MAYBE * Step 1: InnermostRuleRemoval MAYBE + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() U71(tt(),L,N) -> U72(isNat(activate(N)),activate(L)) U72(tt(),L) -> s(length(activate(L))) U81(tt()) -> nil() U91(tt(),IL,M,N) -> U92(isNat(activate(M)),activate(IL),activate(M),activate(N)) U92(tt(),IL,M,N) -> U93(isNat(activate(N)),activate(IL),activate(M),activate(N)) U93(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL))) activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) length(cons(N,L)) -> U71(isNatList(activate(L)),activate(L),N) length(nil()) -> 0() nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> U81(isNatIList(IL)) take(s(M),cons(N,IL)) -> U91(isNatIList(activate(IL)),activate(IL),M,N) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0} / {n__0/0,n__cons/2,n__length/1 ,n__nil/0,n__s/1,n__take/2,n__zeros/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,U42,U51,U52,U61,U62,U71,U72,U81,U91,U92 ,U93,activate,cons,isNat,isNatIList,isNatList,length,nil,s,take,zeros} and constructors {n__0,n__cons ,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. length(cons(N,L)) -> U71(isNatList(activate(L)),activate(L),N) length(nil()) -> 0() take(0(),IL) -> U81(isNatIList(IL)) take(s(M),cons(N,IL)) -> U91(isNatIList(activate(IL)),activate(IL),M,N) All above mentioned rules can be savely removed. * Step 2: DependencyPairs MAYBE + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() U71(tt(),L,N) -> U72(isNat(activate(N)),activate(L)) U72(tt(),L) -> s(length(activate(L))) U81(tt()) -> nil() U91(tt(),IL,M,N) -> U92(isNat(activate(M)),activate(IL),activate(M),activate(N)) U92(tt(),IL,M,N) -> U93(isNat(activate(N)),activate(IL),activate(M),activate(N)) U93(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL))) activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0} / {n__0/0,n__cons/2,n__length/1 ,n__nil/0,n__s/1,n__take/2,n__zeros/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,U42,U51,U52,U61,U62,U71,U72,U81,U91,U92 ,U93,activate,cons,isNat,isNatIList,isNatList,length,nil,s,take,zeros} and constructors {n__0,n__cons ,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 0#() -> c_1() U11#(tt()) -> c_2() U21#(tt()) -> c_3() U31#(tt()) -> c_4() U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U42#(tt()) -> c_6() U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) U52#(tt()) -> c_8() U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U62#(tt()) -> c_10() U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) U81#(tt()) -> c_13(nil#()) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) activate#(X) -> c_17() activate#(n__0()) -> c_18(0#()) activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) activate#(n__length(X)) -> c_20(length#(X)) activate#(n__nil()) -> c_21(nil#()) activate#(n__s(X)) -> c_22(s#(X)) activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) activate#(n__zeros()) -> c_24(zeros#()) cons#(X1,X2) -> c_25() isNat#(n__0()) -> c_26() isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatIList#(n__zeros()) -> c_31() isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__nil()) -> c_33() isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) length#(X) -> c_35() nil#() -> c_36() s#(X) -> c_37() take#(X1,X2) -> c_38() zeros#() -> c_39(cons#(0(),n__zeros()),0#()) zeros#() -> c_40() Weak DPs and mark the set of starting terms. * Step 3: UsableRules MAYBE + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt()) -> c_2() U21#(tt()) -> c_3() U31#(tt()) -> c_4() U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U42#(tt()) -> c_6() U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) U52#(tt()) -> c_8() U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U62#(tt()) -> c_10() U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) U81#(tt()) -> c_13(nil#()) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) activate#(X) -> c_17() activate#(n__0()) -> c_18(0#()) activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) activate#(n__length(X)) -> c_20(length#(X)) activate#(n__nil()) -> c_21(nil#()) activate#(n__s(X)) -> c_22(s#(X)) activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) activate#(n__zeros()) -> c_24(zeros#()) cons#(X1,X2) -> c_25() isNat#(n__0()) -> c_26() isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatIList#(n__zeros()) -> c_31() isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__nil()) -> c_33() isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) length#(X) -> c_35() nil#() -> c_36() s#(X) -> c_37() take#(X1,X2) -> c_38() zeros#() -> c_39(cons#(0(),n__zeros()),0#()) zeros#() -> c_40() - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() U71(tt(),L,N) -> U72(isNat(activate(N)),activate(L)) U72(tt(),L) -> s(length(activate(L))) U81(tt()) -> nil() U91(tt(),IL,M,N) -> U92(isNat(activate(M)),activate(IL),activate(M),activate(N)) U92(tt(),IL,M,N) -> U93(isNat(activate(N)),activate(IL),activate(M),activate(N)) U93(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL))) activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/3,c_10/0,c_11/4,c_12/3 ,c_13/1,c_14/6,c_15/6,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/3 ,c_28/3,c_29/3,c_30/4,c_31/0,c_32/4,c_33/0,c_34/4,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() 0#() -> c_1() U11#(tt()) -> c_2() U21#(tt()) -> c_3() U31#(tt()) -> c_4() U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U42#(tt()) -> c_6() U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) U52#(tt()) -> c_8() U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U62#(tt()) -> c_10() U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) U81#(tt()) -> c_13(nil#()) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) activate#(X) -> c_17() activate#(n__0()) -> c_18(0#()) activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) activate#(n__length(X)) -> c_20(length#(X)) activate#(n__nil()) -> c_21(nil#()) activate#(n__s(X)) -> c_22(s#(X)) activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) activate#(n__zeros()) -> c_24(zeros#()) cons#(X1,X2) -> c_25() isNat#(n__0()) -> c_26() isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatIList#(n__zeros()) -> c_31() isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__nil()) -> c_33() isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) length#(X) -> c_35() nil#() -> c_36() s#(X) -> c_37() take#(X1,X2) -> c_38() zeros#() -> c_39(cons#(0(),n__zeros()),0#()) zeros#() -> c_40() * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt()) -> c_2() U21#(tt()) -> c_3() U31#(tt()) -> c_4() U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U42#(tt()) -> c_6() U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) U52#(tt()) -> c_8() U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U62#(tt()) -> c_10() U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) U81#(tt()) -> c_13(nil#()) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) activate#(X) -> c_17() activate#(n__0()) -> c_18(0#()) activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) activate#(n__length(X)) -> c_20(length#(X)) activate#(n__nil()) -> c_21(nil#()) activate#(n__s(X)) -> c_22(s#(X)) activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) activate#(n__zeros()) -> c_24(zeros#()) cons#(X1,X2) -> c_25() isNat#(n__0()) -> c_26() isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatIList#(n__zeros()) -> c_31() isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__nil()) -> c_33() isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) length#(X) -> c_35() nil#() -> c_36() s#(X) -> c_37() take#(X1,X2) -> c_38() zeros#() -> c_39(cons#(0(),n__zeros()),0#()) zeros#() -> c_40() - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/3,c_10/0,c_11/4,c_12/3 ,c_13/1,c_14/6,c_15/6,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/3 ,c_28/3,c_29/3,c_30/4,c_31/0,c_32/4,c_33/0,c_34/4,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,6,8,10,17,25,26,31,33,35,36,37,38,40} by application of Pre({1,2,3,4,6,8,10,17,25,26,31,33,35,36,37,38,40}) = {5,7,9,11,12,13,14,15,16,18,19,20,21,22,23,24,27,28 ,29,30,32,34,39}. Here rules are labelled as follows: 1: 0#() -> c_1() 2: U11#(tt()) -> c_2() 3: U21#(tt()) -> c_3() 4: U31#(tt()) -> c_4() 5: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) 6: U42#(tt()) -> c_6() 7: U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) 8: U52#(tt()) -> c_8() 9: U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) 10: U62#(tt()) -> c_10() 11: U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)) ,isNat#(activate(N)) ,activate#(N) ,activate#(L)) 12: U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) 13: U81#(tt()) -> c_13(nil#()) 14: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) 15: U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) 16: U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) 17: activate#(X) -> c_17() 18: activate#(n__0()) -> c_18(0#()) 19: activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) 20: activate#(n__length(X)) -> c_20(length#(X)) 21: activate#(n__nil()) -> c_21(nil#()) 22: activate#(n__s(X)) -> c_22(s#(X)) 23: activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) 24: activate#(n__zeros()) -> c_24(zeros#()) 25: cons#(X1,X2) -> c_25() 26: isNat#(n__0()) -> c_26() 27: isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) 28: isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 29: isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) 30: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 31: isNatIList#(n__zeros()) -> c_31() 32: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 33: isNatList#(n__nil()) -> c_33() 34: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 35: length#(X) -> c_35() 36: nil#() -> c_36() 37: s#(X) -> c_37() 38: take#(X1,X2) -> c_38() 39: zeros#() -> c_39(cons#(0(),n__zeros()),0#()) 40: zeros#() -> c_40() * Step 5: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) U81#(tt()) -> c_13(nil#()) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) activate#(n__0()) -> c_18(0#()) activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) activate#(n__length(X)) -> c_20(length#(X)) activate#(n__nil()) -> c_21(nil#()) activate#(n__s(X)) -> c_22(s#(X)) activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) activate#(n__zeros()) -> c_24(zeros#()) isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) zeros#() -> c_39(cons#(0(),n__zeros()),0#()) - Weak DPs: 0#() -> c_1() U11#(tt()) -> c_2() U21#(tt()) -> c_3() U31#(tt()) -> c_4() U42#(tt()) -> c_6() U52#(tt()) -> c_8() U62#(tt()) -> c_10() activate#(X) -> c_17() cons#(X1,X2) -> c_25() isNat#(n__0()) -> c_26() isNatIList#(n__zeros()) -> c_31() isNatList#(n__nil()) -> c_33() length#(X) -> c_35() nil#() -> c_36() s#(X) -> c_37() take#(X1,X2) -> c_38() zeros#() -> c_40() - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/3,c_10/0,c_11/4,c_12/3 ,c_13/1,c_14/6,c_15/6,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/3 ,c_28/3,c_29/3,c_30/4,c_31/0,c_32/4,c_33/0,c_34/4,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {6,10,11,12,13,14,15,23} by application of Pre({6,10,11,12,13,14,15,23}) = {1,2,3,4,5,7,8,9,16,17,18,19,20,21,22}. Here rules are labelled as follows: 1: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) 2: U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) 3: U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) 4: U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)) ,isNat#(activate(N)) ,activate#(N) ,activate#(L)) 5: U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) 6: U81#(tt()) -> c_13(nil#()) 7: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) 8: U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) 9: U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) 10: activate#(n__0()) -> c_18(0#()) 11: activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) 12: activate#(n__length(X)) -> c_20(length#(X)) 13: activate#(n__nil()) -> c_21(nil#()) 14: activate#(n__s(X)) -> c_22(s#(X)) 15: activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) 16: activate#(n__zeros()) -> c_24(zeros#()) 17: isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) 18: isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 19: isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) 20: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 21: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 22: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 23: zeros#() -> c_39(cons#(0(),n__zeros()),0#()) 24: 0#() -> c_1() 25: U11#(tt()) -> c_2() 26: U21#(tt()) -> c_3() 27: U31#(tt()) -> c_4() 28: U42#(tt()) -> c_6() 29: U52#(tt()) -> c_8() 30: U62#(tt()) -> c_10() 31: activate#(X) -> c_17() 32: cons#(X1,X2) -> c_25() 33: isNat#(n__0()) -> c_26() 34: isNatIList#(n__zeros()) -> c_31() 35: isNatList#(n__nil()) -> c_33() 36: length#(X) -> c_35() 37: nil#() -> c_36() 38: s#(X) -> c_37() 39: take#(X1,X2) -> c_38() 40: zeros#() -> c_40() * Step 6: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) activate#(n__zeros()) -> c_24(zeros#()) isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak DPs: 0#() -> c_1() U11#(tt()) -> c_2() U21#(tt()) -> c_3() U31#(tt()) -> c_4() U42#(tt()) -> c_6() U52#(tt()) -> c_8() U62#(tt()) -> c_10() U81#(tt()) -> c_13(nil#()) activate#(X) -> c_17() activate#(n__0()) -> c_18(0#()) activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) activate#(n__length(X)) -> c_20(length#(X)) activate#(n__nil()) -> c_21(nil#()) activate#(n__s(X)) -> c_22(s#(X)) activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) cons#(X1,X2) -> c_25() isNat#(n__0()) -> c_26() isNatIList#(n__zeros()) -> c_31() isNatList#(n__nil()) -> c_33() length#(X) -> c_35() nil#() -> c_36() s#(X) -> c_37() take#(X1,X2) -> c_38() zeros#() -> c_39(cons#(0(),n__zeros()),0#()) zeros#() -> c_40() - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/3,c_10/0,c_11/4,c_12/3 ,c_13/1,c_14/6,c_15/6,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/3 ,c_28/3,c_29/3,c_30/4,c_31/0,c_32/4,c_33/0,c_34/4,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {9} by application of Pre({9}) = {1,2,3,4,5,6,7,8,10,11,12,13,14,15}. Here rules are labelled as follows: 1: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) 2: U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) 3: U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) 4: U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)) ,isNat#(activate(N)) ,activate#(N) ,activate#(L)) 5: U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) 6: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) 7: U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) 8: U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) 9: activate#(n__zeros()) -> c_24(zeros#()) 10: isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) 11: isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 12: isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) 13: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 14: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 15: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 16: 0#() -> c_1() 17: U11#(tt()) -> c_2() 18: U21#(tt()) -> c_3() 19: U31#(tt()) -> c_4() 20: U42#(tt()) -> c_6() 21: U52#(tt()) -> c_8() 22: U62#(tt()) -> c_10() 23: U81#(tt()) -> c_13(nil#()) 24: activate#(X) -> c_17() 25: activate#(n__0()) -> c_18(0#()) 26: activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) 27: activate#(n__length(X)) -> c_20(length#(X)) 28: activate#(n__nil()) -> c_21(nil#()) 29: activate#(n__s(X)) -> c_22(s#(X)) 30: activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) 31: cons#(X1,X2) -> c_25() 32: isNat#(n__0()) -> c_26() 33: isNatIList#(n__zeros()) -> c_31() 34: isNatList#(n__nil()) -> c_33() 35: length#(X) -> c_35() 36: nil#() -> c_36() 37: s#(X) -> c_37() 38: take#(X1,X2) -> c_38() 39: zeros#() -> c_39(cons#(0(),n__zeros()),0#()) 40: zeros#() -> c_40() * Step 7: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak DPs: 0#() -> c_1() U11#(tt()) -> c_2() U21#(tt()) -> c_3() U31#(tt()) -> c_4() U42#(tt()) -> c_6() U52#(tt()) -> c_8() U62#(tt()) -> c_10() U81#(tt()) -> c_13(nil#()) activate#(X) -> c_17() activate#(n__0()) -> c_18(0#()) activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) activate#(n__length(X)) -> c_20(length#(X)) activate#(n__nil()) -> c_21(nil#()) activate#(n__s(X)) -> c_22(s#(X)) activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) activate#(n__zeros()) -> c_24(zeros#()) cons#(X1,X2) -> c_25() isNat#(n__0()) -> c_26() isNatIList#(n__zeros()) -> c_31() isNatList#(n__nil()) -> c_33() length#(X) -> c_35() nil#() -> c_36() s#(X) -> c_37() take#(X1,X2) -> c_38() zeros#() -> c_39(cons#(0(),n__zeros()),0#()) zeros#() -> c_40() - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/3,c_10/0,c_11/4,c_12/3 ,c_13/1,c_14/6,c_15/6,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/3 ,c_28/3,c_29/3,c_30/4,c_31/0,c_32/4,c_33/0,c_34/4,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5,8} by application of Pre({5,8}) = {4,7}. Here rules are labelled as follows: 1: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) 2: U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) 3: U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) 4: U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)) ,isNat#(activate(N)) ,activate#(N) ,activate#(L)) 5: U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) 6: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) 7: U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) 8: U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) 9: isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) 10: isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 11: isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) 12: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 13: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 14: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 15: 0#() -> c_1() 16: U11#(tt()) -> c_2() 17: U21#(tt()) -> c_3() 18: U31#(tt()) -> c_4() 19: U42#(tt()) -> c_6() 20: U52#(tt()) -> c_8() 21: U62#(tt()) -> c_10() 22: U81#(tt()) -> c_13(nil#()) 23: activate#(X) -> c_17() 24: activate#(n__0()) -> c_18(0#()) 25: activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) 26: activate#(n__length(X)) -> c_20(length#(X)) 27: activate#(n__nil()) -> c_21(nil#()) 28: activate#(n__s(X)) -> c_22(s#(X)) 29: activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) 30: activate#(n__zeros()) -> c_24(zeros#()) 31: cons#(X1,X2) -> c_25() 32: isNat#(n__0()) -> c_26() 33: isNatIList#(n__zeros()) -> c_31() 34: isNatList#(n__nil()) -> c_33() 35: length#(X) -> c_35() 36: nil#() -> c_36() 37: s#(X) -> c_37() 38: take#(X1,X2) -> c_38() 39: zeros#() -> c_39(cons#(0(),n__zeros()),0#()) 40: zeros#() -> c_40() * Step 8: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak DPs: 0#() -> c_1() U11#(tt()) -> c_2() U21#(tt()) -> c_3() U31#(tt()) -> c_4() U42#(tt()) -> c_6() U52#(tt()) -> c_8() U62#(tt()) -> c_10() U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) U81#(tt()) -> c_13(nil#()) U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) activate#(X) -> c_17() activate#(n__0()) -> c_18(0#()) activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) activate#(n__length(X)) -> c_20(length#(X)) activate#(n__nil()) -> c_21(nil#()) activate#(n__s(X)) -> c_22(s#(X)) activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) activate#(n__zeros()) -> c_24(zeros#()) cons#(X1,X2) -> c_25() isNat#(n__0()) -> c_26() isNatIList#(n__zeros()) -> c_31() isNatList#(n__nil()) -> c_33() length#(X) -> c_35() nil#() -> c_36() s#(X) -> c_37() take#(X1,X2) -> c_38() zeros#() -> c_39(cons#(0(),n__zeros()),0#()) zeros#() -> c_40() - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/3,c_10/0,c_11/4,c_12/3 ,c_13/1,c_14/6,c_15/6,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/3 ,c_28/3,c_29/3,c_30/4,c_31/0,c_32/4,c_33/0,c_34/4,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):10 -->_2 isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)):9 -->_2 isNatIList#(n__zeros()) -> c_31():33 -->_3 activate#(X) -> c_17():23 -->_1 U42#(tt()) -> c_6():17 2:S:U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):12 -->_2 isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):11 -->_2 isNatList#(n__nil()) -> c_33():34 -->_3 activate#(X) -> c_17():23 -->_1 U52#(tt()) -> c_8():18 3:S:U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):10 -->_2 isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)):9 -->_2 isNatIList#(n__zeros()) -> c_31():33 -->_3 activate#(X) -> c_17():23 -->_1 U62#(tt()) -> c_10():19 4:S:U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)) ,isNat#(activate(N)) ,activate#(N) ,activate#(L)) -->_4 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_4 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_4 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_4 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_4 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_4 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_4 activate#(n__0()) -> c_18(0#()):24 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_1 U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)):20 -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_2 isNat#(n__0()) -> c_26():32 -->_4 activate#(X) -> c_17():23 -->_3 activate#(X) -> c_17():23 5:S:U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) -->_6 activate#(n__zeros()) -> c_24(zeros#()):30 -->_5 activate#(n__zeros()) -> c_24(zeros#()):30 -->_4 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_6 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_5 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_4 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_6 activate#(n__s(X)) -> c_22(s#(X)):28 -->_5 activate#(n__s(X)) -> c_22(s#(X)):28 -->_4 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_6 activate#(n__nil()) -> c_21(nil#()):27 -->_5 activate#(n__nil()) -> c_21(nil#()):27 -->_4 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_6 activate#(n__length(X)) -> c_20(length#(X)):26 -->_5 activate#(n__length(X)) -> c_20(length#(X)):26 -->_4 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_6 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_5 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_4 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_6 activate#(n__0()) -> c_18(0#()):24 -->_5 activate#(n__0()) -> c_18(0#()):24 -->_4 activate#(n__0()) -> c_18(0#()):24 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_1 U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)):6 -->_2 isNat#(n__0()) -> c_26():32 -->_6 activate#(X) -> c_17():23 -->_5 activate#(X) -> c_17():23 -->_4 activate#(X) -> c_17():23 -->_3 activate#(X) -> c_17():23 6:S:U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) -->_6 activate#(n__zeros()) -> c_24(zeros#()):30 -->_5 activate#(n__zeros()) -> c_24(zeros#()):30 -->_4 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_6 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_5 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_4 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_6 activate#(n__s(X)) -> c_22(s#(X)):28 -->_5 activate#(n__s(X)) -> c_22(s#(X)):28 -->_4 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_6 activate#(n__nil()) -> c_21(nil#()):27 -->_5 activate#(n__nil()) -> c_21(nil#()):27 -->_4 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_6 activate#(n__length(X)) -> c_20(length#(X)):26 -->_5 activate#(n__length(X)) -> c_20(length#(X)):26 -->_4 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_6 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_5 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_4 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_6 activate#(n__0()) -> c_18(0#()):24 -->_5 activate#(n__0()) -> c_18(0#()):24 -->_4 activate#(n__0()) -> c_18(0#()):24 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_1 U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)):22 -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_2 isNat#(n__0()) -> c_26():32 -->_6 activate#(X) -> c_17():23 -->_5 activate#(X) -> c_17():23 -->_4 activate#(X) -> c_17():23 -->_3 activate#(X) -> c_17():23 7:S:isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):12 -->_2 isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):11 -->_2 isNatList#(n__nil()) -> c_33():34 -->_3 activate#(X) -> c_17():23 -->_1 U11#(tt()) -> c_2():14 8:S:isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNat#(n__0()) -> c_26():32 -->_3 activate#(X) -> c_17():23 -->_1 U21#(tt()) -> c_3():15 -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 9:S:isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):12 -->_2 isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):11 -->_2 isNatList#(n__nil()) -> c_33():34 -->_3 activate#(X) -> c_17():23 -->_1 U31#(tt()) -> c_4():16 10:S:isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_4 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_4 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_4 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_4 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_4 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_4 activate#(n__0()) -> c_18(0#()):24 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNat#(n__0()) -> c_26():32 -->_4 activate#(X) -> c_17():23 -->_3 activate#(X) -> c_17():23 -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_1 U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)):1 11:S:isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_4 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_4 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_4 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_4 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_4 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_4 activate#(n__0()) -> c_18(0#()):24 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNat#(n__0()) -> c_26():32 -->_4 activate#(X) -> c_17():23 -->_3 activate#(X) -> c_17():23 -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_1 U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)):2 12:S:isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_4 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_4 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_4 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_4 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_4 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_4 activate#(n__0()) -> c_18(0#()):24 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 isNat#(n__0()) -> c_26():32 -->_4 activate#(X) -> c_17():23 -->_3 activate#(X) -> c_17():23 -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_1 U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)):3 13:W:0#() -> c_1() 14:W:U11#(tt()) -> c_2() 15:W:U21#(tt()) -> c_3() 16:W:U31#(tt()) -> c_4() 17:W:U42#(tt()) -> c_6() 18:W:U52#(tt()) -> c_8() 19:W:U62#(tt()) -> c_10() 20:W:U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_1 s#(X) -> c_37():37 -->_2 length#(X) -> c_35():35 -->_3 activate#(X) -> c_17():23 21:W:U81#(tt()) -> c_13(nil#()) -->_1 nil#() -> c_36():36 22:W:U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) -->_4 activate#(n__zeros()) -> c_24(zeros#()):30 -->_3 activate#(n__zeros()) -> c_24(zeros#()):30 -->_2 activate#(n__zeros()) -> c_24(zeros#()):30 -->_4 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_3 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_2 activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)):29 -->_4 activate#(n__s(X)) -> c_22(s#(X)):28 -->_3 activate#(n__s(X)) -> c_22(s#(X)):28 -->_2 activate#(n__s(X)) -> c_22(s#(X)):28 -->_4 activate#(n__nil()) -> c_21(nil#()):27 -->_3 activate#(n__nil()) -> c_21(nil#()):27 -->_2 activate#(n__nil()) -> c_21(nil#()):27 -->_4 activate#(n__length(X)) -> c_20(length#(X)):26 -->_3 activate#(n__length(X)) -> c_20(length#(X)):26 -->_2 activate#(n__length(X)) -> c_20(length#(X)):26 -->_4 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_3 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_2 activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)):25 -->_4 activate#(n__0()) -> c_18(0#()):24 -->_3 activate#(n__0()) -> c_18(0#()):24 -->_2 activate#(n__0()) -> c_18(0#()):24 -->_1 cons#(X1,X2) -> c_25():31 -->_4 activate#(X) -> c_17():23 -->_3 activate#(X) -> c_17():23 -->_2 activate#(X) -> c_17():23 23:W:activate#(X) -> c_17() 24:W:activate#(n__0()) -> c_18(0#()) -->_1 0#() -> c_1():13 25:W:activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) -->_1 cons#(X1,X2) -> c_25():31 26:W:activate#(n__length(X)) -> c_20(length#(X)) -->_1 length#(X) -> c_35():35 27:W:activate#(n__nil()) -> c_21(nil#()) -->_1 nil#() -> c_36():36 28:W:activate#(n__s(X)) -> c_22(s#(X)) -->_1 s#(X) -> c_37():37 29:W:activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) -->_1 take#(X1,X2) -> c_38():38 30:W:activate#(n__zeros()) -> c_24(zeros#()) -->_1 zeros#() -> c_39(cons#(0(),n__zeros()),0#()):39 -->_1 zeros#() -> c_40():40 31:W:cons#(X1,X2) -> c_25() 32:W:isNat#(n__0()) -> c_26() 33:W:isNatIList#(n__zeros()) -> c_31() 34:W:isNatList#(n__nil()) -> c_33() 35:W:length#(X) -> c_35() 36:W:nil#() -> c_36() 37:W:s#(X) -> c_37() 38:W:take#(X1,X2) -> c_38() 39:W:zeros#() -> c_39(cons#(0(),n__zeros()),0#()) -->_1 cons#(X1,X2) -> c_25():31 -->_2 0#() -> c_1():13 40:W:zeros#() -> c_40() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 21: U81#(tt()) -> c_13(nil#()) 22: U93#(tt(),IL,M,N) -> c_16(cons#(activate(N),n__take(activate(M),activate(IL))) ,activate#(N) ,activate#(M) ,activate#(IL)) 20: U72#(tt(),L) -> c_12(s#(length(activate(L))),length#(activate(L)),activate#(L)) 17: U42#(tt()) -> c_6() 14: U11#(tt()) -> c_2() 19: U62#(tt()) -> c_10() 33: isNatIList#(n__zeros()) -> c_31() 16: U31#(tt()) -> c_4() 18: U52#(tt()) -> c_8() 34: isNatList#(n__nil()) -> c_33() 15: U21#(tt()) -> c_3() 23: activate#(X) -> c_17() 32: isNat#(n__0()) -> c_26() 24: activate#(n__0()) -> c_18(0#()) 25: activate#(n__cons(X1,X2)) -> c_19(cons#(X1,X2)) 26: activate#(n__length(X)) -> c_20(length#(X)) 35: length#(X) -> c_35() 27: activate#(n__nil()) -> c_21(nil#()) 36: nil#() -> c_36() 28: activate#(n__s(X)) -> c_22(s#(X)) 37: s#(X) -> c_37() 29: activate#(n__take(X1,X2)) -> c_23(take#(X1,X2)) 38: take#(X1,X2) -> c_38() 30: activate#(n__zeros()) -> c_24(zeros#()) 40: zeros#() -> c_40() 39: zeros#() -> c_39(cons#(0(),n__zeros()),0#()) 13: 0#() -> c_1() 31: cons#(X1,X2) -> c_25() * Step 9: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/3,c_10/0,c_11/4,c_12/3 ,c_13/1,c_14/6,c_15/6,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/3 ,c_28/3,c_29/3,c_30/4,c_31/0,c_32/4,c_33/0,c_34/4,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) -->_2 isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):10 -->_2 isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)):9 2:S:U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) -->_2 isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):12 -->_2 isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):11 3:S:U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)) -->_2 isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):10 -->_2 isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)):9 4:S:U71#(tt(),L,N) -> c_11(U72#(isNat(activate(N)),activate(L)) ,isNat#(activate(N)) ,activate#(N) ,activate#(L)) -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 5:S:U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M)) ,activate#(M) ,activate#(IL) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_1 U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)):6 6:S:U92#(tt(),IL,M,N) -> c_15(U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(IL) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 7:S:isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) -->_2 isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):12 -->_2 isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):11 8:S:isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 9:S:isNatIList#(V) -> c_29(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) -->_2 isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):12 -->_2 isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):11 10:S:isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_1 U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)):1 11:S:isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_1 U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)):2 12:S:isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_28(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__length(V1)) -> c_27(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):7 -->_1 U61#(tt(),V2) -> c_9(U62#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) * Step 10: Decompose MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1 ,U41#/2,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2 ,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/1,c_10/0,c_11/1,c_12/3,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1 ,c_24/1,c_25/0,c_26/0,c_27/1,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71# ,U72#,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} Problem (S) - Strict DPs: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) - Weak DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1 ,U41#/2,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2 ,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0 ,c_9/1,c_10/0,c_11/1,c_12/3,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1 ,c_24/1,c_25/0,c_26/0,c_27/1,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0 ,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71# ,U72#,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} ** Step 10.a:1: PredecessorEstimationCP MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) 7: isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) Consider the set of all dependency pairs 1: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) 2: U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) 3: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) 4: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) 5: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M))) 6: U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) 7: isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) 8: isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) 9: isNatIList#(V) -> c_29(isNatList#(activate(V))) 10: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 11: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 12: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {3,7} These cover all (indirect) predecessors of dependency pairs {3,4,5,6,7} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 10.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1,2}, uargs(c_15) = {1}, uargs(c_27) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_30) = {1,2}, uargs(c_32) = {1,2}, uargs(c_34) = {1,2} Following symbols are considered usable: {0,activate,cons,length,nil,s,take,zeros,0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72#,U81# ,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take#,zeros#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [6] x1 + [0] p(U21) = [1] x1 + [1] p(U31) = [1] x1 + [0] p(U41) = [2] x2 + [2] p(U42) = [2] x1 + [0] p(U51) = [1] x1 + [2] x2 + [2] p(U52) = [2] p(U61) = [1] p(U62) = [1] x1 + [4] p(U71) = [1] x1 + [1] x3 + [0] p(U72) = [2] x1 + [1] x2 + [1] p(U81) = [2] x1 + [1] p(U91) = [4] x1 + [1] x3 + [2] x4 + [2] p(U92) = [4] p(U93) = [1] x1 + [1] x2 + [0] p(activate) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(isNat) = [0] p(isNatIList) = [2] x1 + [0] p(isNatList) = [1] x1 + [2] p(length) = [1] x1 + [2] p(n__0) = [0] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__length) = [1] x1 + [2] p(n__nil) = [0] p(n__s) = [1] x1 + [0] p(n__take) = [1] x1 + [1] x2 + [2] p(n__zeros) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(take) = [1] x1 + [1] x2 + [2] p(tt) = [0] p(zeros) = [0] p(0#) = [2] p(U11#) = [0] p(U21#) = [2] x1 + [0] p(U31#) = [1] p(U41#) = [4] x2 + [0] p(U42#) = [1] p(U51#) = [4] x2 + [0] p(U52#) = [1] p(U61#) = [4] x2 + [5] p(U62#) = [1] p(U71#) = [4] x3 + [7] p(U72#) = [4] x2 + [0] p(U81#) = [0] p(U91#) = [1] x1 + [2] x2 + [5] x3 + [5] x4 + [7] p(U92#) = [4] x4 + [5] p(U93#) = [1] x2 + [1] x4 + [0] p(activate#) = [2] x1 + [0] p(cons#) = [1] x2 + [0] p(isNat#) = [4] x1 + [0] p(isNatIList#) = [4] x1 + [0] p(isNatList#) = [4] x1 + [0] p(length#) = [1] x1 + [4] p(nil#) = [2] p(s#) = [0] p(take#) = [0] p(zeros#) = [4] p(c_1) = [4] p(c_2) = [2] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [6] p(c_12) = [2] x2 + [1] p(c_13) = [1] x1 + [2] p(c_14) = [1] x1 + [1] x2 + [2] p(c_15) = [1] x1 + [5] p(c_16) = [4] x4 + [1] p(c_17) = [1] p(c_18) = [1] x1 + [0] p(c_19) = [4] x1 + [0] p(c_20) = [0] p(c_21) = [1] p(c_22) = [1] x1 + [0] p(c_23) = [1] p(c_24) = [1] x1 + [0] p(c_25) = [4] p(c_26) = [2] p(c_27) = [1] x1 + [7] p(c_28) = [1] x1 + [0] p(c_29) = [1] x1 + [0] p(c_30) = [1] x1 + [1] x2 + [0] p(c_31) = [1] p(c_32) = [1] x1 + [1] x2 + [0] p(c_33) = [1] p(c_34) = [1] x1 + [1] x2 + [3] p(c_35) = [0] p(c_36) = [0] p(c_37) = [1] p(c_38) = [1] p(c_39) = [1] x2 + [1] p(c_40) = [1] Following rules are strictly oriented: U61#(tt(),V2) = [4] V2 + [5] > [4] V2 + [0] = c_9(isNatIList#(activate(V2))) isNat#(n__length(V1)) = [4] V1 + [8] > [4] V1 + [7] = c_27(isNatList#(activate(V1))) Following rules are (at-least) weakly oriented: U41#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = c_5(isNatIList#(activate(V2))) U51#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = c_7(isNatList#(activate(V2))) U71#(tt(),L,N) = [4] N + [7] >= [4] N + [6] = c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) = [2] IL + [5] M + [5] N + [7] >= [4] M + [4] N + [7] = c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) = [4] N + [5] >= [4] N + [5] = c_15(isNat#(activate(N))) isNat#(n__s(V1)) = [4] V1 + [0] >= [4] V1 + [0] = c_28(isNat#(activate(V1))) isNatIList#(V) = [4] V + [0] >= [4] V + [0] = c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [4] V2 + [8] = c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 0() = [0] >= [0] = n__0() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__cons(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(X1,X2) activate(n__length(X)) = [1] X + [2] >= [1] X + [2] = length(X) activate(n__nil()) = [0] >= [0] = nil() activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(X) activate(n__take(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = take(X1,X2) activate(n__zeros()) = [0] >= [0] = zeros() cons(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__cons(X1,X2) length(X) = [1] X + [2] >= [1] X + [2] = n__length(X) nil() = [0] >= [0] = n__nil() s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) take(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__take(X1,X2) zeros() = [0] >= [0] = cons(0(),n__zeros()) zeros() = [0] >= [0] = n__zeros() *** Step 10.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 10.a:1.b:1: PredecessorEstimationCP MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 4: isNatIList#(V) -> c_29(isNatList#(activate(V))) Consider the set of all dependency pairs 1: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) 2: U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) 3: isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) 4: isNatIList#(V) -> c_29(isNatList#(activate(V))) 5: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 7: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 8: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) 9: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) 10: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M))) 11: U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) 12: isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {4} These cover all (indirect) predecessors of dependency pairs {4,9,10,11} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 10.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1,2}, uargs(c_15) = {1}, uargs(c_27) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_30) = {1,2}, uargs(c_32) = {1,2}, uargs(c_34) = {1,2} Following symbols are considered usable: {0,U11,U21,activate,cons,isNat,length,nil,s,take,zeros,0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62# ,U71#,U72#,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take#,zeros#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [3] p(U21) = [1] x1 + [0] p(U31) = [2] p(U41) = [2] x1 + [2] x2 + [0] p(U42) = [0] p(U51) = [2] x2 + [0] p(U52) = [3] x1 + [3] p(U61) = [2] x2 + [0] p(U62) = [0] p(U71) = [4] x2 + [1] p(U72) = [4] x1 + [0] p(U81) = [1] x1 + [0] p(U91) = [1] x1 + [1] x2 + [1] x3 + [1] p(U92) = [1] x2 + [2] x3 + [4] p(U93) = [2] x1 + [0] p(activate) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(isNat) = [4] p(isNatIList) = [7] p(isNatList) = [2] x1 + [1] p(length) = [1] x1 + [0] p(n__0) = [0] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__length) = [1] x1 + [0] p(n__nil) = [0] p(n__s) = [1] x1 + [0] p(n__take) = [1] x1 + [1] x2 + [3] p(n__zeros) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(take) = [1] x1 + [1] x2 + [3] p(tt) = [2] p(zeros) = [0] p(0#) = [1] p(U11#) = [4] p(U21#) = [0] p(U31#) = [0] p(U41#) = [4] x2 + [1] p(U42#) = [1] x1 + [0] p(U51#) = [4] x2 + [0] p(U52#) = [2] p(U61#) = [2] x1 + [4] x2 + [4] p(U62#) = [1] p(U71#) = [5] x3 + [1] p(U72#) = [0] p(U81#) = [1] p(U91#) = [4] x1 + [4] x2 + [6] x3 + [5] x4 + [3] p(U92#) = [1] x2 + [2] x3 + [5] x4 + [2] p(U93#) = [2] x1 + [4] x3 + [1] p(activate#) = [0] p(cons#) = [4] x1 + [4] p(isNat#) = [4] x1 + [0] p(isNatIList#) = [4] x1 + [1] p(isNatList#) = [4] x1 + [0] p(length#) = [1] x1 + [0] p(nil#) = [0] p(s#) = [4] p(take#) = [2] x1 + [4] x2 + [1] p(zeros#) = [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] x1 + [7] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [1] x3 + [0] p(c_13) = [2] x1 + [0] p(c_14) = [1] x1 + [1] x2 + [6] p(c_15) = [1] x1 + [2] p(c_16) = [1] x1 + [1] x4 + [1] p(c_17) = [1] p(c_18) = [4] x1 + [1] p(c_19) = [1] x1 + [0] p(c_20) = [4] x1 + [0] p(c_21) = [4] x1 + [2] p(c_22) = [1] x1 + [0] p(c_23) = [4] x1 + [0] p(c_24) = [4] p(c_25) = [1] p(c_26) = [0] p(c_27) = [1] x1 + [0] p(c_28) = [1] x1 + [0] p(c_29) = [1] x1 + [0] p(c_30) = [1] x1 + [1] x2 + [0] p(c_31) = [1] p(c_32) = [1] x1 + [1] x2 + [0] p(c_33) = [0] p(c_34) = [1] x1 + [1] x2 + [0] p(c_35) = [1] p(c_36) = [0] p(c_37) = [1] p(c_38) = [1] p(c_39) = [1] x2 + [1] p(c_40) = [0] Following rules are strictly oriented: isNatIList#(V) = [4] V + [1] > [4] V + [0] = c_29(isNatList#(activate(V))) Following rules are (at-least) weakly oriented: U41#(tt(),V2) = [4] V2 + [1] >= [4] V2 + [1] = c_5(isNatIList#(activate(V2))) U51#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = c_7(isNatList#(activate(V2))) U61#(tt(),V2) = [4] V2 + [8] >= [4] V2 + [8] = c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) = [5] N + [1] >= [4] N + [0] = c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) = [4] IL + [6] M + [5] N + [11] >= [1] IL + [6] M + [5] N + [8] = c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) = [1] IL + [2] M + [5] N + [2] >= [4] N + [2] = c_15(isNat#(activate(N))) isNat#(n__length(V1)) = [4] V1 + [0] >= [4] V1 + [0] = c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) = [4] V1 + [0] >= [4] V1 + [0] = c_28(isNat#(activate(V1))) isNatIList#(n__cons(V1,V2)) = [4] V1 + [4] V2 + [1] >= [4] V1 + [4] V2 + [1] = c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) = [4] V1 + [4] V2 + [12] >= [4] V1 + [4] V2 + [12] = c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 0() = [0] >= [0] = n__0() U11(tt()) = [3] >= [2] = tt() U21(tt()) = [2] >= [2] = tt() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__cons(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(X1,X2) activate(n__length(X)) = [1] X + [0] >= [1] X + [0] = length(X) activate(n__nil()) = [0] >= [0] = nil() activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(X) activate(n__take(X1,X2)) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = take(X1,X2) activate(n__zeros()) = [0] >= [0] = zeros() cons(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__cons(X1,X2) isNat(n__0()) = [4] >= [2] = tt() isNat(n__length(V1)) = [4] >= [3] = U11(isNatList(activate(V1))) isNat(n__s(V1)) = [4] >= [4] = U21(isNat(activate(V1))) length(X) = [1] X + [0] >= [1] X + [0] = n__length(X) nil() = [0] >= [0] = n__nil() s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) take(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = n__take(X1,X2) zeros() = [0] >= [0] = cons(0(),n__zeros()) zeros() = [0] >= [0] = n__zeros() **** Step 10.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 10.a:1.b:1.b:1: PredecessorEstimationCP MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) 6: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Consider the set of all dependency pairs 1: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) 2: U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) 3: isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) 4: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 5: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 7: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) 8: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) 9: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M))) 10: U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) 11: isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) 12: isNatIList#(V) -> c_29(isNatList#(activate(V))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {3,6} These cover all (indirect) predecessors of dependency pairs {3,6,7,8,9,10} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 10.a:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1,2}, uargs(c_15) = {1}, uargs(c_27) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_30) = {1,2}, uargs(c_32) = {1,2}, uargs(c_34) = {1,2} Following symbols are considered usable: {0,activate,cons,length,nil,s,take,zeros,0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72#,U81# ,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take#,zeros#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [4] x1 + [5] p(U21) = [2] x1 + [0] p(U31) = [2] x1 + [0] p(U41) = [4] x1 + [0] p(U42) = [2] p(U51) = [0] p(U52) = [1] p(U61) = [2] p(U62) = [1] x1 + [1] p(U71) = [1] x1 + [1] x3 + [0] p(U72) = [2] x1 + [0] p(U81) = [1] p(U91) = [1] x1 + [0] p(U92) = [1] x3 + [2] p(U93) = [1] x3 + [1] p(activate) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(isNat) = [0] p(isNatIList) = [5] x1 + [0] p(isNatList) = [1] p(length) = [1] x1 + [0] p(n__0) = [0] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__length) = [1] x1 + [0] p(n__nil) = [4] p(n__s) = [1] x1 + [2] p(n__take) = [1] x1 + [1] x2 + [4] p(n__zeros) = [0] p(nil) = [4] p(s) = [1] x1 + [2] p(take) = [1] x1 + [1] x2 + [4] p(tt) = [0] p(zeros) = [0] p(0#) = [0] p(U11#) = [1] p(U21#) = [1] x1 + [2] p(U31#) = [1] p(U41#) = [2] x2 + [0] p(U42#) = [0] p(U51#) = [2] x2 + [0] p(U52#) = [0] p(U61#) = [2] x2 + [0] p(U62#) = [1] x1 + [0] p(U71#) = [1] x1 + [2] x3 + [0] p(U72#) = [2] x1 + [4] x2 + [0] p(U81#) = [4] x1 + [0] p(U91#) = [4] x2 + [5] x3 + [2] x4 + [0] p(U92#) = [1] x3 + [2] x4 + [0] p(U93#) = [1] x1 + [1] x3 + [0] p(activate#) = [1] x1 + [0] p(cons#) = [0] p(isNat#) = [2] x1 + [0] p(isNatIList#) = [2] x1 + [0] p(isNatList#) = [2] x1 + [0] p(length#) = [1] x1 + [2] p(nil#) = [0] p(s#) = [4] x1 + [2] p(take#) = [1] p(zeros#) = [2] p(c_1) = [1] p(c_2) = [2] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [2] x1 + [1] x2 + [0] p(c_13) = [1] p(c_14) = [1] x1 + [2] x2 + [0] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [1] x2 + [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [2] p(c_20) = [2] x1 + [1] p(c_21) = [1] x1 + [2] p(c_22) = [1] p(c_23) = [4] x1 + [4] p(c_24) = [4] p(c_25) = [4] p(c_26) = [0] p(c_27) = [1] x1 + [0] p(c_28) = [1] x1 + [2] p(c_29) = [1] x1 + [0] p(c_30) = [1] x1 + [1] x2 + [0] p(c_31) = [0] p(c_32) = [1] x1 + [1] x2 + [0] p(c_33) = [2] p(c_34) = [1] x1 + [1] x2 + [7] p(c_35) = [0] p(c_36) = [1] p(c_37) = [0] p(c_38) = [1] p(c_39) = [1] x1 + [2] p(c_40) = [0] Following rules are strictly oriented: isNat#(n__s(V1)) = [2] V1 + [4] > [2] V1 + [2] = c_28(isNat#(activate(V1))) isNatList#(n__take(V1,V2)) = [2] V1 + [2] V2 + [8] > [2] V1 + [2] V2 + [7] = c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U41#(tt(),V2) = [2] V2 + [0] >= [2] V2 + [0] = c_5(isNatIList#(activate(V2))) U51#(tt(),V2) = [2] V2 + [0] >= [2] V2 + [0] = c_7(isNatList#(activate(V2))) U61#(tt(),V2) = [2] V2 + [0] >= [2] V2 + [0] = c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) = [2] N + [0] >= [2] N + [0] = c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) = [4] IL + [5] M + [2] N + [0] >= [5] M + [2] N + [0] = c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) = [1] M + [2] N + [0] >= [2] N + [0] = c_15(isNat#(activate(N))) isNat#(n__length(V1)) = [2] V1 + [0] >= [2] V1 + [0] = c_27(isNatList#(activate(V1))) isNatIList#(V) = [2] V + [0] >= [2] V + [0] = c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) = [2] V1 + [2] V2 + [0] >= [2] V1 + [2] V2 + [0] = c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) = [2] V1 + [2] V2 + [0] >= [2] V1 + [2] V2 + [0] = c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 0() = [0] >= [0] = n__0() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__cons(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(X1,X2) activate(n__length(X)) = [1] X + [0] >= [1] X + [0] = length(X) activate(n__nil()) = [4] >= [4] = nil() activate(n__s(X)) = [1] X + [2] >= [1] X + [2] = s(X) activate(n__take(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = take(X1,X2) activate(n__zeros()) = [0] >= [0] = zeros() cons(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__cons(X1,X2) length(X) = [1] X + [0] >= [1] X + [0] = n__length(X) nil() = [4] >= [4] = n__nil() s(X) = [1] X + [2] >= [1] X + [2] = n__s(X) take(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = n__take(X1,X2) zeros() = [0] >= [0] = cons(0(),n__zeros()) zeros() = [0] >= [0] = n__zeros() ***** Step 10.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 10.a:1.b:1.b:1.b:1: Failure MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 10.b:1: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) - Weak DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3} by application of Pre({1,3}) = {2}. Here rules are labelled as follows: 1: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) 2: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M))) 3: U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) 4: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) 5: U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) 6: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) 7: isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) 8: isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) 9: isNatIList#(V) -> c_29(isNatList#(activate(V))) 10: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 11: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 12: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) ** Step 10.b:2: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) - Weak DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M))) 2: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) 3: U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) 4: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) 5: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) 6: U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) 7: isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) 8: isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) 9: isNatIList#(V) -> c_29(isNatList#(activate(V))) 10: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 11: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 12: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) ** Step 10.b:3: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) U71#(tt(),L,N) -> c_11(isNat#(activate(N))) U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)),isNat#(activate(M))) U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) isNatIList#(V) -> c_29(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) -->_1 isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):10 -->_1 isNatIList#(V) -> c_29(isNatList#(activate(V))):9 2:W:U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) -->_1 isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):12 -->_1 isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):11 3:W:U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) -->_1 isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):10 -->_1 isNatIList#(V) -> c_29(isNatList#(activate(V))):9 4:W:U71#(tt(),L,N) -> c_11(isNat#(activate(N))) -->_1 isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))):8 -->_1 isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))):7 5:W:U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M))) -->_2 isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))):8 -->_2 isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))):7 -->_1 U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))):6 6:W:U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) -->_1 isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))):8 -->_1 isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))):7 7:W:isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) -->_1 isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):12 -->_1 isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):11 8:W:isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) -->_1 isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))):8 -->_1 isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))):7 9:W:isNatIList#(V) -> c_29(isNatList#(activate(V))) -->_1 isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):12 -->_1 isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):11 10:W:isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))):8 -->_2 isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))):7 -->_1 U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))):1 11:W:isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))):8 -->_2 isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))):7 -->_1 U51#(tt(),V2) -> c_7(isNatList#(activate(V2))):2 12:W:isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))):8 -->_2 isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))):7 -->_1 U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: U91#(tt(),IL,M,N) -> c_14(U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) ,isNat#(activate(M))) 6: U92#(tt(),IL,M,N) -> c_15(isNat#(activate(N))) 4: U71#(tt(),L,N) -> c_11(isNat#(activate(N))) 1: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) 10: isNatIList#(n__cons(V1,V2)) -> c_30(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 3: U61#(tt(),V2) -> c_9(isNatIList#(activate(V2))) 12: isNatList#(n__take(V1,V2)) -> c_34(U61#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 9: isNatIList#(V) -> c_29(isNatList#(activate(V))) 7: isNat#(n__length(V1)) -> c_27(isNatList#(activate(V1))) 11: isNatList#(n__cons(V1,V2)) -> c_32(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 2: U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) 8: isNat#(n__s(V1)) -> c_28(isNat#(activate(V1))) ** Step 10.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U31(tt()) -> tt() U41(tt(),V2) -> U42(isNatIList(activate(V2))) U42(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(tt()) -> tt() U61(tt(),V2) -> U62(isNatIList(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__length(X)) -> length(X) activate(n__nil()) -> nil() activate(n__s(X)) -> s(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() cons(X1,X2) -> n__cons(X1,X2) isNat(n__0()) -> tt() isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2)) isNatIList(n__zeros()) -> tt() isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() isNatList(n__take(V1,V2)) -> U61(isNat(activate(V1)),activate(V2)) length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/2,U62/1,U71/3,U72/2,U81/1,U91/4,U92/4,U93/4,activate/1 ,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,take/2,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2 ,U42#/1,U51#/2,U52#/1,U61#/2,U62#/1,U71#/3,U72#/2,U81#/1,U91#/4,U92#/4,U93#/4,activate#/1,cons#/2,isNat#/1 ,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,take#/2,zeros#/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0 ,n__s/1,n__take/2,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/3 ,c_13/1,c_14/2,c_15/1,c_16/4,c_17/0,c_18/1,c_19/1,c_20/1,c_21/1,c_22/1,c_23/1,c_24/1,c_25/0,c_26/0,c_27/1 ,c_28/1,c_29/1,c_30/2,c_31/0,c_32/2,c_33/0,c_34/2,c_35/0,c_36/0,c_37/0,c_38/0,c_39/2,c_40/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,U71#,U72# ,U81#,U91#,U92#,U93#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,take# ,zeros#} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__take,n__zeros,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE