/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. !6220!6220 : [o * o] --> o U11 : [o] --> o U21 : [o * o] --> o U22 : [o] --> o U31 : [o] --> o U41 : [o * o] --> o U42 : [o] --> o U51 : [o * o] --> o U52 : [o] --> o U61 : [o] --> o U71 : [o * o] --> o U72 : [o] --> o U81 : [o] --> o a : [] --> o a!6220!6220!6220!6220 : [o * o] --> o a!6220!6220U11 : [o] --> o a!6220!6220U21 : [o * o] --> o a!6220!6220U22 : [o] --> o a!6220!6220U31 : [o] --> o a!6220!6220U41 : [o * o] --> o a!6220!6220U42 : [o] --> o a!6220!6220U51 : [o * o] --> o a!6220!6220U52 : [o] --> o a!6220!6220U61 : [o] --> o a!6220!6220U71 : [o * o] --> o a!6220!6220U72 : [o] --> o a!6220!6220U81 : [o] --> o a!6220!6220isList : [o] --> o a!6220!6220isNeList : [o] --> o a!6220!6220isNePal : [o] --> o a!6220!6220isPal : [o] --> o a!6220!6220isQid : [o] --> o e : [] --> o i : [] --> o isList : [o] --> o isNeList : [o] --> o isNePal : [o] --> o isPal : [o] --> o isQid : [o] --> o mark : [o] --> o nil : [] --> o o : [] --> o tt : [] --> o u : [] --> o a!6220!6220!6220!6220(!6220!6220(X, Y), Z) => a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220!6220!6220(X, nil) => mark(X) a!6220!6220!6220!6220(nil, X) => mark(X) a!6220!6220U11(tt) => tt a!6220!6220U21(tt, X) => a!6220!6220U22(a!6220!6220isList(X)) a!6220!6220U22(tt) => tt a!6220!6220U31(tt) => tt a!6220!6220U41(tt, X) => a!6220!6220U42(a!6220!6220isNeList(X)) a!6220!6220U42(tt) => tt a!6220!6220U51(tt, X) => a!6220!6220U52(a!6220!6220isList(X)) a!6220!6220U52(tt) => tt a!6220!6220U61(tt) => tt a!6220!6220U71(tt, X) => a!6220!6220U72(a!6220!6220isPal(X)) a!6220!6220U72(tt) => tt a!6220!6220U81(tt) => tt a!6220!6220isList(X) => a!6220!6220U11(a!6220!6220isNeList(X)) a!6220!6220isList(nil) => tt a!6220!6220isList(!6220!6220(X, Y)) => a!6220!6220U21(a!6220!6220isList(X), Y) a!6220!6220isNeList(X) => a!6220!6220U31(a!6220!6220isQid(X)) a!6220!6220isNeList(!6220!6220(X, Y)) => a!6220!6220U41(a!6220!6220isList(X), Y) a!6220!6220isNeList(!6220!6220(X, Y)) => a!6220!6220U51(a!6220!6220isNeList(X), Y) a!6220!6220isNePal(X) => a!6220!6220U61(a!6220!6220isQid(X)) a!6220!6220isNePal(!6220!6220(X, !6220!6220(Y, X))) => a!6220!6220U71(a!6220!6220isQid(X), Y) a!6220!6220isPal(X) => a!6220!6220U81(a!6220!6220isNePal(X)) a!6220!6220isPal(nil) => tt a!6220!6220isQid(a) => tt a!6220!6220isQid(e) => tt a!6220!6220isQid(i) => tt a!6220!6220isQid(o) => tt a!6220!6220isQid(u) => tt mark(!6220!6220(X, Y)) => a!6220!6220!6220!6220(mark(X), mark(Y)) mark(U11(X)) => a!6220!6220U11(mark(X)) mark(U21(X, Y)) => a!6220!6220U21(mark(X), Y) mark(U22(X)) => a!6220!6220U22(mark(X)) mark(isList(X)) => a!6220!6220isList(X) mark(U31(X)) => a!6220!6220U31(mark(X)) mark(U41(X, Y)) => a!6220!6220U41(mark(X), Y) mark(U42(X)) => a!6220!6220U42(mark(X)) mark(isNeList(X)) => a!6220!6220isNeList(X) mark(U51(X, Y)) => a!6220!6220U51(mark(X), Y) mark(U52(X)) => a!6220!6220U52(mark(X)) mark(U61(X)) => a!6220!6220U61(mark(X)) mark(U71(X, Y)) => a!6220!6220U71(mark(X), Y) mark(U72(X)) => a!6220!6220U72(mark(X)) mark(isPal(X)) => a!6220!6220isPal(X) mark(U81(X)) => a!6220!6220U81(mark(X)) mark(isQid(X)) => a!6220!6220isQid(X) mark(isNePal(X)) => a!6220!6220isNePal(X) mark(nil) => nil mark(tt) => tt mark(a) => a mark(e) => e mark(i) => i mark(o) => o mark(u) => u a!6220!6220!6220!6220(X, Y) => !6220!6220(X, Y) a!6220!6220U11(X) => U11(X) a!6220!6220U21(X, Y) => U21(X, Y) a!6220!6220U22(X) => U22(X) a!6220!6220isList(X) => isList(X) a!6220!6220U31(X) => U31(X) a!6220!6220U41(X, Y) => U41(X, Y) a!6220!6220U42(X) => U42(X) a!6220!6220isNeList(X) => isNeList(X) a!6220!6220U51(X, Y) => U51(X, Y) a!6220!6220U52(X) => U52(X) a!6220!6220U61(X) => U61(X) a!6220!6220U71(X, Y) => U71(X, Y) a!6220!6220U72(X) => U72(X) a!6220!6220isPal(X) => isPal(X) a!6220!6220U81(X) => U81(X) a!6220!6220isQid(X) => isQid(X) a!6220!6220isNePal(X) => isNePal(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!6220!6220!6220(!6220!6220(X, Y), Z) >? a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220!6220!6220(X, nil) >? mark(X) a!6220!6220!6220!6220(nil, X) >? mark(X) a!6220!6220U11(tt) >? tt a!6220!6220U21(tt, X) >? a!6220!6220U22(a!6220!6220isList(X)) a!6220!6220U22(tt) >? tt a!6220!6220U31(tt) >? tt a!6220!6220U41(tt, X) >? a!6220!6220U42(a!6220!6220isNeList(X)) a!6220!6220U42(tt) >? tt a!6220!6220U51(tt, X) >? a!6220!6220U52(a!6220!6220isList(X)) a!6220!6220U52(tt) >? tt a!6220!6220U61(tt) >? tt a!6220!6220U71(tt, X) >? a!6220!6220U72(a!6220!6220isPal(X)) a!6220!6220U72(tt) >? tt a!6220!6220U81(tt) >? tt a!6220!6220isList(X) >? a!6220!6220U11(a!6220!6220isNeList(X)) a!6220!6220isList(nil) >? tt a!6220!6220isList(!6220!6220(X, Y)) >? a!6220!6220U21(a!6220!6220isList(X), Y) a!6220!6220isNeList(X) >? a!6220!6220U31(a!6220!6220isQid(X)) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220U41(a!6220!6220isList(X), Y) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220U51(a!6220!6220isNeList(X), Y) a!6220!6220isNePal(X) >? a!6220!6220U61(a!6220!6220isQid(X)) a!6220!6220isNePal(!6220!6220(X, !6220!6220(Y, X))) >? a!6220!6220U71(a!6220!6220isQid(X), Y) a!6220!6220isPal(X) >? a!6220!6220U81(a!6220!6220isNePal(X)) a!6220!6220isPal(nil) >? tt a!6220!6220isQid(a) >? tt a!6220!6220isQid(e) >? tt a!6220!6220isQid(i) >? tt a!6220!6220isQid(o) >? tt a!6220!6220isQid(u) >? tt mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(U11(X)) >? a!6220!6220U11(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(isList(X)) >? a!6220!6220isList(X) mark(U31(X)) >? a!6220!6220U31(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U42(X)) >? a!6220!6220U42(mark(X)) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(U51(X, Y)) >? a!6220!6220U51(mark(X), Y) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U71(X, Y)) >? a!6220!6220U71(mark(X), Y) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(isPal(X)) >? a!6220!6220isPal(X) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220U11(X) >? U11(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220isList(X) >? isList(X) a!6220!6220U31(X) >? U31(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U42(X) >? U42(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220U51(X, Y) >? U51(X, Y) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y) >? U71(X, Y) a!6220!6220U72(X) >? U72(X) a!6220!6220isPal(X) >? isPal(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y1 + 2y0 U11 = \y0.y0 U21 = \y0y1.y0 + 2y1 U22 = \y0.y0 U31 = \y0.2y0 U41 = \y0y1.2y0 + 2y1 U42 = \y0.y0 U51 = \y0y1.y0 + 2y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y0 + y1 U72 = \y0.y0 U81 = \y0.y0 a = 0 a!6220!6220!6220!6220 = \y0y1.y1 + 2y0 a!6220!6220U11 = \y0.y0 a!6220!6220U21 = \y0y1.y0 + 2y1 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0.2y0 a!6220!6220U41 = \y0y1.2y0 + 2y1 a!6220!6220U42 = \y0.y0 a!6220!6220U51 = \y0y1.y0 + 2y1 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.y0 a!6220!6220U71 = \y0y1.y0 + y1 a!6220!6220U72 = \y0.y0 a!6220!6220U81 = \y0.y0 a!6220!6220isList = \y0.2y0 a!6220!6220isNeList = \y0.2y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.y0 a!6220!6220isQid = \y0.y0 e = 0 i = 0 isList = \y0.2y0 isNeList = \y0.2y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 1 o = 0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220!6220!6220(!6220!6220(_x0, _x1), _x2)]] = x2 + 2x1 + 4x0 >= x2 + 2x0 + 2x1 = [[a!6220!6220!6220!6220(mark(_x0), a!6220!6220!6220!6220(mark(_x1), mark(_x2)))]] [[a!6220!6220!6220!6220(_x0, nil)]] = 1 + 2x0 > x0 = [[mark(_x0)]] [[a!6220!6220!6220!6220(nil, _x0)]] = 2 + x0 > x0 = [[mark(_x0)]] [[a!6220!6220U11(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U21(tt, _x0)]] = 2x0 >= 2x0 = [[a!6220!6220U22(a!6220!6220isList(_x0))]] [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U31(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U41(tt, _x0)]] = 2x0 >= 2x0 = [[a!6220!6220U42(a!6220!6220isNeList(_x0))]] [[a!6220!6220U42(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U51(tt, _x0)]] = 2x0 >= 2x0 = [[a!6220!6220U52(a!6220!6220isList(_x0))]] [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U61(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U71(tt, _x0)]] = x0 >= x0 = [[a!6220!6220U72(a!6220!6220isPal(_x0))]] [[a!6220!6220U72(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U81(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220isList(_x0)]] = 2x0 >= 2x0 = [[a!6220!6220U11(a!6220!6220isNeList(_x0))]] [[a!6220!6220isList(nil)]] = 2 > 0 = [[tt]] [[a!6220!6220isList(!6220!6220(_x0, _x1))]] = 2x1 + 4x0 >= 2x0 + 2x1 = [[a!6220!6220U21(a!6220!6220isList(_x0), _x1)]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= 2x0 = [[a!6220!6220U31(a!6220!6220isQid(_x0))]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2x1 + 4x0 >= 2x1 + 4x0 = [[a!6220!6220U41(a!6220!6220isList(_x0), _x1)]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2x1 + 4x0 >= 2x0 + 2x1 = [[a!6220!6220U51(a!6220!6220isNeList(_x0), _x1)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220U61(a!6220!6220isQid(_x0))]] [[a!6220!6220isNePal(!6220!6220(_x0, !6220!6220(_x1, _x0)))]] = 2x1 + 3x0 >= x0 + x1 = [[a!6220!6220U71(a!6220!6220isQid(_x0), _x1)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[a!6220!6220U81(a!6220!6220isNePal(_x0))]] [[a!6220!6220isPal(nil)]] = 1 > 0 = [[tt]] [[a!6220!6220isQid(a)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(e)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(i)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(o)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(u)]] = 0 >= 0 = [[tt]] [[mark(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(U11(_x0))]] = x0 >= x0 = [[a!6220!6220U11(mark(_x0))]] [[mark(U21(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = x0 >= x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(isList(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220isList(_x0)]] [[mark(U31(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U31(mark(_x0))]] [[mark(U41(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(U42(_x0))]] = x0 >= x0 = [[a!6220!6220U42(mark(_x0))]] [[mark(isNeList(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220isNeList(_x0)]] [[mark(U51(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[a!6220!6220U51(mark(_x0), _x1)]] [[mark(U52(_x0))]] = x0 >= x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = x0 >= x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220U71(mark(_x0), _x1)]] [[mark(U72(_x0))]] = x0 >= x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(isPal(_x0))]] = x0 >= x0 = [[a!6220!6220isPal(_x0)]] [[mark(U81(_x0))]] = x0 >= x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(nil)]] = 1 >= 1 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 0 >= 0 = [[a]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220U11(_x0)]] = x0 >= x0 = [[U11(_x0)]] [[a!6220!6220U21(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220isList(_x0)]] = 2x0 >= 2x0 = [[isList(_x0)]] [[a!6220!6220U31(_x0)]] = 2x0 >= 2x0 = [[U31(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U41(_x0, _x1)]] [[a!6220!6220U42(_x0)]] = x0 >= x0 = [[U42(_x0)]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= 2x0 = [[isNeList(_x0)]] [[a!6220!6220U51(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[U51(_x0, _x1)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U71(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[U71(_x0, _x1)]] [[a!6220!6220U72(_x0)]] = x0 >= x0 = [[U72(_x0)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[isPal(_x0)]] [[a!6220!6220U81(_x0)]] = x0 >= x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] We can thus remove the following rules: a!6220!6220!6220!6220(X, nil) => mark(X) a!6220!6220!6220!6220(nil, X) => mark(X) a!6220!6220isList(nil) => tt a!6220!6220isPal(nil) => 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!6220!6220!6220(!6220!6220(X, Y), Z) >? a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220U11(tt) >? tt a!6220!6220U21(tt, X) >? a!6220!6220U22(a!6220!6220isList(X)) a!6220!6220U22(tt) >? tt a!6220!6220U31(tt) >? tt a!6220!6220U41(tt, X) >? a!6220!6220U42(a!6220!6220isNeList(X)) a!6220!6220U42(tt) >? tt a!6220!6220U51(tt, X) >? a!6220!6220U52(a!6220!6220isList(X)) a!6220!6220U52(tt) >? tt a!6220!6220U61(tt) >? tt a!6220!6220U71(tt, X) >? a!6220!6220U72(a!6220!6220isPal(X)) a!6220!6220U72(tt) >? tt a!6220!6220U81(tt) >? tt a!6220!6220isList(X) >? a!6220!6220U11(a!6220!6220isNeList(X)) a!6220!6220isList(!6220!6220(X, Y)) >? a!6220!6220U21(a!6220!6220isList(X), Y) a!6220!6220isNeList(X) >? a!6220!6220U31(a!6220!6220isQid(X)) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220U41(a!6220!6220isList(X), Y) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220U51(a!6220!6220isNeList(X), Y) a!6220!6220isNePal(X) >? a!6220!6220U61(a!6220!6220isQid(X)) a!6220!6220isNePal(!6220!6220(X, !6220!6220(Y, X))) >? a!6220!6220U71(a!6220!6220isQid(X), Y) a!6220!6220isPal(X) >? a!6220!6220U81(a!6220!6220isNePal(X)) a!6220!6220isQid(a) >? tt a!6220!6220isQid(e) >? tt a!6220!6220isQid(i) >? tt a!6220!6220isQid(o) >? tt a!6220!6220isQid(u) >? tt mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(U11(X)) >? a!6220!6220U11(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(isList(X)) >? a!6220!6220isList(X) mark(U31(X)) >? a!6220!6220U31(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U42(X)) >? a!6220!6220U42(mark(X)) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(U51(X, Y)) >? a!6220!6220U51(mark(X), Y) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U71(X, Y)) >? a!6220!6220U71(mark(X), Y) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(isPal(X)) >? a!6220!6220isPal(X) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220U11(X) >? U11(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220isList(X) >? isList(X) a!6220!6220U31(X) >? U31(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U42(X) >? U42(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220U51(X, Y) >? U51(X, Y) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y) >? U71(X, Y) a!6220!6220U72(X) >? U72(X) a!6220!6220isPal(X) >? isPal(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.1 + y1 + 2y0 U11 = \y0.y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y0 + y1 U42 = \y0.y0 U51 = \y0y1.y0 + y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y0 + 2y1 U72 = \y0.2y0 U81 = \y0.y0 a = 0 a!6220!6220!6220!6220 = \y0y1.1 + y1 + 2y0 a!6220!6220U11 = \y0.y0 a!6220!6220U21 = \y0y1.y1 + 2y0 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0.y0 a!6220!6220U41 = \y0y1.y0 + y1 a!6220!6220U42 = \y0.y0 a!6220!6220U51 = \y0y1.y0 + y1 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.y0 a!6220!6220U71 = \y0y1.y0 + 2y1 a!6220!6220U72 = \y0.2y0 a!6220!6220U81 = \y0.y0 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.y0 a!6220!6220isQid = \y0.y0 e = 0 i = 0 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220!6220!6220(!6220!6220(_x0, _x1), _x2)]] = 3 + x2 + 2x1 + 4x0 > 2 + x2 + 2x0 + 2x1 = [[a!6220!6220!6220!6220(mark(_x0), a!6220!6220!6220!6220(mark(_x1), mark(_x2)))]] [[a!6220!6220U11(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U21(tt, _x0)]] = x0 >= x0 = [[a!6220!6220U22(a!6220!6220isList(_x0))]] [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U31(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U41(tt, _x0)]] = x0 >= x0 = [[a!6220!6220U42(a!6220!6220isNeList(_x0))]] [[a!6220!6220U42(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U51(tt, _x0)]] = x0 >= x0 = [[a!6220!6220U52(a!6220!6220isList(_x0))]] [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U61(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U71(tt, _x0)]] = 2x0 >= 2x0 = [[a!6220!6220U72(a!6220!6220isPal(_x0))]] [[a!6220!6220U72(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U81(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[a!6220!6220U11(a!6220!6220isNeList(_x0))]] [[a!6220!6220isList(!6220!6220(_x0, _x1))]] = 1 + x1 + 2x0 > x1 + 2x0 = [[a!6220!6220U21(a!6220!6220isList(_x0), _x1)]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[a!6220!6220U31(a!6220!6220isQid(_x0))]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 1 + x1 + 2x0 > x0 + x1 = [[a!6220!6220U41(a!6220!6220isList(_x0), _x1)]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 1 + x1 + 2x0 > x0 + x1 = [[a!6220!6220U51(a!6220!6220isNeList(_x0), _x1)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220U61(a!6220!6220isQid(_x0))]] [[a!6220!6220isNePal(!6220!6220(_x0, !6220!6220(_x1, _x0)))]] = 2 + 2x1 + 3x0 > x0 + 2x1 = [[a!6220!6220U71(a!6220!6220isQid(_x0), _x1)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[a!6220!6220U81(a!6220!6220isNePal(_x0))]] [[a!6220!6220isQid(a)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(e)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(i)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(o)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(u)]] = 0 >= 0 = [[tt]] [[mark(!6220!6220(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(U11(_x0))]] = x0 >= x0 = [[a!6220!6220U11(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(isList(_x0))]] = x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(U31(_x0))]] = x0 >= x0 = [[a!6220!6220U31(mark(_x0))]] [[mark(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(U42(_x0))]] = x0 >= x0 = [[a!6220!6220U42(mark(_x0))]] [[mark(isNeList(_x0))]] = x0 >= x0 = [[a!6220!6220isNeList(_x0)]] [[mark(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220U51(mark(_x0), _x1)]] [[mark(U52(_x0))]] = x0 >= x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = x0 >= x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U71(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[a!6220!6220U71(mark(_x0), _x1)]] [[mark(U72(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(isPal(_x0))]] = x0 >= x0 = [[a!6220!6220isPal(_x0)]] [[mark(U81(_x0))]] = x0 >= x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 0 >= 0 = [[a]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220U11(_x0)]] = x0 >= x0 = [[U11(_x0)]] [[a!6220!6220U21(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220U31(_x0)]] = x0 >= x0 = [[U31(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[U41(_x0, _x1)]] [[a!6220!6220U42(_x0)]] = x0 >= x0 = [[U42(_x0)]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[isNeList(_x0)]] [[a!6220!6220U51(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[U51(_x0, _x1)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U71(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[U71(_x0, _x1)]] [[a!6220!6220U72(_x0)]] = 2x0 >= 2x0 = [[U72(_x0)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[isPal(_x0)]] [[a!6220!6220U81(_x0)]] = x0 >= x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] We can thus remove the following rules: a!6220!6220!6220!6220(!6220!6220(X, Y), Z) => a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220isList(!6220!6220(X, Y)) => a!6220!6220U21(a!6220!6220isList(X), Y) a!6220!6220isNeList(!6220!6220(X, Y)) => a!6220!6220U41(a!6220!6220isList(X), Y) a!6220!6220isNeList(!6220!6220(X, Y)) => a!6220!6220U51(a!6220!6220isNeList(X), Y) a!6220!6220isNePal(!6220!6220(X, !6220!6220(Y, X))) => a!6220!6220U71(a!6220!6220isQid(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!6220U11(tt) >? tt a!6220!6220U21(tt, X) >? a!6220!6220U22(a!6220!6220isList(X)) a!6220!6220U22(tt) >? tt a!6220!6220U31(tt) >? tt a!6220!6220U41(tt, X) >? a!6220!6220U42(a!6220!6220isNeList(X)) a!6220!6220U42(tt) >? tt a!6220!6220U51(tt, X) >? a!6220!6220U52(a!6220!6220isList(X)) a!6220!6220U52(tt) >? tt a!6220!6220U61(tt) >? tt a!6220!6220U71(tt, X) >? a!6220!6220U72(a!6220!6220isPal(X)) a!6220!6220U72(tt) >? tt a!6220!6220U81(tt) >? tt a!6220!6220isList(X) >? a!6220!6220U11(a!6220!6220isNeList(X)) a!6220!6220isNeList(X) >? a!6220!6220U31(a!6220!6220isQid(X)) a!6220!6220isNePal(X) >? a!6220!6220U61(a!6220!6220isQid(X)) a!6220!6220isPal(X) >? a!6220!6220U81(a!6220!6220isNePal(X)) a!6220!6220isQid(a) >? tt a!6220!6220isQid(e) >? tt a!6220!6220isQid(i) >? tt a!6220!6220isQid(o) >? tt a!6220!6220isQid(u) >? tt mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(U11(X)) >? a!6220!6220U11(mark(X)) mark(U21(X, Y)) >? a!6220!6220U21(mark(X), Y) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(isList(X)) >? a!6220!6220isList(X) mark(U31(X)) >? a!6220!6220U31(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U42(X)) >? a!6220!6220U42(mark(X)) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(U51(X, Y)) >? a!6220!6220U51(mark(X), Y) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U71(X, Y)) >? a!6220!6220U71(mark(X), Y) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(isPal(X)) >? a!6220!6220isPal(X) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220U11(X) >? U11(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220isList(X) >? isList(X) a!6220!6220U31(X) >? U31(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U42(X) >? U42(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220U51(X, Y) >? U51(X, Y) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y) >? U71(X, Y) a!6220!6220U72(X) >? U72(X) a!6220!6220isPal(X) >? isPal(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.2 + y0 + y1 U11 = \y0.y0 U21 = \y0y1.1 + y0 + y1 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y0 + 2y1 U42 = \y0.2y0 U51 = \y0y1.y1 + 2y0 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y0 + y1 U72 = \y0.y0 U81 = \y0.2y0 a = 2 a!6220!6220!6220!6220 = \y0y1.3 + y0 + y1 a!6220!6220U11 = \y0.y0 a!6220!6220U21 = \y0y1.1 + y0 + 2y1 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0.y0 a!6220!6220U41 = \y0y1.y0 + 2y1 a!6220!6220U42 = \y0.2y0 a!6220!6220U51 = \y0y1.y1 + 2y0 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.y0 a!6220!6220U71 = \y0y1.y0 + 2y1 a!6220!6220U72 = \y0.y0 a!6220!6220U81 = \y0.2y0 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.2y0 a!6220!6220isQid = \y0.y0 e = 2 i = 0 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.2y0 isQid = \y0.y0 mark = \y0.2y0 nil = 0 o = 0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220U11(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U21(tt, _x0)]] = 1 + 2x0 > x0 = [[a!6220!6220U22(a!6220!6220isList(_x0))]] [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U31(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U41(tt, _x0)]] = 2x0 >= 2x0 = [[a!6220!6220U42(a!6220!6220isNeList(_x0))]] [[a!6220!6220U42(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U51(tt, _x0)]] = x0 >= x0 = [[a!6220!6220U52(a!6220!6220isList(_x0))]] [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U61(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U71(tt, _x0)]] = 2x0 >= 2x0 = [[a!6220!6220U72(a!6220!6220isPal(_x0))]] [[a!6220!6220U72(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U81(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[a!6220!6220U11(a!6220!6220isNeList(_x0))]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[a!6220!6220U31(a!6220!6220isQid(_x0))]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220U61(a!6220!6220isQid(_x0))]] [[a!6220!6220isPal(_x0)]] = 2x0 >= 2x0 = [[a!6220!6220U81(a!6220!6220isNePal(_x0))]] [[a!6220!6220isQid(a)]] = 2 > 0 = [[tt]] [[a!6220!6220isQid(e)]] = 2 > 0 = [[tt]] [[a!6220!6220isQid(i)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(o)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(u)]] = 0 >= 0 = [[tt]] [[mark(!6220!6220(_x0, _x1))]] = 4 + 2x0 + 2x1 > 3 + 2x0 + 2x1 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(U11(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U11(mark(_x0))]] [[mark(U21(_x0, _x1))]] = 2 + 2x0 + 2x1 > 1 + 2x0 + 2x1 = [[a!6220!6220U21(mark(_x0), _x1)]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(isList(_x0))]] = 2x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(U31(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U31(mark(_x0))]] [[mark(U41(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 + 2x1 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(U42(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220U42(mark(_x0))]] [[mark(isNeList(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNeList(_x0)]] [[mark(U51(_x0, _x1))]] = 2x1 + 4x0 >= x1 + 4x0 = [[a!6220!6220U51(mark(_x0), _x1)]] [[mark(U52(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U71(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U71(mark(_x0), _x1)]] [[mark(U72(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(isPal(_x0))]] = 4x0 >= 2x0 = [[a!6220!6220isPal(_x0)]] [[mark(U81(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 4 > 2 = [[a]] [[mark(e)]] = 4 > 2 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = 3 + x0 + x1 > 2 + x0 + x1 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220U11(_x0)]] = x0 >= x0 = [[U11(_x0)]] [[a!6220!6220U21(_x0, _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220U31(_x0)]] = x0 >= x0 = [[U31(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[U41(_x0, _x1)]] [[a!6220!6220U42(_x0)]] = 2x0 >= 2x0 = [[U42(_x0)]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[isNeList(_x0)]] [[a!6220!6220U51(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[U51(_x0, _x1)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U71(_x0, _x1)]] = x0 + 2x1 >= x0 + x1 = [[U71(_x0, _x1)]] [[a!6220!6220U72(_x0)]] = x0 >= x0 = [[U72(_x0)]] [[a!6220!6220isPal(_x0)]] = 2x0 >= 2x0 = [[isPal(_x0)]] [[a!6220!6220U81(_x0)]] = 2x0 >= 2x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] We can thus remove the following rules: a!6220!6220U21(tt, X) => a!6220!6220U22(a!6220!6220isList(X)) a!6220!6220isQid(a) => tt a!6220!6220isQid(e) => tt mark(!6220!6220(X, Y)) => a!6220!6220!6220!6220(mark(X), mark(Y)) mark(U21(X, Y)) => a!6220!6220U21(mark(X), Y) mark(a) => a mark(e) => e a!6220!6220!6220!6220(X, Y) => !6220!6220(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!6220U11(tt) >? tt a!6220!6220U22(tt) >? tt a!6220!6220U31(tt) >? tt a!6220!6220U41(tt, X) >? a!6220!6220U42(a!6220!6220isNeList(X)) a!6220!6220U42(tt) >? tt a!6220!6220U51(tt, X) >? a!6220!6220U52(a!6220!6220isList(X)) a!6220!6220U52(tt) >? tt a!6220!6220U61(tt) >? tt a!6220!6220U71(tt, X) >? a!6220!6220U72(a!6220!6220isPal(X)) a!6220!6220U72(tt) >? tt a!6220!6220U81(tt) >? tt a!6220!6220isList(X) >? a!6220!6220U11(a!6220!6220isNeList(X)) a!6220!6220isNeList(X) >? a!6220!6220U31(a!6220!6220isQid(X)) a!6220!6220isNePal(X) >? a!6220!6220U61(a!6220!6220isQid(X)) a!6220!6220isPal(X) >? a!6220!6220U81(a!6220!6220isNePal(X)) a!6220!6220isQid(i) >? tt a!6220!6220isQid(o) >? tt a!6220!6220isQid(u) >? tt mark(U11(X)) >? a!6220!6220U11(mark(X)) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(isList(X)) >? a!6220!6220isList(X) mark(U31(X)) >? a!6220!6220U31(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U42(X)) >? a!6220!6220U42(mark(X)) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(U51(X, Y)) >? a!6220!6220U51(mark(X), Y) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U71(X, Y)) >? a!6220!6220U71(mark(X), Y) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(isPal(X)) >? a!6220!6220isPal(X) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(nil) >? nil mark(tt) >? tt mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220U11(X) >? U11(X) a!6220!6220U21(X, Y) >? U21(X, Y) a!6220!6220U22(X) >? U22(X) a!6220!6220isList(X) >? isList(X) a!6220!6220U31(X) >? U31(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U42(X) >? U42(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220U51(X, Y) >? U51(X, Y) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y) >? U71(X, Y) a!6220!6220U72(X) >? U72(X) a!6220!6220isPal(X) >? isPal(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U11 = \y0.y0 U21 = \y0y1.y0 + y1 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y0 + y1 U42 = \y0.y0 U51 = \y0y1.3 + y0 + 2y1 U52 = \y0.2y0 U61 = \y0.y0 U71 = \y0y1.2 + y0 + y1 U72 = \y0.y0 U81 = \y0.y0 a!6220!6220U11 = \y0.y0 a!6220!6220U21 = \y0y1.3 + y0 + y1 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0.y0 a!6220!6220U41 = \y0y1.y0 + y1 a!6220!6220U42 = \y0.y0 a!6220!6220U51 = \y0y1.3 + y0 + 2y1 a!6220!6220U52 = \y0.2y0 a!6220!6220U61 = \y0.y0 a!6220!6220U71 = \y0y1.2 + y0 + y1 a!6220!6220U72 = \y0.y0 a!6220!6220U81 = \y0.y0 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.y0 a!6220!6220isQid = \y0.y0 i = 0 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.2y0 nil = 0 o = 0 tt = 0 u = 2 Using this interpretation, the requirements translate to: [[a!6220!6220U11(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U31(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U41(tt, _x0)]] = x0 >= x0 = [[a!6220!6220U42(a!6220!6220isNeList(_x0))]] [[a!6220!6220U42(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U51(tt, _x0)]] = 3 + 2x0 > 2x0 = [[a!6220!6220U52(a!6220!6220isList(_x0))]] [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U61(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U71(tt, _x0)]] = 2 + x0 > x0 = [[a!6220!6220U72(a!6220!6220isPal(_x0))]] [[a!6220!6220U72(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U81(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[a!6220!6220U11(a!6220!6220isNeList(_x0))]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[a!6220!6220U31(a!6220!6220isQid(_x0))]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220U61(a!6220!6220isQid(_x0))]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[a!6220!6220U81(a!6220!6220isNePal(_x0))]] [[a!6220!6220isQid(i)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(o)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(u)]] = 2 > 0 = [[tt]] [[mark(U11(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U11(mark(_x0))]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(isList(_x0))]] = 2x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(U31(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U31(mark(_x0))]] [[mark(U41(_x0, _x1))]] = 2x0 + 2x1 >= x1 + 2x0 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(U42(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U42(mark(_x0))]] [[mark(isNeList(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNeList(_x0)]] [[mark(U51(_x0, _x1))]] = 6 + 2x0 + 4x1 > 3 + 2x0 + 2x1 = [[a!6220!6220U51(mark(_x0), _x1)]] [[mark(U52(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U71(_x0, _x1))]] = 4 + 2x0 + 2x1 > 2 + x1 + 2x0 = [[a!6220!6220U71(mark(_x0), _x1)]] [[mark(U72(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(isPal(_x0))]] = 2x0 >= x0 = [[a!6220!6220isPal(_x0)]] [[mark(U81(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 4 > 2 = [[u]] [[a!6220!6220U11(_x0)]] = x0 >= x0 = [[U11(_x0)]] [[a!6220!6220U21(_x0, _x1)]] = 3 + x0 + x1 > x0 + x1 = [[U21(_x0, _x1)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220U31(_x0)]] = x0 >= x0 = [[U31(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[U41(_x0, _x1)]] [[a!6220!6220U42(_x0)]] = x0 >= x0 = [[U42(_x0)]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[isNeList(_x0)]] [[a!6220!6220U51(_x0, _x1)]] = 3 + x0 + 2x1 >= 3 + x0 + 2x1 = [[U51(_x0, _x1)]] [[a!6220!6220U52(_x0)]] = 2x0 >= 2x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U71(_x0, _x1)]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[U71(_x0, _x1)]] [[a!6220!6220U72(_x0)]] = x0 >= x0 = [[U72(_x0)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[isPal(_x0)]] [[a!6220!6220U81(_x0)]] = x0 >= x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] We can thus remove the following rules: a!6220!6220U51(tt, X) => a!6220!6220U52(a!6220!6220isList(X)) a!6220!6220U71(tt, X) => a!6220!6220U72(a!6220!6220isPal(X)) a!6220!6220isQid(u) => tt mark(U51(X, Y)) => a!6220!6220U51(mark(X), Y) mark(U71(X, Y)) => a!6220!6220U71(mark(X), Y) mark(u) => u 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!6220U11(tt) >? tt a!6220!6220U22(tt) >? tt a!6220!6220U31(tt) >? tt a!6220!6220U41(tt, X) >? a!6220!6220U42(a!6220!6220isNeList(X)) a!6220!6220U42(tt) >? tt a!6220!6220U52(tt) >? tt a!6220!6220U61(tt) >? tt a!6220!6220U72(tt) >? tt a!6220!6220U81(tt) >? tt a!6220!6220isList(X) >? a!6220!6220U11(a!6220!6220isNeList(X)) a!6220!6220isNeList(X) >? a!6220!6220U31(a!6220!6220isQid(X)) a!6220!6220isNePal(X) >? a!6220!6220U61(a!6220!6220isQid(X)) a!6220!6220isPal(X) >? a!6220!6220U81(a!6220!6220isNePal(X)) a!6220!6220isQid(i) >? tt a!6220!6220isQid(o) >? tt mark(U11(X)) >? a!6220!6220U11(mark(X)) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(isList(X)) >? a!6220!6220isList(X) mark(U31(X)) >? a!6220!6220U31(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U42(X)) >? a!6220!6220U42(mark(X)) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(isPal(X)) >? a!6220!6220isPal(X) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(nil) >? nil mark(tt) >? tt mark(i) >? i mark(o) >? o a!6220!6220U11(X) >? U11(X) a!6220!6220U22(X) >? U22(X) a!6220!6220isList(X) >? isList(X) a!6220!6220U31(X) >? U31(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U42(X) >? U42(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220U51(X, Y) >? U51(X, Y) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U71(X, Y) >? U71(X, Y) a!6220!6220U72(X) >? U72(X) a!6220!6220isPal(X) >? isPal(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U11 = \y0.y0 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.2y0 + 2y1 U42 = \y0.1 + y0 U51 = \y0y1.y0 + y1 U52 = \y0.y0 U61 = \y0.2y0 U71 = \y0y1.y0 + y1 U72 = \y0.y0 U81 = \y0.y0 a!6220!6220U11 = \y0.y0 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0.y0 a!6220!6220U41 = \y0y1.2y0 + 2y1 a!6220!6220U42 = \y0.1 + y0 a!6220!6220U51 = \y0y1.3 + y0 + y1 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.2y0 a!6220!6220U71 = \y0y1.3 + 2y0 + 2y1 a!6220!6220U72 = \y0.y0 a!6220!6220U81 = \y0.y0 a!6220!6220isList = \y0.3 + y0 a!6220!6220isNeList = \y0.1 + y0 a!6220!6220isNePal = \y0.2y0 a!6220!6220isPal = \y0.2y0 a!6220!6220isQid = \y0.y0 i = 2 isList = \y0.2 + y0 isNeList = \y0.1 + y0 isNePal = \y0.2y0 isPal = \y0.2y0 isQid = \y0.y0 mark = \y0.2y0 nil = 0 o = 2 tt = 2 Using this interpretation, the requirements translate to: [[a!6220!6220U11(tt)]] = 2 >= 2 = [[tt]] [[a!6220!6220U22(tt)]] = 2 >= 2 = [[tt]] [[a!6220!6220U31(tt)]] = 2 >= 2 = [[tt]] [[a!6220!6220U41(tt, _x0)]] = 4 + 2x0 > 2 + x0 = [[a!6220!6220U42(a!6220!6220isNeList(_x0))]] [[a!6220!6220U42(tt)]] = 3 > 2 = [[tt]] [[a!6220!6220U52(tt)]] = 2 >= 2 = [[tt]] [[a!6220!6220U61(tt)]] = 4 > 2 = [[tt]] [[a!6220!6220U72(tt)]] = 2 >= 2 = [[tt]] [[a!6220!6220U81(tt)]] = 2 >= 2 = [[tt]] [[a!6220!6220isList(_x0)]] = 3 + x0 > 1 + x0 = [[a!6220!6220U11(a!6220!6220isNeList(_x0))]] [[a!6220!6220isNeList(_x0)]] = 1 + x0 > x0 = [[a!6220!6220U31(a!6220!6220isQid(_x0))]] [[a!6220!6220isNePal(_x0)]] = 2x0 >= 2x0 = [[a!6220!6220U61(a!6220!6220isQid(_x0))]] [[a!6220!6220isPal(_x0)]] = 2x0 >= 2x0 = [[a!6220!6220U81(a!6220!6220isNePal(_x0))]] [[a!6220!6220isQid(i)]] = 2 >= 2 = [[tt]] [[a!6220!6220isQid(o)]] = 2 >= 2 = [[tt]] [[mark(U11(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U11(mark(_x0))]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(isList(_x0))]] = 4 + 2x0 > 3 + x0 = [[a!6220!6220isList(_x0)]] [[mark(U31(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U31(mark(_x0))]] [[mark(U41(_x0, _x1))]] = 4x0 + 4x1 >= 2x1 + 4x0 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(U42(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[a!6220!6220U42(mark(_x0))]] [[mark(isNeList(_x0))]] = 2 + 2x0 > 1 + x0 = [[a!6220!6220isNeList(_x0)]] [[mark(U52(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U72(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(isPal(_x0))]] = 4x0 >= 2x0 = [[a!6220!6220isPal(_x0)]] [[mark(U81(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = 4x0 >= 2x0 = [[a!6220!6220isNePal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 4 > 2 = [[tt]] [[mark(i)]] = 4 > 2 = [[i]] [[mark(o)]] = 4 > 2 = [[o]] [[a!6220!6220U11(_x0)]] = x0 >= x0 = [[U11(_x0)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220isList(_x0)]] = 3 + x0 > 2 + x0 = [[isList(_x0)]] [[a!6220!6220U31(_x0)]] = x0 >= x0 = [[U31(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U41(_x0, _x1)]] [[a!6220!6220U42(_x0)]] = 1 + x0 >= 1 + x0 = [[U42(_x0)]] [[a!6220!6220isNeList(_x0)]] = 1 + x0 >= 1 + x0 = [[isNeList(_x0)]] [[a!6220!6220U51(_x0, _x1)]] = 3 + x0 + x1 > x0 + x1 = [[U51(_x0, _x1)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = 2x0 >= 2x0 = [[U61(_x0)]] [[a!6220!6220U71(_x0, _x1)]] = 3 + 2x0 + 2x1 > x0 + x1 = [[U71(_x0, _x1)]] [[a!6220!6220U72(_x0)]] = x0 >= x0 = [[U72(_x0)]] [[a!6220!6220isPal(_x0)]] = 2x0 >= 2x0 = [[isPal(_x0)]] [[a!6220!6220U81(_x0)]] = x0 >= x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = 2x0 >= 2x0 = [[isNePal(_x0)]] We can thus remove the following rules: a!6220!6220U41(tt, X) => a!6220!6220U42(a!6220!6220isNeList(X)) a!6220!6220U42(tt) => tt a!6220!6220U61(tt) => tt a!6220!6220isList(X) => a!6220!6220U11(a!6220!6220isNeList(X)) a!6220!6220isNeList(X) => a!6220!6220U31(a!6220!6220isQid(X)) mark(isList(X)) => a!6220!6220isList(X) mark(U42(X)) => a!6220!6220U42(mark(X)) mark(isNeList(X)) => a!6220!6220isNeList(X) mark(tt) => tt mark(i) => i mark(o) => o a!6220!6220isList(X) => isList(X) a!6220!6220U51(X, Y) => U51(X, Y) a!6220!6220U71(X, Y) => U71(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!6220U11(tt) >? tt a!6220!6220U22(tt) >? tt a!6220!6220U31(tt) >? tt a!6220!6220U52(tt) >? tt a!6220!6220U72(tt) >? tt a!6220!6220U81(tt) >? tt a!6220!6220isNePal(X) >? a!6220!6220U61(a!6220!6220isQid(X)) a!6220!6220isPal(X) >? a!6220!6220U81(a!6220!6220isNePal(X)) a!6220!6220isQid(i) >? tt a!6220!6220isQid(o) >? tt mark(U11(X)) >? a!6220!6220U11(mark(X)) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X)) >? a!6220!6220U31(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(isPal(X)) >? a!6220!6220isPal(X) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(nil) >? nil a!6220!6220U11(X) >? U11(X) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X) >? U31(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U42(X) >? U42(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U72(X) >? U72(X) a!6220!6220isPal(X) >? isPal(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U11 = \y0.y0 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y1 + 2y0 U42 = \y0.y0 U52 = \y0.y0 U61 = \y0.y0 U72 = \y0.y0 U81 = \y0.y0 a!6220!6220U11 = \y0.y0 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0.y0 a!6220!6220U41 = \y0y1.y1 + 2y0 a!6220!6220U42 = \y0.3 + y0 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.y0 a!6220!6220U72 = \y0.y0 a!6220!6220U81 = \y0.y0 a!6220!6220isNeList = \y0.3 + 2y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.y0 a!6220!6220isQid = \y0.y0 i = 3 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.2y0 nil = 1 o = 3 tt = 0 Using this interpretation, the requirements translate to: [[a!6220!6220U11(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U31(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U72(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U81(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220U61(a!6220!6220isQid(_x0))]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[a!6220!6220U81(a!6220!6220isNePal(_x0))]] [[a!6220!6220isQid(i)]] = 3 > 0 = [[tt]] [[a!6220!6220isQid(o)]] = 3 > 0 = [[tt]] [[mark(U11(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U11(mark(_x0))]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U31(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U31(mark(_x0))]] [[mark(U41(_x0, _x1))]] = 2x1 + 4x0 >= x1 + 4x0 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(U52(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U72(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(isPal(_x0))]] = 2x0 >= x0 = [[a!6220!6220isPal(_x0)]] [[mark(U81(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(nil)]] = 2 > 1 = [[nil]] [[a!6220!6220U11(_x0)]] = x0 >= x0 = [[U11(_x0)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U31(_x0)]] = x0 >= x0 = [[U31(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[U41(_x0, _x1)]] [[a!6220!6220U42(_x0)]] = 3 + x0 > x0 = [[U42(_x0)]] [[a!6220!6220isNeList(_x0)]] = 3 + 2x0 > x0 = [[isNeList(_x0)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U72(_x0)]] = x0 >= x0 = [[U72(_x0)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[isPal(_x0)]] [[a!6220!6220U81(_x0)]] = x0 >= x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] We can thus remove the following rules: a!6220!6220isQid(i) => tt a!6220!6220isQid(o) => tt mark(nil) => nil a!6220!6220U42(X) => U42(X) a!6220!6220isNeList(X) => isNeList(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) >? tt a!6220!6220U22(tt) >? tt a!6220!6220U31(tt) >? tt a!6220!6220U52(tt) >? tt a!6220!6220U72(tt) >? tt a!6220!6220U81(tt) >? tt a!6220!6220isNePal(X) >? a!6220!6220U61(a!6220!6220isQid(X)) a!6220!6220isPal(X) >? a!6220!6220U81(a!6220!6220isNePal(X)) mark(U11(X)) >? a!6220!6220U11(mark(X)) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X)) >? a!6220!6220U31(mark(X)) mark(U41(X, Y)) >? a!6220!6220U41(mark(X), Y) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(isPal(X)) >? a!6220!6220isPal(X) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) a!6220!6220U11(X) >? U11(X) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X) >? U31(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U72(X) >? U72(X) a!6220!6220isPal(X) >? isPal(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U11 = \y0.2 + 2y0 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.1 + y0 + y1 U52 = \y0.2y0 U61 = \y0.y0 U72 = \y0.y0 U81 = \y0.y0 a!6220!6220U11 = \y0.3 + 2y0 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0.y0 a!6220!6220U41 = \y0y1.1 + y0 + 2y1 a!6220!6220U52 = \y0.2y0 a!6220!6220U61 = \y0.y0 a!6220!6220U72 = \y0.y0 a!6220!6220U81 = \y0.y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.3 + y0 a!6220!6220isQid = \y0.y0 isNePal = \y0.y0 isPal = \y0.2 + y0 isQid = \y0.y0 mark = \y0.2y0 tt = 0 Using this interpretation, the requirements translate to: [[a!6220!6220U11(tt)]] = 3 > 0 = [[tt]] [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U31(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U72(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U81(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220U61(a!6220!6220isQid(_x0))]] [[a!6220!6220isPal(_x0)]] = 3 + x0 > x0 = [[a!6220!6220U81(a!6220!6220isNePal(_x0))]] [[mark(U11(_x0))]] = 4 + 4x0 > 3 + 4x0 = [[a!6220!6220U11(mark(_x0))]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U31(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U31(mark(_x0))]] [[mark(U41(_x0, _x1))]] = 2 + 2x0 + 2x1 > 1 + 2x0 + 2x1 = [[a!6220!6220U41(mark(_x0), _x1)]] [[mark(U52(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U72(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(isPal(_x0))]] = 4 + 2x0 > 3 + x0 = [[a!6220!6220isPal(_x0)]] [[mark(U81(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = 2x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[a!6220!6220U11(_x0)]] = 3 + 2x0 > 2 + 2x0 = [[U11(_x0)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U31(_x0)]] = x0 >= x0 = [[U31(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + x1 = [[U41(_x0, _x1)]] [[a!6220!6220U52(_x0)]] = 2x0 >= 2x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U72(_x0)]] = x0 >= x0 = [[U72(_x0)]] [[a!6220!6220isPal(_x0)]] = 3 + x0 > 2 + x0 = [[isPal(_x0)]] [[a!6220!6220U81(_x0)]] = x0 >= x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] We can thus remove the following rules: a!6220!6220U11(tt) => tt a!6220!6220isPal(X) => a!6220!6220U81(a!6220!6220isNePal(X)) mark(U11(X)) => a!6220!6220U11(mark(X)) mark(U41(X, Y)) => a!6220!6220U41(mark(X), Y) mark(isPal(X)) => a!6220!6220isPal(X) a!6220!6220U11(X) => U11(X) a!6220!6220isPal(X) => isPal(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) >? tt a!6220!6220U52(tt) >? tt a!6220!6220U72(tt) >? tt a!6220!6220U81(tt) >? tt a!6220!6220isNePal(X) >? a!6220!6220U61(a!6220!6220isQid(X)) mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U31(X)) >? a!6220!6220U31(mark(X)) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) a!6220!6220U22(X) >? U22(X) a!6220!6220U31(X) >? U31(X) a!6220!6220U41(X, Y) >? U41(X, Y) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U72(X) >? U72(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U22 = \y0.y0 U31 = \y0.2 + y0 U41 = \y0y1.y0 + y1 U52 = \y0.y0 U61 = \y0.y0 U72 = \y0.y0 U81 = \y0.y0 a!6220!6220U22 = \y0.y0 a!6220!6220U31 = \y0.3 + y0 a!6220!6220U41 = \y0y1.3 + 2y0 + 2y1 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.y0 a!6220!6220U72 = \y0.y0 a!6220!6220U81 = \y0.y0 a!6220!6220isNePal = \y0.3 + 2y0 a!6220!6220isQid = \y0.y0 isNePal = \y0.2 + 2y0 isQid = \y0.y0 mark = \y0.2y0 tt = 0 Using this interpretation, the requirements translate to: [[a!6220!6220U22(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U31(tt)]] = 3 > 0 = [[tt]] [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U72(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220U81(tt)]] = 0 >= 0 = [[tt]] [[a!6220!6220isNePal(_x0)]] = 3 + 2x0 > x0 = [[a!6220!6220U61(a!6220!6220isQid(_x0))]] [[mark(U22(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U31(_x0))]] = 4 + 2x0 > 3 + 2x0 = [[a!6220!6220U31(mark(_x0))]] [[mark(U52(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U72(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(U81(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = 4 + 4x0 > 3 + 2x0 = [[a!6220!6220isNePal(_x0)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U31(_x0)]] = 3 + x0 > 2 + x0 = [[U31(_x0)]] [[a!6220!6220U41(_x0, _x1)]] = 3 + 2x0 + 2x1 > x0 + x1 = [[U41(_x0, _x1)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U72(_x0)]] = x0 >= x0 = [[U72(_x0)]] [[a!6220!6220U81(_x0)]] = x0 >= x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = 3 + 2x0 > 2 + 2x0 = [[isNePal(_x0)]] We can thus remove the following rules: a!6220!6220U31(tt) => tt a!6220!6220isNePal(X) => a!6220!6220U61(a!6220!6220isQid(X)) mark(U31(X)) => a!6220!6220U31(mark(X)) mark(isNePal(X)) => a!6220!6220isNePal(X) a!6220!6220U31(X) => U31(X) a!6220!6220U41(X, Y) => U41(X, Y) a!6220!6220isNePal(X) => isNePal(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!6220U52(tt) >? tt a!6220!6220U72(tt) >? tt a!6220!6220U81(tt) >? tt mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) a!6220!6220U22(X) >? U22(X) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U72(X) >? U72(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U22 = \y0.y0 U52 = \y0.y0 U61 = \y0.y0 U72 = \y0.2y0 U81 = \y0.y0 a!6220!6220U22 = \y0.y0 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.y0 a!6220!6220U72 = \y0.2y0 a!6220!6220U81 = \y0.y0 a!6220!6220isQid = \y0.y0 isQid = \y0.y0 mark = \y0.y0 tt = 1 Using this interpretation, the requirements translate to: [[a!6220!6220U22(tt)]] = 1 >= 1 = [[tt]] [[a!6220!6220U52(tt)]] = 1 >= 1 = [[tt]] [[a!6220!6220U72(tt)]] = 2 > 1 = [[tt]] [[a!6220!6220U81(tt)]] = 1 >= 1 = [[tt]] [[mark(U22(_x0))]] = x0 >= x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U52(_x0))]] = x0 >= x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = x0 >= x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U72(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(U81(_x0))]] = x0 >= x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220U22(_x0)]] = x0 >= x0 = [[U22(_x0)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U72(_x0)]] = 2x0 >= 2x0 = [[U72(_x0)]] [[a!6220!6220U81(_x0)]] = x0 >= x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] We can thus remove the following rules: a!6220!6220U72(tt) => 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!6220U22(tt) >? tt a!6220!6220U52(tt) >? tt a!6220!6220U81(tt) >? tt mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(U81(X)) >? a!6220!6220U81(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) a!6220!6220U22(X) >? U22(X) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U72(X) >? U72(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U22 = \y0.1 + y0 U52 = \y0.y0 U61 = \y0.y0 U72 = \y0.2y0 U81 = \y0.1 + y0 a!6220!6220U22 = \y0.2 + y0 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.y0 a!6220!6220U72 = \y0.2y0 a!6220!6220U81 = \y0.1 + y0 a!6220!6220isQid = \y0.2y0 isQid = \y0.y0 mark = \y0.2y0 tt = 1 Using this interpretation, the requirements translate to: [[a!6220!6220U22(tt)]] = 3 > 1 = [[tt]] [[a!6220!6220U52(tt)]] = 1 >= 1 = [[tt]] [[a!6220!6220U81(tt)]] = 2 > 1 = [[tt]] [[mark(U22(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U52(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U72(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(U81(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[a!6220!6220U81(mark(_x0))]] [[mark(isQid(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220U22(_x0)]] = 2 + x0 > 1 + x0 = [[U22(_x0)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U72(_x0)]] = 2x0 >= 2x0 = [[U72(_x0)]] [[a!6220!6220U81(_x0)]] = 1 + x0 >= 1 + x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = 2x0 >= x0 = [[isQid(_x0)]] We can thus remove the following rules: a!6220!6220U22(tt) => tt a!6220!6220U81(tt) => tt mark(U81(X)) => a!6220!6220U81(mark(X)) a!6220!6220U22(X) => U22(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!6220U52(tt) >? tt mark(U22(X)) >? a!6220!6220U22(mark(X)) mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U72(X) >? U72(X) a!6220!6220U81(X) >? U81(X) a!6220!6220isQid(X) >? isQid(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U22 = \y0.3 + 3y0 U52 = \y0.y0 U61 = \y0.2y0 U72 = \y0.y0 U81 = \y0.y0 a!6220!6220U22 = \y0.y0 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.2y0 a!6220!6220U72 = \y0.y0 a!6220!6220U81 = \y0.3 + y0 a!6220!6220isQid = \y0.3 + 2y0 isQid = \y0.3 + 2y0 mark = \y0.y0 tt = 0 Using this interpretation, the requirements translate to: [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[mark(U22(_x0))]] = 3 + 3x0 > x0 = [[a!6220!6220U22(mark(_x0))]] [[mark(U52(_x0))]] = x0 >= x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U72(_x0))]] = x0 >= x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(isQid(_x0))]] = 3 + 2x0 >= 3 + 2x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = 2x0 >= 2x0 = [[U61(_x0)]] [[a!6220!6220U72(_x0)]] = x0 >= x0 = [[U72(_x0)]] [[a!6220!6220U81(_x0)]] = 3 + x0 > x0 = [[U81(_x0)]] [[a!6220!6220isQid(_x0)]] = 3 + 2x0 >= 3 + 2x0 = [[isQid(_x0)]] We can thus remove the following rules: mark(U22(X)) => a!6220!6220U22(mark(X)) a!6220!6220U81(X) => U81(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!6220U52(tt) >? tt mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) mark(U72(X)) >? a!6220!6220U72(mark(X)) mark(isQid(X)) >? a!6220!6220isQid(X) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U72(X) >? U72(X) a!6220!6220isQid(X) >? isQid(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U52 = \y0.y0 U61 = \y0.y0 U72 = \y0.2 + y0 a!6220!6220U52 = \y0.y0 a!6220!6220U61 = \y0.y0 a!6220!6220U72 = \y0.2 + y0 a!6220!6220isQid = \y0.3 + y0 isQid = \y0.3 + y0 mark = \y0.2y0 tt = 0 Using this interpretation, the requirements translate to: [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[mark(U52(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U61(mark(_x0))]] [[mark(U72(_x0))]] = 4 + 2x0 > 2 + 2x0 = [[a!6220!6220U72(mark(_x0))]] [[mark(isQid(_x0))]] = 6 + 2x0 > 3 + x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220U52(_x0)]] = x0 >= x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = x0 >= x0 = [[U61(_x0)]] [[a!6220!6220U72(_x0)]] = 2 + x0 >= 2 + x0 = [[U72(_x0)]] [[a!6220!6220isQid(_x0)]] = 3 + x0 >= 3 + x0 = [[isQid(_x0)]] We can thus remove the following rules: mark(U72(X)) => a!6220!6220U72(mark(X)) mark(isQid(X)) => a!6220!6220isQid(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!6220U52(tt) >? tt mark(U52(X)) >? a!6220!6220U52(mark(X)) mark(U61(X)) >? a!6220!6220U61(mark(X)) a!6220!6220U52(X) >? U52(X) a!6220!6220U61(X) >? U61(X) a!6220!6220U72(X) >? U72(X) a!6220!6220isQid(X) >? isQid(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U52 = \y0.3y0 U61 = \y0.1 + y0 U72 = \y0.y0 a!6220!6220U52 = \y0.3y0 a!6220!6220U61 = \y0.2 + y0 a!6220!6220U72 = \y0.3 + 2y0 a!6220!6220isQid = \y0.3 + y0 isQid = \y0.y0 mark = \y0.3y0 tt = 0 Using this interpretation, the requirements translate to: [[a!6220!6220U52(tt)]] = 0 >= 0 = [[tt]] [[mark(U52(_x0))]] = 9x0 >= 9x0 = [[a!6220!6220U52(mark(_x0))]] [[mark(U61(_x0))]] = 3 + 3x0 > 2 + 3x0 = [[a!6220!6220U61(mark(_x0))]] [[a!6220!6220U52(_x0)]] = 3x0 >= 3x0 = [[U52(_x0)]] [[a!6220!6220U61(_x0)]] = 2 + x0 > 1 + x0 = [[U61(_x0)]] [[a!6220!6220U72(_x0)]] = 3 + 2x0 > x0 = [[U72(_x0)]] [[a!6220!6220isQid(_x0)]] = 3 + x0 > x0 = [[isQid(_x0)]] We can thus remove the following rules: mark(U61(X)) => a!6220!6220U61(mark(X)) a!6220!6220U61(X) => U61(X) a!6220!6220U72(X) => U72(X) a!6220!6220isQid(X) => isQid(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!6220U52(tt) >? tt mark(U52(X)) >? a!6220!6220U52(mark(X)) a!6220!6220U52(X) >? U52(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U52 = \y0.1 + y0 a!6220!6220U52 = \y0.1 + y0 mark = \y0.y0 tt = 0 Using this interpretation, the requirements translate to: [[a!6220!6220U52(tt)]] = 1 > 0 = [[tt]] [[mark(U52(_x0))]] = 1 + x0 >= 1 + x0 = [[a!6220!6220U52(mark(_x0))]] [[a!6220!6220U52(_x0)]] = 1 + x0 >= 1 + x0 = [[U52(_x0)]] We can thus remove the following rules: a!6220!6220U52(tt) => 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]): mark(U52(X)) >? a!6220!6220U52(mark(X)) a!6220!6220U52(X) >? U52(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U52 = \y0.1 + y0 a!6220!6220U52 = \y0.1 + y0 mark = \y0.2y0 Using this interpretation, the requirements translate to: [[mark(U52(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[a!6220!6220U52(mark(_x0))]] [[a!6220!6220U52(_x0)]] = 1 + x0 >= 1 + x0 = [[U52(_x0)]] We can thus remove the following rules: mark(U52(X)) => a!6220!6220U52(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!6220U52(X) >? U52(X) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U52 = \y0.y0 a!6220!6220U52 = \y0.3 + 3y0 Using this interpretation, the requirements translate to: [[a!6220!6220U52(_x0)]] = 3 + 3x0 > x0 = [[U52(_x0)]] We can thus remove the following rules: a!6220!6220U52(X) => U52(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.