WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__eq(s(X),s(Y)) -> a__eq(X,Y) a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(inf(X)) -> a__inf(mark(X)) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq,a__inf,a__length,a__take,mark} and constructors {0 ,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__inf) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2} Following symbols are considered usable: {a__eq,a__inf,a__length,a__take,mark} TcT has computed the following interpretation: p(0) = [1] p(a__eq) = [8] p(a__inf) = [1] x1 + [8] p(a__length) = [1] x1 + [14] p(a__take) = [1] x1 + [1] x2 + [8] p(cons) = [1] x1 + [2] p(eq) = [1] p(false) = [2] p(inf) = [1] x1 + [1] p(length) = [1] x1 + [2] p(mark) = [8] x1 + [8] p(nil) = [0] p(s) = [2] p(take) = [1] x1 + [1] x2 + [2] p(true) = [1] Following rules are strictly oriented: a__eq(X,Y) = [8] > [2] = false() a__eq(X1,X2) = [8] > [1] = eq(X1,X2) a__eq(0(),0()) = [8] > [1] = true() a__inf(X) = [1] X + [8] > [1] X + [2] = cons(X,inf(s(X))) a__inf(X) = [1] X + [8] > [1] X + [1] = inf(X) a__length(X) = [1] X + [14] > [1] X + [2] = length(X) a__length(cons(X,L)) = [1] X + [16] > [2] = s(length(L)) a__length(nil()) = [14] > [1] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [8] > [1] X1 + [1] X2 + [2] = take(X1,X2) a__take(0(),X) = [1] X + [9] > [0] = nil() a__take(s(X),cons(Y,L)) = [1] Y + [12] > [1] Y + [2] = cons(Y,take(X,L)) mark(0()) = [16] > [1] = 0() mark(cons(X1,X2)) = [8] X1 + [24] > [1] X1 + [2] = cons(X1,X2) mark(eq(X1,X2)) = [16] > [8] = a__eq(X1,X2) mark(false()) = [24] > [2] = false() mark(length(X)) = [8] X + [24] > [8] X + [22] = a__length(mark(X)) mark(nil()) = [8] > [0] = nil() mark(s(X)) = [24] > [2] = s(X) mark(true()) = [16] > [1] = true() Following rules are (at-least) weakly oriented: a__eq(s(X),s(Y)) = [8] >= [8] = a__eq(X,Y) mark(inf(X)) = [8] X + [16] >= [8] X + [16] = a__inf(mark(X)) mark(take(X1,X2)) = [8] X1 + [8] X2 + [24] >= [8] X1 + [8] X2 + [24] = a__take(mark(X1),mark(X2)) * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__eq(s(X),s(Y)) -> a__eq(X,Y) mark(inf(X)) -> a__inf(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) - Weak TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq,a__inf,a__length,a__take,mark} and constructors {0 ,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__inf) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__eq) = [0] p(a__inf) = [1] x1 + [6] p(a__length) = [1] x1 + [0] p(a__take) = [1] x1 + [1] x2 + [0] p(cons) = [0] p(eq) = [0] p(false) = [0] p(inf) = [1] x1 + [4] p(length) = [1] x1 + [0] p(mark) = [2] x1 + [0] p(nil) = [1] p(s) = [0] p(take) = [1] x1 + [1] x2 + [0] p(true) = [0] Following rules are strictly oriented: mark(inf(X)) = [2] X + [8] > [2] X + [6] = a__inf(mark(X)) Following rules are (at-least) weakly oriented: a__eq(X,Y) = [0] >= [0] = false() a__eq(X1,X2) = [0] >= [0] = eq(X1,X2) a__eq(0(),0()) = [0] >= [0] = true() a__eq(s(X),s(Y)) = [0] >= [0] = a__eq(X,Y) a__inf(X) = [1] X + [6] >= [0] = cons(X,inf(s(X))) a__inf(X) = [1] X + [6] >= [1] X + [4] = inf(X) a__length(X) = [1] X + [0] >= [1] X + [0] = length(X) a__length(cons(X,L)) = [0] >= [0] = s(length(L)) a__length(nil()) = [1] >= [1] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = take(X1,X2) a__take(0(),X) = [1] X + [1] >= [1] = nil() a__take(s(X),cons(Y,L)) = [0] >= [0] = cons(Y,take(X,L)) mark(0()) = [2] >= [1] = 0() mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(eq(X1,X2)) = [0] >= [0] = a__eq(X1,X2) mark(false()) = [0] >= [0] = false() mark(length(X)) = [2] X + [0] >= [2] X + [0] = a__length(mark(X)) mark(nil()) = [2] >= [1] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(take(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = a__take(mark(X1),mark(X2)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__eq(s(X),s(Y)) -> a__eq(X,Y) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) - Weak TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(inf(X)) -> a__inf(mark(X)) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq,a__inf,a__length,a__take,mark} and constructors {0 ,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__inf) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__eq) = [0] p(a__inf) = [1] x1 + [4] p(a__length) = [1] x1 + [1] p(a__take) = [1] x1 + [1] x2 + [2] p(cons) = [1] x1 + [4] p(eq) = [0] p(false) = [0] p(inf) = [1] x1 + [3] p(length) = [1] x1 + [1] p(mark) = [3] x1 + [0] p(nil) = [0] p(s) = [0] p(take) = [1] x1 + [1] x2 + [1] p(true) = [0] Following rules are strictly oriented: mark(take(X1,X2)) = [3] X1 + [3] X2 + [3] > [3] X1 + [3] X2 + [2] = a__take(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__eq(X,Y) = [0] >= [0] = false() a__eq(X1,X2) = [0] >= [0] = eq(X1,X2) a__eq(0(),0()) = [0] >= [0] = true() a__eq(s(X),s(Y)) = [0] >= [0] = a__eq(X,Y) a__inf(X) = [1] X + [4] >= [1] X + [4] = cons(X,inf(s(X))) a__inf(X) = [1] X + [4] >= [1] X + [3] = inf(X) a__length(X) = [1] X + [1] >= [1] X + [1] = length(X) a__length(cons(X,L)) = [1] X + [5] >= [0] = s(length(L)) a__length(nil()) = [1] >= [0] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [1] = take(X1,X2) a__take(0(),X) = [1] X + [2] >= [0] = nil() a__take(s(X),cons(Y,L)) = [1] Y + [6] >= [1] Y + [4] = cons(Y,take(X,L)) mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [3] X1 + [12] >= [1] X1 + [4] = cons(X1,X2) mark(eq(X1,X2)) = [0] >= [0] = a__eq(X1,X2) mark(false()) = [0] >= [0] = false() mark(inf(X)) = [3] X + [9] >= [3] X + [4] = a__inf(mark(X)) mark(length(X)) = [3] X + [3] >= [3] X + [1] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__eq(s(X),s(Y)) -> a__eq(X,Y) - Weak TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(inf(X)) -> a__inf(mark(X)) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq,a__inf,a__length,a__take,mark} and constructors {0 ,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__inf) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__eq) = [4] x1 + [1] x2 + [5] p(a__inf) = [1] x1 + [6] p(a__length) = [1] x1 + [2] p(a__take) = [1] x1 + [1] x2 + [0] p(cons) = [1] x2 + [1] p(eq) = [1] x1 + [1] x2 + [2] p(false) = [0] p(inf) = [1] x1 + [2] p(length) = [1] x1 + [1] p(mark) = [4] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [2] p(take) = [1] x1 + [1] x2 + [0] p(true) = [0] Following rules are strictly oriented: a__eq(s(X),s(Y)) = [4] X + [1] Y + [15] > [4] X + [1] Y + [5] = a__eq(X,Y) Following rules are (at-least) weakly oriented: a__eq(X,Y) = [4] X + [1] Y + [5] >= [0] = false() a__eq(X1,X2) = [4] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [2] = eq(X1,X2) a__eq(0(),0()) = [5] >= [0] = true() a__inf(X) = [1] X + [6] >= [1] X + [5] = cons(X,inf(s(X))) a__inf(X) = [1] X + [6] >= [1] X + [2] = inf(X) a__length(X) = [1] X + [2] >= [1] X + [1] = length(X) a__length(cons(X,L)) = [1] L + [3] >= [1] L + [3] = s(length(L)) a__length(nil()) = [2] >= [0] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = take(X1,X2) a__take(0(),X) = [1] X + [0] >= [0] = nil() a__take(s(X),cons(Y,L)) = [1] L + [1] X + [3] >= [1] L + [1] X + [1] = cons(Y,take(X,L)) mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [4] X2 + [4] >= [1] X2 + [1] = cons(X1,X2) mark(eq(X1,X2)) = [4] X1 + [4] X2 + [8] >= [4] X1 + [1] X2 + [5] = a__eq(X1,X2) mark(false()) = [0] >= [0] = false() mark(inf(X)) = [4] X + [8] >= [4] X + [6] = a__inf(mark(X)) mark(length(X)) = [4] X + [4] >= [4] X + [2] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [4] X + [8] >= [1] X + [2] = s(X) mark(take(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = a__take(mark(X1),mark(X2)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__eq(s(X),s(Y)) -> a__eq(X,Y) a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(inf(X)) -> a__inf(mark(X)) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq,a__inf,a__length,a__take,mark} and constructors {0 ,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))