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,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) 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,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,s,square ,times} and constructors {0,cons,cons2,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,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(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: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict 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)) 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,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,s,square ,times} and constructors {0,cons,cons2,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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(2ndsneg) = [9] p(2ndspos) = [1] x2 + [0] p(activate) = [6] x1 + [0] p(cons) = [1] x2 + [0] p(cons2) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [1] 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: 2ndsneg(0(),Z) = [9] > [0] = rnil() activate(n__s(X)) = [6] X + [6] > [6] X + [0] = s(activate(X)) Following rules are (at-least) weakly oriented: 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() activate(X) = [6] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [6] X + [0] >= [6] X + [0] = from(activate(X)) from(X) = [1] X + [0] >= [1] X + [1] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) pi(X) = [0] >= [2] = 2ndspos(X,from(0())) plus(0(),Y) = [2] Y + [0] >= [1] Y + [0] = Y s(X) = [1] X + [0] >= [1] X + [1] = n__s(X) square(X) = [0] >= [1] = times(X,X) times(0(),Y) = [1] >= [2] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2ndspos(0(),Z) -> rnil() activate(X) -> X activate(n__from(X)) -> from(activate(X)) 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(n__s(X)) -> s(activate(X)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,s,square ,times} and constructors {0,cons,cons2,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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(2ndsneg) = [1] x1 + [15] p(2ndspos) = [2] x1 + [1] x2 + [0] p(activate) = [1] x1 + [1] p(cons) = [1] x2 + [0] p(cons2) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [7] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [2] p(negrecip) = [1] x1 + [0] p(pi) = [8] x1 + [0] p(plus) = [2] x1 + [2] x2 + [1] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [1] p(s) = [1] x1 + [2] p(square) = [2] x1 + [0] p(times) = [1] x1 + [1] x2 + [13] Following rules are strictly oriented: 2ndspos(0(),Z) = [1] Z + [8] > [1] = rnil() activate(X) = [1] X + [1] > [1] X + [0] = X from(X) = [1] X + [7] > [1] X + [2] = cons(X,n__from(n__s(X))) from(X) = [1] X + [7] > [1] X + [0] = n__from(X) plus(0(),Y) = [2] Y + [9] > [1] Y + [0] = Y times(0(),Y) = [1] Y + [17] > [4] = 0() Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [19] >= [1] = rnil() activate(n__from(X)) = [1] X + [1] >= [1] X + [8] = from(activate(X)) activate(n__s(X)) = [1] X + [3] >= [1] X + [3] = s(activate(X)) pi(X) = [8] X + [0] >= [2] X + [11] = 2ndspos(X,from(0())) s(X) = [1] X + [2] >= [1] X + [2] = n__s(X) square(X) = [2] X + [0] >= [2] X + [13] = times(X,X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__from(X)) -> from(activate(X)) pi(X) -> 2ndspos(X,from(0())) 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__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y times(0(),Y) -> 0() - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,s,square ,times} and constructors {0,cons,cons2,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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {2ndsneg,2ndspos,activate,from,pi,plus,s,square,times} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [2] p(2ndspos) = [1] x2 + [2] p(activate) = [8] x1 + [7] p(cons) = [3] p(cons2) = [2] p(from) = [1] x1 + [3] p(n__from) = [1] x1 + [3] p(n__s) = [1] x1 + [2] p(negrecip) = [1] x1 + [1] p(pi) = [11] p(plus) = [4] x1 + [4] x2 + [1] p(posrecip) = [1] x1 + [2] p(rcons) = [1] p(rnil) = [2] p(s) = [1] x1 + [8] p(square) = [8] x1 + [8] p(times) = [2] x1 + [1] x2 + [3] Following rules are strictly oriented: activate(n__from(X)) = [8] X + [31] > [8] X + [10] = from(activate(X)) pi(X) = [11] > [5] = 2ndspos(X,from(0())) s(X) = [1] X + [8] > [1] X + [2] = n__s(X) square(X) = [8] X + [8] > [3] X + [3] = times(X,X) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [2] >= [2] = rnil() 2ndspos(0(),Z) = [1] Z + [2] >= [2] = rnil() activate(X) = [8] X + [7] >= [1] X + [0] = X activate(n__s(X)) = [8] X + [23] >= [8] X + [15] = s(activate(X)) from(X) = [1] X + [3] >= [3] = cons(X,n__from(n__s(X))) from(X) = [1] X + [3] >= [1] X + [3] = n__from(X) plus(0(),Y) = [4] Y + [1] >= [1] Y + [0] = Y times(0(),Y) = [1] Y + [3] >= [0] = 0() * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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)) 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,from/1,pi/1,plus/2,s/1,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,s,square ,times} and constructors {0,cons,cons2,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))