32(X, Y) >= a!6220!6220U32(X, Y) because U32 = a!6220!6220U32, U32 in Mul, [108] and [109], by (Fun) 119] U33(X) >= a!6220!6220U33(X) because U33 = a!6220!6220U33, U33 in Mul and [120], by (Fun) 120] X >= X by (Meta) 121] U41(X, Y) >= a!6220!6220U41(X, Y) because U41 = a!6220!6220U41, U41 in Mul, [108] and [109], by (Fun) 122] U51(X, Y, Z) >= a!6220!6220U51(X, Y, Z) because U51 = a!6220!6220U51, [108], [109] and [110], by (Fun) 123] plus(X, Y) >= a!6220!6220plus(X, Y) because plus = a!6220!6220plus, [108] and [124], by (Fun) 124] Y >= Y by (Meta) 125] U61(X) >= a!6220!6220U61(X) because U61 = a!6220!6220U61, U61 in Mul and [120], by (Fun) 126] U71(X, Y, Z) >= a!6220!6220U71(X, Y, Z) because U71 = a!6220!6220U71, [108], [124] and [110], by (Fun) 127] x(X, Y) >= a!6220!6220x(X, Y) because x = a!6220!6220x, [108] and [124], by (Fun) 128] and(X, Y) >= a!6220!6220and(X, Y) because and = a!6220!6220and, and in Mul, [108] and [124], by (Fun) 129] X >= X by (Meta) 130] _|_ >= _|_ by (Bot) 131] s(X) >= s(X) because s in Mul and [120], by (Fun) 132] _|_ >= _|_ by (Bot) 133] a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) because a!6220!6220U11 = U11, a!6220!6220U11 in Mul, [108], [124] and [110], by (Fun) 134] a!6220!6220U12(X, Y) >= U12(X, Y) because a!6220!6220U12 = U12, a!6220!6220U12 in Mul, [108] and [124], by (Fun) 135] a!6220!6220isNat(X) >= isNat(X) because a!6220!6220isNat = isNat, a!6220!6220isNat in Mul and [120], by (Fun) 136] X >= X by (Meta) 137] a!6220!6220U21(X, Y) >= U21(X, Y) because a!6220!6220U21 = U21, a!6220!6220U21 in Mul, [108] and [124], by (Fun) 138] X >= X by (Meta) 139] a!6220!6220U31(X, Y, Z) >= U31(X, Y, Z) because a!6220!6220U31 = U31, a!6220!6220U31 in Mul, [108], [124] and [110], by (Fun) 140] a!6220!6220U32(X, Y) >= U32(X, Y) because a!6220!6220U32 = U32, a!6220!6220U32 in Mul, [108] and [124], by (Fun) 141] a!6220!6220U33(X) >= U33(X) because a!6220!6220U33 = U33, a!6220!6220U33 in Mul and [120], by (Fun) 142] a!6220!6220U41(X, Y) >= U41(X, Y) because a!6220!6220U41 = U41, a!6220!6220U41 in Mul, [108] and [124], by (Fun) 143] a!6220!6220U51(X, Y, Z) >= U51(X, Y, Z) because a!6220!6220U51 = U51, [108], [124] and [110], by (Fun) 144] a!6220!6220plus(X, Y) >= plus(X, Y) because a!6220!6220plus = plus, [108] and [124], by (Fun) 145] a!6220!6220U61(X) >= U61(X) because a!6220!6220U61 = U61, a!6220!6220U61 in Mul and [120], by (Fun) 146] a!6220!6220U71(X, Y, Z) >= U71(X, Y, Z) because a!6220!6220U71 = U71, [108], [124] and [110], by (Fun) 147] a!6220!6220x(X, Y) >= x(X, Y) because a!6220!6220x = x, [108] and [124], by (Fun) 148] a!6220!6220and(X, Y) >= and(X, Y) because a!6220!6220and = and, a!6220!6220and in Mul, [108] and [124], by (Fun) 149] X >= X by (Meta) We can thus remove the following rules: a!6220!6220U11(tt, X, Y) => a!6220!6220U12(a!6220!6220isNat(X), Y) a!6220!6220U21(tt, X) => a!6220!6220U22(a!6220!6220isNat(X)) a!6220!6220U32(tt, X) => a!6220!6220U33(a!6220!6220isNat(X)) a!6220!6220U33(tt) => tt a!6220!6220U71(tt, X, Y) => a!6220!6220plus(a!6220!6220x(mark(Y), mark(X)), mark(Y)) a!6220!6220isNat(s(X)) => a!6220!6220U21(a!6220!6220isNatKind(X), X) a!6220!6220isNat(x(X, Y)) => a!6220!6220U31(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) a!6220!6220isNatKind(plus(X, Y)) => a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) a!6220!6220isNatKind(x(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!6220U12(tt, X) >? a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U13(tt) >? tt a!6220!6220U22(tt) >? tt a!6220!6220U31(tt, X, Y) >? a!6220!6220U32(a!6220!6220isNat(X), Y) a!6220!6220U41(tt, X) >? mark(X) a!6220!6220U51(tt, X, Y) >? s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220U61(tt) >? 0 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!6220isNatKind(0) >? tt a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) a!6220!6220plus(X, 0) >? a!6220!6220U41(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) >? a!6220!6220U51(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) a!6220!6220x(X, 0) >? a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X))) a!6220!6220x(X, s(Y)) >? a!6220!6220U71(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, Z)) >? a!6220!6220U31(mark(X), Y, Z) mark(U32(X, Y)) >? a!6220!6220U32(mark(X), Y) mark(U33(X)) >? a!6220!6220U33(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U51(X, Y, Z)) >? a!6220!6220U51(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U71(X, Y, Z)) >? a!6220!6220U71(mark(X), Y, Z) mark(x(X, Y)) >? a!6220!6220x(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, Z) >? U31(X, Y, Z) a!6220!6220U32(X, Y) >? U32(X, Y) a!6220!6220U33(X) >? U33(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U51(X, Y, Z) >? U51(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y, Z) >? U71(X, Y, Z) a!6220!6220x(X, Y) >? x(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]] = _|_ [[U22(x_1)]] = x_1 [[U51(x_1, x_2, x_3)]] = U51(x_2, x_3, x_1) [[a!6220!6220U22(x_1)]] = x_1 [[a!6220!6220U51(x_1, x_2, x_3)]] = a!6220!6220U51(x_2, x_3, x_1) [[a!6220!6220isNat(x_1)]] = x_1 [[a!6220!6220plus(x_1, x_2)]] = a!6220!6220plus(x_2, x_1) [[isNat(x_1)]] = x_1 [[mark(x_1)]] = x_1 [[plus(x_1, x_2)]] = plus(x_2, x_1) [[tt]] = _|_ We choose Lex = {U51, a!6220!6220U51, a!6220!6220plus, plus} and Mul = {U11, U12, U13, U21, U31, U32, U33, U41, U61, U71, a!6220!6220U11, a!6220!6220U12, a!6220!6220U13, a!6220!6220U21, a!6220!6220U31, a!6220!6220U32, a!6220!6220U33, a!6220!6220U41, a!6220!6220U61, a!6220!6220U71, a!6220!6220and, a!6220!6220isNatKind, a!6220!6220x, and, isNatKind, s, x}, and the following precedence: a!6220!6220x = x > U61 = a!6220!6220U61 > U33 = a!6220!6220U33 > U71 = a!6220!6220U71 > U51 = a!6220!6220U51 = a!6220!6220plus = plus > U11 = a!6220!6220U11 > a!6220!6220and = and > a!6220!6220isNatKind = isNatKind > U41 = a!6220!6220U41 > U31 = a!6220!6220U31 > s > U32 = a!6220!6220U32 > U12 = a!6220!6220U12 > U13 = a!6220!6220U13 > 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!6220U12(_|_, X) > a!6220!6220U13(X) a!6220!6220U13(_|_) > _|_ _|_ >= _|_ a!6220!6220U31(_|_, X, Y) >= a!6220!6220U32(X, Y) a!6220!6220U41(_|_, X) > X a!6220!6220U51(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) a!6220!6220U61(_|_) > _|_ a!6220!6220and(_|_, X) >= X _|_ >= _|_ plus(X, Y) > a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) a!6220!6220isNatKind(_|_) >= _|_ a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) a!6220!6220plus(X, _|_) >= a!6220!6220U41(a!6220!6220and(X, isNatKind(X)), X) a!6220!6220plus(X, s(Y)) >= a!6220!6220U51(a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))), Y, X) a!6220!6220x(X, _|_) >= a!6220!6220U61(a!6220!6220and(X, isNatKind(X))) a!6220!6220x(X, s(Y)) > a!6220!6220U71(a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))), Y, X) U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) U12(X, Y) >= a!6220!6220U12(X, Y) X >= X U13(X) >= a!6220!6220U13(X) U21(X, Y) >= a!6220!6220U21(X, Y) X >= X U31(X, Y, Z) >= a!6220!6220U31(X, Y, Z) U32(X, Y) >= a!6220!6220U32(X, Y) U33(X) >= a!6220!6220U33(X) U41(X, Y) >= a!6220!6220U41(X, Y) U51(X, Y, Z) >= a!6220!6220U51(X, Y, Z) plus(X, Y) >= a!6220!6220plus(X, Y) U61(X) >= a!6220!6220U61(X) U71(X, Y, Z) >= a!6220!6220U71(X, Y, Z) x(X, Y) >= a!6220!6220x(X, Y) and(X, Y) >= a!6220!6220and(X, Y) isNatKind(X) >= a!6220!6220isNatKind(X) _|_ >= _|_ s(X) >= s(X) _|_ >= _|_ a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) a!6220!6220U12(X, Y) >= U12(X, Y) X >= X a!6220!6220U13(X) >= U13(X) a!6220!6220U21(X, Y) >= U21(X, Y) X >= X a!6220!6220U31(X, Y, Z) >= U31(X, Y, Z) a!6220!6220U32(X, Y) >= U32(X, Y) a!6220!6220U33(X) >= U33(X) a!6220!6220U41(X, Y) >= U41(X, Y) a!6220!6220U51(X, Y, Z) >= U51(X, Y, Z) a!6220!6220plus(X, Y) >= plus(X, Y) a!6220!6220U61(X) >= U61(X) a!6220!6220U71(X, Y, Z) >= U71(X, Y, Z) a!6220!6220x(X, Y) >= x(X, Y) a!6220!6220and(X, Y) >= and(X, Y) a!6220!6220isNatKind(X) >= isNatKind(X) With these choices, we have: 1] a!6220!6220U12(_|_, X) > a!6220!6220U13(X) because [2], by definition 2] a!6220!6220U12*(_|_, X) >= a!6220!6220U13(X) because a!6220!6220U12 > a!6220!6220U13 and [3], by (Copy) 3] a!6220!6220U12*(_|_, X) >= X because [4], by (Select) 4] X >= X by (Meta) 5] a!6220!6220U13(_|_) > _|_ because [6], by definition 6] a!6220!6220U13*(_|_) >= _|_ by (Bot) 7] _|_ >= _|_ by (Bot) 8] a!6220!6220U31(_|_, X, Y) >= a!6220!6220U32(X, Y) because [9], by (Star) 9] a!6220!6220U31*(_|_, X, Y) >= a!6220!6220U32(X, Y) because a!6220!6220U31 > a!6220!6220U32, [10] and [12], by (Copy) 10] a!6220!6220U31*(_|_, X, Y) >= X because [11], by (Select) 11] X >= X by (Meta) 12] a!6220!6220U31*(_|_, X, Y) >= Y because [4], by (Select) 13] a!6220!6220U41(_|_, X) > X because [14], by definition 14] a!6220!6220U41*(_|_, X) >= X because [15], by (Select) 15] X >= X by (Meta) 16] a!6220!6220U51(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) because [17], by (Star) 17] a!6220!6220U51*(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) because a!6220!6220U51 > s and [18], by (Copy) 18] a!6220!6220U51*(_|_, X, Y) >= a!6220!6220plus(Y, X) because a!6220!6220U51 = a!6220!6220plus, [19], [20], [21] and [22], by (Stat) 19] X >= X by (Meta) 20] Y >= Y by (Meta) 21] a!6220!6220U51*(_|_, X, Y) >= Y because [20], by (Select) 22] a!6220!6220U51*(_|_, X, Y) >= X because [19], by (Select) 23] a!6220!6220U61(_|_) > _|_ because [24], by definition 24] a!6220!6220U61*(_|_) >= _|_ by (Bot) 25] a!6220!6220and(_|_, X) >= X because [26], by (Star) 26] a!6220!6220and*(_|_, X) >= X because [27], by (Select) 27] X >= X by (Meta) 28] _|_ >= _|_ by (Bot) 29] plus(X, Y) > a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) because [30], by definition 30] plus*(X, Y) >= a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) because plus > a!6220!6220U11, [31], [33] and [35], by (Copy) 31] plus*(X, Y) >= a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)) because plus > a!6220!6220and, [32] and [34], by (Copy) 32] plus*(X, Y) >= a!6220!6220isNatKind(X) because plus > a!6220!6220isNatKind and [33], by (Copy) 33] plus*(X, Y) >= X because [11], by (Select) 34] plus*(X, Y) >= isNatKind(Y) because plus > isNatKind and [35], by (Copy) 35] plus*(X, Y) >= Y because [4], by (Select) 36] a!6220!6220isNatKind(_|_) >= _|_ by (Bot) 37] a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) because a!6220!6220isNatKind in Mul and [38], by (Fun) 38] s(X) >= X because [39], by (Star) 39] s*(X) >= X because [11], by (Select) 40] a!6220!6220plus(X, _|_) >= a!6220!6220U41(a!6220!6220and(X, isNatKind(X)), X) because [41], by (Star) 41] a!6220!6220plus*(X, _|_) >= a!6220!6220U41(a!6220!6220and(X, isNatKind(X)), X) because a!6220!6220plus > a!6220!6220U41, [42] and [43], by (Copy) 42] a!6220!6220plus*(X, _|_) >= a!6220!6220and(X, isNatKind(X)) because a!6220!6220plus > a!6220!6220and, [43] and [44], by (Copy) 43] a!6220!6220plus*(X, _|_) >= X because [20], by (Select) 44] a!6220!6220plus*(X, _|_) >= isNatKind(X) because a!6220!6220plus > isNatKind and [43], by (Copy) 45] a!6220!6220plus(X, s(Y)) >= a!6220!6220U51(a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))), Y, X) because [46], by (Star) 46] a!6220!6220plus*(X, s(Y)) >= a!6220!6220U51(a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))), Y, X) because a!6220!6220plus = a!6220!6220U51, [47], [49], [51] and [55], by (Stat) 47] s(Y) > Y because [48], by definition 48] s*(Y) >= Y because [19], by (Select) 49] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))) because a!6220!6220plus > a!6220!6220and, [50] and [54], by (Copy) 50] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(Y, isNatKind(Y)) because a!6220!6220plus > a!6220!6220and, [51] and [53], by (Copy) 51] a!6220!6220plus*(X, s(Y)) >= Y because [52], by (Select) 52] s(Y) >= Y because [48], by (Star) 53] a!6220!6220plus*(X, s(Y)) >= isNatKind(Y) because a!6220!6220plus > isNatKind and [51], by (Copy) 54] a!6220!6220plus*(X, s(Y)) >= and(X, isNatKind(X)) because a!6220!6220plus > and, [55] and [56], by (Copy) 55] a!6220!6220plus*(X, s(Y)) >= X because [20], by (Select) 56] a!6220!6220plus*(X, s(Y)) >= isNatKind(X) because a!6220!6220plus > isNatKind and [55], by (Copy) 57] a!6220!6220x(X, _|_) >= a!6220!6220U61(a!6220!6220and(X, isNatKind(X))) because [58], by (Star) 58] a!6220!6220x*(X, _|_) >= a!6220!6220U61(a!6220!6220and(X, isNatKind(X))) because a!6220!6220x > a!6220!6220U61 and [59], by (Copy) 59] a!6220!6220x*(X, _|_) >= a!6220!6220and(X, isNatKind(X)) because a!6220!6220x > a!6220!6220and, [60] and [61], by (Copy) 60] a!6220!6220x*(X, _|_) >= X because [20], by (Select) 61] a!6220!6220x*(X, _|_) >= isNatKind(X) because a!6220!6220x > isNatKind and [60], by (Copy) 62] a!6220!6220x(X, s(Y)) > a!6220!6220U71(a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))), Y, X) because [63], by definition 63] a!6220!6220x*(X, s(Y)) >= a!6220!6220U71(a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))), Y, X) because a!6220!6220x > a!6220!6220U71, [64], [66] and [69], by (Copy) 64] a!6220!6220x*(X, s(Y)) >= a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))) because a!6220!6220x > a!6220!6220and, [65] and [68], by (Copy) 65] a!6220!6220x*(X, s(Y)) >= a!6220!6220and(Y, isNatKind(Y)) because a!6220!6220x > a!6220!6220and, [66] and [67], by (Copy) 66] a!6220!6220x*(X, s(Y)) >= Y because [52], by (Select) 67] a!6220!6220x*(X, s(Y)) >= isNatKind(Y) because a!6220!6220x > isNatKind and [66], by (Copy) 68] a!6220!6220x*(X, s(Y)) >= and(X, isNatKind(X)) because a!6220!6220x > and, [69] and [70], by (Copy) 69] a!6220!6220x*(X, s(Y)) >= X because [20], by (Select) 70] a!6220!6220x*(X, s(Y)) >= isNatKind(X) because a!6220!6220x > isNatKind and [69], by (Copy) 71] U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) because U11 = a!6220!6220U11, U11 in Mul, [72], [73] and [74], by (Fun) 72] X >= X by (Meta) 73] Y >= Y by (Meta) 74] Z >= Z by (Meta) 75] U12(X, Y) >= a!6220!6220U12(X, Y) because U12 = a!6220!6220U12, U12 in Mul, [72] and [73], by (Fun) 76] X >= X by (Meta) 77] U13(X) >= a!6220!6220U13(X) because U13 = a!6220!6220U13, U13 in Mul and [78], by (Fun) 78] X >= X by (Meta) 79] U21(X, Y) >= a!6220!6220U21(X, Y) because U21 = a!6220!6220U21, U21 in Mul, [72] and [73], by (Fun) 80] X >= X by (Meta) 81] U31(X, Y, Z) >= a!6220!6220U31(X, Y, Z) because U31 = a!6220!6220U31, U31 in Mul, [72], [73] and [74], by (Fun) 82] U32(X, Y) >= a!6220!6220U32(X, Y) because U32 = a!6220!6220U32, U32 in Mul, [72] and [73], by (Fun) 83] U33(X) >= a!6220!6220U33(X) because U33 = a!6220!6220U33, U33 in Mul and [78], by (Fun) 84] U41(X, Y) >= a!6220!6220U41(X, Y) because U41 = a!6220!6220U41, U41 in Mul, [72] and [73], by (Fun) 85] U51(X, Y, Z) >= a!6220!6220U51(X, Y, Z) because U51 = a!6220!6220U51, [72], [73] and [74], by (Fun) 86] plus(X, Y) >= a!6220!6220plus(X, Y) because plus = a!6220!6220plus, [72] and [87], by (Fun) 87] Y >= Y by (Meta) 88] U61(X) >= a!6220!6220U61(X) because U61 = a!6220!6220U61, U61 in Mul and [78], by (Fun) 89] U71(X, Y, Z) >= a!6220!6220U71(X, Y, Z) because U71 = a!6220!6220U71, U71 in Mul, [72], [87] and [74], by (Fun) 90] x(X, Y) >= a!6220!6220x(X, Y) because x = a!6220!6220x, x in Mul, [72] and [87], by (Fun) 91] and(X, Y) >= a!6220!6220and(X, Y) because and = a!6220!6220and, and in Mul, [72] and [87], by (Fun) 92] isNatKind(X) >= a!6220!6220isNatKind(X) because isNatKind = a!6220!6220isNatKind, isNatKind in Mul and [78], by (Fun) 93] _|_ >= _|_ by (Bot) 94] s(X) >= s(X) because s in Mul and [78], by (Fun) 95] _|_ >= _|_ by (Bot) 96] a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) because a!6220!6220U11 = U11, a!6220!6220U11 in Mul, [72], [87] and [74], by (Fun) 97] a!6220!6220U12(X, Y) >= U12(X, Y) because a!6220!6220U12 = U12, a!6220!6220U12 in Mul, [72] and [87], by (Fun) 98] X >= X by (Meta) 99] a!6220!6220U13(X) >= U13(X) because a!6220!6220U13 = U13, a!6220!6220U13 in Mul and [78], by (Fun) 100] a!6220!6220U21(X, Y) >= U21(X, Y) because a!6220!6220U21 = U21, a!6220!6220U21 in Mul, [72] and [87], by (Fun) 101] X >= X by (Meta) 102] a!6220!6220U31(X, Y, Z) >= U31(X, Y, Z) because a!6220!6220U31 = U31, a!6220!6220U31 in Mul, [72], [87] and [74], by (Fun) 103] a!6220!6220U32(X, Y) >= U32(X, Y) because a!6220!6220U32 = U32, a!6220!6220U32 in Mul, [72] and [87], by (Fun) 104] a!6220!6220U33(X) >= U33(X) because a!6220!6220U33 = U33, a!6220!6220U33 in Mul and [78], by (Fun) 105] a!6220!6220U41(X, Y) >= U41(X, Y) because a!6220!6220U41 = U41, a!6220!6220U41 in Mul, [72] and [87], by (Fun) 106] a!6220!6220U51(X, Y, Z) >= U51(X, Y, Z) because a!6220!6220U51 = U51, [72], [87] and [74], by (Fun) 107] a!6220!6220plus(X, Y) >= plus(X, Y) because a!6220!6220plus = plus, [72] and [87], by (Fun) 108] a!6220!6220U61(X) >= U61(X) because a!6220!6220U61 = U61, a!6220!6220U61 in Mul and [78], by (Fun) 109] a!6220!6220U71(X, Y, Z) >= U71(X, Y, Z) because a!6220!6220U71 = U71, a!6220!6220U71 in Mul, [72], [87] and [74], by (Fun) 110] a!6220!6220x(X, Y) >= x(X, Y) because a!6220!6220x = x, a!6220!6220x in Mul, [72] and [87], by (Fun) 111] a!6220!6220and(X, Y) >= and(X, Y) because a!6220!6220and = and, a!6220!6220and in Mul, [72] and [87], by (Fun) 112] a!6220!6220isNatKind(X) >= isNatKind(X) because a!6220!6220isNatKind = isNatKind, a!6220!6220isNatKind in Mul and [78], by (Fun) We can thus remove the following rules: a!6220!6220U12(tt, X) => a!6220!6220U13(a!6220!6220isNat(X)) a!6220!6220U13(tt) => tt a!6220!6220U41(tt, X) => mark(X) a!6220!6220U61(tt) => 0 a!6220!6220isNat(plus(X, Y)) => a!6220!6220U11(a!6220!6220and(a!6220!6220isNatKind(X), isNatKind(Y)), X, Y) a!6220!6220x(X, s(Y)) => a!6220!6220U71(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!6220U22(tt) >? tt a!6220!6220U31(tt, X, Y) >? a!6220!6220U32(a!6220!6220isNat(X), Y) a!6220!6220U51(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!6220isNatKind(0) >? tt a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) a!6220!6220plus(X, 0) >? a!6220!6220U41(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) >? a!6220!6220U51(a!6220!6220and(a!6220!6220and(a!6220!6220isNat(Y), isNatKind(Y)), and(isNat(X), isNatKind(X))), Y, X) a!6220!6220x(X, 0) >? a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(X), isNatKind(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, Z)) >? a!6220!6220U31(mark(X), Y, Z) mark(U32(X, Y)) >? a!6220!6220U32(mark(X), Y) mark(U33(X)) >? a!6220!6220U33(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U51(X, Y, Z)) >? a!6220!6220U51(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U71(X, Y, Z)) >? a!6220!6220U71(mark(X), Y, Z) mark(x(X, Y)) >? a!6220!6220x(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, Z) >? U31(X, Y, Z) a!6220!6220U32(X, Y) >? U32(X, Y) a!6220!6220U33(X) >? U33(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U51(X, Y, Z) >? U51(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y, Z) >? U71(X, Y, Z) a!6220!6220x(X, Y) >? x(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]] = _|_ [[U22(x_1)]] = x_1 [[U33(x_1)]] = x_1 [[U51(x_1, x_2, x_3)]] = U51(x_2, x_3, x_1) [[U61(x_1)]] = x_1 [[a!6220!6220U22(x_1)]] = x_1 [[a!6220!6220U33(x_1)]] = x_1 [[a!6220!6220U51(x_1, x_2, x_3)]] = a!6220!6220U51(x_2, x_3, x_1) [[a!6220!6220U61(x_1)]] = x_1 [[a!6220!6220isNat(x_1)]] = x_1 [[a!6220!6220plus(x_1, x_2)]] = a!6220!6220plus(x_2, x_1) [[isNat(x_1)]] = x_1 [[mark(x_1)]] = x_1 [[plus(x_1, x_2)]] = plus(x_2, x_1) [[tt]] = _|_ We choose Lex = {U51, a!6220!6220U51, a!6220!6220plus, plus} and Mul = {U11, U12, U13, U21, U31, U32, U41, U71, a!6220!6220U11, a!6220!6220U12, a!6220!6220U13, a!6220!6220U21, a!6220!6220U31, a!6220!6220U32, a!6220!6220U41, a!6220!6220U71, a!6220!6220and, a!6220!6220isNatKind, a!6220!6220x, and, isNatKind, s, x}, and the following precedence: U21 = a!6220!6220U21 > U12 = a!6220!6220U12 > a!6220!6220x = x > U31 = a!6220!6220U31 > U51 = a!6220!6220U51 = a!6220!6220plus = plus > s > a!6220!6220and = and > U71 = a!6220!6220U71 > a!6220!6220isNatKind = isNatKind > U11 = a!6220!6220U11 > U41 = a!6220!6220U41 > U32 = a!6220!6220U32 > 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!6220U31(_|_, X, Y) > a!6220!6220U32(X, Y) a!6220!6220U51(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) a!6220!6220and(_|_, X) >= X _|_ >= _|_ a!6220!6220isNatKind(_|_) >= _|_ a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) a!6220!6220plus(X, _|_) > a!6220!6220U41(a!6220!6220and(X, isNatKind(X)), X) a!6220!6220plus(X, s(Y)) > a!6220!6220U51(a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))), Y, X) a!6220!6220x(X, _|_) >= a!6220!6220and(X, isNatKind(X)) U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) U12(X, Y) >= a!6220!6220U12(X, Y) X >= X U13(X) >= a!6220!6220U13(X) U21(X, Y) >= a!6220!6220U21(X, Y) X >= X U31(X, Y, Z) >= a!6220!6220U31(X, Y, Z) U32(X, Y) >= a!6220!6220U32(X, Y) X >= X U41(X, Y) >= a!6220!6220U41(X, Y) U51(X, Y, Z) >= a!6220!6220U51(X, Y, Z) plus(X, Y) >= a!6220!6220plus(X, Y) X >= X U71(X, Y, Z) >= a!6220!6220U71(X, Y, Z) x(X, Y) >= a!6220!6220x(X, Y) and(X, Y) >= a!6220!6220and(X, Y) isNatKind(X) >= a!6220!6220isNatKind(X) _|_ >= _|_ s(X) >= s(X) _|_ >= _|_ a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) a!6220!6220U12(X, Y) >= U12(X, Y) X >= X a!6220!6220U13(X) >= U13(X) a!6220!6220U21(X, Y) >= U21(X, Y) X >= X a!6220!6220U31(X, Y, Z) >= U31(X, Y, Z) a!6220!6220U32(X, Y) >= U32(X, Y) X >= X a!6220!6220U41(X, Y) >= U41(X, Y) a!6220!6220U51(X, Y, Z) >= U51(X, Y, Z) a!6220!6220plus(X, Y) >= plus(X, Y) X >= X a!6220!6220U71(X, Y, Z) >= U71(X, Y, Z) a!6220!6220x(X, Y) >= x(X, Y) a!6220!6220and(X, Y) >= and(X, Y) a!6220!6220isNatKind(X) >= isNatKind(X) With these choices, we have: 1] _|_ >= _|_ by (Bot) 2] a!6220!6220U31(_|_, X, Y) > a!6220!6220U32(X, Y) because [3], by definition 3] a!6220!6220U31*(_|_, X, Y) >= a!6220!6220U32(X, Y) because a!6220!6220U31 > a!6220!6220U32, [4] and [6], by (Copy) 4] a!6220!6220U31*(_|_, X, Y) >= X because [5], by (Select) 5] X >= X by (Meta) 6] a!6220!6220U31*(_|_, X, Y) >= Y because [7], by (Select) 7] Y >= Y by (Meta) 8] a!6220!6220U51(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) because [9], by (Star) 9] a!6220!6220U51*(_|_, X, Y) >= s(a!6220!6220plus(Y, X)) because a!6220!6220U51 > s and [10], by (Copy) 10] a!6220!6220U51*(_|_, X, Y) >= a!6220!6220plus(Y, X) because a!6220!6220U51 = a!6220!6220plus, [11], [12], [13] and [14], by (Stat) 11] X >= X by (Meta) 12] Y >= Y by (Meta) 13] a!6220!6220U51*(_|_, X, Y) >= Y because [12], by (Select) 14] a!6220!6220U51*(_|_, X, Y) >= X because [11], by (Select) 15] a!6220!6220and(_|_, X) >= X because [16], by (Star) 16] a!6220!6220and*(_|_, X) >= X because [17], by (Select) 17] X >= X by (Meta) 18] _|_ >= _|_ by (Bot) 19] a!6220!6220isNatKind(_|_) >= _|_ by (Bot) 20] a!6220!6220isNatKind(s(X)) >= a!6220!6220isNatKind(X) because a!6220!6220isNatKind in Mul and [21], by (Fun) 21] s(X) >= X because [22], by (Star) 22] s*(X) >= X because [5], by (Select) 23] a!6220!6220plus(X, _|_) > a!6220!6220U41(a!6220!6220and(X, isNatKind(X)), X) because [24], by definition 24] a!6220!6220plus*(X, _|_) >= a!6220!6220U41(a!6220!6220and(X, isNatKind(X)), X) because a!6220!6220plus > a!6220!6220U41, [25] and [26], by (Copy) 25] a!6220!6220plus*(X, _|_) >= a!6220!6220and(X, isNatKind(X)) because a!6220!6220plus > a!6220!6220and, [26] and [27], by (Copy) 26] a!6220!6220plus*(X, _|_) >= X because [12], by (Select) 27] a!6220!6220plus*(X, _|_) >= isNatKind(X) because a!6220!6220plus > isNatKind and [26], by (Copy) 28] a!6220!6220plus(X, s(Y)) > a!6220!6220U51(a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))), Y, X) because [29], by definition 29] a!6220!6220plus*(X, s(Y)) >= a!6220!6220U51(a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))), Y, X) because a!6220!6220plus = a!6220!6220U51, [30], [32], [41] and [39], by (Stat) 30] s(Y) > Y because [31], by definition 31] s*(Y) >= Y because [11], by (Select) 32] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(a!6220!6220and(Y, isNatKind(Y)), and(X, isNatKind(X))) because a!6220!6220plus > a!6220!6220and, [33] and [38], by (Copy) 33] a!6220!6220plus*(X, s(Y)) >= a!6220!6220and(Y, isNatKind(Y)) because [34], by (Select) 34] s(Y) >= a!6220!6220and(Y, isNatKind(Y)) because [35], by (Star) 35] s*(Y) >= a!6220!6220and(Y, isNatKind(Y)) because s > a!6220!6220and, [36] and [37], by (Copy) 36] s*(Y) >= Y because [11], by (Select) 37] s*(Y) >= isNatKind(Y) because s > isNatKind and [36], by (Copy) 38] a!6220!6220plus*(X, s(Y)) >= and(X, isNatKind(X)) because a!6220!6220plus > and, [39] and [40], by (Copy) 39] a!6220!6220plus*(X, s(Y)) >= X because [12], by (Select) 40] a!6220!6220plus*(X, s(Y)) >= isNatKind(X) because a!6220!6220plus > isNatKind and [39], by (Copy) 41] a!6220!6220plus*(X, s(Y)) >= Y because [42], by (Select) 42] s(Y) >= Y because [36], by (Star) 43] a!6220!6220x(X, _|_) >= a!6220!6220and(X, isNatKind(X)) because [44], by (Star) 44] a!6220!6220x*(X, _|_) >= a!6220!6220and(X, isNatKind(X)) because a!6220!6220x > a!6220!6220and, [45] and [46], by (Copy) 45] a!6220!6220x*(X, _|_) >= X because [12], by (Select) 46] a!6220!6220x*(X, _|_) >= isNatKind(X) because a!6220!6220x > isNatKind and [45], by (Copy) 47] U11(X, Y, Z) >= a!6220!6220U11(X, Y, Z) because U11 = a!6220!6220U11, U11 in Mul, [48], [49] and [50], by (Fun) 48] X >= X by (Meta) 49] Y >= Y by (Meta) 50] Z >= Z by (Meta) 51] U12(X, Y) >= a!6220!6220U12(X, Y) because U12 = a!6220!6220U12, U12 in Mul, [48] and [49], by (Fun) 52] X >= X by (Meta) 53] U13(X) >= a!6220!6220U13(X) because U13 = a!6220!6220U13, U13 in Mul and [54], by (Fun) 54] X >= X by (Meta) 55] U21(X, Y) >= a!6220!6220U21(X, Y) because U21 = a!6220!6220U21, U21 in Mul, [48] and [49], by (Fun) 56] X >= X by (Meta) 57] U31(X, Y, Z) >= a!6220!6220U31(X, Y, Z) because U31 = a!6220!6220U31, U31 in Mul, [48], [49] and [50], by (Fun) 58] U32(X, Y) >= a!6220!6220U32(X, Y) because U32 = a!6220!6220U32, U32 in Mul, [48] and [49], by (Fun) 59] X >= X by (Meta) 60] U41(X, Y) >= a!6220!6220U41(X, Y) because U41 = a!6220!6220U41, U41 in Mul, [48] and [49], by (Fun) 61] U51(X, Y, Z) >= a!6220!6220U51(X, Y, Z) because U51 = a!6220!6220U51, [48], [49] and [50], by (Fun) 62] plus(X, Y) >= a!6220!6220plus(X, Y) because plus = a!6220!6220plus, [48] and [63], by (Fun) 63] Y >= Y by (Meta) 64] X >= X by (Meta) 65] U71(X, Y, Z) >= a!6220!6220U71(X, Y, Z) because U71 = a!6220!6220U71, U71 in Mul, [48], [63] and [50], by (Fun) 66] x(X, Y) >= a!6220!6220x(X, Y) because x = a!6220!6220x, x in Mul, [48] and [63], by (Fun) 67] and(X, Y) >= a!6220!6220and(X, Y) because and = a!6220!6220and, and in Mul, [48] and [63], by (Fun) 68] isNatKind(X) >= a!6220!6220isNatKind(X) because isNatKind = a!6220!6220isNatKind, isNatKind in Mul and [54], by (Fun) 69] _|_ >= _|_ by (Bot) 70] s(X) >= s(X) because s in Mul and [54], by (Fun) 71] _|_ >= _|_ by (Bot) 72] a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) because a!6220!6220U11 = U11, a!6220!6220U11 in Mul, [48], [63] and [50], by (Fun) 73] a!6220!6220U12(X, Y) >= U12(X, Y) because a!6220!6220U12 = U12, a!6220!6220U12 in Mul, [48] and [63], by (Fun) 74] X >= X by (Meta) 75] a!6220!6220U13(X) >= U13(X) because a!6220!6220U13 = U13, a!6220!6220U13 in Mul and [54], by (Fun) 76] a!6220!6220U21(X, Y) >= U21(X, Y) because a!6220!6220U21 = U21, a!6220!6220U21 in Mul, [48] and [63], by (Fun) 77] X >= X by (Meta) 78] a!6220!6220U31(X, Y, Z) >= U31(X, Y, Z) because a!6220!6220U31 = U31, a!6220!6220U31 in Mul, [48], [63] and [50], by (Fun) 79] a!6220!6220U32(X, Y) >= U32(X, Y) because a!6220!6220U32 = U32, a!6220!6220U32 in Mul, [48] and [63], by (Fun) 80] X >= X by (Meta) 81] a!6220!6220U41(X, Y) >= U41(X, Y) because a!6220!6220U41 = U41, a!6220!6220U41 in Mul, [48] and [63], by (Fun) 82] a!6220!6220U51(X, Y, Z) >= U51(X, Y, Z) because a!6220!6220U51 = U51, [48], [63] and [50], by (Fun) 83] a!6220!6220plus(X, Y) >= plus(X, Y) because a!6220!6220plus = plus, [48] and [63], by (Fun) 84] X >= X by (Meta) 85] a!6220!6220U71(X, Y, Z) >= U71(X, Y, Z) because a!6220!6220U71 = U71, a!6220!6220U71 in Mul, [48], [63] and [50], by (Fun) 86] a!6220!6220x(X, Y) >= x(X, Y) because a!6220!6220x = x, a!6220!6220x in Mul, [48] and [63], by (Fun) 87] a!6220!6220and(X, Y) >= and(X, Y) because a!6220!6220and = and, a!6220!6220and in Mul, [48] and [63], by (Fun) 88] a!6220!6220isNatKind(X) >= isNatKind(X) because a!6220!6220isNatKind = isNatKind, a!6220!6220isNatKind in Mul and [54], by (Fun) We can thus remove the following rules: a!6220!6220U31(tt, X, Y) => a!6220!6220U32(a!6220!6220isNat(X), Y) a!6220!6220plus(X, 0) => a!6220!6220U41(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X)), X) a!6220!6220plus(X, s(Y)) => a!6220!6220U51(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!6220U22(tt) >? tt a!6220!6220U51(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!6220isNatKind(0) >? tt a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) a!6220!6220x(X, 0) >? a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(X), isNatKind(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, Z)) >? a!6220!6220U31(mark(X), Y, Z) mark(U32(X, Y)) >? a!6220!6220U32(mark(X), Y) mark(U33(X)) >? a!6220!6220U33(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U51(X, Y, Z)) >? a!6220!6220U51(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U71(X, Y, Z)) >? a!6220!6220U71(mark(X), Y, Z) mark(x(X, Y)) >? a!6220!6220x(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, Z) >? U31(X, Y, Z) a!6220!6220U32(X, Y) >? U32(X, Y) a!6220!6220U33(X) >? U33(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U51(X, Y, Z) >? U51(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y, Z) >? U71(X, Y, Z) a!6220!6220x(X, Y) >? x(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 = 1 U11 = \y0y1y2.2 + y1 + y2 + 2y0 U12 = \y0y1.y0 + 2y1 U13 = \y0.y0 U21 = \y0y1.y0 + y1 U22 = \y0.y0 U31 = \y0y1y2.2 + y0 + y2 + 2y1 U32 = \y0y1.y0 + y1 U33 = \y0.y0 U41 = \y0y1.y0 + 2y1 U51 = \y0y1y2.y0 + 2y1 + 2y2 U61 = \y0.y0 U71 = \y0y1y2.1 + 2y0 + 2y1 + 2y2 a!6220!6220U11 = \y0y1y2.3 + y2 + 2y0 + 2y1 a!6220!6220U12 = \y0y1.y0 + 2y1 a!6220!6220U13 = \y0.y0 a!6220!6220U21 = \y0y1.y0 + 2y1 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0y1y2.2 + y0 + 2y1 + 2y2 a!6220!6220U32 = \y0y1.y0 + 2y1 a!6220!6220U33 = \y0.y0 a!6220!6220U41 = \y0y1.y0 + 2y1 a!6220!6220U51 = \y0y1y2.y0 + 2y1 + 2y2 a!6220!6220U61 = \y0.y0 a!6220!6220U71 = \y0y1y2.1 + 2y0 + 2y1 + 2y2 a!6220!6220and = \y0y1.y0 + 2y1 a!6220!6220isNat = \y0.y0 a!6220!6220isNatKind = \y0.y0 a!6220!6220plus = \y0y1.y0 + y1 a!6220!6220x = \y0y1.y1 + 3y0 and = \y0y1.y0 + 2y1 isNat = \y0.y0 isNatKind = \y0.y0 mark = \y0.2y0 plus = \y0y1.y0 + y1 s = \y0.y0 tt = 0 x = \y0y1.y1 + 3y0 Using this interpretation, the requirements translate to: [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U51(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[s(a!6220!6220plus(mark(_x1), mark(_x0)))]] [[a!6220!6220and(tt, _x0)]] = 2x0 >= 2x0 = [[mark(_x0)]] [[a!6220!6220isNat(0)]] = 1 > 0 = [[tt]] [[a!6220!6220isNatKind(0)]] = 1 > 0 = [[tt]] [[a!6220!6220isNatKind(s(_x0))]] = x0 >= x0 = [[a!6220!6220isNatKind(_x0)]] [[a!6220!6220x(_x0, 0)]] = 1 + 3x0 > 3x0 = [[a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(_x0), isNatKind(_x0)))]] [[mark(U11(_x0, _x1, _x2))]] = 4 + 2x1 + 2x2 + 4x0 > 3 + x2 + 2x1 + 4x0 = [[a!6220!6220U11(mark(_x0), _x1, _x2)]] [[mark(U12(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 + 2x1 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(isNat(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNat(_x0)]] [[mark(U13(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U13(mark(_x0))]] [[mark(U21(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U31(_x0, _x1, _x2))]] = 4 + 2x0 + 2x2 + 4x1 > 2 + 2x0 + 2x1 + 2x2 = [[a!6220!6220U31(mark(_x0), _x1, _x2)]] [[mark(U32(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U32(mark(_x0), _x1)]] [[mark(U33(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U33(mark(_x0))]] [[mark(U41(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 + 2x1 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(U51(_x0, _x1, _x2))]] = 2x0 + 4x1 + 4x2 >= 2x0 + 2x1 + 2x2 = [[a!6220!6220U51(mark(_x0), _x1, _x2)]] [[mark(plus(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U71(_x0, _x1, _x2))]] = 2 + 4x0 + 4x1 + 4x2 > 1 + 2x1 + 2x2 + 4x0 = [[a!6220!6220U71(mark(_x0), _x1, _x2)]] [[mark(x(_x0, _x1))]] = 2x1 + 6x0 >= 2x1 + 6x0 = [[a!6220!6220x(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 + 2x1 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isNatKind(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(s(_x0))]] = 2x0 >= 2x0 = [[s(mark(_x0))]] [[mark(0)]] = 2 > 1 = [[0]] [[a!6220!6220U11(_x0, _x1, _x2)]] = 3 + x2 + 2x0 + 2x1 > 2 + x1 + x2 + 2x0 = [[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)]] = x0 + 2x1 >= x0 + x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U31(_x0, _x1, _x2)]] = 2 + x0 + 2x1 + 2x2 >= 2 + x0 + x2 + 2x1 = [[U31(_x0, _x1, _x2)]] [[a!6220!6220U32(_x0, _x1)]] = x0 + 2x1 >= x0 + x1 = [[U32(_x0, _x1)]] [[a!6220!6220U33(_x0)]] = x0 >= x0 = [[U33(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[U41(_x0, _x1)]] [[a!6220!6220U51(_x0, _x1, _x2)]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U51(_x0, _x1, _x2)]] [[a!6220!6220plus(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U71(_x0, _x1, _x2)]] = 1 + 2x0 + 2x1 + 2x2 >= 1 + 2x0 + 2x1 + 2x2 = [[U71(_x0, _x1, _x2)]] [[a!6220!6220x(_x0, _x1)]] = x1 + 3x0 >= x1 + 3x0 = [[x(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[and(_x0, _x1)]] [[a!6220!6220isNatKind(_x0)]] = x0 >= x0 = [[isNatKind(_x0)]] We can thus remove the following rules: a!6220!6220isNat(0) => tt a!6220!6220isNatKind(0) => tt a!6220!6220x(X, 0) => a!6220!6220U61(a!6220!6220and(a!6220!6220isNat(X), isNatKind(X))) mark(U11(X, Y, Z)) => a!6220!6220U11(mark(X), Y, Z) mark(U31(X, Y, Z)) => a!6220!6220U31(mark(X), Y, Z) mark(U71(X, Y, Z)) => a!6220!6220U71(mark(X), Y, Z) mark(0) => 0 a!6220!6220U11(X, Y, Z) => U11(X, Y, 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]): a!6220!6220U22(tt) >? tt a!6220!6220U51(tt, X, Y) >? s(a!6220!6220plus(mark(Y), mark(X))) a!6220!6220and(tt, X) >? mark(X) a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) 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(U32(X, Y)) >? a!6220!6220U32(mark(X), Y) mark(U33(X)) >? a!6220!6220U33(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U51(X, Y, Z)) >? a!6220!6220U51(mark(X), Y, Z) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(x(X, Y)) >? a!6220!6220x(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)) 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, Z) >? U31(X, Y, Z) a!6220!6220U32(X, Y) >? U32(X, Y) a!6220!6220U33(X) >? U33(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U51(X, Y, Z) >? U51(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y, Z) >? U71(X, Y, Z) a!6220!6220x(X, Y) >? x(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: U12 = \y0y1.y0 + y1 U13 = \y0.1 + y0 U21 = \y0y1.y0 + y1 U22 = \y0.y0 U31 = \y0y1y2.y0 + y1 + y2 U32 = \y0y1.y0 + y1 U33 = \y0.y0 U41 = \y0y1.y0 + y1 U51 = \y0y1y2.1 + y0 + 2y1 + 2y2 U61 = \y0.y0 U71 = \y0y1y2.y0 + y1 + y2 a!6220!6220U12 = \y0y1.y0 + y1 a!6220!6220U13 = \y0.1 + y0 a!6220!6220U21 = \y0y1.y0 + y1 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0y1y2.3 + 3y0 + 3y1 + 3y2 a!6220!6220U32 = \y0y1.y0 + y1 a!6220!6220U33 = \y0.y0 a!6220!6220U41 = \y0y1.y0 + 2y1 a!6220!6220U51 = \y0y1y2.1 + y0 + 2y1 + 2y2 a!6220!6220U61 = \y0.y0 a!6220!6220U71 = \y0y1y2.3 + 3y0 + 3y1 + 3y2 a!6220!6220and = \y0y1.y0 + 2y1 a!6220!6220isNat = \y0.y0 a!6220!6220isNatKind = \y0.y0 a!6220!6220plus = \y0y1.y0 + y1 a!6220!6220x = \y0y1.y0 + y1 and = \y0y1.y0 + 2y1 isNat = \y0.y0 isNatKind = \y0.y0 mark = \y0.2y0 plus = \y0y1.y0 + y1 s = \y0.y0 tt = 0 x = \y0y1.y0 + y1 Using this interpretation, the requirements translate to: [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U51(tt, _x0, _x1)]] = 1 + 2x0 + 2x1 > 2x0 + 2x1 = [[s(a!6220!6220plus(mark(_x1), mark(_x0)))]] [[a!6220!6220and(tt, _x0)]] = 2x0 >= 2x0 = [[mark(_x0)]] [[a!6220!6220isNatKind(s(_x0))]] = x0 >= x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(U12(_x0, _x1))]] = 2x0 + 2x1 >= x1 + 2x0 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(isNat(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNat(_x0)]] [[mark(U13(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[a!6220!6220U13(mark(_x0))]] [[mark(U21(_x0, _x1))]] = 2x0 + 2x1 >= x1 + 2x0 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U32(_x0, _x1))]] = 2x0 + 2x1 >= x1 + 2x0 = [[a!6220!6220U32(mark(_x0), _x1)]] [[mark(U33(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U33(mark(_x0))]] [[mark(U41(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(U51(_x0, _x1, _x2))]] = 2 + 2x0 + 4x1 + 4x2 > 1 + 2x0 + 2x1 + 2x2 = [[a!6220!6220U51(mark(_x0), _x1, _x2)]] [[mark(plus(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(x(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220x(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 + 2x1 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isNatKind(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(s(_x0))]] = 2x0 >= 2x0 = [[s(mark(_x0))]] [[a!6220!6220U12(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[U12(_x0, _x1)]] [[a!6220!6220isNat(_x0)]] = x0 >= x0 = [[isNat(_x0)]] [[a!6220!6220U13(_x0)]] = 1 + x0 >= 1 + x0 = [[U13(_x0)]] [[a!6220!6220U21(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U31(_x0, _x1, _x2)]] = 3 + 3x0 + 3x1 + 3x2 > x0 + x1 + x2 = [[U31(_x0, _x1, _x2)]] [[a!6220!6220U32(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[U32(_x0, _x1)]] [[a!6220!6220U33(_x0)]] = x0 >= x0 = [[U33(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = x0 + 2x1 >= x0 + x1 = [[U41(_x0, _x1)]] [[a!6220!6220U51(_x0, _x1, _x2)]] = 1 + x0 + 2x1 + 2x2 >= 1 + x0 + 2x1 + 2x2 = [[U51(_x0, _x1, _x2)]] [[a!6220!6220plus(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U71(_x0, _x1, _x2)]] = 3 + 3x0 + 3x1 + 3x2 > x0 + x1 + x2 = [[U71(_x0, _x1, _x2)]] [[a!6220!6220x(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[x(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[and(_x0, _x1)]] [[a!6220!6220isNatKind(_x0)]] = x0 >= x0 = [[isNatKind(_x0)]] We can thus remove the following rules: a!6220!6220U51(tt, X, Y) => s(a!6220!6220plus(mark(Y), mark(X))) mark(U13(X)) => a!6220!6220U13(mark(X)) mark(U51(X, Y, Z)) => a!6220!6220U51(mark(X), Y, Z) a!6220!6220U31(X, Y, Z) => U31(X, Y, Z) a!6220!6220U71(X, Y, Z) => U71(X, Y, 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]): a!6220!6220U22(tt) >? tt a!6220!6220and(tt, X) >? mark(X) a!6220!6220isNatKind(s(X)) >? a!6220!6220isNatKind(X) mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(isNat(X)) >? a!6220!6220isNat(X) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U32(X, Y)) >? a!6220!6220U32(mark(X), Y) mark(U33(X)) >? a!6220!6220U33(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(plus(X, Y)) >? a!6220!6220plus(mark(X), mark(Y)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(x(X, Y)) >? a!6220!6220x(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)) 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!6220U32(X, Y) >? U32(X, Y) a!6220!6220U33(X) >? U33(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U51(X, Y, Z) >? U51(X, Y, Z) a!6220!6220plus(X, Y) >? plus(X, Y) a!6220!6220U61(X) >? U61(X) a!6220!6220x(X, Y) >? x(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: U12 = \y0y1.y0 + y1 U13 = \y0.y0 U21 = \y0y1.y0 + 2y1 U22 = \y0.y0 U32 = \y0y1.y0 + y1 U33 = \y0.y0 U41 = \y0y1.3 + y0 + y1 U51 = \y0y1y2.y0 + y1 + y2 U61 = \y0.1 + y0 a!6220!6220U12 = \y0y1.y0 + 2y1 a!6220!6220U13 = \y0.3 + 3y0 a!6220!6220U21 = \y0y1.y0 + 2y1 a!6220!6220U22 = \y0.y0 a!6220!6220U32 = \y0y1.y0 + 2y1 a!6220!6220U33 = \y0.y0 a!6220!6220U41 = \y0y1.3 + y0 + 2y1 a!6220!6220U51 = \y0y1y2.3 + 3y0 + 3y1 + 3y2 a!6220!6220U61 = \y0.1 + y0 a!6220!6220and = \y0y1.1 + 2y0 + 2y1 a!6220!6220isNat = \y0.1 + 2y0 a!6220!6220isNatKind = \y0.2y0 a!6220!6220plus = \y0y1.3 + y0 + y1 a!6220!6220x = \y0y1.y0 + y1 and = \y0y1.1 + y1 + 2y0 isNat = \y0.1 + 2y0 isNatKind = \y0.2y0 mark = \y0.2y0 plus = \y0y1.2 + y0 + y1 s = \y0.1 + 2y0 tt = 0 x = \y0y1.y0 + y1 Using this interpretation, the requirements translate to: [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220and(tt, _x0)]] = 1 + 2x0 > 2x0 = [[mark(_x0)]] [[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(isNat(_x0))]] = 2 + 4x0 > 1 + 2x0 = [[a!6220!6220isNat(_x0)]] [[mark(U21(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 + 2x1 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U32(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U32(mark(_x0), _x1)]] [[mark(U33(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U33(mark(_x0))]] [[mark(U41(_x0, _x1))]] = 6 + 2x0 + 2x1 > 3 + 2x0 + 2x1 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(plus(_x0, _x1))]] = 4 + 2x0 + 2x1 > 3 + 2x0 + 2x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] [[mark(U61(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(x(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220x(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = 2 + 2x1 + 4x0 > 1 + 2x1 + 4x0 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isNatKind(_x0))]] = 4x0 >= 2x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(s(_x0))]] = 2 + 4x0 > 1 + 4x0 = [[s(mark(_x0))]] [[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)]] = 3 + 3x0 > x0 = [[U13(_x0)]] [[a!6220!6220U21(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U32(_x0, _x1)]] = x0 + 2x1 >= x0 + x1 = [[U32(_x0, _x1)]] [[a!6220!6220U33(_x0)]] = x0 >= x0 = [[U33(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = 3 + x0 + 2x1 >= 3 + x0 + x1 = [[U41(_x0, _x1)]] [[a!6220!6220U51(_x0, _x1, _x2)]] = 3 + 3x0 + 3x1 + 3x2 > x0 + x1 + x2 = [[U51(_x0, _x1, _x2)]] [[a!6220!6220plus(_x0, _x1)]] = 3 + x0 + x1 > 2 + x0 + x1 = [[plus(_x0, _x1)]] [[a!6220!6220U61(_x0)]] = 1 + x0 >= 1 + x0 = [[U61(_x0)]] [[a!6220!6220x(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[x(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = 1 + 2x0 + 2x1 >= 1 + x1 + 2x0 = [[and(_x0, _x1)]] [[a!6220!6220isNatKind(_x0)]] = 2x0 >= 2x0 = [[isNatKind(_x0)]] We can thus remove the following rules: a!6220!6220and(tt, X) => mark(X) a!6220!6220isNatKind(s(X)) => a!6220!6220isNatKind(X) mark(isNat(X)) => a!6220!6220isNat(X) mark(U41(X, Y)) => a!6220!6220U41(mark(X), Y) mark(plus(X, Y)) => a!6220!6220plus(mark(X), mark(Y)) mark(U61(X)) => a!6220!6220U61(mark(X)) mark(and(X, Y)) => a!6220!6220and(mark(X), Y) mark(s(X)) => s(mark(X)) a!6220!6220U13(X) => U13(X) a!6220!6220U51(X, Y, Z) => U51(X, Y, Z) 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]): a!6220!6220U22(tt) >? tt mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U32(X, Y)) >? a!6220!6220U32(mark(X), Y) mark(U33(X)) >? a!6220!6220U33(mark(X)) mark(x(X, Y)) >? a!6220!6220x(mark(X), mark(Y)) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220isNat(X) >? isNat(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220U32(X, Y) >? U32(X, Y) a!6220!6220U33(X) >? U33(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U61(X) >? U61(X) a!6220!6220x(X, Y) >? x(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: U12 = \y0y1.y0 + y1 U21 = \y0y1.y0 + y1 U22 = \y0.y0 U32 = \y0y1.2 + y0 + y1 U33 = \y0.1 + y0 U41 = \y0y1.y0 + y1 U61 = \y0.y0 a!6220!6220U12 = \y0y1.y0 + 2y1 a!6220!6220U21 = \y0y1.y0 + y1 a!6220!6220U22 = \y0.y0 a!6220!6220U32 = \y0y1.3 + y0 + y1 a!6220!6220U33 = \y0.2 + y0 a!6220!6220U41 = \y0y1.3 + y1 + 2y0 a!6220!6220U61 = \y0.3 + 2y0 a!6220!6220and = \y0y1.3 + y0 + 2y1 a!6220!6220isNat = \y0.3 + y0 a!6220!6220isNatKind = \y0.2 + y0 a!6220!6220x = \y0y1.2 + y0 + y1 and = \y0y1.y0 + y1 isNat = \y0.y0 isNatKind = \y0.1 + y0 mark = \y0.2y0 tt = 0 x = \y0y1.2 + y0 + y1 Using this interpretation, the requirements translate to: [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[mark(U12(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(U21(_x0, _x1))]] = 2x0 + 2x1 >= x1 + 2x0 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U32(_x0, _x1))]] = 4 + 2x0 + 2x1 > 3 + x1 + 2x0 = [[a!6220!6220U32(mark(_x0), _x1)]] [[mark(U33(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[a!6220!6220U33(mark(_x0))]] [[mark(x(_x0, _x1))]] = 4 + 2x0 + 2x1 > 2 + 2x0 + 2x1 = [[a!6220!6220x(mark(_x0), mark(_x1))]] [[mark(isNatKind(_x0))]] = 2 + 2x0 >= 2 + x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U12(_x0, _x1)]] = x0 + 2x1 >= x0 + x1 = [[U12(_x0, _x1)]] [[a!6220!6220isNat(_x0)]] = 3 + x0 > x0 = [[isNat(_x0)]] [[a!6220!6220U21(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U32(_x0, _x1)]] = 3 + x0 + x1 > 2 + x0 + x1 = [[U32(_x0, _x1)]] [[a!6220!6220U33(_x0)]] = 2 + x0 > 1 + x0 = [[U33(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = 3 + x1 + 2x0 > x0 + x1 = [[U41(_x0, _x1)]] [[a!6220!6220U61(_x0)]] = 3 + 2x0 > x0 = [[U61(_x0)]] [[a!6220!6220x(_x0, _x1)]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[x(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = 3 + x0 + 2x1 > x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isNatKind(_x0)]] = 2 + x0 > 1 + x0 = [[isNatKind(_x0)]] We can thus remove the following rules: mark(U32(X, Y)) => a!6220!6220U32(mark(X), Y) mark(x(X, Y)) => a!6220!6220x(mark(X), mark(Y)) a!6220!6220isNat(X) => isNat(X) a!6220!6220U32(X, Y) => U32(X, Y) a!6220!6220U33(X) => U33(X) a!6220!6220U41(X, Y) => U41(X, Y) a!6220!6220U61(X) => U61(X) 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!6220U22(tt) >? tt mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U33(X)) >? a!6220!6220U33(mark(X)) mark(isNatKind(X)) >? a!6220!6220isNatKind(X) mark(tt) >? tt a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220x(X, Y) >? x(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U12 = \y0y1.y1 + 2y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.2y0 U33 = \y0.3 + 3y0 a!6220!6220U12 = \y0y1.2y0 + 2y1 a!6220!6220U21 = \y0y1.y1 + 2y0 a!6220!6220U22 = \y0.2y0 a!6220!6220U33 = \y0.y0 a!6220!6220isNatKind = \y0.y0 a!6220!6220x = \y0y1.3 + y1 + 2y0 isNatKind = \y0.3 + y0 mark = \y0.2y0 tt = 2 x = \y0y1.y0 + y1 Using this interpretation, the requirements translate to: [[a!6220!6220U22(tt)]] = 4 > 2 = [[tt]] [[mark(U12(_x0, _x1))]] = 2x1 + 4x0 >= 2x1 + 4x0 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(U21(_x0, _x1))]] = 2x1 + 4x0 >= x1 + 4x0 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U33(_x0))]] = 6 + 6x0 > 2x0 = [[a!6220!6220U33(mark(_x0))]] [[mark(isNatKind(_x0))]] = 6 + 2x0 > x0 = [[a!6220!6220isNatKind(_x0)]] [[mark(tt)]] = 4 > 2 = [[tt]] [[a!6220!6220U12(_x0, _x1)]] = 2x0 + 2x1 >= x1 + 2x0 = [[U12(_x0, _x1)]] [[a!6220!6220U21(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = 2x0 >= 2x0 = [[U22(_x0)]] [[a!6220!6220x(_x0, _x1)]] = 3 + x1 + 2x0 > x0 + x1 = [[x(_x0, _x1)]] We can thus remove the following rules: a!6220!6220U22(tt) => tt mark(U33(X)) => a!6220!6220U33(mark(X)) mark(isNatKind(X)) => a!6220!6220isNatKind(X) mark(tt) => tt a!6220!6220x(X, Y) => x(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(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U12 = \y0y1.y1 + 2y0 U21 = \y0y1.2 + y0 + y1 U22 = \y0.2y0 a!6220!6220U12 = \y0y1.y1 + 2y0 a!6220!6220U21 = \y0y1.2 + y0 + y1 a!6220!6220U22 = \y0.2y0 mark = \y0.2y0 Using this interpretation, the requirements translate to: [[mark(U12(_x0, _x1))]] = 2x1 + 4x0 >= x1 + 4x0 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(U21(_x0, _x1))]] = 4 + 2x0 + 2x1 > 2 + x1 + 2x0 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220U22(mark(_x0))]] [[a!6220!6220U12(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[U12(_x0, _x1)]] [[a!6220!6220U21(_x0, _x1)]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = 2x0 >= 2x0 = [[U22(_x0)]] We can thus remove the following rules: mark(U21(X, Y)) => a!6220!6220U21(mark(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(U22(X)) >? a!6220!6220U22(mark(X)) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U12 = \y0y1.y0 + y1 U21 = \y0y1.y0 + y1 U22 = \y0.y0 a!6220!6220U12 = \y0y1.y0 + y1 a!6220!6220U21 = \y0y1.3 + 2y0 + 2y1 a!6220!6220U22 = \y0.y0 mark = \y0.y0 Using this interpretation, the requirements translate to: [[mark(U12(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(U22(_x0))]] = x0 >= x0 = [[a!6220!6220U22(mark(_x0))]] [[a!6220!6220U12(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[U12(_x0, _x1)]] [[a!6220!6220U21(_x0, _x1)]] = 3 + 2x0 + 2x1 > x0 + x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] We can thus remove the following rules: 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]): mark(U12(X, Y)) >? a!6220!6220U12(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220U22(X) >? U22(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U12 = \y0y1.2 + y0 + y1 U22 = \y0.2y0 a!6220!6220U12 = \y0y1.2 + y0 + y1 a!6220!6220U22 = \y0.2y0 mark = \y0.2y0 Using this interpretation, the requirements translate to: [[mark(U12(_x0, _x1))]] = 4 + 2x0 + 2x1 > 2 + x1 + 2x0 = [[a!6220!6220U12(mark(_x0), _x1)]] [[mark(U22(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220U22(mark(_x0))]] [[a!6220!6220U12(_x0, _x1)]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[U12(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = 2x0 >= 2x0 = [[U22(_x0)]] We can thus remove the following rules: mark(U12(X, Y)) => a!6220!6220U12(mark(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(U22(X)) >? a!6220!6220U22(mark(X)) a!6220!6220U12(X, Y) >? U12(X, Y) a!6220!6220U22(X) >? U22(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U12 = \y0y1.y0 + y1 U22 = \y0.y0 a!6220!6220U12 = \y0y1.3 + y0 + y1 a!6220!6220U22 = \y0.y0 mark = \y0.y0 Using this interpretation, the requirements translate to: [[mark(U22(_x0))]] = x0 >= x0 = [[a!6220!6220U22(mark(_x0))]] [[a!6220!6220U12(_x0, _x1)]] = 3 + x0 + x1 > x0 + x1 = [[U12(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] We can thus remove the following rules: a!6220!6220U12(X, Y) => U12(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(U22(X)) >? a!6220!6220U22(mark(X)) a!6220!6220U22(X) >? U22(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U22 = \y0.1 + y0 a!6220!6220U22 = \y0.1 + y0 mark = \y0.2y0 Using this interpretation, the requirements translate to: [[mark(U22(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[a!6220!6220U22(mark(_x0))]] [[a!6220!6220U22(_x0)]] = 1 + x0 >= 1 + x0 = [[U22(_x0)]] We can thus remove the following rules: mark(U22(X)) => a!6220!6220U22(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!6220U22(X) >? U22(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U22 = \y0.y0 a!6220!6220U22 = \y0.3 + 3y0 Using this interpretation, the requirements translate to: [[a!6220!6220U22(_x0)]] = 3 + 3x0 > x0 = [[U22(_x0)]] We can thus remove the following rules: a!6220!6220U22(X) => U22(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.