WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U13,U14,U15,U16,U21,U22,U23,U31,U32,U41,U51,U52 ,U61,U62,U63,U64,activate,isNat,isNatKind,plus,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 0#() -> c_1() U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_7() U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_10() U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U32#(tt()) -> c_12() U41#(tt()) -> c_13() U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(X) -> c_20() activate#(n__0()) -> c_21(0#()) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__0()) -> c_24() isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__0()) -> c_27() isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N)) plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M)) plus#(X1,X2) -> c_32() s#(X) -> c_33() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U16#(tt()) -> c_7() U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U23#(tt()) -> c_10() U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U32#(tt()) -> c_12() U41#(tt()) -> c_13() U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(X) -> c_20() activate#(n__0()) -> c_21(0#()) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__0()) -> c_24() isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__0()) -> c_27() isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N)) plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M)) plus#(X1,X2) -> c_32() s#(X) -> c_33() - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4 ,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,7,10,12,13,20,24,27,30,31,32,33} by application of Pre({1,7,10,12,13,20,24,27,30,31,32,33}) = {2,3,4,5,6,8,9,11,14,15,16,17,18,19,21,22,23,25,26,28,29}. Here rules are labelled as follows: 1: 0#() -> c_1() 2: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 3: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 4: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 5: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 6: U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 7: U16#(tt()) -> c_7() 8: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 9: U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 10: U23#(tt()) -> c_10() 11: U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 12: U32#(tt()) -> c_12() 13: U41#(tt()) -> c_13() 14: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 15: U52#(tt(),N) -> c_15(activate#(N)) 16: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 17: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 18: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 19: U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 20: activate#(X) -> c_20() 21: activate#(n__0()) -> c_21(0#()) 22: activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 23: activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) 24: isNat#(n__0()) -> c_24() 25: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 26: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 27: isNatKind#(n__0()) -> c_27() 28: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 29: isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) 30: plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N)) 31: plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M)) 32: plus#(X1,X2) -> c_32() 33: s#(X) -> c_33() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(n__0()) -> c_21(0#()) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: 0#() -> c_1() U16#(tt()) -> c_7() U23#(tt()) -> c_10() U32#(tt()) -> c_12() U41#(tt()) -> c_13() activate#(X) -> c_20() isNat#(n__0()) -> c_24() isNatKind#(n__0()) -> c_27() plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N)) plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M)) plus#(X1,X2) -> c_32() s#(X) -> c_33() - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4 ,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {15} by application of Pre({15}) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,19,20,21}. Here rules are labelled as follows: 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 5: U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 7: U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 8: U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) 9: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 10: U52#(tt(),N) -> c_15(activate#(N)) 11: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 12: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 13: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 14: U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 15: activate#(n__0()) -> c_21(0#()) 16: activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 17: activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) 18: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 19: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 20: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 21: isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) 22: 0#() -> c_1() 23: U16#(tt()) -> c_7() 24: U23#(tt()) -> c_10() 25: U32#(tt()) -> c_12() 26: U41#(tt()) -> c_13() 27: activate#(X) -> c_20() 28: isNat#(n__0()) -> c_24() 29: isNatKind#(n__0()) -> c_27() 30: plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N)) 31: plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M)) 32: plus#(X1,X2) -> c_32() 33: s#(X) -> c_33() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: 0#() -> c_1() U16#(tt()) -> c_7() U23#(tt()) -> c_10() U32#(tt()) -> c_12() U41#(tt()) -> c_13() activate#(X) -> c_20() activate#(n__0()) -> c_21(0#()) isNat#(n__0()) -> c_24() isNatKind#(n__0()) -> c_27() plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N)) plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M)) plus#(X1,X2) -> c_32() s#(X) -> c_33() - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4 ,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):2 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):3 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):4 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):5 -->_2 isNat#(n__0()) -> c_24():28 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 5:S:U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 isNat#(n__0()) -> c_24():28 -->_3 activate#(X) -> c_20():26 -->_1 U16#(tt()) -> c_7():22 6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):7 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 7:S:U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 isNat#(n__0()) -> c_24():28 -->_3 activate#(X) -> c_20():26 -->_1 U23#(tt()) -> c_10():23 8:S:U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 isNatKind#(n__0()) -> c_27():29 -->_3 activate#(X) -> c_20():26 -->_1 U32#(tt()) -> c_12():24 9:S:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U52#(tt(),N) -> c_15(activate#(N)):10 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 10:S:U52#(tt(),N) -> c_15(activate#(N)) -->_1 activate#(n__0()) -> c_21(0#()):27 -->_1 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_1 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 activate#(X) -> c_20():26 11:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):12 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 12:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):13 -->_2 isNat#(n__0()) -> c_24():28 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 13:S:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):14 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 14:S:U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 s#(X) -> c_33():33 -->_2 plus#(X1,X2) -> c_32():32 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 15:S:activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_1 plus#(X1,X2) -> c_32():32 -->_3 activate#(X) -> c_20():26 -->_2 activate#(X) -> c_20():26 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 16:S:activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) -->_2 activate#(n__0()) -> c_21(0#()):27 -->_1 s#(X) -> c_33():33 -->_2 activate#(X) -> c_20():26 -->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 17:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_5 activate#(n__0()) -> c_21(0#()):27 -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_2 isNatKind#(n__0()) -> c_27():29 -->_5 activate#(X) -> c_20():26 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):1 18:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 19:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__0()) -> c_21(0#()):27 -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__0()) -> c_27():29 -->_4 activate#(X) -> c_20():26 -->_3 activate#(X) -> c_20():26 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):8 20:S:isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) -->_3 activate#(n__0()) -> c_21(0#()):27 -->_2 isNatKind#(n__0()) -> c_27():29 -->_3 activate#(X) -> c_20():26 -->_1 U41#(tt()) -> c_13():25 -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 21:W:0#() -> c_1() 22:W:U16#(tt()) -> c_7() 23:W:U23#(tt()) -> c_10() 24:W:U32#(tt()) -> c_12() 25:W:U41#(tt()) -> c_13() 26:W:activate#(X) -> c_20() 27:W:activate#(n__0()) -> c_21(0#()) -->_1 0#() -> c_1():21 28:W:isNat#(n__0()) -> c_24() 29:W:isNatKind#(n__0()) -> c_27() 30:W:plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N)) 31:W:plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M)) 32:W:plus#(X1,X2) -> c_32() 33:W:s#(X) -> c_33() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 31: plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M)) 30: plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N)) 22: U16#(tt()) -> c_7() 23: U23#(tt()) -> c_10() 28: isNat#(n__0()) -> c_24() 24: U32#(tt()) -> c_12() 32: plus#(X1,X2) -> c_32() 33: s#(X) -> c_33() 25: U41#(tt()) -> c_13() 26: activate#(X) -> c_20() 29: isNatKind#(n__0()) -> c_27() 27: activate#(n__0()) -> c_21(0#()) 21: 0#() -> c_1() * Step 5: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4 ,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):2 2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):3 3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):4 4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):5 5:S:U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):7 7:S:U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 8:S:U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 9:S:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U52#(tt(),N) -> c_15(activate#(N)):10 10:S:U52#(tt(),N) -> c_15(activate#(N)) -->_1 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_1 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 11:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):12 12:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):18 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):17 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):13 13:S:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):14 14:S:U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 15:S:activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 16:S:activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 17:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):1 18:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 19:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 -->_1 U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):8 20:S:isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)) -->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))) ,isNatKind#(activate(V1)) ,activate#(V1)):20 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):19 -->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16 -->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) * Step 6: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) and a lower component U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) Further, following extension rules are added to the lower component. U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) ** Step 6.a:1: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {9,13} by application of Pre({9,13}) = {8,12}. Here rules are labelled as follows: 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 5: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) 8: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 9: U52#(tt(),N) -> c_15(activate#(N)) 10: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 11: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 12: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 13: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) 14: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 15: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) ** Step 6.a:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) - Weak DPs: U52#(tt(),N) -> c_15(activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {8,11} by application of Pre({8,11}) = {10}. Here rules are labelled as follows: 1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) 4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 5: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) 6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 7: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) 8: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 9: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) 10: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 11: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 12: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) 13: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) 14: U52#(tt(),N) -> c_15(activate#(N)) 15: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) ** Step 6.a:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) - Weak DPs: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) U52#(tt(),N) -> c_15(activate#(N)) U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):2 2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):3 3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):4 4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 -->_1 U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)):5 5:S:U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_1 U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)):7 7:S:U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 8:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):9 9:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):14 -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 10:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):1 11:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 12:W:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) -->_1 U52#(tt(),N) -> c_15(activate#(N)):13 13:W:U52#(tt(),N) -> c_15(activate#(N)) 14:W:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_1 U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)):15 15:W:U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 12: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(N)) 13: U52#(tt(),N) -> c_15(activate#(N)) 14: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N)) ,isNatKind#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 15: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)) ** Step 6.a:4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):2 2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)):3 3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) ,isNatKind#(activate(V2)) ,activate#(V2) ,activate#(V1) ,activate#(V2)) -->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):4 4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 -->_1 U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)):5 5:S:U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_1 U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)):7 7:S:U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 8:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)) ,isNatKind#(activate(M)) ,activate#(M) ,activate#(M) ,activate#(N)) -->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)):9 9:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):11 -->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):10 10:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)) -->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1) ,activate#(V2)):1 11:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)) -->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V1)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) ** Step 6.a:5: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate# ,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [4] p(U12) = [4] x2 + [1] x3 + [7] p(U13) = [4] x1 + [2] x3 + [3] p(U14) = [1] x2 + [5] p(U15) = [1] x1 + [2] x2 + [4] p(U16) = [6] p(U21) = [0] p(U22) = [1] x1 + [1] x2 + [0] p(U23) = [1] p(U31) = [4] p(U32) = [5] x1 + [0] p(U41) = [3] p(U51) = [3] x1 + [1] x2 + [1] p(U52) = [6] x1 + [0] p(U61) = [3] x1 + [5] x2 + [1] x3 + [4] p(U62) = [1] x2 + [0] p(U63) = [2] x1 + [1] x2 + [0] p(U64) = [2] x3 + [0] p(activate) = [0] p(isNat) = [1] x1 + [1] p(isNatKind) = [1] p(n__0) = [0] p(n__plus) = [7] p(n__s) = [1] x1 + [7] p(plus) = [4] x2 + [4] p(s) = [1] p(tt) = [0] p(0#) = [2] p(U11#) = [0] p(U12#) = [0] p(U13#) = [0] p(U14#) = [0] p(U15#) = [0] p(U16#) = [4] x1 + [0] p(U21#) = [0] p(U22#) = [0] p(U23#) = [2] p(U31#) = [0] p(U32#) = [1] p(U41#) = [1] x1 + [0] p(U51#) = [1] x1 + [2] x2 + [1] p(U52#) = [0] p(U61#) = [4] x1 + [4] x2 + [4] x3 + [5] p(U62#) = [1] p(U63#) = [1] x1 + [0] p(U64#) = [1] x1 + [2] x2 + [2] x3 + [0] p(activate#) = [0] p(isNat#) = [0] p(isNatKind#) = [4] x1 + [0] p(plus#) = [1] p(s#) = [1] x1 + [2] p(c_1) = [0] p(c_2) = [4] x1 + [0] p(c_3) = [4] x1 + [0] p(c_4) = [4] x1 + [0] p(c_5) = [4] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [4] x1 + [0] p(c_9) = [4] x1 + [0] p(c_10) = [0] p(c_11) = [1] x2 + [0] p(c_12) = [1] p(c_13) = [1] p(c_14) = [1] x1 + [1] x2 + [1] p(c_15) = [2] x1 + [4] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [1] p(c_18) = [1] x1 + [0] p(c_19) = [1] x1 + [1] p(c_20) = [4] p(c_21) = [4] p(c_22) = [4] x2 + [1] p(c_23) = [0] p(c_24) = [0] p(c_25) = [1] x1 + [0] p(c_26) = [4] x1 + [0] p(c_27) = [0] p(c_28) = [1] x2 + [1] x3 + [1] x4 + [2] p(c_29) = [1] x1 + [4] x2 + [4] p(c_30) = [1] x1 + [1] x2 + [1] p(c_31) = [0] p(c_32) = [0] p(c_33) = [1] Following rules are strictly oriented: U61#(tt(),M,N) = [4] M + [4] N + [5] > [1] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [0] >= [0] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [0] >= [0] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [0] >= [0] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [0] >= [0] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [0] >= [0] = c_6(isNat#(activate(V2))) U21#(tt(),V1) = [0] >= [0] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [0] >= [0] = c_9(isNat#(activate(V1))) U62#(tt(),M,N) = [1] >= [1] = c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [0] >= [0] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [0] >= [0] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) ** Step 6.a:6: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak DPs: U61#(tt(),M,N) -> c_16(U62#(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(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate# ,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [1] p(U11) = [0] p(U12) = [4] x2 + [1] p(U13) = [2] x2 + [6] p(U14) = [1] x1 + [2] p(U15) = [1] x1 + [2] p(U16) = [1] x1 + [6] p(U21) = [1] x1 + [1] x2 + [2] p(U22) = [2] x2 + [4] p(U23) = [4] x1 + [2] p(U31) = [2] p(U32) = [1] x1 + [2] p(U41) = [0] p(U51) = [6] p(U52) = [0] p(U61) = [1] x1 + [6] p(U62) = [2] x1 + [4] x2 + [4] x3 + [4] p(U63) = [1] x1 + [1] x2 + [4] p(U64) = [1] x1 + [1] x2 + [1] x3 + [0] p(activate) = [0] p(isNat) = [0] p(isNatKind) = [0] p(n__0) = [2] p(n__plus) = [1] p(n__s) = [1] x1 + [0] p(plus) = [4] x2 + [4] p(s) = [2] x1 + [2] p(tt) = [0] p(0#) = [1] p(U11#) = [0] p(U12#) = [0] p(U13#) = [0] p(U14#) = [0] p(U15#) = [0] p(U16#) = [1] x1 + [0] p(U21#) = [0] p(U22#) = [0] p(U23#) = [1] x1 + [1] p(U31#) = [1] x1 + [0] p(U32#) = [4] p(U41#) = [1] x1 + [1] p(U51#) = [0] p(U52#) = [1] p(U61#) = [1] x3 + [7] p(U62#) = [2] p(U63#) = [1] x3 + [1] p(U64#) = [1] p(activate#) = [1] x1 + [4] p(isNat#) = [0] p(isNatKind#) = [0] p(plus#) = [1] x2 + [1] p(s#) = [1] p(c_1) = [2] p(c_2) = [1] x1 + [0] p(c_3) = [4] x1 + [0] p(c_4) = [4] x1 + [0] p(c_5) = [1] x1 + [2] x2 + [0] p(c_6) = [4] x1 + [0] p(c_7) = [1] p(c_8) = [4] x1 + [0] p(c_9) = [4] x1 + [0] p(c_10) = [0] p(c_11) = [2] x1 + [4] x2 + [1] p(c_12) = [1] p(c_13) = [0] p(c_14) = [1] x4 + [1] p(c_15) = [1] p(c_16) = [2] x1 + [3] p(c_17) = [1] x1 + [0] p(c_18) = [1] x2 + [1] x4 + [0] p(c_19) = [1] p(c_20) = [2] p(c_21) = [4] p(c_22) = [1] x1 + [1] x2 + [2] p(c_23) = [4] x1 + [0] p(c_24) = [1] p(c_25) = [4] x1 + [0] p(c_26) = [2] x1 + [0] p(c_27) = [2] p(c_28) = [1] x2 + [4] x3 + [0] p(c_29) = [0] p(c_30) = [4] x1 + [0] p(c_31) = [1] x1 + [1] x2 + [2] p(c_32) = [0] p(c_33) = [0] Following rules are strictly oriented: U62#(tt(),M,N) = [2] > [0] = c_17(isNat#(activate(N))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [0] >= [0] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [0] >= [0] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [0] >= [0] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [0] >= [0] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [0] >= [0] = c_6(isNat#(activate(V2))) U21#(tt(),V1) = [0] >= [0] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [0] >= [0] = c_9(isNat#(activate(V1))) U61#(tt(),M,N) = [1] N + [7] >= [7] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) isNat#(n__plus(V1,V2)) = [0] >= [0] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [0] >= [0] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) ** Step 6.a:7: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak DPs: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [2] p(U12) = [4] x1 + [4] p(U13) = [4] x1 + [4] x3 + [6] p(U14) = [4] x3 + [0] p(U15) = [2] x2 + [5] p(U16) = [3] x1 + [5] p(U21) = [6] x2 + [1] p(U22) = [1] x1 + [2] x2 + [3] p(U23) = [2] p(U31) = [4] p(U32) = [0] p(U41) = [0] p(U51) = [1] x2 + [0] p(U52) = [1] x2 + [0] p(U61) = [1] x2 + [1] x3 + [1] p(U62) = [1] x2 + [1] x3 + [1] p(U63) = [1] x2 + [1] x3 + [1] p(U64) = [1] x2 + [1] x3 + [1] p(activate) = [1] x1 + [0] p(isNat) = [2] p(isNatKind) = [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [1] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [1] p(tt) = [0] p(0#) = [4] p(U11#) = [1] x2 + [1] x3 + [0] p(U12#) = [1] x2 + [1] x3 + [0] p(U13#) = [1] x2 + [1] x3 + [0] p(U14#) = [1] x2 + [1] x3 + [0] p(U15#) = [1] x2 + [0] p(U16#) = [1] x1 + [0] p(U21#) = [1] x2 + [0] p(U22#) = [1] x2 + [0] p(U23#) = [2] p(U31#) = [0] p(U32#) = [1] p(U41#) = [4] x1 + [0] p(U51#) = [1] x2 + [0] p(U52#) = [1] x1 + [2] x2 + [4] p(U61#) = [4] x2 + [4] x3 + [3] p(U62#) = [4] x3 + [2] p(U63#) = [2] x3 + [4] p(U64#) = [1] p(activate#) = [1] x1 + [0] p(isNat#) = [1] x1 + [0] p(isNatKind#) = [1] x1 + [1] p(plus#) = [1] p(s#) = [2] x1 + [1] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [2] p(c_14) = [2] x2 + [1] x4 + [1] p(c_15) = [2] x1 + [1] p(c_16) = [1] x1 + [1] p(c_17) = [1] x1 + [2] p(c_18) = [1] x1 + [1] x2 + [2] x3 + [2] x4 + [1] p(c_19) = [1] x1 + [1] x2 + [2] p(c_20) = [1] p(c_21) = [0] p(c_22) = [1] x2 + [4] p(c_23) = [1] x1 + [0] p(c_24) = [1] p(c_25) = [1] x1 + [0] p(c_26) = [1] x1 + [0] p(c_27) = [0] p(c_28) = [1] x1 + [1] x3 + [2] p(c_29) = [4] x2 + [2] p(c_30) = [4] x1 + [1] x2 + [1] p(c_31) = [4] x1 + [1] x2 + [2] p(c_32) = [1] p(c_33) = [1] Following rules are strictly oriented: isNat#(n__s(V1)) = [1] V1 + [1] > [1] V1 + [0] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = c_6(isNat#(activate(V2))) U21#(tt(),V1) = [1] V1 + [0] >= [1] V1 + [0] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [1] V1 + [0] >= [1] V1 + [0] = c_9(isNat#(activate(V1))) U61#(tt(),M,N) = [4] M + [4] N + [3] >= [4] N + [3] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) = [4] N + [2] >= [1] N + [2] = c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) 0() = [0] >= [0] = n__0() U51(tt(),N) = [1] N + [0] >= [1] N + [0] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [1] >= [1] X + [1] = s(activate(X)) plus(N,0()) = [1] N + [0] >= [1] N + [0] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) s(X) = [1] X + [1] >= [1] X + [1] = n__s(X) ** Step 6.a:8: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0,U11,U12,U13,U14,U15,U16,U21,U22,U23,U51,U52,U61,U62,U63,U64,activate,isNat,plus,s,0#,U11#,U12#,U13# ,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind# ,plus#,s#} TcT has computed the following interpretation: p(0) = [1] p(U11) = [0] p(U12) = [0] p(U13) = [0] p(U14) = [0] p(U15) = [0] p(U16) = [0] p(U21) = [0] p(U22) = [0] p(U23) = [0] p(U31) = [1] p(U32) = [1] x1 + [5] p(U41) = [3] p(U51) = [1] x1 + [1] x2 + [0] p(U52) = [1] x2 + [0] p(U61) = [1] x2 + [1] x3 + [2] p(U62) = [1] x2 + [1] x3 + [2] p(U63) = [1] x2 + [1] x3 + [2] p(U64) = [1] x2 + [1] x3 + [2] p(activate) = [1] x1 + [0] p(isNat) = [1] p(isNatKind) = [0] p(n__0) = [1] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [2] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [2] p(tt) = [0] p(0#) = [0] p(U11#) = [6] x2 + [6] x3 + [0] p(U12#) = [6] x2 + [6] x3 + [0] p(U13#) = [6] x2 + [6] x3 + [0] p(U14#) = [6] x2 + [6] x3 + [0] p(U15#) = [6] x2 + [0] p(U16#) = [1] x1 + [0] p(U21#) = [6] x2 + [5] p(U22#) = [6] x2 + [4] p(U23#) = [0] p(U31#) = [4] x1 + [0] p(U32#) = [0] p(U41#) = [1] p(U51#) = [1] p(U52#) = [1] x1 + [0] p(U61#) = [1] x1 + [1] x2 + [6] x3 + [6] p(U62#) = [1] x2 + [6] x3 + [1] p(U63#) = [1] x2 + [0] p(U64#) = [1] x1 + [0] p(activate#) = [1] x1 + [1] p(isNat#) = [6] x1 + [0] p(isNatKind#) = [2] x1 + [0] p(plus#) = [4] x1 + [0] p(s#) = [1] x1 + [1] p(c_1) = [2] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [4] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [4] p(c_13) = [0] p(c_14) = [4] x3 + [1] x4 + [4] p(c_15) = [1] p(c_16) = [1] x1 + [5] p(c_17) = [1] x1 + [1] p(c_18) = [1] x1 + [1] x3 + [1] x5 + [2] p(c_19) = [2] x1 + [4] p(c_20) = [0] p(c_21) = [2] x1 + [1] p(c_22) = [4] x1 + [1] p(c_23) = [4] x1 + [1] p(c_24) = [0] p(c_25) = [1] x1 + [0] p(c_26) = [1] x1 + [5] p(c_27) = [0] p(c_28) = [2] x1 + [1] x2 + [2] x3 + [2] p(c_29) = [4] p(c_30) = [4] p(c_31) = [4] p(c_32) = [0] p(c_33) = [0] Following rules are strictly oriented: U21#(tt(),V1) = [6] V1 + [5] > [6] V1 + [4] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [6] V2 + [0] >= [6] V2 + [0] = c_6(isNat#(activate(V2))) U22#(tt(),V1) = [6] V1 + [4] >= [6] V1 + [4] = c_9(isNat#(activate(V1))) U61#(tt(),M,N) = [1] M + [6] N + [6] >= [1] M + [6] N + [6] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) = [1] M + [6] N + [1] >= [6] N + [1] = c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [6] V1 + [12] >= [6] V1 + [10] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) 0() = [1] >= [1] = n__0() U11(tt(),V1,V2) = [0] >= [0] = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) = [0] >= [0] = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) = [0] >= [0] = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) = [0] >= [0] = U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) = [0] >= [0] = U16(isNat(activate(V2))) U16(tt()) = [0] >= [0] = tt() U21(tt(),V1) = [0] >= [0] = U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) = [0] >= [0] = U23(isNat(activate(V1))) U23(tt()) = [0] >= [0] = tt() U51(tt(),N) = [1] N + [0] >= [1] N + [0] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [1] >= [1] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [2] >= [1] X + [2] = s(activate(X)) isNat(n__0()) = [1] >= [0] = tt() isNat(n__plus(V1,V2)) = [1] >= [0] = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) = [1] >= [0] = U21(isNatKind(activate(V1)),activate(V1)) plus(N,0()) = [1] N + [1] >= [1] N + [1] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) s(X) = [1] X + [2] >= [1] X + [2] = n__s(X) ** Step 6.a:9: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [4] x2 + [0] p(U12) = [2] x1 + [7] x2 + [6] p(U13) = [4] x3 + [0] p(U14) = [1] x1 + [1] x3 + [1] p(U15) = [4] x1 + [4] x2 + [2] p(U16) = [1] x1 + [7] p(U21) = [4] x1 + [4] x2 + [1] p(U22) = [1] x1 + [4] x2 + [4] p(U23) = [3] x1 + [2] p(U31) = [3] p(U32) = [1] p(U41) = [4] x1 + [0] p(U51) = [1] x2 + [2] p(U52) = [1] x2 + [1] p(U61) = [1] x2 + [1] x3 + [6] p(U62) = [1] x2 + [1] x3 + [6] p(U63) = [1] x2 + [1] x3 + [6] p(U64) = [1] x2 + [1] x3 + [6] p(activate) = [1] x1 + [0] p(isNat) = [2] p(isNatKind) = [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [4] p(plus) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [4] p(tt) = [0] p(0#) = [1] p(U11#) = [2] x2 + [2] x3 + [4] p(U12#) = [2] x2 + [2] x3 + [2] p(U13#) = [2] x2 + [2] x3 + [2] p(U14#) = [2] x2 + [2] x3 + [0] p(U15#) = [2] x2 + [0] p(U16#) = [1] x1 + [1] p(U21#) = [2] x2 + [4] p(U22#) = [2] x2 + [0] p(U23#) = [0] p(U31#) = [1] x1 + [1] x2 + [1] p(U32#) = [2] p(U41#) = [4] p(U51#) = [1] x2 + [0] p(U52#) = [0] p(U61#) = [1] x2 + [7] x3 + [0] p(U62#) = [1] x2 + [7] x3 + [0] p(U63#) = [0] p(U64#) = [1] x3 + [0] p(activate#) = [1] p(isNat#) = [2] x1 + [0] p(isNatKind#) = [1] x1 + [1] p(plus#) = [4] x1 + [0] p(s#) = [4] x1 + [1] p(c_1) = [2] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [2] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] x1 + [2] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [4] x1 + [2] x2 + [0] p(c_12) = [4] p(c_13) = [0] p(c_14) = [2] p(c_15) = [2] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [1] x4 + [1] p(c_19) = [1] x2 + [2] p(c_20) = [0] p(c_21) = [1] p(c_22) = [0] p(c_23) = [1] x1 + [1] p(c_24) = [1] p(c_25) = [1] x1 + [0] p(c_26) = [1] x1 + [4] p(c_27) = [1] p(c_28) = [1] x3 + [1] x4 + [4] p(c_29) = [1] x1 + [1] x2 + [0] p(c_30) = [1] p(c_31) = [1] x1 + [0] p(c_32) = [0] p(c_33) = [2] Following rules are strictly oriented: U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [4] > [2] V1 + [2] V2 + [2] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) Following rules are (at-least) weakly oriented: U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [2] >= [2] V1 + [2] V2 + [2] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [2] >= [2] V1 + [2] V2 + [2] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [0] >= [2] V1 + [2] V2 + [0] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [2] V2 + [0] >= [2] V2 + [0] = c_6(isNat#(activate(V2))) U21#(tt(),V1) = [2] V1 + [4] >= [2] V1 + [2] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [2] V1 + [0] >= [2] V1 + [0] = c_9(isNat#(activate(V1))) U61#(tt(),M,N) = [1] M + [7] N + [0] >= [1] M + [7] N + [0] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) = [1] M + [7] N + [0] >= [2] N + [0] = c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [4] >= [2] V1 + [2] V2 + [4] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [2] V1 + [8] >= [2] V1 + [8] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) 0() = [0] >= [0] = n__0() U51(tt(),N) = [1] N + [2] >= [1] N + [1] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [1] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [4] >= [1] X + [4] = s(activate(X)) plus(N,0()) = [1] N + [2] >= [1] N + [2] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__plus(X1,X2) s(X) = [1] X + [4] >= [1] X + [4] = n__s(X) ** Step 6.a:10: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U12) = [4] x1 + [1] x2 + [0] p(U13) = [1] x2 + [1] p(U14) = [1] x1 + [1] x2 + [6] x3 + [0] p(U15) = [4] x1 + [4] x2 + [0] p(U16) = [0] p(U21) = [0] p(U22) = [1] x2 + [2] p(U23) = [1] x1 + [5] p(U31) = [6] p(U32) = [2] x1 + [1] p(U41) = [2] x1 + [0] p(U51) = [1] x2 + [0] p(U52) = [1] x2 + [0] p(U61) = [1] x2 + [1] x3 + [2] p(U62) = [1] x2 + [1] x3 + [2] p(U63) = [1] x2 + [1] x3 + [2] p(U64) = [1] x2 + [1] x3 + [2] p(activate) = [1] x1 + [0] p(isNat) = [0] p(isNatKind) = [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [2] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [2] p(tt) = [0] p(0#) = [2] p(U11#) = [1] x2 + [1] x3 + [0] p(U12#) = [1] x2 + [1] x3 + [0] p(U13#) = [1] x2 + [1] x3 + [0] p(U14#) = [1] x2 + [1] x3 + [0] p(U15#) = [1] x2 + [0] p(U16#) = [1] x1 + [1] p(U21#) = [1] x2 + [1] p(U22#) = [1] x2 + [1] p(U23#) = [4] p(U31#) = [1] x1 + [1] p(U32#) = [1] x1 + [0] p(U41#) = [0] p(U51#) = [1] x1 + [4] x2 + [0] p(U52#) = [1] x1 + [1] x2 + [4] p(U61#) = [2] x3 + [6] p(U62#) = [2] x3 + [2] p(U63#) = [1] x2 + [2] p(U64#) = [2] x3 + [0] p(activate#) = [1] x1 + [1] p(isNat#) = [1] x1 + [0] p(isNatKind#) = [1] p(plus#) = [1] x1 + [1] x2 + [1] p(s#) = [4] x1 + [0] p(c_1) = [2] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [4] p(c_11) = [1] x1 + [2] x2 + [1] p(c_12) = [2] p(c_13) = [1] p(c_14) = [2] x1 + [1] x3 + [1] x4 + [0] p(c_15) = [1] x1 + [4] p(c_16) = [1] x1 + [3] p(c_17) = [2] x1 + [2] p(c_18) = [1] x5 + [0] p(c_19) = [1] x2 + [1] p(c_20) = [1] p(c_21) = [1] p(c_22) = [1] x1 + [0] p(c_23) = [1] p(c_24) = [4] p(c_25) = [1] x1 + [0] p(c_26) = [1] x1 + [1] p(c_27) = [2] p(c_28) = [4] x1 + [1] x3 + [1] x4 + [0] p(c_29) = [1] x2 + [4] p(c_30) = [1] x1 + [1] p(c_31) = [0] p(c_32) = [0] p(c_33) = [0] Following rules are strictly oriented: U22#(tt(),V1) = [1] V1 + [1] > [1] V1 + [0] = c_9(isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = c_6(isNat#(activate(V2))) U21#(tt(),V1) = [1] V1 + [1] >= [1] V1 + [1] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U61#(tt(),M,N) = [2] N + [6] >= [2] N + [5] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) = [2] N + [2] >= [2] N + [2] = c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [1] V1 + [2] >= [1] V1 + [2] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) 0() = [0] >= [0] = n__0() U51(tt(),N) = [1] N + [0] >= [1] N + [0] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [2] >= [1] X + [2] = s(activate(X)) plus(N,0()) = [1] N + [0] >= [1] N + [0] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) s(X) = [1] X + [2] >= [1] X + [2] = n__s(X) ** Step 6.a:11: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [5] p(U11) = [6] p(U12) = [2] x2 + [0] p(U13) = [4] x2 + [0] p(U14) = [2] x2 + [1] x3 + [0] p(U15) = [4] x2 + [3] p(U16) = [2] p(U21) = [0] p(U22) = [4] x1 + [2] x2 + [4] p(U23) = [0] p(U31) = [1] x1 + [1] x2 + [4] p(U32) = [2] x1 + [6] p(U41) = [1] p(U51) = [1] x2 + [3] p(U52) = [1] x2 + [0] p(U61) = [1] x2 + [1] x3 + [6] p(U62) = [1] x2 + [1] x3 + [6] p(U63) = [1] x2 + [1] x3 + [6] p(U64) = [1] x2 + [1] x3 + [6] p(activate) = [1] x1 + [0] p(isNat) = [4] p(isNatKind) = [0] p(n__0) = [5] p(n__plus) = [1] x1 + [1] x2 + [4] p(n__s) = [1] x1 + [2] p(plus) = [1] x1 + [1] x2 + [4] p(s) = [1] x1 + [2] p(tt) = [0] p(0#) = [2] p(U11#) = [2] x2 + [2] x3 + [6] p(U12#) = [2] x2 + [2] x3 + [6] p(U13#) = [2] x2 + [2] x3 + [6] p(U14#) = [2] x2 + [2] x3 + [6] p(U15#) = [2] x2 + [3] p(U16#) = [1] p(U21#) = [2] x2 + [3] p(U22#) = [2] x2 + [3] p(U23#) = [1] x1 + [0] p(U31#) = [4] x1 + [1] p(U32#) = [1] x1 + [1] p(U41#) = [1] x1 + [0] p(U51#) = [4] p(U52#) = [1] p(U61#) = [4] x2 + [4] x3 + [7] p(U62#) = [4] x3 + [3] p(U63#) = [2] x2 + [4] x3 + [1] p(U64#) = [2] x2 + [2] p(activate#) = [1] p(isNat#) = [2] x1 + [3] p(isNatKind#) = [2] p(plus#) = [4] x1 + [1] p(s#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [2] x1 + [1] p(c_12) = [0] p(c_13) = [2] p(c_14) = [2] x4 + [0] p(c_15) = [2] x1 + [4] p(c_16) = [1] x1 + [4] p(c_17) = [1] x1 + [0] p(c_18) = [1] x3 + [1] x4 + [0] p(c_19) = [2] p(c_20) = [4] p(c_21) = [0] p(c_22) = [2] x1 + [0] p(c_23) = [1] x1 + [0] p(c_24) = [1] p(c_25) = [1] x1 + [4] p(c_26) = [1] x1 + [1] p(c_27) = [1] p(c_28) = [1] x1 + [0] p(c_29) = [1] x1 + [1] x2 + [4] p(c_30) = [0] p(c_31) = [2] x1 + [1] x2 + [0] p(c_32) = [2] p(c_33) = [1] Following rules are strictly oriented: isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [11] > [2] V1 + [2] V2 + [10] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [6] >= [2] V1 + [2] V2 + [6] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [6] >= [2] V1 + [2] V2 + [6] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [6] >= [2] V1 + [2] V2 + [6] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [6] >= [2] V1 + [2] V2 + [6] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [2] V2 + [3] >= [2] V2 + [3] = c_6(isNat#(activate(V2))) U21#(tt(),V1) = [2] V1 + [3] >= [2] V1 + [3] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [2] V1 + [3] >= [2] V1 + [3] = c_9(isNat#(activate(V1))) U61#(tt(),M,N) = [4] M + [4] N + [7] >= [4] N + [7] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) = [4] N + [3] >= [2] N + [3] = c_17(isNat#(activate(N))) isNat#(n__s(V1)) = [2] V1 + [7] >= [2] V1 + [4] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) 0() = [5] >= [5] = n__0() U51(tt(),N) = [1] N + [3] >= [1] N + [0] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [5] >= [5] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [2] >= [1] X + [2] = s(activate(X)) plus(N,0()) = [1] N + [9] >= [1] N + [3] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = n__plus(X1,X2) s(X) = [1] X + [2] >= [1] X + [2] = n__s(X) ** Step 6.a:12: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [4] x2 + [1] x3 + [2] p(U12) = [1] x2 + [2] x3 + [0] p(U13) = [1] x1 + [7] p(U14) = [2] x3 + [4] p(U15) = [1] x1 + [1] x2 + [4] p(U16) = [4] x1 + [4] p(U21) = [2] x1 + [2] x2 + [4] p(U22) = [2] x1 + [3] p(U23) = [4] x1 + [3] p(U31) = [1] x1 + [5] p(U32) = [0] p(U41) = [4] x1 + [0] p(U51) = [1] x2 + [0] p(U52) = [1] x2 + [0] p(U61) = [1] x2 + [1] x3 + [3] p(U62) = [1] x2 + [1] x3 + [3] p(U63) = [1] x2 + [1] x3 + [3] p(U64) = [1] x2 + [1] x3 + [3] p(activate) = [1] x1 + [0] p(isNat) = [2] x1 + [0] p(isNatKind) = [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [1] p(plus) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [1] p(tt) = [0] p(0#) = [0] p(U11#) = [2] x2 + [2] x3 + [2] p(U12#) = [2] x2 + [2] x3 + [2] p(U13#) = [2] x2 + [2] x3 + [2] p(U14#) = [2] x2 + [2] x3 + [2] p(U15#) = [2] x2 + [1] p(U16#) = [2] p(U21#) = [2] x2 + [2] p(U22#) = [2] x2 + [2] p(U23#) = [1] p(U31#) = [1] x1 + [1] x2 + [0] p(U32#) = [4] x1 + [1] p(U41#) = [2] p(U51#) = [1] x1 + [1] p(U52#) = [1] x2 + [0] p(U61#) = [1] x2 + [4] x3 + [5] p(U62#) = [4] x3 + [1] p(U63#) = [1] x1 + [1] p(U64#) = [2] x1 + [1] x2 + [0] p(activate#) = [2] x1 + [0] p(isNat#) = [2] x1 + [0] p(isNatKind#) = [2] p(plus#) = [1] p(s#) = [1] x1 + [4] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [1] p(c_6) = [1] x1 + [0] p(c_7) = [2] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [2] p(c_12) = [0] p(c_13) = [1] p(c_14) = [1] x2 + [1] x3 + [2] p(c_15) = [0] p(c_16) = [1] x1 + [0] p(c_17) = [2] x1 + [0] p(c_18) = [2] x1 + [0] p(c_19) = [1] x1 + [0] p(c_20) = [1] p(c_21) = [1] p(c_22) = [4] x1 + [2] x2 + [1] p(c_23) = [1] x1 + [1] p(c_24) = [4] p(c_25) = [1] x1 + [2] p(c_26) = [1] x1 + [0] p(c_27) = [1] p(c_28) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [0] p(c_29) = [1] p(c_30) = [2] x2 + [0] p(c_31) = [1] p(c_32) = [0] p(c_33) = [1] Following rules are strictly oriented: U15#(tt(),V2) = [2] V2 + [1] > [2] V2 + [0] = c_6(isNat#(activate(V2))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [2] >= [2] V1 + [2] V2 + [2] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [2] >= [2] V1 + [2] V2 + [2] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [2] >= [2] V1 + [2] V2 + [2] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [2] >= [2] V1 + [2] V2 + [2] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U21#(tt(),V1) = [2] V1 + [2] >= [2] V1 + [2] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [2] V1 + [2] >= [2] V1 + [0] = c_9(isNat#(activate(V1))) U61#(tt(),M,N) = [1] M + [4] N + [5] >= [4] N + [1] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) = [4] N + [1] >= [4] N + [0] = c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [4] >= [2] V1 + [2] V2 + [4] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [2] V1 + [2] >= [2] V1 + [2] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) 0() = [0] >= [0] = n__0() U51(tt(),N) = [1] N + [0] >= [1] N + [0] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [1] >= [1] X + [1] = s(activate(X)) plus(N,0()) = [1] N + [2] >= [1] N + [0] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__plus(X1,X2) s(X) = [1] X + [1] >= [1] X + [1] = n__s(X) ** Step 6.a:13: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [1] x2 + [4] x3 + [0] p(U12) = [1] x2 + [1] x3 + [0] p(U13) = [2] x2 + [1] x3 + [0] p(U14) = [0] p(U15) = [4] x1 + [0] p(U16) = [1] x1 + [7] p(U21) = [4] x1 + [4] x2 + [0] p(U22) = [2] x1 + [5] p(U23) = [5] x1 + [0] p(U31) = [6] x2 + [0] p(U32) = [1] x1 + [6] p(U41) = [1] p(U51) = [1] x2 + [2] p(U52) = [1] x2 + [2] p(U61) = [1] x2 + [1] x3 + [3] p(U62) = [1] x2 + [1] x3 + [3] p(U63) = [1] x2 + [1] x3 + [3] p(U64) = [1] x2 + [1] x3 + [3] p(activate) = [1] x1 + [0] p(isNat) = [1] x1 + [3] p(isNatKind) = [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [1] p(plus) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [1] p(tt) = [0] p(0#) = [1] p(U11#) = [4] x2 + [4] x3 + [6] p(U12#) = [4] x2 + [4] x3 + [6] p(U13#) = [4] x2 + [4] x3 + [6] p(U14#) = [4] x2 + [4] x3 + [4] p(U15#) = [4] x2 + [2] p(U16#) = [1] x1 + [0] p(U21#) = [4] x2 + [1] p(U22#) = [4] x2 + [1] p(U23#) = [1] p(U31#) = [4] x1 + [1] p(U32#) = [0] p(U41#) = [1] x1 + [1] p(U51#) = [2] x1 + [0] p(U52#) = [1] x1 + [1] x2 + [4] p(U61#) = [1] x1 + [4] x3 + [0] p(U62#) = [4] x3 + [0] p(U63#) = [4] x1 + [1] p(U64#) = [1] x3 + [1] p(activate#) = [1] p(isNat#) = [4] x1 + [0] p(isNatKind#) = [1] p(plus#) = [4] x1 + [1] x2 + [1] p(s#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [4] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [1] p(c_10) = [1] p(c_11) = [1] p(c_12) = [0] p(c_13) = [4] p(c_14) = [1] x2 + [4] x3 + [4] x4 + [0] p(c_15) = [1] x1 + [1] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [1] x3 + [1] x4 + [1] x5 + [1] p(c_19) = [4] x1 + [1] p(c_20) = [0] p(c_21) = [4] x1 + [1] p(c_22) = [1] x1 + [1] p(c_23) = [1] x1 + [4] p(c_24) = [1] p(c_25) = [1] x1 + [2] p(c_26) = [1] x1 + [0] p(c_27) = [0] p(c_28) = [2] x3 + [1] x4 + [1] p(c_29) = [2] x1 + [2] p(c_30) = [2] x2 + [0] p(c_31) = [2] p(c_32) = [1] p(c_33) = [1] Following rules are strictly oriented: U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [6] > [4] V1 + [4] V2 + [4] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [4] > [4] V1 + [4] V2 + [2] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [6] >= [4] V1 + [4] V2 + [6] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [6] >= [4] V1 + [4] V2 + [6] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U15#(tt(),V2) = [4] V2 + [2] >= [4] V2 + [0] = c_6(isNat#(activate(V2))) U21#(tt(),V1) = [4] V1 + [1] >= [4] V1 + [1] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [4] V1 + [1] >= [4] V1 + [1] = c_9(isNat#(activate(V1))) U61#(tt(),M,N) = [4] N + [0] >= [4] N + [0] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) = [4] N + [0] >= [4] N + [0] = c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [4] V2 + [8] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [4] V1 + [4] >= [4] V1 + [1] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) 0() = [0] >= [0] = n__0() U51(tt(),N) = [1] N + [2] >= [1] N + [2] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [2] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [1] >= [1] X + [1] = s(activate(X)) plus(N,0()) = [1] N + [2] >= [1] N + [2] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__plus(X1,X2) s(X) = [1] X + [1] >= [1] X + [1] = n__s(X) ** Step 6.a:14: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [1] p(U11) = [3] x3 + [4] p(U12) = [4] x1 + [1] x2 + [1] x3 + [0] p(U13) = [2] x2 + [1] x3 + [3] p(U14) = [2] x3 + [0] p(U15) = [2] x2 + [4] p(U16) = [2] p(U21) = [0] p(U22) = [2] x1 + [1] p(U23) = [5] p(U31) = [2] p(U32) = [0] p(U41) = [0] p(U51) = [1] x2 + [2] p(U52) = [1] x2 + [2] p(U61) = [1] x2 + [1] x3 + [2] p(U62) = [1] x2 + [1] x3 + [2] p(U63) = [1] x2 + [1] x3 + [2] p(U64) = [1] x2 + [1] x3 + [2] p(activate) = [1] x1 + [0] p(isNat) = [0] p(isNatKind) = [0] p(n__0) = [1] p(n__plus) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [0] p(tt) = [0] p(0#) = [4] p(U11#) = [4] x2 + [4] x3 + [1] p(U12#) = [4] x2 + [4] x3 + [1] p(U13#) = [4] x2 + [4] x3 + [0] p(U14#) = [4] x2 + [4] x3 + [0] p(U15#) = [4] x2 + [0] p(U16#) = [1] x1 + [0] p(U21#) = [4] x2 + [0] p(U22#) = [4] x2 + [0] p(U23#) = [0] p(U31#) = [1] x2 + [4] p(U32#) = [4] p(U41#) = [4] p(U51#) = [1] x2 + [1] p(U52#) = [0] p(U61#) = [4] x2 + [4] x3 + [5] p(U62#) = [2] x2 + [4] x3 + [3] p(U63#) = [1] x2 + [1] p(U64#) = [1] x3 + [1] p(activate#) = [4] x1 + [1] p(isNat#) = [4] x1 + [0] p(isNatKind#) = [1] p(plus#) = [0] p(s#) = [1] p(c_1) = [2] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [2] p(c_12) = [0] p(c_13) = [0] p(c_14) = [2] x1 + [1] x4 + [1] p(c_15) = [0] p(c_16) = [1] x1 + [1] p(c_17) = [1] x1 + [3] p(c_18) = [1] x1 + [4] p(c_19) = [1] x1 + [1] x2 + [0] p(c_20) = [1] p(c_21) = [0] p(c_22) = [2] p(c_23) = [4] p(c_24) = [2] p(c_25) = [1] x1 + [7] p(c_26) = [1] x1 + [0] p(c_27) = [0] p(c_28) = [1] p(c_29) = [1] p(c_30) = [0] p(c_31) = [2] x1 + [4] p(c_32) = [1] p(c_33) = [4] Following rules are strictly oriented: U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [1] > [4] V1 + [4] V2 + [0] = c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [1] >= [4] V1 + [4] V2 + [1] = c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = c_6(isNat#(activate(V2))) U21#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = c_9(isNat#(activate(V1))) U61#(tt(),M,N) = [4] M + [4] N + [5] >= [2] M + [4] N + [4] = c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) = [2] M + [4] N + [3] >= [4] N + [3] = c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [4] V2 + [8] = c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) = [4] V1 + [0] >= [4] V1 + [0] = c_26(U21#(isNatKind(activate(V1)),activate(V1))) 0() = [1] >= [1] = n__0() U51(tt(),N) = [1] N + [2] >= [1] N + [2] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [2] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [1] >= [1] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(activate(X)) plus(N,0()) = [1] N + [3] >= [1] N + [2] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__plus(X1,X2) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) ** Step 6.a:15: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))) U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))) U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))) U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) U15#(tt(),V2) -> c_6(isNat#(activate(V2))) U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))) U22#(tt(),V1) -> c_9(isNat#(activate(V1))) U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))) U62#(tt(),M,N) -> c_17(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))) isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1 ,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) and a lower component activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) Further, following extension rules are added to the lower component. U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) *** Step 6.b:1.a:1: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U15#(tt(),V2) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> activate#(V1) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U51#(tt(),N) -> activate#(N) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> isNatKind#(activate(N)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {13,15,16} by application of Pre({13,15,16}) = {}. Here rules are labelled as follows: 1: U11#(tt(),V1,V2) -> activate#(V1) 2: U11#(tt(),V1,V2) -> activate#(V2) 3: U12#(tt(),V1,V2) -> activate#(V1) 4: U12#(tt(),V1,V2) -> activate#(V2) 5: U13#(tt(),V1,V2) -> activate#(V1) 6: U13#(tt(),V1,V2) -> activate#(V2) 7: U14#(tt(),V1,V2) -> activate#(V1) 8: U14#(tt(),V1,V2) -> activate#(V2) 9: U15#(tt(),V2) -> activate#(V2) 10: U21#(tt(),V1) -> activate#(V1) 11: U22#(tt(),V1) -> activate#(V1) 12: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) 13: U51#(tt(),N) -> activate#(N) 14: U52#(tt(),N) -> activate#(N) 15: U61#(tt(),M,N) -> activate#(M) 16: U61#(tt(),M,N) -> activate#(N) 17: U62#(tt(),M,N) -> activate#(M) 18: U62#(tt(),M,N) -> activate#(N) 19: U63#(tt(),M,N) -> activate#(M) 20: U63#(tt(),M,N) -> activate#(N) 21: U64#(tt(),M,N) -> activate#(M) 22: U64#(tt(),M,N) -> activate#(N) 23: isNat#(n__plus(V1,V2)) -> activate#(V1) 24: isNat#(n__plus(V1,V2)) -> activate#(V2) 25: isNat#(n__s(V1)) -> activate#(V1) 26: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) 27: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) 28: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) 29: U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) 30: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) 31: U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) 32: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) 33: U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) 34: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) 35: U14#(tt(),V1,V2) -> isNat#(activate(V1)) 36: U15#(tt(),V2) -> isNat#(activate(V2)) 37: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) 38: U21#(tt(),V1) -> isNatKind#(activate(V1)) 39: U22#(tt(),V1) -> isNat#(activate(V1)) 40: U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) 41: U51#(tt(),N) -> isNatKind#(activate(N)) 42: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) 43: U61#(tt(),M,N) -> isNatKind#(activate(M)) 44: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) 45: U62#(tt(),M,N) -> isNat#(activate(N)) 46: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) 47: U63#(tt(),M,N) -> isNatKind#(activate(N)) 48: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) 49: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) 50: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) 51: isNat#(n__s(V1)) -> isNatKind#(activate(V1)) *** Step 6.b:1.a:2: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U15#(tt(),V2) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> activate#(V1) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U52#(tt(),N) -> activate#(N) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> isNatKind#(activate(N)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> activate#(V1) 2:S:U11#(tt(),V1,V2) -> activate#(V2) 3:S:U12#(tt(),V1,V2) -> activate#(V1) 4:S:U12#(tt(),V1,V2) -> activate#(V2) 5:S:U13#(tt(),V1,V2) -> activate#(V1) 6:S:U13#(tt(),V1,V2) -> activate#(V2) 7:S:U14#(tt(),V1,V2) -> activate#(V1) 8:S:U14#(tt(),V1,V2) -> activate#(V2) 9:S:U15#(tt(),V2) -> activate#(V2) 10:S:U21#(tt(),V1) -> activate#(V1) 11:S:U22#(tt(),V1) -> activate#(V1) 12:S:U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 13:S:U52#(tt(),N) -> activate#(N) 14:S:U62#(tt(),M,N) -> activate#(M) 15:S:U62#(tt(),M,N) -> activate#(N) 16:S:U63#(tt(),M,N) -> activate#(M) 17:S:U63#(tt(),M,N) -> activate#(N) 18:S:U64#(tt(),M,N) -> activate#(M) 19:S:U64#(tt(),M,N) -> activate#(N) 20:S:isNat#(n__plus(V1,V2)) -> activate#(V1) 21:S:isNat#(n__plus(V1,V2)) -> activate#(V2) 22:S:isNat#(n__s(V1)) -> activate#(V1) 23:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 -->_1 U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)):12 24:S:isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 25:W:U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -->_1 U12#(tt(),V1,V2) -> isNatKind#(activate(V2)):28 -->_1 U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)):27 -->_1 U12#(tt(),V1,V2) -> activate#(V2):4 -->_1 U12#(tt(),V1,V2) -> activate#(V1):3 26:W:U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 27:W:U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -->_1 U13#(tt(),V1,V2) -> isNatKind#(activate(V2)):30 -->_1 U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)):29 -->_1 U13#(tt(),V1,V2) -> activate#(V2):6 -->_1 U13#(tt(),V1,V2) -> activate#(V1):5 28:W:U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 29:W:U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -->_1 U14#(tt(),V1,V2) -> isNat#(activate(V1)):32 -->_1 U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)):31 -->_1 U14#(tt(),V1,V2) -> activate#(V2):8 -->_1 U14#(tt(),V1,V2) -> activate#(V1):7 30:W:U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 31:W:U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -->_1 U15#(tt(),V2) -> isNat#(activate(V2)):33 -->_1 U15#(tt(),V2) -> activate#(V2):9 32:W:U14#(tt(),V1,V2) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48 -->_1 isNat#(n__s(V1)) -> activate#(V1):22 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20 33:W:U15#(tt(),V2) -> isNat#(activate(V2)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48 -->_1 isNat#(n__s(V1)) -> activate#(V1):22 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20 34:W:U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -->_1 U22#(tt(),V1) -> isNat#(activate(V1)):36 -->_1 U22#(tt(),V1) -> activate#(V1):11 35:W:U21#(tt(),V1) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 36:W:U22#(tt(),V1) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48 -->_1 isNat#(n__s(V1)) -> activate#(V1):22 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20 37:W:U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) -->_1 U52#(tt(),N) -> activate#(N):13 38:W:U51#(tt(),N) -> activate#(N) 39:W:U51#(tt(),N) -> isNatKind#(activate(N)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 40:W:U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -->_1 U62#(tt(),M,N) -> isNat#(activate(N)):45 -->_1 U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)):44 -->_1 U62#(tt(),M,N) -> activate#(N):15 -->_1 U62#(tt(),M,N) -> activate#(M):14 41:W:U61#(tt(),M,N) -> activate#(M) 42:W:U61#(tt(),M,N) -> activate#(N) 43:W:U61#(tt(),M,N) -> isNatKind#(activate(M)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 44:W:U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -->_1 U63#(tt(),M,N) -> isNatKind#(activate(N)):47 -->_1 U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)):46 -->_1 U63#(tt(),M,N) -> activate#(N):17 -->_1 U63#(tt(),M,N) -> activate#(M):16 45:W:U62#(tt(),M,N) -> isNat#(activate(N)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48 -->_1 isNat#(n__s(V1)) -> activate#(V1):22 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20 46:W:U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -->_1 U64#(tt(),M,N) -> activate#(N):19 -->_1 U64#(tt(),M,N) -> activate#(M):18 47:W:U63#(tt(),M,N) -> isNatKind#(activate(N)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 48:W:isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -->_1 U11#(tt(),V1,V2) -> isNatKind#(activate(V1)):26 -->_1 U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)):25 -->_1 U11#(tt(),V1,V2) -> activate#(V2):2 -->_1 U11#(tt(),V1,V2) -> activate#(V1):1 49:W:isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 50:W:isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -->_1 U21#(tt(),V1) -> isNatKind#(activate(V1)):35 -->_1 U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)):34 -->_1 U21#(tt(),V1) -> activate#(V1):10 51:W:isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 42: U61#(tt(),M,N) -> activate#(N) 41: U61#(tt(),M,N) -> activate#(M) 38: U51#(tt(),N) -> activate#(N) *** Step 6.b:1.a:3: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U15#(tt(),V2) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> activate#(V1) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) U52#(tt(),N) -> activate#(N) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> isNatKind#(activate(N)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),V1,V2) -> activate#(V1) 2:S:U11#(tt(),V1,V2) -> activate#(V2) 3:S:U12#(tt(),V1,V2) -> activate#(V1) 4:S:U12#(tt(),V1,V2) -> activate#(V2) 5:S:U13#(tt(),V1,V2) -> activate#(V1) 6:S:U13#(tt(),V1,V2) -> activate#(V2) 7:S:U14#(tt(),V1,V2) -> activate#(V1) 8:S:U14#(tt(),V1,V2) -> activate#(V2) 9:S:U15#(tt(),V2) -> activate#(V2) 10:S:U21#(tt(),V1) -> activate#(V1) 11:S:U22#(tt(),V1) -> activate#(V1) 12:S:U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 13:S:U52#(tt(),N) -> activate#(N) 14:S:U62#(tt(),M,N) -> activate#(M) 15:S:U62#(tt(),M,N) -> activate#(N) 16:S:U63#(tt(),M,N) -> activate#(M) 17:S:U63#(tt(),M,N) -> activate#(N) 18:S:U64#(tt(),M,N) -> activate#(M) 19:S:U64#(tt(),M,N) -> activate#(N) 20:S:isNat#(n__plus(V1,V2)) -> activate#(V1) 21:S:isNat#(n__plus(V1,V2)) -> activate#(V2) 22:S:isNat#(n__s(V1)) -> activate#(V1) 23:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 -->_1 U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)):12 24:S:isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 25:W:U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -->_1 U12#(tt(),V1,V2) -> isNatKind#(activate(V2)):28 -->_1 U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)):27 -->_1 U12#(tt(),V1,V2) -> activate#(V2):4 -->_1 U12#(tt(),V1,V2) -> activate#(V1):3 26:W:U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 27:W:U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -->_1 U13#(tt(),V1,V2) -> isNatKind#(activate(V2)):30 -->_1 U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)):29 -->_1 U13#(tt(),V1,V2) -> activate#(V2):6 -->_1 U13#(tt(),V1,V2) -> activate#(V1):5 28:W:U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 29:W:U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -->_1 U14#(tt(),V1,V2) -> isNat#(activate(V1)):32 -->_1 U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)):31 -->_1 U14#(tt(),V1,V2) -> activate#(V2):8 -->_1 U14#(tt(),V1,V2) -> activate#(V1):7 30:W:U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 31:W:U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -->_1 U15#(tt(),V2) -> isNat#(activate(V2)):33 -->_1 U15#(tt(),V2) -> activate#(V2):9 32:W:U14#(tt(),V1,V2) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48 -->_1 isNat#(n__s(V1)) -> activate#(V1):22 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20 33:W:U15#(tt(),V2) -> isNat#(activate(V2)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48 -->_1 isNat#(n__s(V1)) -> activate#(V1):22 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20 34:W:U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -->_1 U22#(tt(),V1) -> isNat#(activate(V1)):36 -->_1 U22#(tt(),V1) -> activate#(V1):11 35:W:U21#(tt(),V1) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 36:W:U22#(tt(),V1) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48 -->_1 isNat#(n__s(V1)) -> activate#(V1):22 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20 37:W:U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) -->_1 U52#(tt(),N) -> activate#(N):13 39:W:U51#(tt(),N) -> isNatKind#(activate(N)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 40:W:U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -->_1 U62#(tt(),M,N) -> isNat#(activate(N)):45 -->_1 U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)):44 -->_1 U62#(tt(),M,N) -> activate#(N):15 -->_1 U62#(tt(),M,N) -> activate#(M):14 43:W:U61#(tt(),M,N) -> isNatKind#(activate(M)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 44:W:U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -->_1 U63#(tt(),M,N) -> isNatKind#(activate(N)):47 -->_1 U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)):46 -->_1 U63#(tt(),M,N) -> activate#(N):17 -->_1 U63#(tt(),M,N) -> activate#(M):16 45:W:U62#(tt(),M,N) -> isNat#(activate(N)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48 -->_1 isNat#(n__s(V1)) -> activate#(V1):22 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20 46:W:U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -->_1 U64#(tt(),M,N) -> activate#(N):19 -->_1 U64#(tt(),M,N) -> activate#(M):18 47:W:U63#(tt(),M,N) -> isNatKind#(activate(N)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 48:W:isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -->_1 U11#(tt(),V1,V2) -> isNatKind#(activate(V1)):26 -->_1 U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)):25 -->_1 U11#(tt(),V1,V2) -> activate#(V2):2 -->_1 U11#(tt(),V1,V2) -> activate#(V1):1 49:W:isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 50:W:isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -->_1 U21#(tt(),V1) -> isNatKind#(activate(V1)):35 -->_1 U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)):34 -->_1 U21#(tt(),V1) -> activate#(V1):10 51:W:isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24 -->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)) ,isNatKind#(activate(V1)) ,activate#(V1) ,activate#(V2)):23 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) *** Step 6.b:1.a:4: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U15#(tt(),V2) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> activate#(V1) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U52#(tt(),N) -> activate#(N) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> isNatKind#(activate(N)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_28) = {1,2}, uargs(c_29) = {1} Following symbols are considered usable: {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate# ,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [4] x2 + [0] p(U12) = [0] p(U13) = [0] p(U14) = [0] p(U15) = [0] p(U16) = [0] p(U21) = [1] x2 + [1] p(U22) = [0] p(U23) = [1] x1 + [0] p(U31) = [0] p(U32) = [3] p(U41) = [0] p(U51) = [2] x1 + [0] p(U52) = [0] p(U61) = [5] p(U62) = [0] p(U63) = [4] x3 + [0] p(U64) = [0] p(activate) = [1] p(isNat) = [2] x1 + [6] p(isNatKind) = [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] p(plus) = [2] p(s) = [1] x1 + [1] p(tt) = [0] p(0#) = [0] p(U11#) = [0] p(U12#) = [0] p(U13#) = [0] p(U14#) = [0] p(U15#) = [0] p(U16#) = [0] p(U21#) = [0] p(U22#) = [0] p(U23#) = [0] p(U31#) = [0] p(U32#) = [0] p(U41#) = [0] p(U51#) = [0] p(U52#) = [0] p(U61#) = [5] p(U62#) = [4] p(U63#) = [4] p(U64#) = [0] p(activate#) = [0] p(isNat#) = [0] p(isNatKind#) = [0] p(plus#) = [0] p(s#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [2] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [2] x1 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [1] x4 + [0] p(c_27) = [0] p(c_28) = [4] x1 + [1] x2 + [0] p(c_29) = [1] x1 + [0] p(c_30) = [1] x1 + [0] p(c_31) = [2] x2 + [0] p(c_32) = [1] p(c_33) = [0] Following rules are strictly oriented: U62#(tt(),M,N) = [4] > [0] = activate#(M) U62#(tt(),M,N) = [4] > [0] = activate#(N) U63#(tt(),M,N) = [4] > [0] = activate#(M) U63#(tt(),M,N) = [4] > [0] = activate#(N) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [0] >= [0] = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [0] >= [0] = activate#(V1) U11#(tt(),V1,V2) = [0] >= [0] = activate#(V2) U11#(tt(),V1,V2) = [0] >= [0] = isNatKind#(activate(V1)) U12#(tt(),V1,V2) = [0] >= [0] = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [0] >= [0] = activate#(V1) U12#(tt(),V1,V2) = [0] >= [0] = activate#(V2) U12#(tt(),V1,V2) = [0] >= [0] = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [0] >= [0] = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [0] >= [0] = activate#(V1) U13#(tt(),V1,V2) = [0] >= [0] = activate#(V2) U13#(tt(),V1,V2) = [0] >= [0] = isNatKind#(activate(V2)) U14#(tt(),V1,V2) = [0] >= [0] = U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) = [0] >= [0] = activate#(V1) U14#(tt(),V1,V2) = [0] >= [0] = activate#(V2) U14#(tt(),V1,V2) = [0] >= [0] = isNat#(activate(V1)) U15#(tt(),V2) = [0] >= [0] = activate#(V2) U15#(tt(),V2) = [0] >= [0] = isNat#(activate(V2)) U21#(tt(),V1) = [0] >= [0] = U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [0] >= [0] = activate#(V1) U21#(tt(),V1) = [0] >= [0] = isNatKind#(activate(V1)) U22#(tt(),V1) = [0] >= [0] = activate#(V1) U22#(tt(),V1) = [0] >= [0] = isNat#(activate(V1)) U31#(tt(),V2) = [0] >= [0] = c_11(isNatKind#(activate(V2))) U51#(tt(),N) = [0] >= [0] = U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) = [0] >= [0] = isNatKind#(activate(N)) U52#(tt(),N) = [0] >= [0] = activate#(N) U61#(tt(),M,N) = [5] >= [4] = U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) = [5] >= [0] = isNatKind#(activate(M)) U62#(tt(),M,N) = [4] >= [4] = U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) = [4] >= [0] = isNat#(activate(N)) U63#(tt(),M,N) = [4] >= [0] = U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [4] >= [0] = isNatKind#(activate(N)) U64#(tt(),M,N) = [0] >= [0] = activate#(M) U64#(tt(),M,N) = [0] >= [0] = activate#(N) isNat#(n__plus(V1,V2)) = [0] >= [0] = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) = [0] >= [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [0] >= [0] = activate#(V2) isNat#(n__plus(V1,V2)) = [0] >= [0] = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [0] >= [0] = U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) = [0] >= [0] = activate#(V1) isNat#(n__s(V1)) = [0] >= [0] = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [0] >= [0] = c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) = [0] >= [0] = c_29(isNatKind#(activate(V1))) *** Step 6.b:1.a:5: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U15#(tt(),V2) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> activate#(V1) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U52#(tt(),N) -> activate#(N) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_28) = {1,2}, uargs(c_29) = {1} Following symbols are considered usable: {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate# ,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U12) = [0] p(U13) = [3] p(U14) = [0] p(U15) = [0] p(U16) = [0] p(U21) = [1] x2 + [0] p(U22) = [0] p(U23) = [0] p(U31) = [0] p(U32) = [0] p(U41) = [3] p(U51) = [0] p(U52) = [2] x1 + [0] p(U61) = [0] p(U62) = [0] p(U63) = [0] p(U64) = [0] p(activate) = [4] p(isNat) = [5] p(isNatKind) = [1] x1 + [2] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [0] p(plus) = [0] p(s) = [0] p(tt) = [0] p(0#) = [0] p(U11#) = [0] p(U12#) = [0] p(U13#) = [0] p(U14#) = [0] p(U15#) = [0] p(U16#) = [0] p(U21#) = [0] p(U22#) = [0] p(U23#) = [0] p(U31#) = [0] p(U32#) = [0] p(U41#) = [0] p(U51#) = [1] p(U52#) = [1] p(U61#) = [1] x1 + [3] p(U62#) = [2] p(U63#) = [2] p(U64#) = [2] p(activate#) = [0] p(isNat#) = [0] p(isNatKind#) = [0] p(plus#) = [0] p(s#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x2 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [2] x1 + [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [4] x1 + [1] x2 + [0] p(c_29) = [1] x1 + [0] p(c_30) = [1] x1 + [1] x2 + [0] p(c_31) = [1] x1 + [0] p(c_32) = [0] p(c_33) = [0] Following rules are strictly oriented: U52#(tt(),N) = [1] > [0] = activate#(N) U64#(tt(),M,N) = [2] > [0] = activate#(M) U64#(tt(),M,N) = [2] > [0] = activate#(N) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [0] >= [0] = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [0] >= [0] = activate#(V1) U11#(tt(),V1,V2) = [0] >= [0] = activate#(V2) U11#(tt(),V1,V2) = [0] >= [0] = isNatKind#(activate(V1)) U12#(tt(),V1,V2) = [0] >= [0] = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [0] >= [0] = activate#(V1) U12#(tt(),V1,V2) = [0] >= [0] = activate#(V2) U12#(tt(),V1,V2) = [0] >= [0] = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [0] >= [0] = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [0] >= [0] = activate#(V1) U13#(tt(),V1,V2) = [0] >= [0] = activate#(V2) U13#(tt(),V1,V2) = [0] >= [0] = isNatKind#(activate(V2)) U14#(tt(),V1,V2) = [0] >= [0] = U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) = [0] >= [0] = activate#(V1) U14#(tt(),V1,V2) = [0] >= [0] = activate#(V2) U14#(tt(),V1,V2) = [0] >= [0] = isNat#(activate(V1)) U15#(tt(),V2) = [0] >= [0] = activate#(V2) U15#(tt(),V2) = [0] >= [0] = isNat#(activate(V2)) U21#(tt(),V1) = [0] >= [0] = U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [0] >= [0] = activate#(V1) U21#(tt(),V1) = [0] >= [0] = isNatKind#(activate(V1)) U22#(tt(),V1) = [0] >= [0] = activate#(V1) U22#(tt(),V1) = [0] >= [0] = isNat#(activate(V1)) U31#(tt(),V2) = [0] >= [0] = c_11(isNatKind#(activate(V2))) U51#(tt(),N) = [1] >= [1] = U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) = [1] >= [0] = isNatKind#(activate(N)) U61#(tt(),M,N) = [3] >= [2] = U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) = [3] >= [0] = isNatKind#(activate(M)) U62#(tt(),M,N) = [2] >= [2] = U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) = [2] >= [0] = activate#(M) U62#(tt(),M,N) = [2] >= [0] = activate#(N) U62#(tt(),M,N) = [2] >= [0] = isNat#(activate(N)) U63#(tt(),M,N) = [2] >= [2] = U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [2] >= [0] = activate#(M) U63#(tt(),M,N) = [2] >= [0] = activate#(N) U63#(tt(),M,N) = [2] >= [0] = isNatKind#(activate(N)) isNat#(n__plus(V1,V2)) = [0] >= [0] = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) = [0] >= [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [0] >= [0] = activate#(V2) isNat#(n__plus(V1,V2)) = [0] >= [0] = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [0] >= [0] = U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) = [0] >= [0] = activate#(V1) isNat#(n__s(V1)) = [0] >= [0] = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [0] >= [0] = c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) = [0] >= [0] = c_29(isNatKind#(activate(V1))) *** Step 6.b:1.a:6: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U15#(tt(),V2) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> activate#(V1) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_28) = {1,2}, uargs(c_29) = {1} Following symbols are considered usable: {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate# ,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U12) = [0] p(U13) = [0] p(U14) = [0] p(U15) = [0] p(U16) = [0] p(U21) = [0] p(U22) = [0] p(U23) = [0] p(U31) = [0] p(U32) = [0] p(U41) = [0] p(U51) = [0] p(U52) = [0] p(U61) = [0] p(U62) = [0] p(U63) = [0] p(U64) = [7] p(activate) = [0] p(isNat) = [0] p(isNatKind) = [0] p(n__0) = [0] p(n__plus) = [1] x2 + [0] p(n__s) = [0] p(plus) = [0] p(s) = [0] p(tt) = [0] p(0#) = [0] p(U11#) = [4] p(U12#) = [4] p(U13#) = [4] p(U14#) = [4] p(U15#) = [4] p(U16#) = [0] p(U21#) = [4] p(U22#) = [4] p(U23#) = [0] p(U31#) = [0] p(U32#) = [0] p(U41#) = [0] p(U51#) = [4] p(U52#) = [3] p(U61#) = [1] x2 + [7] p(U62#) = [7] p(U63#) = [0] p(U64#) = [0] p(activate#) = [0] p(isNat#) = [4] p(isNatKind#) = [0] p(plus#) = [0] p(s#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x3 + [1] p(c_6) = [1] p(c_7) = [2] p(c_8) = [4] x3 + [1] p(c_9) = [4] x1 + [4] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [4] p(c_13) = [1] p(c_14) = [2] x1 + [2] x2 + [4] x3 + [1] x4 + [0] p(c_15) = [1] p(c_16) = [1] x2 + [1] x5 + [4] p(c_17) = [1] x1 + [1] x3 + [2] p(c_18) = [2] x5 + [2] p(c_19) = [1] x1 + [1] p(c_20) = [0] p(c_21) = [0] p(c_22) = [1] x1 + [0] p(c_23) = [1] p(c_24) = [1] p(c_25) = [1] x3 + [1] x4 + [0] p(c_26) = [1] x1 + [1] p(c_27) = [1] p(c_28) = [2] x1 + [4] x2 + [0] p(c_29) = [4] x1 + [0] p(c_30) = [1] x1 + [4] x2 + [4] p(c_31) = [1] x2 + [0] p(c_32) = [1] p(c_33) = [1] Following rules are strictly oriented: U11#(tt(),V1,V2) = [4] > [0] = activate#(V1) U11#(tt(),V1,V2) = [4] > [0] = activate#(V2) U12#(tt(),V1,V2) = [4] > [0] = activate#(V1) U12#(tt(),V1,V2) = [4] > [0] = activate#(V2) U13#(tt(),V1,V2) = [4] > [0] = activate#(V1) U13#(tt(),V1,V2) = [4] > [0] = activate#(V2) U14#(tt(),V1,V2) = [4] > [0] = activate#(V1) U14#(tt(),V1,V2) = [4] > [0] = activate#(V2) U15#(tt(),V2) = [4] > [0] = activate#(V2) U21#(tt(),V1) = [4] > [0] = activate#(V1) U22#(tt(),V1) = [4] > [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [4] > [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [4] > [0] = activate#(V2) isNat#(n__s(V1)) = [4] > [0] = activate#(V1) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [4] >= [4] = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [4] >= [0] = isNatKind#(activate(V1)) U12#(tt(),V1,V2) = [4] >= [4] = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [4] >= [0] = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [4] >= [4] = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [4] >= [0] = isNatKind#(activate(V2)) U14#(tt(),V1,V2) = [4] >= [4] = U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) = [4] >= [4] = isNat#(activate(V1)) U15#(tt(),V2) = [4] >= [4] = isNat#(activate(V2)) U21#(tt(),V1) = [4] >= [4] = U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [4] >= [0] = isNatKind#(activate(V1)) U22#(tt(),V1) = [4] >= [4] = isNat#(activate(V1)) U31#(tt(),V2) = [0] >= [0] = c_11(isNatKind#(activate(V2))) U51#(tt(),N) = [4] >= [3] = U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) = [4] >= [0] = isNatKind#(activate(N)) U52#(tt(),N) = [3] >= [0] = activate#(N) U61#(tt(),M,N) = [1] M + [7] >= [7] = U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) = [1] M + [7] >= [0] = isNatKind#(activate(M)) U62#(tt(),M,N) = [7] >= [0] = U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) = [7] >= [0] = activate#(M) U62#(tt(),M,N) = [7] >= [0] = activate#(N) U62#(tt(),M,N) = [7] >= [4] = isNat#(activate(N)) U63#(tt(),M,N) = [0] >= [0] = U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [0] >= [0] = activate#(M) U63#(tt(),M,N) = [0] >= [0] = activate#(N) U63#(tt(),M,N) = [0] >= [0] = isNatKind#(activate(N)) U64#(tt(),M,N) = [0] >= [0] = activate#(M) U64#(tt(),M,N) = [0] >= [0] = activate#(N) isNat#(n__plus(V1,V2)) = [4] >= [4] = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) = [4] >= [0] = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [4] >= [4] = U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) = [4] >= [0] = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [0] >= [0] = c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) = [0] >= [0] = c_29(isNatKind#(activate(V1))) *** Step 6.b:1.a:7: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_28) = {1,2}, uargs(c_29) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [1] p(U11) = [1] p(U12) = [2] p(U13) = [2] x1 + [2] x3 + [0] p(U14) = [0] p(U15) = [2] x2 + [4] p(U16) = [4] x1 + [4] p(U21) = [4] x2 + [0] p(U22) = [1] x1 + [7] p(U23) = [0] p(U31) = [0] p(U32) = [5] x1 + [5] p(U41) = [4] x1 + [0] p(U51) = [1] x2 + [1] p(U52) = [1] x2 + [0] p(U61) = [1] x2 + [1] x3 + [1] p(U62) = [1] x2 + [1] x3 + [1] p(U63) = [1] x2 + [1] x3 + [1] p(U64) = [1] x2 + [1] x3 + [1] p(activate) = [1] x1 + [0] p(isNat) = [1] x1 + [0] p(isNatKind) = [1] p(n__0) = [1] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [1] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [1] p(tt) = [0] p(0#) = [1] p(U11#) = [6] x2 + [6] x3 + [0] p(U12#) = [6] x2 + [6] x3 + [0] p(U13#) = [6] x2 + [6] x3 + [0] p(U14#) = [6] x2 + [6] x3 + [0] p(U15#) = [6] x2 + [0] p(U16#) = [1] x1 + [0] p(U21#) = [6] x2 + [6] p(U22#) = [6] x2 + [5] p(U23#) = [4] p(U31#) = [6] x2 + [0] p(U32#) = [0] p(U41#) = [1] x1 + [0] p(U51#) = [7] x2 + [2] p(U52#) = [4] x2 + [2] p(U61#) = [1] x1 + [7] x2 + [7] x3 + [1] p(U62#) = [7] x2 + [6] x3 + [1] p(U63#) = [4] x2 + [6] x3 + [1] p(U64#) = [4] x2 + [1] p(activate#) = [0] p(isNat#) = [6] x1 + [0] p(isNatKind#) = [6] x1 + [0] p(plus#) = [2] p(s#) = [1] p(c_1) = [0] p(c_2) = [4] x1 + [1] x2 + [2] x3 + [1] x4 + [0] p(c_3) = [4] x1 + [1] x3 + [1] p(c_4) = [1] x4 + [2] x5 + [1] p(c_5) = [4] x1 + [1] x2 + [1] x4 + [0] p(c_6) = [4] x1 + [1] x2 + [0] p(c_7) = [1] p(c_8) = [1] x4 + [0] p(c_9) = [1] x2 + [1] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [1] p(c_13) = [2] p(c_14) = [1] x1 + [4] x2 + [4] x4 + [0] p(c_15) = [1] x1 + [0] p(c_16) = [4] x2 + [1] x5 + [0] p(c_17) = [1] x2 + [1] x3 + [2] x5 + [0] p(c_18) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] p(c_19) = [0] p(c_20) = [0] p(c_21) = [1] p(c_22) = [2] x1 + [2] p(c_23) = [1] x1 + [2] p(c_24) = [0] p(c_25) = [4] x3 + [4] x5 + [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [1] x1 + [1] x2 + [0] p(c_29) = [1] x1 + [4] p(c_30) = [1] x1 + [2] x2 + [1] p(c_31) = [0] p(c_32) = [1] p(c_33) = [1] Following rules are strictly oriented: isNatKind#(n__s(V1)) = [6] V1 + [6] > [6] V1 + [4] = c_29(isNatKind#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V1) U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V2) U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V1 + [0] = isNatKind#(activate(V1)) U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V1) U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V2) U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V2 + [0] = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V1) U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V2) U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V2 + [0] = isNatKind#(activate(V2)) U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V2 + [0] = U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V1) U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V2) U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0] >= [6] V1 + [0] = isNat#(activate(V1)) U15#(tt(),V2) = [6] V2 + [0] >= [0] = activate#(V2) U15#(tt(),V2) = [6] V2 + [0] >= [6] V2 + [0] = isNat#(activate(V2)) U21#(tt(),V1) = [6] V1 + [6] >= [6] V1 + [5] = U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [6] V1 + [6] >= [0] = activate#(V1) U21#(tt(),V1) = [6] V1 + [6] >= [6] V1 + [0] = isNatKind#(activate(V1)) U22#(tt(),V1) = [6] V1 + [5] >= [0] = activate#(V1) U22#(tt(),V1) = [6] V1 + [5] >= [6] V1 + [0] = isNat#(activate(V1)) U31#(tt(),V2) = [6] V2 + [0] >= [6] V2 + [0] = c_11(isNatKind#(activate(V2))) U51#(tt(),N) = [7] N + [2] >= [4] N + [2] = U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) = [7] N + [2] >= [6] N + [0] = isNatKind#(activate(N)) U52#(tt(),N) = [4] N + [2] >= [0] = activate#(N) U61#(tt(),M,N) = [7] M + [7] N + [1] >= [7] M + [6] N + [1] = U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) = [7] M + [7] N + [1] >= [6] M + [0] = isNatKind#(activate(M)) U62#(tt(),M,N) = [7] M + [6] N + [1] >= [4] M + [6] N + [1] = U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) = [7] M + [6] N + [1] >= [0] = activate#(M) U62#(tt(),M,N) = [7] M + [6] N + [1] >= [0] = activate#(N) U62#(tt(),M,N) = [7] M + [6] N + [1] >= [6] N + [0] = isNat#(activate(N)) U63#(tt(),M,N) = [4] M + [6] N + [1] >= [4] M + [1] = U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [4] M + [6] N + [1] >= [0] = activate#(M) U63#(tt(),M,N) = [4] M + [6] N + [1] >= [0] = activate#(N) U63#(tt(),M,N) = [4] M + [6] N + [1] >= [6] N + [0] = isNatKind#(activate(N)) U64#(tt(),M,N) = [4] M + [1] >= [0] = activate#(M) U64#(tt(),M,N) = [4] M + [1] >= [0] = activate#(N) isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0] >= [0] = activate#(V2) isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0] >= [6] V1 + [0] = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [6] V1 + [6] >= [6] V1 + [6] = U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) = [6] V1 + [6] >= [0] = activate#(V1) isNat#(n__s(V1)) = [6] V1 + [6] >= [6] V1 + [0] = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0] >= [6] V1 + [6] V2 + [0] = c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) 0() = [1] >= [1] = n__0() U51(tt(),N) = [1] N + [1] >= [1] N + [0] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [1] >= [1] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [1] >= [1] X + [1] = s(activate(X)) plus(N,0()) = [1] N + [1] >= [1] N + [1] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) s(X) = [1] X + [1] >= [1] X + [1] = n__s(X) *** Step 6.b:1.a:8: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_28) = {1,2}, uargs(c_29) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [2] x1 + [3] x2 + [4] p(U12) = [2] x1 + [7] p(U13) = [1] p(U14) = [2] x1 + [1] x2 + [4] x3 + [6] p(U15) = [4] x2 + [6] p(U16) = [0] p(U21) = [3] x2 + [4] p(U22) = [2] x1 + [1] x2 + [0] p(U23) = [5] x1 + [7] p(U31) = [2] x1 + [2] p(U32) = [5] p(U41) = [2] p(U51) = [1] x2 + [0] p(U52) = [1] x2 + [0] p(U61) = [1] x2 + [1] x3 + [3] p(U62) = [1] x2 + [1] x3 + [3] p(U63) = [1] x2 + [1] x3 + [3] p(U64) = [1] x2 + [1] x3 + [3] p(activate) = [1] x1 + [0] p(isNat) = [1] p(isNatKind) = [1] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [3] p(n__s) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [3] p(s) = [1] x1 + [0] p(tt) = [0] p(0#) = [1] p(U11#) = [1] x2 + [1] x3 + [4] p(U12#) = [1] x2 + [1] x3 + [4] p(U13#) = [1] x2 + [1] x3 + [4] p(U14#) = [1] x2 + [1] x3 + [3] p(U15#) = [1] x2 + [3] p(U16#) = [0] p(U21#) = [1] x2 + [2] p(U22#) = [1] x2 + [2] p(U23#) = [1] x1 + [0] p(U31#) = [1] x2 + [0] p(U32#) = [4] x1 + [0] p(U41#) = [0] p(U51#) = [4] x2 + [2] p(U52#) = [3] x2 + [0] p(U61#) = [4] x2 + [4] x3 + [2] p(U62#) = [4] x2 + [4] x3 + [2] p(U63#) = [1] x2 + [4] x3 + [2] p(U64#) = [1] x3 + [0] p(activate#) = [0] p(isNat#) = [1] x1 + [2] p(isNatKind#) = [1] x1 + [0] p(plus#) = [4] x1 + [1] p(s#) = [1] x1 + [0] p(c_1) = [1] p(c_2) = [1] x3 + [1] x4 + [1] p(c_3) = [1] x1 + [2] x5 + [0] p(c_4) = [4] x1 + [4] x2 + [1] x4 + [1] p(c_5) = [1] x4 + [2] p(c_6) = [2] x1 + [2] p(c_7) = [2] p(c_8) = [1] x3 + [1] x4 + [0] p(c_9) = [2] x1 + [1] p(c_10) = [4] p(c_11) = [1] x1 + [0] p(c_12) = [1] p(c_13) = [1] p(c_14) = [4] x1 + [4] x3 + [1] p(c_15) = [2] x1 + [2] p(c_16) = [4] x2 + [4] x4 + [1] p(c_17) = [1] x1 + [2] x5 + [0] p(c_18) = [1] x5 + [0] p(c_19) = [1] x1 + [1] p(c_20) = [4] p(c_21) = [1] x1 + [0] p(c_22) = [4] x1 + [2] p(c_23) = [1] p(c_24) = [1] p(c_25) = [0] p(c_26) = [1] x2 + [4] x3 + [0] p(c_27) = [1] p(c_28) = [1] x1 + [1] x2 + [0] p(c_29) = [1] x1 + [0] p(c_30) = [1] x1 + [1] x2 + [0] p(c_31) = [4] p(c_32) = [2] p(c_33) = [0] Following rules are strictly oriented: isNatKind#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3] > [1] V1 + [1] V2 + [0] = c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [4] = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [0] = activate#(V1) U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [0] = activate#(V2) U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [1] V1 + [0] = isNatKind#(activate(V1)) U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [4] = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [0] = activate#(V1) U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [0] = activate#(V2) U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [1] V2 + [0] = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [3] = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [0] = activate#(V1) U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [0] = activate#(V2) U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [4] >= [1] V2 + [0] = isNatKind#(activate(V2)) U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [3] >= [1] V2 + [3] = U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [3] >= [0] = activate#(V1) U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [3] >= [0] = activate#(V2) U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [3] >= [1] V1 + [2] = isNat#(activate(V1)) U15#(tt(),V2) = [1] V2 + [3] >= [0] = activate#(V2) U15#(tt(),V2) = [1] V2 + [3] >= [1] V2 + [2] = isNat#(activate(V2)) U21#(tt(),V1) = [1] V1 + [2] >= [1] V1 + [2] = U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [1] V1 + [2] >= [0] = activate#(V1) U21#(tt(),V1) = [1] V1 + [2] >= [1] V1 + [0] = isNatKind#(activate(V1)) U22#(tt(),V1) = [1] V1 + [2] >= [0] = activate#(V1) U22#(tt(),V1) = [1] V1 + [2] >= [1] V1 + [2] = isNat#(activate(V1)) U31#(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = c_11(isNatKind#(activate(V2))) U51#(tt(),N) = [4] N + [2] >= [3] N + [0] = U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) = [4] N + [2] >= [1] N + [0] = isNatKind#(activate(N)) U52#(tt(),N) = [3] N + [0] >= [0] = activate#(N) U61#(tt(),M,N) = [4] M + [4] N + [2] >= [4] M + [4] N + [2] = U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) = [4] M + [4] N + [2] >= [1] M + [0] = isNatKind#(activate(M)) U62#(tt(),M,N) = [4] M + [4] N + [2] >= [1] M + [4] N + [2] = U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) = [4] M + [4] N + [2] >= [0] = activate#(M) U62#(tt(),M,N) = [4] M + [4] N + [2] >= [0] = activate#(N) U62#(tt(),M,N) = [4] M + [4] N + [2] >= [1] N + [2] = isNat#(activate(N)) U63#(tt(),M,N) = [1] M + [4] N + [2] >= [1] N + [0] = U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [1] M + [4] N + [2] >= [0] = activate#(M) U63#(tt(),M,N) = [1] M + [4] N + [2] >= [0] = activate#(N) U63#(tt(),M,N) = [1] M + [4] N + [2] >= [1] N + [0] = isNatKind#(activate(N)) U64#(tt(),M,N) = [1] N + [0] >= [0] = activate#(M) U64#(tt(),M,N) = [1] N + [0] >= [0] = activate#(N) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [4] = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5] >= [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5] >= [0] = activate#(V2) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [0] = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [1] V1 + [2] >= [1] V1 + [2] = U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) = [1] V1 + [2] >= [0] = activate#(V1) isNat#(n__s(V1)) = [1] V1 + [2] >= [1] V1 + [0] = isNatKind#(activate(V1)) isNatKind#(n__s(V1)) = [1] V1 + [0] >= [1] V1 + [0] = c_29(isNatKind#(activate(V1))) 0() = [0] >= [0] = n__0() U51(tt(),N) = [1] N + [0] >= [1] N + [0] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(activate(X)) plus(N,0()) = [1] N + [3] >= [1] N + [0] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = n__plus(X1,X2) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) *** Step 6.b:1.a:9: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_28) = {1,2}, uargs(c_29) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [6] x1 + [2] x3 + [4] p(U12) = [4] x1 + [4] x3 + [4] p(U13) = [2] p(U14) = [4] x3 + [1] p(U15) = [1] x1 + [2] x2 + [0] p(U16) = [0] p(U21) = [2] x1 + [2] x2 + [2] p(U22) = [4] x1 + [4] x2 + [4] p(U23) = [2] p(U31) = [6] p(U32) = [5] p(U41) = [5] x1 + [3] p(U51) = [1] x2 + [2] p(U52) = [1] x2 + [2] p(U61) = [1] x2 + [1] x3 + [2] p(U62) = [1] x2 + [1] x3 + [2] p(U63) = [1] x2 + [1] x3 + [2] p(U64) = [1] x2 + [1] x3 + [2] p(activate) = [1] x1 + [0] p(isNat) = [0] p(isNatKind) = [1] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [0] p(tt) = [0] p(0#) = [0] p(U11#) = [4] x2 + [4] x3 + [0] p(U12#) = [4] x2 + [4] x3 + [0] p(U13#) = [4] x2 + [4] x3 + [0] p(U14#) = [4] x2 + [4] x3 + [0] p(U15#) = [4] x2 + [0] p(U16#) = [0] p(U21#) = [4] x2 + [0] p(U22#) = [4] x2 + [0] p(U23#) = [1] p(U31#) = [4] x2 + [2] p(U32#) = [1] x1 + [0] p(U41#) = [1] p(U51#) = [6] x2 + [0] p(U52#) = [4] x2 + [0] p(U61#) = [1] x1 + [5] x2 + [4] x3 + [1] p(U62#) = [1] x2 + [4] x3 + [1] p(U63#) = [4] x3 + [0] p(U64#) = [0] p(activate#) = [0] p(isNat#) = [4] x1 + [0] p(isNatKind#) = [4] x1 + [0] p(plus#) = [1] x1 + [1] x2 + [0] p(s#) = [1] p(c_1) = [0] p(c_2) = [1] x1 + [4] x2 + [1] x3 + [1] p(c_3) = [4] x2 + [2] x3 + [1] x5 + [0] p(c_4) = [2] x2 + [1] x3 + [1] x4 + [0] p(c_5) = [1] x1 + [1] x2 + [1] x4 + [2] p(c_6) = [1] x1 + [0] p(c_7) = [4] p(c_8) = [1] x2 + [1] p(c_9) = [1] x1 + [1] x2 + [1] p(c_10) = [2] p(c_11) = [1] x1 + [1] p(c_12) = [1] p(c_13) = [1] p(c_14) = [4] x1 + [1] x2 + [4] x4 + [0] p(c_15) = [1] p(c_16) = [1] x2 + [2] x4 + [2] x5 + [1] p(c_17) = [2] x1 + [2] x3 + [1] x4 + [4] p(c_18) = [1] x1 + [1] x3 + [4] x5 + [1] p(c_19) = [2] x1 + [4] p(c_20) = [0] p(c_21) = [2] x1 + [0] p(c_22) = [2] x2 + [0] p(c_23) = [2] x1 + [4] p(c_24) = [1] p(c_25) = [1] x3 + [0] p(c_26) = [0] p(c_27) = [1] p(c_28) = [1] x1 + [1] x2 + [6] p(c_29) = [1] x1 + [0] p(c_30) = [1] x2 + [0] p(c_31) = [1] x1 + [2] x2 + [1] p(c_32) = [4] p(c_33) = [1] Following rules are strictly oriented: U31#(tt(),V2) = [4] V2 + [2] > [4] V2 + [1] = c_11(isNatKind#(activate(V2))) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [0] = activate#(V1) U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [0] = activate#(V2) U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [0] = isNatKind#(activate(V1)) U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [0] = activate#(V1) U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [0] = activate#(V2) U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [0] = activate#(V1) U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [0] = activate#(V2) U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = isNatKind#(activate(V2)) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [0] = activate#(V1) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [0] = activate#(V2) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [0] = isNat#(activate(V1)) U15#(tt(),V2) = [4] V2 + [0] >= [0] = activate#(V2) U15#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = isNat#(activate(V2)) U21#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [4] V1 + [0] >= [0] = activate#(V1) U21#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = isNatKind#(activate(V1)) U22#(tt(),V1) = [4] V1 + [0] >= [0] = activate#(V1) U22#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = isNat#(activate(V1)) U51#(tt(),N) = [6] N + [0] >= [4] N + [0] = U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) = [6] N + [0] >= [4] N + [0] = isNatKind#(activate(N)) U52#(tt(),N) = [4] N + [0] >= [0] = activate#(N) U61#(tt(),M,N) = [5] M + [4] N + [1] >= [1] M + [4] N + [1] = U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) = [5] M + [4] N + [1] >= [4] M + [0] = isNatKind#(activate(M)) U62#(tt(),M,N) = [1] M + [4] N + [1] >= [4] N + [0] = U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) = [1] M + [4] N + [1] >= [0] = activate#(M) U62#(tt(),M,N) = [1] M + [4] N + [1] >= [0] = activate#(N) U62#(tt(),M,N) = [1] M + [4] N + [1] >= [4] N + [0] = isNat#(activate(N)) U63#(tt(),M,N) = [4] N + [0] >= [0] = U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [4] N + [0] >= [0] = activate#(M) U63#(tt(),M,N) = [4] N + [0] >= [0] = activate#(N) U63#(tt(),M,N) = [4] N + [0] >= [4] N + [0] = isNatKind#(activate(N)) U64#(tt(),M,N) = [0] >= [0] = activate#(M) U64#(tt(),M,N) = [0] >= [0] = activate#(N) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [4] V2 + [0] = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [0] = activate#(V2) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [0] = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [4] V1 + [0] >= [4] V1 + [0] = U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) = [4] V1 + [0] >= [0] = activate#(V1) isNat#(n__s(V1)) = [4] V1 + [0] >= [4] V1 + [0] = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [4] V2 + [8] = c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) = [4] V1 + [0] >= [4] V1 + [0] = c_29(isNatKind#(activate(V1))) 0() = [0] >= [0] = n__0() U51(tt(),N) = [1] N + [2] >= [1] N + [2] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [2] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(activate(X)) plus(N,0()) = [1] N + [2] >= [1] N + [2] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [2] >= [1] M + [1] N + [2] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__plus(X1,X2) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) *** Step 6.b:1.a:10: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U31#(tt(),V2) -> c_11(isNatKind#(activate(V2))) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))) isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.b:1.b:1: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) -->_2 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 2:S:activate#(n__s(X)) -> c_23(activate#(X)) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 3:W:U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -->_1 U12#(tt(),V1,V2) -> isNatKind#(activate(V2)):10 -->_1 U12#(tt(),V1,V2) -> activate#(V2):9 -->_1 U12#(tt(),V1,V2) -> activate#(V1):8 -->_1 U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)):7 4:W:U11#(tt(),V1,V2) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 5:W:U11#(tt(),V1,V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 6:W:U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 7:W:U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -->_1 U13#(tt(),V1,V2) -> isNatKind#(activate(V2)):14 -->_1 U13#(tt(),V1,V2) -> activate#(V2):13 -->_1 U13#(tt(),V1,V2) -> activate#(V1):12 -->_1 U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)):11 8:W:U12#(tt(),V1,V2) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 9:W:U12#(tt(),V1,V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 10:W:U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 11:W:U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -->_1 U14#(tt(),V1,V2) -> isNat#(activate(V1)):18 -->_1 U14#(tt(),V1,V2) -> activate#(V2):17 -->_1 U14#(tt(),V1,V2) -> activate#(V1):16 -->_1 U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)):15 12:W:U13#(tt(),V1,V2) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 13:W:U13#(tt(),V1,V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 14:W:U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 15:W:U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -->_1 U15#(tt(),V2) -> isNat#(activate(V2)):20 -->_1 U15#(tt(),V2) -> activate#(V2):19 16:W:U14#(tt(),V1,V2) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 17:W:U14#(tt(),V1,V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 18:W:U14#(tt(),V1,V2) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):52 -->_1 isNat#(n__s(V1)) -> activate#(V1):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):48 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):47 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):46 19:W:U15#(tt(),V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 20:W:U15#(tt(),V2) -> isNat#(activate(V2)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):52 -->_1 isNat#(n__s(V1)) -> activate#(V1):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):48 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):47 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):46 21:W:U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -->_1 U22#(tt(),V1) -> isNat#(activate(V1)):25 -->_1 U22#(tt(),V1) -> activate#(V1):24 22:W:U21#(tt(),V1) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 23:W:U21#(tt(),V1) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 24:W:U22#(tt(),V1) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 25:W:U22#(tt(),V1) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):52 -->_1 isNat#(n__s(V1)) -> activate#(V1):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):48 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):47 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):46 26:W:U31#(tt(),V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 27:W:U31#(tt(),V2) -> isNatKind#(activate(V2)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 28:W:U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) -->_1 U52#(tt(),N) -> activate#(N):31 29:W:U51#(tt(),N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 30:W:U51#(tt(),N) -> isNatKind#(activate(N)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 31:W:U52#(tt(),N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 32:W:U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -->_1 U62#(tt(),M,N) -> isNat#(activate(N)):39 -->_1 U62#(tt(),M,N) -> activate#(N):38 -->_1 U62#(tt(),M,N) -> activate#(M):37 -->_1 U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)):36 33:W:U61#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 34:W:U61#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 35:W:U61#(tt(),M,N) -> isNatKind#(activate(M)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 36:W:U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -->_1 U63#(tt(),M,N) -> isNatKind#(activate(N)):43 -->_1 U63#(tt(),M,N) -> activate#(N):42 -->_1 U63#(tt(),M,N) -> activate#(M):41 -->_1 U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)):40 37:W:U62#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 38:W:U62#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 39:W:U62#(tt(),M,N) -> isNat#(activate(N)) -->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):52 -->_1 isNat#(n__s(V1)) -> activate#(V1):51 -->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50 -->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):48 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):47 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):46 40:W:U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -->_1 U64#(tt(),M,N) -> activate#(N):45 -->_1 U64#(tt(),M,N) -> activate#(M):44 41:W:U63#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 42:W:U63#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 43:W:U63#(tt(),M,N) -> isNatKind#(activate(N)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 44:W:U64#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 45:W:U64#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 46:W:isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -->_1 U11#(tt(),V1,V2) -> isNatKind#(activate(V1)):6 -->_1 U11#(tt(),V1,V2) -> activate#(V2):5 -->_1 U11#(tt(),V1,V2) -> activate#(V1):4 -->_1 U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)):3 47:W:isNat#(n__plus(V1,V2)) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 48:W:isNat#(n__plus(V1,V2)) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 49:W:isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 50:W:isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -->_1 U21#(tt(),V1) -> isNatKind#(activate(V1)):23 -->_1 U21#(tt(),V1) -> activate#(V1):22 -->_1 U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)):21 51:W:isNat#(n__s(V1)) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 52:W:isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 53:W:isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) -->_1 U31#(tt(),V2) -> isNatKind#(activate(V2)):27 -->_1 U31#(tt(),V2) -> activate#(V2):26 54:W:isNatKind#(n__plus(V1,V2)) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 55:W:isNatKind#(n__plus(V1,V2)) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 56:W:isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 57:W:isNatKind#(n__s(V1)) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_23(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1 58:W:isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58 -->_1 isNatKind#(n__s(V1)) -> activate#(V1):57 -->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55 -->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54 -->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(29,U51#(tt(),N) -> activate#(N)),(33,U61#(tt(),M,N) -> activate#(M)),(34,U61#(tt(),M,N) -> activate#(N))] *** Step 6.b:1.b:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_22) = {1,2}, uargs(c_23) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [2] x1 + [1] x3 + [6] p(U12) = [1] x3 + [0] p(U13) = [5] x1 + [2] x3 + [1] p(U14) = [7] x2 + [4] p(U15) = [2] x1 + [0] p(U16) = [0] p(U21) = [4] x2 + [0] p(U22) = [2] x1 + [0] p(U23) = [1] x1 + [7] p(U31) = [1] x1 + [2] p(U32) = [0] p(U41) = [1] p(U51) = [1] x2 + [0] p(U52) = [1] x2 + [0] p(U61) = [1] x2 + [1] x3 + [1] p(U62) = [1] x2 + [1] x3 + [1] p(U63) = [1] x2 + [1] x3 + [1] p(U64) = [1] x2 + [1] x3 + [1] p(activate) = [1] x1 + [0] p(isNat) = [0] p(isNatKind) = [2] x1 + [1] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [1] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [1] p(tt) = [0] p(0#) = [2] p(U11#) = [2] x2 + [2] x3 + [1] p(U12#) = [2] x2 + [2] x3 + [1] p(U13#) = [2] x2 + [2] x3 + [1] p(U14#) = [2] x2 + [2] x3 + [1] p(U15#) = [2] x2 + [1] p(U16#) = [2] x1 + [1] p(U21#) = [2] x2 + [1] p(U22#) = [2] x2 + [1] p(U23#) = [0] p(U31#) = [2] x2 + [0] p(U32#) = [1] x1 + [1] p(U41#) = [4] x1 + [1] p(U51#) = [1] x1 + [4] x2 + [0] p(U52#) = [2] x2 + [0] p(U61#) = [3] x2 + [4] x3 + [1] p(U62#) = [2] x2 + [2] x3 + [1] p(U63#) = [2] x2 + [2] x3 + [0] p(U64#) = [2] x2 + [2] x3 + [0] p(activate#) = [2] x1 + [0] p(isNat#) = [2] x1 + [1] p(isNatKind#) = [2] x1 + [0] p(plus#) = [1] x2 + [4] p(s#) = [0] p(c_1) = [1] p(c_2) = [4] x2 + [1] x5 + [1] p(c_3) = [1] x1 + [1] x4 + [1] x5 + [0] p(c_4) = [1] x4 + [1] x5 + [1] p(c_5) = [1] x1 + [2] x3 + [1] p(c_6) = [2] x1 + [1] x2 + [1] p(c_7) = [0] p(c_8) = [1] x2 + [4] p(c_9) = [1] p(c_10) = [4] p(c_11) = [4] x1 + [0] p(c_12) = [4] p(c_13) = [0] p(c_14) = [1] x1 + [1] x2 + [1] x3 + [1] p(c_15) = [1] x1 + [2] p(c_16) = [1] x1 + [1] x2 + [2] x5 + [0] p(c_17) = [2] x1 + [1] x2 + [1] x3 + [0] p(c_18) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [2] p(c_22) = [1] x1 + [1] x2 + [0] p(c_23) = [1] x1 + [0] p(c_24) = [2] p(c_25) = [1] x3 + [0] p(c_26) = [1] x1 + [1] x2 + [2] x3 + [1] x4 + [1] p(c_27) = [1] p(c_28) = [1] x1 + [2] x2 + [4] x3 + [2] x4 + [1] p(c_29) = [1] x1 + [0] p(c_30) = [2] x1 + [4] x2 + [1] p(c_31) = [4] x1 + [1] p(c_32) = [1] p(c_33) = [0] Following rules are strictly oriented: activate#(n__s(X)) = [2] X + [2] > [2] X + [0] = c_23(activate#(X)) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V1 + [2] V2 + [1] = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V1 + [0] = activate#(V1) U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V2 + [0] = activate#(V2) U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V1 + [0] = isNatKind#(activate(V1)) U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V1 + [2] V2 + [1] = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V1 + [0] = activate#(V1) U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V2 + [0] = activate#(V2) U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V2 + [0] = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V1 + [2] V2 + [1] = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V1 + [0] = activate#(V1) U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V2 + [0] = activate#(V2) U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V2 + [0] = isNatKind#(activate(V2)) U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V2 + [1] = U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V1 + [0] = activate#(V1) U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V2 + [0] = activate#(V2) U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [1] >= [2] V1 + [1] = isNat#(activate(V1)) U15#(tt(),V2) = [2] V2 + [1] >= [2] V2 + [0] = activate#(V2) U15#(tt(),V2) = [2] V2 + [1] >= [2] V2 + [1] = isNat#(activate(V2)) U21#(tt(),V1) = [2] V1 + [1] >= [2] V1 + [1] = U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [2] V1 + [1] >= [2] V1 + [0] = activate#(V1) U21#(tt(),V1) = [2] V1 + [1] >= [2] V1 + [0] = isNatKind#(activate(V1)) U22#(tt(),V1) = [2] V1 + [1] >= [2] V1 + [0] = activate#(V1) U22#(tt(),V1) = [2] V1 + [1] >= [2] V1 + [1] = isNat#(activate(V1)) U31#(tt(),V2) = [2] V2 + [0] >= [2] V2 + [0] = activate#(V2) U31#(tt(),V2) = [2] V2 + [0] >= [2] V2 + [0] = isNatKind#(activate(V2)) U51#(tt(),N) = [4] N + [0] >= [2] N + [0] = U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) = [4] N + [0] >= [2] N + [0] = isNatKind#(activate(N)) U52#(tt(),N) = [2] N + [0] >= [2] N + [0] = activate#(N) U61#(tt(),M,N) = [3] M + [4] N + [1] >= [2] M + [2] N + [1] = U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) = [3] M + [4] N + [1] >= [2] M + [0] = isNatKind#(activate(M)) U62#(tt(),M,N) = [2] M + [2] N + [1] >= [2] M + [2] N + [0] = U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) = [2] M + [2] N + [1] >= [2] M + [0] = activate#(M) U62#(tt(),M,N) = [2] M + [2] N + [1] >= [2] N + [0] = activate#(N) U62#(tt(),M,N) = [2] M + [2] N + [1] >= [2] N + [1] = isNat#(activate(N)) U63#(tt(),M,N) = [2] M + [2] N + [0] >= [2] M + [2] N + [0] = U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [2] M + [2] N + [0] >= [2] M + [0] = activate#(M) U63#(tt(),M,N) = [2] M + [2] N + [0] >= [2] N + [0] = activate#(N) U63#(tt(),M,N) = [2] M + [2] N + [0] >= [2] N + [0] = isNatKind#(activate(N)) U64#(tt(),M,N) = [2] M + [2] N + [0] >= [2] M + [0] = activate#(M) U64#(tt(),M,N) = [2] M + [2] N + [0] >= [2] N + [0] = activate#(N) activate#(n__plus(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = c_22(activate#(X1),activate#(X2)) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1] >= [2] V1 + [2] V2 + [1] = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1] >= [2] V1 + [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1] >= [2] V2 + [0] = activate#(V2) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1] >= [2] V1 + [0] = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [2] V1 + [3] >= [2] V1 + [1] = U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) = [2] V1 + [3] >= [2] V1 + [0] = activate#(V1) isNat#(n__s(V1)) = [2] V1 + [3] >= [2] V1 + [0] = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [0] >= [2] V2 + [0] = U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [0] >= [2] V1 + [0] = activate#(V1) isNatKind#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [0] >= [2] V2 + [0] = activate#(V2) isNatKind#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [0] >= [2] V1 + [0] = isNatKind#(activate(V1)) isNatKind#(n__s(V1)) = [2] V1 + [2] >= [2] V1 + [0] = activate#(V1) isNatKind#(n__s(V1)) = [2] V1 + [2] >= [2] V1 + [0] = isNatKind#(activate(V1)) 0() = [0] >= [0] = n__0() U51(tt(),N) = [1] N + [0] >= [1] N + [0] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [1] >= [1] X + [1] = s(activate(X)) plus(N,0()) = [1] N + [0] >= [1] N + [0] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) s(X) = [1] X + [1] >= [1] X + [1] = n__s(X) *** Step 6.b:1.b:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_22) = {1,2}, uargs(c_23) = {1} Following symbols are considered usable: {0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41# ,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [2] p(U12) = [1] x1 + [4] x2 + [4] x3 + [4] p(U13) = [4] x3 + [0] p(U14) = [0] p(U15) = [0] p(U16) = [2] x1 + [0] p(U21) = [0] p(U22) = [1] x1 + [4] p(U23) = [1] x1 + [2] p(U31) = [1] p(U32) = [0] p(U41) = [0] p(U51) = [1] x2 + [0] p(U52) = [1] x2 + [0] p(U61) = [1] x2 + [1] x3 + [3] p(U62) = [1] x2 + [1] x3 + [3] p(U63) = [1] x2 + [1] x3 + [3] p(U64) = [1] x2 + [1] x3 + [3] p(activate) = [1] x1 + [0] p(isNat) = [0] p(isNatKind) = [4] x1 + [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [1] p(plus) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [1] p(tt) = [0] p(0#) = [0] p(U11#) = [4] x2 + [4] x3 + [0] p(U12#) = [4] x2 + [4] x3 + [0] p(U13#) = [4] x2 + [4] x3 + [0] p(U14#) = [4] x2 + [4] x3 + [0] p(U15#) = [4] x2 + [0] p(U16#) = [2] x1 + [1] p(U21#) = [4] x2 + [0] p(U22#) = [4] x2 + [0] p(U23#) = [2] x1 + [1] p(U31#) = [4] x2 + [0] p(U32#) = [1] x1 + [1] p(U41#) = [4] p(U51#) = [1] x1 + [5] x2 + [4] p(U52#) = [4] x2 + [2] p(U61#) = [7] x2 + [4] x3 + [4] p(U62#) = [4] x2 + [4] x3 + [4] p(U63#) = [4] x2 + [4] x3 + [0] p(U64#) = [4] x2 + [4] x3 + [0] p(activate#) = [4] x1 + [0] p(isNat#) = [4] x1 + [0] p(isNatKind#) = [4] x1 + [0] p(plus#) = [4] p(s#) = [2] x1 + [0] p(c_1) = [2] p(c_2) = [1] x3 + [1] x4 + [0] p(c_3) = [2] x1 + [4] x2 + [1] x3 + [2] p(c_4) = [2] x3 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [1] x2 + [1] p(c_7) = [1] p(c_8) = [1] x1 + [2] x3 + [0] p(c_9) = [1] x1 + [4] x2 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [1] p(c_12) = [1] p(c_13) = [0] p(c_14) = [2] x2 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [1] x2 + [2] x4 + [2] x5 + [1] p(c_17) = [1] x5 + [4] p(c_18) = [2] x1 + [4] x4 + [1] p(c_19) = [4] x2 + [4] p(c_20) = [2] p(c_21) = [1] x1 + [1] p(c_22) = [1] x1 + [1] x2 + [0] p(c_23) = [1] x1 + [4] p(c_24) = [1] p(c_25) = [4] p(c_26) = [4] x3 + [0] p(c_27) = [1] p(c_28) = [1] x1 + [2] x2 + [0] p(c_29) = [1] x2 + [4] p(c_30) = [4] x1 + [0] p(c_31) = [4] p(c_32) = [0] p(c_33) = [1] Following rules are strictly oriented: activate#(n__plus(X1,X2)) = [4] X1 + [4] X2 + [8] > [4] X1 + [4] X2 + [0] = c_22(activate#(X1),activate#(X2)) Following rules are (at-least) weakly oriented: U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [0] = activate#(V1) U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = activate#(V2) U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [0] = isNatKind#(activate(V1)) U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [0] = activate#(V1) U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = activate#(V2) U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [0] = activate#(V1) U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = activate#(V2) U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = isNatKind#(activate(V2)) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [0] = activate#(V1) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V2 + [0] = activate#(V2) U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0] >= [4] V1 + [0] = isNat#(activate(V1)) U15#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = activate#(V2) U15#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = isNat#(activate(V2)) U21#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = activate#(V1) U21#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = isNatKind#(activate(V1)) U22#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = activate#(V1) U22#(tt(),V1) = [4] V1 + [0] >= [4] V1 + [0] = isNat#(activate(V1)) U31#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = activate#(V2) U31#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = isNatKind#(activate(V2)) U51#(tt(),N) = [5] N + [4] >= [4] N + [2] = U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) = [5] N + [4] >= [4] N + [0] = isNatKind#(activate(N)) U52#(tt(),N) = [4] N + [2] >= [4] N + [0] = activate#(N) U61#(tt(),M,N) = [7] M + [4] N + [4] >= [4] M + [4] N + [4] = U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) = [7] M + [4] N + [4] >= [4] M + [0] = isNatKind#(activate(M)) U62#(tt(),M,N) = [4] M + [4] N + [4] >= [4] M + [4] N + [0] = U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) = [4] M + [4] N + [4] >= [4] M + [0] = activate#(M) U62#(tt(),M,N) = [4] M + [4] N + [4] >= [4] N + [0] = activate#(N) U62#(tt(),M,N) = [4] M + [4] N + [4] >= [4] N + [0] = isNat#(activate(N)) U63#(tt(),M,N) = [4] M + [4] N + [0] >= [4] M + [4] N + [0] = U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [4] M + [4] N + [0] >= [4] M + [0] = activate#(M) U63#(tt(),M,N) = [4] M + [4] N + [0] >= [4] N + [0] = activate#(N) U63#(tt(),M,N) = [4] M + [4] N + [0] >= [4] N + [0] = isNatKind#(activate(N)) U64#(tt(),M,N) = [4] M + [4] N + [0] >= [4] M + [0] = activate#(M) U64#(tt(),M,N) = [4] M + [4] N + [0] >= [4] N + [0] = activate#(N) activate#(n__s(X)) = [4] X + [4] >= [4] X + [4] = c_23(activate#(X)) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [4] V2 + [0] = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V2 + [0] = activate#(V2) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [0] = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [4] V1 + [4] >= [4] V1 + [0] = U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) = [4] V1 + [4] >= [4] V1 + [0] = activate#(V1) isNat#(n__s(V1)) = [4] V1 + [4] >= [4] V1 + [0] = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V2 + [0] = U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [0] = activate#(V1) isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V2 + [0] = activate#(V2) isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8] >= [4] V1 + [0] = isNatKind#(activate(V1)) isNatKind#(n__s(V1)) = [4] V1 + [4] >= [4] V1 + [0] = activate#(V1) isNatKind#(n__s(V1)) = [4] V1 + [4] >= [4] V1 + [0] = isNatKind#(activate(V1)) 0() = [0] >= [0] = n__0() U51(tt(),N) = [1] N + [0] >= [1] N + [0] = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U61(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [1] >= [1] X + [1] = s(activate(X)) plus(N,0()) = [1] N + [2] >= [1] N + [0] = U51(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [3] >= [1] M + [1] N + [3] = U61(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__plus(X1,X2) s(X) = [1] X + [1] >= [1] X + [1] = n__s(X) *** Step 6.b:1.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U51#(tt(),N) -> isNatKind#(activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U61#(tt(),M,N) -> isNatKind#(activate(M)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_23(activate#(X)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3 ,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2 ,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1 ,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4 ,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1 ,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32# ,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus ,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))