WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/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,U21,U31,U41,U42,activate,isNat,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(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(X) -> c_8() activate#(n__0()) -> c_9(0#()) activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)) isNat#(n__0()) -> c_12() isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) plus#(N,0()) -> c_15(U31#(isNat(N),N),isNat#(N)) plus#(N,s(M)) -> c_16(U41#(isNat(M),M,N),isNat#(M)) plus#(X1,X2) -> c_17() s#(X) -> c_18() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(X) -> c_8() activate#(n__0()) -> c_9(0#()) activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)) isNat#(n__0()) -> c_12() isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) plus#(N,0()) -> c_15(U31#(isNat(N),N),isNat#(N)) plus#(N,s(M)) -> c_16(U41#(isNat(M),M,N),isNat#(M)) plus#(X1,X2) -> c_17() s#(X) -> c_18() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/4,c_8/0,c_9/1,c_10/3,c_11/2,c_12/0,c_13/4,c_14/3,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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 {1,3,4,8,12,15,16,17,18} by application of Pre({1,3,4,8,12,15,16,17,18}) = {2,5,6,7,9,10,11,13,14}. Here rules are labelled as follows: 1: 0#() -> c_1() 2: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 3: U12#(tt()) -> c_3() 4: U21#(tt()) -> c_4() 5: U31#(tt(),N) -> c_5(activate#(N)) 6: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 7: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 8: activate#(X) -> c_8() 9: activate#(n__0()) -> c_9(0#()) 10: activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 11: activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)) 12: isNat#(n__0()) -> c_12() 13: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 14: isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 15: plus#(N,0()) -> c_15(U31#(isNat(N),N),isNat#(N)) 16: plus#(N,s(M)) -> c_16(U41#(isNat(M),M,N),isNat#(M)) 17: plus#(X1,X2) -> c_17() 18: s#(X) -> c_18() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(n__0()) -> c_9(0#()) activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) - Weak DPs: 0#() -> c_1() U12#(tt()) -> c_3() U21#(tt()) -> c_4() activate#(X) -> c_8() isNat#(n__0()) -> c_12() plus#(N,0()) -> c_15(U31#(isNat(N),N),isNat#(N)) plus#(N,s(M)) -> c_16(U41#(isNat(M),M,N),isNat#(M)) plus#(X1,X2) -> c_17() s#(X) -> c_18() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/4,c_8/0,c_9/1,c_10/3,c_11/2,c_12/0,c_13/4,c_14/3,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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 {5} by application of Pre({5}) = {1,2,3,4,6,7,8,9}. Here rules are labelled as follows: 1: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 2: U31#(tt(),N) -> c_5(activate#(N)) 3: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 4: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 5: activate#(n__0()) -> c_9(0#()) 6: activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 7: activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)) 8: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 9: isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 10: 0#() -> c_1() 11: U12#(tt()) -> c_3() 12: U21#(tt()) -> c_4() 13: activate#(X) -> c_8() 14: isNat#(n__0()) -> c_12() 15: plus#(N,0()) -> c_15(U31#(isNat(N),N),isNat#(N)) 16: plus#(N,s(M)) -> c_16(U41#(isNat(M),M,N),isNat#(M)) 17: plus#(X1,X2) -> c_17() 18: s#(X) -> c_18() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) - Weak DPs: 0#() -> c_1() U12#(tt()) -> c_3() U21#(tt()) -> c_4() activate#(X) -> c_8() activate#(n__0()) -> c_9(0#()) isNat#(n__0()) -> c_12() plus#(N,0()) -> c_15(U31#(isNat(N),N),isNat#(N)) plus#(N,s(M)) -> c_16(U41#(isNat(M),M,N),isNat#(M)) plus#(X1,X2) -> c_17() s#(X) -> c_18() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/4,c_8/0,c_9/1,c_10/3,c_11/2,c_12/0,c_13/4,c_14/3,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_3 activate#(n__0()) -> c_9(0#()):13 -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_2 isNat#(n__0()) -> c_12():14 -->_3 activate#(X) -> c_8():12 -->_1 U12#(tt()) -> c_3():10 2:S:U31#(tt(),N) -> c_5(activate#(N)) -->_1 activate#(n__0()) -> c_9(0#()):13 -->_1 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_1 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_1 activate#(X) -> c_8():12 3:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__0()) -> c_9(0#()):13 -->_4 activate#(n__0()) -> c_9(0#()):13 -->_3 activate#(n__0()) -> c_9(0#()):13 -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_5 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_4 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_5 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_4 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):4 -->_2 isNat#(n__0()) -> c_12():14 -->_5 activate#(X) -> c_8():12 -->_4 activate#(X) -> c_8():12 -->_3 activate#(X) -> c_8():12 4:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) -->_4 activate#(n__0()) -> c_9(0#()):13 -->_3 activate#(n__0()) -> c_9(0#()):13 -->_4 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_4 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_1 s#(X) -> c_18():18 -->_2 plus#(X1,X2) -> c_17():17 -->_4 activate#(X) -> c_8():12 -->_3 activate#(X) -> c_8():12 5:S:activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__0()) -> c_9(0#()):13 -->_2 activate#(n__0()) -> c_9(0#()):13 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_2 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_1 plus#(X1,X2) -> c_17():17 -->_3 activate#(X) -> c_8():12 -->_2 activate#(X) -> c_8():12 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 6:S:activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)) -->_2 activate#(n__0()) -> c_9(0#()):13 -->_1 s#(X) -> c_18():18 -->_2 activate#(X) -> c_8():12 -->_2 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_2 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 7:S:isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__0()) -> c_9(0#()):13 -->_3 activate#(n__0()) -> c_9(0#()):13 -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__0()) -> c_12():14 -->_4 activate#(X) -> c_8():12 -->_3 activate#(X) -> c_8():12 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_4 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_4 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_1 U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):1 8:S:isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_3 activate#(n__0()) -> c_9(0#()):13 -->_2 isNat#(n__0()) -> c_12():14 -->_3 activate#(X) -> c_8():12 -->_1 U21#(tt()) -> c_4():11 -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 9:W:0#() -> c_1() 10:W:U12#(tt()) -> c_3() 11:W:U21#(tt()) -> c_4() 12:W:activate#(X) -> c_8() 13:W:activate#(n__0()) -> c_9(0#()) -->_1 0#() -> c_1():9 14:W:isNat#(n__0()) -> c_12() 15:W:plus#(N,0()) -> c_15(U31#(isNat(N),N),isNat#(N)) 16:W:plus#(N,s(M)) -> c_16(U41#(isNat(M),M,N),isNat#(M)) 17:W:plus#(X1,X2) -> c_17() 18:W:s#(X) -> c_18() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 16: plus#(N,s(M)) -> c_16(U41#(isNat(M),M,N),isNat#(M)) 15: plus#(N,0()) -> c_15(U31#(isNat(N),N),isNat#(N)) 10: U12#(tt()) -> c_3() 17: plus#(X1,X2) -> c_17() 18: s#(X) -> c_18() 11: U21#(tt()) -> c_4() 12: activate#(X) -> c_8() 14: isNat#(n__0()) -> c_12() 13: activate#(n__0()) -> c_9(0#()) 9: 0#() -> c_1() * Step 5: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/4,c_8/0,c_9/1,c_10/3,c_11/2,c_12/0,c_13/4,c_14/3,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 2:S:U31#(tt(),N) -> c_5(activate#(N)) -->_1 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_1 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 3:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_5 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_4 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_5 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_4 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):4 4:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) -->_4 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_4 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 5:S:activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_2 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 6:S:activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_2 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 7:S:isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_4 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_4 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 -->_1 U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):1 8:S:isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_3 activate#(n__s(X)) -> c_11(s#(activate(X)),activate#(X)):6 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(activate#(X)) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) * Step 6: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(activate#(X)) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/4,c_14/2,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):8 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 2:S:U31#(tt(),N) -> c_5(activate#(N)) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 3:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_5 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_4 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_3 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_5 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 -->_4 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 -->_3 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 -->_1 U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)):4 4:S:U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) -->_2 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_1 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 5:S:activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) -->_2 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_1 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 6:S:activate#(n__s(X)) -> c_11(activate#(X)) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 7:S:isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):8 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_4 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_3 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_4 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 -->_3 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 -->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)):1 8:S:isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):8 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 activate#(n__s(X)) -> c_11(activate#(X)):6 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):5 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). [(2,U31#(tt(),N) -> c_5(activate#(N)))] * Step 7: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(activate#(X)) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/4,c_14/2,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) and a lower component activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(activate#(X)) Further, following extension rules are added to the lower component. U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) ** Step 7.a:1: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/4,c_14/2,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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 {3} by application of Pre({3}) = {2}. Here rules are labelled as follows: 1: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 2: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 3: U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) 4: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 5: isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) ** Step 7.a:2: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) - Weak DPs: U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/4,c_14/2,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):4 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):3 2:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):4 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):3 -->_1 U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)):5 3:S:isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):4 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):3 -->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)):1 4:S:isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):4 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):3 5:W:U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) ** Step 7.a:3: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/4,c_14/2,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):4 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):3 2:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):4 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):3 3:S:isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):4 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):3 -->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)):1 4:S:isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):4 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U41#(tt(),M,N) -> c_6(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1))) ** Step 7.a:4: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U41#(tt(),M,N) -> c_6(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/2,c_14/1,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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_6) = {1}, uargs(c_13) = {1,2}, uargs(c_14) = {1} Following symbols are considered usable: {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus#,s#} TcT has computed the following interpretation: p(0) = [2] p(U11) = [2] x1 + [14] p(U12) = [0] p(U21) = [2] x1 + [4] p(U31) = [2] x1 + [4] x2 + [0] p(U41) = [8] x2 + [8] p(U42) = [2] x1 + [0] p(activate) = [2] p(isNat) = [0] p(n__0) = [4] p(n__plus) = [1] x2 + [0] p(n__s) = [0] p(plus) = [8] x1 + [4] x2 + [0] p(s) = [6] p(tt) = [0] p(0#) = [2] p(U11#) = [0] p(U12#) = [1] x1 + [1] p(U21#) = [1] x1 + [4] p(U31#) = [1] x1 + [1] p(U41#) = [1] x3 + [12] p(U42#) = [4] x1 + [1] x2 + [1] x3 + [1] p(activate#) = [1] x1 + [2] p(isNat#) = [0] p(plus#) = [2] x2 + [0] p(s#) = [8] x1 + [0] p(c_1) = [1] p(c_2) = [8] x1 + [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [2] x1 + [1] p(c_6) = [2] x1 + [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [8] p(c_10) = [1] x1 + [0] p(c_11) = [4] p(c_12) = [2] p(c_13) = [2] x1 + [2] x2 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [4] p(c_16) = [1] x2 + [1] p(c_17) = [2] p(c_18) = [2] Following rules are strictly oriented: U41#(tt(),M,N) = [1] N + [12] > [0] = c_6(isNat#(activate(N))) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [0] >= [0] = c_2(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) = [0] >= [0] = c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) = [0] >= [0] = c_14(isNat#(activate(V1))) ** Step 7.a:5: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1))) - Weak DPs: U41#(tt(),M,N) -> c_6(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/2,c_14/1,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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_6) = {1}, uargs(c_13) = {1,2}, uargs(c_14) = {1} Following symbols are considered usable: {0,U31,U41,U42,activate,plus,s,0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [10] x2 + [14] p(U12) = [1] x1 + [0] p(U21) = [0] p(U31) = [1] x2 + [0] p(U41) = [1] x2 + [1] x3 + [1] p(U42) = [1] x2 + [1] x3 + [1] p(activate) = [1] x1 + [0] p(isNat) = [8] 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#) = [1] p(U11#) = [8] x2 + [0] p(U12#) = [1] p(U21#) = [2] x1 + [1] p(U31#) = [1] x1 + [2] x2 + [0] p(U41#) = [1] x1 + [9] x3 + [2] p(U42#) = [1] x2 + [1] x3 + [4] p(activate#) = [1] x1 + [0] p(isNat#) = [8] x1 + [0] p(plus#) = [1] x1 + [1] x2 + [0] p(s#) = [1] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [8] p(c_4) = [1] p(c_5) = [4] x1 + [1] p(c_6) = [1] x1 + [1] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [4] p(c_10) = [2] x2 + [2] p(c_11) = [1] x1 + [2] p(c_12) = [0] p(c_13) = [1] x1 + [1] x2 + [0] p(c_14) = [1] x1 + [6] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] p(c_18) = [1] Following rules are strictly oriented: isNat#(n__s(V1)) = [8] V1 + [8] > [8] V1 + [6] = c_14(isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [8] V2 + [0] >= [8] V2 + [0] = c_2(isNat#(activate(V2))) U41#(tt(),M,N) = [9] N + [2] >= [8] N + [1] = c_6(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [8] V1 + [8] V2 + [0] >= [8] V1 + [8] V2 + [0] = c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 0() = [0] >= [0] = n__0() U31(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U41(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U42(isNat(activate(N)),activate(M),activate(N)) U42(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] = U31(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U41(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 7.a:6: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U41#(tt(),M,N) -> c_6(isNat#(activate(N))) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/2,c_14/1,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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_6) = {1}, uargs(c_13) = {1,2}, uargs(c_14) = {1} Following symbols are considered usable: {0,U31,U41,U42,activate,plus,s,0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus#,s#} TcT has computed the following interpretation: p(0) = [3] p(U11) = [1] x1 + [3] p(U12) = [8] p(U21) = [13] p(U31) = [1] x2 + [0] p(U41) = [1] x2 + [1] x3 + [12] p(U42) = [1] x2 + [1] x3 + [12] p(activate) = [1] x1 + [0] p(isNat) = [0] p(n__0) = [3] p(n__plus) = [1] x1 + [1] x2 + [8] p(n__s) = [1] x1 + [4] p(plus) = [1] x1 + [1] x2 + [8] p(s) = [1] x1 + [4] p(tt) = [0] p(0#) = [8] p(U11#) = [1] x2 + [0] p(U12#) = [1] x1 + [1] p(U21#) = [1] x1 + [8] p(U31#) = [2] x1 + [2] p(U41#) = [4] x1 + [1] x3 + [13] p(U42#) = [8] x2 + [1] p(activate#) = [1] x1 + [1] p(isNat#) = [1] x1 + [0] p(plus#) = [2] x2 + [1] p(s#) = [2] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] x1 + [1] p(c_7) = [1] x2 + [2] p(c_8) = [8] p(c_9) = [1] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [1] p(c_12) = [1] p(c_13) = [1] x1 + [1] x2 + [6] p(c_14) = [1] x1 + [4] p(c_15) = [1] x1 + [4] x2 + [1] p(c_16) = [4] x1 + [8] x2 + [4] p(c_17) = [2] p(c_18) = [4] Following rules are strictly oriented: isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8] > [1] V1 + [1] V2 + [6] = c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = c_2(isNat#(activate(V2))) U41#(tt(),M,N) = [1] N + [13] >= [1] N + [1] = c_6(isNat#(activate(N))) isNat#(n__s(V1)) = [1] V1 + [4] >= [1] V1 + [4] = c_14(isNat#(activate(V1))) 0() = [3] >= [3] = n__0() U31(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U41(tt(),M,N) = [1] M + [1] N + [12] >= [1] M + [1] N + [12] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1] M + [1] N + [12] >= [1] M + [1] N + [12] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [3] >= [3] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [4] >= [1] X + [4] = s(activate(X)) plus(N,0()) = [1] N + [11] >= [1] N + [0] = U31(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [12] >= [1] M + [1] N + [12] = U41(isNat(M),M,N) plus(X1,X2) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = n__plus(X1,X2) s(X) = [1] X + [4] >= [1] X + [4] = n__s(X) ** Step 7.a:7: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) - Weak DPs: U41#(tt(),M,N) -> c_6(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/2,c_14/1,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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_6) = {1}, uargs(c_13) = {1,2}, uargs(c_14) = {1} Following symbols are considered usable: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s,0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [6] p(U12) = [1] x1 + [0] p(U21) = [6] p(U31) = [1] x2 + [3] p(U41) = [1] x2 + [1] x3 + [4] p(U42) = [1] x2 + [1] x3 + [4] p(activate) = [1] x1 + [0] p(isNat) = [6] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [4] p(n__s) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [4] p(s) = [1] x1 + [0] p(tt) = [4] p(0#) = [4] p(U11#) = [4] x1 + [6] x2 + [0] p(U12#) = [1] p(U21#) = [1] p(U31#) = [2] p(U41#) = [3] x1 + [1] x2 + [6] x3 + [8] p(U42#) = [1] x1 + [1] x2 + [2] x3 + [2] p(activate#) = [0] p(isNat#) = [6] x1 + [4] p(plus#) = [1] x2 + [1] p(s#) = [1] x1 + [2] p(c_1) = [0] p(c_2) = [1] x1 + [8] p(c_3) = [1] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [10] p(c_7) = [1] x2 + [2] p(c_8) = [4] p(c_9) = [2] x1 + [2] p(c_10) = [1] x2 + [1] p(c_11) = [1] x1 + [1] p(c_12) = [0] p(c_13) = [1] x1 + [1] x2 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [2] x2 + [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] Following rules are strictly oriented: U11#(tt(),V2) = [6] V2 + [16] > [6] V2 + [12] = c_2(isNat#(activate(V2))) Following rules are (at-least) weakly oriented: U41#(tt(),M,N) = [1] M + [6] N + [20] >= [6] N + [14] = c_6(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [28] >= [6] V1 + [6] V2 + [28] = c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) = [6] V1 + [4] >= [6] V1 + [4] = c_14(isNat#(activate(V1))) 0() = [0] >= [0] = n__0() U11(tt(),V2) = [6] >= [6] = U12(isNat(activate(V2))) U12(tt()) = [4] >= [4] = tt() U21(tt()) = [6] >= [4] = tt() U31(tt(),N) = [1] N + [3] >= [1] N + [0] = activate(N) U41(tt(),M,N) = [1] M + [1] N + [4] >= [1] M + [1] N + [4] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1] M + [1] N + [4] >= [1] M + [1] N + [4] = 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 + [4] >= [1] X1 + [1] X2 + [4] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(activate(X)) isNat(n__0()) = [6] >= [4] = tt() isNat(n__plus(V1,V2)) = [6] >= [6] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [6] >= [6] = U21(isNat(activate(V1))) plus(N,0()) = [1] N + [4] >= [1] N + [3] = U31(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [4] >= [1] M + [1] N + [4] = U41(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 + [0] >= [1] X + [0] = n__s(X) ** Step 7.a:8: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U41#(tt(),M,N) -> c_6(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_14(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/2,c_14/1,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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 7.b:1: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(activate#(X)) - Weak DPs: U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/4,c_14/2,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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_10(activate#(X1),activate#(X2)) -->_2 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 2:S:activate#(n__s(X)) -> c_11(activate#(X)) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 3:W:U11#(tt(),V2) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 4:W:U11#(tt(),V2) -> isNat#(activate(V2)) -->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):16 -->_1 isNat#(n__s(V1)) -> activate#(V1):15 -->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):14 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):13 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):12 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):11 5:W:U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) -->_1 U42#(tt(),M,N) -> activate#(N):10 -->_1 U42#(tt(),M,N) -> activate#(M):9 6:W:U41#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 7:W:U41#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 8:W:U41#(tt(),M,N) -> isNat#(activate(N)) -->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):16 -->_1 isNat#(n__s(V1)) -> activate#(V1):15 -->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):14 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):13 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):12 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):11 9:W:U42#(tt(),M,N) -> activate#(M) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 10:W:U42#(tt(),M,N) -> activate#(N) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 11:W:isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) -->_1 U11#(tt(),V2) -> isNat#(activate(V2)):4 -->_1 U11#(tt(),V2) -> activate#(V2):3 12:W:isNat#(n__plus(V1,V2)) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 13:W:isNat#(n__plus(V1,V2)) -> activate#(V2) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 14:W:isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):16 -->_1 isNat#(n__s(V1)) -> activate#(V1):15 -->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):14 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):13 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):12 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):11 15:W:isNat#(n__s(V1)) -> activate#(V1) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):2 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):1 16:W:isNat#(n__s(V1)) -> isNat#(activate(V1)) -->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):16 -->_1 isNat#(n__s(V1)) -> activate#(V1):15 -->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):14 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):13 -->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):12 -->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):11 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). [(6,U41#(tt(),M,N) -> activate#(M)),(7,U41#(tt(),M,N) -> activate#(N))] ** Step 7.b:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(activate#(X)) - Weak DPs: U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U41#(tt(),M,N) -> isNat#(activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/4,c_14/2,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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_10) = {1,2}, uargs(c_11) = {1} Following symbols are considered usable: {0,U31,U41,U42,activate,plus,s,0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus#,s#} TcT has computed the following interpretation: p(0) = [6] p(U11) = [0] p(U12) = [3] p(U21) = [1] p(U31) = [1] x2 + [6] p(U41) = [1] x2 + [1] x3 + [1] p(U42) = [1] x2 + [1] x3 + [1] p(activate) = [1] x1 + [0] p(isNat) = [0] p(n__0) = [6] 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 + [1] p(U12#) = [2] p(U21#) = [4] p(U31#) = [4] x1 + [1] p(U41#) = [1] x2 + [4] x3 + [5] p(U42#) = [1] x2 + [1] x3 + [3] p(activate#) = [1] x1 + [0] p(isNat#) = [2] x1 + [1] p(plus#) = [2] x1 + [1] p(s#) = [2] x1 + [4] p(c_1) = [0] p(c_2) = [1] x2 + [4] p(c_3) = [1] p(c_4) = [4] p(c_5) = [0] p(c_6) = [1] x2 + [1] p(c_7) = [4] x1 + [1] p(c_8) = [2] p(c_9) = [4] x1 + [1] p(c_10) = [1] x1 + [1] x2 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] p(c_13) = [4] x1 + [4] x2 + [1] x3 + [1] p(c_14) = [4] x1 + [4] p(c_15) = [4] x1 + [0] p(c_16) = [1] x1 + [1] p(c_17) = [1] p(c_18) = [1] Following rules are strictly oriented: activate#(n__s(X)) = [1] X + [1] > [1] X + [0] = c_11(activate#(X)) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [2] V2 + [1] >= [1] V2 + [0] = activate#(V2) U11#(tt(),V2) = [2] V2 + [1] >= [2] V2 + [1] = isNat#(activate(V2)) U41#(tt(),M,N) = [1] M + [4] N + [5] >= [1] M + [1] N + [3] = U42#(isNat(activate(N)),activate(M),activate(N)) U41#(tt(),M,N) = [1] M + [4] N + [5] >= [2] N + [1] = isNat#(activate(N)) U42#(tt(),M,N) = [1] M + [1] N + [3] >= [1] M + [0] = activate#(M) U42#(tt(),M,N) = [1] M + [1] N + [3] >= [1] N + [0] = activate#(N) activate#(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = c_10(activate#(X1),activate#(X2)) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1] >= [2] V2 + [1] = U11#(isNat(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1] >= [1] V1 + [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1] >= [1] V2 + [0] = activate#(V2) isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1] >= [2] V1 + [1] = isNat#(activate(V1)) isNat#(n__s(V1)) = [2] V1 + [3] >= [1] V1 + [0] = activate#(V1) isNat#(n__s(V1)) = [2] V1 + [3] >= [2] V1 + [1] = isNat#(activate(V1)) 0() = [6] >= [6] = n__0() U31(tt(),N) = [1] N + [6] >= [1] N + [0] = activate(N) U41(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U42(isNat(activate(N)),activate(M),activate(N)) U42(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()) = [6] >= [6] = 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 + [6] >= [1] N + [6] = U31(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = U41(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 7.b:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) - Weak DPs: U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U41#(tt(),M,N) -> isNat#(activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) activate#(n__s(X)) -> c_11(activate#(X)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/4,c_14/2,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,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_10) = {1,2}, uargs(c_11) = {1} Following symbols are considered usable: {0,U31,U41,U42,activate,plus,s,0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus#,s#} TcT has computed the following interpretation: p(0) = [1] p(U11) = [0] p(U12) = [1] p(U21) = [0] p(U31) = [1] x2 + [3] p(U41) = [1] x2 + [1] x3 + [4] p(U42) = [1] x2 + [1] x3 + [4] p(activate) = [1] x1 + [0] p(isNat) = [0] p(n__0) = [1] p(n__plus) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [2] p(plus) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [2] p(tt) = [0] p(0#) = [0] p(U11#) = [1] x2 + [3] p(U12#) = [1] p(U21#) = [1] x1 + [4] p(U31#) = [2] x1 + [0] p(U41#) = [1] x2 + [2] x3 + [1] p(U42#) = [1] x2 + [2] x3 + [1] p(activate#) = [1] x1 + [0] p(isNat#) = [1] x1 + [1] p(plus#) = [4] p(s#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [0] p(c_6) = [2] x3 + [4] x4 + [2] p(c_7) = [2] p(c_8) = [2] p(c_9) = [2] x1 + [1] p(c_10) = [1] x1 + [1] x2 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [0] p(c_13) = [1] x1 + [4] x2 + [0] p(c_14) = [1] x2 + [0] p(c_15) = [0] p(c_16) = [1] x2 + [0] p(c_17) = [1] p(c_18) = [1] Following rules are strictly oriented: activate#(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2] > [1] X1 + [1] X2 + [0] = c_10(activate#(X1),activate#(X2)) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [1] V2 + [3] >= [1] V2 + [0] = activate#(V2) U11#(tt(),V2) = [1] V2 + [3] >= [1] V2 + [1] = isNat#(activate(V2)) U41#(tt(),M,N) = [1] M + [2] N + [1] >= [1] M + [2] N + [1] = U42#(isNat(activate(N)),activate(M),activate(N)) U41#(tt(),M,N) = [1] M + [2] N + [1] >= [1] N + [1] = isNat#(activate(N)) U42#(tt(),M,N) = [1] M + [2] N + [1] >= [1] M + [0] = activate#(M) U42#(tt(),M,N) = [1] M + [2] N + [1] >= [1] N + [0] = activate#(N) activate#(n__s(X)) = [1] X + [2] >= [1] X + [1] = c_11(activate#(X)) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V2 + [3] = U11#(isNat(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [0] = activate#(V1) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V2 + [0] = activate#(V2) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [1] = isNat#(activate(V1)) isNat#(n__s(V1)) = [1] V1 + [3] >= [1] V1 + [0] = activate#(V1) isNat#(n__s(V1)) = [1] V1 + [3] >= [1] V1 + [1] = isNat#(activate(V1)) 0() = [1] >= [1] = n__0() U31(tt(),N) = [1] N + [3] >= [1] N + [0] = activate(N) U41(tt(),M,N) = [1] M + [1] N + [4] >= [1] M + [1] N + [4] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1] M + [1] N + [4] >= [1] M + [1] N + [4] = 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 + [2] >= [1] X + [2] = s(activate(X)) plus(N,0()) = [1] N + [3] >= [1] N + [3] = U31(isNat(N),N) plus(N,s(M)) = [1] M + [1] N + [4] >= [1] M + [1] N + [4] = U41(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 + [2] >= [1] X + [2] = n__s(X) ** Step 7.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U41#(tt(),M,N) -> isNat#(activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_11(activate#(X)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(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(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0 ,c_5/1,c_6/5,c_7/2,c_8/0,c_9/1,c_10/2,c_11/1,c_12/0,c_13/4,c_14/2,c_15/2,c_16/2,c_17/0,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))