/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 a : [] --> o a!6220!6220!6220!6220 : [o * o] --> o a!6220!6220and : [o * o] --> o a!6220!6220isList : [o] --> o a!6220!6220isNeList : [o] --> o a!6220!6220isNePal : [o] --> o a!6220!6220isPal : [o] --> o a!6220!6220isQid : [o] --> o 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 tt : [] --> o u : [] --> o a!6220!6220!6220!6220(!6220!6220(X, Y), Z) => a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220!6220!6220(X, nil) => mark(X) a!6220!6220!6220!6220(nil, X) => mark(X) a!6220!6220and(tt, X) => mark(X) a!6220!6220isList(X) => a!6220!6220isNeList(X) a!6220!6220isList(nil) => tt a!6220!6220isList(!6220!6220(X, Y)) => a!6220!6220and(a!6220!6220isList(X), isList(Y)) a!6220!6220isNeList(X) => a!6220!6220isQid(X) a!6220!6220isNeList(!6220!6220(X, Y)) => a!6220!6220and(a!6220!6220isList(X), isNeList(Y)) a!6220!6220isNeList(!6220!6220(X, Y)) => a!6220!6220and(a!6220!6220isNeList(X), isList(Y)) a!6220!6220isNePal(X) => a!6220!6220isQid(X) a!6220!6220isNePal(!6220!6220(X, !6220!6220(Y, X))) => a!6220!6220and(a!6220!6220isQid(X), isPal(Y)) a!6220!6220isPal(X) => a!6220!6220isNePal(X) a!6220!6220isPal(nil) => tt a!6220!6220isQid(a) => tt a!6220!6220isQid(e) => tt a!6220!6220isQid(i) => tt a!6220!6220isQid(o) => tt a!6220!6220isQid(u) => tt mark(!6220!6220(X, Y)) => a!6220!6220!6220!6220(mark(X), mark(Y)) mark(and(X, Y)) => a!6220!6220and(mark(X), Y) mark(isList(X)) => a!6220!6220isList(X) mark(isNeList(X)) => a!6220!6220isNeList(X) mark(isQid(X)) => a!6220!6220isQid(X) mark(isNePal(X)) => a!6220!6220isNePal(X) mark(isPal(X)) => a!6220!6220isPal(X) mark(nil) => nil mark(tt) => tt mark(a) => a mark(e) => e mark(i) => i mark(o) => o mark(u) => u a!6220!6220!6220!6220(X, Y) => !6220!6220(X, Y) a!6220!6220and(X, Y) => and(X, Y) a!6220!6220isList(X) => isList(X) a!6220!6220isNeList(X) => isNeList(X) a!6220!6220isQid(X) => isQid(X) a!6220!6220isNePal(X) => isNePal(X) a!6220!6220isPal(X) => isPal(X) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220!6220!6220(!6220!6220(X, Y), Z) >? a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220!6220!6220(X, nil) >? mark(X) a!6220!6220!6220!6220(nil, X) >? mark(X) a!6220!6220and(tt, X) >? mark(X) a!6220!6220isList(X) >? a!6220!6220isNeList(X) a!6220!6220isList(nil) >? tt a!6220!6220isList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isList(Y)) a!6220!6220isNeList(X) >? a!6220!6220isQid(X) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isNeList(Y)) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isNeList(X), isList(Y)) a!6220!6220isNePal(X) >? a!6220!6220isQid(X) a!6220!6220isNePal(!6220!6220(X, !6220!6220(Y, X))) >? a!6220!6220and(a!6220!6220isQid(X), isPal(Y)) a!6220!6220isPal(X) >? a!6220!6220isNePal(X) a!6220!6220isPal(nil) >? tt a!6220!6220isQid(a) >? tt a!6220!6220isQid(e) >? tt a!6220!6220isQid(i) >? tt a!6220!6220isQid(o) >? tt a!6220!6220isQid(u) >? tt mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isList(X)) >? a!6220!6220isList(X) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(isPal(X)) >? a!6220!6220isPal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isList(X) >? isList(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) a!6220!6220isPal(X) >? isPal(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 a = 2 a!6220!6220!6220!6220 = \y0y1.1 + y0 + y1 a!6220!6220and = \y0y1.y0 + y1 a!6220!6220isList = \y0.1 + y0 a!6220!6220isNeList = \y0.y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.1 + y0 a!6220!6220isQid = \y0.y0 and = \y0y1.y0 + y1 e = 0 i = 0 isList = \y0.1 + y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.1 + y0 isQid = \y0.y0 mark = \y0.y0 nil = 2 o = 1 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220!6220!6220(!6220!6220(_x0, _x1), _x2)]] = 2 + x0 + x1 + x2 >= 2 + x0 + x1 + x2 = [[a!6220!6220!6220!6220(mark(_x0), a!6220!6220!6220!6220(mark(_x1), mark(_x2)))]] [[a!6220!6220!6220!6220(_x0, nil)]] = 3 + x0 > x0 = [[mark(_x0)]] [[a!6220!6220!6220!6220(nil, _x0)]] = 3 + x0 > x0 = [[mark(_x0)]] [[a!6220!6220and(tt, _x0)]] = x0 >= x0 = [[mark(_x0)]] [[a!6220!6220isList(_x0)]] = 1 + x0 > x0 = [[a!6220!6220isNeList(_x0)]] [[a!6220!6220isList(nil)]] = 3 > 0 = [[tt]] [[a!6220!6220isList(!6220!6220(_x0, _x1))]] = 2 + x0 + x1 >= 2 + x0 + x1 = [[a!6220!6220and(a!6220!6220isList(_x0), isList(_x1))]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[a!6220!6220and(a!6220!6220isList(_x0), isNeList(_x1))]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[a!6220!6220and(a!6220!6220isNeList(_x0), isList(_x1))]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isNePal(!6220!6220(_x0, !6220!6220(_x1, _x0)))]] = 2 + x1 + 2x0 > 1 + x0 + x1 = [[a!6220!6220and(a!6220!6220isQid(_x0), isPal(_x1))]] [[a!6220!6220isPal(_x0)]] = 1 + x0 > x0 = [[a!6220!6220isNePal(_x0)]] [[a!6220!6220isPal(nil)]] = 3 > 0 = [[tt]] [[a!6220!6220isQid(a)]] = 2 > 0 = [[tt]] [[a!6220!6220isQid(e)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(i)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(o)]] = 1 > 0 = [[tt]] [[a!6220!6220isQid(u)]] = 0 >= 0 = [[tt]] [[mark(!6220!6220(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isList(_x0))]] = 1 + x0 >= 1 + x0 = [[a!6220!6220isList(_x0)]] [[mark(isNeList(_x0))]] = x0 >= x0 = [[a!6220!6220isNeList(_x0)]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(isPal(_x0))]] = 1 + x0 >= 1 + x0 = [[a!6220!6220isPal(_x0)]] [[mark(nil)]] = 2 >= 2 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 2 >= 2 = [[a]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 1 >= 1 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = 1 + x0 >= 1 + x0 = [[isList(_x0)]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[isNeList(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] [[a!6220!6220isPal(_x0)]] = 1 + x0 >= 1 + x0 = [[isPal(_x0)]] We can thus remove the following rules: a!6220!6220!6220!6220(X, nil) => mark(X) a!6220!6220!6220!6220(nil, X) => mark(X) a!6220!6220isList(X) => a!6220!6220isNeList(X) a!6220!6220isList(nil) => tt a!6220!6220isNePal(!6220!6220(X, !6220!6220(Y, X))) => a!6220!6220and(a!6220!6220isQid(X), isPal(Y)) a!6220!6220isPal(X) => a!6220!6220isNePal(X) a!6220!6220isPal(nil) => tt a!6220!6220isQid(a) => tt a!6220!6220isQid(o) => tt We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220!6220!6220(!6220!6220(X, Y), Z) >? a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220and(tt, X) >? mark(X) a!6220!6220isList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isList(Y)) a!6220!6220isNeList(X) >? a!6220!6220isQid(X) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isNeList(Y)) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isNeList(X), isList(Y)) a!6220!6220isNePal(X) >? a!6220!6220isQid(X) a!6220!6220isQid(e) >? tt a!6220!6220isQid(i) >? tt a!6220!6220isQid(u) >? tt mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isList(X)) >? a!6220!6220isList(X) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(isPal(X)) >? a!6220!6220isPal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isList(X) >? isList(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) a!6220!6220isPal(X) >? isPal(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 a!6220!6220!6220!6220 = \y0y1.y1 + 2y0 a!6220!6220and = \y0y1.y0 + y1 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.2y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.y0 a!6220!6220isQid = \y0.y0 and = \y0y1.y0 + y1 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 = 0 tt = 0 u = 1 Using this interpretation, the requirements translate to: [[a!6220!6220!6220!6220(!6220!6220(_x0, _x1), _x2)]] = x2 + 2x1 + 4x0 >= x2 + 2x0 + 2x1 = [[a!6220!6220!6220!6220(mark(_x0), a!6220!6220!6220!6220(mark(_x1), mark(_x2)))]] [[a!6220!6220and(tt, _x0)]] = x0 >= x0 = [[mark(_x0)]] [[a!6220!6220isList(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x0 + x1 = [[a!6220!6220and(a!6220!6220isList(_x0), isList(_x1))]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2x1 + 4x0 >= x0 + 2x1 = [[a!6220!6220and(a!6220!6220isList(_x0), isNeList(_x1))]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2x1 + 4x0 >= x1 + 2x0 = [[a!6220!6220and(a!6220!6220isNeList(_x0), isList(_x1))]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isQid(e)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(i)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(u)]] = 1 > 0 = [[tt]] [[mark(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isList(_x0))]] = x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(isNeList(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220isNeList(_x0)]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(isPal(_x0))]] = x0 >= x0 = [[a!6220!6220isPal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 0 >= 0 = [[a]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 1 >= 1 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= 2x0 = [[isNeList(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[isPal(_x0)]] We can thus remove the following rules: a!6220!6220isQid(u) => tt We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220!6220!6220(!6220!6220(X, Y), Z) >? a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220and(tt, X) >? mark(X) a!6220!6220isList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isList(Y)) a!6220!6220isNeList(X) >? a!6220!6220isQid(X) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isNeList(Y)) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isNeList(X), isList(Y)) a!6220!6220isNePal(X) >? a!6220!6220isQid(X) a!6220!6220isQid(e) >? tt a!6220!6220isQid(i) >? tt mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isList(X)) >? a!6220!6220isList(X) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(isPal(X)) >? a!6220!6220isPal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isList(X) >? isList(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) a!6220!6220isPal(X) >? isPal(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 a!6220!6220!6220!6220 = \y0y1.y1 + 2y0 a!6220!6220and = \y0y1.y1 + 2y0 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.2y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.2y0 a!6220!6220isQid = \y0.y0 and = \y0y1.y1 + 2y0 e = 0 i = 1 isList = \y0.y0 isNeList = \y0.2y0 isNePal = \y0.y0 isPal = \y0.2y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220!6220!6220(!6220!6220(_x0, _x1), _x2)]] = x2 + 2x1 + 4x0 >= x2 + 2x0 + 2x1 = [[a!6220!6220!6220!6220(mark(_x0), a!6220!6220!6220!6220(mark(_x1), mark(_x2)))]] [[a!6220!6220and(tt, _x0)]] = x0 >= x0 = [[mark(_x0)]] [[a!6220!6220isList(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220and(a!6220!6220isList(_x0), isList(_x1))]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2x1 + 4x0 >= 2x0 + 2x1 = [[a!6220!6220and(a!6220!6220isList(_x0), isNeList(_x1))]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2x1 + 4x0 >= x1 + 4x0 = [[a!6220!6220and(a!6220!6220isNeList(_x0), isList(_x1))]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isQid(e)]] = 0 >= 0 = [[tt]] [[a!6220!6220isQid(i)]] = 1 > 0 = [[tt]] [[mark(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isList(_x0))]] = x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(isNeList(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220isNeList(_x0)]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(isPal(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220isPal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 0 >= 0 = [[a]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 1 >= 1 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[and(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= 2x0 = [[isNeList(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] [[a!6220!6220isPal(_x0)]] = 2x0 >= 2x0 = [[isPal(_x0)]] We can thus remove the following rules: a!6220!6220isQid(i) => tt We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220!6220!6220(!6220!6220(X, Y), Z) >? a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220and(tt, X) >? mark(X) a!6220!6220isList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isList(Y)) a!6220!6220isNeList(X) >? a!6220!6220isQid(X) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isNeList(Y)) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isNeList(X), isList(Y)) a!6220!6220isNePal(X) >? a!6220!6220isQid(X) a!6220!6220isQid(e) >? tt mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isList(X)) >? a!6220!6220isList(X) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(isPal(X)) >? a!6220!6220isPal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isList(X) >? isList(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) a!6220!6220isPal(X) >? isPal(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 a!6220!6220!6220!6220 = \y0y1.y1 + 2y0 a!6220!6220and = \y0y1.y0 + y1 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.2y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.y0 a!6220!6220isQid = \y0.y0 and = \y0y1.y0 + y1 e = 2 i = 0 isList = \y0.y0 isNeList = \y0.2y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 tt = 1 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220!6220!6220(!6220!6220(_x0, _x1), _x2)]] = x2 + 2x1 + 4x0 >= x2 + 2x0 + 2x1 = [[a!6220!6220!6220!6220(mark(_x0), a!6220!6220!6220!6220(mark(_x1), mark(_x2)))]] [[a!6220!6220and(tt, _x0)]] = 1 + x0 > x0 = [[mark(_x0)]] [[a!6220!6220isList(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x0 + x1 = [[a!6220!6220and(a!6220!6220isList(_x0), isList(_x1))]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2x1 + 4x0 >= x0 + 2x1 = [[a!6220!6220and(a!6220!6220isList(_x0), isNeList(_x1))]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2x1 + 4x0 >= x1 + 2x0 = [[a!6220!6220and(a!6220!6220isNeList(_x0), isList(_x1))]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isQid(e)]] = 2 > 1 = [[tt]] [[mark(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isList(_x0))]] = x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(isNeList(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220isNeList(_x0)]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(isPal(_x0))]] = x0 >= x0 = [[a!6220!6220isPal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 1 >= 1 = [[tt]] [[mark(a)]] = 0 >= 0 = [[a]] [[mark(e)]] = 2 >= 2 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= 2x0 = [[isNeList(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[isPal(_x0)]] We can thus remove the following rules: a!6220!6220and(tt, X) => mark(X) a!6220!6220isQid(e) => tt We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220!6220!6220(!6220!6220(X, Y), Z) >? a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220isList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isList(Y)) a!6220!6220isNeList(X) >? a!6220!6220isQid(X) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isList(X), isNeList(Y)) a!6220!6220isNeList(!6220!6220(X, Y)) >? a!6220!6220and(a!6220!6220isNeList(X), isList(Y)) a!6220!6220isNePal(X) >? a!6220!6220isQid(X) mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isList(X)) >? a!6220!6220isList(X) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(isPal(X)) >? a!6220!6220isPal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isList(X) >? isList(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) a!6220!6220isPal(X) >? isPal(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 a = 0 a!6220!6220!6220!6220 = \y0y1.1 + y0 + y1 a!6220!6220and = \y0y1.y0 + y1 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.2y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.y0 a!6220!6220isQid = \y0.y0 and = \y0y1.y0 + y1 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 = 0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220!6220!6220(!6220!6220(_x0, _x1), _x2)]] = 2 + x0 + x1 + x2 >= 2 + x0 + x1 + x2 = [[a!6220!6220!6220!6220(mark(_x0), a!6220!6220!6220!6220(mark(_x1), mark(_x2)))]] [[a!6220!6220isList(!6220!6220(_x0, _x1))]] = 1 + x0 + x1 > x0 + x1 = [[a!6220!6220and(a!6220!6220isList(_x0), isList(_x1))]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2 + 2x0 + 2x1 > x0 + 2x1 = [[a!6220!6220and(a!6220!6220isList(_x0), isNeList(_x1))]] [[a!6220!6220isNeList(!6220!6220(_x0, _x1))]] = 2 + 2x0 + 2x1 > x1 + 2x0 = [[a!6220!6220and(a!6220!6220isNeList(_x0), isList(_x1))]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(!6220!6220(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isList(_x0))]] = x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(isNeList(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220isNeList(_x0)]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(isPal(_x0))]] = x0 >= x0 = [[a!6220!6220isPal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 0 >= 0 = [[a]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220isNeList(_x0)]] = 2x0 >= 2x0 = [[isNeList(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[isPal(_x0)]] We can thus remove the following rules: a!6220!6220isList(!6220!6220(X, Y)) => a!6220!6220and(a!6220!6220isList(X), isList(Y)) a!6220!6220isNeList(!6220!6220(X, Y)) => a!6220!6220and(a!6220!6220isList(X), isNeList(Y)) a!6220!6220isNeList(!6220!6220(X, Y)) => a!6220!6220and(a!6220!6220isNeList(X), isList(Y)) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220!6220!6220(!6220!6220(X, Y), Z) >? a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220isNeList(X) >? a!6220!6220isQid(X) a!6220!6220isNePal(X) >? a!6220!6220isQid(X) mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isList(X)) >? a!6220!6220isList(X) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(isPal(X)) >? a!6220!6220isPal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isList(X) >? isList(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) a!6220!6220isPal(X) >? isPal(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 a!6220!6220!6220!6220 = \y0y1.y1 + 2y0 a!6220!6220and = \y0y1.y0 + y1 a!6220!6220isList = \y0.1 + y0 a!6220!6220isNeList = \y0.y0 a!6220!6220isNePal = \y0.1 + y0 a!6220!6220isPal = \y0.y0 a!6220!6220isQid = \y0.y0 and = \y0y1.y0 + y1 e = 0 i = 0 isList = \y0.1 + y0 isNeList = \y0.y0 isNePal = \y0.1 + y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220!6220!6220(!6220!6220(_x0, _x1), _x2)]] = x2 + 2x1 + 4x0 >= x2 + 2x0 + 2x1 = [[a!6220!6220!6220!6220(mark(_x0), a!6220!6220!6220!6220(mark(_x1), mark(_x2)))]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = 1 + x0 > x0 = [[a!6220!6220isQid(_x0)]] [[mark(!6220!6220(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isList(_x0))]] = 1 + x0 >= 1 + x0 = [[a!6220!6220isList(_x0)]] [[mark(isNeList(_x0))]] = x0 >= x0 = [[a!6220!6220isNeList(_x0)]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = 1 + x0 >= 1 + x0 = [[a!6220!6220isNePal(_x0)]] [[mark(isPal(_x0))]] = x0 >= x0 = [[a!6220!6220isPal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 0 >= 0 = [[a]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = 1 + x0 >= 1 + x0 = [[isList(_x0)]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[isNeList(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = 1 + x0 >= 1 + x0 = [[isNePal(_x0)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[isPal(_x0)]] We can thus remove the following rules: a!6220!6220isNePal(X) => a!6220!6220isQid(X) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220!6220!6220(!6220!6220(X, Y), Z) >? a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) a!6220!6220isNeList(X) >? a!6220!6220isQid(X) mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isList(X)) >? a!6220!6220isList(X) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(isPal(X)) >? a!6220!6220isPal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isList(X) >? isList(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) a!6220!6220isPal(X) >? isPal(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 a!6220!6220!6220!6220 = \y0y1.2 + y1 + 2y0 a!6220!6220and = \y0y1.y0 + y1 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.y0 a!6220!6220isNePal = \y0.y0 a!6220!6220isPal = \y0.y0 a!6220!6220isQid = \y0.y0 and = \y0y1.y0 + y1 e = 0 i = 0 isList = \y0.y0 isNeList = \y0.y0 isNePal = \y0.y0 isPal = \y0.y0 isQid = \y0.y0 mark = \y0.y0 nil = 0 o = 0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220!6220!6220(!6220!6220(_x0, _x1), _x2)]] = 6 + x2 + 2x1 + 4x0 > 4 + x2 + 2x0 + 2x1 = [[a!6220!6220!6220!6220(mark(_x0), a!6220!6220!6220!6220(mark(_x1), mark(_x2)))]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(!6220!6220(_x0, _x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isList(_x0))]] = x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(isNeList(_x0))]] = x0 >= x0 = [[a!6220!6220isNeList(_x0)]] [[mark(isQid(_x0))]] = x0 >= x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = x0 >= x0 = [[a!6220!6220isNePal(_x0)]] [[mark(isPal(_x0))]] = x0 >= x0 = [[a!6220!6220isPal(_x0)]] [[mark(nil)]] = 0 >= 0 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 0 >= 0 = [[a]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220isNeList(_x0)]] = x0 >= x0 = [[isNeList(_x0)]] [[a!6220!6220isQid(_x0)]] = x0 >= x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = x0 >= x0 = [[isNePal(_x0)]] [[a!6220!6220isPal(_x0)]] = x0 >= x0 = [[isPal(_x0)]] We can thus remove the following rules: a!6220!6220!6220!6220(!6220!6220(X, Y), Z) => a!6220!6220!6220!6220(mark(X), a!6220!6220!6220!6220(mark(Y), mark(Z))) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220isNeList(X) >? a!6220!6220isQid(X) mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(and(X, Y)) >? a!6220!6220and(mark(X), Y) mark(isList(X)) >? a!6220!6220isList(X) mark(isNeList(X)) >? a!6220!6220isNeList(X) mark(isQid(X)) >? a!6220!6220isQid(X) mark(isNePal(X)) >? a!6220!6220isNePal(X) mark(isPal(X)) >? a!6220!6220isPal(X) mark(nil) >? nil mark(tt) >? tt mark(a) >? a mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isList(X) >? isList(X) a!6220!6220isNeList(X) >? isNeList(X) a!6220!6220isQid(X) >? isQid(X) a!6220!6220isNePal(X) >? isNePal(X) a!6220!6220isPal(X) >? 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 = 2 a!6220!6220!6220!6220 = \y0y1.y0 + y1 a!6220!6220and = \y0y1.1 + y1 + 2y0 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.3 + 2y0 a!6220!6220isNePal = \y0.1 + y0 a!6220!6220isPal = \y0.2 + y0 a!6220!6220isQid = \y0.3 + 2y0 and = \y0y1.1 + y1 + 2y0 e = 0 i = 0 isList = \y0.y0 isNeList = \y0.2 + y0 isNePal = \y0.1 + y0 isPal = \y0.2 + y0 isQid = \y0.2 + y0 mark = \y0.2y0 nil = 1 o = 0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220isNeList(_x0)]] = 3 + 2x0 >= 3 + 2x0 = [[a!6220!6220isQid(_x0)]] [[mark(!6220!6220(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(and(_x0, _x1))]] = 2 + 2x1 + 4x0 > 1 + x1 + 4x0 = [[a!6220!6220and(mark(_x0), _x1)]] [[mark(isList(_x0))]] = 2x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(isNeList(_x0))]] = 4 + 2x0 > 3 + 2x0 = [[a!6220!6220isNeList(_x0)]] [[mark(isQid(_x0))]] = 4 + 2x0 > 3 + 2x0 = [[a!6220!6220isQid(_x0)]] [[mark(isNePal(_x0))]] = 2 + 2x0 > 1 + x0 = [[a!6220!6220isNePal(_x0)]] [[mark(isPal(_x0))]] = 4 + 2x0 > 2 + x0 = [[a!6220!6220isPal(_x0)]] [[mark(nil)]] = 2 > 1 = [[nil]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(a)]] = 4 > 2 = [[a]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[and(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220isNeList(_x0)]] = 3 + 2x0 > 2 + x0 = [[isNeList(_x0)]] [[a!6220!6220isQid(_x0)]] = 3 + 2x0 > 2 + x0 = [[isQid(_x0)]] [[a!6220!6220isNePal(_x0)]] = 1 + x0 >= 1 + x0 = [[isNePal(_x0)]] [[a!6220!6220isPal(_x0)]] = 2 + x0 >= 2 + x0 = [[isPal(_x0)]] We can thus remove the following rules: mark(and(X, Y)) => a!6220!6220and(mark(X), Y) mark(isNeList(X)) => a!6220!6220isNeList(X) mark(isQid(X)) => a!6220!6220isQid(X) mark(isNePal(X)) => a!6220!6220isNePal(X) mark(isPal(X)) => a!6220!6220isPal(X) mark(nil) => nil mark(a) => a a!6220!6220isNeList(X) => isNeList(X) a!6220!6220isQid(X) => isQid(X) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): a!6220!6220isNeList(X) >? a!6220!6220isQid(X) mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(isList(X)) >? a!6220!6220isList(X) mark(tt) >? tt mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220and(X, Y) >? and(X, Y) a!6220!6220isList(X) >? isList(X) a!6220!6220isNePal(X) >? isNePal(X) a!6220!6220isPal(X) >? 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!6220!6220!6220!6220 = \y0y1.y0 + y1 a!6220!6220and = \y0y1.3 + y0 + y1 a!6220!6220isList = \y0.y0 a!6220!6220isNeList = \y0.3 + y0 a!6220!6220isNePal = \y0.3 + 2y0 a!6220!6220isPal = \y0.3 + y0 a!6220!6220isQid = \y0.y0 and = \y0y1.y0 + y1 e = 0 i = 0 isList = \y0.y0 isNePal = \y0.y0 isPal = \y0.y0 mark = \y0.y0 o = 0 tt = 0 u = 0 Using this interpretation, the requirements translate to: [[a!6220!6220isNeList(_x0)]] = 3 + x0 > x0 = [[a!6220!6220isQid(_x0)]] [[mark(!6220!6220(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(isList(_x0))]] = x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(tt)]] = 0 >= 0 = [[tt]] [[mark(e)]] = 0 >= 0 = [[e]] [[mark(i)]] = 0 >= 0 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 0 >= 0 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220and(_x0, _x1)]] = 3 + x0 + x1 > x0 + x1 = [[and(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] [[a!6220!6220isNePal(_x0)]] = 3 + 2x0 > x0 = [[isNePal(_x0)]] [[a!6220!6220isPal(_x0)]] = 3 + x0 > x0 = [[isPal(_x0)]] We can thus remove the following rules: a!6220!6220isNeList(X) => a!6220!6220isQid(X) a!6220!6220and(X, Y) => and(X, Y) a!6220!6220isNePal(X) => isNePal(X) a!6220!6220isPal(X) => isPal(X) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): mark(!6220!6220(X, Y)) >? a!6220!6220!6220!6220(mark(X), mark(Y)) mark(isList(X)) >? a!6220!6220isList(X) mark(tt) >? tt mark(e) >? e mark(i) >? i mark(o) >? o mark(u) >? u a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220isList(X) >? isList(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!6220!6220!6220!6220 = \y0y1.1 + y0 + 2y1 a!6220!6220isList = \y0.y0 e = 1 i = 1 isList = \y0.y0 mark = \y0.2y0 o = 0 tt = 1 u = 1 Using this interpretation, the requirements translate to: [[mark(!6220!6220(_x0, _x1))]] = 2 + 2x0 + 4x1 > 1 + 2x0 + 4x1 = [[a!6220!6220!6220!6220(mark(_x0), mark(_x1))]] [[mark(isList(_x0))]] = 2x0 >= x0 = [[a!6220!6220isList(_x0)]] [[mark(tt)]] = 2 > 1 = [[tt]] [[mark(e)]] = 2 > 1 = [[e]] [[mark(i)]] = 2 > 1 = [[i]] [[mark(o)]] = 0 >= 0 = [[o]] [[mark(u)]] = 2 > 1 = [[u]] [[a!6220!6220!6220!6220(_x0, _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = x0 >= x0 = [[isList(_x0)]] We can thus remove the following rules: mark(!6220!6220(X, Y)) => a!6220!6220!6220!6220(mark(X), mark(Y)) mark(tt) => tt mark(e) => e mark(i) => i mark(u) => 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]): mark(isList(X)) >? a!6220!6220isList(X) mark(o) >? o a!6220!6220!6220!6220(X, Y) >? !6220!6220(X, Y) a!6220!6220isList(X) >? isList(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!6220!6220!6220!6220 = \y0y1.3 + y0 + y1 a!6220!6220isList = \y0.2 + 2y0 isList = \y0.1 + y0 mark = \y0.3 + 3y0 o = 0 Using this interpretation, the requirements translate to: [[mark(isList(_x0))]] = 6 + 3x0 > 2 + 2x0 = [[a!6220!6220isList(_x0)]] [[mark(o)]] = 3 > 0 = [[o]] [[a!6220!6220!6220!6220(_x0, _x1)]] = 3 + x0 + x1 > x0 + x1 = [[!6220!6220(_x0, _x1)]] [[a!6220!6220isList(_x0)]] = 2 + 2x0 > 1 + x0 = [[isList(_x0)]] We can thus remove the following rules: mark(isList(X)) => a!6220!6220isList(X) mark(o) => o a!6220!6220!6220!6220(X, Y) => !6220!6220(X, Y) a!6220!6220isList(X) => isList(X) All rules were succesfully removed. Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold. +++ Citations +++ [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012.