WORST_CASE(?,O(n^1)) * Step 1: Sum WORST_CASE(?,O(n^1)) + 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(X1,X2) activate(n__s(X)) -> s(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: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DependencyPairs WORST_CASE(?,O(n^1)) + 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(X1,X2) activate(n__s(X)) -> s(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#(X1,X2)) activate#(n__s(X)) -> c_11(s#(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 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + 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#(X1,X2)) activate#(n__s(X)) -> c_11(s#(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(X1,X2) activate(n__s(X)) -> s(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_10/1,c_11/1,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#(X1,X2)) 11: activate#(n__s(X)) -> c_11(s#(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 4: PredecessorEstimation WORST_CASE(?,O(n^1)) + 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#(X1,X2)) activate#(n__s(X)) -> c_11(s#(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(X1,X2) activate(n__s(X)) -> s(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_10/1,c_11/1,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 {5,6,7} by application of Pre({5,6,7}) = {1,2,3,4,8,9}. 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#(X1,X2)) 7: activate#(n__s(X)) -> c_11(s#(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 5: PredecessorEstimation WORST_CASE(?,O(n^1)) + 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)) 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#()) activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) 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(X1,X2) activate(n__s(X)) -> s(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_10/1,c_11/1,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 {2,4} by application of Pre({2,4}) = {3}. 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: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 6: isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 7: 0#() -> c_1() 8: U12#(tt()) -> c_3() 9: U21#(tt()) -> c_4() 10: activate#(X) -> c_8() 11: activate#(n__0()) -> c_9(0#()) 12: activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) 13: activate#(n__s(X)) -> c_11(s#(X)) 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 6: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),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(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) - Weak DPs: 0#() -> c_1() U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),N) -> c_5(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#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) 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(X1,X2) activate(n__s(X)) -> s(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_10/1,c_11/1,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__s(X)) -> c_11(s#(X)):13 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_3 activate#(n__0()) -> c_9(0#()):11 -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),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 -->_2 isNat#(n__0()) -> c_12():14 -->_3 activate#(X) -> c_8():10 -->_1 U12#(tt()) -> c_3():6 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)) -->_5 activate#(n__s(X)) -> c_11(s#(X)):13 -->_4 activate#(n__s(X)) -> c_11(s#(X)):13 -->_3 activate#(n__s(X)) -> c_11(s#(X)):13 -->_5 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_4 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_5 activate#(n__0()) -> c_9(0#()):11 -->_4 activate#(n__0()) -> c_9(0#()):11 -->_3 activate#(n__0()) -> c_9(0#()):11 -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):9 -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),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 -->_2 isNat#(n__0()) -> c_12():14 -->_5 activate#(X) -> c_8():10 -->_4 activate#(X) -> c_8():10 -->_3 activate#(X) -> c_8():10 3:S:isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__s(X)) -> c_11(s#(X)):13 -->_3 activate#(n__s(X)) -> c_11(s#(X)):13 -->_4 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_4 activate#(n__0()) -> c_9(0#()):11 -->_3 activate#(n__0()) -> c_9(0#()):11 -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):4 -->_2 isNat#(n__0()) -> c_12():14 -->_4 activate#(X) -> c_8():10 -->_3 activate#(X) -> c_8():10 -->_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(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):1 4:S:isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_3 activate#(n__s(X)) -> c_11(s#(X)):13 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_3 activate#(n__0()) -> c_9(0#()):11 -->_2 isNat#(n__0()) -> c_12():14 -->_3 activate#(X) -> c_8():10 -->_1 U21#(tt()) -> c_4():7 -->_2 isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))),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 5:W:0#() -> c_1() 6:W:U12#(tt()) -> c_3() 7:W:U21#(tt()) -> c_4() 8:W:U31#(tt(),N) -> c_5(activate#(N)) -->_1 activate#(n__s(X)) -> c_11(s#(X)):13 -->_1 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_1 activate#(n__0()) -> c_9(0#()):11 -->_1 activate#(X) -> c_8():10 9:W: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#(X)):13 -->_3 activate#(n__s(X)) -> c_11(s#(X)):13 -->_4 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_3 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_4 activate#(n__0()) -> c_9(0#()):11 -->_3 activate#(n__0()) -> c_9(0#()):11 -->_1 s#(X) -> c_18():18 -->_2 plus#(X1,X2) -> c_17():17 -->_4 activate#(X) -> c_8():10 -->_3 activate#(X) -> c_8():10 10:W:activate#(X) -> c_8() 11:W:activate#(n__0()) -> c_9(0#()) -->_1 0#() -> c_1():5 12:W:activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_17():17 13:W:activate#(n__s(X)) -> c_11(s#(X)) -->_1 s#(X) -> c_18():18 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)) 8: U31#(tt(),N) -> c_5(activate#(N)) 9: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 6: U12#(tt()) -> c_3() 7: U21#(tt()) -> c_4() 10: activate#(X) -> c_8() 14: isNat#(n__0()) -> c_12() 11: activate#(n__0()) -> c_9(0#()) 5: 0#() -> c_1() 12: activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) 17: plus#(X1,X2) -> c_17() 13: activate#(n__s(X)) -> c_11(s#(X)) 18: s#(X) -> c_18() * Step 7: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),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(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(X1,X2) activate(n__s(X)) -> s(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_10/1,c_11/1,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)):4 -->_2 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(U21#(isNat(activate(V1))),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(U21#(isNat(activate(V1))),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(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):1 4: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)):4 -->_2 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 8: 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(X1,X2) activate(n__s(X)) -> s(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_7/4,c_8/0,c_9/1,c_10/1,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] x2 + [12] p(U12) = [8] p(U21) = [1] x1 + [11] p(U31) = [1] x2 + [8] p(U41) = [2] p(U42) = [10] x3 + [0] p(activate) = [2] p(isNat) = [0] p(n__0) = [0] p(n__plus) = [1] x2 + [2] p(n__s) = [0] p(plus) = [1] x2 + [9] p(s) = [7] p(tt) = [0] p(0#) = [1] p(U11#) = [0] p(U12#) = [4] x1 + [1] p(U21#) = [1] x1 + [1] p(U31#) = [1] x2 + [1] p(U41#) = [6] p(U42#) = [2] x1 + [0] p(activate#) = [1] x1 + [8] p(isNat#) = [0] p(plus#) = [1] x1 + [8] x2 + [1] p(s#) = [4] x1 + [1] p(c_1) = [4] p(c_2) = [2] x1 + [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [0] p(c_6) = [8] x1 + [2] p(c_7) = [4] x3 + [1] x4 + [2] p(c_8) = [1] p(c_9) = [1] x1 + [2] p(c_10) = [1] x1 + [2] p(c_11) = [2] x1 + [0] p(c_12) = [1] p(c_13) = [8] x1 + [8] x2 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x2 + [0] p(c_16) = [1] x1 + [1] x2 + [4] p(c_17) = [4] p(c_18) = [1] Following rules are strictly oriented: U41#(tt(),M,N) = [6] > [2] = 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 9: NaturalPI 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(X1,X2) activate(n__s(X)) -> s(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_7/4,c_8/0,c_9/1,c_10/1,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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): 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) = x1 p(U12) = 1 p(U21) = 1 p(U31) = x2 p(U41) = 5 + 4*x1 + x2 + x3 p(U42) = 7 + 2*x1 + x2 + x3 p(activate) = x1 p(isNat) = 1 p(n__0) = 0 p(n__plus) = 4 + x1 + x2 p(n__s) = 5 + x1 p(plus) = 4 + x1 + x2 p(s) = 5 + x1 p(tt) = 1 p(0#) = 2 p(U11#) = 2*x2 p(U12#) = 0 p(U21#) = x1 p(U31#) = x1 p(U41#) = x1 + 5*x3 p(U42#) = 4 + x1 + x3 p(activate#) = 2 p(isNat#) = 2*x1 p(plus#) = 4 p(s#) = 1 p(c_1) = 2 p(c_2) = x1 p(c_3) = 0 p(c_4) = 1 p(c_5) = 1 p(c_6) = x1 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 1 p(c_11) = 1 p(c_12) = 1 p(c_13) = 2 + x1 + x2 p(c_14) = 6 + x1 p(c_15) = 0 p(c_16) = 1 + x1 p(c_17) = 0 p(c_18) = 0 Following rules are strictly oriented: isNat#(n__plus(V1,V2)) = 8 + 2*V1 + 2*V2 > 2 + 2*V1 + 2*V2 = c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) = 10 + 2*V1 > 6 + 2*V1 = c_14(isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = 2*V2 >= 2*V2 = c_2(isNat#(activate(V2))) U41#(tt(),M,N) = 1 + 5*N >= 2*N = c_6(isNat#(activate(N))) 0() = 0 >= 0 = n__0() U11(tt(),V2) = 1 >= 1 = U12(isNat(activate(V2))) U12(tt()) = 1 >= 1 = tt() U21(tt()) = 1 >= 1 = tt() U31(tt(),N) = N >= N = activate(N) U41(tt(),M,N) = 9 + M + N >= 9 + M + N = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = 9 + M + N >= 9 + M + N = s(plus(activate(N),activate(M))) activate(X) = X >= X = X activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 4 + X1 + X2 >= 4 + X1 + X2 = plus(X1,X2) activate(n__s(X)) = 5 + X >= 5 + X = s(X) isNat(n__0()) = 1 >= 1 = tt() isNat(n__plus(V1,V2)) = 1 >= 1 = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = 1 >= 1 = U21(isNat(activate(V1))) plus(N,0()) = 4 + N >= N = U31(isNat(N),N) plus(N,s(M)) = 9 + M + N >= 9 + M + N = U41(isNat(M),M,N) plus(X1,X2) = 4 + X1 + X2 >= 4 + X1 + X2 = n__plus(X1,X2) s(X) = 5 + X >= 5 + X = n__s(X) * Step 10: NaturalPI 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(X1,X2) activate(n__s(X)) -> s(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_7/4,c_8/0,c_9/1,c_10/1,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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): 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) = 1 p(U11) = x1 p(U12) = 0 p(U21) = 7 p(U31) = 3 + x2 p(U41) = 2 + x2 + x3 p(U42) = 2 + x2 + x3 p(activate) = x1 p(isNat) = 0 p(n__0) = 1 p(n__plus) = 2 + x1 + x2 p(n__s) = x1 p(plus) = 2 + x1 + x2 p(s) = x1 p(tt) = 1 p(0#) = 0 p(U11#) = 1 + x2 p(U12#) = 2 + x1 p(U21#) = 2*x1 p(U31#) = 2*x1 + 4*x2 p(U41#) = 5 + 3*x1 + x2 + 5*x3 p(U42#) = 4*x2 + x3 p(activate#) = 0 p(isNat#) = x1 p(plus#) = 0 p(s#) = 1 p(c_1) = 0 p(c_2) = x1 p(c_3) = 0 p(c_4) = 0 p(c_5) = 4 p(c_6) = 5 + x1 p(c_7) = x2 p(c_8) = 1 p(c_9) = 1 p(c_10) = 1 + x1 p(c_11) = 1 p(c_12) = 0 p(c_13) = x1 + x2 p(c_14) = x1 p(c_15) = x2 p(c_16) = 1 p(c_17) = 0 p(c_18) = 0 Following rules are strictly oriented: U11#(tt(),V2) = 1 + V2 > V2 = c_2(isNat#(activate(V2))) Following rules are (at-least) weakly oriented: U41#(tt(),M,N) = 8 + M + 5*N >= 5 + N = c_6(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = 2 + V1 + V2 >= 1 + V1 + V2 = c_13(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) = V1 >= V1 = c_14(isNat#(activate(V1))) 0() = 1 >= 1 = n__0() U31(tt(),N) = 3 + N >= N = activate(N) U41(tt(),M,N) = 2 + M + N >= 2 + M + N = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = 2 + M + N >= 2 + M + N = s(plus(activate(N),activate(M))) activate(X) = X >= X = X activate(n__0()) = 1 >= 1 = 0() activate(n__plus(X1,X2)) = 2 + X1 + X2 >= 2 + X1 + X2 = plus(X1,X2) activate(n__s(X)) = X >= X = s(X) plus(N,0()) = 3 + N >= 3 + N = U31(isNat(N),N) plus(N,s(M)) = 2 + M + N >= 2 + M + N = U41(isNat(M),M,N) plus(X1,X2) = 2 + X1 + X2 >= 2 + X1 + X2 = n__plus(X1,X2) s(X) = X >= X = n__s(X) * Step 11: 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(X1,X2) activate(n__s(X)) -> s(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_7/4,c_8/0,c_9/1,c_10/1,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). WORST_CASE(?,O(n^1))