WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,U21,U22,activate,plus,x} and constructors {0,s ,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(X) -> c_5() plus#(N,0()) -> c_6() plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,0()) -> c_8() x#(N,s(M)) -> c_9(U21#(tt(),M,N)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(X) -> c_5() plus#(N,0()) -> c_6() plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,0()) -> c_8() x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/3,c_2/3,c_3/3,c_4/5,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5,6,8} by application of Pre({5,6,8}) = {1,2,3,4}. Here rules are labelled as follows: 1: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) 2: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) 3: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) 4: U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 5: activate#(X) -> c_5() 6: plus#(N,0()) -> c_6() 7: plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) 8: x#(N,0()) -> c_8() 9: x#(N,s(M)) -> c_9(U21#(tt(),M,N)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak DPs: activate#(X) -> c_5() plus#(N,0()) -> c_6() x#(N,0()) -> c_8() - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/3,c_2/3,c_3/3,c_4/5,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)):2 -->_3 activate#(X) -> c_5():7 -->_2 activate#(X) -> c_5():7 2:S:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):5 -->_1 plus#(N,0()) -> c_6():8 -->_3 activate#(X) -> c_5():7 -->_2 activate#(X) -> c_5():7 3:S:U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) -->_1 U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)):4 -->_3 activate#(X) -> c_5():7 -->_2 activate#(X) -> c_5():7 4:S:U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 x#(N,s(M)) -> c_9(U21#(tt(),M,N)):6 -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):5 -->_2 x#(N,0()) -> c_8():9 -->_1 plus#(N,0()) -> c_6():8 -->_5 activate#(X) -> c_5():7 -->_4 activate#(X) -> c_5():7 -->_3 activate#(X) -> c_5():7 5:S:plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):1 6:S:x#(N,s(M)) -> c_9(U21#(tt(),M,N)) -->_1 U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):3 7:W:activate#(X) -> c_5() 8:W:plus#(N,0()) -> c_6() 9:W:x#(N,0()) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: x#(N,0()) -> c_8() 7: activate#(X) -> c_5() 8: plus#(N,0()) -> c_6() * Step 4: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/3,c_2/3,c_3/3,c_4/5,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)):2 2:S:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):5 3:S:U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) -->_1 U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)):4 4:S:U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 x#(N,s(M)) -> c_9(U21#(tt(),M,N)):6 -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):5 5:S:plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):1 6:S:x#(N,s(M)) -> c_9(U21#(tt(),M,N)) -->_1 U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) * Step 5: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,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(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) - Weak DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2 ,x#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} Problem (S) - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2 ,x#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} ** Step 5.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) - Weak DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 5: plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) Consider the set of all dependency pairs 1: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) 2: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) 3: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) 4: U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) 5: plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) 6: x#(N,s(M)) -> c_9(U21#(tt(),M,N)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {5} These cover all (indirect) predecessors of dependency pairs {1,2,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 5.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) - Weak DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1,2}, uargs(c_7) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {activate,U11#,U12#,U21#,U22#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = 0 p(U11) = 0 p(U12) = x2*x3 + x3^2 p(U21) = x3^2 p(U22) = 1 + x2^2 p(activate) = x1 p(plus) = 1 + x1 + x1*x2 + x2 + x2^2 p(s) = 1 + x1 p(tt) = 1 p(x) = 0 p(U11#) = 1 + x2 p(U12#) = 1 + x1*x2 p(U21#) = 1 + x1*x3 + x2 + x2*x3 + x3 + x3^2 p(U22#) = 1 + x1*x2 + x1*x3 + x2*x3 + x3 + x3^2 p(activate#) = 0 p(plus#) = 1 + x2 p(x#) = x1 + x1*x2 + x1^2 + x2 p(c_1) = x1 p(c_2) = x1 p(c_3) = x1 p(c_4) = x1 + x2 p(c_5) = 0 p(c_6) = 0 p(c_7) = x1 p(c_8) = 0 p(c_9) = x1 Following rules are strictly oriented: plus#(N,s(M)) = 2 + M > 1 + M = c_7(U11#(tt(),M,N)) Following rules are (at-least) weakly oriented: U11#(tt(),M,N) = 1 + M >= 1 + M = c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) = 1 + M >= 1 + M = c_2(plus#(activate(N),activate(M))) U21#(tt(),M,N) = 1 + M + M*N + 2*N + N^2 >= 1 + M + M*N + 2*N + N^2 = c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) = 1 + M + M*N + 2*N + N^2 >= 1 + M + M*N + 2*N + N^2 = c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) x#(N,s(M)) = 1 + M + M*N + 2*N + N^2 >= 1 + M + M*N + 2*N + N^2 = c_9(U21#(tt(),M,N)) activate(X) = X >= X = X *** Step 5.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) - Weak DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))):2 2:W:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):5 3:W:U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) -->_1 U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))):4 4:W:U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) -->_2 x#(N,s(M)) -> c_9(U21#(tt(),M,N)):6 -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):5 5:W:plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))):1 6:W:x#(N,s(M)) -> c_9(U21#(tt(),M,N)) -->_1 U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) 6: x#(N,s(M)) -> c_9(U21#(tt(),M,N)) 4: U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) 1: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) 5: plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) 2: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) *** Step 5.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) -->_1 U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))):2 2:S:U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):6 -->_2 x#(N,s(M)) -> c_9(U21#(tt(),M,N)):3 3:S:x#(N,s(M)) -> c_9(U21#(tt(),M,N)) -->_1 U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))):1 4:W:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))):5 5:W:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):6 6:W:plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) 5: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) 4: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) -->_1 U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))):2 2:S:U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) -->_2 x#(N,s(M)) -> c_9(U21#(tt(),M,N)):3 3:S:x#(N,s(M)) -> c_9(U21#(tt(),M,N)) -->_1 U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) ** Step 5.b:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) ** Step 5.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) 3: x#(N,s(M)) -> c_9(U21#(tt(),M,N)) Consider the set of all dependency pairs 1: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) 2: U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) 3: x#(N,s(M)) -> c_9(U21#(tt(),M,N)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2,3} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 5.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {activate,U11#,U12#,U21#,U22#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = [1] p(U11) = [1] x3 + [1] p(U12) = [2] x3 + [1] p(U21) = [1] x1 + [8] x3 + [1] p(U22) = [1] x2 + [1] p(activate) = [1] x1 + [0] p(plus) = [1] x1 + [1] p(s) = [1] x1 + [2] p(tt) = [8] p(x) = [1] x1 + [0] p(U11#) = [0] p(U12#) = [1] x1 + [2] x2 + [8] x3 + [1] p(U21#) = [2] x1 + [8] x2 + [7] p(U22#) = [1] x1 + [8] x2 + [11] p(activate#) = [0] p(plus#) = [1] x1 + [4] p(x#) = [8] x2 + [12] p(c_1) = [8] x1 + [2] p(c_2) = [2] p(c_3) = [1] x1 + [4] p(c_4) = [1] x1 + [4] p(c_5) = [0] p(c_6) = [2] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [4] Following rules are strictly oriented: U22#(tt(),M,N) = [8] M + [19] > [8] M + [16] = c_4(x#(activate(N),activate(M))) x#(N,s(M)) = [8] M + [28] > [8] M + [27] = c_9(U21#(tt(),M,N)) Following rules are (at-least) weakly oriented: U21#(tt(),M,N) = [8] M + [23] >= [8] M + [23] = c_3(U22#(tt(),activate(M),activate(N))) activate(X) = [1] X + [0] >= [1] X + [0] = X *** Step 5.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) - Weak DPs: U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) -->_1 U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))):2 2:W:U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) -->_1 x#(N,s(M)) -> c_9(U21#(tt(),M,N)):3 3:W:x#(N,s(M)) -> c_9(U21#(tt(),M,N)) -->_1 U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) 3: x#(N,s(M)) -> c_9(U21#(tt(),M,N)) 2: U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) *** Step 5.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))