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: NaturalMI 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: 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(2ndspos) = {2}, uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square,times} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [8] x2 + [8] p(2ndspos) = [1] x2 + [0] p(activate) = [1] x1 + [12] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(n__cons) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [9] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [2] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [9] p(square) = [2] x1 + [0] p(times) = [2] x1 + [0] Following rules are strictly oriented: 2ndsneg(0(),Z) = [8] Z + [8] > [0] = rnil() activate(X) = [1] X + [12] > [1] X + [0] = X Following rules are (at-least) weakly oriented: 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() activate(n__cons(X1,X2)) = [1] X1 + [12] >= [1] X1 + [12] = cons(activate(X1),X2) activate(n__from(X)) = [1] X + [12] >= [1] X + [12] = from(activate(X)) activate(n__s(X)) = [1] X + [21] >= [1] X + [21] = s(activate(X)) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = n__cons(X1,X2) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) pi(X) = [0] >= [0] = 2ndspos(X,from(0())) plus(0(),Y) = [2] Y + [0] >= [1] Y + [0] = Y s(X) = [1] X + [9] >= [1] X + [9] = n__s(X) square(X) = [2] X + [0] >= [2] X + [0] = times(X,X) times(0(),Y) = [0] >= [0] = 0() * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2ndspos(0(),Z) -> rnil() 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() - Weak TRS: 2ndsneg(0(),Z) -> rnil() activate(X) -> X - 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(2ndspos) = {2}, uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(2ndsneg) = [2] x1 + [14] p(2ndspos) = [4] x1 + [1] x2 + [8] p(activate) = [2] x1 + [2] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(n__cons) = [1] x1 + [0] p(n__from) = [1] x1 + [8] p(n__s) = [1] x1 + [1] p(negrecip) = [0] p(pi) = [4] x1 + [12] p(plus) = [2] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [0] p(times) = [0] Following rules are strictly oriented: 2ndspos(0(),Z) = [1] Z + [16] > [0] = rnil() activate(n__from(X)) = [2] X + [18] > [2] X + [2] = from(activate(X)) activate(n__s(X)) = [2] X + [4] > [2] X + [2] = s(activate(X)) pi(X) = [4] X + [12] > [4] X + [10] = 2ndspos(X,from(0())) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [18] >= [0] = rnil() activate(X) = [2] X + [2] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [2] X1 + [2] >= [2] X1 + [2] = cons(activate(X1),X2) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = n__cons(X1,X2) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [8] = n__from(X) plus(0(),Y) = [2] Y + [0] >= [1] Y + [0] = Y s(X) = [1] X + [0] >= [1] X + [1] = n__s(X) square(X) = [0] >= [0] = times(X,X) times(0(),Y) = [0] >= [2] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__cons(X1,X2)) -> cons(activate(X1),X2) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y s(X) -> n__s(X) square(X) -> times(X,X) times(0(),Y) -> 0() - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndspos(0(),Z) -> rnil() activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) pi(X) -> 2ndspos(X,from(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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(2ndspos) = {2}, uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [1] x2 + [0] p(2ndspos) = [1] x2 + [0] p(activate) = [2] x1 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(n__cons) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [2] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [0] p(times) = [1] Following rules are strictly oriented: times(0(),Y) = [1] > [0] = 0() Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [1] Z + [0] >= [0] = rnil() 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [2] X1 + [0] >= [2] X1 + [0] = cons(activate(X1),X2) activate(n__from(X)) = [2] X + [0] >= [2] X + [0] = from(activate(X)) activate(n__s(X)) = [2] X + [0] >= [2] X + [0] = s(activate(X)) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = n__cons(X1,X2) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) pi(X) = [0] >= [0] = 2ndspos(X,from(0())) plus(0(),Y) = [2] Y + [0] >= [1] Y + [0] = Y s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) square(X) = [0] >= [1] = times(X,X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__cons(X1,X2)) -> cons(activate(X1),X2) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y s(X) -> n__s(X) square(X) -> times(X,X) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndspos(0(),Z) -> rnil() activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) pi(X) -> 2ndspos(X,from(0())) 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(2ndspos) = {2}, uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [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(n__cons) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [2] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [1] p(times) = [0] Following rules are strictly oriented: square(X) = [1] > [0] = times(X,X) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [0] >= [0] = rnil() 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [2] X1 + [0] >= [2] X1 + [0] = cons(activate(X1),X2) activate(n__from(X)) = [2] X + [0] >= [2] X + [0] = from(activate(X)) activate(n__s(X)) = [2] X + [0] >= [2] X + [0] = s(activate(X)) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = n__cons(X1,X2) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) pi(X) = [0] >= [0] = 2ndspos(X,from(0())) plus(0(),Y) = [2] Y + [0] >= [1] Y + [0] = Y s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) times(0(),Y) = [0] >= [0] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__cons(X1,X2)) -> cons(activate(X1),X2) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y s(X) -> n__s(X) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndspos(0(),Z) -> rnil() activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) pi(X) -> 2ndspos(X,from(0())) 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(2ndspos) = {2}, uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [3] p(2ndsneg) = [3] x1 + [15] p(2ndspos) = [1] x2 + [0] p(activate) = [1] x1 + [9] p(cons) = [1] x1 + [9] p(from) = [1] x1 + [5] p(n__cons) = [1] x1 + [0] p(n__from) = [1] x1 + [10] p(n__s) = [1] x1 + [8] p(negrecip) = [1] x1 + [0] p(pi) = [8] p(plus) = [2] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [8] x1 + [3] p(times) = [3] Following rules are strictly oriented: cons(X1,X2) = [1] X1 + [9] > [1] X1 + [0] = n__cons(X1,X2) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [24] >= [0] = rnil() 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() activate(X) = [1] X + [9] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X1 + [9] >= [1] X1 + [18] = cons(activate(X1),X2) activate(n__from(X)) = [1] X + [19] >= [1] X + [14] = from(activate(X)) activate(n__s(X)) = [1] X + [17] >= [1] X + [9] = s(activate(X)) from(X) = [1] X + [5] >= [1] X + [9] = cons(X,n__from(n__s(X))) from(X) = [1] X + [5] >= [1] X + [10] = n__from(X) pi(X) = [8] >= [8] = 2ndspos(X,from(0())) plus(0(),Y) = [2] Y + [0] >= [1] Y + [0] = Y s(X) = [1] X + [0] >= [1] X + [8] = n__s(X) square(X) = [8] X + [3] >= [3] = times(X,X) times(0(),Y) = [3] >= [3] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__cons(X1,X2)) -> cons(activate(X1),X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y s(X) -> n__s(X) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndspos(0(),Z) -> rnil() activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) pi(X) -> 2ndspos(X,from(0())) 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(2ndspos) = {2}, uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(2ndsneg) = [9] x1 + [0] p(2ndspos) = [8] x1 + [1] x2 + [1] p(activate) = [3] x1 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [7] p(n__cons) = [1] x1 + [0] p(n__from) = [1] x1 + [6] p(n__s) = [1] x1 + [1] p(negrecip) = [1] x1 + [0] p(pi) = [8] x1 + [9] p(plus) = [10] x1 + [2] x2 + [0] p(posrecip) = [0] p(rcons) = [1] p(rnil) = [9] p(s) = [1] x1 + [3] p(square) = [3] x1 + [8] p(times) = [8] Following rules are strictly oriented: from(X) = [1] X + [7] > [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [7] > [1] X + [6] = n__from(X) plus(0(),Y) = [2] Y + [10] > [1] Y + [0] = Y s(X) = [1] X + [3] > [1] X + [1] = n__s(X) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [9] >= [9] = rnil() 2ndspos(0(),Z) = [1] Z + [9] >= [9] = rnil() activate(X) = [3] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [3] X1 + [0] >= [3] X1 + [0] = cons(activate(X1),X2) activate(n__from(X)) = [3] X + [18] >= [3] X + [7] = from(activate(X)) activate(n__s(X)) = [3] X + [3] >= [3] X + [3] = s(activate(X)) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = n__cons(X1,X2) pi(X) = [8] X + [9] >= [8] X + [9] = 2ndspos(X,from(0())) square(X) = [3] X + [8] >= [8] = times(X,X) times(0(),Y) = [8] >= [1] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__cons(X1,X2)) -> cons(activate(X1),X2) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndspos(0(),Z) -> rnil() activate(X) -> X 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(2ndspos) = {2}, uargs(cons) = {1}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [1] x2 + [0] p(activate) = [3] x1 + [0] p(cons) = [1] x1 + [6] p(from) = [1] x1 + [6] p(n__cons) = [1] x1 + [5] p(n__from) = [1] x1 + [2] p(n__s) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [6] p(plus) = [2] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [0] p(times) = [0] Following rules are strictly oriented: activate(n__cons(X1,X2)) = [3] X1 + [15] > [3] X1 + [6] = cons(activate(X1),X2) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [0] >= [0] = rnil() 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() activate(X) = [3] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [3] X + [6] >= [3] X + [6] = from(activate(X)) activate(n__s(X)) = [3] X + [0] >= [3] X + [0] = s(activate(X)) cons(X1,X2) = [1] X1 + [6] >= [1] X1 + [5] = n__cons(X1,X2) from(X) = [1] X + [6] >= [1] X + [6] = cons(X,n__from(n__s(X))) from(X) = [1] X + [6] >= [1] X + [2] = n__from(X) pi(X) = [6] >= [6] = 2ndspos(X,from(0())) plus(0(),Y) = [2] Y + [0] >= [1] Y + [0] = Y s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) square(X) = [0] >= [0] = times(X,X) times(0(),Y) = [0] >= [0] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))