WORST_CASE(?,O(n^2)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^2)) + 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(X1,X2) activate(n__s(X)) -> s(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^2)) + 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(X1,X2) activate(n__s(X)) -> s(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#(X1,X2)) activate#(n__s(X)) -> c_23(s#(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^2)) + 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#(X1,X2)) activate#(n__s(X)) -> c_23(s#(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(X1,X2) activate(n__s(X)) -> s(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_22/1,c_23/1 ,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(X1,X2) activate(n__s(X)) -> s(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#(X1,X2)) activate#(n__s(X)) -> c_23(s#(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^2)) + 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#(X1,X2)) activate#(n__s(X)) -> c_23(s#(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(X1,X2) activate(n__s(X)) -> s(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_22/1,c_23/1 ,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#(X1,X2)) 23: activate#(n__s(X)) -> c_23(s#(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^2)) + 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#(X1,X2)) activate#(n__s(X)) -> c_23(s#(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(X1,X2) activate(n__s(X)) -> s(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_22/1,c_23/1 ,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 {15,16,17} by application of Pre({15,16,17}) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,18,19,20,21}. 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#(X1,X2)) 17: activate#(n__s(X)) -> c_23(s#(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: PredecessorEstimation WORST_CASE(?,O(n^2)) + 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)) 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#()) activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)) activate#(n__s(X)) -> c_23(s#(X)) 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(X1,X2) activate(n__s(X)) -> s(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_22/1,c_23/1 ,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 {10,14} by application of Pre({10,14}) = {9,13}. 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: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 16: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 17: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 18: isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) 19: 0#() -> c_1() 20: U16#(tt()) -> c_7() 21: U23#(tt()) -> c_10() 22: U32#(tt()) -> c_12() 23: U41#(tt()) -> c_13() 24: activate#(X) -> c_20() 25: activate#(n__0()) -> c_21(0#()) 26: activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)) 27: activate#(n__s(X)) -> c_23(s#(X)) 28: isNat#(n__0()) -> c_24() 29: isNatKind#(n__0()) -> c_27() 30: plus#(X1,X2) -> c_30() 31: s#(X) -> c_31() * Step 7: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + 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)) 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)) 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() U52#(tt(),N) -> c_15(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#(X1,X2)) activate#(n__s(X)) -> c_23(s#(X)) 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(X1,X2) activate(n__s(X)) -> s(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_22/1,c_23/1 ,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__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_5 activate#(n__0()) -> c_21(0#()):25 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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():24 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 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__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_5 activate#(n__0()) -> c_21(0#()):25 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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():24 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 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__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_5 activate#(n__0()) -> c_21(0#()):25 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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():24 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):14 -->_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)):13 -->_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():24 -->_3 activate#(X) -> c_20():24 5:S:U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):14 -->_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)):13 -->_2 isNat#(n__0()) -> c_24():28 -->_3 activate#(X) -> c_20():24 -->_1 U16#(tt()) -> c_7():18 6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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():24 -->_3 activate#(X) -> c_20():24 7:S:U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):14 -->_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)):13 -->_2 isNat#(n__0()) -> c_24():28 -->_3 activate#(X) -> c_20():24 -->_1 U23#(tt()) -> c_10():19 8:S:U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 -->_2 isNatKind#(n__0()) -> c_27():29 -->_3 activate#(X) -> c_20():24 -->_1 U32#(tt()) -> c_12():20 9:S:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_1 U52#(tt(),N) -> c_15(activate#(N)):22 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 10: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__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_5 activate#(n__0()) -> c_21(0#()):25 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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)):11 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():24 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 11: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__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_5 activate#(n__0()) -> c_21(0#()):25 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):14 -->_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)):13 -->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):12 -->_2 isNat#(n__0()) -> c_24():28 -->_5 activate#(X) -> c_20():24 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 12: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__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_5 activate#(n__0()) -> c_21(0#()):25 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_1 U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):23 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():24 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 13: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__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_5 activate#(n__0()) -> c_21(0#()):25 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():24 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 -->_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 14:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 15:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 -->_1 U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):8 16:S:isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_2 isNatKind#(n__0()) -> c_27():29 -->_3 activate#(X) -> c_20():24 -->_1 U41#(tt()) -> c_13():21 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 17:W:0#() -> c_1() 18:W:U16#(tt()) -> c_7() 19:W:U23#(tt()) -> c_10() 20:W:U32#(tt()) -> c_12() 21:W:U41#(tt()) -> c_13() 22:W:U52#(tt(),N) -> c_15(activate#(N)) -->_1 activate#(n__s(X)) -> c_23(s#(X)):27 -->_1 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_1 activate#(n__0()) -> c_21(0#()):25 -->_1 activate#(X) -> c_20():24 23:W: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#(X)):27 -->_3 activate#(n__s(X)) -> c_23(s#(X)):27 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)):26 -->_4 activate#(n__0()) -> c_21(0#()):25 -->_3 activate#(n__0()) -> c_21(0#()):25 -->_1 s#(X) -> c_31():31 -->_2 plus#(X1,X2) -> c_30():30 -->_4 activate#(X) -> c_20():24 -->_3 activate#(X) -> c_20():24 24:W:activate#(X) -> c_20() 25:W:activate#(n__0()) -> c_21(0#()) -->_1 0#() -> c_1():17 26:W:activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_30():30 27:W:activate#(n__s(X)) -> c_23(s#(X)) -->_1 s#(X) -> c_31():31 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. 23: U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 22: U52#(tt(),N) -> c_15(activate#(N)) 18: U16#(tt()) -> c_7() 19: U23#(tt()) -> c_10() 28: isNat#(n__0()) -> c_24() 20: U32#(tt()) -> c_12() 21: U41#(tt()) -> c_13() 24: activate#(X) -> c_20() 29: isNatKind#(n__0()) -> c_27() 25: activate#(n__0()) -> c_21(0#()) 17: 0#() -> c_1() 26: activate#(n__plus(X1,X2)) -> c_22(plus#(X1,X2)) 30: plus#(X1,X2) -> c_30() 27: activate#(n__s(X)) -> c_23(s#(X)) 31: s#(X) -> c_31() * Step 8: SimplifyRHS WORST_CASE(?,O(n^2)) + 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)) 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)) 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(X1,X2) activate(n__s(X)) -> s(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_22/1,c_23/1 ,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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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)):14 -->_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)):13 -->_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)):14 -->_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)):13 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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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)):14 -->_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)):13 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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 10: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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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)):11 11: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)):14 -->_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)):13 -->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):12 12: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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 13: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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):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 14: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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 15: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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 -->_1 U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):8 16: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)):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):15 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)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) * Step 9: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak DPs: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0 ,c_8/2,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1 ,c_23/1,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0 ,c_8/2,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1 ,c_23/1,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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 9.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak DPs: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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: 8: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) 13: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 14: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 16: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) Consider the set of all dependency pairs 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1))) 8: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) 9: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) 10: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 11: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 12: U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) 13: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 14: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 15: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 16: isNatKind#(n__s(V1)) -> c_29(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 {8,13,14,16} These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5,6,7,8,9,10,11,12,13,14,16} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 9.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak DPs: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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_2) = {1,2}, uargs(c_3) = {1,2}, uargs(c_4) = {1,2}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1,2}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1,2}, uargs(c_17) = {1,2}, uargs(c_18) = {1}, uargs(c_25) = {1,2}, uargs(c_26) = {1,2}, uargs(c_28) = {1,2}, uargs(c_29) = {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) = 1 p(U11) = x1*x3 + x3 p(U12) = 1 p(U13) = x2 p(U14) = 1 + x1 + x1*x2 + x1^2 + x2*x3 + x3 + x3^2 p(U15) = 0 p(U16) = 0 p(U21) = x1*x2 p(U22) = x1*x2 + x1^2 p(U23) = x1^2 p(U31) = x2 p(U32) = x1 p(U41) = x1 p(U51) = 0 p(U52) = 0 p(U61) = 0 p(U62) = 0 p(U63) = 0 p(U64) = 0 p(activate) = x1 p(isNat) = 0 p(isNatKind) = x1 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#) = x1*x3 + x2 + x2*x3 + x2^2 + x3 + x3^2 p(U12#) = x1*x3 + x2^2 + x3 + x3^2 p(U13#) = x2^2 + x3 + x3^2 p(U14#) = x2^2 + x3^2 p(U15#) = x2^2 p(U16#) = 0 p(U21#) = x2 + x2^2 p(U22#) = x2^2 p(U23#) = 0 p(U31#) = 1 + x2 p(U32#) = 0 p(U41#) = 0 p(U51#) = x1*x2 + x2 + x2^2 p(U52#) = 0 p(U61#) = 1 + x1*x2 + x1*x3 + x1^2 + x2 + x2*x3 + x3^2 p(U62#) = 1 + x2 + x2*x3 + x3 + x3^2 p(U63#) = x3 p(U64#) = 0 p(activate#) = 0 p(isNat#) = x1^2 p(isNatKind#) = x1 p(plus#) = 0 p(s#) = 0 p(c_1) = 0 p(c_2) = x1 + x2 p(c_3) = x1 + x2 p(c_4) = x1 + x2 p(c_5) = x1 + x2 p(c_6) = x1 p(c_7) = 0 p(c_8) = x1 + x2 p(c_9) = x1 p(c_10) = 0 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = x1 p(c_15) = 0 p(c_16) = x1 + x2 p(c_17) = x1 + x2 p(c_18) = x1 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = 0 p(c_24) = 0 p(c_25) = x1 + x2 p(c_26) = x1 + x2 p(c_27) = 0 p(c_28) = x1 + x2 p(c_29) = x1 p(c_30) = 0 p(c_31) = 0 Following rules are strictly oriented: U31#(tt(),V2) = 1 + V2 > V2 = c_11(isNatKind#(activate(V2))) isNat#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2 > 2*V1 + 2*V1*V2 + V1^2 + V2 + V2^2 = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) isNat#(n__s(V1)) = 1 + 2*V1 + V1^2 > 2*V1 + V1^2 = c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) = 1 + V1 > V1 = c_29(isNatKind#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = V1 + V1*V2 + V1^2 + 2*V2 + V2^2 >= V1 + V1*V2 + V1^2 + V2 + V2^2 = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) = V1^2 + 2*V2 + V2^2 >= V1^2 + 2*V2 + V2^2 = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) = V1^2 + V2 + V2^2 >= V1^2 + V2 + V2^2 = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) = V1^2 + V2^2 >= V1^2 + V2^2 = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = V2^2 >= V2^2 = c_6(isNat#(activate(V2))) U21#(tt(),V1) = V1 + V1^2 >= V1 + V1^2 = c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) = V1^2 >= V1^2 = c_9(isNat#(activate(V1))) U51#(tt(),N) = 2*N + N^2 >= N = c_14(isNatKind#(activate(N))) U61#(tt(),M,N) = 2 + 2*M + M*N + N + N^2 >= 1 + 2*M + M*N + N + N^2 = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) = 1 + M + M*N + N + N^2 >= N + N^2 = c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) = N >= N = c_18(isNatKind#(activate(N))) isNatKind#(n__plus(V1,V2)) = 1 + V1 + V2 >= 1 + V1 + V2 = c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 0() = 1 >= 1 = n__0() U31(tt(),V2) = V2 >= V2 = U32(isNatKind(activate(V2))) U32(tt()) = 1 >= 1 = tt() U41(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(X1,X2) activate(n__s(X)) = 1 + X >= 1 + X = s(X) isNatKind(n__0()) = 1 >= 1 = tt() isNatKind(n__plus(V1,V2)) = 1 + V1 + V2 >= V2 = U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = 1 + V1 >= V1 = U41(isNatKind(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 9.a:1.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)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) - Weak DPs: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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 9.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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: 1: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) Consider the set of all dependency pairs 1: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 2: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 3: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 4: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 5: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: U15#(tt(),V2) -> c_6(isNat#(activate(V2))) 7: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 8: U22#(tt(),V1) -> c_9(isNat#(activate(V1))) 9: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) 10: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) 11: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 12: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 13: U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) 14: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 15: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 16: isNatKind#(n__s(V1)) -> c_29(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 {1} These cover all (indirect) predecessors of dependency pairs {1,9,10,11,12,13} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 9.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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_2) = {1,2}, uargs(c_3) = {1,2}, uargs(c_4) = {1,2}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1,2}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1,2}, uargs(c_17) = {1,2}, uargs(c_18) = {1}, uargs(c_25) = {1,2}, uargs(c_26) = {1,2}, uargs(c_28) = {1,2}, uargs(c_29) = {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) = 1 p(U11) = 1 + x1 + x3 + x3^2 p(U12) = x1*x2 + x2 + x2^2 + x3 p(U13) = x1*x3 p(U14) = 1 + x1*x3 + x1^2 + x3^2 p(U15) = x1*x2 + x2 p(U16) = 1 + x1^2 p(U21) = x2 + x2^2 p(U22) = 1 + x1^2 + x2^2 p(U23) = 0 p(U31) = 1 p(U32) = 1 p(U41) = x1 p(U51) = 0 p(U52) = 0 p(U61) = 0 p(U62) = 0 p(U63) = 0 p(U64) = 0 p(activate) = x1 p(isNat) = 0 p(isNatKind) = x1 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*x3 + x2^2 + x3 + x3^2 p(U12#) = x1*x3 + x2^2 + x3 + x3^2 p(U13#) = x2^2 + x3 + x3^2 p(U14#) = x2^2 + x3^2 p(U15#) = x2^2 p(U16#) = 0 p(U21#) = x2 + x2^2 p(U22#) = x2^2 p(U23#) = 0 p(U31#) = x2 p(U32#) = 0 p(U41#) = 0 p(U51#) = 1 + x2 p(U52#) = 0 p(U61#) = 1 + x1*x2 + x2 + x3 + x3^2 p(U62#) = 1 + x3 + x3^2 p(U63#) = 1 + x3 p(U64#) = 0 p(activate#) = 0 p(isNat#) = x1^2 p(isNatKind#) = x1 p(plus#) = 0 p(s#) = 0 p(c_1) = 0 p(c_2) = x1 + x2 p(c_3) = x1 + x2 p(c_4) = x1 + x2 p(c_5) = x1 + x2 p(c_6) = x1 p(c_7) = 0 p(c_8) = x1 + x2 p(c_9) = x1 p(c_10) = 0 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 1 + x1 p(c_15) = 0 p(c_16) = x1 + x2 p(c_17) = x1 + x2 p(c_18) = 1 + x1 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = 0 p(c_24) = 0 p(c_25) = 1 + x1 + x2 p(c_26) = x1 + x2 p(c_27) = 0 p(c_28) = x1 + x2 p(c_29) = x1 p(c_30) = 0 p(c_31) = 0 Following rules are strictly oriented: isNatKind#(n__plus(V1,V2)) = 1 + V1 + V2 > V1 + V2 = c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = V1 + V1*V2 + V1^2 + V2 + V2^2 >= V1 + V1*V2 + V1^2 + V2 + V2^2 = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) = V1^2 + 2*V2 + V2^2 >= V1^2 + 2*V2 + V2^2 = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) = V1^2 + V2 + V2^2 >= V1^2 + V2 + V2^2 = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) = V1^2 + V2^2 >= V1^2 + V2^2 = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = V2^2 >= V2^2 = c_6(isNat#(activate(V2))) U21#(tt(),V1) = V1 + V1^2 >= V1 + V1^2 = c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) = V1^2 >= V1^2 = c_9(isNat#(activate(V1))) U31#(tt(),V2) = V2 >= V2 = c_11(isNatKind#(activate(V2))) U51#(tt(),N) = 1 + N >= 1 + N = c_14(isNatKind#(activate(N))) U61#(tt(),M,N) = 1 + 2*M + N + N^2 >= 1 + M + N + N^2 = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) = 1 + N + N^2 >= 1 + N + N^2 = c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) = 1 + N >= 1 + N = c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2 >= 1 + 2*V1 + V1*V2 + V1^2 + V2 + V2^2 = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) isNat#(n__s(V1)) = 1 + 2*V1 + V1^2 >= 2*V1 + V1^2 = c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) = 1 + V1 >= V1 = c_29(isNatKind#(activate(V1))) 0() = 1 >= 1 = n__0() U31(tt(),V2) = 1 >= 1 = U32(isNatKind(activate(V2))) U32(tt()) = 1 >= 1 = tt() U41(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(X1,X2) activate(n__s(X)) = 1 + X >= 1 + X = s(X) isNatKind(n__0()) = 1 >= 1 = tt() isNatKind(n__plus(V1,V2)) = 1 + V1 + V2 >= 1 = U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = 1 + V1 >= V1 = U41(isNatKind(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 9.a:1.b:1.a:2: Assumption 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))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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 9.a:1.b:1.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)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):2 2:W:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):3 3:W:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_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)),isNatKind#(activate(V1))):14 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):13 -->_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)),isNatKind#(activate(V1))):14 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):13 6:W:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_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)),isNatKind#(activate(V1))):14 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):13 8:W:U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 9:W:U51#(tt(),N) -> c_14(isNatKind#(activate(N))) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 10:W:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):11 11:W:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):14 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):13 -->_1 U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))):12 12:W:U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 13:W:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):1 14:W:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):6 15:W:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))):8 16:W:isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 11: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 12: U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) 9: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 13: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1))) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 14: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(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)),isNatKind#(activate(V2))) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 16: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) 15: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 8: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) **** Step 9.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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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 9.b:1: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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(isNatKind#(activate(N))) 2: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 3: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 4: U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) 5: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 6: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 7: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 8: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 9: U15#(tt(),V2) -> c_6(isNat#(activate(V2))) 10: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 11: U22#(tt(),V1) -> c_9(isNat#(activate(V1))) 12: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) 13: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 14: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 15: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 16: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) ** Step 9.b:2: 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))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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))) 2: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 3: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 4: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 5: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 6: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 7: U15#(tt(),V2) -> c_6(isNat#(activate(V2))) 8: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 9: U22#(tt(),V1) -> c_9(isNat#(activate(V1))) 10: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) 11: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) 12: U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) 13: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 14: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 15: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 16: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) ** Step 9.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))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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))) 2: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 3: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 4: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 5: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: U15#(tt(),V2) -> c_6(isNat#(activate(V2))) 7: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 8: U22#(tt(),V1) -> c_9(isNat#(activate(V1))) 9: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) 10: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) 11: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 12: U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) 13: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 14: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 15: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 16: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) ** Step 9.b:4: 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))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(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)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> c_14(isNatKind#(activate(N))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):2 2:W:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):3 3:W:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_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)),isNatKind#(activate(V1))):14 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):13 -->_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)),isNatKind#(activate(V1))):14 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):13 6:W:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_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)),isNatKind#(activate(V1))):14 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):13 8:W:U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 9:W:U51#(tt(),N) -> c_14(isNatKind#(activate(N))) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 10:W:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):11 11:W:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):14 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):13 -->_1 U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))):12 12:W:U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 13:W:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):1 14:W:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):6 15:W:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 -->_1 U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))):8 16:W:isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))):16 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):15 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 11: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 12: U63#(tt(),M,N) -> c_18(isNatKind#(activate(N))) 9: U51#(tt(),N) -> c_14(isNatKind#(activate(N))) 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 13: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1))) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 14: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(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)),isNatKind#(activate(V2))) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 16: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) 15: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 8: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) ** Step 9.b:5: 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(X1,X2) activate(n__s(X)) -> s(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_2/2,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/2 ,c_9/1,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/2,c_17/2,c_18/1,c_19/4,c_20/0,c_21/1,c_22/1,c_23/1 ,c_24/0,c_25/2,c_26/2,c_27/0,c_28/2,c_29/1,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^2))