WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) a__zprimes() -> zprimes() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(s(X)) -> s(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + 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(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__nats) = [1] x1 + [0] p(a__sieve) = [1] x1 + [0] p(a__zprimes) = [0] p(cons) = [1] x1 + [0] p(filter) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [1] p(nats) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sieve) = [1] x1 + [0] p(zprimes) = [0] Following rules are strictly oriented: mark(0()) = [1] > [0] = 0() mark(zprimes()) = [1] > [0] = a__zprimes() Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [0] >= [0] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [0] >= [1] X + [1] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1] N + [0] >= [1] N + [1] = cons(mark(N),nats(s(N))) a__nats(X) = [1] X + [0] >= [1] X + [0] = nats(X) a__sieve(X) = [1] X + [0] >= [1] X + [0] = sieve(X) a__sieve(cons(0(),Y)) = [0] >= [0] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1] N + [0] >= [1] N + [1] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [0] >= [0] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [0] >= [0] = zprimes() mark(cons(X1,X2)) = [1] X1 + [1] >= [1] X1 + [1] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [3] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1] X + [1] >= [1] X + [1] = a__nats(mark(X)) mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(mark(X)) mark(sieve(X)) = [1] X + [1] >= [1] X + [1] = a__sieve(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) a__zprimes() -> zprimes() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(s(X)) -> s(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) - Weak TRS: mark(0()) -> 0() mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + 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(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__nats) = [1] x1 + [0] p(a__sieve) = [1] x1 + [0] p(a__zprimes) = [0] p(cons) = [1] x1 + [0] p(filter) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(nats) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sieve) = [1] x1 + [7] p(zprimes) = [0] Following rules are strictly oriented: mark(sieve(X)) = [1] X + [7] > [1] X + [0] = a__sieve(mark(X)) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [0] >= [0] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [0] >= [1] X + [0] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1] N + [0] >= [1] N + [0] = cons(mark(N),nats(s(N))) a__nats(X) = [1] X + [0] >= [1] X + [0] = nats(X) a__sieve(X) = [1] X + [0] >= [1] X + [7] = sieve(X) a__sieve(cons(0(),Y)) = [0] >= [0] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1] N + [0] >= [1] N + [0] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [0] >= [0] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [0] >= [0] = zprimes() mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1] X + [0] >= [1] X + [0] = a__nats(mark(X)) mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) mark(zprimes()) = [0] >= [0] = a__zprimes() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) a__zprimes() -> zprimes() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: mark(0()) -> 0() mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + 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(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [3] p(a__nats) = [1] x1 + [0] p(a__sieve) = [1] x1 + [0] p(a__zprimes) = [1] p(cons) = [1] x1 + [0] p(filter) = [1] x1 + [1] x2 + [1] x3 + [7] p(mark) = [1] x1 + [1] p(nats) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sieve) = [1] x1 + [0] p(zprimes) = [0] Following rules are strictly oriented: a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [8] > [5] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [3] > [1] X + [1] = cons(mark(X),filter(Y,N,M)) a__zprimes() = [1] > [0] = zprimes() mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [8] > [1] X1 + [1] X2 + [1] X3 + [6] = a__filter(mark(X1),mark(X2),mark(X3)) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [3] >= [1] X1 + [1] X2 + [1] X3 + [7] = filter(X1,X2,X3) a__nats(N) = [1] N + [0] >= [1] N + [1] = cons(mark(N),nats(s(N))) a__nats(X) = [1] X + [0] >= [1] X + [0] = nats(X) a__sieve(X) = [1] X + [0] >= [1] X + [0] = sieve(X) a__sieve(cons(0(),Y)) = [5] >= [5] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1] N + [0] >= [1] N + [1] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [1] >= [5] = a__sieve(a__nats(s(s(0())))) mark(0()) = [6] >= [5] = 0() mark(cons(X1,X2)) = [1] X1 + [1] >= [1] X1 + [1] = cons(mark(X1),X2) mark(nats(X)) = [1] X + [1] >= [1] X + [1] = a__nats(mark(X)) mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(mark(X)) mark(sieve(X)) = [1] X + [1] >= [1] X + [1] = a__sieve(mark(X)) mark(zprimes()) = [1] >= [1] = a__zprimes() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__nats(N) -> cons(mark(N),nats(s(N))) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nats(X)) -> a__nats(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__zprimes() -> zprimes() mark(0()) -> 0() mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + 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(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__nats) = [1] x1 + [0] p(a__sieve) = [1] x1 + [0] p(a__zprimes) = [3] p(cons) = [1] x1 + [4] p(filter) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(nats) = [1] x1 + [6] p(s) = [1] x1 + [3] p(sieve) = [1] x1 + [0] p(zprimes) = [3] Following rules are strictly oriented: mark(nats(X)) = [1] X + [6] > [1] X + [0] = a__nats(mark(X)) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [5] >= [5] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [7] >= [1] X + [4] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1] N + [0] >= [1] N + [4] = cons(mark(N),nats(s(N))) a__nats(X) = [1] X + [0] >= [1] X + [6] = nats(X) a__sieve(X) = [1] X + [0] >= [1] X + [0] = sieve(X) a__sieve(cons(0(),Y)) = [5] >= [5] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1] N + [7] >= [1] N + [7] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [3] >= [7] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [3] >= [3] = zprimes() mark(0()) = [1] >= [1] = 0() mark(cons(X1,X2)) = [1] X1 + [4] >= [1] X1 + [4] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__filter(mark(X1),mark(X2),mark(X3)) mark(s(X)) = [1] X + [3] >= [1] X + [3] = s(mark(X)) mark(sieve(X)) = [1] X + [0] >= [1] X + [0] = a__sieve(mark(X)) mark(zprimes()) = [3] >= [3] = a__zprimes() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__nats(N) -> cons(mark(N),nats(s(N))) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__zprimes() -> zprimes() mark(0()) -> 0() mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + 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(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [7] p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__nats) = [1] x1 + [4] p(a__sieve) = [1] x1 + [0] p(a__zprimes) = [4] p(cons) = [1] x1 + [1] p(filter) = [1] x1 + [1] x2 + [1] x3 + [7] p(mark) = [1] x1 + [1] p(nats) = [1] x1 + [4] p(s) = [1] x1 + [2] p(sieve) = [1] x1 + [0] p(zprimes) = [4] Following rules are strictly oriented: a__nats(N) = [1] N + [4] > [1] N + [2] = cons(mark(N),nats(s(N))) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [7] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [8] >= [8] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [3] >= [1] X + [2] = cons(mark(X),filter(Y,N,M)) a__nats(X) = [1] X + [4] >= [1] X + [4] = nats(X) a__sieve(X) = [1] X + [0] >= [1] X + [0] = sieve(X) a__sieve(cons(0(),Y)) = [8] >= [8] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1] N + [3] >= [1] N + [4] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [4] >= [15] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [4] >= [4] = zprimes() mark(0()) = [8] >= [7] = 0() mark(cons(X1,X2)) = [1] X1 + [2] >= [1] X1 + [2] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [8] >= [1] X1 + [1] X2 + [1] X3 + [3] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1] X + [5] >= [1] X + [5] = a__nats(mark(X)) mark(s(X)) = [1] X + [3] >= [1] X + [3] = s(mark(X)) mark(sieve(X)) = [1] X + [1] >= [1] X + [1] = a__sieve(mark(X)) mark(zprimes()) = [5] >= [4] = a__zprimes() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__zprimes() -> zprimes() mark(0()) -> 0() mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + 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(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [6] p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__nats) = [1] x1 + [5] p(a__sieve) = [1] x1 + [1] p(a__zprimes) = [0] p(cons) = [1] x1 + [0] p(filter) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(nats) = [1] x1 + [5] p(s) = [1] x1 + [1] p(sieve) = [1] x1 + [6] p(zprimes) = [0] Following rules are strictly oriented: a__sieve(cons(0(),Y)) = [7] > [6] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1] N + [2] > [1] N + [1] = cons(s(mark(N)),sieve(filter(Y,N,N))) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [6] >= [6] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [1] >= [1] X + [0] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1] N + [5] >= [1] N + [0] = cons(mark(N),nats(s(N))) a__nats(X) = [1] X + [5] >= [1] X + [5] = nats(X) a__sieve(X) = [1] X + [1] >= [1] X + [6] = sieve(X) a__zprimes() = [0] >= [14] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [0] >= [0] = zprimes() mark(0()) = [6] >= [6] = 0() mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1] X + [5] >= [1] X + [5] = a__nats(mark(X)) mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(mark(X)) mark(sieve(X)) = [1] X + [6] >= [1] X + [1] = a__sieve(mark(X)) mark(zprimes()) = [0] >= [0] = a__zprimes() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) a__zprimes() -> a__sieve(a__nats(s(s(0())))) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> zprimes() mark(0()) -> 0() mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + 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(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [1] p(a__nats) = [1] x1 + [0] p(a__sieve) = [1] x1 + [0] p(a__zprimes) = [1] p(cons) = [1] x1 + [0] p(filter) = [1] x1 + [1] x2 + [1] x3 + [2] p(mark) = [1] x1 + [0] p(nats) = [1] x1 + [4] p(s) = [1] x1 + [0] p(sieve) = [1] x1 + [1] p(zprimes) = [1] Following rules are strictly oriented: a__zprimes() = [1] > [0] = a__sieve(a__nats(s(s(0())))) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [2] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [1] >= [0] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [1] >= [1] X + [0] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1] N + [0] >= [1] N + [0] = cons(mark(N),nats(s(N))) a__nats(X) = [1] X + [0] >= [1] X + [4] = nats(X) a__sieve(X) = [1] X + [0] >= [1] X + [1] = sieve(X) a__sieve(cons(0(),Y)) = [0] >= [0] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1] N + [0] >= [1] N + [0] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [1] >= [1] = zprimes() mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [2] >= [1] X1 + [1] X2 + [1] X3 + [1] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1] X + [4] >= [1] X + [0] = a__nats(mark(X)) mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) mark(sieve(X)) = [1] X + [1] >= [1] X + [0] = a__sieve(mark(X)) mark(zprimes()) = [1] >= [1] = a__zprimes() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) a__zprimes() -> zprimes() mark(0()) -> 0() mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__filter,a__nats,a__sieve,a__zprimes,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__filter) = [1 1 0] [1 1 0] [1 0 1] [0] [0 1 1] x1 + [0 1 1] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 0] [0 0 1] [0] p(a__nats) = [1 1 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__sieve) = [1 1 0] [1] [0 1 0] x1 + [1] [0 0 0] [0] p(a__zprimes) = [1] [1] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(filter) = [1 1 0] [1 1 0] [1 0 1] [0] [0 1 1] x1 + [0 1 1] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(nats) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(sieve) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(zprimes) = [0] [1] [0] Following rules are strictly oriented: a__sieve(X) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] > [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = sieve(X) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1 1 0] [1 1 0] [1 0 1] [0] [0 1 1] X1 + [0 1 1] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 0] [0 0 1] [0] >= [1 1 0] [1 1 0] [1 0 1] [0] [0 1 1] X1 + [0 1 1] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1 0 1] [1 1 0] [0] [0 1 0] M + [0 1 0] X + [0] [0 0 1] [0 0 0] [0] >= [0] [0] [0] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1 0 1] [1 1 0] [1 1 0] [0] [0 1 0] M + [0 1 0] N + [0 1 0] X + [0] [0 0 1] [0 0 0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1 1 1] [0] [0 1 0] N + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] N + [0] [0 0 0] [0] = cons(mark(N),nats(s(N))) a__nats(X) = [1 1 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = nats(X) a__sieve(cons(0(),Y)) = [1] [1] [0] >= [0] [0] [0] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1 1 0] [1] [0 1 0] N + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] N + [0] [0 0 0] [0] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [1] [1] [0] >= [1] [1] [0] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [1] [1] [0] >= [0] [1] [0] = zprimes() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1 2 1] [1 2 1] [1 1 1] [0] [0 1 1] X1 + [0 1 1] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] >= [1 2 0] [1 2 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__nats(mark(X)) mark(s(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = s(mark(X)) mark(sieve(X)) = [1 2 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [1 2 0] [1] [0 1 0] X + [1] [0 0 0] [0] = a__sieve(mark(X)) mark(zprimes()) = [1] [1] [0] >= [1] [1] [0] = a__zprimes() * Step 9: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__nats(X) -> nats(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) a__zprimes() -> zprimes() mark(0()) -> 0() mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__filter,a__nats,a__sieve,a__zprimes,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__filter) = [1 1 0] [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0 0 1] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] p(a__nats) = [1 1 1] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(a__sieve) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(a__zprimes) = [1] [1] [1] p(cons) = [1 1 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(filter) = [1 1 0] [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0 0 1] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] p(mark) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(nats) = [1 1 1] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(s) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(sieve) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(zprimes) = [1] [1] [1] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 1 1] [1] [0 0 1] X1 + [1] [0 0 1] [1] > [1 1 1] [0] [0 0 1] X1 + [1] [0 0 1] [1] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1 1 0] [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0 0 1] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0 0 1] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1 1 0] [1 1 1] [1] [0 0 1] M + [0 0 1] X + [1] [0 0 1] [0 0 1] [1] >= [0] [1] [1] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1 1 0] [1 1 1] [1 1 1] [1] [0 0 1] M + [0 0 1] N + [0 0 1] X + [1] [0 0 1] [0 0 1] [0 0 1] [1] >= [1 1 1] [0] [0 0 1] X + [1] [0 0 1] [1] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1 1 1] [0] [0 0 1] N + [1] [0 0 1] [1] >= [1 1 1] [0] [0 0 1] N + [1] [0 0 1] [1] = cons(mark(N),nats(s(N))) a__nats(X) = [1 1 1] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [1] = nats(X) a__sieve(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = sieve(X) a__sieve(cons(0(),Y)) = [1] [1] [1] >= [0] [1] [1] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1 1 2] [1] [0 0 1] N + [1] [0 0 1] [1] >= [1 1 2] [0] [0 0 1] N + [1] [0 0 1] [1] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [1] [1] [1] >= [1] [1] [1] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [1] [1] [1] >= [1] [1] [1] = zprimes() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(filter(X1,X2,X3)) = [1 1 1] [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0 0 1] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0 0 1] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1 1 2] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 2] [0] [0 0 1] X + [1] [0 0 1] [1] = a__nats(mark(X)) mark(s(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = s(mark(X)) mark(sieve(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = a__sieve(mark(X)) mark(zprimes()) = [2] [1] [1] >= [1] [1] [1] = a__zprimes() * Step 10: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__nats(X) -> nats(X) mark(s(X)) -> s(mark(X)) - Weak TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) a__zprimes() -> zprimes() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__filter,a__nats,a__sieve,a__zprimes,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__filter) = [1 0 1] [1 0 0] [1 0 0] [1] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(a__nats) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__sieve) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(a__zprimes) = [0] [0] [1] p(cons) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(filter) = [1 0 1] [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nats) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(sieve) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(zprimes) = [0] [0] [1] Following rules are strictly oriented: a__filter(X1,X2,X3) = [1 0 1] [1 0 0] [1 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] > [1 0 1] [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] = filter(X1,X2,X3) Following rules are (at-least) weakly oriented: a__filter(cons(X,Y),0(),M) = [1 0 0] [1 1 1] [1] [0 0 0] M + [0 0 0] X + [0] [0 0 1] [0 0 1] [1] >= [0] [0] [0] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1 0 0] [1 0 0] [1 1 1] [1] [0 0 0] M + [0 0 0] N + [0 0 0] X + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [0] = cons(mark(N),nats(s(N))) a__nats(X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = nats(X) a__sieve(X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = sieve(X) a__sieve(cons(0(),Y)) = [0] [0] [1] >= [0] [0] [0] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [0] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [0] [0] [1] >= [0] [0] [1] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [0] [0] [1] >= [0] [0] [1] = zprimes() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 1 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1 0 2] [1 0 1] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [1 0 2] [1 0 1] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = a__nats(mark(X)) mark(s(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) mark(sieve(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [1] = a__sieve(mark(X)) mark(zprimes()) = [1] [0] [1] >= [0] [0] [1] = a__zprimes() * Step 11: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__nats(X) -> nats(X) mark(s(X)) -> s(mark(X)) - Weak TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) a__zprimes() -> zprimes() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__filter,a__nats,a__sieve,a__zprimes,mark} TcT has computed the following interpretation: p(0) = [0] [1] [0] p(a__filter) = [1 0 1] [1 1 0] [1 1 0] [1] [0 0 1] x1 + [0 0 1] x2 + [0 0 0] x3 + [1] [0 0 1] [0 0 1] [0 0 1] [1] p(a__nats) = [1 0 1] [1] [0 0 1] x1 + [0] [0 0 1] [1] p(a__sieve) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(a__zprimes) = [1] [1] [1] p(cons) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(filter) = [1 0 1] [1 1 0] [1 1 0] [1] [0 0 1] x1 + [0 0 1] x2 + [0 0 0] x3 + [1] [0 0 1] [0 0 1] [0 0 1] [1] p(mark) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(nats) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(sieve) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(zprimes) = [1] [1] [1] Following rules are strictly oriented: a__nats(X) = [1 0 1] [1] [0 0 1] X + [0] [0 0 1] [1] > [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [1] = nats(X) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1 0 1] [1 1 0] [1 1 0] [1] [0 0 1] X1 + [0 0 1] X2 + [0 0 0] X3 + [1] [0 0 1] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 1 0] [1 1 0] [1] [0 0 1] X1 + [0 0 1] X2 + [0 0 0] X3 + [1] [0 0 1] [0 0 1] [0 0 1] [1] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1 1 0] [1 0 1] [2] [0 0 0] M + [0 0 1] X + [1] [0 0 1] [0 0 1] [1] >= [0] [0] [0] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1 1 0] [1 0 0] [1 0 1] [1] [0 0 0] M + [0 0 1] N + [0 0 1] X + [1] [0 0 1] [0 0 1] [0 0 1] [1] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1 0 1] [1] [0 0 1] N + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 1] N + [0] [0 0 1] [0] = cons(mark(N),nats(s(N))) a__sieve(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = sieve(X) a__sieve(cons(0(),Y)) = [0] [0] [0] >= [0] [0] [0] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1 0 1] [0] [0 0 1] N + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] N + [0] [0 0 1] [0] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [1] [1] [1] >= [1] [1] [1] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [1] [1] [1] >= [1] [1] [1] = zprimes() mark(0()) = [0] [1] [0] >= [0] [1] [0] = 0() mark(cons(X1,X2)) = [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1 0 2] [1 1 1] [1 1 1] [2] [0 0 1] X1 + [0 0 1] X2 + [0 0 0] X3 + [1] [0 0 1] [0 0 1] [0 0 1] [1] >= [1 0 2] [1 1 1] [1 1 1] [1] [0 0 1] X1 + [0 0 1] X2 + [0 0 0] X3 + [1] [0 0 1] [0 0 1] [0 0 1] [1] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1 0 2] [1] [0 0 1] X + [0] [0 0 1] [1] >= [1 0 2] [1] [0 0 1] X + [0] [0 0 1] [1] = a__nats(mark(X)) mark(s(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) mark(sieve(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = a__sieve(mark(X)) mark(zprimes()) = [2] [1] [1] >= [1] [1] [1] = a__zprimes() * Step 12: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(s(X)) -> s(mark(X)) - Weak TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) a__zprimes() -> zprimes() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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(a__filter) = {1,2,3}, uargs(a__nats) = {1}, uargs(a__sieve) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__filter,a__nats,a__sieve,a__zprimes,mark} TcT has computed the following interpretation: p(0) = [0] [0] p(a__filter) = [1 1] x1 + [1 0] x2 + [1 0] x3 + [4] [0 1] [0 1] [0 1] [4] p(a__nats) = [1 1] x1 + [1] [0 1] [0] p(a__sieve) = [1 2] x1 + [0] [0 1] [0] p(a__zprimes) = [7] [3] p(cons) = [1 0] x1 + [0] [0 1] [0] p(filter) = [1 1] x1 + [1 0] x2 + [1 0] x3 + [2] [0 1] [0 1] [0 1] [4] p(mark) = [1 1] x1 + [1] [0 1] [0] p(nats) = [1 1] x1 + [1] [0 1] [0] p(s) = [1 0] x1 + [0] [0 1] [1] p(sieve) = [1 2] x1 + [0] [0 1] [0] p(zprimes) = [3] [3] Following rules are strictly oriented: mark(s(X)) = [1 1] X + [2] [0 1] [1] > [1 1] X + [1] [0 1] [1] = s(mark(X)) Following rules are (at-least) weakly oriented: a__filter(X1,X2,X3) = [1 1] X1 + [1 0] X2 + [1 0] X3 + [4] [0 1] [0 1] [0 1] [4] >= [1 1] X1 + [1 0] X2 + [1 0] X3 + [2] [0 1] [0 1] [0 1] [4] = filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) = [1 0] M + [1 1] X + [4] [0 1] [0 1] [4] >= [0] [0] = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = [1 0] M + [1 0] N + [1 1] X + [4] [0 1] [0 1] [0 1] [5] >= [1 1] X + [1] [0 1] [0] = cons(mark(X),filter(Y,N,M)) a__nats(N) = [1 1] N + [1] [0 1] [0] >= [1 1] N + [1] [0 1] [0] = cons(mark(N),nats(s(N))) a__nats(X) = [1 1] X + [1] [0 1] [0] >= [1 1] X + [1] [0 1] [0] = nats(X) a__sieve(X) = [1 2] X + [0] [0 1] [0] >= [1 2] X + [0] [0 1] [0] = sieve(X) a__sieve(cons(0(),Y)) = [0] [0] >= [0] [0] = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = [1 2] N + [2] [0 1] [1] >= [1 1] N + [1] [0 1] [1] = cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() = [7] [3] >= [7] [2] = a__sieve(a__nats(s(s(0())))) a__zprimes() = [7] [3] >= [3] [3] = zprimes() mark(0()) = [1] [0] >= [0] [0] = 0() mark(cons(X1,X2)) = [1 1] X1 + [1] [0 1] [0] >= [1 1] X1 + [1] [0 1] [0] = cons(mark(X1),X2) mark(filter(X1,X2,X3)) = [1 2] X1 + [1 1] X2 + [1 1] X3 + [7] [0 1] [0 1] [0 1] [4] >= [1 2] X1 + [1 1] X2 + [1 1] X3 + [7] [0 1] [0 1] [0 1] [4] = a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) = [1 2] X + [2] [0 1] [0] >= [1 2] X + [2] [0 1] [0] = a__nats(mark(X)) mark(sieve(X)) = [1 3] X + [1] [0 1] [0] >= [1 3] X + [1] [0 1] [0] = a__sieve(mark(X)) mark(zprimes()) = [7] [3] >= [7] [3] = a__zprimes() * Step 13: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__nats(N) -> cons(mark(N),nats(s(N))) a__nats(X) -> nats(X) a__sieve(X) -> sieve(X) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) a__zprimes() -> zprimes() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(nats(X)) -> a__nats(mark(X)) mark(s(X)) -> s(mark(X)) mark(sieve(X)) -> a__sieve(mark(X)) mark(zprimes()) -> a__zprimes() - Signature: {a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes ,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))