MAYBE * Step 1: WeightGap MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(X) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) nats(N) -> cons(N,n__nats(s(N))) nats(X) -> n__nats(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N))) zprimes() -> sieve(nats(s(s(0())))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,filter,nats,sieve,zprimes} and constructors {0 ,cons,n__filter,n__nats,n__sieve,s} + 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) = {2}, uargs(filter) = {1}, uargs(n__filter) = {1}, uargs(n__sieve) = {1}, uargs(sieve) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [4] p(cons) = [1] x2 + [3] p(filter) = [1] x1 + [0] p(n__filter) = [1] x1 + [0] p(n__nats) = [0] p(n__sieve) = [1] x1 + [0] p(nats) = [0] p(s) = [1] p(sieve) = [1] x1 + [0] p(zprimes) = [0] Following rules are strictly oriented: activate(X) = [1] X + [4] > [1] X + [0] = X activate(n__filter(X1,X2,X3)) = [1] X1 + [4] > [1] X1 + [0] = filter(X1,X2,X3) activate(n__nats(X)) = [4] > [0] = nats(X) activate(n__sieve(X)) = [1] X + [4] > [1] X + [0] = sieve(X) Following rules are (at-least) weakly oriented: filter(X1,X2,X3) = [1] X1 + [0] >= [1] X1 + [0] = n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) = [1] Y + [3] >= [1] Y + [7] = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = [1] Y + [3] >= [1] Y + [7] = cons(X,n__filter(activate(Y),N,M)) nats(N) = [0] >= [3] = cons(N,n__nats(s(N))) nats(X) = [0] >= [0] = n__nats(X) sieve(X) = [1] X + [0] >= [1] X + [0] = n__sieve(X) sieve(cons(0(),Y)) = [1] Y + [3] >= [1] Y + [7] = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = [1] Y + [3] >= [1] Y + [7] = cons(s(N),n__sieve(filter(activate(Y),N,N))) zprimes() = [0] >= [0] = sieve(nats(s(s(0())))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap MAYBE + Considered Problem: - Strict TRS: filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) nats(N) -> cons(N,n__nats(s(N))) nats(X) -> n__nats(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N))) zprimes() -> sieve(nats(s(s(0())))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(X) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,filter,nats,sieve,zprimes} and constructors {0 ,cons,n__filter,n__nats,n__sieve,s} + 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) = {2}, uargs(filter) = {1}, uargs(n__filter) = {1}, uargs(n__sieve) = {1}, uargs(sieve) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [6] p(cons) = [1] x1 + [1] x2 + [0] p(filter) = [1] x1 + [0] p(n__filter) = [1] x1 + [4] p(n__nats) = [1] x1 + [0] p(n__sieve) = [1] x1 + [3] p(nats) = [1] x1 + [0] p(s) = [4] p(sieve) = [1] x1 + [1] p(zprimes) = [6] Following rules are strictly oriented: zprimes() = [6] > [5] = sieve(nats(s(s(0())))) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [6] >= [1] X + [0] = X activate(n__filter(X1,X2,X3)) = [1] X1 + [10] >= [1] X1 + [0] = filter(X1,X2,X3) activate(n__nats(X)) = [1] X + [6] >= [1] X + [0] = nats(X) activate(n__sieve(X)) = [1] X + [9] >= [1] X + [1] = sieve(X) filter(X1,X2,X3) = [1] X1 + [0] >= [1] X1 + [4] = n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) = [1] X + [1] Y + [0] >= [1] Y + [10] = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = [1] X + [1] Y + [0] >= [1] X + [1] Y + [10] = cons(X,n__filter(activate(Y),N,M)) nats(N) = [1] N + [0] >= [1] N + [4] = cons(N,n__nats(s(N))) nats(X) = [1] X + [0] >= [1] X + [0] = n__nats(X) sieve(X) = [1] X + [1] >= [1] X + [3] = n__sieve(X) sieve(cons(0(),Y)) = [1] Y + [1] >= [1] Y + [9] = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = [1] Y + [5] >= [1] Y + [13] = cons(s(N),n__sieve(filter(activate(Y),N,N))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap MAYBE + Considered Problem: - Strict TRS: filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) nats(N) -> cons(N,n__nats(s(N))) nats(X) -> n__nats(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(X) zprimes() -> sieve(nats(s(s(0())))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,filter,nats,sieve,zprimes} and constructors {0 ,cons,n__filter,n__nats,n__sieve,s} + 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) = {2}, uargs(filter) = {1}, uargs(n__filter) = {1}, uargs(n__sieve) = {1}, uargs(sieve) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [3] p(cons) = [1] x1 + [1] x2 + [0] p(filter) = [1] x1 + [0] p(n__filter) = [1] x1 + [0] p(n__nats) = [1] x1 + [0] p(n__sieve) = [1] x1 + [0] p(nats) = [1] x1 + [3] p(s) = [1] p(sieve) = [1] x1 + [1] p(zprimes) = [5] Following rules are strictly oriented: nats(N) = [1] N + [3] > [1] N + [1] = cons(N,n__nats(s(N))) nats(X) = [1] X + [3] > [1] X + [0] = n__nats(X) sieve(X) = [1] X + [1] > [1] X + [0] = n__sieve(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [3] >= [1] X + [0] = X activate(n__filter(X1,X2,X3)) = [1] X1 + [3] >= [1] X1 + [0] = filter(X1,X2,X3) activate(n__nats(X)) = [1] X + [3] >= [1] X + [3] = nats(X) activate(n__sieve(X)) = [1] X + [3] >= [1] X + [1] = sieve(X) filter(X1,X2,X3) = [1] X1 + [0] >= [1] X1 + [0] = n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) = [1] X + [1] Y + [0] >= [1] Y + [3] = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = [1] X + [1] Y + [0] >= [1] X + [1] Y + [3] = cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) = [1] Y + [1] >= [1] Y + [3] = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = [1] Y + [2] >= [1] Y + [4] = cons(s(N),n__sieve(filter(activate(Y),N,N))) zprimes() = [5] >= [5] = sieve(nats(s(s(0())))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap MAYBE + Considered Problem: - Strict TRS: filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(X) nats(N) -> cons(N,n__nats(s(N))) nats(X) -> n__nats(X) sieve(X) -> n__sieve(X) zprimes() -> sieve(nats(s(s(0())))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,filter,nats,sieve,zprimes} and constructors {0 ,cons,n__filter,n__nats,n__sieve,s} + 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) = {2}, uargs(filter) = {1}, uargs(n__filter) = {1}, uargs(n__sieve) = {1}, uargs(sieve) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [0] p(filter) = [1] x1 + [2] p(n__filter) = [1] x1 + [0] p(n__nats) = [0] p(n__sieve) = [1] x1 + [0] p(nats) = [2] p(s) = [0] p(sieve) = [1] x1 + [0] p(zprimes) = [4] Following rules are strictly oriented: filter(X1,X2,X3) = [1] X1 + [2] > [1] X1 + [0] = n__filter(X1,X2,X3) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__filter(X1,X2,X3)) = [1] X1 + [2] >= [1] X1 + [2] = filter(X1,X2,X3) activate(n__nats(X)) = [2] >= [2] = nats(X) activate(n__sieve(X)) = [1] X + [2] >= [1] X + [0] = sieve(X) filter(cons(X,Y),0(),M) = [1] Y + [2] >= [1] Y + [2] = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = [1] Y + [2] >= [1] Y + [2] = cons(X,n__filter(activate(Y),N,M)) nats(N) = [2] >= [0] = cons(N,n__nats(s(N))) nats(X) = [2] >= [0] = n__nats(X) sieve(X) = [1] X + [0] >= [1] X + [0] = n__sieve(X) sieve(cons(0(),Y)) = [1] Y + [0] >= [1] Y + [2] = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = [1] Y + [0] >= [1] Y + [4] = cons(s(N),n__sieve(filter(activate(Y),N,N))) zprimes() = [4] >= [2] = sieve(nats(s(s(0())))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: Failure MAYBE + Considered Problem: - Strict TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(X) filter(X1,X2,X3) -> n__filter(X1,X2,X3) nats(N) -> cons(N,n__nats(s(N))) nats(X) -> n__nats(X) sieve(X) -> n__sieve(X) zprimes() -> sieve(nats(s(s(0())))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,filter,nats,sieve,zprimes} and constructors {0 ,cons,n__filter,n__nats,n__sieve,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE