/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. !6220!6220 : [o * o] --> o a : [] --> o active : [o] --> o and : [o * 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(and(tt, X)) => mark(X) active(isList(X)) => mark(isNeList(X)) active(isList(nil)) => mark(tt) active(isList(!6220!6220(X, Y))) => mark(and(isList(X), isList(Y))) active(isNeList(X)) => mark(isQid(X)) active(isNeList(!6220!6220(X, Y))) => mark(and(isList(X), isNeList(Y))) active(isNeList(!6220!6220(X, Y))) => mark(and(isNeList(X), isList(Y))) active(isNePal(X)) => mark(isQid(X)) active(isNePal(!6220!6220(X, !6220!6220(Y, X)))) => mark(and(isQid(X), isPal(Y))) active(isPal(X)) => mark(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(and(X, Y)) => and(active(X), Y) !6220!6220(mark(X), Y) => mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) => mark(!6220!6220(X, Y)) and(mark(X), Y) => mark(and(X, Y)) proper(!6220!6220(X, Y)) => !6220!6220(proper(X), proper(Y)) proper(nil) => ok(nil) proper(and(X, Y)) => and(proper(X), proper(Y)) proper(tt) => ok(tt) proper(isList(X)) => isList(proper(X)) proper(isNeList(X)) => isNeList(proper(X)) proper(isQid(X)) => isQid(proper(X)) proper(isNePal(X)) => isNePal(proper(X)) proper(isPal(X)) => isPal(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)) and(ok(X), ok(Y)) => ok(and(X, Y)) isList(ok(X)) => ok(isList(X)) isNeList(ok(X)) => ok(isNeList(X)) isQid(ok(X)) => ok(isQid(X)) isNePal(ok(X)) => ok(isNePal(X)) isPal(ok(X)) => ok(isPal(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(and(tt, X)) >? mark(X) active(isList(X)) >? mark(isNeList(X)) active(isList(nil)) >? mark(tt) active(isList(!6220!6220(X, Y))) >? mark(and(isList(X), isList(Y))) active(isNeList(X)) >? mark(isQid(X)) active(isNeList(!6220!6220(X, Y))) >? mark(and(isList(X), isNeList(Y))) active(isNeList(!6220!6220(X, Y))) >? mark(and(isNeList(X), isList(Y))) active(isNePal(X)) >? mark(isQid(X)) active(isNePal(!6220!6220(X, !6220!6220(Y, X)))) >? mark(and(isQid(X), isPal(Y))) active(isPal(X)) >? mark(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(and(X, Y)) >? and(active(X), Y) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) and(mark(X), Y) >? mark(and(X, Y)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(tt) >? ok(tt) proper(isList(X)) >? isList(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(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)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 a = 0 active = \y0.y0 and = \y0y1.y1 + 2y0 e = 0 i = 1 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(and(tt, _x0))]] = x0 >= x0 = [[mark(_x0)]] [[active(isList(_x0))]] = x0 >= x0 = [[mark(isNeList(_x0))]] [[active(isList(nil))]] = 0 >= 0 = [[mark(tt)]] [[active(isList(!6220!6220(_x0, _x1)))]] = x1 + 2x0 >= x1 + 2x0 = [[mark(and(isList(_x0), isList(_x1)))]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(isQid(_x0))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = x1 + 2x0 >= x1 + 2x0 = [[mark(and(isList(_x0), isNeList(_x1)))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = x1 + 2x0 >= x1 + 2x0 = [[mark(and(isNeList(_x0), isList(_x1)))]] [[active(isNePal(_x0))]] = 2x0 >= x0 = [[mark(isQid(_x0))]] [[active(isNePal(!6220!6220(_x0, !6220!6220(_x1, _x0))))]] = 4x1 + 6x0 >= 2x0 + 2x1 = [[mark(and(isQid(_x0), isPal(_x1)))]] [[active(isPal(_x0))]] = 2x0 >= 2x0 = [[mark(isNePal(_x0))]] [[active(isPal(nil))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(a))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(e))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(i))]] = 1 > 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(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[and(active(_x0), _x1)]] [[!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))]] [[and(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(and(_x0, _x1))]] [[proper(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[and(proper(_x0), proper(_x1))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 2x0 >= 2x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = 2x0 >= 2x0 = [[isPal(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))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_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(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(!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(and(tt, X)) >? mark(X) active(isList(X)) >? mark(isNeList(X)) active(isList(nil)) >? mark(tt) active(isList(!6220!6220(X, Y))) >? mark(and(isList(X), isList(Y))) active(isNeList(X)) >? mark(isQid(X)) active(isNeList(!6220!6220(X, Y))) >? mark(and(isList(X), isNeList(Y))) active(isNeList(!6220!6220(X, Y))) >? mark(and(isNeList(X), isList(Y))) active(isNePal(X)) >? mark(isQid(X)) active(isNePal(!6220!6220(X, !6220!6220(Y, X)))) >? mark(and(isQid(X), isPal(Y))) active(isPal(X)) >? mark(isNePal(X)) active(isPal(nil)) >? mark(tt) active(isQid(a)) >? mark(tt) active(isQid(e)) >? 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(and(X, Y)) >? and(active(X), Y) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) and(mark(X), Y) >? mark(and(X, Y)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(tt) >? ok(tt) proper(isList(X)) >? isList(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(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)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 + 3y0 a = 1 active = \y0.y0 and = \y0y1.y1 + 2y0 e = 0 i = 1 isList = \y0.1 + 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(!6220!6220(!6220!6220(_x0, _x1), _x2))]] = 8 + x2 + 3x1 + 9x0 > 4 + x2 + 3x0 + 3x1 = [[mark(!6220!6220(_x0, !6220!6220(_x1, _x2)))]] [[active(!6220!6220(_x0, nil))]] = 2 + 3x0 > x0 = [[mark(_x0)]] [[active(!6220!6220(nil, _x0))]] = 2 + x0 > x0 = [[mark(_x0)]] [[active(and(tt, _x0))]] = x0 >= x0 = [[mark(_x0)]] [[active(isList(_x0))]] = 1 + x0 > x0 = [[mark(isNeList(_x0))]] [[active(isList(nil))]] = 1 > 0 = [[mark(tt)]] [[active(isList(!6220!6220(_x0, _x1)))]] = 3 + x1 + 3x0 >= 3 + x1 + 2x0 = [[mark(and(isList(_x0), isList(_x1)))]] [[active(isNeList(_x0))]] = x0 >= x0 = [[mark(isQid(_x0))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = 2 + x1 + 3x0 >= 2 + x1 + 2x0 = [[mark(and(isList(_x0), isNeList(_x1)))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = 2 + x1 + 3x0 > 1 + x1 + 2x0 = [[mark(and(isNeList(_x0), isList(_x1)))]] [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(isQid(_x0))]] [[active(isNePal(!6220!6220(_x0, !6220!6220(_x1, _x0))))]] = 4 + 3x1 + 4x0 > x1 + 2x0 = [[mark(and(isQid(_x0), isPal(_x1)))]] [[active(isPal(_x0))]] = x0 >= x0 = [[mark(isNePal(_x0))]] [[active(isPal(nil))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(a))]] = 1 > 0 = [[mark(tt)]] [[active(isQid(e))]] = 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 + 3x0 >= 2 + x1 + 3x0 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = 2 + x1 + 3x0 >= 2 + x1 + 3x0 = [[!6220!6220(_x0, active(_x1))]] [[active(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[and(active(_x0), _x1)]] [[!6220!6220(mark(_x0), _x1)]] = 2 + x1 + 3x0 >= 2 + x1 + 3x0 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = 2 + x1 + 3x0 >= 2 + x1 + 3x0 = [[mark(!6220!6220(_x0, _x1))]] [[and(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(and(_x0, _x1))]] [[proper(!6220!6220(_x0, _x1))]] = 2 + x1 + 3x0 >= 2 + x1 + 3x0 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[and(proper(_x0), proper(_x1))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(isList(_x0))]] = 1 + x0 >= 1 + x0 = [[isList(proper(_x0))]] [[proper(isNeList(_x0))]] = x0 >= x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = x0 >= x0 = [[isPal(proper(_x0))]] [[proper(a)]] = 1 >= 1 = [[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))]] = 2 + x1 + 3x0 >= 2 + x1 + 3x0 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_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(isList(X)) => mark(isNeList(X)) active(isList(nil)) => mark(tt) active(isNeList(!6220!6220(X, Y))) => mark(and(isNeList(X), isList(Y))) active(isNePal(!6220!6220(X, !6220!6220(Y, X)))) => mark(and(isQid(X), isPal(Y))) 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(and(tt, X)) >? mark(X) active(isList(!6220!6220(X, Y))) >? mark(and(isList(X), isList(Y))) active(isNeList(X)) >? mark(isQid(X)) active(isNeList(!6220!6220(X, Y))) >? mark(and(isList(X), isNeList(Y))) active(isNePal(X)) >? mark(isQid(X)) active(isPal(X)) >? mark(isNePal(X)) active(isPal(nil)) >? mark(tt) active(isQid(e)) >? 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(and(X, Y)) >? and(active(X), Y) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) and(mark(X), Y) >? mark(and(X, Y)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(tt) >? ok(tt) proper(isList(X)) >? isList(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(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)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 a = 0 active = \y0.y0 and = \y0y1.y0 + 2y1 e = 0 i = 0 isList = \y0.y0 isNeList = \y0.2y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 2 ok = \y0.y0 proper = \y0.y0 top = \y0.2y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(and(tt, _x0))]] = 2x0 >= x0 = [[mark(_x0)]] [[active(isList(!6220!6220(_x0, _x1)))]] = x0 + 2x1 >= x0 + 2x1 = [[mark(and(isList(_x0), isList(_x1)))]] [[active(isNeList(_x0))]] = 2x0 >= x0 = [[mark(isQid(_x0))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = 2x0 + 4x1 >= x0 + 4x1 = [[mark(and(isList(_x0), isNeList(_x1)))]] [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(isQid(_x0))]] [[active(isPal(_x0))]] = x0 >= x0 = [[mark(isNePal(_x0))]] [[active(isPal(nil))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(e))]] = 0 >= 0 = [[mark(tt)]] [[active(isQid(o))]] = 2 > 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(and(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[and(active(_x0), _x1)]] [[!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))]] [[and(mark(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[mark(and(_x0, _x1))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(and(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[and(proper(_x0), proper(_x1))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(isNeList(_x0))]] = 2x0 >= 2x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = x0 >= x0 = [[isPal(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 + 2x1 >= x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_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(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(and(tt, X)) >? mark(X) active(isList(!6220!6220(X, Y))) >? mark(and(isList(X), isList(Y))) active(isNeList(X)) >? mark(isQid(X)) active(isNeList(!6220!6220(X, Y))) >? mark(and(isList(X), isNeList(Y))) active(isNePal(X)) >? mark(isQid(X)) active(isPal(X)) >? mark(isNePal(X)) active(isPal(nil)) >? mark(tt) active(isQid(e)) >? 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(and(X, Y)) >? and(active(X), Y) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) and(mark(X), Y) >? mark(and(X, Y)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(tt) >? ok(tt) proper(isList(X)) >? isList(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(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)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 a = 0 active = \y0.y0 and = \y0y1.y0 + y1 e = 0 i = 0 isList = \y0.2y0 isNeList = \y0.1 + y0 isNePal = \y0.y0 isPal = \y0.1 + y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 1 ok = \y0.y0 proper = \y0.y0 top = \y0.y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(and(tt, _x0))]] = x0 >= x0 = [[mark(_x0)]] [[active(isList(!6220!6220(_x0, _x1)))]] = 4 + 2x1 + 4x0 > 2x0 + 2x1 = [[mark(and(isList(_x0), isList(_x1)))]] [[active(isNeList(_x0))]] = 1 + x0 > x0 = [[mark(isQid(_x0))]] [[active(isNeList(!6220!6220(_x0, _x1)))]] = 3 + x1 + 2x0 > 1 + x1 + 2x0 = [[mark(and(isList(_x0), isNeList(_x1)))]] [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(isQid(_x0))]] [[active(isPal(_x0))]] = 1 + x0 > x0 = [[mark(isNePal(_x0))]] [[active(isPal(nil))]] = 1 > 0 = [[mark(tt)]] [[active(isQid(e))]] = 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(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[and(active(_x0), _x1)]] [[!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))]] [[and(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[mark(and(_x0, _x1))]] [[proper(!6220!6220(_x0, _x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[and(proper(_x0), proper(_x1))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(isList(_x0))]] = 2x0 >= 2x0 = [[isList(proper(_x0))]] [[proper(isNeList(_x0))]] = 1 + x0 >= 1 + x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = 1 + x0 >= 1 + x0 = [[isPal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 0 >= 0 = [[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))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = 1 + x0 >= 1 + x0 = [[ok(isPal(_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(isList(!6220!6220(X, Y))) => mark(and(isList(X), isList(Y))) active(isNeList(X)) => mark(isQid(X)) active(isNeList(!6220!6220(X, Y))) => mark(and(isList(X), isNeList(Y))) active(isPal(X)) => mark(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(and(tt, X)) >? mark(X) active(isNePal(X)) >? mark(isQid(X)) active(isQid(e)) >? 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(and(X, Y)) >? and(active(X), Y) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) and(mark(X), Y) >? mark(and(X, Y)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(tt) >? ok(tt) proper(isList(X)) >? isList(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(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)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 a = 0 active = \y0.y0 and = \y0y1.1 + y0 + y1 e = 0 i = 1 isList = \y0.y0 isNeList = \y0.1 + 2y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 1 o = 0 ok = \y0.y0 proper = \y0.y0 top = \y0.2y0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[active(and(tt, _x0))]] = 1 + x0 > x0 = [[mark(_x0)]] [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(isQid(_x0))]] [[active(isQid(e))]] = 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(and(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[and(active(_x0), _x1)]] [[!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))]] [[and(mark(_x0), _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[mark(and(_x0, _x1))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 1 >= 1 = [[ok(nil)]] [[proper(and(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[and(proper(_x0), proper(_x1))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(isList(_x0))]] = x0 >= x0 = [[isList(proper(_x0))]] [[proper(isNeList(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = x0 >= x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = x0 >= x0 = [[isPal(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 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = x0 >= x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_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(and(tt, X)) => mark(X) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): active(isNePal(X)) >? mark(isQid(X)) active(isQid(e)) >? 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(and(X, Y)) >? and(active(X), Y) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) and(mark(X), Y) >? mark(and(X, Y)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(tt) >? ok(tt) proper(isList(X)) >? isList(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(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)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 a = 0 active = \y0.y0 and = \y0y1.y1 + 2y0 e = 0 i = 0 isList = \y0.2y0 isNeList = \y0.2y0 isNePal = \y0.2 + 2y0 isPal = \y0.2y0 isQid = \y0.2 + 2y0 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(isNePal(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[mark(isQid(_x0))]] [[active(isQid(e))]] = 2 > 0 = [[mark(tt)]] [[active(isQid(u))]] = 2 > 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(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[and(active(_x0), _x1)]] [[!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))]] [[and(mark(_x0), _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[mark(and(_x0, _x1))]] [[proper(!6220!6220(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[and(proper(_x0), proper(_x1))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(isList(_x0))]] = 2x0 >= 2x0 = [[isList(proper(_x0))]] [[proper(isNeList(_x0))]] = 2x0 >= 2x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = 2x0 >= 2x0 = [[isPal(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))]] [[and(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_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) 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(isNePal(X)) >? mark(isQid(X)) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(and(X, Y)) >? and(active(X), Y) !6220!6220(mark(X), Y) >? mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) >? mark(!6220!6220(X, Y)) and(mark(X), Y) >? mark(and(X, Y)) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(tt) >? ok(tt) proper(isList(X)) >? isList(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(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)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 + 3y0 + 3y1 a = 0 active = \y0.y0 and = \y0y1.y1 + 2y0 e = 0 i = 0 isList = \y0.2y0 isNeList = \y0.2y0 isNePal = \y0.2 + y0 isPal = \y0.2y0 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(isNePal(_x0))]] = 2 + x0 >= 2 + x0 = [[mark(isQid(_x0))]] [[active(!6220!6220(_x0, _x1))]] = 2 + 3x0 + 3x1 >= 2 + 3x0 + 3x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = 2 + 3x0 + 3x1 >= 2 + 3x0 + 3x1 = [[!6220!6220(_x0, active(_x1))]] [[active(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[and(active(_x0), _x1)]] [[!6220!6220(mark(_x0), _x1)]] = 8 + 3x0 + 3x1 > 4 + 3x0 + 3x1 = [[mark(!6220!6220(_x0, _x1))]] [[!6220!6220(_x0, mark(_x1))]] = 8 + 3x0 + 3x1 > 4 + 3x0 + 3x1 = [[mark(!6220!6220(_x0, _x1))]] [[and(mark(_x0), _x1)]] = 4 + x1 + 2x0 > 2 + x1 + 2x0 = [[mark(and(_x0, _x1))]] [[proper(!6220!6220(_x0, _x1))]] = 2 + 3x0 + 3x1 >= 2 + 3x0 + 3x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 0 >= 0 = [[ok(nil)]] [[proper(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[and(proper(_x0), proper(_x1))]] [[proper(tt)]] = 0 >= 0 = [[ok(tt)]] [[proper(isList(_x0))]] = 2x0 >= 2x0 = [[isList(proper(_x0))]] [[proper(isNeList(_x0))]] = 2x0 >= 2x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = x0 >= x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 2 + x0 >= 2 + x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = 2x0 >= 2x0 = [[isPal(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 + 3x0 + 3x1 >= 2 + 3x0 + 3x1 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2 + x0 >= 2 + x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isPal(_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: !6220!6220(mark(X), Y) => mark(!6220!6220(X, Y)) !6220!6220(X, mark(Y)) => mark(!6220!6220(X, Y)) and(mark(X), Y) => mark(and(X, Y)) 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(isNePal(X)) >? mark(isQid(X)) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(and(X, Y)) >? and(active(X), Y) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(nil) >? ok(nil) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(tt) >? ok(tt) proper(isList(X)) >? isList(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(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)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 a = 0 active = \y0.y0 and = \y0y1.2y0 + 2y1 e = 1 i = 1 isList = \y0.2y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 1 o = 1 ok = \y0.y0 proper = \y0.3y0 top = \y0.y0 tt = 2 u = 0 Using this interpretation, the requirements translate to: [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(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(and(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[and(active(_x0), _x1)]] [[proper(!6220!6220(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(nil)]] = 3 > 1 = [[ok(nil)]] [[proper(and(_x0, _x1))]] = 6x0 + 6x1 >= 6x0 + 6x1 = [[and(proper(_x0), proper(_x1))]] [[proper(tt)]] = 6 > 2 = [[ok(tt)]] [[proper(isList(_x0))]] = 6x0 >= 6x0 = [[isList(proper(_x0))]] [[proper(isNeList(_x0))]] = 3x0 >= 3x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = 3x0 >= 3x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 3x0 >= 3x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = 3x0 >= 3x0 = [[isPal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(e)]] = 3 > 1 = [[ok(e)]] [[proper(i)]] = 3 > 1 = [[ok(i)]] [[proper(o)]] = 3 > 1 = [[ok(o)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = x0 >= x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[top(ok(_x0))]] = x0 >= x0 = [[top(active(_x0))]] We can thus remove the following rules: proper(nil) => ok(nil) proper(tt) => ok(tt) 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(isNePal(X)) >? mark(isQid(X)) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(and(X, Y)) >? and(active(X), Y) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(isList(X)) >? isList(proper(X)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(a) >? ok(a) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 a = 0 active = \y0.y0 and = \y0y1.y1 + 2y0 isList = \y0.1 + 2y0 isNeList = \y0.2y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 ok = \y0.y0 proper = \y0.3y0 top = \y0.2y0 u = 0 Using this interpretation, the requirements translate to: [[active(isNePal(_x0))]] = x0 >= x0 = [[mark(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(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[and(active(_x0), _x1)]] [[proper(!6220!6220(_x0, _x1))]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(and(_x0, _x1))]] = 3x1 + 6x0 >= 3x1 + 6x0 = [[and(proper(_x0), proper(_x1))]] [[proper(isList(_x0))]] = 3 + 6x0 > 1 + 6x0 = [[isList(proper(_x0))]] [[proper(isNeList(_x0))]] = 6x0 >= 6x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = 3x0 >= 3x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 3x0 >= 3x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = 3x0 >= 3x0 = [[isPal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = x0 + x1 >= x0 + x1 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = x1 + 2x0 >= x1 + 2x0 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = 1 + 2x0 >= 1 + 2x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = x0 >= x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = x0 >= x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_x0))]] [[top(ok(_x0))]] = 2x0 >= 2x0 = [[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(isNePal(X)) >? mark(isQid(X)) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(and(X, Y)) >? and(active(X), Y) proper(!6220!6220(X, Y)) >? !6220!6220(proper(X), proper(Y)) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(a) >? ok(a) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 + 2y1 a = 0 active = \y0.y0 and = \y0y1.2y0 + 2y1 isList = \y0.3y0 isNeList = \y0.2y0 isNePal = \y0.2y0 isPal = \y0.y0 isQid = \y0.2y0 mark = \y0.y0 ok = \y0.y0 proper = \y0.3y0 top = \y0.2y0 u = 0 Using this interpretation, the requirements translate to: [[active(isNePal(_x0))]] = 2x0 >= 2x0 = [[mark(isQid(_x0))]] [[active(!6220!6220(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(and(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[and(active(_x0), _x1)]] [[proper(!6220!6220(_x0, _x1))]] = 3 + 3x0 + 6x1 > 1 + 3x0 + 6x1 = [[!6220!6220(proper(_x0), proper(_x1))]] [[proper(and(_x0, _x1))]] = 6x0 + 6x1 >= 6x0 + 6x1 = [[and(proper(_x0), proper(_x1))]] [[proper(isNeList(_x0))]] = 6x0 >= 6x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = 6x0 >= 6x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 6x0 >= 6x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = 3x0 >= 3x0 = [[isPal(proper(_x0))]] [[proper(a)]] = 0 >= 0 = [[ok(a)]] [[proper(u)]] = 0 >= 0 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = 3x0 >= 3x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 2x0 >= 2x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = x0 >= x0 = [[ok(isPal(_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)) 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(isNePal(X)) >? mark(isQid(X)) active(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) active(!6220!6220(X, Y)) >? !6220!6220(X, active(Y)) active(and(X, Y)) >? and(active(X), Y) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(a) >? ok(a) proper(u) >? ok(u) !6220!6220(ok(X), ok(Y)) >? ok(!6220!6220(X, Y)) and(ok(X), ok(Y)) >? ok(and(X, Y)) isList(ok(X)) >? ok(isList(X)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(X)) isPal(ok(X)) >? ok(isPal(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 + y1 + 2y0 a = 0 active = \y0.1 + 2y0 and = \y0y1.3 + y0 + 2y1 isList = \y0.3y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 ok = \y0.3 + 3y0 proper = \y0.3 + 3y0 top = \y0.3y0 u = 0 Using this interpretation, the requirements translate to: [[active(isNePal(_x0))]] = 1 + 2x0 > x0 = [[mark(isQid(_x0))]] [[active(!6220!6220(_x0, _x1))]] = 3 + 2x1 + 4x0 >= 3 + x1 + 4x0 = [[!6220!6220(active(_x0), _x1)]] [[active(!6220!6220(_x0, _x1))]] = 3 + 2x1 + 4x0 > 2 + 2x0 + 2x1 = [[!6220!6220(_x0, active(_x1))]] [[active(and(_x0, _x1))]] = 7 + 2x0 + 4x1 > 4 + 2x0 + 2x1 = [[and(active(_x0), _x1)]] [[proper(and(_x0, _x1))]] = 12 + 3x0 + 6x1 >= 12 + 3x0 + 6x1 = [[and(proper(_x0), proper(_x1))]] [[proper(isNeList(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[isPal(proper(_x0))]] [[proper(a)]] = 3 >= 3 = [[ok(a)]] [[proper(u)]] = 3 >= 3 = [[ok(u)]] [[!6220!6220(ok(_x0), ok(_x1))]] = 10 + 3x1 + 6x0 > 6 + 3x1 + 6x0 = [[ok(!6220!6220(_x0, _x1))]] [[and(ok(_x0), ok(_x1))]] = 12 + 3x0 + 6x1 >= 12 + 3x0 + 6x1 = [[ok(and(_x0, _x1))]] [[isList(ok(_x0))]] = 9 + 9x0 > 3 + 9x0 = [[ok(isList(_x0))]] [[isNeList(ok(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = 3 + 3x0 >= 3 + 3x0 = [[ok(isPal(_x0))]] [[top(ok(_x0))]] = 9 + 9x0 > 3 + 6x0 = [[top(active(_x0))]] We can thus remove the following rules: active(isNePal(X)) => mark(isQid(X)) active(!6220!6220(X, Y)) => !6220!6220(X, active(Y)) active(and(X, Y)) => and(active(X), Y) !6220!6220(ok(X), ok(Y)) => ok(!6220!6220(X, Y)) isList(ok(X)) => ok(isList(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) proper(and(X, Y)) >? and(proper(X), proper(Y)) proper(isNeList(X)) >? isNeList(proper(X)) proper(isQid(X)) >? isQid(proper(X)) proper(isNePal(X)) >? isNePal(proper(X)) proper(isPal(X)) >? isPal(proper(X)) proper(a) >? ok(a) proper(u) >? ok(u) and(ok(X), ok(Y)) >? ok(and(X, Y)) isNeList(ok(X)) >? ok(isNeList(X)) isQid(ok(X)) >? ok(isQid(X)) isNePal(ok(X)) >? ok(isNePal(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.y0 + y1 a = 3 active = \y0.3y0 and = \y0y1.3 + y1 + 2y0 isNeList = \y0.3 + 2y0 isNePal = \y0.3 + 2y0 isPal = \y0.3 + 2y0 isQid = \y0.3 + 2y0 ok = \y0.1 + y0 proper = \y0.2 + 3y0 u = 3 Using this interpretation, the requirements translate to: [[active(!6220!6220(_x0, _x1))]] = 3x0 + 3x1 >= x1 + 3x0 = [[!6220!6220(active(_x0), _x1)]] [[proper(and(_x0, _x1))]] = 11 + 3x1 + 6x0 > 9 + 3x1 + 6x0 = [[and(proper(_x0), proper(_x1))]] [[proper(isNeList(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[isNeList(proper(_x0))]] [[proper(isQid(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[isQid(proper(_x0))]] [[proper(isNePal(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[isNePal(proper(_x0))]] [[proper(isPal(_x0))]] = 11 + 6x0 > 7 + 6x0 = [[isPal(proper(_x0))]] [[proper(a)]] = 11 > 4 = [[ok(a)]] [[proper(u)]] = 11 > 4 = [[ok(u)]] [[and(ok(_x0), ok(_x1))]] = 6 + x1 + 2x0 > 4 + x1 + 2x0 = [[ok(and(_x0, _x1))]] [[isNeList(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(isNeList(_x0))]] [[isQid(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(isQid(_x0))]] [[isNePal(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(isNePal(_x0))]] [[isPal(ok(_x0))]] = 5 + 2x0 > 4 + 2x0 = [[ok(isPal(_x0))]] We can thus remove the following rules: proper(and(X, Y)) => and(proper(X), proper(Y)) proper(isNeList(X)) => isNeList(proper(X)) proper(isQid(X)) => isQid(proper(X)) proper(isNePal(X)) => isNePal(proper(X)) proper(isPal(X)) => isPal(proper(X)) proper(a) => ok(a) proper(u) => ok(u) and(ok(X), ok(Y)) => ok(and(X, Y)) isNeList(ok(X)) => ok(isNeList(X)) isQid(ok(X)) => ok(isQid(X)) isNePal(ok(X)) => ok(isNePal(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(!6220!6220(X, Y)) >? !6220!6220(active(X), Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: !6220!6220 = \y0y1.1 + y0 + y1 active = \y0.3y0 Using this interpretation, the requirements translate to: [[active(!6220!6220(_x0, _x1))]] = 3 + 3x0 + 3x1 > 1 + x1 + 3x0 = [[!6220!6220(active(_x0), _x1)]] We can thus remove the following rules: active(!6220!6220(X, Y)) => !6220!6220(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.