WORST_CASE(?,O(n^4)) * Step 1: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(U11) = [0] p(a__U11) = [1] x1 + [4] p(a__and) = [1] x1 + [0] p(a__isNat) = [0] p(a__isNatIList) = [0] p(a__isNatList) = [0] p(a__length) = [1] x1 + [0] p(a__zeros) = [0] p(and) = [0] p(cons) = [1] x1 + [0] p(isNat) = [0] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [1] x1 + [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__U11(X1,X2) = [1] X1 + [4] > [0] = U11(X1,X2) a__U11(tt(),L) = [4] > [0] = s(a__length(mark(L))) Following rules are (at-least) weakly oriented: a__and(X1,X2) = [1] X1 + [0] >= [0] = and(X1,X2) a__and(tt(),X) = [0] >= [0] = mark(X) a__isNat(X) = [0] >= [0] = isNat(X) a__isNat(0()) = [0] >= [0] = tt() a__isNat(length(V1)) = [0] >= [0] = a__isNatList(V1) a__isNat(s(V1)) = [0] >= [0] = a__isNat(V1) a__isNatIList(V) = [0] >= [0] = a__isNatList(V) a__isNatIList(X) = [0] >= [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0] >= [0] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [0] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [0] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] N + [0] >= [4] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [0] >= [2] = 0() a__zeros() = [0] >= [2] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [2] = 0() mark(U11(X1,X2)) = [0] >= [4] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [0] >= [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(isNat(X)) = [0] >= [0] = a__isNat(X) mark(isNatIList(X)) = [0] >= [0] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) mark(length(X)) = [0] >= [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(a__U11) = [1] x1 + [0] p(a__and) = [1] x1 + [7] p(a__isNat) = [0] p(a__isNatIList) = [0] p(a__isNatList) = [0] p(a__length) = [1] x1 + [0] p(a__zeros) = [0] p(and) = [0] p(cons) = [1] x1 + [0] p(isNat) = [0] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [1] x1 + [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__and(X1,X2) = [1] X1 + [7] > [0] = and(X1,X2) a__and(tt(),X) = [7] > [0] = mark(X) Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1] X1 + [0] >= [0] = U11(X1,X2) a__U11(tt(),L) = [0] >= [0] = s(a__length(mark(L))) a__isNat(X) = [0] >= [0] = isNat(X) a__isNat(0()) = [0] >= [0] = tt() a__isNat(length(V1)) = [0] >= [0] = a__isNatList(V1) a__isNat(s(V1)) = [0] >= [0] = a__isNat(V1) a__isNatIList(V) = [0] >= [0] = a__isNatList(V) a__isNatIList(X) = [0] >= [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0] >= [7] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [0] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [7] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] N + [0] >= [7] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [0] >= [0] = 0() a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(U11(X1,X2)) = [0] >= [0] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [0] >= [7] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(isNat(X)) = [0] >= [0] = a__isNat(X) mark(isNatIList(X)) = [0] >= [0] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) mark(length(X)) = [0] >= [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [3] p(a__U11) = [1] x1 + [3] p(a__and) = [1] x1 + [1] p(a__isNat) = [4] p(a__isNatIList) = [0] p(a__isNatList) = [0] p(a__length) = [1] x1 + [3] p(a__zeros) = [0] p(and) = [1] x1 + [1] p(cons) = [1] x1 + [0] p(isNat) = [0] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [1] x1 + [0] p(mark) = [0] p(nil) = [1] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__isNat(X) = [4] > [0] = isNat(X) a__isNat(0()) = [4] > [0] = tt() a__isNat(length(V1)) = [4] > [0] = a__isNatList(V1) a__length(X) = [1] X + [3] > [1] X + [0] = length(X) a__length(nil()) = [4] > [0] = 0() Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1] X1 + [3] >= [1] X1 + [3] = U11(X1,X2) a__U11(tt(),L) = [3] >= [3] = s(a__length(mark(L))) a__and(X1,X2) = [1] X1 + [1] >= [1] X1 + [1] = and(X1,X2) a__and(tt(),X) = [1] >= [0] = mark(X) a__isNat(s(V1)) = [4] >= [4] = a__isNat(V1) a__isNatIList(V) = [0] >= [0] = a__isNatList(V) a__isNatIList(X) = [0] >= [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0] >= [5] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [0] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [5] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [0] >= [0] = tt() a__length(cons(N,L)) = [1] N + [3] >= [4] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(U11(X1,X2)) = [0] >= [3] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [0] >= [1] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(isNat(X)) = [0] >= [4] = a__isNat(X) mark(isNatIList(X)) = [0] >= [0] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) mark(length(X)) = [0] >= [3] = a__length(mark(X)) mark(nil()) = [0] >= [1] = nil() mark(s(X)) = [0] >= [0] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__length(X) -> length(X) a__length(nil()) -> 0() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(U11) = [1] x1 + [5] p(a__U11) = [1] x1 + [5] p(a__and) = [1] x1 + [0] p(a__isNat) = [0] p(a__isNatIList) = [4] p(a__isNatList) = [0] p(a__length) = [1] x1 + [1] p(a__zeros) = [1] p(and) = [1] x1 + [0] p(cons) = [1] x1 + [5] p(isNat) = [0] p(isNatIList) = [1] p(isNatList) = [1] p(length) = [1] x1 + [1] p(mark) = [0] p(nil) = [4] p(s) = [1] x1 + [4] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__isNatIList(V) = [4] > [0] = a__isNatList(V) a__isNatIList(X) = [4] > [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [4] > [0] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [4] > [0] = tt() a__length(cons(N,L)) = [1] N + [6] > [5] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__zeros() = [1] > [0] = zeros() Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1] X1 + [5] >= [1] X1 + [5] = U11(X1,X2) a__U11(tt(),L) = [5] >= [5] = s(a__length(mark(L))) a__and(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = and(X1,X2) a__and(tt(),X) = [0] >= [0] = mark(X) a__isNat(X) = [0] >= [0] = isNat(X) a__isNat(0()) = [0] >= [0] = tt() a__isNat(length(V1)) = [0] >= [0] = a__isNatList(V1) a__isNat(s(V1)) = [0] >= [0] = a__isNat(V1) a__isNatList(X) = [0] >= [1] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [0] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [1] >= [1] X + [1] = length(X) a__length(nil()) = [5] >= [5] = 0() a__zeros() = [1] >= [10] = cons(0(),zeros()) mark(0()) = [0] >= [5] = 0() mark(U11(X1,X2)) = [0] >= [5] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [0] >= [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [5] = cons(mark(X1),X2) mark(isNat(X)) = [0] >= [0] = a__isNat(X) mark(isNatIList(X)) = [0] >= [4] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) mark(length(X)) = [0] >= [1] = a__length(mark(X)) mark(nil()) = [0] >= [4] = nil() mark(s(X)) = [0] >= [4] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() mark(zeros()) = [0] >= [1] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__zeros() -> cons(0(),zeros()) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> zeros() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(U11) = [1] x1 + [4] p(a__U11) = [1] x1 + [4] p(a__and) = [1] x1 + [0] p(a__isNat) = [2] p(a__isNatIList) = [2] p(a__isNatList) = [2] p(a__length) = [1] x1 + [3] p(a__zeros) = [2] p(and) = [1] x1 + [0] p(cons) = [1] x1 + [6] p(isNat) = [2] p(isNatIList) = [0] p(isNatList) = [1] p(length) = [1] p(mark) = [2] p(nil) = [1] p(s) = [1] x1 + [1] p(tt) = [2] p(zeros) = [2] Following rules are strictly oriented: a__isNatList(X) = [2] > [1] = isNatList(X) mark(nil()) = [2] > [1] = nil() Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1] X1 + [4] >= [1] X1 + [4] = U11(X1,X2) a__U11(tt(),L) = [6] >= [6] = s(a__length(mark(L))) a__and(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = and(X1,X2) a__and(tt(),X) = [2] >= [2] = mark(X) a__isNat(X) = [2] >= [2] = isNat(X) a__isNat(0()) = [2] >= [2] = tt() a__isNat(length(V1)) = [2] >= [2] = a__isNatList(V1) a__isNat(s(V1)) = [2] >= [2] = a__isNat(V1) a__isNatIList(V) = [2] >= [2] = a__isNatList(V) a__isNatIList(X) = [2] >= [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [2] >= [2] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [2] >= [2] = tt() a__isNatList(cons(V1,V2)) = [2] >= [2] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [2] >= [2] = tt() a__length(X) = [1] X + [3] >= [1] = length(X) a__length(cons(N,L)) = [1] N + [9] >= [6] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [4] >= [4] = 0() a__zeros() = [2] >= [10] = cons(0(),zeros()) a__zeros() = [2] >= [2] = zeros() mark(0()) = [2] >= [4] = 0() mark(U11(X1,X2)) = [2] >= [6] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [2] >= [2] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [2] >= [8] = cons(mark(X1),X2) mark(isNat(X)) = [2] >= [2] = a__isNat(X) mark(isNatIList(X)) = [2] >= [2] = a__isNatIList(X) mark(isNatList(X)) = [2] >= [2] = a__isNatList(X) mark(length(X)) = [2] >= [5] = a__length(mark(X)) mark(s(X)) = [2] >= [3] = s(mark(X)) mark(tt()) = [2] >= [2] = tt() mark(zeros()) = [2] >= [2] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__zeros() -> cons(0(),zeros()) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> zeros() mark(nil()) -> nil() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(a__U11) = [1] x1 + [4] p(a__and) = [1] x1 + [0] p(a__isNat) = [0] p(a__isNatIList) = [0] p(a__isNatList) = [0] p(a__length) = [1] x1 + [4] p(a__zeros) = [1] p(and) = [1] x1 + [0] p(cons) = [1] x1 + [0] p(isNat) = [0] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__zeros() = [1] > [0] = cons(0(),zeros()) Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1] X1 + [4] >= [0] = U11(X1,X2) a__U11(tt(),L) = [4] >= [4] = s(a__length(mark(L))) a__and(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = and(X1,X2) a__and(tt(),X) = [0] >= [0] = mark(X) a__isNat(X) = [0] >= [0] = isNat(X) a__isNat(0()) = [0] >= [0] = tt() a__isNat(length(V1)) = [0] >= [0] = a__isNatList(V1) a__isNat(s(V1)) = [0] >= [0] = a__isNat(V1) a__isNatIList(V) = [0] >= [0] = a__isNatList(V) a__isNatIList(X) = [0] >= [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0] >= [0] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [0] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [0] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [4] >= [0] = length(X) a__length(cons(N,L)) = [1] N + [4] >= [4] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [4] >= [0] = 0() a__zeros() = [1] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(U11(X1,X2)) = [0] >= [4] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [0] >= [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(isNat(X)) = [0] >= [0] = a__isNat(X) mark(isNatIList(X)) = [0] >= [0] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) mark(length(X)) = [0] >= [4] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() mark(zeros()) = [0] >= [1] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(nil()) -> nil() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(a__U11) = [1] x1 + [1] p(a__and) = [1] x1 + [0] p(a__isNat) = [4] p(a__isNatIList) = [4] p(a__isNatList) = [4] p(a__length) = [1] x1 + [0] p(a__zeros) = [5] p(and) = [0] p(cons) = [1] x1 + [5] p(isNat) = [0] p(isNatIList) = [4] p(isNatList) = [1] p(length) = [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] Following rules are strictly oriented: a__isNatList(nil()) = [4] > [0] = tt() Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1] X1 + [1] >= [0] = U11(X1,X2) a__U11(tt(),L) = [1] >= [0] = s(a__length(mark(L))) a__and(X1,X2) = [1] X1 + [0] >= [0] = and(X1,X2) a__and(tt(),X) = [0] >= [0] = mark(X) a__isNat(X) = [4] >= [0] = isNat(X) a__isNat(0()) = [4] >= [0] = tt() a__isNat(length(V1)) = [4] >= [4] = a__isNatList(V1) a__isNat(s(V1)) = [4] >= [4] = a__isNat(V1) a__isNatIList(V) = [4] >= [4] = a__isNatList(V) a__isNatIList(X) = [4] >= [4] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [4] >= [4] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [4] >= [0] = tt() a__isNatList(X) = [4] >= [1] = isNatList(X) a__isNatList(cons(V1,V2)) = [4] >= [4] = a__and(a__isNat(V1),isNatList(V2)) a__length(X) = [1] X + [0] >= [0] = length(X) a__length(cons(N,L)) = [1] N + [5] >= [5] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [0] >= [0] = 0() a__zeros() = [5] >= [5] = cons(0(),zeros()) a__zeros() = [5] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(U11(X1,X2)) = [0] >= [1] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [0] >= [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [5] = cons(mark(X1),X2) mark(isNat(X)) = [0] >= [4] = a__isNat(X) mark(isNatIList(X)) = [0] >= [4] = a__isNatIList(X) mark(isNatList(X)) = [0] >= [4] = a__isNatList(X) mark(length(X)) = [0] >= [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() mark(zeros()) = [0] >= [5] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(nil()) -> nil() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [0] p(a__U11) = [1] x1 + [0] p(a__and) = [1] x1 + [0] p(a__isNat) = [1] p(a__isNatIList) = [1] p(a__isNatList) = [1] p(a__length) = [1] x1 + [0] p(a__zeros) = [1] p(and) = [1] x1 + [0] p(cons) = [1] x1 + [1] p(isNat) = [1] p(isNatIList) = [1] p(isNatList) = [1] p(length) = [0] p(mark) = [1] p(nil) = [1] p(s) = [1] x1 + [0] p(tt) = [1] p(zeros) = [0] Following rules are strictly oriented: mark(0()) = [1] > [0] = 0() Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = U11(X1,X2) a__U11(tt(),L) = [1] >= [1] = s(a__length(mark(L))) a__and(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = and(X1,X2) a__and(tt(),X) = [1] >= [1] = mark(X) a__isNat(X) = [1] >= [1] = isNat(X) a__isNat(0()) = [1] >= [1] = tt() a__isNat(length(V1)) = [1] >= [1] = a__isNatList(V1) a__isNat(s(V1)) = [1] >= [1] = a__isNat(V1) a__isNatIList(V) = [1] >= [1] = a__isNatList(V) a__isNatIList(X) = [1] >= [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1] >= [1] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [1] >= [1] = tt() a__isNatList(X) = [1] >= [1] = isNatList(X) a__isNatList(cons(V1,V2)) = [1] >= [1] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [1] >= [1] = tt() a__length(X) = [1] X + [0] >= [0] = length(X) a__length(cons(N,L)) = [1] N + [1] >= [1] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [1] >= [0] = 0() a__zeros() = [1] >= [1] = cons(0(),zeros()) a__zeros() = [1] >= [0] = zeros() mark(U11(X1,X2)) = [1] >= [1] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [1] >= [1] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1] >= [2] = cons(mark(X1),X2) mark(isNat(X)) = [1] >= [1] = a__isNat(X) mark(isNatIList(X)) = [1] >= [1] = a__isNatIList(X) mark(isNatList(X)) = [1] >= [1] = a__isNatList(X) mark(length(X)) = [1] >= [1] = a__length(mark(X)) mark(nil()) = [1] >= [1] = nil() mark(s(X)) = [1] >= [1] = s(mark(X)) mark(tt()) = [1] >= [1] = tt() mark(zeros()) = [1] >= [1] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(nil()) -> nil() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] p(a__U11) = [1] x1 + [1] p(a__and) = [1] x1 + [4] p(a__isNat) = [0] p(a__isNatIList) = [4] p(a__isNatList) = [0] p(a__length) = [1] x1 + [0] p(a__zeros) = [5] p(and) = [0] p(cons) = [1] x1 + [5] p(isNat) = [0] p(isNatIList) = [0] p(isNatList) = [0] p(length) = [0] p(mark) = [1] p(nil) = [0] p(s) = [1] x1 + [0] p(tt) = [0] p(zeros) = [1] Following rules are strictly oriented: mark(isNat(X)) = [1] > [0] = a__isNat(X) mark(isNatList(X)) = [1] > [0] = a__isNatList(X) mark(tt()) = [1] > [0] = tt() Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1] X1 + [1] >= [1] = U11(X1,X2) a__U11(tt(),L) = [1] >= [1] = s(a__length(mark(L))) a__and(X1,X2) = [1] X1 + [4] >= [0] = and(X1,X2) a__and(tt(),X) = [4] >= [1] = mark(X) a__isNat(X) = [0] >= [0] = isNat(X) a__isNat(0()) = [0] >= [0] = tt() a__isNat(length(V1)) = [0] >= [0] = a__isNatList(V1) a__isNat(s(V1)) = [0] >= [0] = a__isNat(V1) a__isNatIList(V) = [4] >= [0] = a__isNatList(V) a__isNatIList(X) = [4] >= [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [4] >= [4] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [4] >= [0] = tt() a__isNatList(X) = [0] >= [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] >= [4] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [0] >= [0] = tt() a__length(X) = [1] X + [0] >= [0] = length(X) a__length(cons(N,L)) = [1] N + [5] >= [5] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [0] >= [0] = 0() a__zeros() = [5] >= [5] = cons(0(),zeros()) a__zeros() = [5] >= [1] = zeros() mark(0()) = [1] >= [0] = 0() mark(U11(X1,X2)) = [1] >= [2] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [1] >= [5] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1] >= [6] = cons(mark(X1),X2) mark(isNatIList(X)) = [1] >= [4] = a__isNatIList(X) mark(length(X)) = [1] >= [1] = a__length(mark(X)) mark(nil()) = [1] >= [0] = nil() mark(s(X)) = [1] >= [1] = s(mark(X)) mark(zeros()) = [1] >= [5] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 10: WeightGap WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNatIList(X)) -> a__isNatIList(X) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [1] x2 + [0] p(a__U11) = [1] x1 + [1] x2 + [1] p(a__and) = [1] x1 + [1] x2 + [0] p(a__isNat) = [4] p(a__isNatIList) = [1] x1 + [4] p(a__isNatList) = [3] p(a__length) = [1] x1 + [3] p(a__zeros) = [7] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [4] p(isNat) = [3] p(isNatIList) = [1] x1 + [4] p(isNatList) = [2] p(length) = [1] x1 + [0] p(mark) = [1] x1 + [1] p(nil) = [3] p(s) = [1] x1 + [0] p(tt) = [3] p(zeros) = [3] Following rules are strictly oriented: mark(isNatIList(X)) = [1] X + [5] > [1] X + [4] = a__isNatIList(X) Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [0] = U11(X1,X2) a__U11(tt(),L) = [1] L + [4] >= [1] L + [4] = s(a__length(mark(L))) a__and(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(tt(),X) = [1] X + [3] >= [1] X + [1] = mark(X) a__isNat(X) = [4] >= [3] = isNat(X) a__isNat(0()) = [4] >= [3] = tt() a__isNat(length(V1)) = [4] >= [3] = a__isNatList(V1) a__isNat(s(V1)) = [4] >= [4] = a__isNat(V1) a__isNatIList(V) = [1] V + [4] >= [3] = a__isNatList(V) a__isNatIList(X) = [1] X + [4] >= [1] X + [4] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V2 + [8] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [7] >= [3] = tt() a__isNatList(X) = [3] >= [2] = isNatList(X) a__isNatList(cons(V1,V2)) = [3] >= [6] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [3] >= [3] = tt() a__length(X) = [1] X + [3] >= [1] X + [0] = length(X) a__length(cons(N,L)) = [1] L + [1] N + [7] >= [1] L + [7] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [6] >= [0] = 0() a__zeros() = [7] >= [7] = cons(0(),zeros()) a__zeros() = [7] >= [3] = zeros() mark(0()) = [1] >= [0] = 0() mark(U11(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [2] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = cons(mark(X1),X2) mark(isNat(X)) = [4] >= [4] = a__isNat(X) mark(isNatList(X)) = [3] >= [3] = a__isNatList(X) mark(length(X)) = [1] X + [1] >= [1] X + [4] = a__length(mark(X)) mark(nil()) = [4] >= [3] = nil() mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(mark(X)) mark(tt()) = [4] >= [3] = tt() mark(zeros()) = [4] >= [7] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 11: NaturalMI WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(s(X)) -> s(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U11) = [1 0 0] [0 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__and) = [1 0 0] [0 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__isNat) = [0] [0] [0] p(a__isNatIList) = [1] [0] [1] p(a__isNatList) = [0] [0] [0] p(a__length) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(a__zeros) = [1] [0] [1] p(and) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 1 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(isNat) = [0] [0] [0] p(isNatIList) = [0] [0] [1] p(isNatList) = [0] [0] [0] p(length) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(mark) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tt) = [0] [0] [0] p(zeros) = [0] [0] [1] Following rules are strictly oriented: mark(U11(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] > [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__U11(mark(X1),X2) mark(length(X)) = [0 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] > [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__length(mark(X)) Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1 0 0] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U11(X1,X2) a__U11(tt(),L) = [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = s(a__length(mark(L))) a__and(X1,X2) = [1 0 0] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = and(X1,X2) a__and(tt(),X) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__isNat(X) = [0] [0] [0] >= [0] [0] [0] = isNat(X) a__isNat(0()) = [0] [0] [0] >= [0] [0] [0] = tt() a__isNat(length(V1)) = [0] [0] [0] >= [0] [0] [0] = a__isNatList(V1) a__isNat(s(V1)) = [0] [0] [0] >= [0] [0] [0] = a__isNat(V1) a__isNatIList(V) = [1] [0] [1] >= [0] [0] [0] = a__isNatList(V) a__isNatIList(X) = [1] [0] [1] >= [0] [0] [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1] [0] [1] >= [1] [0] [1] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [1] [0] [1] >= [0] [0] [0] = tt() a__isNatList(X) = [0] [0] [0] >= [0] [0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] [0] [0] >= [0] [0] [0] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [0] [0] [0] >= [0] [0] [0] = tt() a__length(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = length(X) a__length(cons(N,L)) = [0 0 1] [1 1 0] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [1] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [0] [0] [1] >= [0] [0] [0] = 0() a__zeros() = [1] [0] [1] >= [1] [0] [1] = cons(0(),zeros()) a__zeros() = [1] [0] [1] >= [0] [0] [1] = zeros() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(and(X1,X2)) = [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = cons(mark(X1),X2) mark(isNat(X)) = [0] [0] [0] >= [0] [0] [0] = a__isNat(X) mark(isNatIList(X)) = [1] [0] [1] >= [1] [0] [1] = a__isNatIList(X) mark(isNatList(X)) = [0] [0] [0] >= [0] [0] [0] = a__isNatList(X) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) mark(tt()) = [0] [0] [0] >= [0] [0] [0] = tt() mark(zeros()) = [1] [0] [1] >= [1] [0] [1] = a__zeros() * Step 12: NaturalMI WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__U11) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__and) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__isNat) = [0] [0] [1] p(a__isNatIList) = [0] [0] [1] p(a__isNatList) = [0] [0] [1] p(a__length) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(a__zeros) = [0] [0] [1] p(and) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(isNat) = [0] [0] [0] p(isNatIList) = [0] [0] [0] p(isNatList) = [0] [0] [0] p(length) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(mark) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(nil) = [0] [0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tt) = [0] [0] [1] p(zeros) = [0] [0] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] > [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = U11(X1,X2) a__U11(tt(),L) = [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [2] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [2] = s(a__length(mark(L))) a__and(X1,X2) = [1 0 0] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = and(X1,X2) a__and(tt(),X) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = mark(X) a__isNat(X) = [0] [0] [1] >= [0] [0] [0] = isNat(X) a__isNat(0()) = [0] [0] [1] >= [0] [0] [1] = tt() a__isNat(length(V1)) = [0] [0] [1] >= [0] [0] [1] = a__isNatList(V1) a__isNat(s(V1)) = [0] [0] [1] >= [0] [0] [1] = a__isNat(V1) a__isNatIList(V) = [0] [0] [1] >= [0] [0] [1] = a__isNatList(V) a__isNatIList(X) = [0] [0] [1] >= [0] [0] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0] [0] [1] >= [0] [0] [1] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [0] [0] [1] >= [0] [0] [1] = tt() a__isNatList(X) = [0] [0] [1] >= [0] [0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0] [0] [1] >= [0] [0] [1] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [0] [0] [1] >= [0] [0] [1] = tt() a__length(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = length(X) a__length(cons(N,L)) = [0 0 1] [1 0 0] [0] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [0] [0 0 0] L + [0] [0 0 1] [2] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [0] [0] [1] >= [0] [0] [0] = 0() a__zeros() = [0] [0] [1] >= [0] [0] [1] = cons(0(),zeros()) a__zeros() = [0] [0] [1] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [1] >= [0] [0] [0] = 0() mark(U11(X1,X2)) = [0 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [2] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__and(mark(X1),X2) mark(isNat(X)) = [0] [0] [1] >= [0] [0] [1] = a__isNat(X) mark(isNatIList(X)) = [0] [0] [1] >= [0] [0] [1] = a__isNatIList(X) mark(isNatList(X)) = [0] [0] [1] >= [0] [0] [1] = a__isNatList(X) mark(length(X)) = [0 0 1] [1] [0 0 0] X + [0] [0 0 1] [2] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [2] = a__length(mark(X)) mark(nil()) = [0] [0] [1] >= [0] [0] [0] = nil() mark(s(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = s(mark(X)) mark(tt()) = [1] [0] [2] >= [0] [0] [1] = tt() mark(zeros()) = [0] [0] [1] >= [0] [0] [1] = a__zeros() * Step 13: NaturalMI WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] p(U11) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] p(a__U11) = [1 0] x1 + [1 4] x2 + [0] [0 1] [0 1] [0] p(a__and) = [1 0] x1 + [1 4] x2 + [0] [0 1] [0 1] [0] p(a__isNat) = [0 1] x1 + [0] [0 0] [2] p(a__isNatIList) = [0 1] x1 + [0] [0 0] [2] p(a__isNatList) = [0 1] x1 + [0] [0 0] [2] p(a__length) = [1 0] x1 + [0] [0 1] [0] p(a__zeros) = [0] [2] p(and) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] p(cons) = [1 1] x1 + [1 5] x2 + [0] [0 1] [0 1] [2] p(isNat) = [0 1] x1 + [0] [0 0] [0] p(isNatIList) = [0 1] x1 + [0] [0 0] [0] p(isNatList) = [0 1] x1 + [0] [0 0] [0] p(length) = [1 0] x1 + [0] [0 1] [0] p(mark) = [1 4] x1 + [0] [0 1] [2] p(nil) = [7] [0] p(s) = [1 0] x1 + [0] [0 1] [0] p(tt) = [0] [2] p(zeros) = [0] [0] Following rules are strictly oriented: a__isNatList(cons(V1,V2)) = [0 1] V1 + [0 1] V2 + [2] [0 0] [0 0] [2] > [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [2] = a__and(a__isNat(V1),isNatList(V2)) Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1 0] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] >= [1 0] X1 + [1 0] X2 + [0] [0 1] [0 1] [0] = U11(X1,X2) a__U11(tt(),L) = [1 4] L + [0] [0 1] [2] >= [1 4] L + [0] [0 1] [2] = s(a__length(mark(L))) a__and(X1,X2) = [1 0] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] >= [1 0] X1 + [1 0] X2 + [0] [0 1] [0 1] [0] = and(X1,X2) a__and(tt(),X) = [1 4] X + [0] [0 1] [2] >= [1 4] X + [0] [0 1] [2] = mark(X) a__isNat(X) = [0 1] X + [0] [0 0] [2] >= [0 1] X + [0] [0 0] [0] = isNat(X) a__isNat(0()) = [0] [2] >= [0] [2] = tt() a__isNat(length(V1)) = [0 1] V1 + [0] [0 0] [2] >= [0 1] V1 + [0] [0 0] [2] = a__isNatList(V1) a__isNat(s(V1)) = [0 1] V1 + [0] [0 0] [2] >= [0 1] V1 + [0] [0 0] [2] = a__isNat(V1) a__isNatIList(V) = [0 1] V + [0] [0 0] [2] >= [0 1] V + [0] [0 0] [2] = a__isNatList(V) a__isNatIList(X) = [0 1] X + [0] [0 0] [2] >= [0 1] X + [0] [0 0] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 1] V1 + [0 1] V2 + [2] [0 0] [0 0] [2] >= [0 1] V1 + [0 1] V2 + [0] [0 0] [0 0] [2] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [0] [2] >= [0] [2] = tt() a__isNatList(X) = [0 1] X + [0] [0 0] [2] >= [0 1] X + [0] [0 0] [0] = isNatList(X) a__isNatList(nil()) = [0] [2] >= [0] [2] = tt() a__length(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = length(X) a__length(cons(N,L)) = [1 5] L + [1 1] N + [0] [0 1] [0 1] [2] >= [1 5] L + [0 1] N + [0] [0 1] [0 0] [2] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [7] [0] >= [0] [0] = 0() a__zeros() = [0] [2] >= [0] [2] = cons(0(),zeros()) a__zeros() = [0] [2] >= [0] [0] = zeros() mark(0()) = [0] [2] >= [0] [0] = 0() mark(U11(X1,X2)) = [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [2] >= [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [2] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [2] >= [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [2] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1 5] X1 + [1 9] X2 + [8] [0 1] [0 1] [4] >= [1 5] X1 + [1 5] X2 + [2] [0 1] [0 1] [4] = cons(mark(X1),X2) mark(isNat(X)) = [0 1] X + [0] [0 0] [2] >= [0 1] X + [0] [0 0] [2] = a__isNat(X) mark(isNatIList(X)) = [0 1] X + [0] [0 0] [2] >= [0 1] X + [0] [0 0] [2] = a__isNatIList(X) mark(isNatList(X)) = [0 1] X + [0] [0 0] [2] >= [0 1] X + [0] [0 0] [2] = a__isNatList(X) mark(length(X)) = [1 4] X + [0] [0 1] [2] >= [1 4] X + [0] [0 1] [2] = a__length(mark(X)) mark(nil()) = [7] [2] >= [7] [0] = nil() mark(s(X)) = [1 4] X + [0] [0 1] [2] >= [1 4] X + [0] [0 1] [2] = s(mark(X)) mark(tt()) = [8] [4] >= [0] [2] = tt() mark(zeros()) = [0] [2] >= [0] [2] = a__zeros() * Step 14: NaturalMI WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(U11) = [1 1 0] [1 1 0] [0] [0 0 0] x1 + [0 1 1] x2 + [1] [0 0 1] [0 0 1] [1] p(a__U11) = [1 1 0] [1 1 1] [0] [0 0 0] x1 + [0 1 1] x2 + [1] [0 0 1] [0 0 1] [1] p(a__and) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 1] [0] p(a__isNat) = [0 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] p(a__isNatIList) = [0 1 1] [1] [0 1 1] x1 + [1] [0 0 0] [0] p(a__isNatList) = [0 1 1] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(a__length) = [1 1 0] [0] [0 1 1] x1 + [0] [0 0 1] [1] p(a__zeros) = [0] [1] [0] p(and) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [1 1 1] [0] [0 1 0] x1 + [0 1 1] x2 + [1] [0 0 1] [0 0 1] [0] p(isNat) = [0 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(isNatIList) = [0 1 1] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(isNatList) = [0 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(length) = [1 1 0] [0] [0 1 1] x1 + [0] [0 0 1] [1] p(mark) = [1 0 1] [1] [0 1 0] x1 + [1] [0 0 1] [0] p(nil) = [1] [0] [1] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(tt) = [1] [1] [0] p(zeros) = [0] [0] [0] Following rules are strictly oriented: mark(zeros()) = [1] [1] [0] > [0] [1] [0] = a__zeros() Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1 1 0] [1 1 1] [0] [0 0 0] X1 + [0 1 1] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 0] [1 1 0] [0] [0 0 0] X1 + [0 1 1] X2 + [1] [0 0 1] [0 0 1] [1] = U11(X1,X2) a__U11(tt(),L) = [1 1 1] [2] [0 1 1] L + [1] [0 0 1] [1] >= [1 1 1] [2] [0 1 1] L + [1] [0 0 1] [1] = s(a__length(mark(L))) a__and(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = and(X1,X2) a__and(tt(),X) = [1 0 1] [1] [0 1 0] X + [1] [0 0 1] [0] >= [1 0 1] [1] [0 1 0] X + [1] [0 0 1] [0] = mark(X) a__isNat(X) = [0 1 0] [1] [0 0 0] X + [1] [0 0 0] [0] >= [0 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = isNat(X) a__isNat(0()) = [1] [1] [0] >= [1] [1] [0] = tt() a__isNat(length(V1)) = [0 1 1] [1] [0 0 0] V1 + [1] [0 0 0] [0] >= [0 1 1] [0] [0 0 0] V1 + [1] [0 0 0] [0] = a__isNatList(V1) a__isNat(s(V1)) = [0 1 0] [1] [0 0 0] V1 + [1] [0 0 0] [0] >= [0 1 0] [1] [0 0 0] V1 + [1] [0 0 0] [0] = a__isNat(V1) a__isNatIList(V) = [0 1 1] [1] [0 1 1] V + [1] [0 0 0] [0] >= [0 1 1] [0] [0 0 0] V + [1] [0 0 0] [0] = a__isNatList(V) a__isNatIList(X) = [0 1 1] [1] [0 1 1] X + [1] [0 0 0] [0] >= [0 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 1 1] [0 1 2] [2] [0 1 1] V1 + [0 1 2] V2 + [2] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 1 1] [1] [0 0 0] V1 + [0 1 1] V2 + [2] [0 0 0] [0 0 0] [0] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [1] [1] [0] >= [1] [1] [0] = tt() a__isNatList(X) = [0 1 1] [0] [0 0 0] X + [1] [0 0 0] [0] >= [0 1 1] [0] [0 0 0] X + [0] [0 0 0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 1 1] [0 1 2] [1] [0 0 0] V1 + [0 0 0] V2 + [1] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 1 1] [1] [0 0 0] V1 + [0 0 0] V2 + [1] [0 0 0] [0 0 0] [0] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [1] [1] [0] >= [1] [1] [0] = tt() a__length(X) = [1 1 0] [0] [0 1 1] X + [0] [0 0 1] [1] >= [1 1 0] [0] [0 1 1] X + [0] [0 0 1] [1] = length(X) a__length(cons(N,L)) = [1 2 2] [1 1 0] [1] [0 1 2] L + [0 1 1] N + [1] [0 0 1] [0 0 1] [1] >= [1 2 2] [0 1 0] [1] [0 1 1] L + [0 0 0] N + [1] [0 0 1] [0 0 0] [1] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [1] [1] [2] >= [0] [0] [0] = 0() a__zeros() = [0] [1] [0] >= [0] [1] [0] = cons(0(),zeros()) a__zeros() = [0] [1] [0] >= [0] [0] [0] = zeros() mark(0()) = [1] [1] [0] >= [0] [0] [0] = 0() mark(U11(X1,X2)) = [1 1 1] [1 1 1] [2] [0 0 0] X1 + [0 1 1] X2 + [2] [0 0 1] [0 0 1] [1] >= [1 1 1] [1 1 1] [2] [0 0 0] X1 + [0 1 1] X2 + [1] [0 0 1] [0 0 1] [1] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [1 0 1] [1 0 1] [1] [0 0 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 1] [1] [0 0 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1 0 1] [1 1 2] [1] [0 1 0] X1 + [0 1 1] X2 + [2] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 1 1] [1] [0 1 0] X1 + [0 1 1] X2 + [2] [0 0 1] [0 0 1] [0] = cons(mark(X1),X2) mark(isNat(X)) = [0 1 0] [1] [0 0 0] X + [1] [0 0 0] [0] >= [0 1 0] [1] [0 0 0] X + [1] [0 0 0] [0] = a__isNat(X) mark(isNatIList(X)) = [0 1 1] [1] [0 1 1] X + [2] [0 0 0] [0] >= [0 1 1] [1] [0 1 1] X + [1] [0 0 0] [0] = a__isNatIList(X) mark(isNatList(X)) = [0 1 1] [1] [0 0 0] X + [1] [0 0 0] [0] >= [0 1 1] [0] [0 0 0] X + [1] [0 0 0] [0] = a__isNatList(X) mark(length(X)) = [1 1 1] [2] [0 1 1] X + [1] [0 0 1] [1] >= [1 1 1] [2] [0 1 1] X + [1] [0 0 1] [1] = a__length(mark(X)) mark(nil()) = [3] [1] [1] >= [1] [0] [1] = nil() mark(s(X)) = [1 0 1] [1] [0 1 0] X + [1] [0 0 1] [0] >= [1 0 1] [1] [0 1 0] X + [1] [0 0 1] [0] = s(mark(X)) mark(tt()) = [2] [2] [0] >= [1] [1] [0] = tt() * Step 15: NaturalMI WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 4, 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(U11) = [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 1 1 0] x2 + [1] [0 0 1 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] p(a__U11) = [1 0 0 0] [0 1 0 0] [0] [0 1 0 0] x1 + [0 1 1 0] x2 + [1] [0 0 1 0] [0 0 0 1] [1] [0 0 0 1] [0 0 0 1] [1] p(a__and) = [1 0 0 0] [0 1 0 0] [0] [0 1 0 0] x1 + [0 1 1 0] x2 + [0] [0 0 1 0] [0 0 0 1] [0] [0 0 0 1] [0 0 0 1] [0] p(a__isNat) = [0 0 0 1] [0] [0 0 0 1] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] p(a__isNatIList) = [0 1 0 1] [0] [0 1 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] p(a__isNatList) = [0 0 0 1] [0] [0 0 0 1] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [1] p(a__length) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 1] [1] p(a__zeros) = [0] [1] [1] [1] p(and) = [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 1 1 0] x2 + [0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [0] p(cons) = [1 0 0 1] [0 1 0 1] [0] [0 1 0 1] x1 + [0 1 1 1] x2 + [1] [0 0 1 0] [0 0 0 1] [1] [0 0 0 1] [0 0 0 1] [1] p(isNat) = [0 0 0 1] [0] [0 0 0 1] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(isNatIList) = [0 1 0 0] [0] [0 1 0 1] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [0] p(isNatList) = [0 0 0 0] [0] [0 0 0 1] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(length) = [0 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [1] p(mark) = [0 1 0 0] [0] [0 1 1 0] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] p(nil) = [0] [0] [0] [1] p(s) = [1 0 0 0] [0] [0 1 0 0] x1 + [1] [0 0 1 0] [0] [0 0 0 1] [0] p(tt) = [0] [1] [1] [1] p(zeros) = [0] [0] [0] [0] Following rules are strictly oriented: mark(s(X)) = [0 1 0 0] [1] [0 1 1 0] X + [2] [0 0 0 1] [1] [0 0 0 1] [1] > [0 1 0 0] [0] [0 1 1 0] X + [2] [0 0 0 1] [1] [0 0 0 1] [1] = s(mark(X)) Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1 0 0 0] [0 1 0 0] [0] [0 1 0 0] X1 + [0 1 1 0] X2 + [1] [0 0 1 0] [0 0 0 1] [1] [0 0 0 1] [0 0 0 1] [1] >= [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 1 1 0] X2 + [1] [0 0 1 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [1] = U11(X1,X2) a__U11(tt(),L) = [0 1 0 0] [0] [0 1 1 0] L + [2] [0 0 0 1] [2] [0 0 0 1] [2] >= [0 1 0 0] [0] [0 1 1 0] L + [2] [0 0 0 1] [2] [0 0 0 1] [2] = s(a__length(mark(L))) a__and(X1,X2) = [1 0 0 0] [0 1 0 0] [0] [0 1 0 0] X1 + [0 1 1 0] X2 + [0] [0 0 1 0] [0 0 0 1] [0] [0 0 0 1] [0 0 0 1] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 1 1 0] X2 + [0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 1] [0 0 0 1] [0] = and(X1,X2) a__and(tt(),X) = [0 1 0 0] [0] [0 1 1 0] X + [1] [0 0 0 1] [1] [0 0 0 1] [1] >= [0 1 0 0] [0] [0 1 1 0] X + [1] [0 0 0 1] [1] [0 0 0 1] [1] = mark(X) a__isNat(X) = [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = isNat(X) a__isNat(0()) = [0] [1] [1] [1] >= [0] [1] [1] [1] = tt() a__isNat(length(V1)) = [0 0 0 1] [1] [0 0 0 1] V1 + [2] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] V1 + [0] [0 0 0 0] [1] [0 0 0 0] [1] = a__isNatList(V1) a__isNat(s(V1)) = [0 0 0 1] [0] [0 0 0 1] V1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] V1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] = a__isNat(V1) a__isNatIList(V) = [0 1 0 1] [0] [0 1 0 1] V + [1] [0 0 0 1] [1] [0 0 0 1] [1] >= [0 0 0 1] [0] [0 0 0 1] V + [0] [0 0 0 0] [1] [0 0 0 0] [1] = a__isNatList(V) a__isNatIList(X) = [0 1 0 1] [0] [0 1 0 1] X + [1] [0 0 0 1] [1] [0 0 0 1] [1] >= [0 1 0 0] [0] [0 1 0 1] X + [0] [0 0 0 1] [0] [0 0 0 1] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 1 0 2] [0 1 1 2] [2] [0 1 0 2] V1 + [0 1 1 2] V2 + [3] [0 0 0 1] [0 0 0 1] [2] [0 0 0 1] [0 0 0 1] [2] >= [0 0 0 1] [0 1 0 1] [0] [0 0 0 1] V1 + [0 1 0 2] V2 + [1] [0 0 0 0] [0 0 0 1] [1] [0 0 0 0] [0 0 0 1] [1] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [0] [1] [1] [1] >= [0] [1] [1] [1] = tt() a__isNatList(X) = [0 0 0 1] [0] [0 0 0 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 0 0] [0] [0 0 0 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 0 0 1] [0 0 0 1] [1] [0 0 0 1] V1 + [0 0 0 1] V2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [1] >= [0 0 0 1] [0 0 0 1] [0] [0 0 0 1] V1 + [0 0 0 1] V2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [1] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [1] [1] [1] [1] >= [0] [1] [1] [1] = tt() a__length(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [1] [0 0 0 1] [1] >= [0 0 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [1] = length(X) a__length(cons(N,L)) = [0 1 0 1] [1 0 0 1] [0] [0 1 1 1] L + [0 1 0 1] N + [1] [0 0 0 1] [0 0 1 0] [2] [0 0 0 1] [0 0 0 1] [2] >= [0 1 0 1] [0 0 0 1] [0] [0 1 1 1] L + [0 0 0 1] N + [1] [0 0 0 1] [0 0 0 0] [2] [0 0 0 1] [0 0 0 0] [2] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [0] [0] [1] [2] >= [0] [0] [0] [0] = 0() a__zeros() = [0] [1] [1] [1] >= [0] [1] [1] [1] = cons(0(),zeros()) a__zeros() = [0] [1] [1] [1] >= [0] [0] [0] [0] = zeros() mark(0()) = [0] [1] [1] [1] >= [0] [0] [0] [0] = 0() mark(U11(X1,X2)) = [0 1 0 0] [0 1 1 0] [1] [0 1 1 0] X1 + [0 1 1 0] X2 + [2] [0 0 0 1] [0 0 0 1] [2] [0 0 0 1] [0 0 0 1] [2] >= [0 1 0 0] [0 1 0 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [2] [0 0 0 1] [0 0 0 1] [2] [0 0 0 1] [0 0 0 1] [2] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [0 1 0 0] [0 1 1 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [1] [0 0 0 1] [0 0 0 1] [1] [0 0 0 1] [0 0 0 1] [1] >= [0 1 0 0] [0 1 0 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [1] [0 0 0 1] [0 0 0 1] [1] [0 0 0 1] [0 0 0 1] [1] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0 1 0 1] [0 1 1 1] [1] [0 1 1 1] X1 + [0 1 1 2] X2 + [3] [0 0 0 1] [0 0 0 1] [2] [0 0 0 1] [0 0 0 1] [2] >= [0 1 0 1] [0 1 0 1] [1] [0 1 1 1] X1 + [0 1 1 1] X2 + [3] [0 0 0 1] [0 0 0 1] [2] [0 0 0 1] [0 0 0 1] [2] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 0 0] [1] [0 0 0 0] [1] = a__isNat(X) mark(isNatIList(X)) = [0 1 0 1] [0] [0 1 0 2] X + [1] [0 0 0 1] [1] [0 0 0 1] [1] >= [0 1 0 1] [0] [0 1 0 1] X + [1] [0 0 0 1] [1] [0 0 0 1] [1] = a__isNatIList(X) mark(isNatList(X)) = [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 0 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] = a__isNatList(X) mark(length(X)) = [0 1 0 0] [0] [0 1 1 0] X + [1] [0 0 0 1] [2] [0 0 0 1] [2] >= [0 1 0 0] [0] [0 1 1 0] X + [1] [0 0 0 1] [2] [0 0 0 1] [2] = a__length(mark(X)) mark(nil()) = [0] [1] [2] [2] >= [0] [0] [0] [1] = nil() mark(tt()) = [1] [3] [2] [2] >= [0] [1] [1] [1] = tt() mark(zeros()) = [0] [1] [1] [1] >= [0] [1] [1] [1] = a__zeros() * Step 16: NaturalMI WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) mark(and(X1,X2)) -> a__and(mark(X1),X2) - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 4, 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(U11) = [0 0 0 0] [0 0 0 0] [0] [0 1 1 0] x1 + [0 1 1 0] x2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [1] p(a__U11) = [1 0 0 0] [0 1 0 0] [0] [0 1 1 0] x1 + [0 1 1 1] x2 + [0] [0 0 0 0] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [1] p(a__and) = [1 0 0 0] [0 1 0 0] [0] [0 1 0 0] x1 + [0 1 0 1] x2 + [1] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] p(a__isNat) = [0 0 0 1] [0] [0 0 0 1] x1 + [1] [0 0 1 0] [1] [0 0 0 0] [1] p(a__isNatIList) = [0 0 1 1] [0] [0 0 1 1] x1 + [1] [0 0 1 0] [1] [0 0 0 0] [1] p(a__isNatList) = [0 0 0 1] [0] [0 0 0 1] x1 + [1] [0 0 1 0] [0] [0 0 0 0] [1] p(a__length) = [1 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [1] p(a__zeros) = [0] [1] [1] [1] p(and) = [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [1] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] p(cons) = [1 0 0 1] [0 1 0 1] [0] [0 1 0 1] x1 + [0 1 1 1] x2 + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 1] [0 0 0 1] [1] p(isNat) = [0 0 0 0] [0] [0 0 0 1] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(isNatIList) = [0 0 1 0] [0] [0 0 1 1] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(isNatList) = [0 0 0 0] [0] [0 0 0 1] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(length) = [0 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [1] p(mark) = [0 1 0 0] [0] [0 1 0 1] x1 + [1] [0 0 1 0] [1] [0 0 0 1] [1] p(nil) = [0] [0] [1] [0] p(s) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] p(tt) = [0] [1] [1] [1] p(zeros) = [0] [0] [0] [0] Following rules are strictly oriented: mark(and(X1,X2)) = [0 1 0 0] [0 1 0 0] [1] [0 1 0 1] X1 + [0 1 0 1] X2 + [2] [0 0 1 0] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [1] > [0 1 0 0] [0 1 0 0] [0] [0 1 0 1] X1 + [0 1 0 1] X2 + [2] [0 0 1 0] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [1] = a__and(mark(X1),X2) Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1 0 0 0] [0 1 0 0] [0] [0 1 1 0] X1 + [0 1 1 1] X2 + [0] [0 0 0 0] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [1] >= [0 0 0 0] [0 0 0 0] [0] [0 1 1 0] X1 + [0 1 1 0] X2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [1] = U11(X1,X2) a__U11(tt(),L) = [0 1 0 0] [0] [0 1 1 1] L + [2] [0 0 1 0] [1] [0 0 0 1] [2] >= [0 1 0 0] [0] [0 1 1 1] L + [2] [0 0 1 0] [1] [0 0 0 1] [2] = s(a__length(mark(L))) a__and(X1,X2) = [1 0 0 0] [0 1 0 0] [0] [0 1 0 0] X1 + [0 1 0 1] X2 + [1] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [1] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] = and(X1,X2) a__and(tt(),X) = [0 1 0 0] [0] [0 1 0 1] X + [2] [0 0 1 0] [1] [0 0 0 1] [1] >= [0 1 0 0] [0] [0 1 0 1] X + [1] [0 0 1 0] [1] [0 0 0 1] [1] = mark(X) a__isNat(X) = [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 1 0] [1] [0 0 0 0] [1] >= [0 0 0 0] [0] [0 0 0 1] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] = isNat(X) a__isNat(0()) = [0] [1] [1] [1] >= [0] [1] [1] [1] = tt() a__isNat(length(V1)) = [0 0 0 1] [1] [0 0 0 1] V1 + [2] [0 0 1 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] V1 + [1] [0 0 1 0] [0] [0 0 0 0] [1] = a__isNatList(V1) a__isNat(s(V1)) = [0 0 0 1] [0] [0 0 0 1] V1 + [1] [0 0 1 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] V1 + [1] [0 0 1 0] [1] [0 0 0 0] [1] = a__isNat(V1) a__isNatIList(V) = [0 0 1 1] [0] [0 0 1 1] V + [1] [0 0 1 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] V + [1] [0 0 1 0] [0] [0 0 0 0] [1] = a__isNatList(V) a__isNatIList(X) = [0 0 1 1] [0] [0 0 1 1] X + [1] [0 0 1 0] [1] [0 0 0 0] [1] >= [0 0 1 0] [0] [0 0 1 1] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [0 0 1 1] [0 0 1 2] [2] [0 0 1 1] V1 + [0 0 1 2] V2 + [3] [0 0 1 0] [0 0 1 1] [2] [0 0 0 0] [0 0 0 0] [1] >= [0 0 0 1] [0 0 1 1] [0] [0 0 0 1] V1 + [0 0 1 1] V2 + [2] [0 0 1 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [1] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [0] [1] [1] [1] >= [0] [1] [1] [1] = tt() a__isNatList(X) = [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 1 0] [0] [0 0 0 0] [1] >= [0 0 0 0] [0] [0 0 0 1] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 0 0 1] [0 0 0 1] [1] [0 0 0 1] V1 + [0 0 0 1] V2 + [2] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [1] >= [0 0 0 1] [0 0 0 1] [0] [0 0 0 1] V1 + [0 0 0 1] V2 + [2] [0 0 1 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [1] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [0] [1] [1] [1] >= [0] [1] [1] [1] = tt() a__length(X) = [1 0 0 0] [0] [0 1 1 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [1] >= [0 0 0 0] [0] [0 1 1 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [1] = length(X) a__length(cons(N,L)) = [0 1 0 1] [1 0 0 1] [0] [0 1 2 2] L + [0 1 1 1] N + [2] [0 0 1 1] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [2] >= [0 1 0 1] [0 0 0 1] [0] [0 1 2 2] L + [0 0 1 1] N + [2] [0 0 1 0] [0 0 0 0] [1] [0 0 0 1] [0 0 0 0] [2] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [0] [1] [1] [1] >= [0] [0] [0] [0] = 0() a__zeros() = [0] [1] [1] [1] >= [0] [1] [1] [1] = cons(0(),zeros()) a__zeros() = [0] [1] [1] [1] >= [0] [0] [0] [0] = zeros() mark(0()) = [0] [1] [1] [1] >= [0] [0] [0] [0] = 0() mark(U11(X1,X2)) = [0 1 1 0] [0 1 1 0] [0] [0 1 1 1] X1 + [0 1 1 1] X2 + [2] [0 0 0 0] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [2] >= [0 1 0 0] [0 1 0 0] [0] [0 1 1 1] X1 + [0 1 1 1] X2 + [2] [0 0 0 0] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [2] = a__U11(mark(X1),X2) mark(cons(X1,X2)) = [0 1 0 1] [0 1 1 1] [1] [0 1 0 2] X1 + [0 1 1 2] X2 + [3] [0 0 1 0] [0 0 1 1] [2] [0 0 0 1] [0 0 0 1] [2] >= [0 1 0 1] [0 1 0 1] [1] [0 1 0 2] X1 + [0 1 1 1] X2 + [3] [0 0 1 0] [0 0 1 1] [2] [0 0 0 1] [0 0 0 1] [2] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 1 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 1 0] [1] [0 0 0 0] [1] = a__isNat(X) mark(isNatIList(X)) = [0 0 1 1] [0] [0 0 1 1] X + [1] [0 0 1 0] [1] [0 0 0 0] [1] >= [0 0 1 1] [0] [0 0 1 1] X + [1] [0 0 1 0] [1] [0 0 0 0] [1] = a__isNatIList(X) mark(isNatList(X)) = [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 1 0] [1] [0 0 0 0] [1] >= [0 0 0 1] [0] [0 0 0 1] X + [1] [0 0 1 0] [0] [0 0 0 0] [1] = a__isNatList(X) mark(length(X)) = [0 1 1 0] [0] [0 1 1 1] X + [2] [0 0 1 0] [1] [0 0 0 1] [2] >= [0 1 0 0] [0] [0 1 1 1] X + [2] [0 0 1 0] [1] [0 0 0 1] [2] = a__length(mark(X)) mark(nil()) = [0] [1] [2] [1] >= [0] [0] [1] [0] = nil() mark(s(X)) = [0 1 0 0] [0] [0 1 0 1] X + [1] [0 0 1 0] [1] [0 0 0 1] [1] >= [0 1 0 0] [0] [0 1 0 1] X + [1] [0 0 1 0] [1] [0 0 0 1] [1] = s(mark(X)) mark(tt()) = [1] [3] [2] [2] >= [0] [1] [1] [1] = tt() mark(zeros()) = [0] [1] [1] [1] >= [0] [1] [1] [1] = a__zeros() * Step 17: NaturalMI WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: a__isNat(s(V1)) -> a__isNat(V1) - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 4, 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__U11) = {1}, uargs(a__and) = {1}, uargs(a__length) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList,a__length,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(U11) = [1 0 0 0] [1 1 1 1] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [1] [0 0 1 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 1] [0] p(a__U11) = [1 0 0 0] [1 1 1 1] [1] [0 1 0 0] x1 + [0 1 0 0] x2 + [1] [0 0 1 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 1] [0] p(a__and) = [1 0 0 0] [1 1 0 0] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [0] [0 0 0 1] [0 0 1 0] [0] [0 0 0 0] [0 0 0 1] [0] p(a__isNat) = [0 0 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 1] [1] [0 0 0 1] [1] p(a__isNatIList) = [1 0 1 1] [1] [0 0 0 1] x1 + [0] [0 0 0 1] [1] [0 0 0 1] [1] p(a__isNatList) = [0 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 1] [1] [0 0 0 1] [0] p(a__length) = [1 0 1 1] [1] [0 1 0 0] x1 + [1] [0 0 1 0] [0] [0 0 0 1] [0] p(a__zeros) = [0] [0] [1] [0] p(and) = [1 0 0 0] [1 0 0 0] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [0] [0 0 0 1] [0 0 1 0] [0] [0 0 0 0] [0 0 0 1] [0] p(cons) = [1 0 0 0] [1 1 1 0] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [0] [0 0 1 1] [0 0 1 1] [1] [0 0 0 1] [0 0 0 1] [0] p(isNat) = [0 0 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [1] p(isNatIList) = [1 0 1 1] [1] [0 0 0 1] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [1] p(isNatList) = [0 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [0] p(length) = [1 0 1 1] [1] [0 1 0 0] x1 + [1] [0 0 1 0] [0] [0 0 0 1] [0] p(mark) = [1 1 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 1] [0] p(nil) = [0] [0] [1] [1] p(s) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 1] [0] p(tt) = [1] [0] [1] [1] p(zeros) = [0] [0] [0] [0] Following rules are strictly oriented: a__isNat(s(V1)) = [0 0 1 0] [2] [0 0 0 0] V1 + [0] [0 0 0 1] [1] [0 0 0 1] [1] > [0 0 1 0] [1] [0 0 0 0] V1 + [0] [0 0 0 1] [1] [0 0 0 1] [1] = a__isNat(V1) Following rules are (at-least) weakly oriented: a__U11(X1,X2) = [1 0 0 0] [1 1 1 1] [1] [0 1 0 0] X1 + [0 1 0 0] X2 + [1] [0 0 1 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 1] [0] >= [1 0 0 0] [1 1 1 1] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [1] [0 0 1 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 1] [0] = U11(X1,X2) a__U11(tt(),L) = [1 1 1 1] [2] [0 1 0 0] L + [1] [0 0 1 0] [2] [0 0 0 1] [0] >= [1 1 1 1] [2] [0 1 0 0] L + [1] [0 0 1 0] [2] [0 0 0 1] [0] = s(a__length(mark(L))) a__and(X1,X2) = [1 0 0 0] [1 1 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 0 1] [0 0 1 0] [0] [0 0 0 0] [0 0 0 1] [0] >= [1 0 0 0] [1 0 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 0 1] [0 0 1 0] [0] [0 0 0 0] [0 0 0 1] [0] = and(X1,X2) a__and(tt(),X) = [1 1 0 0] [1] [0 1 0 0] X + [0] [0 0 1 0] [1] [0 0 0 1] [0] >= [1 1 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [1] [0 0 0 1] [0] = mark(X) a__isNat(X) = [0 0 1 0] [1] [0 0 0 0] X + [0] [0 0 0 1] [1] [0 0 0 1] [1] >= [0 0 1 0] [1] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 1] [1] = isNat(X) a__isNat(0()) = [1] [0] [1] [1] >= [1] [0] [1] [1] = tt() a__isNat(length(V1)) = [0 0 1 0] [1] [0 0 0 0] V1 + [0] [0 0 0 1] [1] [0 0 0 1] [1] >= [0 0 1 0] [0] [0 0 0 0] V1 + [0] [0 0 0 1] [1] [0 0 0 1] [0] = a__isNatList(V1) a__isNatIList(V) = [1 0 1 1] [1] [0 0 0 1] V + [0] [0 0 0 1] [1] [0 0 0 1] [1] >= [0 0 1 0] [0] [0 0 0 0] V + [0] [0 0 0 1] [1] [0 0 0 1] [0] = a__isNatList(V) a__isNatIList(X) = [1 0 1 1] [1] [0 0 0 1] X + [0] [0 0 0 1] [1] [0 0 0 1] [1] >= [1 0 1 1] [1] [0 0 0 1] X + [0] [0 0 0 1] [0] [0 0 0 1] [1] = isNatIList(X) a__isNatIList(cons(V1,V2)) = [1 0 1 2] [1 1 2 2] [2] [0 0 0 1] V1 + [0 0 0 1] V2 + [0] [0 0 0 1] [0 0 0 1] [1] [0 0 0 1] [0 0 0 1] [1] >= [0 0 1 0] [1 0 1 2] [2] [0 0 0 0] V1 + [0 0 0 1] V2 + [0] [0 0 0 1] [0 0 0 1] [1] [0 0 0 0] [0 0 0 1] [1] = a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) = [1] [0] [1] [1] >= [1] [0] [1] [1] = tt() a__isNatList(X) = [0 0 1 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [1] [0 0 0 1] [0] >= [0 0 1 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [0] [0 0 0 1] [0] = isNatList(X) a__isNatList(cons(V1,V2)) = [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] V1 + [0 0 0 0] V2 + [0] [0 0 0 1] [0 0 0 1] [1] [0 0 0 1] [0 0 0 1] [0] >= [0 0 1 0] [0 0 1 0] [1] [0 0 0 0] V1 + [0 0 0 0] V2 + [0] [0 0 0 1] [0 0 0 1] [1] [0 0 0 0] [0 0 0 1] [0] = a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) = [1] [0] [2] [1] >= [1] [0] [1] [1] = tt() a__length(X) = [1 0 1 1] [1] [0 1 0 0] X + [1] [0 0 1 0] [0] [0 0 0 1] [0] >= [1 0 1 1] [1] [0 1 0 0] X + [1] [0 0 1 0] [0] [0 0 0 1] [0] = length(X) a__length(cons(N,L)) = [1 1 2 2] [1 0 1 2] [2] [0 1 0 0] L + [0 1 0 0] N + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 1] [0 0 0 1] [0] >= [1 1 2 1] [0 0 1 0] [2] [0 1 0 0] L + [0 0 0 0] N + [1] [0 0 1 1] [0 0 0 1] [1] [0 0 0 1] [0 0 0 0] [0] = a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) = [3] [1] [1] [1] >= [0] [0] [0] [0] = 0() a__zeros() = [0] [0] [1] [0] >= [0] [0] [1] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [1] [0] >= [0] [0] [0] [0] = zeros() mark(0()) = [0] [0] [1] [0] >= [0] [0] [0] [0] = 0() mark(U11(X1,X2)) = [1 1 0 0] [1 2 1 1] [1] [0 1 0 0] X1 + [0 1 0 0] X2 + [1] [0 0 1 0] [0 0 1 0] [2] [0 0 0 0] [0 0 0 1] [0] >= [1 1 0 0] [1 1 1 1] [1] [0 1 0 0] X1 + [0 1 0 0] X2 + [1] [0 0 1 0] [0 0 1 0] [2] [0 0 0 0] [0 0 0 1] [0] = a__U11(mark(X1),X2) mark(and(X1,X2)) = [1 1 0 0] [1 1 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 0 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 1] [0] >= [1 1 0 0] [1 1 0 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 0 1] [0 0 1 0] [0] [0 0 0 0] [0 0 0 1] [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [1 1 0 0] [1 2 1 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 1 1] [0 0 1 1] [2] [0 0 0 1] [0 0 0 1] [0] >= [1 1 0 0] [1 1 1 0] [0] [0 1 0 0] X1 + [0 1 0 0] X2 + [0] [0 0 1 1] [0 0 1 1] [2] [0 0 0 1] [0 0 0 1] [0] = cons(mark(X1),X2) mark(isNat(X)) = [0 0 1 0] [1] [0 0 0 0] X + [0] [0 0 0 1] [1] [0 0 0 1] [1] >= [0 0 1 0] [1] [0 0 0 0] X + [0] [0 0 0 1] [1] [0 0 0 1] [1] = a__isNat(X) mark(isNatIList(X)) = [1 0 1 2] [1] [0 0 0 1] X + [0] [0 0 0 1] [1] [0 0 0 1] [1] >= [1 0 1 1] [1] [0 0 0 1] X + [0] [0 0 0 1] [1] [0 0 0 1] [1] = a__isNatIList(X) mark(isNatList(X)) = [0 0 1 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [1] [0 0 0 1] [0] >= [0 0 1 0] [0] [0 0 0 0] X + [0] [0 0 0 1] [1] [0 0 0 1] [0] = a__isNatList(X) mark(length(X)) = [1 1 1 1] [2] [0 1 0 0] X + [1] [0 0 1 0] [1] [0 0 0 1] [0] >= [1 1 1 1] [2] [0 1 0 0] X + [1] [0 0 1 0] [1] [0 0 0 1] [0] = a__length(mark(X)) mark(nil()) = [0] [0] [2] [1] >= [0] [0] [1] [1] = nil() mark(s(X)) = [1 1 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [2] [0 0 0 1] [0] >= [1 1 0 0] [0] [0 1 0 0] X + [0] [0 0 1 0] [2] [0 0 0 1] [0] = s(mark(X)) mark(tt()) = [1] [0] [2] [1] >= [1] [0] [1] [1] = tt() mark(zeros()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = a__zeros() * Step 18: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(X) -> isNatIList(X) a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatIList(zeros()) -> tt() a__isNatList(X) -> isNatList(X) a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__isNatList(nil()) -> tt() a__length(X) -> length(X) a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) a__length(nil()) -> 0() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(zeros()) -> a__zeros() - Signature: {a__U11/2,a__and/2,a__isNat/1,a__isNatIList/1,a__isNatList/1,a__length/1,a__zeros/0,mark/1} / {0/0,U11/2 ,and/2,cons/2,isNat/1,isNatIList/1,isNatList/1,length/1,nil/0,s/1,tt/0,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__and,a__isNat,a__isNatIList,a__isNatList ,a__length,a__zeros,mark} and constructors {0,U11,and,cons,isNat,isNatIList,isNatList,length,nil,s,tt,zeros} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^4))