WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(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: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: a__eq(x,y){x -> s(x),y -> s(y)} = a__eq(s(x),s(y)) ->^+ a__eq(x,y) = C[a__eq(x,y) = a__eq(x,y){}] ** Step 1.b:1: NaturalPI 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): 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) = 0 p(a__inf) = x1 p(a__length) = 3 + x1 p(a__take) = x1 + x2 p(cons) = x1 p(eq) = 0 p(false) = 0 p(inf) = x1 p(length) = 1 + x1 p(mark) = 4*x1 p(nil) = 0 p(s) = 0 p(take) = x1 + x2 p(true) = 0 Following rules are strictly oriented: a__length(X) = 3 + X > 1 + X = length(X) a__length(cons(X,L)) = 3 + X > 0 = s(length(L)) a__length(nil()) = 3 > 1 = 0() a__take(0(),X) = 1 + X > 0 = nil() mark(0()) = 4 > 1 = 0() mark(length(X)) = 4 + 4*X > 3 + 4*X = a__length(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) = X >= X = cons(X,inf(s(X))) a__inf(X) = X >= X = inf(X) a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) a__take(s(X),cons(Y,L)) = Y >= Y = cons(Y,take(X,L)) mark(cons(X1,X2)) = 4*X1 >= X1 = cons(X1,X2) mark(eq(X1,X2)) = 0 >= 0 = a__eq(X1,X2) mark(false()) = 0 >= 0 = false() mark(inf(X)) = 4*X >= 4*X = a__inf(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(s(X)) = 0 >= 0 = s(X) mark(take(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__take(mark(X1),mark(X2)) mark(true()) = 0 >= 0 = true() ** Step 1.b:2: 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__take(X1,X2) -> take(X1,X2) a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) 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(nil()) -> nil() mark(s(X)) -> s(X) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(true()) -> true() - Weak TRS: a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(0(),X) -> nil() mark(0()) -> 0() mark(length(X)) -> a__length(mark(X)) - 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) = [0] p(a__eq) = [0] p(a__inf) = [1] x1 + [0] p(a__length) = [1] x1 + [0] p(a__take) = [1] x1 + [1] x2 + [3] p(cons) = [0] p(eq) = [0] p(false) = [0] p(inf) = [1] x1 + [0] p(length) = [1] x1 + [0] p(mark) = [4] x1 + [0] p(nil) = [0] p(s) = [0] p(take) = [1] x1 + [1] x2 + [2] p(true) = [0] Following rules are strictly oriented: a__take(X1,X2) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [2] = take(X1,X2) a__take(s(X),cons(Y,L)) = [3] > [0] = cons(Y,take(X,L)) mark(take(X1,X2)) = [4] X1 + [4] X2 + [8] > [4] X1 + [4] X2 + [3] = 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 + [0] >= [0] = cons(X,inf(s(X))) a__inf(X) = [1] X + [0] >= [1] X + [0] = 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()) = [0] >= [0] = 0() a__take(0(),X) = [1] X + [3] >= [0] = nil() mark(0()) = [0] >= [0] = 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(inf(X)) = [4] X + [0] >= [4] X + [0] = a__inf(mark(X)) mark(length(X)) = [4] X + [0] >= [4] X + [0] = a__length(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(true()) = [0] >= [0] = true() ** Step 1.b:3: 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) 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(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Weak TRS: 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(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) - 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) = [0] p(a__eq) = [15] p(a__inf) = [1] x1 + [6] p(a__length) = [1] x1 + [7] p(a__take) = [1] x1 + [1] x2 + [3] p(cons) = [1] p(eq) = [8] p(false) = [1] p(inf) = [1] x1 + [3] p(length) = [1] x1 + [4] p(mark) = [2] x1 + [1] p(nil) = [2] p(s) = [1] p(take) = [1] x1 + [1] x2 + [2] p(true) = [5] Following rules are strictly oriented: a__eq(X,Y) = [15] > [1] = false() a__eq(X1,X2) = [15] > [8] = eq(X1,X2) a__eq(0(),0()) = [15] > [5] = true() a__inf(X) = [1] X + [6] > [1] = cons(X,inf(s(X))) a__inf(X) = [1] X + [6] > [1] X + [3] = inf(X) mark(cons(X1,X2)) = [3] > [1] = cons(X1,X2) mark(eq(X1,X2)) = [17] > [15] = a__eq(X1,X2) mark(false()) = [3] > [1] = false() mark(nil()) = [5] > [2] = nil() mark(s(X)) = [3] > [1] = s(X) mark(true()) = [11] > [5] = true() Following rules are (at-least) weakly oriented: a__eq(s(X),s(Y)) = [15] >= [15] = a__eq(X,Y) a__length(X) = [1] X + [7] >= [1] X + [4] = length(X) a__length(cons(X,L)) = [8] >= [1] = s(length(L)) a__length(nil()) = [9] >= [0] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [2] = take(X1,X2) a__take(0(),X) = [1] X + [3] >= [2] = nil() a__take(s(X),cons(Y,L)) = [5] >= [1] = cons(Y,take(X,L)) mark(0()) = [1] >= [0] = 0() mark(inf(X)) = [2] X + [7] >= [2] X + [7] = a__inf(mark(X)) mark(length(X)) = [2] X + [9] >= [2] X + [8] = a__length(mark(X)) mark(take(X1,X2)) = [2] X1 + [2] X2 + [5] >= [2] X1 + [2] X2 + [5] = a__take(mark(X1),mark(X2)) ** Step 1.b:4: NaturalPI 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)) - 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(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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): 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) = 0 p(a__inf) = 1 + x1 p(a__length) = 2 + x1 p(a__take) = x1 + x2 p(cons) = 0 p(eq) = 0 p(false) = 0 p(inf) = 1 + x1 p(length) = 2 + x1 p(mark) = 4*x1 p(nil) = 1 p(s) = 0 p(take) = x1 + x2 p(true) = 0 Following rules are strictly oriented: mark(inf(X)) = 4 + 4*X > 1 + 4*X = 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 >= 0 = cons(X,inf(s(X))) a__inf(X) = 1 + X >= 1 + X = inf(X) a__length(X) = 2 + X >= 2 + X = length(X) a__length(cons(X,L)) = 2 >= 0 = s(length(L)) a__length(nil()) = 3 >= 1 = 0() a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) a__take(0(),X) = 1 + X >= 1 = nil() a__take(s(X),cons(Y,L)) = 0 >= 0 = cons(Y,take(X,L)) mark(0()) = 4 >= 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)) = 8 + 4*X >= 2 + 4*X = a__length(mark(X)) mark(nil()) = 4 >= 1 = nil() mark(s(X)) = 0 >= 0 = s(X) mark(take(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__take(mark(X1),mark(X2)) mark(true()) = 0 >= 0 = true() ** Step 1.b:5: NaturalPI 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): 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) = 0 p(a__eq) = 1 + 4*x1 p(a__inf) = 4 + x1 p(a__length) = 7 + x1 p(a__take) = 5 + x1 + x2 p(cons) = x2 p(eq) = x1 p(false) = 0 p(inf) = 1 + x1 p(length) = 3 + x1 p(mark) = 2 + 4*x1 p(nil) = 2 p(s) = 3 + x1 p(take) = 2 + x1 + x2 p(true) = 0 Following rules are strictly oriented: a__eq(s(X),s(Y)) = 13 + 4*X > 1 + 4*X = a__eq(X,Y) Following rules are (at-least) weakly oriented: a__eq(X,Y) = 1 + 4*X >= 0 = false() a__eq(X1,X2) = 1 + 4*X1 >= X1 = eq(X1,X2) a__eq(0(),0()) = 1 >= 0 = true() a__inf(X) = 4 + X >= 4 + X = cons(X,inf(s(X))) a__inf(X) = 4 + X >= 1 + X = inf(X) a__length(X) = 7 + X >= 3 + X = length(X) a__length(cons(X,L)) = 7 + L >= 6 + L = s(length(L)) a__length(nil()) = 9 >= 0 = 0() a__take(X1,X2) = 5 + X1 + X2 >= 2 + X1 + X2 = take(X1,X2) a__take(0(),X) = 5 + X >= 2 = nil() a__take(s(X),cons(Y,L)) = 8 + L + X >= 2 + L + X = cons(Y,take(X,L)) mark(0()) = 2 >= 0 = 0() mark(cons(X1,X2)) = 2 + 4*X2 >= X2 = cons(X1,X2) mark(eq(X1,X2)) = 2 + 4*X1 >= 1 + 4*X1 = a__eq(X1,X2) mark(false()) = 2 >= 0 = false() mark(inf(X)) = 6 + 4*X >= 6 + 4*X = a__inf(mark(X)) mark(length(X)) = 14 + 4*X >= 9 + 4*X = a__length(mark(X)) mark(nil()) = 10 >= 2 = nil() mark(s(X)) = 14 + 4*X >= 3 + X = s(X) mark(take(X1,X2)) = 10 + 4*X1 + 4*X2 >= 9 + 4*X1 + 4*X2 = a__take(mark(X1),mark(X2)) mark(true()) = 2 >= 0 = true() ** Step 1.b:6: 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(Omega(n^1),O(n^1))