WORST_CASE(?,O(n^3)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U13,U14,U15,U16,U21,U22,U23,U31,U32,U41,U51,U52 ,U61,U62,U63,U64,activate,isNat,isNatKind,plus,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U13,U14,U15,U16,U21,U22,U23,U31,U32,U41,U51,U52 ,U61,U62,U63,U64,activate,isNat,isNatKind,plus,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 0#() -> c_1() U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_7() U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_10() U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U32#(tt()) -> c_12() U41#(tt()) -> c_13() U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(X) -> c_20() activate#(n__0()) -> c_21(0#()) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__0()) -> c_24() isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__0()) -> c_27() isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) plus#(X1,X2) -> c_30() s#(X) -> c_31() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_7() U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_10() U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U32#(tt()) -> c_12() U41#(tt()) -> c_13() U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(X) -> c_20() activate#(n__0()) -> c_21(0#()) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__0()) -> c_24() isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__0()) -> c_27() isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) plus#(X1,X2) -> c_30() s#(X) -> c_31() - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4 ,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) 0#() -> c_1() U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_7() U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_10() U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U32#(tt()) -> c_12() U41#(tt()) -> c_13() U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(X) -> c_20() activate#(n__0()) -> c_21(0#()) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__0()) -> c_24() isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__0()) -> c_27() isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) plus#(X1,X2) -> c_30() s#(X) -> c_31() * Step 4: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_7() U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_10() U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U32#(tt()) -> c_12() U41#(tt()) -> c_13() U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(X) -> c_20() activate#(n__0()) -> c_21(0#()) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__0()) -> c_24() isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__0()) -> c_27() isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) plus#(X1,X2) -> c_30() s#(X) -> c_31() - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4 ,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,7,10,12,13,20,24,27,30,31} by application of Pre({1,7,10,12,13,20,24,27,30,31}) = {2,3,4,5,6,8,9,11,14,15,16,17,18,19,21,22,23,25,26,28,29}. Here rules are labelled as follows: 1: 0#() -> c_1() 2: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 3: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 4: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 5: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 6: U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 7: U16#(tt()) -> c_7() 8: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 9: U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 10: U23#(tt()) -> c_10() 11: U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 12: U32#(tt()) -> c_12() 13: U41#(tt()) -> c_13() 14: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 15: U52#(tt(),N) -> c_15(activate#(N)) 16: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 17: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 18: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 19: U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 20: activate#(X) -> c_20() 21: activate#(n__0()) -> c_21(0#()) 22: activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 23: activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) 24: isNat#(n__0()) -> c_24() 25: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 26: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 27: isNatKind#(n__0()) -> c_27() 28: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 29: isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) 30: plus#(X1,X2) -> c_30() 31: s#(X) -> c_31() * Step 5: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(n__0()) -> c_21(0#()) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: 0#() -> c_1() U16#(tt()) -> c_7() U23#(tt()) -> c_10() U32#(tt()) -> c_12() U41#(tt()) -> c_13() activate#(X) -> c_20() isNat#(n__0()) -> c_24() isNatKind#(n__0()) -> c_27() plus#(X1,X2) -> c_30() s#(X) -> c_31() - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4 ,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {15} by application of Pre({15}) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,19,20,21}. Here rules are labelled as follows: 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 5: U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 7: U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 8: U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 9: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 10: U52#(tt(),N) -> c_15(activate#(N)) 11: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 12: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 13: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 14: U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 15: activate#(n__0()) -> c_21(0#()) 16: activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 17: activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) 18: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 19: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 20: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 21: isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) 22: 0#() -> c_1() 23: U16#(tt()) -> c_7() 24: U23#(tt()) -> c_10() 25: U32#(tt()) -> c_12() 26: U41#(tt()) -> c_13() 27: activate#(X) -> c_20() 28: isNat#(n__0()) -> c_24() 29: isNatKind#(n__0()) -> c_27() 30: plus#(X1,X2) -> c_30() 31: s#(X) -> c_31() * Step 6: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: 0#() -> c_1() U16#(tt()) -> c_7() U23#(tt()) -> c_10() U32#(tt()) -> c_12() U41#(tt()) -> c_13() activate#(X) -> c_20() activate#(n__0()) -> c_21(0#()) isNat#(n__0()) -> c_24() isNatKind#(n__0()) -> c_27() plus#(X1,X2) -> c_30() s#(X) -> c_31() - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4 ,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):2 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):3 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):4 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):5 -->_2 isNat#(n__0()) -> c_24():28 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 5:S:U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 isNat#(n__0()) -> c_24():28 -->_3 activate#(X) -> c_20():26 -->_1 U16#(tt()) -> c_7():22 6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):7 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 7:S:U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 isNat#(n__0()) -> c_24():28 -->_3 activate#(X) -> c_20():26 -->_1 U23#(tt()) -> c_10():23 8:S:U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 isNatKind#(n__0()) -> c_27():29 -->_3 activate#(X) -> c_20():26 -->_1 U32#(tt()) -> c_12():24 9:S:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U52#(tt(),N) -> c_15(activate#(N)):10 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 10:S:U52#(tt(),N) -> c_15(activate#(N)) -->_1 activate#(n__0()) -> c_21(0#()):27 -->_1 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_1 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 activate#(X) -> c_20():26 11:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):12 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 12:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):13 -->_2 isNat#(n__0()) -> c_24():28 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 13:S:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):14 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 14:S:U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 s#(X) -> c_31():31 -->_2 plus#(X1,X2) -> c_30():30 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 15:S:activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_1 plus#(X1,X2) -> c_30():30 -->_3 activate#(X) -> c_20():26 -->_2 activate#(X) -> c_20():26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 16:S:activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) -->_2 activate#(n__0()) -> c_21(0#()):27 -->_1 s#(X) -> c_31():31 -->_2 activate#(X) -> c_20():26 -->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 17:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):1 18:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 19:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):8 20:S:isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__0()) -> c_27():29 -->_3 activate#(X) -> c_20():26 -->_1 U41#(tt()) -> c_13():25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 21:W:0#() -> c_1() 22:W:U16#(tt()) -> c_7() 23:W:U23#(tt()) -> c_10() 24:W:U32#(tt()) -> c_12() 25:W:U41#(tt()) -> c_13() 26:W:activate#(X) -> c_20() 27:W:activate#(n__0()) -> c_21(0#()) -->_1 0#() -> c_1():21 28:W:isNat#(n__0()) -> c_24() 29:W:isNatKind#(n__0()) -> c_27() 30:W:plus#(X1,X2) -> c_30() 31:W:s#(X) -> c_31() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 22: U16#(tt()) -> c_7() 23: U23#(tt()) -> c_10() 28: isNat#(n__0()) -> c_24() 24: U32#(tt()) -> c_12() 30: plus#(X1,X2) -> c_30() 31: s#(X) -> c_31() 25: U41#(tt()) -> c_13() 26: activate#(X) -> c_20() 29: isNatKind#(n__0()) -> c_27() 27: activate#(n__0()) -> c_21(0#()) 21: 0#() -> c_1() * Step 7: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4 ,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):2 2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):3 3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):4 4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):5 5:S:U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):7 7:S:U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 8:S:U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 9:S:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U52#(tt(),N) -> c_15(activate#(N)):10 10:S:U52#(tt(),N) -> c_15(activate#(N)) -->_1 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_1 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 11:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):12 12:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):13 13:S:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):14 14:S:U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 15:S:activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 16:S:activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 17:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):1 18:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 19:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):8 20:S:isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) * Step 8: Decompose WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,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: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0 ,c_8/4,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2 ,c_23/1,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31# ,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0 ,n__plus,n__s,tt} Problem (S) - Strict DPs: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0 ,c_8/4,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2 ,c_23/1,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31# ,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0 ,n__plus,n__s,tt} ** Step 8.a:1: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) and a lower component U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) Further, following extension rules are added to the lower component. U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) *** Step 8.a:1.a:1: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {9,13} by application of Pre({9,13}) = {8,12}. Here rules are labelled as follows: 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 5: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) 8: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 9: U52#(tt(),N) -> c_15(activate#(N)) 10: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 11: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 12: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 13: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) 14: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 15: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) *** Step 8.a:1.a:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) - Weak DPs: U52#(tt(),N) -> c_15(activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {8,11} by application of Pre({8,11}) = {10}. Here rules are labelled as follows: 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 5: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) 8: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 9: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 10: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 11: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 12: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 13: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 14: U52#(tt(),N) -> c_15(activate#(N)) 15: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) *** Step 8.a:1.a:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) - Weak DPs: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):2 2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):3 3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):4 4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 -->_1 U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)):5 5:S:U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_1 U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)):7 7:S:U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 8:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):9 9:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):14 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 10:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):1 11:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 12:W:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_1 U52#(tt(),N) -> c_15(activate#(N)):13 13:W:U52#(tt(),N) -> c_15(activate#(N)) 14:W:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_1 U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)):15 15:W:U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 12: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 13: U52#(tt(),N) -> c_15(activate#(N)) 14: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 15: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) *** Step 8.a:1.a:4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):2 2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):3 3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):4 4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 -->_1 U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)):5 5:S:U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_1 U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)):7 7:S:U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 8:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):9 9:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 10:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):1 11:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) *** Step 8.a:1.a:5: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) 9: U62#(tt(),M,N) -> c_17(isNat#(activate(N))) Consider the set of all dependency pairs 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 5: U15#(tt(),V2) -> c_6(isNat#(activate(V2))) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1))) 8: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) 9: U62#(tt(),M,N) -> c_17(isNat#(activate(N))) 10: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 11: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) Processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {4,6,9} These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5,6,7,8,9,10,11} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 8.a:1.a:5.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 2, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0,U31,U32,U41,activate,isNatKind,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] [0] p(U11) = [0] [0] p(U12) = [0 1] x3 + [0] [0 0] [0] p(U13) = [0 1] x1 + [0] [0 1] [0] p(U14) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0] [0 0] [0 1] [0 1] [0] p(U15) = [0] [0] p(U16) = [1] [0] p(U21) = [0 0] x1 + [0] [0 1] [1] p(U22) = [1 0] x1 + [0] [1 0] [0] p(U23) = [0] [1] p(U31) = [0 0] x1 + [0] [0 1] [0] p(U32) = [0 0] x1 + [0] [0 1] [0] p(U41) = [1] [1] p(U51) = [0] [0] p(U52) = [0] [0] p(U61) = [0] [0] p(U62) = [0] [0] p(U63) = [0] [0] p(U64) = [0] [0] p(activate) = [1 0] x1 + [0] [0 1] [0] p(isNat) = [0] [1] p(isNatKind) = [1 0] x1 + [0] [0 0] [1] p(n__0) = [0] [0] p(n__plus) = [0 0] x1 + [0 0] x2 + [1] [0 1] [0 1] [1] p(n__s) = [0 0] x1 + [1] [0 1] [1] p(plus) = [0 0] x1 + [0 0] x2 + [1] [0 1] [0 1] [1] p(s) = [0 0] x1 + [1] [0 1] [1] p(tt) = [0] [1] p(0#) = [0] [0] p(U11#) = [0 1] x1 + [0 1] x2 + [0 1] x3 + [0] [0 0] [0 0] [0 0] [0] p(U12#) = [0 1] x2 + [0 1] x3 + [1] [0 0] [0 0] [1] p(U13#) = [0 1] x1 + [0 1] x2 + [0 1] x3 + [0] [0 0] [0 0] [0 0] [1] p(U14#) = [0 0] x1 + [0 1] x2 + [0 1] x3 + [1] [0 1] [0 0] [0 0] [1] p(U15#) = [0 1] x2 + [0] [0 0] [1] p(U16#) = [0] [0] p(U21#) = [0 1] x2 + [1] [0 0] [0] p(U22#) = [0 1] x2 + [0] [0 0] [1] p(U23#) = [0] [0] p(U31#) = [0] [0] p(U32#) = [0] [0] p(U41#) = [0] [0] p(U51#) = [0] [0] p(U52#) = [0] [0] p(U61#) = [0 1] x3 + [1] [0 0] [1] p(U62#) = [0 0] x2 + [0 1] x3 + [1] [1 0] [1 1] [0] p(U63#) = [0] [0] p(U64#) = [0] [0] p(activate#) = [0] [0] p(isNat#) = [0 1] x1 + [0] [0 0] [0] p(isNatKind#) = [0] [0] p(plus#) = [0] [0] p(s#) = [0] [0] p(c_1) = [0] [0] p(c_2) = [1 0] x1 + [0] [0 0] [0] p(c_3) = [1 0] x1 + [0] [0 0] [1] p(c_4) = [1 0] x1 + [0] [0 0] [0] p(c_5) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_6) = [1 0] x1 + [0] [0 0] [1] p(c_7) = [0] [0] p(c_8) = [1 0] x1 + [0] [0 0] [0] p(c_9) = [1 0] x1 + [0] [0 0] [0] p(c_10) = [0] [0] p(c_11) = [0] [0] p(c_12) = [0] [0] p(c_13) = [0] [0] p(c_14) = [0] [0] p(c_15) = [0] [0] p(c_16) = [1 0] x1 + [0] [0 0] [0] p(c_17) = [1 0] x1 + [0] [0 0] [0] p(c_18) = [0] [0] p(c_19) = [0] [0] p(c_20) = [0] [0] p(c_21) = [0] [0] p(c_22) = [0] [0] p(c_23) = [0] [0] p(c_24) = [0] [0] p(c_25) = [1 0] x1 + [0] [0 0] [0] p(c_26) = [1 0] x1 + [0] [0 0] [0] p(c_27) = [0] [0] p(c_28) = [0] [0] p(c_29) = [0] [0] p(c_30) = [0] [0] p(c_31) = [0] [0] Following rules are strictly oriented: U14#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [2] > [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U21#(tt(),V1) = [0 1] V1 + [1] [0 0] [0] > [0 1] V1 + [0] [0 0] [0] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U62#(tt(),M,N) = [0 0] M + [0 1] N + [1] [1 0] [1 1] [0] > [0 1] N + [0] [0 0] [0] = c_17(isNat#(activate(N))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [1] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U15#(tt(),V2) = [0 1] V2 + [0] [0 0] [1] >= [0 1] V2 + [0] [0 0] [1] = c_6(isNat#(activate(V2))) U22#(tt(),V1) = [0 1] V1 + [0] [0 0] [1] >= [0 1] V1 + [0] [0 0] [0] = c_9(isNat#(activate(V1))) U61#(tt(),M,N) = [0 1] N + [1] [0 0] [1] >= [0 1] N + [1] [0 0] [0] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) isNat#(n__plus(V1,V2)) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [0 1] V1 + [1] [0 0] [0] >= [0 1] V1 + [1] [0 0] [0] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) 0() = [0] [0] >= [0] [0] = n__0() U31(tt(),V2) = [0] [1] >= [0] [1] = U32(isNatKind(activate(V2))) U32(tt()) = [0] [1] >= [0] [1] = tt() U41(tt()) = [1] [1] >= [0] [1] = tt() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [0] [0] >= [0] [0] = 0() activate(n__plus(X1,X2)) = [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] >= [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [0 0] X + [1] [0 1] [1] >= [0 0] X + [1] [0 1] [1] = s(activate(X)) isNatKind(n__0()) = [0] [1] >= [0] [1] = tt() isNatKind(n__plus(V1,V2)) = [1] [1] >= [0] [1] = U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = [1] [1] >= [1] [1] = U41(isNatKind(activate(V1))) plus(X1,X2) = [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] >= [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] = n__plus(X1,X2) s(X) = [0 0] X + [1] [0 1] [1] >= [0 0] X + [1] [0 1] [1] = n__s(X) **** Step 8.a:1.a:5.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak DPs: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 8.a:1.a:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))):2 2:W:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))):3 3:W:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):4 4:W:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))):11 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):10 -->_1 U15#(tt(),V2) -> c_6(isNat#(activate(V2))):5 5:W:U15#(tt(),V2) -> c_6(isNat#(activate(V2))) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):10 6:W:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) -->_1 U22#(tt(),V1) -> c_9(isNat#(activate(V1))):7 7:W:U22#(tt(),V1) -> c_9(isNat#(activate(V1))) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):10 8:W:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) -->_1 U62#(tt(),M,N) -> c_17(isNat#(activate(N))):9 9:W:U62#(tt(),M,N) -> c_17(isNat#(activate(N))) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):10 10:W:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))):1 11:W:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) 9: U62#(tt(),M,N) -> c_17(isNat#(activate(N))) 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 10: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1))) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) 11: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) 5: U15#(tt(),V2) -> c_6(isNat#(activate(V2))) 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) **** Step 8.a:1.a:5.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 8.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) 3: activate#(n__s(X)) -> c_23(activate#(X)) 4: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 5: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) Consider the set of all dependency pairs 1: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) 2: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) 3: activate#(n__s(X)) -> c_23(activate#(X)) 4: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 5: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) 6: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) 7: U11#(tt(),V1,V2) -> activate#(V1) 8: U11#(tt(),V1,V2) -> activate#(V2) 9: U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) 10: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) 11: U12#(tt(),V1,V2) -> activate#(V1) 12: U12#(tt(),V1,V2) -> activate#(V2) 13: U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) 14: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) 15: U13#(tt(),V1,V2) -> activate#(V1) 16: U13#(tt(),V1,V2) -> activate#(V2) 17: U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) 18: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) 19: U14#(tt(),V1,V2) -> activate#(V1) 20: U14#(tt(),V1,V2) -> activate#(V2) 21: U14#(tt(),V1,V2) -> isNat#(activate(V1)) 22: U15#(tt(),V2) -> activate#(V2) 23: U15#(tt(),V2) -> isNat#(activate(V2)) 24: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) 25: U21#(tt(),V1) -> activate#(V1) 26: U21#(tt(),V1) -> isNatKind#(activate(V1)) 27: U22#(tt(),V1) -> activate#(V1) 28: U22#(tt(),V1) -> isNat#(activate(V1)) 29: U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) 30: U51#(tt(),N) -> activate#(N) 31: U51#(tt(),N) -> isNatKind#(activate(N)) 32: U52#(tt(),N) -> activate#(N) 33: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) 34: U61#(tt(),M,N) -> activate#(M) 35: U61#(tt(),M,N) -> activate#(N) 36: U61#(tt(),M,N) -> isNatKind#(activate(M)) 37: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) 38: U62#(tt(),M,N) -> activate#(M) 39: U62#(tt(),M,N) -> activate#(N) 40: U62#(tt(),M,N) -> isNat#(activate(N)) 41: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) 42: U63#(tt(),M,N) -> activate#(M) 43: U63#(tt(),M,N) -> activate#(N) 44: U63#(tt(),M,N) -> isNatKind#(activate(N)) 45: U64#(tt(),M,N) -> activate#(M) 46: U64#(tt(),M,N) -> activate#(N) 47: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) 48: isNat#(n__plus(V1,V2)) -> activate#(V1) 49: isNat#(n__plus(V1,V2)) -> activate#(V2) 50: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) 51: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) 52: isNat#(n__s(V1)) -> activate#(V1) 53: isNat#(n__s(V1)) -> isNatKind#(activate(V1)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {2,3,4,5} These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 8.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_11) = {1,2}, uargs(c_22) = {1,2}, uargs(c_23) = {1}, uargs(c_28) = {1,2,3,4}, uargs(c_29) = {1,2} Following symbols are considered usable: {0,U11,U12,U13,U14,U15,U16,U21,U22,U23,activate,isNat,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22# ,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = 1 p(U11) = 1 + x3 p(U12) = 1 + x3 p(U13) = 1 + x3 p(U14) = 1 + x3 p(U15) = x2 p(U16) = x1 p(U21) = x2 p(U22) = x2 p(U23) = x1 p(U31) = x1*x2 + x2 p(U32) = 0 p(U41) = 0 p(U51) = 0 p(U52) = 0 p(U61) = 0 p(U62) = 0 p(U63) = 0 p(U64) = 0 p(activate) = x1 p(isNat) = x1 p(isNatKind) = 0 p(n__0) = 1 p(n__plus) = 1 + x1 + x2 p(n__s) = 1 + x1 p(plus) = 1 + x1 + x2 p(s) = 1 + x1 p(tt) = 1 p(0#) = 0 p(U11#) = x2 + x2^2 + x3 + x3^2 p(U12#) = x2 + x2^2 + x3 + x3^2 p(U13#) = x2 + x2^2 + x3 + x3^2 p(U14#) = x2 + x2^2 + x3 + x3^2 p(U15#) = x2 + x2^2 p(U16#) = 0 p(U21#) = x2 + x2^2 p(U22#) = x2 + x2^2 p(U23#) = 0 p(U31#) = x2 + x2^2 p(U32#) = 0 p(U41#) = 0 p(U51#) = x2 + x2^2 p(U52#) = x2 p(U61#) = 1 + x2 + x2*x3 + x2^2 + x3 + x3^2 p(U62#) = 1 + x2 + x2*x3 + x3 + x3^2 p(U63#) = x1*x2 + x2 + x3 + x3^2 p(U64#) = x2 + x3 p(activate#) = x1 p(isNat#) = x1^2 p(isNatKind#) = x1^2 p(plus#) = 0 p(s#) = 0 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = x1 + x2 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = x1 + x2 p(c_23) = x1 p(c_24) = 0 p(c_25) = 0 p(c_26) = 0 p(c_27) = 0 p(c_28) = x1 + x2 + x3 + x4 p(c_29) = x1 + x2 p(c_30) = 0 p(c_31) = 0 Following rules are strictly oriented: activate#(n__plus(X1,X2)) = 1 + X1 + X2 > X1 + X2 = c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) = 1 + X > X = c_23(activate#(X)) isNatKind#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2 > V1 + V1^2 + 2*V2 + V2^2 = c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)) isNatKind#(n__s(V1)) = 1 + 2*V1 + V1^2 > V1 + V1^2 = c_29(isNatKind#(activate(V1)),activate#(V1)) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V1 + V1^2 + V2 + V2^2 = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V1 = activate#(V1) U11#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V2 = activate#(V2) U11#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V1^2 = isNatKind#(activate(V1)) U12#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V1 + V1^2 + V2 + V2^2 = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V1 = activate#(V1) U12#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V2 = activate#(V2) U12#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V2^2 = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V1 + V1^2 + V2 + V2^2 = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V1 = activate#(V1) U13#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V2 = activate#(V2) U13#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V2^2 = isNatKind#(activate(V2)) U14#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V2 + V2^2 = U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V1 = activate#(V1) U14#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V2 = activate#(V2) U14#(tt(),V1,V2) = V1 + V1^2 + V2 + V2^2 >= V1^2 = isNat#(activate(V1)) U15#(tt(),V2) = V2 + V2^2 >= V2 = activate#(V2) U15#(tt(),V2) = V2 + V2^2 >= V2^2 = isNat#(activate(V2)) U21#(tt(),V1) = V1 + V1^2 >= V1 + V1^2 = U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = V1 + V1^2 >= V1 = activate#(V1) U21#(tt(),V1) = V1 + V1^2 >= V1^2 = isNatKind#(activate(V1)) U22#(tt(),V1) = V1 + V1^2 >= V1 = activate#(V1) U22#(tt(),V1) = V1 + V1^2 >= V1^2 = isNat#(activate(V1)) U31#(tt(),V2) = V2 + V2^2 >= V2 + V2^2 = c_11(isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) = N + N^2 >= N = U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) = N + N^2 >= N = activate#(N) U51#(tt(),N) = N + N^2 >= N^2 = isNatKind#(activate(N)) U52#(tt(),N) = N >= N = activate#(N) U61#(tt(),M,N) = 1 + M + M*N + M^2 + N + N^2 >= 1 + M + M*N + N + N^2 = U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) = 1 + M + M*N + M^2 + N + N^2 >= M = activate#(M) U61#(tt(),M,N) = 1 + M + M*N + M^2 + N + N^2 >= N = activate#(N) U61#(tt(),M,N) = 1 + M + M*N + M^2 + N + N^2 >= M^2 = isNatKind#(activate(M)) U62#(tt(),M,N) = 1 + M + M*N + N + N^2 >= M + M*N + N + N^2 = U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) = 1 + M + M*N + N + N^2 >= M = activate#(M) U62#(tt(),M,N) = 1 + M + M*N + N + N^2 >= N = activate#(N) U62#(tt(),M,N) = 1 + M + M*N + N + N^2 >= N^2 = isNat#(activate(N)) U63#(tt(),M,N) = 2*M + N + N^2 >= M + N = U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = 2*M + N + N^2 >= M = activate#(M) U63#(tt(),M,N) = 2*M + N + N^2 >= N = activate#(N) U63#(tt(),M,N) = 2*M + N + N^2 >= N^2 = isNatKind#(activate(N)) U64#(tt(),M,N) = M + N >= M = activate#(M) U64#(tt(),M,N) = M + N >= N = activate#(N) isNat#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2 >= V1 + V1^2 + V2 + V2^2 = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2 >= V1 = activate#(V1) isNat#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2 >= V2 = activate#(V2) isNat#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2 >= V1^2 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = 1 + 2*V1 + V1^2 >= V1 + V1^2 = U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) = 1 + 2*V1 + V1^2 >= V1 = activate#(V1) isNat#(n__s(V1)) = 1 + 2*V1 + V1^2 >= V1^2 = isNatKind#(activate(V1)) 0() = 1 >= 1 = n__0() U11(tt(),V1,V2) = 1 + V2 >= 1 + V2 = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) = 1 + V2 >= 1 + V2 = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) = 1 + V2 >= 1 + V2 = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) = 1 + V2 >= V2 = U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) = V2 >= V2 = U16(isNat(activate(V2))) U16(tt()) = 1 >= 1 = tt() U21(tt(),V1) = V1 >= V1 = U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) = V1 >= V1 = U23(isNat(activate(V1))) U23(tt()) = 1 >= 1 = tt() activate(X) = X >= X = X activate(n__0()) = 1 >= 1 = 0() activate(n__plus(X1,X2)) = 1 + X1 + X2 >= 1 + X1 + X2 = plus(activate(X1),activate(X2)) activate(n__s(X)) = 1 + X >= 1 + X = s(activate(X)) isNat(n__0()) = 1 >= 1 = tt() isNat(n__plus(V1,V2)) = 1 + V1 + V2 >= 1 + V2 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) = 1 + V1 >= V1 = U21(isNatKind(activate(V1)),activate(V1)) plus(X1,X2) = 1 + X1 + X2 >= 1 + X1 + X2 = n__plus(X1,X2) s(X) = 1 + X >= 1 + X = n__s(X) **** Step 8.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 8.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -->_1 U12#(tt(),V1,V2) -> isNatKind#(activate(V2)):8 -->_1 U12#(tt(),V1,V2) -> activate#(V2):7 -->_1 U12#(tt(),V1,V2) -> activate#(V1):6 -->_1 U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)):5 2:W:U11#(tt(),V1,V2) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 3:W:U11#(tt(),V1,V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 4:W:U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 5:W:U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -->_1 U13#(tt(),V1,V2) -> isNatKind#(activate(V2)):12 -->_1 U13#(tt(),V1,V2) -> activate#(V2):11 -->_1 U13#(tt(),V1,V2) -> activate#(V1):10 -->_1 U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)):9 6:W:U12#(tt(),V1,V2) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 7:W:U12#(tt(),V1,V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 8:W:U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 9:W:U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -->_1 U14#(tt(),V1,V2) -> isNat#(activate(V1)):16 -->_1 U14#(tt(),V1,V2) -> activate#(V2):15 -->_1 U14#(tt(),V1,V2) -> activate#(V1):14 -->_1 U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)):13 10:W:U13#(tt(),V1,V2) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 11:W:U13#(tt(),V1,V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 12:W:U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 13:W:U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -->_1 U15#(tt(),V2) -> isNat#(activate(V2)):18 -->_1 U15#(tt(),V2) -> activate#(V2):17 14:W:U14#(tt(),V1,V2) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 15:W:U14#(tt(),V1,V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 16:W:U14#(tt(),V1,V2) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> activate#(V1):50 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):48 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):47 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):46 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):45 17:W:U15#(tt(),V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 18:W:U15#(tt(),V2) -> isNat#(activate(V2)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> activate#(V1):50 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):48 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):47 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):46 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):45 19:W:U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -->_1 U22#(tt(),V1) -> isNat#(activate(V1)):23 -->_1 U22#(tt(),V1) -> activate#(V1):22 20:W:U21#(tt(),V1) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 21:W:U21#(tt(),V1) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 22:W:U22#(tt(),V1) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 23:W:U22#(tt(),V1) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> activate#(V1):50 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):48 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):47 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):46 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):45 24:W:U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 -->_2 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 25:W:U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) -->_1 U52#(tt(),N) -> activate#(N):28 26:W:U51#(tt(),N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 27:W:U51#(tt(),N) -> isNatKind#(activate(N)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 28:W:U52#(tt(),N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 29:W:U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -->_1 U62#(tt(),M,N) -> isNat#(activate(N)):36 -->_1 U62#(tt(),M,N) -> activate#(N):35 -->_1 U62#(tt(),M,N) -> activate#(M):34 -->_1 U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)):33 30:W:U61#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 31:W:U61#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 32:W:U61#(tt(),M,N) -> isNatKind#(activate(M)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 33:W:U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -->_1 U63#(tt(),M,N) -> isNatKind#(activate(N)):40 -->_1 U63#(tt(),M,N) -> activate#(N):39 -->_1 U63#(tt(),M,N) -> activate#(M):38 -->_1 U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)):37 34:W:U62#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 35:W:U62#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 36:W:U62#(tt(),M,N) -> isNat#(activate(N)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> activate#(V1):50 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):48 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):47 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):46 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):45 37:W:U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -->_1 U64#(tt(),M,N) -> activate#(N):42 -->_1 U64#(tt(),M,N) -> activate#(M):41 38:W:U63#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 39:W:U63#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 40:W:U63#(tt(),M,N) -> isNatKind#(activate(N)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 41:W:U64#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 42:W:U64#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 43:W:activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) -->_2 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 44:W:activate#(n__s(X)) -> c_23(activate#(X)) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 45:W:isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -->_1 U11#(tt(),V1,V2) -> isNatKind#(activate(V1)):4 -->_1 U11#(tt(),V1,V2) -> activate#(V2):3 -->_1 U11#(tt(),V1,V2) -> activate#(V1):2 -->_1 U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)):1 46:W:isNat#(n__plus(V1,V2)) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 47:W:isNat#(n__plus(V1,V2)) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 48:W:isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 49:W:isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -->_1 U21#(tt(),V1) -> isNatKind#(activate(V1)):21 -->_1 U21#(tt(),V1) -> activate#(V1):20 -->_1 U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)):19 50:W:isNat#(n__s(V1)) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 51:W:isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 52:W:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 -->_1 U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)):24 53:W:isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):53 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):52 -->_2 activate#(n__s(X)) -> c_23(activate#(X)):44 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):43 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 32: U61#(tt(),M,N) -> isNatKind#(activate(M)) 31: U61#(tt(),M,N) -> activate#(N) 30: U61#(tt(),M,N) -> activate#(M) 29: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) 33: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) 37: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) 41: U64#(tt(),M,N) -> activate#(M) 42: U64#(tt(),M,N) -> activate#(N) 38: U63#(tt(),M,N) -> activate#(M) 39: U63#(tt(),M,N) -> activate#(N) 40: U63#(tt(),M,N) -> isNatKind#(activate(N)) 34: U62#(tt(),M,N) -> activate#(M) 35: U62#(tt(),M,N) -> activate#(N) 36: U62#(tt(),M,N) -> isNat#(activate(N)) 27: U51#(tt(),N) -> isNatKind#(activate(N)) 26: U51#(tt(),N) -> activate#(N) 25: U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) 28: U52#(tt(),N) -> activate#(N) 1: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) 45: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) 23: U22#(tt(),V1) -> isNat#(activate(V1)) 19: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) 49: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) 18: U15#(tt(),V2) -> isNat#(activate(V2)) 13: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) 9: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) 5: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) 16: U14#(tt(),V1,V2) -> isNat#(activate(V1)) 17: U15#(tt(),V2) -> activate#(V2) 14: U14#(tt(),V1,V2) -> activate#(V1) 15: U14#(tt(),V1,V2) -> activate#(V2) 22: U22#(tt(),V1) -> activate#(V1) 2: U11#(tt(),V1,V2) -> activate#(V1) 3: U11#(tt(),V1,V2) -> activate#(V2) 4: U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) 46: isNat#(n__plus(V1,V2)) -> activate#(V1) 47: isNat#(n__plus(V1,V2)) -> activate#(V2) 48: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) 20: U21#(tt(),V1) -> activate#(V1) 21: U21#(tt(),V1) -> isNatKind#(activate(V1)) 50: isNat#(n__s(V1)) -> activate#(V1) 51: isNat#(n__s(V1)) -> isNatKind#(activate(V1)) 10: U13#(tt(),V1,V2) -> activate#(V1) 11: U13#(tt(),V1,V2) -> activate#(V2) 12: U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) 6: U12#(tt(),V1,V2) -> activate#(V1) 7: U12#(tt(),V1,V2) -> activate#(V2) 8: U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) 53: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) 52: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 24: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) 44: activate#(n__s(X)) -> c_23(activate#(X)) 43: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) **** Step 8.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 8.b:1: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,6} by application of Pre({2,6}) = {1,5}. Here rules are labelled as follows: 1: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 2: U52#(tt(),N) -> c_15(activate#(N)) 3: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 4: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 5: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 6: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) 7: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 8: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 9: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 10: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 11: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) 12: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 13: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) 14: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) 15: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) 16: activate#(n__s(X)) -> c_23(activate#(X)) 17: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 18: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 19: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 20: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) ** Step 8.b:2: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U52#(tt(),N) -> c_15(activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4} by application of Pre({1,4}) = {3}. Here rules are labelled as follows: 1: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 2: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 3: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 4: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 5: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 6: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 7: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 8: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 9: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) 10: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 11: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) 12: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) 13: U52#(tt(),N) -> c_15(activate#(N)) 14: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) 15: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) 16: activate#(n__s(X)) -> c_23(activate#(X)) 17: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 18: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 19: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 20: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) ** Step 8.b:3: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {1}. Here rules are labelled as follows: 1: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 2: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 3: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 4: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 5: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 6: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 7: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) 8: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 9: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) 10: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) 11: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 12: U52#(tt(),N) -> c_15(activate#(N)) 13: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 14: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) 15: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) 16: activate#(n__s(X)) -> c_23(activate#(X)) 17: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 18: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 19: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 20: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) ** Step 8.b:4: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,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: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 2: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 3: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 4: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 5: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 6: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) 7: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 8: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) 9: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) 10: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 11: U52#(tt(),N) -> c_15(activate#(N)) 12: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 13: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 14: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) 15: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) 16: activate#(n__s(X)) -> c_23(activate#(X)) 17: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 18: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 19: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 20: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) ** Step 8.b:5: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):2 2:W:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):3 3:W:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):4 4:W:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)):5 5:W:U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_2 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 6:W:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)):7 7:W:U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_2 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 8:W:U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_2 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 9:W:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U52#(tt(),N) -> c_15(activate#(N)):10 10:W:U52#(tt(),N) -> c_15(activate#(N)) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 11:W:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):12 12:W:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_5 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):13 13:W:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)):14 14:W:U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) -->_2 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_1 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 15:W:activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) -->_2 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_1 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 16:W:activate#(n__s(X)) -> c_23(activate#(X)) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 17:W:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):1 18:W:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 19:W:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 -->_1 U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)):8 20:W:isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):20 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_2 activate#(n__s(X)) -> c_23(activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):15 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 12: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 13: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 14: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) 9: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 10: U52#(tt(),N) -> c_15(activate#(N)) 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 17: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 18: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 5: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 20: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) 19: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 8: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) 16: activate#(n__s(X)) -> c_23(activate#(X)) 15: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) ** Step 8.b:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))