/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. !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 active : [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 ok : [o] --> o proper : [o] --> o top : [o] --> o tt : [] --> o u : [] --> o active(!6220!6220(!6220!6220(X, Y), Z)) => mark(!6220!6220(X, !6220!6220(Y, Z))) active(!6220!6220(X, nil)) => mark(X) active(!6220!6220(nil, X)) => mark(X) active(U11(tt)) => mark(tt) active(U21(tt, X)) => mark(U22(isList(X))) active(U22(tt)) => mark(tt) active(U31(tt)) => mark(tt) active(U41(tt, X)) => mark(U42(isNeList(X))) active(U42(tt)) => mark(tt) active(U51(tt, X)) => mark(U52(isList(X))) active(U52(tt)) => mark(tt) active(U61(tt)) => mark(tt) active(U71(tt, X)) => mark(U72(isPal(X))) active(U72(tt)) => mark(tt) active(U81(tt)) => mark(tt) active(isList(X)) => mark(U11(isNeList(X))) active(isList(nil)) => mark(tt) active(isList(!6220!6220(X, Y))) => mark(U21(isList(X), Y)) active(isNeList(X)) => mark(U31(isQid(X))) active(isNeList(!6220!6220(X, Y))) => mark(U41(isList(X), Y)) active(isNeList(!6220!6220(X, Y))) => mark(U51(isNeList(X), Y)) active(isNePal(X)) => mark(U61(isQid(X))) active(isNePal(!6220!6220(X, !6220!6220(Y, X)))) => mark(U71(isQid(X), Y)) active(isPal(X)) => mark(U81(isNePal(X))) active(isPal(nil)) => mark(tt) active(isQid(a)) => mark(tt) active(isQid(e)) => mark(tt) active(isQid(i)) => mark(tt) active(isQid(o)) => mark(tt) active(isQid(u)) => mark(tt) active(!6220!6220(X, Y)) => !6220!6220(active(X), Y) active(!6220!6220(X, Y)) => !6220!6220(X, active(Y)) active(U11(X)) => U11(active(X)) active(U21(X, Y)) => U21(active(X), Y) active(U22(X)) => U22(active(X)) active(U31(X)) => U31(active(X)) active(U41(X, Y)) => U41(active(X), Y) active(U42(X)) => U42(active(X)) active(U51(X, Y)) => U51(active(X), Y) active(U52(X)) => U52(active(X)) active(U61(X)) => U61(active(X)) active(U71(X, Y)) => U71(active(X), Y) active(U72(X)) => U72(active(X)) active(U81(X)) => U81(active(X)) !6220!6220(mark(X), Y) => mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) => mark(!6220!6220(X, Y)) U11(mark(X)) => mark(U11(X)) U21(mark(X), Y) => mark(U21(X, Y)) U22(mark(X)) => mark(U22(X)) U31(mark(X)) => mark(U31(X)) U41(mark(X), Y) => mark(U41(X, Y)) U42(mark(X)) => mark(U42(X)) U51(mark(X), Y) => mark(U51(X, Y)) U52(mark(X)) => mark(U52(X)) U61(mark(X)) => mark(U61(X)) U71(mark(X), Y) => mark(U71(X, Y)) U72(mark(X)) => mark(U72(X)) U81(mark(X)) => mark(U81(X)) proper(!6220!6220(X, Y)) => !6220!6220(proper(X), proper(Y)) proper(nil) => ok(nil) proper(U11(X)) => U11(proper(X)) proper(tt) => ok(tt) proper(U21(X, Y)) => U21(proper(X), proper(Y)) proper(U22(X)) => U22(proper(X)) proper(isList(X)) => isList(proper(X)) proper(U31(X)) => U31(proper(X)) proper(U41(X, Y)) => U41(proper(X), proper(Y)) proper(U42(X)) => U42(proper(X)) proper(isNeList(X)) => isNeList(proper(X)) proper(U51(X, Y)) => U51(proper(X), proper(Y)) proper(U52(X)) => U52(proper(X)) proper(U61(X)) => U61(proper(X)) proper(U71(X, Y)) => U71(proper(X), proper(Y)) proper(U72(X)) => U72(proper(X)) proper(isPal(X)) => isPal(proper(X)) proper(U81(X)) => U81(proper(X)) proper(isQid(X)) => isQid(proper(X)) proper(isNePal(X)) => isNePal(proper(X)) proper(a) => ok(a) proper(e) => ok(e) proper(i) => ok(i) proper(o) => ok(o) proper(u) => ok(u) !6220!6220(ok(X), ok(Y)) => ok(!6220!6220(X, Y)) U11(ok(X)) => ok(U11(X)) U21(ok(X), ok(Y)) => ok(U21(X, Y)) U22(ok(X)) => ok(U22(X)) isList(ok(X)) => ok(isList(X)) U31(ok(X)) => ok(U31(X)) U41(ok(X), ok(Y)) => ok(U41(X, Y)) U42(ok(X)) => ok(U42(X)) isNeList(ok(X)) => ok(isNeList(X)) U51(ok(X), ok(Y)) => ok(U51(X, Y)) U52(ok(X)) => ok(U52(X)) U61(ok(X)) => ok(U61(X)) U71(ok(X), ok(Y)) => ok(U71(X, Y)) U72(ok(X)) => ok(U72(X)) isPal(ok(X)) => ok(isPal(X)) U81(ok(X)) => ok(U81(X)) isQid(ok(X)) => ok(isQid(X)) isNePal(ok(X)) => ok(isNePal(X)) top(mark(X)) => top(proper(X)) top(ok(X)) => top(active(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]): active(!6220!6220(!6220!6220(X, Y), Z)) >? mark(!6220!6220(X, !6220!6220(Y, Z))) active(!6220!6220(X, nil)) >? mark(X) active(!6220!6220(nil, X)) >? mark(X) active(U11(tt)) >? mark(tt) active(U21(tt, X)) >? mark(U22(isList(X))) active(U22(tt)) >? mark(tt) active(U31(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U51(tt, X)) >? mark(U52(isList(X))) active(U52(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(U71(tt, X)) >? mark(U72(isPal(X))) active(U72(tt)) >? mark(tt) active(U81(tt)) >? mark(tt) active(isList(X)) >? mark(U11(isNeList(X))) active(isList(nil)) >? mark(tt) active(isList(!6220!6220(X, Y))) >? mark(U21(isList(X), Y)) active(isNeList(X)) >? mark(U31(isQid(X))) active(isNeList(!6220!6220(X, Y))) >? mark(U41(isList(X), Y)) active(isNeList(!6220!6220(X, Y))) >? mark(U51(isNeList(X), Y)) active(isNePal(X)) >? mark(U61(isQid(X))) active(isNePal(!6220!6220(X, !6220!6220(Y, X)))) >? mark(U71(isQid(X), Y)) active(isPal(X)) >? mark(U81(isNePal(X))) active(isPal(nil)) >? mark(tt) active(isQid(a)) >? mark(tt) active(isQid(e)) >? mark(tt) active(isQid(i)) >? mark(tt) active(isQid(o)) >? mark(tt) active(isQid(u)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(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 + y1 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y1 + 2y0 U42 = \y0.y0 U51 = \y0y1.y1 + 2y0 U52 = \y0.y0 U61 = \y0.2y0 U71 = \y0y1.2y0 + 2y1 U72 = \y0.y0 U81 = \y0.y0 a = 0 active = \y0.y0 e = 1 i = 0 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.2y0 isPal = \y0.2y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(!6220!6220(!6220!6220(_x0, _x1), _x2))]] = x2 + 2x1 + 4x0 >= x2 + 2x0 + 2x1 = [[mark(!6220!6220(_x0, !6220!6220(_x1, _x2)))]] [[active(!6220!6220(_x0, nil))]] = 2x0 >= x0 = [[mark(_x0)]] [[active(!6220!6220(nil, _x0))]] = x0 >= x0 = [[mark(_x0)]] [[active(U11(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U21(tt, _x0))]] = x0 >= x0 = [[mark(U22(isList(_x0)))]] [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U31(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = x0 >= x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U51(tt, _x0))]] = x0 >= x0 = [[mark(U52(isList(_x0)))]] [[active(U52(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U71(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U72(isPal(_x0)))]] [[active(U72(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U81(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(isList(_x0))]] = x0 >= x0 = [[mark(U11(isNeList(_x0)))]] [[active(isList(nil))]] = 0 >= 0 = [[mark(tt)]] [[active(isList(!6220!6220(_x0, _x1)))]] = x1 + 2x0 >= x0 + x1 = [[mark(U21(isList(_x0), _x1))]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U41(isList(_x0), _x1))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U51(isNeList(_x0), _x1))]] [[active(isNePal(_x0))]] = 2x0 >= 2x0 = [[mark(U61(isQid(_x0)))]] [[active(isNePal(!6220!6220(_x0, !6220!6220(_x1, _x0))))]] = 4x1 + 6x0 >= 2x0 + 2x1 = [[mark(U71(isQid(_x0), _x1))]] [[active(isPal(_x0))]] = 2x0 >= 2x0 = [[mark(U81(isNePal(_x0)))]] [[active(isPal(nil))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(a))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(e))]] = 1 > 0 = [[mark(tt)]] [[active(isQid(i))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(o))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(u))]] = 0 >= 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 2x0 >= 2x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = x0 >= x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = x0 >= x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = x0 >= x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = x0 >= x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = x0 >= x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = x0 >= x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = x0 >= x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = x0 >= x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = 2x0 >= 2x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = x0 >= x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 2x0 >= 2x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = x0 >= x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 2x0 >= 2x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 1 >= 1 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = x0 >= x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = x0 >= x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: active(isQid(e)) => mark(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]): active(!6220!6220(!6220!6220(X, Y), Z)) >? mark(!6220!6220(X, !6220!6220(Y, Z))) active(!6220!6220(X, nil)) >? mark(X) active(!6220!6220(nil, X)) >? mark(X) active(U11(tt)) >? mark(tt) active(U21(tt, X)) >? mark(U22(isList(X))) active(U22(tt)) >? mark(tt) active(U31(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U51(tt, X)) >? mark(U52(isList(X))) active(U52(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(U71(tt, X)) >? mark(U72(isPal(X))) active(U72(tt)) >? mark(tt) active(U81(tt)) >? mark(tt) active(isList(X)) >? mark(U11(isNeList(X))) active(isList(nil)) >? mark(tt) active(isList(!6220!6220(X, Y))) >? mark(U21(isList(X), Y)) active(isNeList(X)) >? mark(U31(isQid(X))) active(isNeList(!6220!6220(X, Y))) >? mark(U41(isList(X), Y)) active(isNeList(!6220!6220(X, Y))) >? mark(U51(isNeList(X), Y)) active(isNePal(X)) >? mark(U61(isQid(X))) active(isNePal(!6220!6220(X, !6220!6220(Y, X)))) >? mark(U71(isQid(X), Y)) active(isPal(X)) >? mark(U81(isNePal(X))) active(isPal(nil)) >? mark(tt) active(isQid(a)) >? mark(tt) active(isQid(i)) >? mark(tt) active(isQid(o)) >? mark(tt) active(isQid(u)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.2 + y1 + 2y0 U11 = \y0.y0 U21 = \y0y1.1 + 2y0 + 2y1 U22 = \y0.y0 U31 = \y0.2y0 U41 = \y0y1.y0 + 2y1 U42 = \y0.y0 U51 = \y0y1.1 + y0 + 2y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.1 + y0 + 2y1 U72 = \y0.y0 U81 = \y0.2y0 a = 0 active = \y0.y0 e = 0 i = 0 isList = \y0.1 + 2y0 isNeList = \y0.2y0 isNePal = \y0.y0 isPal = \y0.2y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.2y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(!6220!6220(!6220!6220(_x0, _x1), _x2))]] = 6 + x2 + 2x1 + 4x0 > 4 + x2 + 2x0 + 2x1 = [[mark(!6220!6220(_x0, !6220!6220(_x1, _x2)))]] [[active(!6220!6220(_x0, nil))]] = 2 + 2x0 > x0 = [[mark(_x0)]] [[active(!6220!6220(nil, _x0))]] = 2 + x0 > x0 = [[mark(_x0)]] [[active(U11(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U21(tt, _x0))]] = 1 + 2x0 >= 1 + 2x0 = [[mark(U22(isList(_x0)))]] [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U31(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U51(tt, _x0))]] = 1 + 2x0 >= 1 + 2x0 = [[mark(U52(isList(_x0)))]] [[active(U52(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U71(tt, _x0))]] = 1 + 2x0 > 2x0 = [[mark(U72(isPal(_x0)))]] [[active(U72(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U81(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(isList(_x0))]] = 1 + 2x0 > 2x0 = [[mark(U11(isNeList(_x0)))]] [[active(isList(nil))]] = 1 > 0 = [[mark(tt)]] [[active(isList(!6220!6220(_x0, _x1)))]] = 5 + 2x1 + 4x0 > 3 + 2x1 + 4x0 = [[mark(U21(isList(_x0), _x1))]] [[active(isNeList(_x0))]] = 2x0 >= 2x0 = [[mark(U31(isQid(_x0)))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = 4 + 2x1 + 4x0 > 1 + 2x0 + 2x1 = [[mark(U41(isList(_x0), _x1))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = 4 + 2x1 + 4x0 > 1 + 2x0 + 2x1 = [[mark(U51(isNeList(_x0), _x1))]] [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(U61(isQid(_x0)))]] [[active(isNePal(!6220!6220(_x0, !6220!6220(_x1, _x0))))]] = 4 + 2x1 + 3x0 > 1 + x0 + 2x1 = [[mark(U71(isQid(_x0), _x1))]] [[active(isPal(_x0))]] = 2x0 >= 2x0 = [[mark(U81(isNePal(_x0)))]] [[active(isPal(nil))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(a))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(i))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(o))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(u))]] = 0 >= 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = 1 + 2x0 + 2x1 >= 1 + 2x0 + 2x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = 2x0 >= 2x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = x0 >= x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = 1 + 2x0 + 2x1 >= 1 + 2x0 + 2x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = x0 >= x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = x0 >= x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = x0 >= x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = x0 >= x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = x0 >= x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = 1 + 2x0 + 2x1 >= 1 + 2x0 + 2x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = x0 >= x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = 2x0 >= 2x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = x0 >= x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = 2x0 >= 2x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = x0 >= x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = x0 >= x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 2x0 >= 2x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 2x0 >= 2x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = 1 + 2x0 + 2x1 >= 1 + 2x0 + 2x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = x0 >= x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = 2x0 >= 2x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = 2x0 >= 2x0 = [[top(active(_x0))]] We can thus remove the following rules: active(!6220!6220(!6220!6220(X, Y), Z)) => mark(!6220!6220(X, !6220!6220(Y, Z))) active(!6220!6220(X, nil)) => mark(X) active(!6220!6220(nil, X)) => mark(X) active(U71(tt, X)) => mark(U72(isPal(X))) active(isList(X)) => mark(U11(isNeList(X))) active(isList(nil)) => mark(tt) active(isList(!6220!6220(X, Y))) => mark(U21(isList(X), Y)) active(isNeList(!6220!6220(X, Y))) => mark(U41(isList(X), Y)) active(isNeList(!6220!6220(X, Y))) => mark(U51(isNeList(X), Y)) active(isNePal(!6220!6220(X, !6220!6220(Y, X)))) => mark(U71(isQid(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]): active(U11(tt)) >? mark(tt) active(U21(tt, X)) >? mark(U22(isList(X))) active(U22(tt)) >? mark(tt) active(U31(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U51(tt, X)) >? mark(U52(isList(X))) active(U52(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(U72(tt)) >? mark(tt) active(U81(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(isNePal(X)) >? mark(U61(isQid(X))) active(isPal(X)) >? mark(U81(isNePal(X))) active(isPal(nil)) >? mark(tt) active(isQid(a)) >? mark(tt) active(isQid(i)) >? mark(tt) active(isQid(o)) >? mark(tt) active(isQid(u)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.2y0 U21 = \y0y1.y0 + 2y1 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y1 + 2y0 U42 = \y0.y0 U51 = \y0y1.y0 + 2y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.1 + y0 + y1 U72 = \y0.1 + y0 U81 = \y0.1 + 2y0 a = 0 active = \y0.y0 e = 0 i = 0 isList = \y0.2y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.2 + 2y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(U11(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U21(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U22(isList(_x0)))]] [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U31(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = x0 >= x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U51(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U52(isList(_x0)))]] [[active(U52(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U72(tt))]] = 1 > 0 = [[mark(tt)]] [[active(U81(tt))]] = 1 > 0 = [[mark(tt)]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(U61(isQid(_x0)))]] [[active(isPal(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[mark(U81(isNePal(_x0)))]] [[active(isPal(nil))]] = 2 > 0 = [[mark(tt)]] [[active(isQid(a))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(i))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(o))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(u))]] = 0 >= 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 2x0 >= 2x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 1 + x0 >= 1 + x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = x0 >= x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = x0 >= x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = x0 >= x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = 1 + x0 >= 1 + x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = 2x0 >= 2x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = x0 >= x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = 2x0 >= 2x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = x0 >= x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = x0 >= x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 1 + x0 >= 1 + x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = x0 >= x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: active(U72(tt)) => mark(tt) active(U81(tt)) => mark(tt) active(isPal(X)) => mark(U81(isNePal(X))) active(isPal(nil)) => mark(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]): active(U11(tt)) >? mark(tt) active(U21(tt, X)) >? mark(U22(isList(X))) active(U22(tt)) >? mark(tt) active(U31(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U51(tt, X)) >? mark(U52(isList(X))) active(U52(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(isNePal(X)) >? mark(U61(isQid(X))) active(isQid(a)) >? mark(tt) active(isQid(i)) >? mark(tt) active(isQid(o)) >? mark(tt) active(isQid(u)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + 2y1 U11 = \y0.y0 U21 = \y0y1.y0 + y1 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y0 + y1 U42 = \y0.y0 U51 = \y0y1.1 + y1 + 2y0 U52 = \y0.y0 U61 = \y0.2y0 U71 = \y0y1.y0 + y1 U72 = \y0.y0 U81 = \y0.2y0 a = 0 active = \y0.y0 e = 0 i = 0 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.2y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(U11(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U21(tt, _x0))]] = x0 >= x0 = [[mark(U22(isList(_x0)))]] [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U31(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = x0 >= x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U51(tt, _x0))]] = 1 + x0 > x0 = [[mark(U52(isList(_x0)))]] [[active(U52(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(isNePal(_x0))]] = 2x0 >= 2x0 = [[mark(U61(isQid(_x0)))]] [[active(isQid(a))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(i))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(o))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(u))]] = 0 >= 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 2x0 >= 2x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = x0 >= x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = x0 >= x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = x0 >= x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = x0 >= x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = x0 >= x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = x0 >= x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = x0 >= x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = 2x0 >= 2x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = x0 >= x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = x0 >= x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 2x0 >= 2x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 2x0 >= 2x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = x0 >= x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = x0 >= x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: active(U51(tt, X)) => mark(U52(isList(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]): active(U11(tt)) >? mark(tt) active(U21(tt, X)) >? mark(U22(isList(X))) active(U22(tt)) >? mark(tt) active(U31(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U52(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(isNePal(X)) >? mark(U61(isQid(X))) active(isQid(a)) >? mark(tt) active(isQid(i)) >? mark(tt) active(isQid(o)) >? mark(tt) active(isQid(u)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.2y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.y0 U31 = \y0.2 + y0 U41 = \y0y1.2 + y1 + 2y0 U42 = \y0.y0 U51 = \y0y1.y0 + y1 U52 = \y0.1 + y0 U61 = \y0.2y0 U71 = \y0y1.y0 + y1 U72 = \y0.y0 U81 = \y0.y0 a = 1 active = \y0.y0 e = 0 i = 0 isList = \y0.y0 isNeList = \y0.2 + y0 isNePal = \y0.2y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 1 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(U11(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U21(tt, _x0))]] = x0 >= x0 = [[mark(U22(isList(_x0)))]] [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U31(tt))]] = 2 > 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = 2 + x0 >= 2 + x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U52(tt))]] = 1 > 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(isNeList(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(U31(isQid(_x0)))]] [[active(isNePal(_x0))]] = 2x0 >= 2x0 = [[mark(U61(isQid(_x0)))]] [[active(isQid(a))]] = 1 > 0 = [[mark(tt)]] [[active(isQid(i))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(o))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(u))]] = 0 >= 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 2x0 >= 2x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = 2 + x0 >= 2 + x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = 1 + x0 >= 1 + x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 2x0 >= 2x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = x0 >= x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = x0 >= x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = 1 + x0 >= 1 + x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = x0 >= x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = x0 >= x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 1 >= 1 = [[ok(nil)]] [[proper(U11(_x0))]] = 2x0 >= 2x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = x0 >= x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = 2 + x0 >= 2 + x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = x0 >= x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = 2 + x0 >= 2 + x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = 1 + x0 >= 1 + x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = 2x0 >= 2x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = x0 >= x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = x0 >= x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = x0 >= x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 2x0 >= 2x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 1 >= 1 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = 2 + x0 >= 2 + x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 2 + x0 >= 2 + x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = x0 >= x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = x0 >= x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: active(U31(tt)) => mark(tt) active(U52(tt)) => mark(tt) active(isQid(a)) => mark(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]): active(U11(tt)) >? mark(tt) active(U21(tt, X)) >? mark(U22(isList(X))) active(U22(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(isNePal(X)) >? mark(U61(isQid(X))) active(isQid(i)) >? mark(tt) active(isQid(o)) >? mark(tt) active(isQid(u)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.2y0 + 2y1 U11 = \y0.2y0 U21 = \y0y1.2y0 + 2y1 U22 = \y0.2y0 U31 = \y0.y0 U41 = \y0y1.2y0 + 2y1 U42 = \y0.2y0 U51 = \y0y1.y0 + y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y1 + 2y0 U72 = \y0.2y0 U81 = \y0.y0 a = 0 active = \y0.y0 e = 0 i = 0 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.1 + y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 0 u = 2 Using this interpretation, the requirements translate to: [[active(U11(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U21(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U22(isList(_x0)))]] [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(U61(isQid(_x0)))]] [[active(isQid(i))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(o))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(u))]] = 2 > 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 2x0 >= 2x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 2x0 >= 2x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = x0 >= x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = x0 >= x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = 2x0 >= 2x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = 2x0 >= 2x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 2x0 >= 2x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = x0 >= x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 2x0 >= 2x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 1 + x0 >= 1 + x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = x0 >= x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 2 >= 2 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = x0 >= x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: active(isQid(u)) => mark(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]): active(U11(tt)) >? mark(tt) active(U21(tt, X)) >? mark(U22(isList(X))) active(U22(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(isNePal(X)) >? mark(U61(isQid(X))) active(isQid(i)) >? mark(tt) active(isQid(o)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.y0 U21 = \y0y1.y0 + y1 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.2y0 + 2y1 U42 = \y0.2y0 U51 = \y0y1.y0 + y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y1 + 2y0 U72 = \y0.2y0 U81 = \y0.2y0 a = 0 active = \y0.y0 e = 0 i = 0 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.2y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 2 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(U11(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U21(tt, _x0))]] = x0 >= x0 = [[mark(U22(isList(_x0)))]] [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(U61(isQid(_x0)))]] [[active(isQid(i))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(o))]] = 2 > 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = x0 >= x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = x0 >= x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = x0 >= x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = x0 >= x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = x0 >= x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 2x0 >= 2x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = x0 >= x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 2x0 >= 2x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 2x0 >= 2x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 2x0 >= 2x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 2 >= 2 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = x0 >= x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: active(isQid(o)) => mark(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]): active(U11(tt)) >? mark(tt) active(U21(tt, X)) >? mark(U22(isList(X))) active(U22(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(isNePal(X)) >? mark(U61(isQid(X))) active(isQid(i)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + 2y1 U11 = \y0.2 + y0 U21 = \y0y1.2y0 + 2y1 U22 = \y0.y0 U31 = \y0.2y0 U41 = \y0y1.2y0 + 2y1 U42 = \y0.y0 U51 = \y0y1.y0 + y1 U52 = \y0.y0 U61 = \y0.2y0 U71 = \y0y1.y0 + y1 U72 = \y0.2y0 U81 = \y0.y0 a = 0 active = \y0.y0 e = 0 i = 0 isList = \y0.2y0 isNeList = \y0.2y0 isNePal = \y0.2y0 isPal = \y0.2y0 isQid = \y0.y0 mark = \y0.y0 nil = 3 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(U11(tt))]] = 2 > 0 = [[mark(tt)]] [[active(U21(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U22(isList(_x0)))]] [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(isNeList(_x0))]] = 2x0 >= 2x0 = [[mark(U31(isQid(_x0)))]] [[active(isNePal(_x0))]] = 2x0 >= 2x0 = [[mark(U61(isQid(_x0)))]] [[active(isQid(i))]] = 0 >= 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 2 + x0 >= 2 + x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = 2x0 >= 2x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 2x0 >= 2x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = x0 >= x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = x0 >= x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = x0 >= x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 3 >= 3 = [[ok(nil)]] [[proper(U11(_x0))]] = 2 + x0 >= 2 + x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = x0 >= x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = 2x0 >= 2x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = 2x0 >= 2x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = x0 >= x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = 2x0 >= 2x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = 2x0 >= 2x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 2x0 >= 2x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 2x0 >= 2x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = x0 >= x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 2x0 >= 2x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2 + x0 >= 2 + x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = x0 >= x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: active(U11(tt)) => mark(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]): active(U21(tt, X)) >? mark(U22(isList(X))) active(U22(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(isNePal(X)) >? mark(U61(isQid(X))) active(isQid(i)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.y0 U21 = \y0y1.1 + y0 + 2y1 U22 = \y0.2y0 U31 = \y0.y0 U41 = \y0y1.y1 + 2y0 U42 = \y0.y0 U51 = \y0y1.y0 + y1 U52 = \y0.2y0 U61 = \y0.y0 U71 = \y0y1.y1 + 2y0 U72 = \y0.2y0 U81 = \y0.2y0 a = 0 active = \y0.y0 e = 0 i = 0 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.1 + y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.2y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(U21(tt, _x0))]] = 1 + 2x0 > 2x0 = [[mark(U22(isList(_x0)))]] [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = x0 >= x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(isNePal(_x0))]] = 1 + x0 > x0 = [[mark(U61(isQid(_x0)))]] [[active(isQid(i))]] = 0 >= 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 2x0 >= 2x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = 2x0 >= 2x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = x0 >= x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = x0 >= x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = x0 >= x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = x0 >= x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = 2x0 >= 2x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = x0 >= x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = 2x0 >= 2x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = x0 >= x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 2x0 >= 2x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = x0 >= x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 2x0 >= 2x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 1 + x0 >= 1 + x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = 2x0 >= 2x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = 2x0 >= 2x0 = [[top(active(_x0))]] We can thus remove the following rules: active(U21(tt, X)) => mark(U22(isList(X))) active(isNePal(X)) => mark(U61(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]): active(U22(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(isQid(i)) >? mark(tt) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + 2y1 U11 = \y0.y0 U21 = \y0y1.y0 + y1 U22 = \y0.2y0 U31 = \y0.y0 U41 = \y0y1.y0 + 2y1 U42 = \y0.2y0 U51 = \y0y1.y1 + 2y0 U52 = \y0.2y0 U61 = \y0.y0 U71 = \y0y1.1 + y1 + 2y0 U72 = \y0.y0 U81 = \y0.2y0 a = 0 active = \y0.y0 e = 0 i = 1 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.2y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(U22(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U41(tt, _x0))]] = 2x0 >= 2x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(U61(tt))]] = 0 >= 0 = [[mark(tt)]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(isQid(i))]] = 1 > 0 = [[mark(tt)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 2x0 >= 2x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = 2x0 >= 2x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = x0 >= x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = x0 >= x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = x0 >= x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = x0 >= x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = 2x0 >= 2x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 2x0 >= 2x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = 2x0 >= 2x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = x0 >= x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = x0 >= x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = x0 >= x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 2x0 >= 2x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 1 >= 1 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = x0 >= x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = 2x0 >= 2x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = 2x0 >= 2x0 = [[top(active(_x0))]] We can thus remove the following rules: active(isQid(i)) => mark(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]): active(U22(tt)) >? mark(tt) active(U41(tt, X)) >? mark(U42(isNeList(X))) active(U42(tt)) >? mark(tt) active(U61(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.2y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.2y0 U31 = \y0.y0 U41 = \y0y1.y1 + 2y0 U42 = \y0.1 + y0 U51 = \y0y1.3 + y0 + y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y0 + y1 U72 = \y0.y0 U81 = \y0.2y0 a = 0 active = \y0.y0 e = 1 i = 0 isList = \y0.2y0 isNeList = \y0.y0 isNePal = \y0.1 + y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 1 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 2 u = 0 Using this interpretation, the requirements translate to: [[active(U22(tt))]] = 4 > 2 = [[mark(tt)]] [[active(U41(tt, _x0))]] = 4 + x0 > 1 + x0 = [[mark(U42(isNeList(_x0)))]] [[active(U42(tt))]] = 3 > 2 = [[mark(tt)]] [[active(U61(tt))]] = 2 >= 2 = [[mark(tt)]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 2x0 >= 2x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 2x0 >= 2x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 1 + x0 >= 1 + x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = 3 + x0 + x1 >= 3 + x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = 1 + x0 >= 1 + x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = 3 + x0 + x1 >= 3 + x0 + x1 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = x0 >= x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = x0 >= x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = 2x0 >= 2x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 2 >= 2 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = 2x0 >= 2x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = 2x0 >= 2x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 1 + x0 >= 1 + x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = 3 + x0 + x1 >= 3 + x0 + x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = x0 >= x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = x0 >= x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = x0 >= x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 2x0 >= 2x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 1 + x0 >= 1 + x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 1 >= 1 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 1 >= 1 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = 3 + x0 + x1 >= 3 + x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = x0 >= x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = x0 >= x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: active(U22(tt)) => mark(tt) active(U41(tt, X)) => mark(U42(isNeList(X))) active(U42(tt)) => mark(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]): active(U61(tt)) >? mark(tt) active(isNeList(X)) >? mark(U31(isQid(X))) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + 2y1 U11 = \y0.2y0 U21 = \y0y1.y0 + y1 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y0 + y1 U42 = \y0.2y0 U51 = \y0y1.y0 + y1 U52 = \y0.y0 U61 = \y0.2 + y0 U71 = \y0y1.y0 + y1 U72 = \y0.y0 U81 = \y0.y0 a = 0 active = \y0.y0 e = 0 i = 0 isList = \y0.2y0 isNeList = \y0.y0 isNePal = \y0.2y0 isPal = \y0.2y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(U61(tt))]] = 2 > 0 = [[mark(tt)]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 2x0 >= 2x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 2 + x0 >= 2 + x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = x0 >= x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = x0 >= x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = x0 >= x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = 2x0 >= 2x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = x0 >= x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = 2x0 >= 2x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 2x0 >= 2x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = 2 + x0 >= 2 + x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = x0 >= x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 2x0 >= 2x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = x0 >= x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 2x0 >= 2x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2 + x0 >= 2 + x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = x0 >= x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = x0 >= x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: active(U61(tt)) => mark(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]): active(isNeList(X)) >? mark(U31(isQid(X))) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U21(mark(X), Y) >? mark(U21(X, Y)) U22(mark(X)) >? mark(U22(X)) U31(mark(X)) >? mark(U31(X)) U41(mark(X), Y) >? mark(U41(X, Y)) U42(mark(X)) >? mark(U42(X)) U51(mark(X), Y) >? mark(U51(X, Y)) U52(mark(X)) >? mark(U52(X)) U61(mark(X)) >? mark(U61(X)) U71(mark(X), Y) >? mark(U71(X, Y)) U72(mark(X)) >? mark(U72(X)) U81(mark(X)) >? mark(U81(X)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(mark(X)) >? top(proper(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.2y0 U31 = \y0.y0 U41 = \y0y1.y1 + 2y0 U42 = \y0.2y0 U51 = \y0y1.y1 + 2y0 U52 = \y0.y0 U61 = \y0.2 + 3y0 U71 = \y0y1.y0 + y1 U72 = \y0.2y0 U81 = \y0.2y0 a = 0 active = \y0.y0 e = 1 i = 0 isList = \y0.y0 isNeList = \y0.2 + y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.2 + y0 nil = 0 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.2y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(isNeList(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(U31(isQid(_x0)))]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 2x0 >= 2x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 2 + 3x0 >= 2 + 3x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(U11(_x0))]] [[U21(mark(_x0), _x1)]] = 4 + x1 + 2x0 > 2 + x1 + 2x0 = [[mark(U21(_x0, _x1))]] [[U22(mark(_x0))]] = 4 + 2x0 > 2 + 2x0 = [[mark(U22(_x0))]] [[U31(mark(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(U31(_x0))]] [[U41(mark(_x0), _x1)]] = 4 + x1 + 2x0 > 2 + x1 + 2x0 = [[mark(U41(_x0, _x1))]] [[U42(mark(_x0))]] = 4 + 2x0 > 2 + 2x0 = [[mark(U42(_x0))]] [[U51(mark(_x0), _x1)]] = 4 + x1 + 2x0 > 2 + x1 + 2x0 = [[mark(U51(_x0, _x1))]] [[U52(mark(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(U52(_x0))]] [[U61(mark(_x0))]] = 8 + 3x0 > 4 + 3x0 = [[mark(U61(_x0))]] [[U71(mark(_x0), _x1)]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[mark(U71(_x0, _x1))]] [[U72(mark(_x0))]] = 4 + 2x0 > 2 + 2x0 = [[mark(U72(_x0))]] [[U81(mark(_x0))]] = 4 + 2x0 > 2 + 2x0 = [[mark(U81(_x0))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(U11(_x0))]] = x0 >= x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = 2x0 >= 2x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = x0 >= x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 2x0 >= 2x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = 2 + x0 >= 2 + x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = x0 >= x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = 2 + 3x0 >= 2 + 3x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 2x0 >= 2x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = x0 >= x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 2x0 >= 2x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 1 >= 1 = [[ok(e)]] [[proper(i)]] = 0 >= 0 = [[ok(i)]] [[proper(o)]] = 0 >= 0 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 2 + x0 >= 2 + x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2 + 3x0 >= 2 + 3x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(mark(_x0))]] = 4 + 2x0 > 2x0 = [[top(proper(_x0))]] [[top(ok(_x0))]] = 2x0 >= 2x0 = [[top(active(_x0))]] We can thus remove the following rules: U21(mark(X), Y) => mark(U21(X, Y)) U22(mark(X)) => mark(U22(X)) U41(mark(X), Y) => mark(U41(X, Y)) U42(mark(X)) => mark(U42(X)) U51(mark(X), Y) => mark(U51(X, Y)) U61(mark(X)) => mark(U61(X)) U72(mark(X)) => mark(U72(X)) U81(mark(X)) => mark(U81(X)) top(mark(X)) => top(proper(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]): active(isNeList(X)) >? mark(U31(isQid(X))) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U31(mark(X)) >? mark(U31(X)) U52(mark(X)) >? mark(U52(X)) U71(mark(X), Y) >? mark(U71(X, Y)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(U11(X)) >? U11(proper(X)) proper(tt) >? ok(tt) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U22(X)) >? U22(proper(X)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U52(X)) >? U52(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(a) >? ok(a) proper(e) >? ok(e) proper(i) >? ok(i) proper(o) >? ok(o) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.1 + y0 + y1 U11 = \y0.y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.1 + y0 U31 = \y0.y0 U41 = \y0y1.y0 + y1 U42 = \y0.y0 U51 = \y0y1.y0 + y1 U52 = \y0.3 + 2y0 U61 = \y0.2y0 U71 = \y0y1.y0 + y1 U72 = \y0.2y0 U81 = \y0.2y0 a = 3 active = \y0.y0 e = 3 i = 3 isList = \y0.2y0 isNeList = \y0.y0 isNePal = \y0.2y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 3 o = 3 ok = \y0.y0 proper = \y0.3y0 top = \y0.2y0 tt = 2 u = 0 Using this interpretation, the requirements translate to: [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(!6220!6220(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 1 + x0 >= 1 + x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = 3 + 2x0 >= 3 + 2x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 2x0 >= 2x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = x0 >= x0 = [[mark(U11(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U52(mark(_x0))]] = 3 + 2x0 >= 3 + 2x0 = [[mark(U52(_x0))]] [[U71(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U71(_x0, _x1))]] [[proper(!6220!6220(_x0, _x1))]] = 3 + 3x0 + 3x1 > 1 + 3x0 + 3x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 9 > 3 = [[ok(nil)]] [[proper(U11(_x0))]] = 3x0 >= 3x0 = [[U11(proper(_x0))]] [[proper(tt)]] = 6 > 2 = [[ok(tt)]] [[proper(U21(_x0, _x1))]] = 3x1 + 6x0 >= 3x1 + 6x0 = [[U21(proper(_x0), proper(_x1))]] [[proper(U22(_x0))]] = 3 + 3x0 > 1 + 3x0 = [[U22(proper(_x0))]] [[proper(isList(_x0))]] = 6x0 >= 6x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = 3x0 >= 3x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 3x0 >= 3x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = 3x0 >= 3x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U52(_x0))]] = 9 + 6x0 > 3 + 6x0 = [[U52(proper(_x0))]] [[proper(U61(_x0))]] = 6x0 >= 6x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 6x0 >= 6x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 3x0 >= 3x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 6x0 >= 6x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = 3x0 >= 3x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 6x0 >= 6x0 = [[isNePal(proper(_x0))]] [[proper(a)]] = 9 > 3 = [[ok(a)]] [[proper(e)]] = 9 > 3 = [[ok(e)]] [[proper(i)]] = 9 > 3 = [[ok(i)]] [[proper(o)]] = 9 > 3 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = 3 + 2x0 >= 3 + 2x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = 2x0 >= 2x0 = [[top(active(_x0))]] We can thus remove the following rules: proper(!6220!6220(X, Y)) => !6220!6220(proper(X), proper(Y)) proper(nil) => ok(nil) proper(tt) => ok(tt) proper(U22(X)) => U22(proper(X)) proper(U52(X)) => U52(proper(X)) proper(a) => ok(a) proper(e) => ok(e) proper(i) => ok(i) proper(o) => ok(o) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): active(isNeList(X)) >? mark(U31(isQid(X))) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U31(mark(X)) >? mark(U31(X)) U52(mark(X)) >? mark(U52(X)) U71(mark(X), Y) >? mark(U71(X, Y)) proper(U11(X)) >? U11(proper(X)) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U51(X, Y)) >? U51(proper(X), proper(Y)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.2y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.3 + y0 U31 = \y0.y0 U41 = \y0y1.y0 + y1 U42 = \y0.2y0 U51 = \y0y1.1 + y0 + 2y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y0 + y1 U72 = \y0.y0 U81 = \y0.y0 active = \y0.y0 isList = \y0.2y0 isNeList = \y0.y0 isNePal = \y0.2y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 ok = \y0.y0 proper = \y0.3y0 top = \y0.y0 u = 3 Using this interpretation, the requirements translate to: [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(U31(isQid(_x0)))]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 2x0 >= 2x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 3 + x0 >= 3 + x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U11(_x0))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U71(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(U71(_x0, _x1))]] [[proper(U11(_x0))]] = 6x0 >= 6x0 = [[U11(proper(_x0))]] [[proper(U21(_x0, _x1))]] = 3x1 + 6x0 >= 3x1 + 6x0 = [[U21(proper(_x0), proper(_x1))]] [[proper(isList(_x0))]] = 6x0 >= 6x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = 3x0 >= 3x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 6x0 >= 6x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = 3x0 >= 3x0 = [[isNeList(proper(_x0))]] [[proper(U51(_x0, _x1))]] = 3 + 3x0 + 6x1 > 1 + 3x0 + 6x1 = [[U51(proper(_x0), proper(_x1))]] [[proper(U61(_x0))]] = 3x0 >= 3x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 3x0 >= 3x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 3x0 >= 3x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 3x0 >= 3x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = 3x0 >= 3x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 6x0 >= 6x0 = [[isNePal(proper(_x0))]] [[proper(u)]] = 9 > 3 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 3 + x0 >= 3 + x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = x0 >= x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: proper(U51(X, Y)) => U51(proper(X), proper(Y)) proper(u) => ok(u) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): active(isNeList(X)) >? mark(U31(isQid(X))) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) U11(mark(X)) >? mark(U11(X)) U31(mark(X)) >? mark(U31(X)) U52(mark(X)) >? mark(U52(X)) U71(mark(X), Y) >? mark(U71(X, Y)) proper(U11(X)) >? U11(proper(X)) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + 2y1 U11 = \y0.2y0 U21 = \y0y1.y0 + y1 U22 = \y0.2y0 U31 = \y0.y0 U41 = \y0y1.y0 + 2y1 U42 = \y0.y0 U51 = \y0y1.2y0 + 2y1 U52 = \y0.y0 U61 = \y0.2y0 U71 = \y0y1.y0 + 2y1 U72 = \y0.2y0 U81 = \y0.y0 active = \y0.y0 isList = \y0.y0 isNeList = \y0.1 + 2y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.2y0 mark = \y0.1 + y0 ok = \y0.y0 proper = \y0.3y0 top = \y0.y0 Using this interpretation, the requirements translate to: [[active(isNeList(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[mark(U31(isQid(_x0)))]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 2x0 >= 2x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 2x0 >= 2x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 2x0 >= 2x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = 2 + x0 + 2x1 > 1 + x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[U11(mark(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[mark(U11(_x0))]] [[U31(mark(_x0))]] = 1 + x0 >= 1 + x0 = [[mark(U31(_x0))]] [[U52(mark(_x0))]] = 1 + x0 >= 1 + x0 = [[mark(U52(_x0))]] [[U71(mark(_x0), _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[mark(U71(_x0, _x1))]] [[proper(U11(_x0))]] = 6x0 >= 6x0 = [[U11(proper(_x0))]] [[proper(U21(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(isList(_x0))]] = 3x0 >= 3x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = 3x0 >= 3x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 3x0 + 6x1 >= 3x0 + 6x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 3x0 >= 3x0 = [[U42(proper(_x0))]] [[proper(isNeList(_x0))]] = 3 + 6x0 > 1 + 6x0 = [[isNeList(proper(_x0))]] [[proper(U61(_x0))]] = 6x0 >= 6x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 3x0 + 6x1 >= 3x0 + 6x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 6x0 >= 6x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 3x0 >= 3x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 3x0 >= 3x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = 6x0 >= 6x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 3x0 >= 3x0 = [[isNePal(proper(_x0))]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: !6220!6220(X, mark(Y)) => mark(!6220!6220(X, Y)) U11(mark(X)) => mark(U11(X)) proper(isNeList(X)) => isNeList(proper(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]): active(isNeList(X)) >? mark(U31(isQid(X))) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) U31(mark(X)) >? mark(U31(X)) U52(mark(X)) >? mark(U52(X)) U71(mark(X), Y) >? mark(U71(X, Y)) proper(U11(X)) >? U11(proper(X)) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(isList(X)) >? isList(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(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 + y1 U22 = \y0.2y0 U31 = \y0.2y0 U41 = \y0y1.y1 + 2y0 U42 = \y0.2y0 U51 = \y0y1.y1 + 2y0 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y0 + 2y1 U72 = \y0.2y0 U81 = \y0.y0 active = \y0.y0 isList = \y0.1 + y0 isNeList = \y0.3y0 isNePal = \y0.2y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 ok = \y0.y0 proper = \y0.3y0 top = \y0.y0 Using this interpretation, the requirements translate to: [[active(isNeList(_x0))]] = 3x0 >= 2x0 = [[mark(U31(isQid(_x0)))]] [[active(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 2x0 >= 2x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = 2x0 >= 2x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(!6220!6220(_x0, _x1))]] [[U31(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U31(_x0))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U71(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(U71(_x0, _x1))]] [[proper(U11(_x0))]] = 3x0 >= 3x0 = [[U11(proper(_x0))]] [[proper(U21(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(isList(_x0))]] = 3 + 3x0 > 1 + 3x0 = [[isList(proper(_x0))]] [[proper(U31(_x0))]] = 6x0 >= 6x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 3x1 + 6x0 >= 3x1 + 6x0 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 6x0 >= 6x0 = [[U42(proper(_x0))]] [[proper(U61(_x0))]] = 3x0 >= 3x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 3x0 + 6x1 >= 3x0 + 6x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 6x0 >= 6x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 3x0 >= 3x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 3x0 >= 3x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = 3x0 >= 3x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 6x0 >= 6x0 = [[isNePal(proper(_x0))]] [[!6220!6220(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: proper(isList(X)) => isList(proper(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]): active(isNeList(X)) >? mark(U31(isQid(X))) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) U31(mark(X)) >? mark(U31(X)) U52(mark(X)) >? mark(U52(X)) U71(mark(X), Y) >? mark(U71(X, Y)) proper(U11(X)) >? U11(proper(X)) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(U72(X)) >? U72(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + 2y1 U11 = \y0.y0 U21 = \y0y1.y0 + y1 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y0 + 2y1 U42 = \y0.y0 U51 = \y0y1.y0 + y1 U52 = \y0.y0 U61 = \y0.2y0 U71 = \y0y1.y0 + 2y1 U72 = \y0.1 + 2y0 U81 = \y0.y0 active = \y0.y0 isList = \y0.3y0 isNeList = \y0.2 + 3y0 isNePal = \y0.y0 isPal = \y0.2y0 isQid = \y0.2y0 mark = \y0.y0 ok = \y0.y0 proper = \y0.3y0 top = \y0.2y0 Using this interpretation, the requirements translate to: [[active(isNeList(_x0))]] = 2 + 3x0 > 2x0 = [[mark(U31(isQid(_x0)))]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 2x0 >= 2x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U71(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(U71(_x0, _x1))]] [[proper(U11(_x0))]] = 3x0 >= 3x0 = [[U11(proper(_x0))]] [[proper(U21(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[U21(proper(_x0), proper(_x1))]] [[proper(U31(_x0))]] = 3x0 >= 3x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 3x0 + 6x1 >= 3x0 + 6x1 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 3x0 >= 3x0 = [[U42(proper(_x0))]] [[proper(U61(_x0))]] = 6x0 >= 6x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 3x0 + 6x1 >= 3x0 + 6x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(U72(_x0))]] = 3 + 6x0 > 1 + 6x0 = [[U72(proper(_x0))]] [[proper(isPal(_x0))]] = 6x0 >= 6x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 3x0 >= 3x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = 6x0 >= 6x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 3x0 >= 3x0 = [[isNePal(proper(_x0))]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = x0 >= x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 2 + 3x0 >= 2 + 3x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = 2x0 >= 2x0 = [[top(active(_x0))]] We can thus remove the following rules: active(isNeList(X)) => mark(U31(isQid(X))) proper(U72(X)) => U72(proper(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]): active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) U31(mark(X)) >? mark(U31(X)) U52(mark(X)) >? mark(U52(X)) U71(mark(X), Y) >? mark(U71(X, Y)) proper(U11(X)) >? U11(proper(X)) proper(U21(X, Y)) >? U21(proper(X), proper(Y)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(U61(X)) >? U61(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.y0 U21 = \y0y1.2 + y1 + 2y0 U22 = \y0.2y0 U31 = \y0.y0 U41 = \y0y1.y1 + 2y0 U42 = \y0.2y0 U51 = \y0y1.1 + y0 + y1 U52 = \y0.2y0 U61 = \y0.1 + y0 U71 = \y0y1.y1 + 2y0 U72 = \y0.2y0 U81 = \y0.y0 active = \y0.y0 isList = \y0.3y0 isNeList = \y0.3y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.1 + y0 mark = \y0.y0 ok = \y0.y0 proper = \y0.3y0 top = \y0.2y0 Using this interpretation, the requirements translate to: [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 2x0 >= 2x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = 2x0 >= 2x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 1 + x0 >= 1 + x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U52(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U52(_x0))]] [[U71(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U71(_x0, _x1))]] [[proper(U11(_x0))]] = 3x0 >= 3x0 = [[U11(proper(_x0))]] [[proper(U21(_x0, _x1))]] = 6 + 3x1 + 6x0 > 2 + 3x1 + 6x0 = [[U21(proper(_x0), proper(_x1))]] [[proper(U31(_x0))]] = 3x0 >= 3x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 3x1 + 6x0 >= 3x1 + 6x0 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 6x0 >= 6x0 = [[U42(proper(_x0))]] [[proper(U61(_x0))]] = 3 + 3x0 > 1 + 3x0 = [[U61(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 3x1 + 6x0 >= 3x1 + 6x0 = [[U71(proper(_x0), proper(_x1))]] [[proper(isPal(_x0))]] = 3x0 >= 3x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 3x0 >= 3x0 = [[U81(proper(_x0))]] [[proper(isQid(_x0))]] = 3 + 3x0 > 1 + 3x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 3x0 >= 3x0 = [[isNePal(proper(_x0))]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = 2x0 >= 2x0 = [[top(active(_x0))]] We can thus remove the following rules: proper(U21(X, Y)) => U21(proper(X), proper(Y)) proper(U61(X)) => U61(proper(X)) proper(isQid(X)) => isQid(proper(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]): active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) U31(mark(X)) >? mark(U31(X)) U52(mark(X)) >? mark(U52(X)) U71(mark(X), Y) >? mark(U71(X, Y)) proper(U11(X)) >? U11(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U41(X, Y)) >? U41(proper(X), proper(Y)) proper(U42(X)) >? U42(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + 2y1 U11 = \y0.y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.1 + y1 + 2y0 U42 = \y0.2y0 U51 = \y0y1.y0 + y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y1 + 2y0 U72 = \y0.2y0 U81 = \y0.y0 active = \y0.y0 isList = \y0.3y0 isNeList = \y0.1 + 3y0 isNePal = \y0.y0 isPal = \y0.2y0 isQid = \y0.3y0 mark = \y0.y0 ok = \y0.y0 proper = \y0.3y0 top = \y0.y0 Using this interpretation, the requirements translate to: [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 2x0 >= 2x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[U31(mark(_x0))]] = x0 >= x0 = [[mark(U31(_x0))]] [[U52(mark(_x0))]] = x0 >= x0 = [[mark(U52(_x0))]] [[U71(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(U71(_x0, _x1))]] [[proper(U11(_x0))]] = 3x0 >= 3x0 = [[U11(proper(_x0))]] [[proper(U31(_x0))]] = 3x0 >= 3x0 = [[U31(proper(_x0))]] [[proper(U41(_x0, _x1))]] = 3 + 3x1 + 6x0 > 1 + 3x1 + 6x0 = [[U41(proper(_x0), proper(_x1))]] [[proper(U42(_x0))]] = 6x0 >= 6x0 = [[U42(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 3x1 + 6x0 >= 3x1 + 6x0 = [[U71(proper(_x0), proper(_x1))]] [[proper(isPal(_x0))]] = 6x0 >= 6x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 3x0 >= 3x0 = [[U81(proper(_x0))]] [[proper(isNePal(_x0))]] = 3x0 >= 3x0 = [[isNePal(proper(_x0))]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = x0 >= x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = x0 >= x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 1 + 3x0 >= 1 + 3x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = x0 >= x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: proper(U41(X, Y)) => U41(proper(X), proper(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]): active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) U31(mark(X)) >? mark(U31(X)) U52(mark(X)) >? mark(U52(X)) U71(mark(X), Y) >? mark(U71(X, Y)) proper(U11(X)) >? U11(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U42(X)) >? U42(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.y0 U21 = \y0y1.y1 + 2y0 U22 = \y0.y0 U31 = \y0.y0 U41 = \y0y1.y0 + y1 U42 = \y0.y0 U51 = \y0y1.y0 + 2y1 U52 = \y0.2y0 U61 = \y0.y0 U71 = \y0y1.2y0 + 2y1 U72 = \y0.y0 U81 = \y0.y0 active = \y0.y0 isList = \y0.3y0 isNeList = \y0.3y0 isNePal = \y0.y0 isPal = \y0.2y0 isQid = \y0.3y0 mark = \y0.2 + y0 ok = \y0.2y0 proper = \y0.3y0 top = \y0.y0 Using this interpretation, the requirements translate to: [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = x0 >= x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = x0 >= x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = 2x0 >= 2x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = x0 >= x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U31(mark(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(U31(_x0))]] [[U52(mark(_x0))]] = 4 + 2x0 > 2 + 2x0 = [[mark(U52(_x0))]] [[U71(mark(_x0), _x1)]] = 4 + 2x0 + 2x1 > 2 + 2x0 + 2x1 = [[mark(U71(_x0, _x1))]] [[proper(U11(_x0))]] = 3x0 >= 3x0 = [[U11(proper(_x0))]] [[proper(U31(_x0))]] = 3x0 >= 3x0 = [[U31(proper(_x0))]] [[proper(U42(_x0))]] = 3x0 >= 3x0 = [[U42(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 6x0 + 6x1 >= 6x0 + 6x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(isPal(_x0))]] = 6x0 >= 6x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 3x0 >= 3x0 = [[U81(proper(_x0))]] [[proper(isNePal(_x0))]] = 3x0 >= 3x0 = [[isNePal(proper(_x0))]] [[!6220!6220(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = 2x1 + 4x0 >= 2x1 + 4x0 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 6x0 >= 6x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 6x0 >= 6x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = 2x0 + 4x1 >= 2x0 + 4x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = 4x0 >= 4x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = 4x0 + 4x1 >= 4x0 + 4x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 4x0 >= 4x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = 6x0 >= 6x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = 2x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: U52(mark(X)) => mark(U52(X)) U71(mark(X), Y) => mark(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]): active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) U31(mark(X)) >? mark(U31(X)) proper(U11(X)) >? U11(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U42(X)) >? U42(proper(X)) proper(U71(X, Y)) >? U71(proper(X), proper(Y)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.y0 + y1 U11 = \y0.2y0 U21 = \y0y1.2y0 + 2y1 U22 = \y0.y0 U31 = \y0.2y0 U41 = \y0y1.y0 + y1 U42 = \y0.2y0 U51 = \y0y1.y0 + 2y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.2 + y0 + 2y1 U72 = \y0.y0 U81 = \y0.2y0 active = \y0.y0 isList = \y0.3y0 isNeList = \y0.3y0 isNePal = \y0.y0 isPal = \y0.2y0 isQid = \y0.3y0 mark = \y0.y0 ok = \y0.y0 proper = \y0.3y0 top = \y0.y0 Using this interpretation, the requirements translate to: [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 2x0 >= 2x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = 2x0 >= 2x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 2x0 >= 2x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = x0 >= x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(!6220!6220(_x0, _x1))]] [[U31(mark(_x0))]] = 2x0 >= 2x0 = [[mark(U31(_x0))]] [[proper(U11(_x0))]] = 6x0 >= 6x0 = [[U11(proper(_x0))]] [[proper(U31(_x0))]] = 6x0 >= 6x0 = [[U31(proper(_x0))]] [[proper(U42(_x0))]] = 6x0 >= 6x0 = [[U42(proper(_x0))]] [[proper(U71(_x0, _x1))]] = 6 + 3x0 + 6x1 > 2 + 3x0 + 6x1 = [[U71(proper(_x0), proper(_x1))]] [[proper(isPal(_x0))]] = 6x0 >= 6x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 6x0 >= 6x0 = [[U81(proper(_x0))]] [[proper(isNePal(_x0))]] = 3x0 >= 3x0 = [[isNePal(proper(_x0))]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = x0 >= x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = x0 >= x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = x0 >= x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = 2 + x0 + 2x1 >= 2 + x0 + 2x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = x0 >= x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2x0 >= 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: proper(U71(X, Y)) => U71(proper(X), proper(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]): active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) U31(mark(X)) >? mark(U31(X)) proper(U11(X)) >? U11(proper(X)) proper(U31(X)) >? U31(proper(X)) proper(U42(X)) >? U42(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) U11(ok(X)) >? ok(U11(X)) U21(ok(X), ok(Y)) >? ok(U21(X, Y)) U22(ok(X)) >? ok(U22(X)) isList(ok(X)) >? ok(isList(X)) U31(ok(X)) >? ok(U31(X)) U41(ok(X), ok(Y)) >? ok(U41(X, Y)) U42(ok(X)) >? ok(U42(X)) isNeList(ok(X)) >? ok(isNeList(X)) U51(ok(X), ok(Y)) >? ok(U51(X, Y)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U71(ok(X), ok(Y)) >? ok(U71(X, Y)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) U81(ok(X)) >? ok(U81(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) top(ok(X)) >? top(active(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.2y0 + 2y1 U11 = \y0.1 + y0 U21 = \y0y1.3 + y0 + y1 U22 = \y0.y0 U31 = \y0.1 + y0 U41 = \y0y1.y0 + y1 U42 = \y0.y0 U51 = \y0y1.1 + 2y0 + 2y1 U52 = \y0.y0 U61 = \y0.y0 U71 = \y0y1.y0 + y1 U72 = \y0.3 + y0 U81 = \y0.2y0 active = \y0.y0 isList = \y0.3y0 isNeList = \y0.3y0 isNePal = \y0.2y0 isPal = \y0.y0 isQid = \y0.3y0 mark = \y0.y0 ok = \y0.1 + y0 proper = \y0.3y0 top = \y0.y0 Using this interpretation, the requirements translate to: [[active(!6220!6220(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 1 + x0 >= 1 + x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = 3 + x0 + x1 >= 3 + x0 + x1 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = x0 >= x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = 1 + x0 >= 1 + x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = x0 >= x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = 1 + 2x0 + 2x1 >= 1 + 2x0 + 2x1 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = x0 >= x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = x0 >= x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 3 + x0 >= 3 + x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2x0 >= 2x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(!6220!6220(_x0, _x1))]] [[U31(mark(_x0))]] = 1 + x0 >= 1 + x0 = [[mark(U31(_x0))]] [[proper(U11(_x0))]] = 3 + 3x0 > 1 + 3x0 = [[U11(proper(_x0))]] [[proper(U31(_x0))]] = 3 + 3x0 > 1 + 3x0 = [[U31(proper(_x0))]] [[proper(U42(_x0))]] = 3x0 >= 3x0 = [[U42(proper(_x0))]] [[proper(isPal(_x0))]] = 3x0 >= 3x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 6x0 >= 6x0 = [[U81(proper(_x0))]] [[proper(isNePal(_x0))]] = 6x0 >= 6x0 = [[isNePal(proper(_x0))]] [[!6220!6220(ok(_x0), ok(_x1))]] = 4 + 2x0 + 2x1 > 1 + 2x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[U11(ok(_x0))]] = 2 + x0 >= 2 + x0 = [[ok(U11(_x0))]] [[U21(ok(_x0), ok(_x1))]] = 5 + x0 + x1 > 4 + x0 + x1 = [[ok(U21(_x0, _x1))]] [[U22(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(U22(_x0))]] [[isList(ok(_x0))]] = 3 + 3x0 > 1 + 3x0 = [[ok(isList(_x0))]] [[U31(ok(_x0))]] = 2 + x0 >= 2 + x0 = [[ok(U31(_x0))]] [[U41(ok(_x0), ok(_x1))]] = 2 + x0 + x1 > 1 + x0 + x1 = [[ok(U41(_x0, _x1))]] [[U42(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(U42(_x0))]] [[isNeList(ok(_x0))]] = 3 + 3x0 > 1 + 3x0 = [[ok(isNeList(_x0))]] [[U51(ok(_x0), ok(_x1))]] = 5 + 2x0 + 2x1 > 2 + 2x0 + 2x1 = [[ok(U51(_x0, _x1))]] [[U52(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(U61(_x0))]] [[U71(ok(_x0), ok(_x1))]] = 2 + x0 + x1 > 1 + x0 + x1 = [[ok(U71(_x0, _x1))]] [[U72(ok(_x0))]] = 4 + x0 >= 4 + x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(isPal(_x0))]] [[U81(ok(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[ok(U81(_x0))]] [[isQid(ok(_x0))]] = 3 + 3x0 > 1 + 3x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[ok(isNePal(_x0))]] [[top(ok(_x0))]] = 1 + x0 > x0 = [[top(active(_x0))]] We can thus remove the following rules: proper(U11(X)) => U11(proper(X)) proper(U31(X)) => U31(proper(X)) !6220!6220(ok(X), ok(Y)) => ok(!6220!6220(X, Y)) U21(ok(X), ok(Y)) => ok(U21(X, Y)) isList(ok(X)) => ok(isList(X)) U41(ok(X), ok(Y)) => ok(U41(X, Y)) isNeList(ok(X)) => ok(isNeList(X)) U51(ok(X), ok(Y)) => ok(U51(X, Y)) U71(ok(X), ok(Y)) => ok(U71(X, Y)) U81(ok(X)) => ok(U81(X)) isQid(ok(X)) => ok(isQid(X)) isNePal(ok(X)) => ok(isNePal(X)) top(ok(X)) => top(active(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]): active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(U11(X)) >? U11(active(X)) active(U21(X, Y)) >? U21(active(X), Y) active(U22(X)) >? U22(active(X)) active(U31(X)) >? U31(active(X)) active(U41(X, Y)) >? U41(active(X), Y) active(U42(X)) >? U42(active(X)) active(U51(X, Y)) >? U51(active(X), Y) active(U52(X)) >? U52(active(X)) active(U61(X)) >? U61(active(X)) active(U71(X, Y)) >? U71(active(X), Y) active(U72(X)) >? U72(active(X)) active(U81(X)) >? U81(active(X)) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) U31(mark(X)) >? mark(U31(X)) proper(U42(X)) >? U42(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(U81(X)) >? U81(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) U11(ok(X)) >? ok(U11(X)) U22(ok(X)) >? ok(U22(X)) U31(ok(X)) >? ok(U31(X)) U42(ok(X)) >? ok(U42(X)) U52(ok(X)) >? ok(U52(X)) U61(ok(X)) >? ok(U61(X)) U72(ok(X)) >? ok(U72(X)) isPal(ok(X)) >? ok(isPal(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.3 + y1 + 2y0 U11 = \y0.3 + 2y0 U21 = \y0y1.y0 + y1 U22 = \y0.3 + 2y0 U31 = \y0.3 + 2y0 U41 = \y0y1.y0 + y1 U42 = \y0.3 + 2y0 U51 = \y0y1.y0 + y1 U52 = \y0.3 + 2y0 U61 = \y0.3 + 2y0 U71 = \y0y1.y0 + y1 U72 = \y0.3 + 2y0 U81 = \y0.y0 active = \y0.2 + 3y0 isNePal = \y0.y0 isPal = \y0.3 + 2y0 mark = \y0.2 + y0 ok = \y0.1 + y0 proper = \y0.2 + 3y0 Using this interpretation, the requirements translate to: [[active(!6220!6220(_x0, _x1))]] = 11 + 3x1 + 6x0 > 7 + x1 + 6x0 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = 11 + 3x1 + 6x0 > 5 + 2x0 + 3x1 = [[!6220!6220(_x0, active(_x1))]] [[active(U11(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[U11(active(_x0))]] [[active(U21(_x0, _x1))]] = 2 + 3x0 + 3x1 >= 2 + x1 + 3x0 = [[U21(active(_x0), _x1)]] [[active(U22(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[U22(active(_x0))]] [[active(U31(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[U31(active(_x0))]] [[active(U41(_x0, _x1))]] = 2 + 3x0 + 3x1 >= 2 + x1 + 3x0 = [[U41(active(_x0), _x1)]] [[active(U42(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[U42(active(_x0))]] [[active(U51(_x0, _x1))]] = 2 + 3x0 + 3x1 >= 2 + x1 + 3x0 = [[U51(active(_x0), _x1)]] [[active(U52(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[U52(active(_x0))]] [[active(U61(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[U61(active(_x0))]] [[active(U71(_x0, _x1))]] = 2 + 3x0 + 3x1 >= 2 + x1 + 3x0 = [[U71(active(_x0), _x1)]] [[active(U72(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[U72(active(_x0))]] [[active(U81(_x0))]] = 2 + 3x0 >= 2 + 3x0 = [[U81(active(_x0))]] [[!6220!6220(mark(_x0), _x1)]] = 7 + x1 + 2x0 > 5 + x1 + 2x0 = [[mark(!6220!6220(_x0, _x1))]] [[U31(mark(_x0))]] = 7 + 2x0 > 5 + 2x0 = [[mark(U31(_x0))]] [[proper(U42(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[U42(proper(_x0))]] [[proper(isPal(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[isPal(proper(_x0))]] [[proper(U81(_x0))]] = 2 + 3x0 >= 2 + 3x0 = [[U81(proper(_x0))]] [[proper(isNePal(_x0))]] = 2 + 3x0 >= 2 + 3x0 = [[isNePal(proper(_x0))]] [[U11(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(U11(_x0))]] [[U22(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(U22(_x0))]] [[U31(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(U31(_x0))]] [[U42(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(U42(_x0))]] [[U52(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(U52(_x0))]] [[U61(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(U61(_x0))]] [[U72(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(U72(_x0))]] [[isPal(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(isPal(_x0))]] We can thus remove the following rules: active(!6220!6220(X, Y)) => !6220!6220(active(X), Y) active(!6220!6220(X, Y)) => !6220!6220(X, active(Y)) active(U11(X)) => U11(active(X)) active(U22(X)) => U22(active(X)) active(U31(X)) => U31(active(X)) active(U42(X)) => U42(active(X)) active(U52(X)) => U52(active(X)) active(U61(X)) => U61(active(X)) active(U72(X)) => U72(active(X)) !6220!6220(mark(X), Y) => mark(!6220!6220(X, Y)) U31(mark(X)) => mark(U31(X)) proper(U42(X)) => U42(proper(X)) proper(isPal(X)) => isPal(proper(X)) U11(ok(X)) => ok(U11(X)) U22(ok(X)) => ok(U22(X)) U31(ok(X)) => ok(U31(X)) U42(ok(X)) => ok(U42(X)) U52(ok(X)) => ok(U52(X)) U61(ok(X)) => ok(U61(X)) U72(ok(X)) => ok(U72(X)) isPal(ok(X)) => ok(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]): active(U21(X, Y)) >? U21(active(X), Y) active(U41(X, Y)) >? U41(active(X), Y) active(U51(X, Y)) >? U51(active(X), Y) active(U71(X, Y)) >? U71(active(X), Y) active(U81(X)) >? U81(active(X)) proper(U81(X)) >? U81(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U21 = \y0y1.y0 + y1 U41 = \y0y1.y0 + y1 U51 = \y0y1.y0 + y1 U71 = \y0y1.y0 + y1 U81 = \y0.2 + y0 active = \y0.3y0 isNePal = \y0.y0 proper = \y0.3y0 Using this interpretation, the requirements translate to: [[active(U21(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U21(active(_x0), _x1)]] [[active(U41(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U41(active(_x0), _x1)]] [[active(U51(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U51(active(_x0), _x1)]] [[active(U71(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U71(active(_x0), _x1)]] [[active(U81(_x0))]] = 6 + 3x0 > 2 + 3x0 = [[U81(active(_x0))]] [[proper(U81(_x0))]] = 6 + 3x0 > 2 + 3x0 = [[U81(proper(_x0))]] [[proper(isNePal(_x0))]] = 3x0 >= 3x0 = [[isNePal(proper(_x0))]] We can thus remove the following rules: active(U81(X)) => U81(active(X)) proper(U81(X)) => U81(proper(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]): active(U21(X, Y)) >? U21(active(X), Y) active(U41(X, Y)) >? U41(active(X), Y) active(U51(X, Y)) >? U51(active(X), Y) active(U71(X, Y)) >? U71(active(X), Y) proper(isNePal(X)) >? isNePal(proper(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U21 = \y0y1.y0 + y1 U41 = \y0y1.y0 + y1 U51 = \y0y1.y0 + y1 U71 = \y0y1.y0 + y1 active = \y0.3y0 isNePal = \y0.2 + y0 proper = \y0.3y0 Using this interpretation, the requirements translate to: [[active(U21(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U21(active(_x0), _x1)]] [[active(U41(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U41(active(_x0), _x1)]] [[active(U51(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U51(active(_x0), _x1)]] [[active(U71(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U71(active(_x0), _x1)]] [[proper(isNePal(_x0))]] = 6 + 3x0 > 2 + 3x0 = [[isNePal(proper(_x0))]] We can thus remove the following rules: proper(isNePal(X)) => isNePal(proper(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]): active(U21(X, Y)) >? U21(active(X), Y) active(U41(X, Y)) >? U41(active(X), Y) active(U51(X, Y)) >? U51(active(X), Y) active(U71(X, Y)) >? U71(active(X), Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U21 = \y0y1.2 + y0 + y1 U41 = \y0y1.y0 + y1 U51 = \y0y1.y0 + y1 U71 = \y0y1.y0 + y1 active = \y0.3y0 Using this interpretation, the requirements translate to: [[active(U21(_x0, _x1))]] = 6 + 3x0 + 3x1 > 2 + x1 + 3x0 = [[U21(active(_x0), _x1)]] [[active(U41(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U41(active(_x0), _x1)]] [[active(U51(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U51(active(_x0), _x1)]] [[active(U71(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U71(active(_x0), _x1)]] We can thus remove the following rules: active(U21(X, Y)) => U21(active(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]): active(U41(X, Y)) >? U41(active(X), Y) active(U51(X, Y)) >? U51(active(X), Y) active(U71(X, Y)) >? U71(active(X), Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U41 = \y0y1.2 + y0 + y1 U51 = \y0y1.y0 + y1 U71 = \y0y1.y0 + y1 active = \y0.3y0 Using this interpretation, the requirements translate to: [[active(U41(_x0, _x1))]] = 6 + 3x0 + 3x1 > 2 + x1 + 3x0 = [[U41(active(_x0), _x1)]] [[active(U51(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U51(active(_x0), _x1)]] [[active(U71(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U71(active(_x0), _x1)]] We can thus remove the following rules: active(U41(X, Y)) => U41(active(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]): active(U51(X, Y)) >? U51(active(X), Y) active(U71(X, Y)) >? U71(active(X), Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U51 = \y0y1.2 + y0 + y1 U71 = \y0y1.y0 + y1 active = \y0.3y0 Using this interpretation, the requirements translate to: [[active(U51(_x0, _x1))]] = 6 + 3x0 + 3x1 > 2 + x1 + 3x0 = [[U51(active(_x0), _x1)]] [[active(U71(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[U71(active(_x0), _x1)]] We can thus remove the following rules: active(U51(X, Y)) => U51(active(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]): active(U71(X, Y)) >? U71(active(X), Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U71 = \y0y1.1 + y0 + y1 active = \y0.3y0 Using this interpretation, the requirements translate to: [[active(U71(_x0, _x1))]] = 3 + 3x0 + 3x1 > 1 + x1 + 3x0 = [[U71(active(_x0), _x1)]] We can thus remove the following rules: active(U71(X, Y)) => U71(active(X), Y) All rules were succesfully removed. Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold. +++ Citations +++ [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012.