* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3))
U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3))
U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3))
U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3))
U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3))
U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3))
U105(mark(X1),X2) -> mark(U105(X1,X2))
U105(ok(X1),ok(X2)) -> ok(U105(X1,X2))
U106(mark(X)) -> mark(U106(X))
U106(ok(X)) -> ok(U106(X))
U11(mark(X1),X2) -> mark(U11(X1,X2))
U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
U111(mark(X1),X2,X3) -> mark(U111(X1,X2,X3))
U111(ok(X1),ok(X2),ok(X3)) -> ok(U111(X1,X2,X3))
U112(mark(X1),X2,X3) -> mark(U112(X1,X2,X3))
U112(ok(X1),ok(X2),ok(X3)) -> ok(U112(X1,X2,X3))
U113(mark(X1),X2,X3) -> mark(U113(X1,X2,X3))
U113(ok(X1),ok(X2),ok(X3)) -> ok(U113(X1,X2,X3))
U114(mark(X1),X2) -> mark(U114(X1,X2))
U114(ok(X1),ok(X2)) -> ok(U114(X1,X2))
U12(mark(X1),X2) -> mark(U12(X1,X2))
U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
U121(mark(X1),X2) -> mark(U121(X1,X2))
U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
U122(mark(X)) -> mark(U122(X))
U122(ok(X)) -> ok(U122(X))
U13(mark(X)) -> mark(U13(X))
U13(ok(X)) -> ok(U13(X))
U131(mark(X1),X2,X3,X4) -> mark(U131(X1,X2,X3,X4))
U131(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U131(X1,X2,X3,X4))
U132(mark(X1),X2,X3,X4) -> mark(U132(X1,X2,X3,X4))
U132(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U132(X1,X2,X3,X4))
U133(mark(X1),X2,X3,X4) -> mark(U133(X1,X2,X3,X4))
U133(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U133(X1,X2,X3,X4))
U134(mark(X1),X2,X3,X4) -> mark(U134(X1,X2,X3,X4))
U134(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U134(X1,X2,X3,X4))
U135(mark(X1),X2,X3,X4) -> mark(U135(X1,X2,X3,X4))
U135(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U135(X1,X2,X3,X4))
U136(mark(X1),X2,X3,X4) -> mark(U136(X1,X2,X3,X4))
U136(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U136(X1,X2,X3,X4))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X1),X2) -> mark(U22(X1,X2))
U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
U23(mark(X)) -> mark(U23(X))
U23(ok(X)) -> ok(U23(X))
U31(mark(X1),X2) -> mark(U31(X1,X2))
U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
U32(mark(X1),X2) -> mark(U32(X1,X2))
U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
U33(mark(X)) -> mark(U33(X))
U33(ok(X)) -> ok(U33(X))
U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
U42(mark(X1),X2,X3) -> mark(U42(X1,X2,X3))
U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
U43(mark(X1),X2,X3) -> mark(U43(X1,X2,X3))
U43(ok(X1),ok(X2),ok(X3)) -> ok(U43(X1,X2,X3))
U44(mark(X1),X2,X3) -> mark(U44(X1,X2,X3))
U44(ok(X1),ok(X2),ok(X3)) -> ok(U44(X1,X2,X3))
U45(mark(X1),X2) -> mark(U45(X1,X2))
U45(ok(X1),ok(X2)) -> ok(U45(X1,X2))
U46(mark(X)) -> mark(U46(X))
U46(ok(X)) -> ok(U46(X))
U51(mark(X1),X2) -> mark(U51(X1,X2))
U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
U52(mark(X)) -> mark(U52(X))
U52(ok(X)) -> ok(U52(X))
U61(mark(X1),X2) -> mark(U61(X1,X2))
U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
U62(mark(X)) -> mark(U62(X))
U62(ok(X)) -> ok(U62(X))
U71(mark(X)) -> mark(U71(X))
U71(ok(X)) -> ok(U71(X))
U81(mark(X)) -> mark(U81(X))
U81(ok(X)) -> ok(U81(X))
U91(mark(X1),X2,X3) -> mark(U91(X1,X2,X3))
U91(ok(X1),ok(X2),ok(X3)) -> ok(U91(X1,X2,X3))
U92(mark(X1),X2,X3) -> mark(U92(X1,X2,X3))
U92(ok(X1),ok(X2),ok(X3)) -> ok(U92(X1,X2,X3))
U93(mark(X1),X2,X3) -> mark(U93(X1,X2,X3))
U93(ok(X1),ok(X2),ok(X3)) -> ok(U93(X1,X2,X3))
U94(mark(X1),X2,X3) -> mark(U94(X1,X2,X3))
U94(ok(X1),ok(X2),ok(X3)) -> ok(U94(X1,X2,X3))
U95(mark(X1),X2) -> mark(U95(X1,X2))
U95(ok(X1),ok(X2)) -> ok(U95(X1,X2))
U96(mark(X)) -> mark(U96(X))
U96(ok(X)) -> ok(U96(X))
active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3)
active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3)
active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3)
active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
active(U105(X1,X2)) -> U105(active(X1),X2)
active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
active(U106(X)) -> U106(active(X))
active(U106(tt())) -> mark(tt())
active(U11(X1,X2)) -> U11(active(X1),X2)
active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
active(U111(X1,X2,X3)) -> U111(active(X1),X2,X3)
active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
active(U112(X1,X2,X3)) -> U112(active(X1),X2,X3)
active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
active(U113(X1,X2,X3)) -> U113(active(X1),X2,X3)
active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
active(U114(X1,X2)) -> U114(active(X1),X2)
active(U114(tt(),L)) -> mark(s(length(L)))
active(U12(X1,X2)) -> U12(active(X1),X2)
active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
active(U121(X1,X2)) -> U121(active(X1),X2)
active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(X)) -> U122(active(X))
active(U122(tt())) -> mark(nil())
active(U13(X)) -> U13(active(X))
active(U13(tt())) -> mark(tt())
active(U131(X1,X2,X3,X4)) -> U131(active(X1),X2,X3,X4)
active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
active(U132(X1,X2,X3,X4)) -> U132(active(X1),X2,X3,X4)
active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
active(U133(X1,X2,X3,X4)) -> U133(active(X1),X2,X3,X4)
active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
active(U134(X1,X2,X3,X4)) -> U134(active(X1),X2,X3,X4)
active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
active(U135(X1,X2,X3,X4)) -> U135(active(X1),X2,X3,X4)
active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
active(U136(X1,X2,X3,X4)) -> U136(active(X1),X2,X3,X4)
active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
active(U21(X1,X2)) -> U21(active(X1),X2)
active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
active(U22(X1,X2)) -> U22(active(X1),X2)
active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
active(U23(X)) -> U23(active(X))
active(U23(tt())) -> mark(tt())
active(U31(X1,X2)) -> U31(active(X1),X2)
active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
active(U32(X1,X2)) -> U32(active(X1),X2)
active(U32(tt(),V)) -> mark(U33(isNatList(V)))
active(U33(X)) -> U33(active(X))
active(U33(tt())) -> mark(tt())
active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
active(U42(X1,X2,X3)) -> U42(active(X1),X2,X3)
active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
active(U43(X1,X2,X3)) -> U43(active(X1),X2,X3)
active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
active(U44(X1,X2,X3)) -> U44(active(X1),X2,X3)
active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
active(U45(X1,X2)) -> U45(active(X1),X2)
active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
active(U46(X)) -> U46(active(X))
active(U46(tt())) -> mark(tt())
active(U51(X1,X2)) -> U51(active(X1),X2)
active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(X)) -> U52(active(X))
active(U52(tt())) -> mark(tt())
active(U61(X1,X2)) -> U61(active(X1),X2)
active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(X)) -> U62(active(X))
active(U62(tt())) -> mark(tt())
active(U71(X)) -> U71(active(X))
active(U71(tt())) -> mark(tt())
active(U81(X)) -> U81(active(X))
active(U81(tt())) -> mark(tt())
active(U91(X1,X2,X3)) -> U91(active(X1),X2,X3)
active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
active(U92(X1,X2,X3)) -> U92(active(X1),X2,X3)
active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
active(U93(X1,X2,X3)) -> U93(active(X1),X2,X3)
active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
active(U94(X1,X2,X3)) -> U94(active(X1),X2,X3)
active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
active(U95(X1,X2)) -> U95(active(X1),X2)
active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
active(U96(X)) -> U96(active(X))
active(U96(tt())) -> mark(tt())
active(cons(X1,X2)) -> cons(active(X1),X2)
active(isNat(0())) -> mark(tt())
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
active(isNatIList(zeros())) -> mark(tt())
active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
active(isNatIListKind(nil())) -> mark(tt())
active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
active(isNatIListKind(zeros())) -> mark(tt())
active(isNatKind(0())) -> mark(tt())
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
active(isNatList(nil())) -> mark(tt())
active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
active(length(X)) -> length(active(X))
active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
active(length(nil())) -> mark(0())
active(s(X)) -> s(active(X))
active(take(X1,X2)) -> take(X1,active(X2))
active(take(X1,X2)) -> take(active(X1),X2)
active(take(0(),IL)) -> mark(U121(isNatIList(IL),IL))
active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
active(zeros()) -> mark(cons(0(),zeros()))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
isNat(ok(X)) -> ok(isNat(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
isNatList(ok(X)) -> ok(isNatList(X))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(0()) -> ok(0())
proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3))
proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3))
proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3))
proper(U105(X1,X2)) -> U105(proper(X1),proper(X2))
proper(U106(X)) -> U106(proper(X))
proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
proper(U111(X1,X2,X3)) -> U111(proper(X1),proper(X2),proper(X3))
proper(U112(X1,X2,X3)) -> U112(proper(X1),proper(X2),proper(X3))
proper(U113(X1,X2,X3)) -> U113(proper(X1),proper(X2),proper(X3))
proper(U114(X1,X2)) -> U114(proper(X1),proper(X2))
proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(U131(X1,X2,X3,X4)) -> U131(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U132(X1,X2,X3,X4)) -> U132(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U133(X1,X2,X3,X4)) -> U133(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U134(X1,X2,X3,X4)) -> U134(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U135(X1,X2,X3,X4)) -> U135(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U136(X1,X2,X3,X4)) -> U136(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
proper(U43(X1,X2,X3)) -> U43(proper(X1),proper(X2),proper(X3))
proper(U44(X1,X2,X3)) -> U44(proper(X1),proper(X2),proper(X3))
proper(U45(X1,X2)) -> U45(proper(X1),proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1,X2,X3)) -> U91(proper(X1),proper(X2),proper(X3))
proper(U92(X1,X2,X3)) -> U92(proper(X1),proper(X2),proper(X3))
proper(U93(X1,X2,X3)) -> U93(proper(X1),proper(X2),proper(X3))
proper(U94(X1,X2,X3)) -> U94(proper(X1),proper(X2),proper(X3))
proper(U95(X1,X2)) -> U95(proper(X1),proper(X2))
proper(U96(X)) -> U96(proper(X))
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil()) -> ok(nil())
proper(s(X)) -> s(proper(X))
proper(take(X1,X2)) -> take(proper(X1),proper(X2))
proper(tt()) -> ok(tt())
proper(zeros()) -> ok(zeros())
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
take(X1,mark(X2)) -> mark(take(X1,X2))
take(mark(X1),X2) -> mark(take(X1,X2))
take(ok(X1),ok(X2)) -> ok(take(X1,X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{U101/3,U102/3,U103/3,U104/3,U105/2,U106/1,U11/2,U111/3,U112/3,U113/3,U114/2,U12/2,U121/2,U122/1,U13/1
,U131/4,U132/4,U133/4,U134/4,U135/4,U136/4,U21/2,U22/2,U23/1,U31/2,U32/2,U33/1,U41/3,U42/3,U43/3,U44/3,U45/2
,U46/1,U51/2,U52/1,U61/2,U62/1,U71/1,U81/1,U91/3,U92/3,U93/3,U94/3,U95/2,U96/1,active/1,cons/2,isNat/1
,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,proper/1,s/1,take/2,top/1} / {0/0,mark/1
,nil/0,ok/1,tt/0,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U101,U102,U103,U104,U105,U106,U11,U111,U112,U113,U114,U12
,U121,U122,U13,U131,U132,U133,U134,U135,U136,U21,U22,U23,U31,U32,U33,U41,U42,U43,U44,U45,U46,U51,U52,U61,U62
,U71,U81,U91,U92,U93,U94,U95,U96,active,cons,isNat,isNatIList,isNatIListKind,isNatKind,isNatList,length
,proper,s,take,top} and constructors {0,mark,nil,ok,tt,zeros}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3))
U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3))
U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3))
U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3))
U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3))
U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3))
U105(mark(X1),X2) -> mark(U105(X1,X2))
U105(ok(X1),ok(X2)) -> ok(U105(X1,X2))
U106(mark(X)) -> mark(U106(X))
U106(ok(X)) -> ok(U106(X))
U11(mark(X1),X2) -> mark(U11(X1,X2))
U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
U111(mark(X1),X2,X3) -> mark(U111(X1,X2,X3))
U111(ok(X1),ok(X2),ok(X3)) -> ok(U111(X1,X2,X3))
U112(mark(X1),X2,X3) -> mark(U112(X1,X2,X3))
U112(ok(X1),ok(X2),ok(X3)) -> ok(U112(X1,X2,X3))
U113(mark(X1),X2,X3) -> mark(U113(X1,X2,X3))
U113(ok(X1),ok(X2),ok(X3)) -> ok(U113(X1,X2,X3))
U114(mark(X1),X2) -> mark(U114(X1,X2))
U114(ok(X1),ok(X2)) -> ok(U114(X1,X2))
U12(mark(X1),X2) -> mark(U12(X1,X2))
U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
U121(mark(X1),X2) -> mark(U121(X1,X2))
U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
U122(mark(X)) -> mark(U122(X))
U122(ok(X)) -> ok(U122(X))
U13(mark(X)) -> mark(U13(X))
U13(ok(X)) -> ok(U13(X))
U131(mark(X1),X2,X3,X4) -> mark(U131(X1,X2,X3,X4))
U131(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U131(X1,X2,X3,X4))
U132(mark(X1),X2,X3,X4) -> mark(U132(X1,X2,X3,X4))
U132(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U132(X1,X2,X3,X4))
U133(mark(X1),X2,X3,X4) -> mark(U133(X1,X2,X3,X4))
U133(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U133(X1,X2,X3,X4))
U134(mark(X1),X2,X3,X4) -> mark(U134(X1,X2,X3,X4))
U134(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U134(X1,X2,X3,X4))
U135(mark(X1),X2,X3,X4) -> mark(U135(X1,X2,X3,X4))
U135(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U135(X1,X2,X3,X4))
U136(mark(X1),X2,X3,X4) -> mark(U136(X1,X2,X3,X4))
U136(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U136(X1,X2,X3,X4))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X1),X2) -> mark(U22(X1,X2))
U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
U23(mark(X)) -> mark(U23(X))
U23(ok(X)) -> ok(U23(X))
U31(mark(X1),X2) -> mark(U31(X1,X2))
U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
U32(mark(X1),X2) -> mark(U32(X1,X2))
U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
U33(mark(X)) -> mark(U33(X))
U33(ok(X)) -> ok(U33(X))
U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
U42(mark(X1),X2,X3) -> mark(U42(X1,X2,X3))
U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
U43(mark(X1),X2,X3) -> mark(U43(X1,X2,X3))
U43(ok(X1),ok(X2),ok(X3)) -> ok(U43(X1,X2,X3))
U44(mark(X1),X2,X3) -> mark(U44(X1,X2,X3))
U44(ok(X1),ok(X2),ok(X3)) -> ok(U44(X1,X2,X3))
U45(mark(X1),X2) -> mark(U45(X1,X2))
U45(ok(X1),ok(X2)) -> ok(U45(X1,X2))
U46(mark(X)) -> mark(U46(X))
U46(ok(X)) -> ok(U46(X))
U51(mark(X1),X2) -> mark(U51(X1,X2))
U51(ok(X1),ok(X2)) -> ok(U51(X1,X2))
U52(mark(X)) -> mark(U52(X))
U52(ok(X)) -> ok(U52(X))
U61(mark(X1),X2) -> mark(U61(X1,X2))
U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
U62(mark(X)) -> mark(U62(X))
U62(ok(X)) -> ok(U62(X))
U71(mark(X)) -> mark(U71(X))
U71(ok(X)) -> ok(U71(X))
U81(mark(X)) -> mark(U81(X))
U81(ok(X)) -> ok(U81(X))
U91(mark(X1),X2,X3) -> mark(U91(X1,X2,X3))
U91(ok(X1),ok(X2),ok(X3)) -> ok(U91(X1,X2,X3))
U92(mark(X1),X2,X3) -> mark(U92(X1,X2,X3))
U92(ok(X1),ok(X2),ok(X3)) -> ok(U92(X1,X2,X3))
U93(mark(X1),X2,X3) -> mark(U93(X1,X2,X3))
U93(ok(X1),ok(X2),ok(X3)) -> ok(U93(X1,X2,X3))
U94(mark(X1),X2,X3) -> mark(U94(X1,X2,X3))
U94(ok(X1),ok(X2),ok(X3)) -> ok(U94(X1,X2,X3))
U95(mark(X1),X2) -> mark(U95(X1,X2))
U95(ok(X1),ok(X2)) -> ok(U95(X1,X2))
U96(mark(X)) -> mark(U96(X))
U96(ok(X)) -> ok(U96(X))
active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3)
active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3)
active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3)
active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
active(U105(X1,X2)) -> U105(active(X1),X2)
active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
active(U106(X)) -> U106(active(X))
active(U106(tt())) -> mark(tt())
active(U11(X1,X2)) -> U11(active(X1),X2)
active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
active(U111(X1,X2,X3)) -> U111(active(X1),X2,X3)
active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
active(U112(X1,X2,X3)) -> U112(active(X1),X2,X3)
active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
active(U113(X1,X2,X3)) -> U113(active(X1),X2,X3)
active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
active(U114(X1,X2)) -> U114(active(X1),X2)
active(U114(tt(),L)) -> mark(s(length(L)))
active(U12(X1,X2)) -> U12(active(X1),X2)
active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
active(U121(X1,X2)) -> U121(active(X1),X2)
active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(X)) -> U122(active(X))
active(U122(tt())) -> mark(nil())
active(U13(X)) -> U13(active(X))
active(U13(tt())) -> mark(tt())
active(U131(X1,X2,X3,X4)) -> U131(active(X1),X2,X3,X4)
active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
active(U132(X1,X2,X3,X4)) -> U132(active(X1),X2,X3,X4)
active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
active(U133(X1,X2,X3,X4)) -> U133(active(X1),X2,X3,X4)
active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
active(U134(X1,X2,X3,X4)) -> U134(active(X1),X2,X3,X4)
active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
active(U135(X1,X2,X3,X4)) -> U135(active(X1),X2,X3,X4)
active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
active(U136(X1,X2,X3,X4)) -> U136(active(X1),X2,X3,X4)
active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
active(U21(X1,X2)) -> U21(active(X1),X2)
active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
active(U22(X1,X2)) -> U22(active(X1),X2)
active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
active(U23(X)) -> U23(active(X))
active(U23(tt())) -> mark(tt())
active(U31(X1,X2)) -> U31(active(X1),X2)
active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
active(U32(X1,X2)) -> U32(active(X1),X2)
active(U32(tt(),V)) -> mark(U33(isNatList(V)))
active(U33(X)) -> U33(active(X))
active(U33(tt())) -> mark(tt())
active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
active(U42(X1,X2,X3)) -> U42(active(X1),X2,X3)
active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
active(U43(X1,X2,X3)) -> U43(active(X1),X2,X3)
active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
active(U44(X1,X2,X3)) -> U44(active(X1),X2,X3)
active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
active(U45(X1,X2)) -> U45(active(X1),X2)
active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
active(U46(X)) -> U46(active(X))
active(U46(tt())) -> mark(tt())
active(U51(X1,X2)) -> U51(active(X1),X2)
active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(X)) -> U52(active(X))
active(U52(tt())) -> mark(tt())
active(U61(X1,X2)) -> U61(active(X1),X2)
active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(X)) -> U62(active(X))
active(U62(tt())) -> mark(tt())
active(U71(X)) -> U71(active(X))
active(U71(tt())) -> mark(tt())
active(U81(X)) -> U81(active(X))
active(U81(tt())) -> mark(tt())
active(U91(X1,X2,X3)) -> U91(active(X1),X2,X3)
active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
active(U92(X1,X2,X3)) -> U92(active(X1),X2,X3)
active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
active(U93(X1,X2,X3)) -> U93(active(X1),X2,X3)
active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
active(U94(X1,X2,X3)) -> U94(active(X1),X2,X3)
active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
active(U95(X1,X2)) -> U95(active(X1),X2)
active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
active(U96(X)) -> U96(active(X))
active(U96(tt())) -> mark(tt())
active(cons(X1,X2)) -> cons(active(X1),X2)
active(isNat(0())) -> mark(tt())
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
active(isNatIList(zeros())) -> mark(tt())
active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
active(isNatIListKind(nil())) -> mark(tt())
active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
active(isNatIListKind(zeros())) -> mark(tt())
active(isNatKind(0())) -> mark(tt())
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
active(isNatList(nil())) -> mark(tt())
active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
active(length(X)) -> length(active(X))
active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
active(length(nil())) -> mark(0())
active(s(X)) -> s(active(X))
active(take(X1,X2)) -> take(X1,active(X2))
active(take(X1,X2)) -> take(active(X1),X2)
active(take(0(),IL)) -> mark(U121(isNatIList(IL),IL))
active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
active(zeros()) -> mark(cons(0(),zeros()))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
isNat(ok(X)) -> ok(isNat(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
isNatList(ok(X)) -> ok(isNatList(X))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(0()) -> ok(0())
proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
proper(U102(X1,X2,X3)) -> U102(proper(X1),proper(X2),proper(X3))
proper(U103(X1,X2,X3)) -> U103(proper(X1),proper(X2),proper(X3))
proper(U104(X1,X2,X3)) -> U104(proper(X1),proper(X2),proper(X3))
proper(U105(X1,X2)) -> U105(proper(X1),proper(X2))
proper(U106(X)) -> U106(proper(X))
proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
proper(U111(X1,X2,X3)) -> U111(proper(X1),proper(X2),proper(X3))
proper(U112(X1,X2,X3)) -> U112(proper(X1),proper(X2),proper(X3))
proper(U113(X1,X2,X3)) -> U113(proper(X1),proper(X2),proper(X3))
proper(U114(X1,X2)) -> U114(proper(X1),proper(X2))
proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(U131(X1,X2,X3,X4)) -> U131(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U132(X1,X2,X3,X4)) -> U132(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U133(X1,X2,X3,X4)) -> U133(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U134(X1,X2,X3,X4)) -> U134(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U135(X1,X2,X3,X4)) -> U135(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U136(X1,X2,X3,X4)) -> U136(proper(X1),proper(X2),proper(X3),proper(X4))
proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
proper(U22(X1,X2)) -> U22(proper(X1),proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
proper(U43(X1,X2,X3)) -> U43(proper(X1),proper(X2),proper(X3))
proper(U44(X1,X2,X3)) -> U44(proper(X1),proper(X2),proper(X3))
proper(U45(X1,X2)) -> U45(proper(X1),proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1,X2)) -> U51(proper(X1),proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1,X2,X3)) -> U91(proper(X1),proper(X2),proper(X3))
proper(U92(X1,X2,X3)) -> U92(proper(X1),proper(X2),proper(X3))
proper(U93(X1,X2,X3)) -> U93(proper(X1),proper(X2),proper(X3))
proper(U94(X1,X2,X3)) -> U94(proper(X1),proper(X2),proper(X3))
proper(U95(X1,X2)) -> U95(proper(X1),proper(X2))
proper(U96(X)) -> U96(proper(X))
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil()) -> ok(nil())
proper(s(X)) -> s(proper(X))
proper(take(X1,X2)) -> take(proper(X1),proper(X2))
proper(tt()) -> ok(tt())
proper(zeros()) -> ok(zeros())
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
take(X1,mark(X2)) -> mark(take(X1,X2))
take(mark(X1),X2) -> mark(take(X1,X2))
take(ok(X1),ok(X2)) -> ok(take(X1,X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{U101/3,U102/3,U103/3,U104/3,U105/2,U106/1,U11/2,U111/3,U112/3,U113/3,U114/2,U12/2,U121/2,U122/1,U13/1
,U131/4,U132/4,U133/4,U134/4,U135/4,U136/4,U21/2,U22/2,U23/1,U31/2,U32/2,U33/1,U41/3,U42/3,U43/3,U44/3,U45/2
,U46/1,U51/2,U52/1,U61/2,U62/1,U71/1,U81/1,U91/3,U92/3,U93/3,U94/3,U95/2,U96/1,active/1,cons/2,isNat/1
,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,proper/1,s/1,take/2,top/1} / {0/0,mark/1
,nil/0,ok/1,tt/0,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U101,U102,U103,U104,U105,U106,U11,U111,U112,U113,U114,U12
,U121,U122,U13,U131,U132,U133,U134,U135,U136,U21,U22,U23,U31,U32,U33,U41,U42,U43,U44,U45,U46,U51,U52,U61,U62
,U71,U81,U91,U92,U93,U94,U95,U96,active,cons,isNat,isNatIList,isNatIListKind,isNatKind,isNatList,length
,proper,s,take,top} and constructors {0,mark,nil,ok,tt,zeros}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
U101(x,y,z){x -> mark(x)} =
U101(mark(x),y,z) ->^+ mark(U101(x,y,z))
= C[U101(x,y,z) = U101(x,y,z){}]
WORST_CASE(Omega(n^1),?)