WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) head(cons(X,Y)) -> X sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) All above mentioned rules can be savely removed. * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + 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(cons) = {1}, uargs(filter) = {1,2}, uargs(from) = {1}, uargs(s) = {1}, uargs(sieve) = {1} Following symbols are considered usable: {activate,cons,filter,from,head,if,primes,s,sieve,tail} TcT has computed the following interpretation: p(0) = [0] p(activate) = [8] x1 + [2] p(cons) = [1] x1 + [0] p(divides) = [1] p(false) = [0] p(filter) = [1] x1 + [1] x2 + [1] p(from) = [1] x1 + [2] p(head) = [8] x1 + [1] p(if) = [1] x1 + [8] x2 + [8] x3 + [3] p(n__cons) = [1] x1 + [0] p(n__filter) = [1] x1 + [1] x2 + [1] p(n__from) = [1] x1 + [2] p(n__s) = [1] x1 + [0] p(n__sieve) = [1] x1 + [2] p(primes) = [13] p(s) = [1] x1 + [0] p(sieve) = [1] x1 + [8] p(tail) = [8] x1 + [2] p(true) = [9] Following rules are strictly oriented: activate(X) = [8] X + [2] > [1] X + [0] = X activate(n__filter(X1,X2)) = [8] X1 + [8] X2 + [10] > [8] X1 + [8] X2 + [5] = filter(activate(X1),activate(X2)) activate(n__from(X)) = [8] X + [18] > [8] X + [4] = from(activate(X)) activate(n__sieve(X)) = [8] X + [18] > [8] X + [10] = sieve(activate(X)) from(X) = [1] X + [2] > [1] X + [0] = cons(X,n__from(n__s(X))) if(false(),X,Y) = [8] X + [8] Y + [3] > [8] Y + [2] = activate(Y) if(true(),X,Y) = [8] X + [8] Y + [12] > [8] X + [2] = activate(X) primes() = [13] > [10] = sieve(from(s(s(0())))) sieve(X) = [1] X + [8] > [1] X + [2] = n__sieve(X) Following rules are (at-least) weakly oriented: activate(n__cons(X1,X2)) = [8] X1 + [2] >= [8] X1 + [2] = cons(activate(X1),X2) activate(n__s(X)) = [8] X + [2] >= [8] X + [2] = s(activate(X)) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = n__cons(X1,X2) filter(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = n__filter(X1,X2) from(X) = [1] X + [2] >= [1] X + [2] = n__from(X) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) * Step 3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) from(X) -> cons(X,n__from(n__s(X))) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + 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(cons) = {1}, uargs(filter) = {1,2}, uargs(from) = {1}, uargs(s) = {1}, uargs(sieve) = {1} Following symbols are considered usable: {activate,cons,filter,from,head,if,primes,s,sieve,tail} TcT has computed the following interpretation: p(0) = [1] p(activate) = [8] x1 + [0] p(cons) = [1] x1 + [0] p(divides) = [1] x2 + [0] p(false) = [0] p(filter) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [0] p(head) = [1] x1 + [1] p(if) = [4] x1 + [9] x2 + [8] x3 + [12] p(n__cons) = [1] x1 + [0] p(n__filter) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [1] p(n__sieve) = [1] x1 + [0] p(primes) = [15] p(s) = [1] x1 + [7] p(sieve) = [1] x1 + [0] p(tail) = [1] x1 + [2] p(true) = [4] Following rules are strictly oriented: activate(n__s(X)) = [8] X + [8] > [8] X + [7] = s(activate(X)) s(X) = [1] X + [7] > [1] X + [1] = n__s(X) Following rules are (at-least) weakly oriented: activate(X) = [8] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [8] X1 + [0] >= [8] X1 + [0] = cons(activate(X1),X2) activate(n__filter(X1,X2)) = [8] X1 + [8] X2 + [0] >= [8] X1 + [8] X2 + [0] = filter(activate(X1),activate(X2)) activate(n__from(X)) = [8] X + [0] >= [8] X + [0] = from(activate(X)) activate(n__sieve(X)) = [8] X + [0] >= [8] X + [0] = sieve(activate(X)) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = n__cons(X1,X2) filter(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__filter(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) if(false(),X,Y) = [9] X + [8] Y + [12] >= [8] Y + [0] = activate(Y) if(true(),X,Y) = [9] X + [8] Y + [28] >= [8] X + [0] = activate(X) primes() = [15] >= [15] = sieve(from(s(s(0())))) sieve(X) = [1] X + [0] >= [1] X + [0] = n__sieve(X) * 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) filter(X1,X2) -> n__filter(X1,X2) from(X) -> n__from(X) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) from(X) -> cons(X,n__from(n__s(X))) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + 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(cons) = {1}, uargs(filter) = {1,2}, uargs(from) = {1}, uargs(s) = {1}, uargs(sieve) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [3] p(activate) = [1] x1 + [0] p(cons) = [1] x1 + [0] p(divides) = [1] x1 + [1] x2 + [0] p(false) = [0] p(filter) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [0] p(head) = [4] p(if) = [4] x2 + [4] x3 + [1] p(n__cons) = [1] x1 + [1] p(n__filter) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [2] p(n__s) = [1] x1 + [0] p(n__sieve) = [1] x1 + [4] p(primes) = [7] p(s) = [1] x1 + [0] p(sieve) = [1] x1 + [4] p(tail) = [0] p(true) = [2] Following rules are strictly oriented: activate(n__cons(X1,X2)) = [1] X1 + [1] > [1] X1 + [0] = cons(activate(X1),X2) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__filter(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = filter(activate(X1),activate(X2)) activate(n__from(X)) = [1] X + [2] >= [1] X + [0] = from(activate(X)) activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(activate(X)) activate(n__sieve(X)) = [1] X + [4] >= [1] X + [4] = sieve(activate(X)) cons(X1,X2) = [1] X1 + [0] >= [1] X1 + [1] = n__cons(X1,X2) filter(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__filter(X1,X2) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [2] = n__from(X) if(false(),X,Y) = [4] X + [4] Y + [1] >= [1] Y + [0] = activate(Y) if(true(),X,Y) = [4] X + [4] Y + [1] >= [1] X + [0] = activate(X) primes() = [7] >= [7] = sieve(from(s(s(0())))) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) sieve(X) = [1] X + [4] >= [1] X + [4] = n__sieve(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> n__from(X) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) from(X) -> cons(X,n__from(n__s(X))) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + 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(cons) = {1}, uargs(filter) = {1,2}, uargs(from) = {1}, uargs(s) = {1}, uargs(sieve) = {1} Following symbols are considered usable: {activate,cons,filter,from,head,if,primes,s,sieve,tail} TcT has computed the following interpretation: p(0) = [0] p(activate) = [4] x1 + [0] p(cons) = [1] x1 + [1] p(divides) = [1] x2 + [1] p(false) = [8] p(filter) = [1] x1 + [1] x2 + [8] p(from) = [1] x1 + [8] p(head) = [0] p(if) = [4] x2 + [4] x3 + [14] p(n__cons) = [1] x1 + [1] p(n__filter) = [1] x1 + [1] x2 + [5] p(n__from) = [1] x1 + [2] p(n__s) = [1] x1 + [1] p(n__sieve) = [1] x1 + [0] p(primes) = [12] p(s) = [1] x1 + [2] p(sieve) = [1] x1 + [0] p(tail) = [1] x1 + [1] p(true) = [8] Following rules are strictly oriented: filter(X1,X2) = [1] X1 + [1] X2 + [8] > [1] X1 + [1] X2 + [5] = n__filter(X1,X2) from(X) = [1] X + [8] > [1] X + [2] = n__from(X) Following rules are (at-least) weakly oriented: activate(X) = [4] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [4] X1 + [4] >= [4] X1 + [1] = cons(activate(X1),X2) activate(n__filter(X1,X2)) = [4] X1 + [4] X2 + [20] >= [4] X1 + [4] X2 + [8] = filter(activate(X1),activate(X2)) activate(n__from(X)) = [4] X + [8] >= [4] X + [8] = from(activate(X)) activate(n__s(X)) = [4] X + [4] >= [4] X + [2] = s(activate(X)) activate(n__sieve(X)) = [4] X + [0] >= [4] X + [0] = sieve(activate(X)) cons(X1,X2) = [1] X1 + [1] >= [1] X1 + [1] = n__cons(X1,X2) from(X) = [1] X + [8] >= [1] X + [1] = cons(X,n__from(n__s(X))) if(false(),X,Y) = [4] X + [4] Y + [14] >= [4] Y + [0] = activate(Y) if(true(),X,Y) = [4] X + [4] Y + [14] >= [4] X + [0] = activate(X) primes() = [12] >= [12] = sieve(from(s(s(0())))) s(X) = [1] X + [2] >= [1] X + [1] = n__s(X) sieve(X) = [1] X + [0] >= [1] X + [0] = n__sieve(X) * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: cons(X1,X2) -> n__cons(X1,X2) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + 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(cons) = {1}, uargs(filter) = {1,2}, uargs(from) = {1}, uargs(s) = {1}, uargs(sieve) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [4] x1 + [0] p(cons) = [1] x1 + [4] p(divides) = [1] x1 + [1] x2 + [0] p(false) = [0] p(filter) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [4] p(head) = [0] p(if) = [4] x2 + [4] x3 + [0] p(n__cons) = [1] x1 + [2] p(n__filter) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [1] p(n__s) = [1] x1 + [0] p(n__sieve) = [1] x1 + [0] p(primes) = [5] p(s) = [1] x1 + [0] p(sieve) = [1] x1 + [0] p(tail) = [0] p(true) = [0] Following rules are strictly oriented: cons(X1,X2) = [1] X1 + [4] > [1] X1 + [2] = n__cons(X1,X2) Following rules are (at-least) weakly oriented: activate(X) = [4] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [4] X1 + [8] >= [4] X1 + [4] = cons(activate(X1),X2) activate(n__filter(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = filter(activate(X1),activate(X2)) activate(n__from(X)) = [4] X + [4] >= [4] X + [4] = from(activate(X)) activate(n__s(X)) = [4] X + [0] >= [4] X + [0] = s(activate(X)) activate(n__sieve(X)) = [4] X + [0] >= [4] X + [0] = sieve(activate(X)) filter(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__filter(X1,X2) from(X) = [1] X + [4] >= [1] X + [4] = cons(X,n__from(n__s(X))) from(X) = [1] X + [4] >= [1] X + [1] = n__from(X) if(false(),X,Y) = [4] X + [4] Y + [0] >= [4] Y + [0] = activate(Y) if(true(),X,Y) = [4] X + [4] Y + [0] >= [4] X + [0] = activate(X) primes() = [5] >= [4] = sieve(from(s(s(0())))) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) sieve(X) = [1] X + [0] >= [1] X + [0] = n__sieve(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))