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(),L,N) -> U62(isNat(activate(N)),activate(L)) U62(tt(),L) -> s(length(activate(L))) 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__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() length(X) -> n__length(X) length(cons(N,L)) -> U61(isNatList(activate(L)),activate(L),N) length(nil()) -> 0() nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,U42,U51,U52,U61,U62,activate,cons,isNat ,isNatIList,isNatList,length,nil,s,zeros} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__zeros,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. length(cons(N,L)) -> U61(isNatList(activate(L)),activate(L),N) length(nil()) -> 0() 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(),L,N) -> U62(isNat(activate(N)),activate(L)) U62(tt(),L) -> s(length(activate(L))) 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,U42,U51,U52,U61,U62,activate,cons,isNat ,isNatIList,isNatList,length,nil,s,zeros} and constructors {n__0,n__cons,n__length,n__nil,n__s,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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) activate#(X) -> c_11() activate#(n__0()) -> c_12(0#()) activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) activate#(n__length(X)) -> c_14(length#(X)) activate#(n__nil()) -> c_15(nil#()) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__zeros()) -> c_17(zeros#()) cons#(X1,X2) -> c_18() isNat#(n__0()) -> c_19() isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatIList#(n__zeros()) -> c_24() isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__nil()) -> c_26() length#(X) -> c_27() nil#() -> c_28() s#(X) -> c_29() zeros#() -> c_30(cons#(0(),n__zeros()),0#()) zeros#() -> c_31() 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) activate#(X) -> c_11() activate#(n__0()) -> c_12(0#()) activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) activate#(n__length(X)) -> c_14(length#(X)) activate#(n__nil()) -> c_15(nil#()) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__zeros()) -> c_17(zeros#()) cons#(X1,X2) -> c_18() isNat#(n__0()) -> c_19() isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatIList#(n__zeros()) -> c_24() isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__nil()) -> c_26() length#(X) -> c_27() nil#() -> c_28() s#(X) -> c_29() zeros#() -> c_30(cons#(0(),n__zeros()),0#()) zeros#() -> c_31() - 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(),L,N) -> U62(isNat(activate(N)),activate(L)) U62(tt(),L) -> s(length(activate(L))) 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_9/4,c_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,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() 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) activate#(X) -> c_11() activate#(n__0()) -> c_12(0#()) activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) activate#(n__length(X)) -> c_14(length#(X)) activate#(n__nil()) -> c_15(nil#()) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__zeros()) -> c_17(zeros#()) cons#(X1,X2) -> c_18() isNat#(n__0()) -> c_19() isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatIList#(n__zeros()) -> c_24() isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__nil()) -> c_26() length#(X) -> c_27() nil#() -> c_28() s#(X) -> c_29() zeros#() -> c_30(cons#(0(),n__zeros()),0#()) zeros#() -> c_31() * 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) activate#(X) -> c_11() activate#(n__0()) -> c_12(0#()) activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) activate#(n__length(X)) -> c_14(length#(X)) activate#(n__nil()) -> c_15(nil#()) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__zeros()) -> c_17(zeros#()) cons#(X1,X2) -> c_18() isNat#(n__0()) -> c_19() isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatIList#(n__zeros()) -> c_24() isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__nil()) -> c_26() length#(X) -> c_27() nil#() -> c_28() s#(X) -> c_29() zeros#() -> c_30(cons#(0(),n__zeros()),0#()) zeros#() -> c_31() - 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() 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_9/4,c_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,6,8,11,18,19,24,26,27,28,29,31} by application of Pre({1,2,3,4,6,8,11,18,19,24,26,27,28,29,31}) = {5,7,9,10,12,13,14,15,16,17,20,21,22,23,25,30}. 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) 10: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) 11: activate#(X) -> c_11() 12: activate#(n__0()) -> c_12(0#()) 13: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) 14: activate#(n__length(X)) -> c_14(length#(X)) 15: activate#(n__nil()) -> c_15(nil#()) 16: activate#(n__s(X)) -> c_16(s#(X)) 17: activate#(n__zeros()) -> c_17(zeros#()) 18: cons#(X1,X2) -> c_18() 19: isNat#(n__0()) -> c_19() 20: isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) 21: isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 22: isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) 23: isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 24: isNatIList#(n__zeros()) -> c_24() 25: isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 26: isNatList#(n__nil()) -> c_26() 27: length#(X) -> c_27() 28: nil#() -> c_28() 29: s#(X) -> c_29() 30: zeros#() -> c_30(cons#(0(),n__zeros()),0#()) 31: zeros#() -> c_31() * 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) activate#(n__0()) -> c_12(0#()) activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) activate#(n__length(X)) -> c_14(length#(X)) activate#(n__nil()) -> c_15(nil#()) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__zeros()) -> c_17(zeros#()) isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) zeros#() -> c_30(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() activate#(X) -> c_11() cons#(X1,X2) -> c_18() isNat#(n__0()) -> c_19() isNatIList#(n__zeros()) -> c_24() isNatList#(n__nil()) -> c_26() length#(X) -> c_27() nil#() -> c_28() s#(X) -> c_29() zeros#() -> c_31() - 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() 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_9/4,c_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5,6,7,8,9,16} by application of Pre({5,6,7,8,9,16}) = {1,2,3,4,10,11,12,13,14,15}. 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) 4: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) 5: activate#(n__0()) -> c_12(0#()) 6: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) 7: activate#(n__length(X)) -> c_14(length#(X)) 8: activate#(n__nil()) -> c_15(nil#()) 9: activate#(n__s(X)) -> c_16(s#(X)) 10: activate#(n__zeros()) -> c_17(zeros#()) 11: isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) 12: isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 13: isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) 14: isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 15: isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 16: zeros#() -> c_30(cons#(0(),n__zeros()),0#()) 17: 0#() -> c_1() 18: U11#(tt()) -> c_2() 19: U21#(tt()) -> c_3() 20: U31#(tt()) -> c_4() 21: U42#(tt()) -> c_6() 22: U52#(tt()) -> c_8() 23: activate#(X) -> c_11() 24: cons#(X1,X2) -> c_18() 25: isNat#(n__0()) -> c_19() 26: isNatIList#(n__zeros()) -> c_24() 27: isNatList#(n__nil()) -> c_26() 28: length#(X) -> c_27() 29: nil#() -> c_28() 30: s#(X) -> c_29() 31: zeros#() -> c_31() * 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) activate#(n__zeros()) -> c_17(zeros#()) isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_25(U51#(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() activate#(X) -> c_11() activate#(n__0()) -> c_12(0#()) activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) activate#(n__length(X)) -> c_14(length#(X)) activate#(n__nil()) -> c_15(nil#()) activate#(n__s(X)) -> c_16(s#(X)) cons#(X1,X2) -> c_18() isNat#(n__0()) -> c_19() isNatIList#(n__zeros()) -> c_24() isNatList#(n__nil()) -> c_26() length#(X) -> c_27() nil#() -> c_28() s#(X) -> c_29() zeros#() -> c_30(cons#(0(),n__zeros()),0#()) zeros#() -> c_31() - 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() 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_9/4,c_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5} by application of Pre({5}) = {1,2,3,4,6,7,8,9,10}. 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) 4: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) 5: activate#(n__zeros()) -> c_17(zeros#()) 6: isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) 7: isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 8: isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) 9: isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 10: isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 11: 0#() -> c_1() 12: U11#(tt()) -> c_2() 13: U21#(tt()) -> c_3() 14: U31#(tt()) -> c_4() 15: U42#(tt()) -> c_6() 16: U52#(tt()) -> c_8() 17: activate#(X) -> c_11() 18: activate#(n__0()) -> c_12(0#()) 19: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) 20: activate#(n__length(X)) -> c_14(length#(X)) 21: activate#(n__nil()) -> c_15(nil#()) 22: activate#(n__s(X)) -> c_16(s#(X)) 23: cons#(X1,X2) -> c_18() 24: isNat#(n__0()) -> c_19() 25: isNatIList#(n__zeros()) -> c_24() 26: isNatList#(n__nil()) -> c_26() 27: length#(X) -> c_27() 28: nil#() -> c_28() 29: s#(X) -> c_29() 30: zeros#() -> c_30(cons#(0(),n__zeros()),0#()) 31: zeros#() -> c_31() * 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_25(U51#(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() activate#(X) -> c_11() activate#(n__0()) -> c_12(0#()) activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) activate#(n__length(X)) -> c_14(length#(X)) activate#(n__nil()) -> c_15(nil#()) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__zeros()) -> c_17(zeros#()) cons#(X1,X2) -> c_18() isNat#(n__0()) -> c_19() isNatIList#(n__zeros()) -> c_24() isNatList#(n__nil()) -> c_26() length#(X) -> c_27() nil#() -> c_28() s#(X) -> c_29() zeros#() -> c_30(cons#(0(),n__zeros()),0#()) zeros#() -> c_31() - 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() 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_9/4,c_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,n__zeros,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4} by application of Pre({4}) = {3}. 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) 4: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) 5: isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) 6: isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 7: isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) 8: isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 9: isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 10: 0#() -> c_1() 11: U11#(tt()) -> c_2() 12: U21#(tt()) -> c_3() 13: U31#(tt()) -> c_4() 14: U42#(tt()) -> c_6() 15: U52#(tt()) -> c_8() 16: activate#(X) -> c_11() 17: activate#(n__0()) -> c_12(0#()) 18: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) 19: activate#(n__length(X)) -> c_14(length#(X)) 20: activate#(n__nil()) -> c_15(nil#()) 21: activate#(n__s(X)) -> c_16(s#(X)) 22: activate#(n__zeros()) -> c_17(zeros#()) 23: cons#(X1,X2) -> c_18() 24: isNat#(n__0()) -> c_19() 25: isNatIList#(n__zeros()) -> c_24() 26: isNatList#(n__nil()) -> c_26() 27: length#(X) -> c_27() 28: nil#() -> c_28() 29: s#(X) -> c_29() 30: zeros#() -> c_30(cons#(0(),n__zeros()),0#()) 31: zeros#() -> c_31() * 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_25(U51#(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(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) activate#(X) -> c_11() activate#(n__0()) -> c_12(0#()) activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) activate#(n__length(X)) -> c_14(length#(X)) activate#(n__nil()) -> c_15(nil#()) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__zeros()) -> c_17(zeros#()) cons#(X1,X2) -> c_18() isNat#(n__0()) -> c_19() isNatIList#(n__zeros()) -> c_24() isNatList#(n__nil()) -> c_26() length#(X) -> c_27() nil#() -> c_28() s#(X) -> c_29() zeros#() -> c_30(cons#(0(),n__zeros()),0#()) zeros#() -> c_31() - 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() 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_9/4,c_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,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_17(zeros#()):22 -->_3 activate#(n__s(X)) -> c_16(s#(X)):21 -->_3 activate#(n__nil()) -> c_15(nil#()):20 -->_3 activate#(n__length(X)) -> c_14(length#(X)):19 -->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_3 activate#(n__0()) -> c_12(0#()):17 -->_2 isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)):6 -->_2 isNatIList#(n__zeros()) -> c_24():25 -->_3 activate#(X) -> c_11():16 -->_1 U42#(tt()) -> c_6():13 2:S:U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) -->_3 activate#(n__zeros()) -> c_17(zeros#()):22 -->_3 activate#(n__s(X)) -> c_16(s#(X)):21 -->_3 activate#(n__nil()) -> c_15(nil#()):20 -->_3 activate#(n__length(X)) -> c_14(length#(X)):19 -->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_3 activate#(n__0()) -> c_12(0#()):17 -->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):8 -->_2 isNatList#(n__nil()) -> c_26():26 -->_3 activate#(X) -> c_11():16 -->_1 U52#(tt()) -> c_8():14 3:S:U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)) ,isNat#(activate(N)) ,activate#(N) ,activate#(L)) -->_4 activate#(n__zeros()) -> c_17(zeros#()):22 -->_3 activate#(n__zeros()) -> c_17(zeros#()):22 -->_4 activate#(n__s(X)) -> c_16(s#(X)):21 -->_3 activate#(n__s(X)) -> c_16(s#(X)):21 -->_4 activate#(n__nil()) -> c_15(nil#()):20 -->_3 activate#(n__nil()) -> c_15(nil#()):20 -->_4 activate#(n__length(X)) -> c_14(length#(X)):19 -->_3 activate#(n__length(X)) -> c_14(length#(X)):19 -->_4 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_4 activate#(n__0()) -> c_12(0#()):17 -->_3 activate#(n__0()) -> c_12(0#()):17 -->_1 U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)):15 -->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5 -->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4 -->_2 isNat#(n__0()) -> c_19():24 -->_4 activate#(X) -> c_11():16 -->_3 activate#(X) -> c_11():16 4:S:isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) -->_3 activate#(n__zeros()) -> c_17(zeros#()):22 -->_3 activate#(n__s(X)) -> c_16(s#(X)):21 -->_3 activate#(n__nil()) -> c_15(nil#()):20 -->_3 activate#(n__length(X)) -> c_14(length#(X)):19 -->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_3 activate#(n__0()) -> c_12(0#()):17 -->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):8 -->_2 isNatList#(n__nil()) -> c_26():26 -->_3 activate#(X) -> c_11():16 -->_1 U11#(tt()) -> c_2():10 5:S:isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_3 activate#(n__zeros()) -> c_17(zeros#()):22 -->_3 activate#(n__s(X)) -> c_16(s#(X)):21 -->_3 activate#(n__nil()) -> c_15(nil#()):20 -->_3 activate#(n__length(X)) -> c_14(length#(X)):19 -->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_3 activate#(n__0()) -> c_12(0#()):17 -->_2 isNat#(n__0()) -> c_19():24 -->_3 activate#(X) -> c_11():16 -->_1 U21#(tt()) -> c_3():11 -->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5 -->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4 6:S:isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) -->_3 activate#(n__zeros()) -> c_17(zeros#()):22 -->_3 activate#(n__s(X)) -> c_16(s#(X)):21 -->_3 activate#(n__nil()) -> c_15(nil#()):20 -->_3 activate#(n__length(X)) -> c_14(length#(X)):19 -->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_3 activate#(n__0()) -> c_12(0#()):17 -->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):8 -->_2 isNatList#(n__nil()) -> c_26():26 -->_3 activate#(X) -> c_11():16 -->_1 U31#(tt()) -> c_4():12 7:S:isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__zeros()) -> c_17(zeros#()):22 -->_3 activate#(n__zeros()) -> c_17(zeros#()):22 -->_4 activate#(n__s(X)) -> c_16(s#(X)):21 -->_3 activate#(n__s(X)) -> c_16(s#(X)):21 -->_4 activate#(n__nil()) -> c_15(nil#()):20 -->_3 activate#(n__nil()) -> c_15(nil#()):20 -->_4 activate#(n__length(X)) -> c_14(length#(X)):19 -->_3 activate#(n__length(X)) -> c_14(length#(X)):19 -->_4 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_4 activate#(n__0()) -> c_12(0#()):17 -->_3 activate#(n__0()) -> c_12(0#()):17 -->_2 isNat#(n__0()) -> c_19():24 -->_4 activate#(X) -> c_11():16 -->_3 activate#(X) -> c_11():16 -->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5 -->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4 -->_1 U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)):1 8:S:isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__zeros()) -> c_17(zeros#()):22 -->_3 activate#(n__zeros()) -> c_17(zeros#()):22 -->_4 activate#(n__s(X)) -> c_16(s#(X)):21 -->_3 activate#(n__s(X)) -> c_16(s#(X)):21 -->_4 activate#(n__nil()) -> c_15(nil#()):20 -->_3 activate#(n__nil()) -> c_15(nil#()):20 -->_4 activate#(n__length(X)) -> c_14(length#(X)):19 -->_3 activate#(n__length(X)) -> c_14(length#(X)):19 -->_4 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_4 activate#(n__0()) -> c_12(0#()):17 -->_3 activate#(n__0()) -> c_12(0#()):17 -->_2 isNat#(n__0()) -> c_19():24 -->_4 activate#(X) -> c_11():16 -->_3 activate#(X) -> c_11():16 -->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5 -->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4 -->_1 U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)):2 9:W:0#() -> c_1() 10:W:U11#(tt()) -> c_2() 11:W:U21#(tt()) -> c_3() 12:W:U31#(tt()) -> c_4() 13:W:U42#(tt()) -> c_6() 14:W:U52#(tt()) -> c_8() 15:W:U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) -->_3 activate#(n__zeros()) -> c_17(zeros#()):22 -->_3 activate#(n__s(X)) -> c_16(s#(X)):21 -->_3 activate#(n__nil()) -> c_15(nil#()):20 -->_3 activate#(n__length(X)) -> c_14(length#(X)):19 -->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18 -->_3 activate#(n__0()) -> c_12(0#()):17 -->_1 s#(X) -> c_29():29 -->_2 length#(X) -> c_27():27 -->_3 activate#(X) -> c_11():16 16:W:activate#(X) -> c_11() 17:W:activate#(n__0()) -> c_12(0#()) -->_1 0#() -> c_1():9 18:W:activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) -->_1 cons#(X1,X2) -> c_18():23 19:W:activate#(n__length(X)) -> c_14(length#(X)) -->_1 length#(X) -> c_27():27 20:W:activate#(n__nil()) -> c_15(nil#()) -->_1 nil#() -> c_28():28 21:W:activate#(n__s(X)) -> c_16(s#(X)) -->_1 s#(X) -> c_29():29 22:W:activate#(n__zeros()) -> c_17(zeros#()) -->_1 zeros#() -> c_30(cons#(0(),n__zeros()),0#()):30 -->_1 zeros#() -> c_31():31 23:W:cons#(X1,X2) -> c_18() 24:W:isNat#(n__0()) -> c_19() 25:W:isNatIList#(n__zeros()) -> c_24() 26:W:isNatList#(n__nil()) -> c_26() 27:W:length#(X) -> c_27() 28:W:nil#() -> c_28() 29:W:s#(X) -> c_29() 30:W:zeros#() -> c_30(cons#(0(),n__zeros()),0#()) -->_1 cons#(X1,X2) -> c_18():23 -->_2 0#() -> c_1():9 31:W:zeros#() -> c_31() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)) 13: U42#(tt()) -> c_6() 25: isNatIList#(n__zeros()) -> c_24() 12: U31#(tt()) -> c_4() 10: U11#(tt()) -> c_2() 14: U52#(tt()) -> c_8() 26: isNatList#(n__nil()) -> c_26() 11: U21#(tt()) -> c_3() 16: activate#(X) -> c_11() 24: isNat#(n__0()) -> c_19() 17: activate#(n__0()) -> c_12(0#()) 18: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)) 19: activate#(n__length(X)) -> c_14(length#(X)) 27: length#(X) -> c_27() 20: activate#(n__nil()) -> c_15(nil#()) 28: nil#() -> c_28() 21: activate#(n__s(X)) -> c_16(s#(X)) 29: s#(X) -> c_29() 22: activate#(n__zeros()) -> c_17(zeros#()) 31: zeros#() -> c_31() 30: zeros#() -> c_30(cons#(0(),n__zeros()),0#()) 9: 0#() -> c_1() 23: cons#(X1,X2) -> c_18() * 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(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L)) isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatList#(n__cons(V1,V2)) -> c_25(U51#(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() 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_9/4,c_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,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_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)):6 2:S:U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)) -->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):8 3:S:U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)) ,isNat#(activate(N)) ,activate#(N) ,activate#(L)) -->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5 -->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4 4:S:isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)) -->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):8 5:S:isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5 -->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4 6:S:isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)) -->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):8 7:S:isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5 -->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4 -->_1 U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)):1 8:S:isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5 -->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4 -->_1 U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)):2 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(),L,N) -> c_9(isNat#(activate(N))) isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_21(isNat#(activate(V1))) isNatIList#(V) -> c_22(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) * Step 10: UsableRules MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),L,N) -> c_9(isNat#(activate(N))) isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_21(isNat#(activate(V1))) isNatIList#(V) -> c_22(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_25(U51#(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() 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__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() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/1,c_22/1,c_23/2,c_24/0,c_25/2 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,n__zeros,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(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__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))) isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),L,N) -> c_9(isNat#(activate(N))) isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_21(isNat#(activate(V1))) isNatIList#(V) -> c_22(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) * Step 11: NaturalMI MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) U61#(tt(),L,N) -> c_9(isNat#(activate(N))) isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_21(isNat#(activate(V1))) isNatIList#(V) -> c_22(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(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__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))) isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/1,c_22/1,c_23/2,c_24/0,c_25/2 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,n__zeros,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_20) = {1}, uargs(c_21) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1,2}, uargs(c_25) = {1,2} Following symbols are considered usable: {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#,cons#,isNat#,isNatIList#,isNatList#,length# ,nil#,s#,zeros#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U21) = [0] p(U31) = [0] p(U41) = [0] p(U42) = [0] p(U51) = [0] p(U52) = [0] p(U61) = [0] p(U62) = [0] p(activate) = [4] p(cons) = [0] p(isNat) = [4] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [0] p(n__0) = [0] p(n__cons) = [0] p(n__length) = [0] p(n__nil) = [0] p(n__s) = [1] p(n__zeros) = [4] p(nil) = [4] p(s) = [1] p(tt) = [0] p(zeros) = [1] p(0#) = [4] p(U11#) = [1] p(U21#) = [1] p(U31#) = [1] p(U41#) = [0] p(U42#) = [2] x1 + [1] p(U51#) = [0] p(U52#) = [1] p(U61#) = [1] x2 + [1] x3 + [6] p(U62#) = [2] x2 + [1] p(activate#) = [0] p(cons#) = [0] p(isNat#) = [0] p(isNatIList#) = [0] p(isNatList#) = [0] p(length#) = [0] p(nil#) = [0] p(s#) = [2] x1 + [4] p(zeros#) = [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [2] p(c_4) = [0] p(c_5) = [4] x1 + [0] p(c_6) = [1] p(c_7) = [4] x1 + [0] p(c_8) = [1] p(c_9) = [2] x1 + [2] p(c_10) = [4] x3 + [1] p(c_11) = [1] p(c_12) = [0] p(c_13) = [1] x1 + [2] p(c_14) = [1] p(c_15) = [0] p(c_16) = [2] x1 + [0] p(c_17) = [1] p(c_18) = [0] p(c_19) = [1] p(c_20) = [1] x1 + [0] p(c_21) = [1] x1 + [0] p(c_22) = [1] x1 + [0] p(c_23) = [4] x1 + [1] x2 + [0] p(c_24) = [0] p(c_25) = [4] x1 + [2] x2 + [0] p(c_26) = [1] p(c_27) = [1] p(c_28) = [1] p(c_29) = [0] p(c_30) = [1] x2 + [0] p(c_31) = [1] Following rules are strictly oriented: U61#(tt(),L,N) = [1] L + [1] N + [6] > [2] = c_9(isNat#(activate(N))) Following rules are (at-least) weakly oriented: U41#(tt(),V2) = [0] >= [0] = c_5(isNatIList#(activate(V2))) U51#(tt(),V2) = [0] >= [0] = c_7(isNatList#(activate(V2))) isNat#(n__length(V1)) = [0] >= [0] = c_20(isNatList#(activate(V1))) isNat#(n__s(V1)) = [0] >= [0] = c_21(isNat#(activate(V1))) isNatIList#(V) = [0] >= [0] = c_22(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) = [0] >= [0] = c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) = [0] >= [0] = c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) * Step 12: NaturalMI MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_21(isNat#(activate(V1))) isNatIList#(V) -> c_22(isNatList#(activate(V))) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),L,N) -> c_9(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(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__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))) isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/1,c_22/1,c_23/2,c_24/0,c_25/2 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,n__zeros,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_20) = {1}, uargs(c_21) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1,2}, uargs(c_25) = {1,2} Following symbols are considered usable: {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#,cons#,isNat#,isNatIList#,isNatList#,length# ,nil#,s#,zeros#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [4] x1 + [4] p(U21) = [0] p(U31) = [4] p(U41) = [1] x2 + [2] p(U42) = [1] p(U51) = [0] p(U52) = [1] x1 + [0] p(U61) = [0] p(U62) = [4] x1 + [1] p(activate) = [3] x1 + [0] p(cons) = [2] x1 + [1] x2 + [5] p(isNat) = [4] x1 + [4] p(isNatIList) = [2] x1 + [1] p(isNatList) = [1] x1 + [2] p(length) = [1] x1 + [1] p(n__0) = [0] p(n__cons) = [0] p(n__length) = [1] p(n__nil) = [2] p(n__s) = [1] p(n__zeros) = [1] p(nil) = [4] p(s) = [0] p(tt) = [0] p(zeros) = [0] p(0#) = [4] p(U11#) = [2] x1 + [0] p(U21#) = [1] x1 + [1] p(U31#) = [2] p(U41#) = [4] p(U42#) = [1] x1 + [4] p(U51#) = [0] p(U52#) = [1] x1 + [1] p(U61#) = [1] x1 + [4] x3 + [4] p(U62#) = [1] x1 + [0] p(activate#) = [0] p(cons#) = [4] x1 + [4] p(isNat#) = [0] p(isNatIList#) = [4] p(isNatList#) = [0] p(length#) = [1] p(nil#) = [0] p(s#) = [2] x1 + [2] p(zeros#) = [0] p(c_1) = [2] p(c_2) = [1] p(c_3) = [2] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] x1 + [4] p(c_10) = [1] x1 + [1] p(c_11) = [0] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [4] p(c_14) = [2] p(c_15) = [0] p(c_16) = [4] x1 + [0] p(c_17) = [4] x1 + [0] p(c_18) = [1] p(c_19) = [1] p(c_20) = [4] x1 + [0] p(c_21) = [4] x1 + [0] p(c_22) = [1] x1 + [3] p(c_23) = [1] x1 + [1] x2 + [0] p(c_24) = [0] p(c_25) = [4] x1 + [2] x2 + [0] p(c_26) = [0] p(c_27) = [2] p(c_28) = [0] p(c_29) = [0] p(c_30) = [4] x1 + [4] x2 + [0] p(c_31) = [1] Following rules are strictly oriented: isNatIList#(V) = [4] > [3] = c_22(isNatList#(activate(V))) Following rules are (at-least) weakly oriented: U41#(tt(),V2) = [4] >= [4] = c_5(isNatIList#(activate(V2))) U51#(tt(),V2) = [0] >= [0] = c_7(isNatList#(activate(V2))) U61#(tt(),L,N) = [4] N + [4] >= [4] = c_9(isNat#(activate(N))) isNat#(n__length(V1)) = [0] >= [0] = c_20(isNatList#(activate(V1))) isNat#(n__s(V1)) = [0] >= [0] = c_21(isNat#(activate(V1))) isNatIList#(n__cons(V1,V2)) = [4] >= [4] = c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) = [0] >= [0] = c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) * Step 13: Failure MAYBE + Considered Problem: - Strict DPs: U41#(tt(),V2) -> c_5(isNatIList#(activate(V2))) U51#(tt(),V2) -> c_7(isNatList#(activate(V2))) isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1))) isNat#(n__s(V1)) -> c_21(isNat#(activate(V1))) isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U61#(tt(),L,N) -> c_9(isNat#(activate(N))) isNatIList#(V) -> c_22(isNatList#(activate(V))) - Weak TRS: 0() -> n__0() U11(tt()) -> tt() U21(tt()) -> tt() U51(tt(),V2) -> U52(isNatList(activate(V2))) U52(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__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))) isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2)) isNatList(n__nil()) -> tt() length(X) -> n__length(X) nil() -> n__nil() s(X) -> n__s(X) 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,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1 ,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2 ,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2 ,n__length/1,n__nil/0,n__s/1,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_10/3 ,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/1,c_22/1,c_23/2,c_24/0,c_25/2 ,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate# ,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil ,n__s,n__zeros,tt} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE