WORST_CASE(?,O(n^2)) * Step 1: InnermostRuleRemoval 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: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),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(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#(X1,X2) -> c_15() s#(X) -> c_16() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),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#(X1,X2) -> c_15() s#(X) -> c_16() - 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(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_15/0,c_16/0} - 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: UsableRules + Details: We replace rewrite rules by usable rules: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) 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#(X1,X2) -> c_15() s#(X) -> c_16() * Step 4: 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#(X1,X2) -> c_15() s#(X) -> c_16() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) 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_15/0,c_16/0} - 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} by application of Pre({1,3,4,8,12,15,16}) = {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#(X1,X2) -> c_15() 16: s#(X) -> c_16() * Step 5: 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#(X1,X2) -> c_15() s#(X) -> c_16() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) 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_15/0,c_16/0} - 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#(X1,X2) -> c_15() 16: s#(X) -> c_16() * Step 6: 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#(X1,X2) -> c_15() s#(X) -> c_16() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) 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_15/0,c_16/0} - 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_16():16 -->_2 plus#(X1,X2) -> c_15():15 -->_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_15():15 -->_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_16():16 -->_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#(X1,X2) -> c_15() 16:W:s#(X) -> c_16() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: U12#(tt()) -> c_3() 15: plus#(X1,X2) -> c_15() 16: s#(X) -> c_16() 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 7: 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() 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(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_15/0,c_16/0} - 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 8: 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() 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(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_15/0,c_16/0} - 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 9: Decompose 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() 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(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_15/0,c_16/0} - 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: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 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 DPs: 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)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) 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_15/0,c_16/0} - 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} Problem (S) - Strict DPs: 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)) - Weak DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 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() 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(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_15/0,c_16/0} - 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} ** Step 9.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 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 DPs: 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)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) 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_15/0,c_16/0} - 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 6: activate#(n__s(X)) -> c_11(activate#(X)) 7: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 8: isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) Consider the set of all dependency pairs 1: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 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(activate#(N),activate#(M)) 5: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) 6: activate#(n__s(X)) -> c_11(activate#(X)) 7: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 8: isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) Processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {6,7,8} These cover all (indirect) predecessors of dependency pairs {1,3,4,6,7,8} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 9.a:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 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 DPs: 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)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) 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_15/0,c_16/0} - 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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_6) = {1,2,3,4,5}, uargs(c_7) = {1,2}, uargs(c_10) = {1,2}, uargs(c_11) = {1}, uargs(c_13) = {1,2,3,4}, uargs(c_14) = {1,2} Following symbols are considered usable: {0,activate,plus,s,0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus#,s#} TcT has computed the following interpretation: p(0) = [2] [1] p(U11) = [1 0] x1 + [0 3] x2 + [0] [0 0] [0 3] [0] p(U12) = [0 0] x1 + [1] [2 0] [0] p(U21) = [0 0] x1 + [1] [1 1] [2] p(U31) = [0 0] x1 + [0 1] x2 + [0] [2 0] [0 0] [0] p(U41) = [1 2] x2 + [0 0] x3 + [0] [0 0] [0 2] [0] p(U42) = [0 0] x1 + [0 0] x3 + [0] [1 0] [2 0] [0] p(activate) = [1 0] x1 + [0] [0 1] [0] p(isNat) = [2 0] x1 + [1] [0 0] [0] p(n__0) = [2] [1] p(n__plus) = [1 2] x1 + [1 3] x2 + [1] [0 1] [0 1] [0] p(n__s) = [1 2] x1 + [2] [0 1] [2] p(plus) = [1 2] x1 + [1 3] x2 + [1] [0 1] [0 1] [0] p(s) = [1 2] x1 + [2] [0 1] [2] p(tt) = [0] [0] p(0#) = [1] [0] p(U11#) = [1 1] x2 + [0] [0 0] [3] p(U12#) = [0] [0] p(U21#) = [0 0] x1 + [0] [2 0] [0] p(U31#) = [1 2] x1 + [0 2] x2 + [2] [1 0] [0 0] [0] p(U41#) = [0 3] x2 + [2 3] x3 + [2] [2 2] [1 3] [2] p(U42#) = [0 1] x2 + [0 1] x3 + [0] [0 3] [1 0] [1] p(activate#) = [0 1] x1 + [0] [0 1] [0] p(isNat#) = [1 0] x1 + [0] [2 0] [2] p(plus#) = [0 1] x1 + [0 1] x2 + [0] [2 1] [0 2] [1] p(s#) = [0 1] x1 + [2] [0 0] [2] p(c_1) = [0] [0] p(c_2) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [3] p(c_3) = [0] [0] p(c_4) = [2] [2] p(c_5) = [1] [0] p(c_6) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [2 0] x4 + [1 0] x5 + [1] [0 0] [0 0] [3 0] [2 0] [0 0] [2] p(c_7) = [1 0] x1 + [1 0] x2 + [0] [0 0] [2 1] [0] p(c_8) = [2] [2] p(c_9) = [0 0] x1 + [0] [1 0] [0] p(c_10) = [1 0] x1 + [1 0] x2 + [0] [0 1] [1 0] [0] p(c_11) = [1 0] x1 + [0] [0 1] [2] p(c_12) = [0] [2] p(c_13) = [1 0] x1 + [1 0] x2 + [2 0] x3 + [2 0] x4 + [0] [2 0] [0 1] [2 2] [1 0] [2] p(c_14) = [1 0] x1 + [2 0] x2 + [0] [2 0] [0 1] [0] p(c_15) = [0] [0] p(c_16) = [0] [1] Following rules are strictly oriented: activate#(n__s(X)) = [0 1] X + [2] [0 1] [2] > [0 1] X + [0] [0 1] [2] = c_11(activate#(X)) isNat#(n__plus(V1,V2)) = [1 2] V1 + [1 3] V2 + [1] [2 4] [2 6] [4] > [1 2] V1 + [1 3] V2 + [0] [2 4] [2 3] [4] = c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)) isNat#(n__s(V1)) = [1 2] V1 + [2] [2 4] [6] > [1 2] V1 + [0] [2 1] [0] = c_14(isNat#(activate(V1)),activate#(V1)) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [1 1] V2 + [0] [0 0] [3] >= [1 1] V2 + [0] [0 0] [3] = c_2(isNat#(activate(V2)),activate#(V2)) U41#(tt(),M,N) = [0 3] M + [2 3] N + [2] [2 2] [1 3] [2] >= [0 3] M + [1 3] N + [1] [0 2] [0 3] [2] = c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) = [0 1] M + [0 1] N + [0] [0 3] [1 0] [1] >= [0 1] M + [0 1] N + [0] [0 3] [0 0] [0] = c_7(activate#(N),activate#(M)) activate#(n__plus(X1,X2)) = [0 1] X1 + [0 1] X2 + [0] [0 1] [0 1] [0] >= [0 1] X1 + [0 1] X2 + [0] [0 1] [0 1] [0] = c_10(activate#(X1),activate#(X2)) 0() = [2] [1] >= [2] [1] = n__0() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [2] [1] >= [2] [1] = 0() activate(n__plus(X1,X2)) = [1 2] X1 + [1 3] X2 + [1] [0 1] [0 1] [0] >= [1 2] X1 + [1 3] X2 + [1] [0 1] [0 1] [0] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1 2] X + [2] [0 1] [2] >= [1 2] X + [2] [0 1] [2] = s(activate(X)) plus(X1,X2) = [1 2] X1 + [1 3] X2 + [1] [0 1] [0 1] [0] >= [1 2] X1 + [1 3] X2 + [1] [0 1] [0 1] [0] = n__plus(X1,X2) s(X) = [1 2] X + [2] [0 1] [2] >= [1 2] X + [2] [0 1] [2] = n__s(X) *** Step 9.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) - Weak DPs: 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__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() 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(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_15/0,c_16/0} - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 9.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) - Weak 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__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() 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(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_15/0,c_16/0} - 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) Consider the set of all dependency pairs 1: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) 2: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 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(activate#(N),activate#(M)) 5: activate#(n__s(X)) -> c_11(activate#(X)) 6: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 7: isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) Processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 9.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) - Weak 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__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() 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(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_15/0,c_16/0} - 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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_6) = {1,2,3,4,5}, uargs(c_7) = {1,2}, uargs(c_10) = {1,2}, uargs(c_11) = {1}, uargs(c_13) = {1,2,3,4}, uargs(c_14) = {1,2} Following symbols are considered usable: {0,activate,plus,s,0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus#,s#} TcT has computed the following interpretation: p(0) = [0] [0] p(U11) = [2 0] x2 + [0] [1 2] [0] p(U12) = [2 2] x1 + [1] [2 0] [0] p(U21) = [0 1] x1 + [1] [0 0] [1] p(U31) = [1 0] x1 + [0] [2 0] [2] p(U41) = [0 0] x1 + [0 0] x2 + [0] [0 2] [1 0] [0] p(U42) = [0] [0] p(activate) = [1 0] x1 + [0] [0 1] [0] p(isNat) = [0] [0] p(n__0) = [0] [0] p(n__plus) = [1 2] x1 + [1 2] x2 + [2] [0 1] [0 1] [1] p(n__s) = [1 2] x1 + [0] [0 1] [2] p(plus) = [1 2] x1 + [1 2] x2 + [2] [0 1] [0 1] [1] p(s) = [1 2] x1 + [0] [0 1] [2] p(tt) = [0] [2] p(0#) = [0] [0] p(U11#) = [1 1] x2 + [0] [0 3] [2] p(U12#) = [1] [0] p(U21#) = [0 0] x1 + [0] [1 2] [0] p(U31#) = [0] [0] p(U41#) = [0 2] x1 + [2 3] x2 + [3 3] x3 + [3] [0 2] [0 2] [2 1] [3] p(U42#) = [0 2] x2 + [2 1] x3 + [1] [2 0] [3 0] [0] p(activate#) = [0 1] x1 + [0] [0 0] [2] p(isNat#) = [1 0] x1 + [0] [0 1] [0] p(plus#) = [0] [1] p(s#) = [2] [0] p(c_1) = [0] [2] p(c_2) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [1] p(c_3) = [2] [0] p(c_4) = [2] [1] p(c_5) = [1] [0] p(c_6) = [1 0] x1 + [1 0] x2 + [1 1] x3 + [1 2] x4 + [1 0] x5 + [0] [0 0] [2 1] [0 2] [0 0] [0 0] [3] p(c_7) = [1 0] x1 + [2 0] x2 + [0] [0 0] [0 0] [0] p(c_8) = [1] [0] p(c_9) = [0 2] x1 + [0] [1 0] [1] p(c_10) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 1] [0] p(c_11) = [1 1] x1 + [0] [0 0] [2] p(c_12) = [2] [0] p(c_13) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1 0] x4 + [0] [0 0] [0 0] [0 0] [1 0] [1] p(c_14) = [1 0] x1 + [2 0] x2 + [0] [0 0] [1 1] [0] p(c_15) = [1] [0] p(c_16) = [0] [1] Following rules are strictly oriented: activate#(n__plus(X1,X2)) = [0 1] X1 + [0 1] X2 + [1] [0 0] [0 0] [2] > [0 1] X1 + [0 1] X2 + [0] [0 0] [0 0] [2] = c_10(activate#(X1),activate#(X2)) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [1 1] V2 + [0] [0 3] [2] >= [1 1] V2 + [0] [0 0] [1] = c_2(isNat#(activate(V2)),activate#(V2)) U41#(tt(),M,N) = [2 3] M + [3 3] N + [7] [0 2] [2 1] [7] >= [0 3] M + [3 3] N + [7] [0 0] [2 1] [7] = c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U42#(tt(),M,N) = [0 2] M + [2 1] N + [1] [2 0] [3 0] [0] >= [0 2] M + [0 1] N + [0] [0 0] [0 0] [0] = c_7(activate#(N),activate#(M)) activate#(n__s(X)) = [0 1] X + [2] [0 0] [2] >= [0 1] X + [2] [0 0] [2] = c_11(activate#(X)) isNat#(n__plus(V1,V2)) = [1 2] V1 + [1 2] V2 + [2] [0 1] [0 1] [1] >= [1 2] V1 + [1 2] V2 + [2] [0 0] [0 1] [1] = c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)) isNat#(n__s(V1)) = [1 2] V1 + [0] [0 1] [2] >= [1 2] V1 + [0] [0 1] [2] = c_14(isNat#(activate(V1)),activate#(V1)) 0() = [0] [0] >= [0] [0] = n__0() activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [0] [0] >= [0] [0] = 0() activate(n__plus(X1,X2)) = [1 2] X1 + [1 2] X2 + [2] [0 1] [0 1] [1] >= [1 2] X1 + [1 2] X2 + [2] [0 1] [0 1] [1] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1 2] X + [0] [0 1] [2] >= [1 2] X + [0] [0 1] [2] = s(activate(X)) plus(X1,X2) = [1 2] X1 + [1 2] X2 + [2] [0 1] [0 1] [1] >= [1 2] X1 + [1 2] X2 + [2] [0 1] [0 1] [1] = n__plus(X1,X2) s(X) = [1 2] X + [0] [0 1] [2] >= [1 2] X + [0] [0 1] [2] = n__s(X) **** Step 9.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak 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() 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(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_15/0,c_16/0} - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 9.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),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() 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(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_15/0,c_16/0} - 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:W:U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):7 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):6 -->_2 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 2:W: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)):7 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):6 -->_5 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_4 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_3 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_5 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_4 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_3 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_1 U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)):3 3:W:U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) -->_2 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 4:W:activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) -->_2 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 5:W:activate#(n__s(X)) -> c_11(activate#(X)) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 6:W: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)):7 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):6 -->_4 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_3 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_4 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_3 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)):1 7:W:isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):7 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):6 -->_2 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 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)) 1: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 6: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 7: isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) 5: activate#(n__s(X)) -> c_11(activate#(X)) 4: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) **** Step 9.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) 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_15/0,c_16/0} - 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 9.b:1: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 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)) - Weak DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 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() 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(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_15/0,c_16/0} - 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 {2} by application of Pre({2}) = {1}. Here rules are labelled as follows: 1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 2: U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) 3: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 4: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) 5: activate#(n__s(X)) -> c_11(activate#(X)) 6: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 7: isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) ** Step 9.b:2: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) - Weak DPs: 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__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() 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(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_15/0,c_16/0} - 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} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 2: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 3: U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) 4: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) 5: activate#(n__s(X)) -> c_11(activate#(X)) 6: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 7: isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) ** Step 9.b:3: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak 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() 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(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_15/0,c_16/0} - 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:W:U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):7 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):6 -->_2 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 2:W: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)):7 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):6 -->_5 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_4 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_3 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_5 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_4 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_3 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_1 U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)):3 3:W:U42#(tt(),M,N) -> c_7(activate#(N),activate#(M)) -->_2 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 4:W:activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) -->_2 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 5:W:activate#(n__s(X)) -> c_11(activate#(X)) -->_1 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_1 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 6:W: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)):7 -->_2 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):6 -->_4 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_3 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_4 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_3 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 -->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)):1 7:W:isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) -->_1 isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)):7 -->_1 isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):6 -->_2 activate#(n__s(X)) -> c_11(activate#(X)):5 -->_2 activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 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)) 1: U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)) 6: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 7: isNat#(n__s(V1)) -> c_14(isNat#(activate(V1)),activate#(V1)) 5: activate#(n__s(X)) -> c_11(activate#(X)) 4: activate#(n__plus(X1,X2)) -> c_10(activate#(X1),activate#(X2)) ** Step 9.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) 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_15/0,c_16/0} - 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))