WORST_CASE(?,O(n^2)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U101(tt(),M,N) -> U102(isNatKind(activate(M)),activate(M),activate(N)) U102(tt(),M,N) -> U103(isNat(activate(N)),activate(M),activate(N)) U103(tt(),M,N) -> U104(isNatKind(activate(N)),activate(M),activate(N)) U104(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() U71(tt(),N) -> U72(isNatKind(activate(N)),activate(N)) U72(tt(),N) -> activate(N) U81(tt(),M,N) -> U82(isNatKind(activate(M)),activate(M),activate(N)) U82(tt(),M,N) -> U83(isNat(activate(N)),activate(M),activate(N)) U83(tt(),M,N) -> U84(isNatKind(activate(N)),activate(M),activate(N)) U84(tt(),M,N) -> s(plus(activate(N),activate(M))) U91(tt(),N) -> U92(isNatKind(activate(N))) U92(tt()) -> 0() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(N,0()) -> U71(isNat(N),N) plus(N,s(M)) -> U81(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0()) -> U91(isNat(N),N) x(N,s(M)) -> U101(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U101,U102,U103,U104,U11,U12,U13,U14,U15,U16,U21,U22,U23 ,U31,U32,U33,U34,U35,U36,U41,U42,U51,U61,U62,U71,U72,U81,U82,U83,U84,U91,U92,activate,isNat,isNatKind,plus,s ,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. plus(N,0()) -> U71(isNat(N),N) plus(N,s(M)) -> U81(isNat(M),M,N) x(N,0()) -> U91(isNat(N),N) x(N,s(M)) -> U101(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() U101(tt(),M,N) -> U102(isNatKind(activate(M)),activate(M),activate(N)) U102(tt(),M,N) -> U103(isNat(activate(N)),activate(M),activate(N)) U103(tt(),M,N) -> U104(isNatKind(activate(N)),activate(M),activate(N)) U104(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() U71(tt(),N) -> U72(isNatKind(activate(N)),activate(N)) U72(tt(),N) -> activate(N) U81(tt(),M,N) -> U82(isNatKind(activate(M)),activate(M),activate(N)) U82(tt(),M,N) -> U83(isNat(activate(N)),activate(M),activate(N)) U83(tt(),M,N) -> U84(isNatKind(activate(N)),activate(M),activate(N)) U84(tt(),M,N) -> s(plus(activate(N),activate(M))) U91(tt(),N) -> U92(isNatKind(activate(N))) U92(tt()) -> 0() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U101,U102,U103,U104,U11,U12,U13,U14,U15,U16,U21,U22,U23 ,U31,U32,U33,U34,U35,U36,U41,U42,U51,U61,U62,U71,U72,U81,U82,U83,U84,U91,U92,activate,isNat,isNatKind,plus,s ,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 0#() -> c_1() U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_11() U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_14() U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U36#(tt()) -> c_20() U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U42#(tt()) -> c_22() U51#(tt()) -> c_23() U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U62#(tt()) -> c_25() U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U72#(tt(),N) -> c_27(activate#(N)) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) U92#(tt()) -> c_33(0#()) activate#(X) -> c_34() activate#(n__0()) -> c_35(0#()) activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) activate#(n__s(X)) -> c_37(s#(X)) activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) isNat#(n__0()) -> c_39() isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNatKind#(n__0()) -> c_43() isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) plus#(X1,X2) -> c_47() s#(X) -> c_48() x#(X1,X2) -> c_49() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 0#() -> c_1() U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_11() U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_14() U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U36#(tt()) -> c_20() U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U42#(tt()) -> c_22() U51#(tt()) -> c_23() U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U62#(tt()) -> c_25() U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U72#(tt(),N) -> c_27(activate#(N)) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) U92#(tt()) -> c_33(0#()) activate#(X) -> c_34() activate#(n__0()) -> c_35(0#()) activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) activate#(n__s(X)) -> c_37(s#(X)) activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) isNat#(n__0()) -> c_39() isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNatKind#(n__0()) -> c_43() isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) plus#(X1,X2) -> c_47() s#(X) -> c_48() x#(X1,X2) -> c_49() - Weak TRS: 0() -> n__0() U101(tt(),M,N) -> U102(isNatKind(activate(M)),activate(M),activate(N)) U102(tt(),M,N) -> U103(isNat(activate(N)),activate(M),activate(N)) U103(tt(),M,N) -> U104(isNatKind(activate(N)),activate(M),activate(N)) U104(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() U71(tt(),N) -> U72(isNatKind(activate(N)),activate(N)) U72(tt(),N) -> activate(N) U81(tt(),M,N) -> U82(isNatKind(activate(M)),activate(M),activate(N)) U82(tt(),M,N) -> U83(isNat(activate(N)),activate(M),activate(N)) U83(tt(),M,N) -> U84(isNatKind(activate(N)),activate(M),activate(N)) U84(tt(),M,N) -> s(plus(activate(N),activate(M))) U91(tt(),N) -> U92(isNatKind(activate(N))) U92(tt()) -> 0() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3 ,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0 ,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5 ,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) 0#() -> c_1() U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_11() U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_14() U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U36#(tt()) -> c_20() U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U42#(tt()) -> c_22() U51#(tt()) -> c_23() U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U62#(tt()) -> c_25() U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U72#(tt(),N) -> c_27(activate#(N)) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) U92#(tt()) -> c_33(0#()) activate#(X) -> c_34() activate#(n__0()) -> c_35(0#()) activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) activate#(n__s(X)) -> c_37(s#(X)) activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) isNat#(n__0()) -> c_39() isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNatKind#(n__0()) -> c_43() isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) plus#(X1,X2) -> c_47() s#(X) -> c_48() x#(X1,X2) -> c_49() * Step 4: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 0#() -> c_1() U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_11() U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_14() U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U36#(tt()) -> c_20() U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U42#(tt()) -> c_22() U51#(tt()) -> c_23() U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U62#(tt()) -> c_25() U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U72#(tt(),N) -> c_27(activate#(N)) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) U92#(tt()) -> c_33(0#()) activate#(X) -> c_34() activate#(n__0()) -> c_35(0#()) activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) activate#(n__s(X)) -> c_37(s#(X)) activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) isNat#(n__0()) -> c_39() isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNatKind#(n__0()) -> c_43() isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) plus#(X1,X2) -> c_47() s#(X) -> c_48() x#(X1,X2) -> c_49() - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3 ,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0 ,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5 ,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,11,14,20,22,23,25,34,39,43,47,48,49} by application of Pre({1,11,14,20,22,23,25,34,39,43,47,48,49}) = {2,3,4,5,6,7,8,9,10,12,13,15,16,17,18,19,21,24,26,27,28,29 ,30,31,32,33,35,36,37,38,40,41,42,44,45,46}. Here rules are labelled as follows: 1: 0#() -> c_1() 2: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 3: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 4: U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 5: U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 6: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 7: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 8: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 9: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 10: U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 11: U16#(tt()) -> c_11() 12: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 13: U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 14: U23#(tt()) -> c_14() 15: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 16: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 17: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 18: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 19: U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 20: U36#(tt()) -> c_20() 21: U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 22: U42#(tt()) -> c_22() 23: U51#(tt()) -> c_23() 24: U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 25: U62#(tt()) -> c_25() 26: U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 27: U72#(tt(),N) -> c_27(activate#(N)) 28: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 29: U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 30: U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 31: U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 32: U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) 33: U92#(tt()) -> c_33(0#()) 34: activate#(X) -> c_34() 35: activate#(n__0()) -> c_35(0#()) 36: activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) 37: activate#(n__s(X)) -> c_37(s#(X)) 38: activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) 39: isNat#(n__0()) -> c_39() 40: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 41: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 42: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 43: isNatKind#(n__0()) -> c_43() 44: isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 45: isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) 46: isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 47: plus#(X1,X2) -> c_47() 48: s#(X) -> c_48() 49: x#(X1,X2) -> c_49() * Step 5: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U72#(tt(),N) -> c_27(activate#(N)) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) U92#(tt()) -> c_33(0#()) activate#(n__0()) -> c_35(0#()) activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) activate#(n__s(X)) -> c_37(s#(X)) activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak DPs: 0#() -> c_1() U16#(tt()) -> c_11() U23#(tt()) -> c_14() U36#(tt()) -> c_20() U42#(tt()) -> c_22() U51#(tt()) -> c_23() U62#(tt()) -> c_25() activate#(X) -> c_34() isNat#(n__0()) -> c_39() isNatKind#(n__0()) -> c_43() plus#(X1,X2) -> c_47() s#(X) -> c_48() x#(X1,X2) -> c_49() - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3 ,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0 ,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5 ,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {26,27,28,29,30} by application of Pre({26,27,28,29,30}) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,31,32,33,34,35 ,36}. Here rules are labelled as follows: 1: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 2: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 3: U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 4: U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 5: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 6: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 7: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 8: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 9: U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 10: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 11: U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 12: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 13: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 14: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 15: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 16: U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 17: U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 18: U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 19: U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 20: U72#(tt(),N) -> c_27(activate#(N)) 21: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 22: U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 23: U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 24: U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 25: U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) 26: U92#(tt()) -> c_33(0#()) 27: activate#(n__0()) -> c_35(0#()) 28: activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) 29: activate#(n__s(X)) -> c_37(s#(X)) 30: activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) 31: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 32: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 33: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 34: isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 35: isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) 36: isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 37: 0#() -> c_1() 38: U16#(tt()) -> c_11() 39: U23#(tt()) -> c_14() 40: U36#(tt()) -> c_20() 41: U42#(tt()) -> c_22() 42: U51#(tt()) -> c_23() 43: U62#(tt()) -> c_25() 44: activate#(X) -> c_34() 45: isNat#(n__0()) -> c_39() 46: isNatKind#(n__0()) -> c_43() 47: plus#(X1,X2) -> c_47() 48: s#(X) -> c_48() 49: x#(X1,X2) -> c_49() * Step 6: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U72#(tt(),N) -> c_27(activate#(N)) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak DPs: 0#() -> c_1() U16#(tt()) -> c_11() U23#(tt()) -> c_14() U36#(tt()) -> c_20() U42#(tt()) -> c_22() U51#(tt()) -> c_23() U62#(tt()) -> c_25() U92#(tt()) -> c_33(0#()) activate#(X) -> c_34() activate#(n__0()) -> c_35(0#()) activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) activate#(n__s(X)) -> c_37(s#(X)) activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) isNat#(n__0()) -> c_39() isNatKind#(n__0()) -> c_43() plus#(X1,X2) -> c_47() s#(X) -> c_48() x#(X1,X2) -> c_49() - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3 ,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0 ,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5 ,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,20,24} by application of Pre({4,20,24}) = {3,19,23}. Here rules are labelled as follows: 1: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 2: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 3: U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 4: U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 5: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 6: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 7: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 8: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 9: U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 10: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 11: U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 12: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 13: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 14: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 15: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 16: U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 17: U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 18: U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 19: U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 20: U72#(tt(),N) -> c_27(activate#(N)) 21: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 22: U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 23: U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 24: U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 25: U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) 26: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 27: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 28: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 29: isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 30: isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) 31: isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 32: 0#() -> c_1() 33: U16#(tt()) -> c_11() 34: U23#(tt()) -> c_14() 35: U36#(tt()) -> c_20() 36: U42#(tt()) -> c_22() 37: U51#(tt()) -> c_23() 38: U62#(tt()) -> c_25() 39: U92#(tt()) -> c_33(0#()) 40: activate#(X) -> c_34() 41: activate#(n__0()) -> c_35(0#()) 42: activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) 43: activate#(n__s(X)) -> c_37(s#(X)) 44: activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) 45: isNat#(n__0()) -> c_39() 46: isNatKind#(n__0()) -> c_43() 47: plus#(X1,X2) -> c_47() 48: s#(X) -> c_48() 49: x#(X1,X2) -> c_49() * Step 7: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak DPs: 0#() -> c_1() U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) U16#(tt()) -> c_11() U23#(tt()) -> c_14() U36#(tt()) -> c_20() U42#(tt()) -> c_22() U51#(tt()) -> c_23() U62#(tt()) -> c_25() U72#(tt(),N) -> c_27(activate#(N)) U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U92#(tt()) -> c_33(0#()) activate#(X) -> c_34() activate#(n__0()) -> c_35(0#()) activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) activate#(n__s(X)) -> c_37(s#(X)) activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) isNat#(n__0()) -> c_39() isNatKind#(n__0()) -> c_43() plus#(X1,X2) -> c_47() s#(X) -> c_48() x#(X1,X2) -> c_49() - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3 ,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0 ,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5 ,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):2 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 2:S:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_1 U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):3 -->_2 isNat#(n__0()) -> c_39():45 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 3:S:U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_1 U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)):30 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 4:S:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):5 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 5:S:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):6 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 6:S:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 7:S:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_1 U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):8 -->_2 isNat#(n__0()) -> c_39():45 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 8:S:U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_2 isNat#(n__0()) -> c_39():45 -->_3 activate#(X) -> c_34():40 -->_1 U16#(tt()) -> c_11():31 9:S:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):10 -->_2 isNatKind#(n__0()) -> c_43():46 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 10:S:U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_2 isNat#(n__0()) -> c_39():45 -->_3 activate#(X) -> c_34():40 -->_1 U23#(tt()) -> c_14():32 11:S:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):12 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 12:S:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):13 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 13:S:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):14 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 14:S:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_1 U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):15 -->_2 isNat#(n__0()) -> c_39():45 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 15:S:U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_2 isNat#(n__0()) -> c_39():45 -->_3 activate#(X) -> c_34():40 -->_1 U36#(tt()) -> c_20():33 16:S:U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_2 isNatKind#(n__0()) -> c_43():46 -->_3 activate#(X) -> c_34():40 -->_1 U42#(tt()) -> c_22():34 17:S:U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_2 isNatKind#(n__0()) -> c_43():46 -->_3 activate#(X) -> c_34():40 -->_1 U62#(tt()) -> c_25():36 18:S:U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_1 U72#(tt(),N) -> c_27(activate#(N)):37 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_2 isNatKind#(n__0()) -> c_43():46 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 19:S:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):20 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 20:S:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_1 U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):21 -->_2 isNat#(n__0()) -> c_39():45 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 21:S:U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_1 U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):38 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 22:S:U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_1 U92#(tt()) -> c_33(0#()):39 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_2 isNatKind#(n__0()) -> c_43():46 -->_3 activate#(X) -> c_34():40 23:S:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 -->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):4 24:S:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_2 isNatKind#(n__0()) -> c_43():46 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 -->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):9 25:S:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_2 isNatKind#(n__0()) -> c_43():46 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 -->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):11 26:S:isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__0()) -> c_43():46 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):16 27:S:isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__0()) -> c_43():46 -->_3 activate#(X) -> c_34():40 -->_1 U51#(tt()) -> c_23():35 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 28:S:isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 isNatKind#(n__0()) -> c_43():46 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):17 29:W:0#() -> c_1() 30:W:U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_5 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_5 activate#(n__0()) -> c_35(0#()):41 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_2 x#(X1,X2) -> c_49():49 -->_1 plus#(X1,X2) -> c_47():47 -->_5 activate#(X) -> c_34():40 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 31:W:U16#(tt()) -> c_11() 32:W:U23#(tt()) -> c_14() 33:W:U36#(tt()) -> c_20() 34:W:U42#(tt()) -> c_22() 35:W:U51#(tt()) -> c_23() 36:W:U62#(tt()) -> c_25() 37:W:U72#(tt(),N) -> c_27(activate#(N)) -->_1 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_1 activate#(n__s(X)) -> c_37(s#(X)):43 -->_1 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_1 activate#(n__0()) -> c_35(0#()):41 -->_1 activate#(X) -> c_34():40 38:W:U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) -->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44 -->_4 activate#(n__s(X)) -> c_37(s#(X)):43 -->_3 activate#(n__s(X)) -> c_37(s#(X)):43 -->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42 -->_4 activate#(n__0()) -> c_35(0#()):41 -->_3 activate#(n__0()) -> c_35(0#()):41 -->_1 s#(X) -> c_48():48 -->_2 plus#(X1,X2) -> c_47():47 -->_4 activate#(X) -> c_34():40 -->_3 activate#(X) -> c_34():40 39:W:U92#(tt()) -> c_33(0#()) -->_1 0#() -> c_1():29 40:W:activate#(X) -> c_34() 41:W:activate#(n__0()) -> c_35(0#()) -->_1 0#() -> c_1():29 42:W:activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_47():47 43:W:activate#(n__s(X)) -> c_37(s#(X)) -->_1 s#(X) -> c_48():48 44:W:activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) -->_1 x#(X1,X2) -> c_49():49 45:W:isNat#(n__0()) -> c_39() 46:W:isNatKind#(n__0()) -> c_43() 47:W:plus#(X1,X2) -> c_47() 48:W:s#(X) -> c_48() 49:W:x#(X1,X2) -> c_49() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 39: U92#(tt()) -> c_33(0#()) 38: U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 37: U72#(tt(),N) -> c_27(activate#(N)) 30: U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 33: U36#(tt()) -> c_20() 32: U23#(tt()) -> c_14() 31: U16#(tt()) -> c_11() 45: isNat#(n__0()) -> c_39() 36: U62#(tt()) -> c_25() 34: U42#(tt()) -> c_22() 35: U51#(tt()) -> c_23() 40: activate#(X) -> c_34() 46: isNatKind#(n__0()) -> c_43() 41: activate#(n__0()) -> c_35(0#()) 29: 0#() -> c_1() 42: activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)) 47: plus#(X1,X2) -> c_47() 43: activate#(n__s(X)) -> c_37(s#(X)) 48: s#(X) -> c_48() 44: activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)) 49: x#(X1,X2) -> c_49() * Step 8: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3 ,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0 ,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5 ,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):2 2:S:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_1 U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):3 3:S:U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 4:S:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):5 5:S:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):6 6:S:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 7:S:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_1 U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):8 8:S:U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 9:S:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):10 10:S:U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 11:S:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):12 12:S:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):13 13:S:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):14 14:S:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_1 U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):15 15:S:U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 16:S:U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 17:S:U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 18:S:U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 19:S:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):20 20:S:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):23 -->_1 U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):21 21:S:U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 22:S:U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 23:S:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):4 24:S:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):9 25:S:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):11 26:S:isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):16 27:S:isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 28:S:isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):28 -->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):26 -->_1 U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):17 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) * Step 9: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1 ,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2 ,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,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: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) - Weak DPs: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1 ,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3 ,U14#/3,U15#/2,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1 ,U61#/2,U62#/1,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2 ,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0 ,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1 ,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15# ,U16#,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84# ,U91#,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} Problem (S) - Strict DPs: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1 ,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3 ,U14#/3,U15#/2,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1 ,U61#/2,U62#/1,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2 ,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0 ,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1 ,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15# ,U16#,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84# ,U91#,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} ** Step 9.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) - Weak DPs: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1 ,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2 ,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,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: 26: isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))) 27: isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) 28: isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) Consider the set of all dependency pairs 1: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 2: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 3: U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) 4: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 5: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 6: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 7: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 8: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 9: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 10: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 11: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 12: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) 13: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) 14: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 15: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 16: U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) 17: U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) 18: U71#(tt(),N) -> c_26(isNatKind#(activate(N))) 19: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 20: U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 21: U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) 22: U91#(tt(),N) -> c_32(isNatKind#(activate(N))) 23: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 24: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 25: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 26: isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 27: isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) 28: isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),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 {26,27,28} These cover all (indirect) predecessors of dependency pairs {1,2,3,16,17,18,19,20,21,22,26,27,28} 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: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) - Weak DPs: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1 ,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2 ,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,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}, uargs(c_6) = {1,2}, uargs(c_7) = {1,2}, uargs(c_8) = {1,2}, uargs(c_9) = {1,2}, uargs(c_10) = {1}, uargs(c_12) = {1,2}, uargs(c_13) = {1}, uargs(c_15) = {1,2}, uargs(c_16) = {1,2}, uargs(c_17) = {1,2}, uargs(c_18) = {1,2}, uargs(c_19) = {1}, uargs(c_21) = {1}, uargs(c_24) = {1}, uargs(c_26) = {1}, uargs(c_28) = {1,2}, uargs(c_29) = {1,2}, uargs(c_30) = {1}, uargs(c_32) = {1}, uargs(c_40) = {1,2}, uargs(c_41) = {1,2}, uargs(c_42) = {1,2}, uargs(c_44) = {1,2}, uargs(c_45) = {1}, uargs(c_46) = {1,2} Following symbols are considered usable: {0,U41,U42,U51,U61,U62,activate,isNatKind,plus,s,x,0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15# ,U16#,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84# ,U91#,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = 1 p(U101) = 0 p(U102) = 0 p(U103) = 0 p(U104) = 0 p(U11) = x1 p(U12) = 1 + x3 p(U13) = 0 p(U14) = x2 + x3 p(U15) = 1 p(U16) = x1 p(U21) = 0 p(U22) = 0 p(U23) = x1 + x1^2 p(U31) = x1*x2 + x2 p(U32) = x3 p(U33) = x1*x2 + x2*x3 + x3 p(U34) = x1^2 + x3 + x3^2 p(U35) = x1*x2 + x2 p(U36) = x1^2 p(U41) = x2 p(U42) = x1 p(U51) = 1 p(U61) = x1 p(U62) = 1 p(U71) = 0 p(U72) = 0 p(U81) = 0 p(U82) = 0 p(U83) = 0 p(U84) = 0 p(U91) = 0 p(U92) = 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(n__x) = 1 + x1 + x2 p(plus) = 1 + x1 + x2 p(s) = 1 + x1 p(tt) = 1 p(x) = 1 + x1 + x2 p(0#) = 0 p(U101#) = 1 + x1 + x1*x2 + x2 + x2*x3 + x3 + x3^2 p(U102#) = 1 + x2 + x3 + x3^2 p(U103#) = 1 + x3 p(U104#) = 0 p(U11#) = 1 + x1*x3 + x2 + x2*x3 + x2^2 + x3^2 p(U12#) = 1 + x1*x3 + x2^2 + x3 + x3^2 p(U13#) = 1 + x2^2 + x3 + x3^2 p(U14#) = 1 + x2^2 + x3^2 p(U15#) = x2^2 p(U16#) = 0 p(U21#) = 1 + x2 + x2^2 p(U22#) = 1 + x2^2 p(U23#) = 0 p(U31#) = 1 + x1*x3 + x2 + x2*x3 + x2^2 + x3 + x3^2 p(U32#) = x1*x3 + x2^2 + x3 + x3^2 p(U33#) = x2^2 + x3 + x3^2 p(U34#) = x2^2 + x3^2 p(U35#) = x2^2 p(U36#) = 0 p(U41#) = x2 p(U42#) = 0 p(U51#) = 0 p(U61#) = x2 p(U62#) = 0 p(U71#) = x2 p(U72#) = 0 p(U81#) = 1 + x1*x2 + x2 + x2*x3 + x2^2 + x3 + x3^2 p(U82#) = x1 + x1^2 + x2*x3 + x3 + x3^2 p(U83#) = x3 p(U84#) = 0 p(U91#) = x1*x2 p(U92#) = 0 p(activate#) = 0 p(isNat#) = x1^2 p(isNatKind#) = x1 p(plus#) = 0 p(s#) = 0 p(x#) = 0 p(c_1) = 0 p(c_2) = 1 + x1 + x2 p(c_3) = x1 + x2 p(c_4) = 1 + x1 p(c_5) = 0 p(c_6) = x1 + x2 p(c_7) = x1 + x2 p(c_8) = x1 + x2 p(c_9) = 1 + x1 + x2 p(c_10) = x1 p(c_11) = 0 p(c_12) = x1 + x2 p(c_13) = 1 + x1 p(c_14) = 0 p(c_15) = 1 + x1 + x2 p(c_16) = x1 + x2 p(c_17) = x1 + x2 p(c_18) = x1 + x2 p(c_19) = x1 p(c_20) = 0 p(c_21) = x1 p(c_22) = 0 p(c_23) = 0 p(c_24) = x1 p(c_25) = 0 p(c_26) = x1 p(c_27) = 0 p(c_28) = 1 + x1 + x2 p(c_29) = x1 + x2 p(c_30) = x1 p(c_31) = 0 p(c_32) = x1 p(c_33) = 0 p(c_34) = 0 p(c_35) = 0 p(c_36) = 0 p(c_37) = 0 p(c_38) = 0 p(c_39) = 0 p(c_40) = x1 + x2 p(c_41) = x1 + x2 p(c_42) = x1 + x2 p(c_43) = 0 p(c_44) = x1 + x2 p(c_45) = x1 p(c_46) = x1 + x2 p(c_47) = 0 p(c_48) = 0 p(c_49) = 0 Following rules are strictly oriented: isNatKind#(n__plus(V1,V2)) = 1 + V1 + V2 > V1 + V2 = c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) = 1 + V1 > V1 = c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) = 1 + V1 + V2 > V1 + V2 = c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) Following rules are (at-least) weakly oriented: U101#(tt(),M,N) = 2 + 2*M + M*N + N + N^2 >= 2 + 2*M + N + N^2 = c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U102#(tt(),M,N) = 1 + M + N + N^2 >= 1 + N + N^2 = c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) = 1 + N >= 1 + N = c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) = 1 + V1 + V1*V2 + V1^2 + V2 + V2^2 >= 1 + V1 + V1*V2 + V1^2 + V2 + V2^2 = c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) = 1 + V1^2 + 2*V2 + V2^2 >= 1 + V1^2 + 2*V2 + V2^2 = c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) = 1 + V1^2 + V2 + V2^2 >= 1 + V1^2 + V2 + V2^2 = c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) = 1 + V1^2 + V2^2 >= 1 + V1^2 + V2^2 = c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = V2^2 >= V2^2 = c_10(isNat#(activate(V2))) U21#(tt(),V1) = 1 + V1 + V1^2 >= 1 + V1 + V1^2 = c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) = 1 + V1^2 >= 1 + V1^2 = c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) = 1 + V1 + V1*V2 + V1^2 + 2*V2 + V2^2 >= 1 + V1 + V1*V2 + V1^2 + V2 + V2^2 = c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) = V1^2 + 2*V2 + V2^2 >= V1^2 + 2*V2 + V2^2 = c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) = V1^2 + V2 + V2^2 >= V1^2 + V2 + V2^2 = c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) = V1^2 + V2^2 >= V1^2 + V2^2 = c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) = V2^2 >= V2^2 = c_19(isNat#(activate(V2))) U41#(tt(),V2) = V2 >= V2 = c_21(isNatKind#(activate(V2))) U61#(tt(),V2) = V2 >= V2 = c_24(isNatKind#(activate(V2))) U71#(tt(),N) = N >= N = c_26(isNatKind#(activate(N))) U81#(tt(),M,N) = 1 + 2*M + M*N + M^2 + N + N^2 >= 1 + 2*M + M*N + M^2 + N + N^2 = c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) = 2 + M*N + N + N^2 >= N + N^2 = c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) = N >= N = c_30(isNatKind#(activate(N))) U91#(tt(),N) = N >= N = c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2 >= 1 + 2*V1 + 2*V1*V2 + V1^2 + V2^2 = c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) isNat#(n__s(V1)) = 1 + 2*V1 + V1^2 >= 1 + 2*V1 + V1^2 = c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2 >= 1 + 2*V1 + 2*V1*V2 + V1^2 + V2 + V2^2 = c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 0() = 1 >= 1 = n__0() U41(tt(),V2) = V2 >= V2 = U42(isNatKind(activate(V2))) U42(tt()) = 1 >= 1 = tt() U51(tt()) = 1 >= 1 = tt() U61(tt(),V2) = 1 >= 1 = U62(isNatKind(activate(V2))) U62(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) activate(n__x(X1,X2)) = 1 + X1 + X2 >= 1 + X1 + X2 = x(X1,X2) isNatKind(n__0()) = 1 >= 1 = tt() isNatKind(n__plus(V1,V2)) = 1 + V1 + V2 >= V2 = U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = 1 + V1 >= 1 = U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) = 1 + V1 + V2 >= V1 = U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) = 1 + X1 + X2 >= 1 + X1 + X2 = n__plus(X1,X2) s(X) = 1 + X >= 1 + X = n__s(X) x(X1,X2) = 1 + X1 + X2 >= 1 + X1 + X2 = n__x(X1,X2) *** Step 9.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) - Weak DPs: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1 ,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2 ,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 9.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1 ,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2 ,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):2 2:W:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):23 -->_1 U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))):3 3:W:U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 4:W:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):5 5:W:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):6 6:W:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 7:W:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):23 -->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):8 8:W:U15#(tt(),V2) -> c_10(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):25 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):23 9:W:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):10 10:W:U22#(tt(),V1) -> c_13(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):25 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):23 11:W:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):12 12:W:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):13 13:W:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):14 14:W:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):23 -->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):15 15:W:U35#(tt(),V2) -> c_19(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):25 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):23 16:W:U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 17:W:U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 18:W:U71#(tt(),N) -> c_26(isNatKind#(activate(N))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 19:W:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):20 20:W:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):25 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):23 -->_1 U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))):21 21:W:U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 22:W:U91#(tt(),N) -> c_32(isNatKind#(activate(N))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 23:W:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):4 24:W:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):9 25:W:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):11 26:W:isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))):16 27:W:isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 28:W:isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))):17 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 22: U91#(tt(),N) -> c_32(isNatKind#(activate(N))) 19: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 20: U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 21: U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) 18: U71#(tt(),N) -> c_26(isNatKind#(activate(N))) 1: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 2: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 3: U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) 25: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 15: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 14: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 13: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) 12: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) 11: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 10: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 9: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 24: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 8: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 7: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 5: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 4: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 23: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 28: isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 27: isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) 26: isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 17: U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) 16: U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) *** Step 9.a: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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1 ,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2 ,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 9.b:1: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1 ,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2 ,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,15,18,19} by application of Pre({2,15,18,19}) = {1,17}. Here rules are labelled as follows: 1: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 2: U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) 3: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) 4: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 5: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) 6: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 7: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 8: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 9: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 10: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 11: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) 12: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) 13: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 14: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 15: U71#(tt(),N) -> c_26(isNatKind#(activate(N))) 16: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 17: U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) 18: U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) 19: U91#(tt(),N) -> c_32(isNatKind#(activate(N))) 20: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 21: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) 22: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) 23: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) 24: U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) 25: U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) 26: isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 27: isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) 28: isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) ** Step 9.b:2: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) U71#(tt(),N) -> c_26(isNatKind#(activate(N))) U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) U91#(tt(),N) -> c_32(isNatKind#(activate(N))) isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1 ,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2 ,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) -->_1 U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))):20 -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 2:S:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):3 3:S:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):4 4:S:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 5:S:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 -->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):6 6:S:U15#(tt(),V2) -> c_10(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 7:S:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):8 8:S:U22#(tt(),V1) -> c_13(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 9:S:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):10 10:S:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):11 11:S:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):12 12:S:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 -->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):13 13:S:U35#(tt(),V2) -> c_19(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 14:S:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):15 15:S:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) -->_1 U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))):24 -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 16:S:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):2 17:S:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):7 18:S:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):9 19:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):1 20:W:U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 21:W:U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 22:W:U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 23:W:U71#(tt(),N) -> c_26(isNatKind#(activate(N))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 24:W:U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 25:W:U91#(tt(),N) -> c_32(isNatKind#(activate(N))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 26:W:isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))):21 27:W:isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) -->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 28:W:isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) -->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):28 -->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27 -->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1))):26 -->_1 U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))):22 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 25: U91#(tt(),N) -> c_32(isNatKind#(activate(N))) 23: U71#(tt(),N) -> c_26(isNatKind#(activate(N))) 24: U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))) 20: U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))) 28: isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 27: isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))) 26: isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 22: U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))) 21: U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))) ** Step 9.b:3: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1 ,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2 ,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 2:S:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):3 3:S:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):4 4:S:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 5:S:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 -->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):6 6:S:U15#(tt(),V2) -> c_10(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 7:S:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):8 8:S:U22#(tt(),V1) -> c_13(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 9:S:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):10 10:S:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))):11 11:S:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2))) -->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):12 12:S:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 -->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):13 13:S:U35#(tt(),V2) -> c_19(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 14:S:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) -->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):15 15:S:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):18 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):16 16:S:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):2 17:S:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))) -->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):7 18:S:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))) -->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1))):9 19:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M))) -->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) ** Step 9.b:4: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,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: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1 ,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3 ,U14#/3,U15#/2,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1 ,U61#/2,U62#/1,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1 ,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0 ,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1 ,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15# ,U16#,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84# ,U91#,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} Problem (S) - Strict DPs: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1 ,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3 ,U14#/3,U15#/2,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1 ,U61#/2,U62#/1,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1 ,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0 ,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1 ,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15# ,U16#,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84# ,U91#,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} *** Step 9.b:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 12: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Consider the set of all dependency pairs 1: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 2: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 3: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 4: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 5: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 7: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 8: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 9: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 10: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 11: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 12: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 13: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 14: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 15: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) 16: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 17: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 18: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 19: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) Processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1,12} These cover all (indirect) predecessors of dependency pairs {1,12,13,14,15,19} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 9.b:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1,2}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1,2}, uargs(c_19) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_40) = {1}, uargs(c_41) = {1}, uargs(c_42) = {1} Following symbols are considered usable: {0,activate,plus,s,x,0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91#,U92#,activate#,isNat# ,isNatKind#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] [0] p(U101) = [0] [0] p(U102) = [0] [0] p(U103) = [0] [0] p(U104) = [0] [0] p(U11) = [0 0] x1 + [0] [0 1] [0] p(U12) = [0 0] x2 + [1 0] x3 + [1] [1 0] [0 0] [0] p(U13) = [1] [0] p(U14) = [0 1] x1 + [0] [0 0] [0] p(U15) = [0] [0] p(U16) = [0] [0] p(U21) = [0] [1] p(U22) = [0 0] x1 + [0] [0 1] [1] p(U23) = [0] [1] p(U31) = [0] [0] p(U32) = [0 0] x1 + [1 0] x2 + [0 1] x3 + [0] [0 1] [0 0] [0 1] [1] p(U33) = [0] [0] p(U34) = [0 1] x1 + [0 0] x3 + [0] [0 0] [1 0] [0] p(U35) = [0] [0] p(U36) = [0] [1] p(U41) = [1 0] x2 + [1] [1 0] [0] p(U42) = [0] [1] p(U51) = [0 1] x1 + [0] [0 0] [1] p(U61) = [0 0] x2 + [0] [0 1] [0] p(U62) = [1] [1] p(U71) = [0] [0] p(U72) = [0] [0] p(U81) = [0] [0] p(U82) = [0] [0] p(U83) = [0] [0] p(U84) = [0] [0] p(U91) = [0] [0] p(U92) = [0] [0] p(activate) = [1 0] x1 + [0] [0 1] [0] p(isNat) = [0 0] x1 + [0] [0 1] [0] p(isNatKind) = [0 0] x1 + [0] [0 1] [0] p(n__0) = [0] [0] p(n__plus) = [1 1] x1 + [1 1] x2 + [0] [0 0] [0 0] [1] p(n__s) = [1 1] x1 + [0] [0 0] [0] p(n__x) = [1 0] x1 + [1 1] x2 + [1] [0 0] [0 0] [0] p(plus) = [1 1] x1 + [1 1] x2 + [0] [0 0] [0 0] [1] p(s) = [1 1] x1 + [0] [0 0] [0] p(tt) = [0] [0] p(x) = [1 0] x1 + [1 1] x2 + [1] [0 0] [0 0] [0] p(0#) = [0] [0] p(U101#) = [1 1] x3 + [1] [1 0] [1] p(U102#) = [1 0] x3 + [1] [1 0] [1] p(U103#) = [0] [0] p(U104#) = [0] [0] p(U11#) = [1 1] x2 + [1 1] x3 + [0] [1 0] [0 0] [1] p(U12#) = [1 1] x2 + [1 1] x3 + [0] [0 1] [0 1] [0] p(U13#) = [1 1] x2 + [1 1] x3 + [0] [0 0] [0 0] [0] p(U14#) = [1 0] x2 + [1 1] x3 + [0] [0 1] [0 0] [0] p(U15#) = [1 1] x2 + [0] [0 0] [0] p(U16#) = [0] [0] p(U21#) = [1 1] x2 + [0] [0 0] [0] p(U22#) = [1 0] x2 + [0] [0 1] [0] p(U23#) = [0] [0] p(U31#) = [1 0] x2 + [1 1] x3 + [1] [0 0] [1 0] [1] p(U32#) = [1 0] x2 + [1 1] x3 + [1] [0 0] [0 0] [0] p(U33#) = [1 0] x2 + [1 1] x3 + [1] [0 0] [1 1] [1] p(U34#) = [1 0] x2 + [1 1] x3 + [1] [0 0] [1 1] [1] p(U35#) = [1 0] x2 + [0] [1 0] [1] p(U36#) = [0] [0] p(U41#) = [0] [0] p(U42#) = [0] [0] p(U51#) = [0] [0] p(U61#) = [0] [0] p(U62#) = [0] [0] p(U71#) = [0] [0] p(U72#) = [0] [0] p(U81#) = [1 1] x3 + [1] [0 0] [1] p(U82#) = [1 0] x3 + [0] [1 0] [1] p(U83#) = [0] [0] p(U84#) = [0] [0] p(U91#) = [0] [0] p(U92#) = [0] [0] p(activate#) = [0] [0] p(isNat#) = [1 0] x1 + [0] [0 0] [0] p(isNatKind#) = [0] [0] p(plus#) = [0] [0] p(s#) = [0] [0] p(x#) = [0] [0] p(c_1) = [0] [0] p(c_2) = [1 0] x1 + [0] [0 1] [0] p(c_3) = [1 0] x1 + [0] [0 0] [1] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [1 0] x1 + [0] [0 0] [0] p(c_7) = [1 0] x1 + [0] [0 0] [0] p(c_8) = [1 1] x1 + [0] [0 0] [0] p(c_9) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_10) = [1 0] x1 + [0] [0 0] [0] p(c_11) = [0] [0] p(c_12) = [1 1] x1 + [0] [0 0] [0] p(c_13) = [1 0] x1 + [0] [0 0] [0] p(c_14) = [0] [0] p(c_15) = [1 0] x1 + [0] [0 0] [0] p(c_16) = [1 0] x1 + [0] [0 0] [0] p(c_17) = [1 0] x1 + [0] [0 1] [0] p(c_18) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] p(c_19) = [1 0] x1 + [0] [1 0] [1] p(c_20) = [0] [0] p(c_21) = [0] [0] p(c_22) = [0] [0] p(c_23) = [0] [0] p(c_24) = [0] [0] p(c_25) = [0] [0] p(c_26) = [0] [0] p(c_27) = [0] [0] p(c_28) = [1 0] x1 + [1] [0 0] [0] p(c_29) = [1 0] x1 + [0] [0 0] [0] p(c_30) = [0] [0] p(c_31) = [0] [0] p(c_32) = [0] [0] p(c_33) = [0] [0] p(c_34) = [0] [0] p(c_35) = [0] [0] p(c_36) = [0] [0] p(c_37) = [0] [0] p(c_38) = [0] [0] p(c_39) = [0] [0] p(c_40) = [1 0] x1 + [0] [0 0] [0] p(c_41) = [1 0] x1 + [0] [0 0] [0] p(c_42) = [1 0] x1 + [0] [0 0] [0] p(c_43) = [0] [0] p(c_44) = [0] [0] p(c_45) = [0] [0] p(c_46) = [0] [0] p(c_47) = [0] [0] p(c_48) = [0] [0] p(c_49) = [0] [0] Following rules are strictly oriented: U102#(tt(),M,N) = [1 0] N + [1] [1 0] [1] > [1 0] N + [0] [0 0] [1] = c_3(isNat#(activate(N))) U34#(tt(),V1,V2) = [1 0] V1 + [1 1] V2 + [1] [0 0] [1 1] [1] > [1 0] V1 + [1 0] V2 + [0] [0 0] [1 0] [1] = c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U101#(tt(),M,N) = [1 1] N + [1] [1 0] [1] >= [1 0] N + [1] [1 0] [1] = c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U11#(tt(),V1,V2) = [1 1] V1 + [1 1] V2 + [0] [1 0] [0 0] [1] >= [1 1] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] = c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [1 1] V1 + [1 1] V2 + [0] [0 1] [0 1] [0] >= [1 1] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] = c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [1 1] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] >= [1 1] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] = c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [1 0] V1 + [1 1] V2 + [0] [0 1] [0 0] [0] >= [1 0] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] = c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [1 1] V2 + [0] [0 0] [0] >= [1 0] V2 + [0] [0 0] [0] = c_10(isNat#(activate(V2))) U21#(tt(),V1) = [1 1] V1 + [0] [0 0] [0] >= [1 1] V1 + [0] [0 0] [0] = c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [1 0] V1 + [0] [0 1] [0] >= [1 0] V1 + [0] [0 0] [0] = c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) = [1 0] V1 + [1 1] V2 + [1] [0 0] [1 0] [1] >= [1 0] V1 + [1 1] V2 + [1] [0 0] [0 0] [0] = c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) = [1 0] V1 + [1 1] V2 + [1] [0 0] [0 0] [0] >= [1 0] V1 + [1 1] V2 + [1] [0 0] [0 0] [0] = c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) = [1 0] V1 + [1 1] V2 + [1] [0 0] [1 1] [1] >= [1 0] V1 + [1 1] V2 + [1] [0 0] [1 1] [1] = c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U35#(tt(),V2) = [1 0] V2 + [0] [1 0] [1] >= [1 0] V2 + [0] [1 0] [1] = c_19(isNat#(activate(V2))) U81#(tt(),M,N) = [1 1] N + [1] [0 0] [1] >= [1 0] N + [1] [0 0] [0] = c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) = [1 0] N + [0] [1 0] [1] >= [1 0] N + [0] [0 0] [0] = c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [1 1] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] >= [1 1] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] = c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [1 1] V1 + [0] [0 0] [0] >= [1 1] V1 + [0] [0 0] [0] = c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) = [1 0] V1 + [1 1] V2 + [1] [0 0] [0 0] [0] >= [1 0] V1 + [1 1] V2 + [1] [0 0] [0 0] [0] = c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 0() = [0] [0] >= [0] [0] = n__0() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [0] [0] >= [0] [0] = 0() activate(n__plus(X1,X2)) = [1 1] X1 + [1 1] X2 + [0] [0 0] [0 0] [1] >= [1 1] X1 + [1 1] X2 + [0] [0 0] [0 0] [1] = plus(X1,X2) activate(n__s(X)) = [1 1] X + [0] [0 0] [0] >= [1 1] X + [0] [0 0] [0] = s(X) activate(n__x(X1,X2)) = [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] >= [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] = x(X1,X2) plus(X1,X2) = [1 1] X1 + [1 1] X2 + [0] [0 0] [0 0] [1] >= [1 1] X1 + [1 1] X2 + [0] [0 0] [0 0] [1] = n__plus(X1,X2) s(X) = [1 1] X + [0] [0 0] [0] >= [1 1] X + [0] [0 0] [0] = n__s(X) x(X1,X2) = [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] >= [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] = n__x(X1,X2) **** Step 9.b:4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 9.b:4.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 6: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) Consider the set of all dependency pairs 1: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 2: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 3: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 4: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 5: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 6: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 7: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 8: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 9: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 10: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 11: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 12: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 13: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 14: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) 15: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 16: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 17: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 18: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 19: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) Processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1,6} These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5,6,7,14,15,18,19} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 9.b:4.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1,2}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1,2}, uargs(c_19) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_40) = {1}, uargs(c_41) = {1}, uargs(c_42) = {1} Following symbols are considered usable: {0,activate,plus,s,x,0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91#,U92#,activate#,isNat# ,isNatKind#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] [1] p(U101) = [0] [0] p(U102) = [0] [0] p(U103) = [0] [0] p(U104) = [0] [0] p(U11) = [1] [0] p(U12) = [1 0] x2 + [0] [1 1] [0] p(U13) = [1 0] x2 + [0] [0 0] [1] p(U14) = [0 0] x2 + [1 1] x3 + [1] [0 1] [0 0] [0] p(U15) = [0] [1] p(U16) = [0] [1] p(U21) = [0] [0] p(U22) = [1 0] x2 + [1] [0 0] [0] p(U23) = [0] [0] p(U31) = [0 0] x2 + [0] [1 0] [1] p(U32) = [1 0] x2 + [1] [0 0] [1] p(U33) = [0 0] x2 + [0 0] x3 + [1] [1 0] [1 0] [0] p(U34) = [1 0] x3 + [0] [0 0] [0] p(U35) = [1] [1] p(U36) = [0 1] x1 + [1] [0 0] [0] p(U41) = [1] [0] p(U42) = [0] [0] p(U51) = [0] [0] p(U61) = [0 1] x2 + [0] [0 0] [1] p(U62) = [0] [0] p(U71) = [0] [0] p(U72) = [0] [0] p(U81) = [0] [0] p(U82) = [0] [0] p(U83) = [0] [0] p(U84) = [0] [0] p(U91) = [0] [0] p(U92) = [0] [0] p(activate) = [1 0] x1 + [0] [0 1] [0] p(isNat) = [0] [1] p(isNatKind) = [0] [0] p(n__0) = [0] [1] p(n__plus) = [0 0] x1 + [0 0] x2 + [1] [0 1] [0 1] [1] p(n__s) = [0 0] x1 + [1] [0 1] [1] p(n__x) = [0 0] x1 + [0 0] x2 + [1] [0 1] [0 1] [0] p(plus) = [0 0] x1 + [0 0] x2 + [1] [0 1] [0 1] [1] p(s) = [0 0] x1 + [1] [0 1] [1] p(tt) = [0] [0] p(x) = [0 0] x1 + [0 0] x2 + [1] [0 1] [0 1] [0] p(0#) = [0] [0] p(U101#) = [0 0] x2 + [1 1] x3 + [0] [1 1] [1 1] [1] p(U102#) = [0 0] x2 + [1 1] x3 + [0] [1 0] [0 1] [1] p(U103#) = [0] [0] p(U104#) = [0] [0] p(U11#) = [0 1] x2 + [0 1] x3 + [1] [1 1] [0 1] [1] p(U12#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [0 0] [1] p(U13#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [0 0] [0] p(U14#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [0 0] [1] p(U15#) = [0 1] x2 + [0] [1 0] [1] p(U16#) = [0] [0] p(U21#) = [0 1] x2 + [1] [1 1] [1] p(U22#) = [0 1] x2 + [0] [0 1] [0] p(U23#) = [0] [0] p(U31#) = [0 1] x2 + [0 1] x3 + [0] [0 1] [1 0] [0] p(U32#) = [0 1] x2 + [0 1] x3 + [0] [0 1] [0 0] [0] p(U33#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [1 0] [0] p(U34#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [0 1] [1] p(U35#) = [0 1] x2 + [0] [0 1] [0] p(U36#) = [0] [0] p(U41#) = [0] [0] p(U42#) = [0] [0] p(U51#) = [0] [0] p(U61#) = [0] [0] p(U62#) = [0] [0] p(U71#) = [0] [0] p(U72#) = [0] [0] p(U81#) = [1 1] x3 + [1] [1 0] [1] p(U82#) = [0 0] x2 + [0 1] x3 + [1] [0 1] [1 0] [1] p(U83#) = [0] [0] p(U84#) = [0] [0] p(U91#) = [0] [0] p(U92#) = [0] [0] p(activate#) = [0] [0] p(isNat#) = [0 1] x1 + [0] [0 0] [1] p(isNatKind#) = [0] [0] p(plus#) = [0] [0] p(s#) = [0] [0] p(x#) = [0] [0] p(c_1) = [0] [0] p(c_2) = [1 0] x1 + [0] [1 0] [0] p(c_3) = [1 0] x1 + [0] [0 0] [0] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [1 0] x1 + [0] [1 0] [1] p(c_7) = [1 0] x1 + [0] [0 0] [1] p(c_8) = [1 0] x1 + [0] [0 0] [0] p(c_9) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [1] p(c_10) = [1 0] x1 + [0] [0 1] [0] p(c_11) = [0] [0] p(c_12) = [1 0] x1 + [0] [0 1] [1] p(c_13) = [1 0] x1 + [0] [0 0] [0] p(c_14) = [0] [0] p(c_15) = [1 0] x1 + [0] [0 1] [0] p(c_16) = [1 0] x1 + [0] [0 0] [0] p(c_17) = [1 0] x1 + [0] [0 0] [0] p(c_18) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] p(c_19) = [1 0] x1 + [0] [1 0] [0] p(c_20) = [0] [0] p(c_21) = [0] [0] p(c_22) = [0] [0] p(c_23) = [0] [0] p(c_24) = [0] [0] p(c_25) = [0] [0] p(c_26) = [0] [0] p(c_27) = [0] [0] p(c_28) = [1 0] x1 + [0] [0 0] [0] p(c_29) = [1 0] x1 + [0] [0 0] [0] p(c_30) = [0] [0] p(c_31) = [0] [0] p(c_32) = [0] [0] p(c_33) = [0] [0] p(c_34) = [0] [0] p(c_35) = [0] [0] p(c_36) = [0] [0] p(c_37) = [0] [0] p(c_38) = [0] [0] p(c_39) = [0] [0] p(c_40) = [1 0] x1 + [0] [0 0] [0] p(c_41) = [1 0] x1 + [0] [0 0] [1] p(c_42) = [1 0] x1 + [0] [0 0] [0] p(c_43) = [0] [0] p(c_44) = [0] [0] p(c_45) = [0] [0] p(c_46) = [0] [0] p(c_47) = [0] [0] p(c_48) = [0] [0] p(c_49) = [0] [0] Following rules are strictly oriented: U11#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [1 1] [0 1] [1] > [0 1] V1 + [0 1] V2 + [0] [0 1] [0 1] [1] = c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U21#(tt(),V1) = [0 1] V1 + [1] [1 1] [1] > [0 1] V1 + [0] [0 1] [1] = c_12(U22#(isNatKind(activate(V1)),activate(V1))) Following rules are (at-least) weakly oriented: U101#(tt(),M,N) = [0 0] M + [1 1] N + [0] [1 1] [1 1] [1] >= [1 1] N + [0] [1 1] [0] = c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) = [0 0] M + [1 1] N + [0] [1 0] [0 1] [1] >= [0 1] N + [0] [0 0] [0] = c_3(isNat#(activate(N))) U12#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] = c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] = c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [0 1] V2 + [0] [1 0] [1] >= [0 1] V2 + [0] [0 0] [1] = c_10(isNat#(activate(V2))) U22#(tt(),V1) = [0 1] V1 + [0] [0 1] [0] >= [0 1] V1 + [0] [0 0] [0] = c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 1] [1 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 1] [0 0] [0] = c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 1] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [1 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 1] [1] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 1] [0] = c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) = [0 1] V2 + [0] [0 1] [0] >= [0 1] V2 + [0] [0 1] [0] = c_19(isNat#(activate(V2))) U81#(tt(),M,N) = [1 1] N + [1] [1 0] [1] >= [0 1] N + [1] [0 0] [0] = c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) = [0 0] M + [0 1] N + [1] [0 1] [1 0] [1] >= [0 1] N + [0] [0 0] [0] = c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] = c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [0 1] V1 + [1] [0 0] [1] >= [0 1] V1 + [1] [0 0] [1] = c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 0() = [0] [1] >= [0] [1] = n__0() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [0] [1] >= [0] [1] = 0() activate(n__plus(X1,X2)) = [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] >= [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] = plus(X1,X2) activate(n__s(X)) = [0 0] X + [1] [0 1] [1] >= [0 0] X + [1] [0 1] [1] = s(X) activate(n__x(X1,X2)) = [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [0] >= [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [0] = x(X1,X2) plus(X1,X2) = [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] >= [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] = n__plus(X1,X2) s(X) = [0 0] X + [1] [0 1] [1] >= [0 0] X + [1] [0 1] [1] = n__s(X) x(X1,X2) = [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [0] >= [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [0] = n__x(X1,X2) ***** Step 9.b:4.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 9.b:4.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) Consider the set of all dependency pairs 1: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 2: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 3: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 4: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 5: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 6: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 7: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) 8: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 9: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 10: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 11: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 12: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 13: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 14: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 15: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 16: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 17: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 18: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 19: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) Processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {3} These cover all (indirect) predecessors of dependency pairs {3,7,8,16,17,18,19} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 9.b:4.a:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1,2}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1,2}, uargs(c_19) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_40) = {1}, uargs(c_41) = {1}, uargs(c_42) = {1} Following symbols are considered usable: {0,activate,plus,s,x,0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91#,U92#,activate#,isNat# ,isNatKind#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] [1] p(U101) = [0] [0] p(U102) = [0] [0] p(U103) = [0] [0] p(U104) = [0] [0] p(U11) = [0] [1] p(U12) = [1 0] x3 + [0] [0 0] [0] p(U13) = [0 0] x2 + [0] [0 1] [0] p(U14) = [0 0] x2 + [0 0] x3 + [0] [0 1] [0 1] [1] p(U15) = [1] [1] p(U16) = [1] [1] p(U21) = [1] [0] p(U22) = [0] [1] p(U23) = [0 1] x1 + [0] [0 0] [0] p(U31) = [1 1] x2 + [0] [0 0] [1] p(U32) = [1 0] x2 + [0] [1 0] [0] p(U33) = [1] [1] p(U34) = [1] [0] p(U35) = [0 0] x1 + [0] [1 1] [1] p(U36) = [0 1] x1 + [0] [1 1] [1] p(U41) = [1] [0] p(U42) = [1] [1] p(U51) = [0] [0] p(U61) = [1] [1] p(U62) = [1] [0] p(U71) = [0] [0] p(U72) = [0] [0] p(U81) = [0] [0] p(U82) = [0] [0] p(U83) = [0] [0] p(U84) = [0] [0] p(U91) = [0] [0] p(U92) = [0] [0] p(activate) = [1 0] x1 + [0] [0 1] [0] p(isNat) = [0 0] x1 + [1] [0 1] [1] p(isNatKind) = [0] [0] p(n__0) = [0] [1] p(n__plus) = [0 0] x1 + [0 0] x2 + [1] [0 1] [0 1] [1] p(n__s) = [0 0] x1 + [0] [0 1] [0] p(n__x) = [0 0] x1 + [0 0] x2 + [0] [0 1] [0 1] [1] p(plus) = [0 0] x1 + [0 0] x2 + [1] [0 1] [0 1] [1] p(s) = [0 0] x1 + [0] [0 1] [0] p(tt) = [0] [0] p(x) = [0 0] x1 + [0 0] x2 + [0] [0 1] [0 1] [1] p(0#) = [0] [0] p(U101#) = [1 1] x2 + [1 1] x3 + [1] [1 0] [1 1] [0] p(U102#) = [1 0] x2 + [0 1] x3 + [0] [0 0] [1 0] [1] p(U103#) = [0] [0] p(U104#) = [0] [0] p(U11#) = [0 1] x2 + [0 1] x3 + [1] [0 0] [0 0] [1] p(U12#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [0 0] [1] p(U13#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [0 0] [0] p(U14#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [0 0] [0] p(U15#) = [0 1] x2 + [0] [0 1] [0] p(U16#) = [0] [0] p(U21#) = [0 1] x2 + [0] [0 0] [0] p(U22#) = [0 1] x2 + [0] [0 0] [1] p(U23#) = [0] [0] p(U31#) = [0 1] x2 + [0 1] x3 + [1] [0 0] [0 0] [1] p(U32#) = [0 1] x2 + [0 1] x3 + [1] [0 0] [0 0] [0] p(U33#) = [0 1] x2 + [0 1] x3 + [1] [0 0] [0 0] [1] p(U34#) = [0 1] x2 + [0 1] x3 + [0] [0 1] [0 0] [0] p(U35#) = [0 1] x2 + [0] [0 1] [0] p(U36#) = [0] [0] p(U41#) = [0] [0] p(U42#) = [0] [0] p(U51#) = [0] [0] p(U61#) = [0] [0] p(U62#) = [0] [0] p(U71#) = [0] [0] p(U72#) = [0] [0] p(U81#) = [1 0] x2 + [1 1] x3 + [1] [1 1] [0 1] [1] p(U82#) = [0 0] x2 + [1 1] x3 + [1] [1 0] [0 0] [1] p(U83#) = [0] [0] p(U84#) = [0] [0] p(U91#) = [0] [0] p(U92#) = [0] [0] p(activate#) = [0] [0] p(isNat#) = [0 1] x1 + [0] [1 0] [0] p(isNatKind#) = [0] [0] p(plus#) = [0] [0] p(s#) = [0] [0] p(x#) = [0] [0] p(c_1) = [0] [0] p(c_2) = [1 0] x1 + [1] [1 0] [0] p(c_3) = [1 0] x1 + [0] [0 0] [1] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [1 0] x1 + [0] [0 0] [1] p(c_7) = [1 0] x1 + [0] [0 0] [1] p(c_8) = [1 0] x1 + [0] [0 0] [0] p(c_9) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_10) = [1 0] x1 + [0] [0 0] [0] p(c_11) = [0] [0] p(c_12) = [1 0] x1 + [0] [0 0] [0] p(c_13) = [1 0] x1 + [0] [0 0] [1] p(c_14) = [0] [0] p(c_15) = [1 0] x1 + [0] [0 0] [0] p(c_16) = [1 0] x1 + [0] [0 0] [0] p(c_17) = [1 0] x1 + [0] [0 0] [0] p(c_18) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_19) = [1 0] x1 + [0] [0 0] [0] p(c_20) = [0] [0] p(c_21) = [0] [0] p(c_22) = [0] [0] p(c_23) = [0] [0] p(c_24) = [0] [0] p(c_25) = [0] [0] p(c_26) = [0] [0] p(c_27) = [0] [0] p(c_28) = [1 0] x1 + [0] [0 0] [0] p(c_29) = [1 1] x1 + [0] [0 0] [0] p(c_30) = [0] [0] p(c_31) = [0] [0] p(c_32) = [0] [0] p(c_33) = [0] [0] p(c_34) = [0] [0] p(c_35) = [0] [0] p(c_36) = [0] [0] p(c_37) = [0] [0] p(c_38) = [0] [0] p(c_39) = [0] [0] p(c_40) = [1 0] x1 + [0] [0 1] [0] p(c_41) = [1 0] x1 + [0] [0 0] [0] p(c_42) = [1 0] x1 + [0] [0 0] [0] p(c_43) = [0] [0] p(c_44) = [0] [0] p(c_45) = [0] [0] p(c_46) = [0] [0] p(c_47) = [0] [0] p(c_48) = [0] [0] p(c_49) = [0] [0] Following rules are strictly oriented: U33#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [1] > [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) Following rules are (at-least) weakly oriented: U101#(tt(),M,N) = [1 1] M + [1 1] N + [1] [1 0] [1 1] [0] >= [1 0] M + [0 1] N + [1] [1 0] [0 1] [0] = c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) = [1 0] M + [0 1] N + [0] [0 0] [1 0] [1] >= [0 1] N + [0] [0 0] [1] = c_3(isNat#(activate(N))) U11#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] = c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] = c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [0 1] V2 + [0] [0 1] [0] >= [0 1] V2 + [0] [0 0] [0] = c_10(isNat#(activate(V2))) U21#(tt(),V1) = [0 1] V1 + [0] [0 0] [0] >= [0 1] V1 + [0] [0 0] [0] = c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [0 1] V1 + [0] [0 0] [1] >= [0 1] V1 + [0] [0 0] [1] = c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] = c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] = c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 1] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) = [0 1] V2 + [0] [0 1] [0] >= [0 1] V2 + [0] [0 0] [0] = c_19(isNat#(activate(V2))) U81#(tt(),M,N) = [1 0] M + [1 1] N + [1] [1 1] [0 1] [1] >= [1 1] N + [1] [0 0] [0] = c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) = [0 0] M + [1 1] N + [1] [1 0] [0 0] [1] >= [1 1] N + [0] [0 0] [0] = c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [1] = c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [0 1] V1 + [0] [0 0] [0] >= [0 1] V1 + [0] [0 0] [0] = c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) = [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [1] [0 0] [0 0] [0] = c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 0() = [0] [1] >= [0] [1] = n__0() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [0] [1] >= [0] [1] = 0() activate(n__plus(X1,X2)) = [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] >= [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] = plus(X1,X2) activate(n__s(X)) = [0 0] X + [0] [0 1] [0] >= [0 0] X + [0] [0 1] [0] = s(X) activate(n__x(X1,X2)) = [0 0] X1 + [0 0] X2 + [0] [0 1] [0 1] [1] >= [0 0] X1 + [0 0] X2 + [0] [0 1] [0 1] [1] = x(X1,X2) plus(X1,X2) = [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] >= [0 0] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] = n__plus(X1,X2) s(X) = [0 0] X + [0] [0 1] [0] >= [0 0] X + [0] [0 1] [0] = n__s(X) x(X1,X2) = [0 0] X1 + [0 0] X2 + [0] [0 1] [0 1] [1] >= [0 0] X1 + [0 0] X2 + [0] [0 1] [0 1] [1] = n__x(X1,X2) ****** Step 9.b:4.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 9.b:4.a:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 5: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) Consider the set of all dependency pairs 1: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 2: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 3: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 4: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 5: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 6: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) 7: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 8: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 9: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 10: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 11: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 12: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 13: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 14: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 15: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 16: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 17: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 18: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 19: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) Processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {5} These cover all (indirect) predecessors of dependency pairs {1,2,5,6,7,15,16,17,18,19} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 9.b:4.a:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1,2}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1,2}, uargs(c_19) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_40) = {1}, uargs(c_41) = {1}, uargs(c_42) = {1} Following symbols are considered usable: {0,activate,plus,s,x,0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91#,U92#,activate#,isNat# ,isNatKind#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] [0] p(U101) = [0] [0] p(U102) = [0] [0] p(U103) = [0] [0] p(U104) = [0] [0] p(U11) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [1] [0 1] [0 1] [0 0] [0] p(U12) = [0 0] x1 + [0 0] x2 + [0] [1 0] [1 1] [0] p(U13) = [0 0] x3 + [0] [1 0] [0] p(U14) = [1 0] x1 + [0 0] x3 + [1] [1 0] [0 1] [1] p(U15) = [1] [0] p(U16) = [0] [0] p(U21) = [0] [1] p(U22) = [0] [0] p(U23) = [0] [0] p(U31) = [0 0] x1 + [1] [1 0] [1] p(U32) = [1 0] x1 + [1 0] x2 + [0 0] x3 + [1] [0 0] [0 0] [1 0] [1] p(U33) = [1 1] x1 + [1] [1 0] [0] p(U34) = [0 0] x1 + [0 1] x3 + [0] [1 0] [0 0] [0] p(U35) = [0] [1] p(U36) = [0] [0] p(U41) = [0] [0] p(U42) = [0 1] x1 + [0] [0 0] [0] p(U51) = [0 0] x1 + [0] [1 0] [1] p(U61) = [1 0] x1 + [0] [1 0] [1] p(U62) = [1 0] x1 + [0] [0 0] [0] p(U71) = [0] [0] p(U72) = [0] [0] p(U81) = [0] [0] p(U82) = [0] [0] p(U83) = [0] [0] p(U84) = [0] [0] p(U91) = [0] [0] p(U92) = [0] [0] p(activate) = [1 0] x1 + [0] [0 1] [0] p(isNat) = [0] [0] p(isNatKind) = [0 0] x1 + [1] [0 1] [0] p(n__0) = [0] [0] p(n__plus) = [1 1] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(n__s) = [1 0] x1 + [0] [0 0] [0] p(n__x) = [1 0] x1 + [1 1] x2 + [1] [0 0] [0 0] [0] p(plus) = [1 1] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(s) = [1 0] x1 + [0] [0 0] [0] p(tt) = [0] [0] p(x) = [1 0] x1 + [1 1] x2 + [1] [0 0] [0 0] [0] p(0#) = [0] [0] p(U101#) = [0 0] x2 + [1 1] x3 + [1] [1 1] [0 0] [0] p(U102#) = [1 0] x3 + [0] [0 0] [0] p(U103#) = [0] [0] p(U104#) = [0] [0] p(U11#) = [1 0] x2 + [1 0] x3 + [0] [0 0] [0 0] [1] p(U12#) = [1 0] x2 + [1 0] x3 + [0] [1 0] [0 0] [0] p(U13#) = [1 0] x2 + [1 0] x3 + [0] [0 0] [1 1] [0] p(U14#) = [1 0] x2 + [1 0] x3 + [0] [0 0] [0 0] [1] p(U15#) = [1 0] x2 + [0] [0 0] [1] p(U16#) = [0] [0] p(U21#) = [1 0] x2 + [0] [0 0] [0] p(U22#) = [1 0] x2 + [0] [0 0] [0] p(U23#) = [0] [0] p(U31#) = [1 0] x2 + [1 0] x3 + [0] [0 0] [0 1] [0] p(U32#) = [1 0] x2 + [1 0] x3 + [0] [0 0] [0 0] [1] p(U33#) = [1 0] x2 + [1 0] x3 + [0] [0 0] [0 0] [0] p(U34#) = [1 0] x2 + [1 0] x3 + [0] [0 1] [0 1] [1] p(U35#) = [1 0] x2 + [0] [0 0] [1] p(U36#) = [0] [0] p(U41#) = [0] [0] p(U42#) = [0] [0] p(U51#) = [0] [0] p(U61#) = [0] [0] p(U62#) = [0] [0] p(U71#) = [0] [0] p(U72#) = [0] [0] p(U81#) = [0 0] x2 + [1 0] x3 + [1] [1 1] [0 1] [0] p(U82#) = [1 0] x3 + [0] [0 0] [0] p(U83#) = [0] [0] p(U84#) = [0] [0] p(U91#) = [0] [0] p(U92#) = [0] [0] p(activate#) = [0] [0] p(isNat#) = [1 0] x1 + [0] [0 0] [0] p(isNatKind#) = [0] [0] p(plus#) = [0] [0] p(s#) = [0] [0] p(x#) = [0] [0] p(c_1) = [0] [0] p(c_2) = [1 0] x1 + [1] [0 0] [0] p(c_3) = [1 0] x1 + [0] [0 0] [0] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [1 0] x1 + [0] [0 0] [0] p(c_7) = [1 0] x1 + [0] [0 0] [0] p(c_8) = [1 0] x1 + [0] [0 0] [0] p(c_9) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_10) = [1 0] x1 + [0] [0 0] [1] p(c_11) = [0] [0] p(c_12) = [1 0] x1 + [0] [0 0] [0] p(c_13) = [1 0] x1 + [0] [0 0] [0] p(c_14) = [0] [0] p(c_15) = [1 0] x1 + [0] [0 0] [0] p(c_16) = [1 0] x1 + [0] [0 0] [0] p(c_17) = [1 0] x1 + [0] [0 0] [0] p(c_18) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] p(c_19) = [1 0] x1 + [0] [0 0] [1] p(c_20) = [0] [0] p(c_21) = [0] [0] p(c_22) = [0] [0] p(c_23) = [0] [0] p(c_24) = [0] [0] p(c_25) = [0] [0] p(c_26) = [0] [0] p(c_27) = [0] [0] p(c_28) = [1 0] x1 + [0] [0 0] [0] p(c_29) = [1 0] x1 + [0] [0 0] [0] p(c_30) = [0] [0] p(c_31) = [0] [0] p(c_32) = [0] [0] p(c_33) = [0] [0] p(c_34) = [0] [0] p(c_35) = [0] [0] p(c_36) = [0] [0] p(c_37) = [0] [0] p(c_38) = [0] [0] p(c_39) = [0] [0] p(c_40) = [1 0] x1 + [0] [0 0] [0] p(c_41) = [1 0] x1 + [0] [0 0] [0] p(c_42) = [1 1] x1 + [0] [0 0] [0] p(c_43) = [0] [0] p(c_44) = [0] [0] p(c_45) = [0] [0] p(c_46) = [0] [0] p(c_47) = [0] [0] p(c_48) = [0] [0] p(c_49) = [0] [0] Following rules are strictly oriented: isNat#(n__x(V1,V2)) = [1 0] V1 + [1 1] V2 + [1] [0 0] [0 0] [0] > [1 0] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] = c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) Following rules are (at-least) weakly oriented: U101#(tt(),M,N) = [0 0] M + [1 1] N + [1] [1 1] [0 0] [0] >= [1 0] N + [1] [0 0] [0] = c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) = [1 0] N + [0] [0 0] [0] >= [1 0] N + [0] [0 0] [0] = c_3(isNat#(activate(N))) U11#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [1 0] [0 0] [0] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 0] [1 1] [0] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [1 0] V2 + [0] [0 0] [1] >= [1 0] V2 + [0] [0 0] [1] = c_10(isNat#(activate(V2))) U21#(tt(),V1) = [1 0] V1 + [0] [0 0] [0] >= [1 0] V1 + [0] [0 0] [0] = c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [1 0] V1 + [0] [0 0] [0] >= [1 0] V1 + [0] [0 0] [0] = c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 0] [0 1] [0] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 1] [0 1] [1] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] = c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) = [1 0] V2 + [0] [0 0] [1] >= [1 0] V2 + [0] [0 0] [1] = c_19(isNat#(activate(V2))) U81#(tt(),M,N) = [0 0] M + [1 0] N + [1] [1 1] [0 1] [0] >= [1 0] N + [0] [0 0] [0] = c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) = [1 0] N + [0] [0 0] [0] >= [1 0] N + [0] [0 0] [0] = c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [1 1] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [1 0] V1 + [0] [0 0] [0] >= [1 0] V1 + [0] [0 0] [0] = c_41(U21#(isNatKind(activate(V1)),activate(V1))) 0() = [0] [0] >= [0] [0] = n__0() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [0] [0] >= [0] [0] = 0() activate(n__plus(X1,X2)) = [1 1] X1 + [1 0] X2 + [0] [0 0] [0 0] [0] >= [1 1] X1 + [1 0] X2 + [0] [0 0] [0 0] [0] = plus(X1,X2) activate(n__s(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = s(X) activate(n__x(X1,X2)) = [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] >= [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] = x(X1,X2) plus(X1,X2) = [1 1] X1 + [1 0] X2 + [0] [0 0] [0 0] [0] >= [1 1] X1 + [1 0] X2 + [0] [0 0] [0 0] [0] = n__plus(X1,X2) s(X) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = n__s(X) x(X1,X2) = [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] >= [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] = n__x(X1,X2) ******* Step 9.b:4.a:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 9.b:4.a:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) Consider the set of all dependency pairs 1: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 2: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 3: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) 4: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 5: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 6: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 7: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 8: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 9: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 10: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 11: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 12: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 13: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 14: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 15: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 16: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 17: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 18: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) 19: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) Processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,3,4,5,6,7,8,9,17,18} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******** Step 9.b:4.a:1.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1,2}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1,2}, uargs(c_19) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_40) = {1}, uargs(c_41) = {1}, uargs(c_42) = {1} Following symbols are considered usable: {0,activate,plus,s,x,0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91#,U92#,activate#,isNat# ,isNatKind#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] [0] p(U101) = [0] [0] p(U102) = [0] [0] p(U103) = [0] [0] p(U104) = [0] [0] p(U11) = [0 1] x3 + [0] [0 0] [0] p(U12) = [0 1] x1 + [0] [0 0] [1] p(U13) = [0 1] x1 + [0] [0 0] [0] p(U14) = [0] [0] p(U15) = [1] [0] p(U16) = [0] [1] p(U21) = [0] [0] p(U22) = [0] [0] p(U23) = [0] [1] p(U31) = [1] [0] p(U32) = [0] [1] p(U33) = [0] [0] p(U34) = [0 0] x2 + [0 0] x3 + [1] [1 0] [1 0] [0] p(U35) = [1] [0] p(U36) = [0] [0] p(U41) = [0] [0] p(U42) = [0 0] x1 + [0] [0 1] [0] p(U51) = [0] [1] p(U61) = [0] [0] p(U62) = [0] [0] p(U71) = [0] [0] p(U72) = [0] [0] p(U81) = [0] [0] p(U82) = [0] [0] p(U83) = [0] [0] p(U84) = [0] [0] p(U91) = [0] [0] p(U92) = [0] [0] p(activate) = [1 0] x1 + [0] [0 1] [0] p(isNat) = [1] [1] p(isNatKind) = [0] [1] p(n__0) = [0] [0] p(n__plus) = [1 0] x1 + [1 1] x2 + [1] [0 0] [0 0] [0] p(n__s) = [1 1] x1 + [1] [0 0] [0] p(n__x) = [1 1] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(plus) = [1 0] x1 + [1 1] x2 + [1] [0 0] [0 0] [0] p(s) = [1 1] x1 + [1] [0 0] [0] p(tt) = [0] [0] p(x) = [1 1] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(0#) = [0] [0] p(U101#) = [0 1] x2 + [1 1] x3 + [1] [1 0] [1 0] [0] p(U102#) = [1 0] x3 + [1] [0 1] [0] p(U103#) = [0] [0] p(U104#) = [0] [0] p(U11#) = [1 0] x2 + [1 1] x3 + [0] [0 0] [0 0] [0] p(U12#) = [1 0] x2 + [1 0] x3 + [0] [1 0] [0 0] [0] p(U13#) = [1 0] x2 + [1 0] x3 + [0] [0 0] [0 0] [1] p(U14#) = [1 0] x2 + [1 0] x3 + [0] [0 0] [1 1] [1] p(U15#) = [1 0] x2 + [0] [0 1] [0] p(U16#) = [0] [0] p(U21#) = [1 1] x2 + [1] [0 0] [0] p(U22#) = [1 0] x2 + [0] [0 0] [0] p(U23#) = [0] [0] p(U31#) = [1 1] x2 + [1 0] x3 + [0] [0 0] [0 0] [1] p(U32#) = [1 1] x2 + [1 0] x3 + [0] [0 1] [0 0] [0] p(U33#) = [1 1] x2 + [1 0] x3 + [0] [0 0] [0 0] [1] p(U34#) = [1 0] x2 + [1 0] x3 + [0] [0 1] [0 0] [0] p(U35#) = [1 0] x2 + [0] [1 1] [0] p(U36#) = [0] [0] p(U41#) = [0] [0] p(U42#) = [0] [0] p(U51#) = [0] [0] p(U61#) = [0] [0] p(U62#) = [0] [0] p(U71#) = [0] [0] p(U72#) = [0] [0] p(U81#) = [0 0] x2 + [1 1] x3 + [0] [1 0] [0 0] [0] p(U82#) = [0 0] x2 + [1 0] x3 + [0] [1 0] [0 0] [0] p(U83#) = [0] [0] p(U84#) = [0] [0] p(U91#) = [0] [0] p(U92#) = [0] [0] p(activate#) = [0] [0] p(isNat#) = [1 0] x1 + [0] [0 0] [1] p(isNatKind#) = [0] [0] p(plus#) = [0] [0] p(s#) = [0] [0] p(x#) = [0] [0] p(c_1) = [0] [0] p(c_2) = [1 1] x1 + [0] [0 0] [0] p(c_3) = [1 0] x1 + [1] [0 0] [0] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [1 0] x1 + [0] [0 0] [0] p(c_7) = [1 0] x1 + [0] [0 0] [0] p(c_8) = [1 0] x1 + [0] [0 0] [0] p(c_9) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] p(c_10) = [1 0] x1 + [0] [0 0] [0] p(c_11) = [0] [0] p(c_12) = [1 0] x1 + [1] [0 0] [0] p(c_13) = [1 0] x1 + [0] [0 0] [0] p(c_14) = [0] [0] p(c_15) = [1 0] x1 + [0] [0 0] [1] p(c_16) = [1 0] x1 + [0] [0 0] [0] p(c_17) = [1 1] x1 + [0] [0 0] [1] p(c_18) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_19) = [1 0] x1 + [0] [0 0] [0] p(c_20) = [0] [0] p(c_21) = [0] [0] p(c_22) = [0] [0] p(c_23) = [0] [0] p(c_24) = [0] [0] p(c_25) = [0] [0] p(c_26) = [0] [0] p(c_27) = [0] [0] p(c_28) = [1 0] x1 + [0] [0 0] [0] p(c_29) = [1 0] x1 + [0] [0 0] [0] p(c_30) = [0] [0] p(c_31) = [0] [0] p(c_32) = [0] [0] p(c_33) = [0] [0] p(c_34) = [0] [0] p(c_35) = [0] [0] p(c_36) = [0] [0] p(c_37) = [0] [0] p(c_38) = [0] [0] p(c_39) = [0] [0] p(c_40) = [1 0] x1 + [0] [0 0] [0] p(c_41) = [1 0] x1 + [0] [0 0] [0] p(c_42) = [1 0] x1 + [0] [0 1] [0] p(c_43) = [0] [0] p(c_44) = [0] [0] p(c_45) = [0] [0] p(c_46) = [0] [0] p(c_47) = [0] [0] p(c_48) = [0] [0] p(c_49) = [0] [0] Following rules are strictly oriented: isNat#(n__plus(V1,V2)) = [1 0] V1 + [1 1] V2 + [1] [0 0] [0 0] [1] > [1 0] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] = c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) Following rules are (at-least) weakly oriented: U101#(tt(),M,N) = [0 1] M + [1 1] N + [1] [1 0] [1 0] [0] >= [1 1] N + [1] [0 0] [0] = c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) = [1 0] N + [1] [0 1] [0] >= [1 0] N + [1] [0 0] [0] = c_3(isNat#(activate(N))) U11#(tt(),V1,V2) = [1 0] V1 + [1 1] V2 + [0] [0 0] [0 0] [0] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [1 0] [0 0] [0] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 0] [1 1] [1] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 1] [0] = c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [1 0] V2 + [0] [0 1] [0] >= [1 0] V2 + [0] [0 0] [0] = c_10(isNat#(activate(V2))) U21#(tt(),V1) = [1 1] V1 + [1] [0 0] [0] >= [1 0] V1 + [1] [0 0] [0] = c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [1 0] V1 + [0] [0 0] [0] >= [1 0] V1 + [0] [0 0] [0] = c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) = [1 1] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] >= [1 1] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] = c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) = [1 1] V1 + [1 0] V2 + [0] [0 1] [0 0] [0] >= [1 1] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) = [1 1] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] >= [1 1] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] = c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) = [1 0] V1 + [1 0] V2 + [0] [0 1] [0 0] [0] >= [1 0] V1 + [1 0] V2 + [0] [0 0] [0 0] [0] = c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) = [1 0] V2 + [0] [1 1] [0] >= [1 0] V2 + [0] [0 0] [0] = c_19(isNat#(activate(V2))) U81#(tt(),M,N) = [0 0] M + [1 1] N + [0] [1 0] [0 0] [0] >= [1 0] N + [0] [0 0] [0] = c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) = [0 0] M + [1 0] N + [0] [1 0] [0 0] [0] >= [1 0] N + [0] [0 0] [0] = c_29(isNat#(activate(N))) isNat#(n__s(V1)) = [1 1] V1 + [1] [0 0] [1] >= [1 1] V1 + [1] [0 0] [0] = c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) = [1 1] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] >= [1 1] V1 + [1 0] V2 + [0] [0 0] [0 0] [1] = c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 0() = [0] [0] >= [0] [0] = n__0() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [0] [0] >= [0] [0] = 0() activate(n__plus(X1,X2)) = [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] >= [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] = plus(X1,X2) activate(n__s(X)) = [1 1] X + [1] [0 0] [0] >= [1 1] X + [1] [0 0] [0] = s(X) activate(n__x(X1,X2)) = [1 1] X1 + [1 0] X2 + [0] [0 0] [0 0] [0] >= [1 1] X1 + [1 0] X2 + [0] [0 0] [0 0] [0] = x(X1,X2) plus(X1,X2) = [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] >= [1 0] X1 + [1 1] X2 + [1] [0 0] [0 0] [0] = n__plus(X1,X2) s(X) = [1 1] X + [1] [0 0] [0] >= [1 1] X + [1] [0 0] [0] = n__s(X) x(X1,X2) = [1 1] X1 + [1 0] X2 + [0] [0 0] [0 0] [0] >= [1 1] X1 + [1 0] X2 + [0] [0 0] [0 0] [0] = n__x(X1,X2) ******** Step 9.b:4.a:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 9.b:4.a:1.b:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) Consider the set of all dependency pairs 1: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 2: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) 3: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 4: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 5: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 6: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 7: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 8: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 9: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 10: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 11: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 12: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 13: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 14: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 15: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 16: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 17: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) 18: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 19: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) Processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2,3,9,10,16,17} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ********* Step 9.b:4.a:1.b:1.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1,2}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1,2}, uargs(c_19) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_40) = {1}, uargs(c_41) = {1}, uargs(c_42) = {1} Following symbols are considered usable: {0,U41,U42,U51,U61,U62,activate,isNatKind,plus,s,x,0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15# ,U16#,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84# ,U91#,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [1] [1] p(U101) = [0] [0] p(U102) = [0] [0] p(U103) = [0] [0] p(U104) = [0] [0] p(U11) = [0 0] x1 + [1] [1 0] [1] p(U12) = [1 0] x1 + [0 0] x2 + [0] [1 0] [1 0] [1] p(U13) = [0 0] x1 + [0 0] x3 + [0] [1 0] [0 1] [1] p(U14) = [0 0] x1 + [0] [1 0] [0] p(U15) = [0 0] x2 + [1] [0 1] [1] p(U16) = [0 0] x1 + [1] [1 0] [1] p(U21) = [0] [0] p(U22) = [1 0] x1 + [0] [1 0] [0] p(U23) = [1 0] x1 + [0] [0 0] [0] p(U31) = [1 0] x2 + [0 0] x3 + [0] [0 0] [0 1] [0] p(U32) = [0 0] x1 + [1 0] x2 + [0] [1 0] [0 0] [0] p(U33) = [1 0] x2 + [0] [1 0] [0] p(U34) = [0 1] x3 + [0] [0 0] [0] p(U35) = [0] [0] p(U36) = [1 0] x1 + [0] [1 0] [1] p(U41) = [1 0] x1 + [0] [0 0] [0] p(U42) = [0] [0] p(U51) = [1 0] x1 + [0] [0 0] [0] p(U61) = [1] [0] p(U62) = [1] [0] p(U71) = [0] [0] p(U72) = [0] [0] p(U81) = [0] [0] p(U82) = [0] [0] p(U83) = [0] [0] p(U84) = [0] [0] p(U91) = [0] [0] p(U92) = [0] [0] p(activate) = [1 0] x1 + [0] [0 1] [0] p(isNat) = [1 1] x1 + [1] [0 0] [0] p(isNatKind) = [1] [0] p(n__0) = [1] [1] p(n__plus) = [0 0] x1 + [0 0] x2 + [0] [0 1] [0 1] [0] p(n__s) = [0 0] x1 + [0] [0 1] [1] p(n__x) = [0 1] x1 + [0 0] x2 + [1] [0 1] [0 1] [1] p(plus) = [0 0] x1 + [0 0] x2 + [0] [0 1] [0 1] [0] p(s) = [0 0] x1 + [0] [0 1] [1] p(tt) = [0] [0] p(x) = [0 1] x1 + [0 0] x2 + [1] [0 1] [0 1] [1] p(0#) = [0] [0] p(U101#) = [0 1] x2 + [1 1] x3 + [1] [0 1] [1 1] [1] p(U102#) = [0 1] x2 + [1 1] x3 + [1] [1 0] [0 0] [0] p(U103#) = [0] [0] p(U104#) = [0] [0] p(U11#) = [0 1] x2 + [0 1] x3 + [0] [1 0] [0 0] [0] p(U12#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [1 0] [0] p(U13#) = [0 1] x2 + [0 1] x3 + [0] [0 1] [0 1] [1] p(U14#) = [0 1] x2 + [0 1] x3 + [0] [0 0] [0 0] [1] p(U15#) = [0 1] x2 + [0] [0 0] [1] p(U16#) = [0] [0] p(U21#) = [0 1] x2 + [0] [1 0] [1] p(U22#) = [0 1] x2 + [0] [0 1] [1] p(U23#) = [0] [0] p(U31#) = [0 1] x2 + [0 1] x3 + [1] [0 1] [0 0] [1] p(U32#) = [0 1] x2 + [0 1] x3 + [1] [0 1] [0 0] [1] p(U33#) = [1 0] x1 + [0 1] x2 + [0 1] x3 + [0] [1 0] [0 1] [0 0] [0] p(U34#) = [0 1] x2 + [0 1] x3 + [0] [0 1] [1 1] [0] p(U35#) = [0 1] x2 + [0] [0 0] [0] p(U36#) = [0] [0] p(U41#) = [0] [0] p(U42#) = [0] [0] p(U51#) = [0] [0] p(U61#) = [0] [0] p(U62#) = [0] [0] p(U71#) = [0] [0] p(U72#) = [0] [0] p(U81#) = [1 1] x3 + [1] [0 0] [0] p(U82#) = [0 0] x1 + [1 1] x3 + [0] [1 0] [0 1] [1] p(U83#) = [0] [0] p(U84#) = [0] [0] p(U91#) = [0] [0] p(U92#) = [0] [0] p(activate#) = [0] [0] p(isNat#) = [0 1] x1 + [0] [1 0] [1] p(isNatKind#) = [0] [0] p(plus#) = [0] [0] p(s#) = [0] [0] p(x#) = [0] [0] p(c_1) = [0] [0] p(c_2) = [1 0] x1 + [0] [1 0] [0] p(c_3) = [1 1] x1 + [0] [0 0] [0] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [1 0] x1 + [0] [0 0] [0] p(c_7) = [1 0] x1 + [0] [0 0] [0] p(c_8) = [1 0] x1 + [0] [0 0] [1] p(c_9) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_10) = [1 0] x1 + [0] [0 0] [1] p(c_11) = [0] [0] p(c_12) = [1 0] x1 + [0] [0 0] [0] p(c_13) = [1 0] x1 + [0] [1 0] [0] p(c_14) = [0] [0] p(c_15) = [1 0] x1 + [0] [0 1] [0] p(c_16) = [1 0] x1 + [0] [0 1] [0] p(c_17) = [1 0] x1 + [0] [0 0] [0] p(c_18) = [1 0] x1 + [1 0] x2 + [0] [0 0] [1 0] [0] p(c_19) = [1 0] x1 + [0] [0 0] [0] p(c_20) = [0] [0] p(c_21) = [0] [0] p(c_22) = [0] [0] p(c_23) = [0] [0] p(c_24) = [0] [0] p(c_25) = [0] [0] p(c_26) = [0] [0] p(c_27) = [0] [0] p(c_28) = [1 0] x1 + [0] [0 0] [0] p(c_29) = [1 0] x1 + [0] [1 0] [1] p(c_30) = [0] [0] p(c_31) = [0] [0] p(c_32) = [0] [0] p(c_33) = [0] [0] p(c_34) = [0] [0] p(c_35) = [0] [0] p(c_36) = [0] [0] p(c_37) = [0] [0] p(c_38) = [0] [0] p(c_39) = [0] [0] p(c_40) = [1 0] x1 + [0] [0 0] [1] p(c_41) = [1 0] x1 + [0] [0 0] [1] p(c_42) = [1 0] x1 + [0] [0 1] [1] p(c_43) = [0] [0] p(c_44) = [0] [0] p(c_45) = [0] [0] p(c_46) = [0] [0] p(c_47) = [0] [0] p(c_48) = [0] [0] p(c_49) = [0] [0] Following rules are strictly oriented: isNat#(n__s(V1)) = [0 1] V1 + [1] [0 0] [1] > [0 1] V1 + [0] [0 0] [1] = c_41(U21#(isNatKind(activate(V1)),activate(V1))) Following rules are (at-least) weakly oriented: U101#(tt(),M,N) = [0 1] M + [1 1] N + [1] [0 1] [1 1] [1] >= [0 1] M + [1 1] N + [1] [0 1] [1 1] [1] = c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) = [0 1] M + [1 1] N + [1] [1 0] [0 0] [0] >= [1 1] N + [1] [0 0] [0] = c_3(isNat#(activate(N))) U11#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [1 0] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [1 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 1] [0 1] [1] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] = c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [0 1] V2 + [0] [0 0] [1] >= [0 1] V2 + [0] [0 0] [1] = c_10(isNat#(activate(V2))) U21#(tt(),V1) = [0 1] V1 + [0] [1 0] [1] >= [0 1] V1 + [0] [0 0] [0] = c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [0 1] V1 + [0] [0 1] [1] >= [0 1] V1 + [0] [0 1] [0] = c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 1] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [1] [0 1] [0 0] [1] = c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [1] [0 1] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [1] [0 1] [0 0] [1] = c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 1] [0 0] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [0] = c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) = [0 1] V1 + [0 1] V2 + [0] [0 1] [1 1] [0] >= [0 1] V1 + [0 1] V2 + [0] [0 1] [0 0] [0] = c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) = [0 1] V2 + [0] [0 0] [0] >= [0 1] V2 + [0] [0 0] [0] = c_19(isNat#(activate(V2))) U81#(tt(),M,N) = [1 1] N + [1] [0 0] [0] >= [1 1] N + [0] [0 0] [0] = c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) = [1 1] N + [0] [0 1] [1] >= [0 1] N + [0] [0 1] [1] = c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [1] = c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__x(V1,V2)) = [0 1] V1 + [0 1] V2 + [1] [0 1] [0 0] [2] >= [0 1] V1 + [0 1] V2 + [1] [0 1] [0 0] [2] = c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 0() = [1] [1] >= [1] [1] = n__0() U41(tt(),V2) = [0] [0] >= [0] [0] = U42(isNatKind(activate(V2))) U42(tt()) = [0] [0] >= [0] [0] = tt() U51(tt()) = [0] [0] >= [0] [0] = tt() U61(tt(),V2) = [1] [0] >= [1] [0] = U62(isNatKind(activate(V2))) U62(tt()) = [1] [0] >= [0] [0] = tt() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [1] [1] >= [1] [1] = 0() activate(n__plus(X1,X2)) = [0 0] X1 + [0 0] X2 + [0] [0 1] [0 1] [0] >= [0 0] X1 + [0 0] X2 + [0] [0 1] [0 1] [0] = plus(X1,X2) activate(n__s(X)) = [0 0] X + [0] [0 1] [1] >= [0 0] X + [0] [0 1] [1] = s(X) activate(n__x(X1,X2)) = [0 1] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] >= [0 1] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] = x(X1,X2) isNatKind(n__0()) = [1] [0] >= [0] [0] = tt() isNatKind(n__plus(V1,V2)) = [1] [0] >= [1] [0] = U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = [1] [0] >= [1] [0] = U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) = [1] [0] >= [1] [0] = U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) = [0 0] X1 + [0 0] X2 + [0] [0 1] [0 1] [0] >= [0 0] X1 + [0 0] X2 + [0] [0 1] [0 1] [0] = n__plus(X1,X2) s(X) = [0 0] X + [0] [0 1] [1] >= [0 0] X + [0] [0 1] [1] = n__s(X) x(X1,X2) = [0 1] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] >= [0 1] X1 + [0 0] X2 + [1] [0 1] [0 1] [1] = n__x(X1,X2) ********* Step 9.b:4.a:1.b:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ********* Step 9.b:4.a:1.b:1.b:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) -->_1 U102#(tt(),M,N) -> c_3(isNat#(activate(N))):2 2:W:U102#(tt(),M,N) -> c_3(isNat#(activate(N))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 3:W:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))):4 4:W:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))):5 5:W:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):6 6:W:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 -->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):7 7:W:U15#(tt(),V2) -> c_10(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 8:W:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) -->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):9 9:W:U22#(tt(),V1) -> c_13(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 10:W:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))):11 11:W:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))):12 12:W:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):13 13:W:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 -->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):14 14:W:U35#(tt(),V2) -> c_19(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 15:W:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) -->_1 U82#(tt(),M,N) -> c_29(isNat#(activate(N))):16 16:W:U82#(tt(),M,N) -> c_29(isNat#(activate(N))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 17:W:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))):3 18:W:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) -->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))):8 19:W:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 16: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) 1: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) 2: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 19: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 14: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 13: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 12: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 11: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 10: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 9: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 8: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 18: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 7: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 6: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 5: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 4: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 3: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 17: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) ********* Step 9.b:4.a:1.b:1.b:1.b:1.b: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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 9.b:4.b:1: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,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: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 2: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) 3: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) 4: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 5: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 6: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 7: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 8: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 9: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 10: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 11: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 12: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 13: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 14: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 15: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 16: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 17: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 18: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 19: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) *** Step 9.b:4.b:2: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,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: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 2: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) 3: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 4: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 5: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 6: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 7: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 8: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 9: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 10: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 11: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 12: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 13: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 14: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 15: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 16: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) 17: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 18: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 19: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) *** Step 9.b:4.b:3: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) U102#(tt(),M,N) -> c_3(isNat#(activate(N))) U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_10(isNat#(activate(V2))) U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_13(isNat#(activate(V1))) U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U35#(tt(),V2) -> c_19(isNat#(activate(V2))) U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) U82#(tt(),M,N) -> c_29(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) - 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) -->_1 U102#(tt(),M,N) -> c_3(isNat#(activate(N))):2 2:W:U102#(tt(),M,N) -> c_3(isNat#(activate(N))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 3:W:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))):4 4:W:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))):5 5:W:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):6 6:W:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 -->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):7 7:W:U15#(tt(),V2) -> c_10(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 8:W:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) -->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):9 9:W:U22#(tt(),V1) -> c_13(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 10:W:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))):11 11:W:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))):12 12:W:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) -->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):13 13:W:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 -->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):14 14:W:U35#(tt(),V2) -> c_19(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 15:W:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) -->_1 U82#(tt(),M,N) -> c_29(isNat#(activate(N))):16 16:W:U82#(tt(),M,N) -> c_29(isNat#(activate(N))) -->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19 -->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18 -->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17 17:W:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))):3 18:W:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) -->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))):8 19:W:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) -->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N))) 16: U82#(tt(),M,N) -> c_29(isNat#(activate(N))) 1: U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N))) 2: U102#(tt(),M,N) -> c_3(isNat#(activate(N))) 19: isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))) 14: U35#(tt(),V2) -> c_19(isNat#(activate(V2))) 13: U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 12: U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))) 11: U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))) 10: U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))) 9: U22#(tt(),V1) -> c_13(isNat#(activate(V1))) 8: U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))) 18: isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))) 7: U15#(tt(),V2) -> c_10(isNat#(activate(V2))) 6: U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 5: U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) 4: U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) 3: U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) 17: isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) *** Step 9.b:4.b:4: 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(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2)) U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2)) U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2)) U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2)) U35(tt(),V2) -> U36(isNat(activate(V2))) U36(tt()) -> tt() U41(tt(),V2) -> U42(isNatKind(activate(V2))) U42(tt()) -> tt() U51(tt()) -> tt() U61(tt(),V2) -> U62(isNatKind(activate(V2))) U62(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) 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)) isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1))) isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3 ,U34/3,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,activate/1 ,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/0,U101#/3,U102#/3,U103#/3,U104#/3,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2 ,U16#/1,U21#/2,U22#/2,U23#/1,U31#/3,U32#/3,U33#/3,U34#/3,U35#/2,U36#/1,U41#/2,U42#/1,U51#/1,U61#/2,U62#/1 ,U71#/2,U72#/2,U81#/3,U82#/3,U83#/3,U84#/3,U91#/2,U92#/1,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1 ,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1 ,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0 ,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1 ,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U101#,U102#,U103#,U104#,U11#,U12#,U13#,U14#,U15#,U16# ,U21#,U22#,U23#,U31#,U32#,U33#,U34#,U35#,U36#,U41#,U42#,U51#,U61#,U62#,U71#,U72#,U81#,U82#,U83#,U84#,U91# ,U92#,activate#,isNat#,isNatKind#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))