WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) s(X) -> n__s(X) square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square ,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. 2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndspos(0(),Z) -> rnil() activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y s(X) -> n__s(X) square(X) -> times(X,X) times(0(),Y) -> 0() - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square ,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_2() activate#(X) -> c_3() activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) cons#(X1,X2) -> c_7() from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() pi#(X) -> c_10(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_11() s#(X) -> c_12() square#(X) -> c_13(times#(X,X)) times#(0(),Y) -> c_14() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_2() activate#(X) -> c_3() activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) cons#(X1,X2) -> c_7() from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() pi#(X) -> c_10(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_11() s#(X) -> c_12() square#(X) -> c_13(times#(X,X)) times#(0(),Y) -> c_14() - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndspos(0(),Z) -> rnil() activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y s(X) -> n__s(X) square(X) -> times(X,X) times(0(),Y) -> 0() - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_2() activate#(X) -> c_3() activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) cons#(X1,X2) -> c_7() from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() pi#(X) -> c_10(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_11() s#(X) -> c_12() square#(X) -> c_13(times#(X,X)) times#(0(),Y) -> c_14() * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_2() activate#(X) -> c_3() activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) cons#(X1,X2) -> c_7() from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() pi#(X) -> c_10(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_11() s#(X) -> c_12() square#(X) -> c_13(times#(X,X)) times#(0(),Y) -> c_14() - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1}, uargs(2ndspos#) = {2}, uargs(cons#) = {1}, uargs(from#) = {1}, uargs(s#) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1}, uargs(c_13) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [2] x1 + [5] p(cons) = [1] x1 + [5] p(from) = [1] x1 + [7] p(n__cons) = [1] x1 + [3] p(n__from) = [1] x1 + [4] p(n__s) = [1] x1 + [2] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [3] p(square) = [0] p(times) = [0] p(2ndsneg#) = [0] p(2ndspos#) = [1] x2 + [0] p(activate#) = [2] x1 + [0] p(cons#) = [1] x1 + [0] p(from#) = [1] x1 + [0] p(pi#) = [0] p(plus#) = [0] p(s#) = [1] x1 + [0] p(square#) = [0] p(times#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] Following rules are strictly oriented: activate#(n__cons(X1,X2)) = [2] X1 + [6] > [2] X1 + [5] = c_4(cons#(activate(X1),X2)) activate#(n__from(X)) = [2] X + [8] > [2] X + [5] = c_5(from#(activate(X))) activate(X) = [2] X + [5] > [1] X + [0] = X activate(n__cons(X1,X2)) = [2] X1 + [11] > [2] X1 + [10] = cons(activate(X1),X2) activate(n__from(X)) = [2] X + [13] > [2] X + [12] = from(activate(X)) activate(n__s(X)) = [2] X + [9] > [2] X + [8] = s(activate(X)) cons(X1,X2) = [1] X1 + [5] > [1] X1 + [3] = n__cons(X1,X2) from(X) = [1] X + [7] > [1] X + [5] = cons(X,n__from(n__s(X))) from(X) = [1] X + [7] > [1] X + [4] = n__from(X) s(X) = [1] X + [3] > [1] X + [2] = n__s(X) Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = [0] >= [0] = c_1() 2ndspos#(0(),Z) = [1] Z + [0] >= [0] = c_2() activate#(X) = [2] X + [0] >= [0] = c_3() activate#(n__s(X)) = [2] X + [4] >= [2] X + [5] = c_6(s#(activate(X))) cons#(X1,X2) = [1] X1 + [0] >= [0] = c_7() from#(X) = [1] X + [0] >= [1] X + [0] = c_8(cons#(X,n__from(n__s(X)))) from#(X) = [1] X + [0] >= [0] = c_9() pi#(X) = [0] >= [7] = c_10(2ndspos#(X,from(0()))) plus#(0(),Y) = [0] >= [0] = c_11() s#(X) = [1] X + [0] >= [0] = c_12() square#(X) = [0] >= [0] = c_13(times#(X,X)) times#(0(),Y) = [0] >= [0] = c_14() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_2() activate#(X) -> c_3() activate#(n__s(X)) -> c_6(s#(activate(X))) cons#(X1,X2) -> c_7() from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() pi#(X) -> c_10(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_11() s#(X) -> c_12() square#(X) -> c_13(times#(X,X)) times#(0(),Y) -> c_14() - Weak DPs: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,9,10,12} by application of Pre({1,2,3,9,10,12}) = {4,8,11}. Here rules are labelled as follows: 1: 2ndsneg#(0(),Z) -> c_1() 2: 2ndspos#(0(),Z) -> c_2() 3: activate#(X) -> c_3() 4: activate#(n__s(X)) -> c_6(s#(activate(X))) 5: cons#(X1,X2) -> c_7() 6: from#(X) -> c_8(cons#(X,n__from(n__s(X)))) 7: from#(X) -> c_9() 8: pi#(X) -> c_10(2ndspos#(X,from(0()))) 9: plus#(0(),Y) -> c_11() 10: s#(X) -> c_12() 11: square#(X) -> c_13(times#(X,X)) 12: times#(0(),Y) -> c_14() 13: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) 14: activate#(n__from(X)) -> c_5(from#(activate(X))) * Step 6: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__s(X)) -> c_6(s#(activate(X))) cons#(X1,X2) -> c_7() from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() pi#(X) -> c_10(2ndspos#(X,from(0()))) square#(X) -> c_13(times#(X,X)) - Weak DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_2() activate#(X) -> c_3() activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) plus#(0(),Y) -> c_11() s#(X) -> c_12() times#(0(),Y) -> c_14() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,6} by application of Pre({1,5,6}) = {}. Here rules are labelled as follows: 1: activate#(n__s(X)) -> c_6(s#(activate(X))) 2: cons#(X1,X2) -> c_7() 3: from#(X) -> c_8(cons#(X,n__from(n__s(X)))) 4: from#(X) -> c_9() 5: pi#(X) -> c_10(2ndspos#(X,from(0()))) 6: square#(X) -> c_13(times#(X,X)) 7: 2ndsneg#(0(),Z) -> c_1() 8: 2ndspos#(0(),Z) -> c_2() 9: activate#(X) -> c_3() 10: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) 11: activate#(n__from(X)) -> c_5(from#(activate(X))) 12: plus#(0(),Y) -> c_11() 13: s#(X) -> c_12() 14: times#(0(),Y) -> c_14() * Step 7: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cons#(X1,X2) -> c_7() from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() - Weak DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_2() activate#(X) -> c_3() activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) pi#(X) -> c_10(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_11() s#(X) -> c_12() square#(X) -> c_13(times#(X,X)) times#(0(),Y) -> c_14() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cons#(X1,X2) -> c_7() 2:S:from#(X) -> c_8(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():1 3:S:from#(X) -> c_9() 4:W:2ndsneg#(0(),Z) -> c_1() 5:W:2ndspos#(0(),Z) -> c_2() 6:W:activate#(X) -> c_3() 7:W:activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) -->_1 cons#(X1,X2) -> c_7():1 8:W:activate#(n__from(X)) -> c_5(from#(activate(X))) -->_1 from#(X) -> c_9():3 -->_1 from#(X) -> c_8(cons#(X,n__from(n__s(X)))):2 9:W:activate#(n__s(X)) -> c_6(s#(activate(X))) -->_1 s#(X) -> c_12():12 10:W:pi#(X) -> c_10(2ndspos#(X,from(0()))) -->_1 2ndspos#(0(),Z) -> c_2():5 11:W:plus#(0(),Y) -> c_11() 12:W:s#(X) -> c_12() 13:W:square#(X) -> c_13(times#(X,X)) -->_1 times#(0(),Y) -> c_14():14 14:W:times#(0(),Y) -> c_14() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: square#(X) -> c_13(times#(X,X)) 14: times#(0(),Y) -> c_14() 11: plus#(0(),Y) -> c_11() 10: pi#(X) -> c_10(2ndspos#(X,from(0()))) 9: activate#(n__s(X)) -> c_6(s#(activate(X))) 12: s#(X) -> c_12() 6: activate#(X) -> c_3() 5: 2ndspos#(0(),Z) -> c_2() 4: 2ndsneg#(0(),Z) -> c_1() * Step 8: Decompose WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cons#(X1,X2) -> c_7() from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() - Weak DPs: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + 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: cons#(X1,X2) -> c_7() - Weak DPs: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} Problem (S) - Strict DPs: from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() - Weak DPs: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) cons#(X1,X2) -> c_7() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} ** Step 8.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cons#(X1,X2) -> c_7() - Weak DPs: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cons#(X1,X2) -> c_7() 2:W:from#(X) -> c_8(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():1 3:W:from#(X) -> c_9() 7:W:activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) -->_1 cons#(X1,X2) -> c_7():1 8:W:activate#(n__from(X)) -> c_5(from#(activate(X))) -->_1 from#(X) -> c_8(cons#(X,n__from(n__s(X)))):2 -->_1 from#(X) -> c_9():3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: from#(X) -> c_9() ** Step 8.a:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cons#(X1,X2) -> c_7() - Weak DPs: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) from#(X) -> c_8(cons#(X,n__from(n__s(X)))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:cons#(X1,X2) -> c_7() 2:W:from#(X) -> c_8(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():1 7:W:activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) -->_1 cons#(X1,X2) -> c_7():1 8:W:activate#(n__from(X)) -> c_5(from#(activate(X))) -->_1 from#(X) -> c_8(cons#(X,n__from(n__s(X)))):2 The dependency graph contains no loops, we remove all dependency pairs. ** Step 8.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() - Weak DPs: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_5(from#(activate(X))) cons#(X1,X2) -> c_7() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:from#(X) -> c_8(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():5 2:S:from#(X) -> c_9() 3:W:activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) -->_1 cons#(X1,X2) -> c_7():5 4:W:activate#(n__from(X)) -> c_5(from#(activate(X))) -->_1 from#(X) -> c_9():2 -->_1 from#(X) -> c_8(cons#(X,n__from(n__s(X)))):1 5:W:cons#(X1,X2) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: activate#(n__cons(X1,X2)) -> c_4(cons#(activate(X1),X2)) 5: cons#(X1,X2) -> c_7() ** Step 8.b:2: SimplifyRHS WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_8(cons#(X,n__from(n__s(X)))) from#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_5(from#(activate(X))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:from#(X) -> c_8(cons#(X,n__from(n__s(X)))) 2:S:from#(X) -> c_9() 4:W:activate#(n__from(X)) -> c_5(from#(activate(X))) -->_1 from#(X) -> c_9():2 -->_1 from#(X) -> c_8(cons#(X,n__from(n__s(X)))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: from#(X) -> c_8() ** Step 8.b:3: Decompose WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_8() from#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_5(from#(activate(X))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + 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: from#(X) -> c_8() - Weak DPs: activate#(n__from(X)) -> c_5(from#(activate(X))) from#(X) -> c_9() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} Problem (S) - Strict DPs: from#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_5(from#(activate(X))) from#(X) -> c_8() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} *** Step 8.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_8() - Weak DPs: activate#(n__from(X)) -> c_5(from#(activate(X))) from#(X) -> c_9() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:from#(X) -> c_8() 2:W:from#(X) -> c_9() 3:W:activate#(n__from(X)) -> c_5(from#(activate(X))) -->_1 from#(X) -> c_8():1 -->_1 from#(X) -> c_9():2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: from#(X) -> c_9() *** Step 8.b:3.a:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_8() - Weak DPs: activate#(n__from(X)) -> c_5(from#(activate(X))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:from#(X) -> c_8() 3:W:activate#(n__from(X)) -> c_5(from#(activate(X))) -->_1 from#(X) -> c_8():1 The dependency graph contains no loops, we remove all dependency pairs. *** Step 8.b:3.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 8.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_5(from#(activate(X))) from#(X) -> c_8() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:from#(X) -> c_9() 2:W:activate#(n__from(X)) -> c_5(from#(activate(X))) -->_1 from#(X) -> c_8():3 -->_1 from#(X) -> c_9():1 3:W:from#(X) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: from#(X) -> c_8() *** Step 8.b:3.b:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_5(from#(activate(X))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:from#(X) -> c_9() 2:W:activate#(n__from(X)) -> c_5(from#(activate(X))) -->_1 from#(X) -> c_9():1 The dependency graph contains no loops, we remove all dependency pairs. *** Step 8.b:3.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))