WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(x){x -> add(x,y)} = mark(add(x,y)) ->^+ a__add(mark(x),mark(y)) = C[mark(x) = mark(x){}] ** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = 0 p(a__add) = x1 + x2 p(a__from) = 1 + x1 p(a__fst) = x1 + x2 p(a__len) = x1 p(add) = x1 + x2 p(cons) = x1 p(from) = 1 + x1 p(fst) = x1 + x2 p(len) = x1 p(mark) = x1 p(nil) = 0 p(s) = 0 Following rules are strictly oriented: a__from(X) = 1 + X > X = cons(mark(X),from(s(X))) Following rules are (at-least) weakly oriented: a__add(X1,X2) = X1 + X2 >= X1 + X2 = add(X1,X2) a__add(0(),X) = X >= X = mark(X) a__add(s(X),Y) = Y >= 0 = s(add(X,Y)) a__from(X) = 1 + X >= 1 + X = from(X) a__fst(X1,X2) = X1 + X2 >= X1 + X2 = fst(X1,X2) a__fst(0(),Z) = Z >= 0 = nil() a__fst(s(X),cons(Y,Z)) = Y >= Y = cons(mark(Y),fst(X,Z)) a__len(X) = X >= X = len(X) a__len(cons(X,Z)) = X >= 0 = s(len(Z)) a__len(nil()) = 0 >= 0 = 0() mark(0()) = 0 >= 0 = 0() mark(add(X1,X2)) = X1 + X2 >= X1 + X2 = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(from(X)) = 1 + X >= 1 + X = a__from(mark(X)) mark(fst(X1,X2)) = X1 + X2 >= X1 + X2 = a__fst(mark(X1),mark(X2)) mark(len(X)) = X >= X = a__len(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(s(X)) = 0 >= 0 = s(X) ** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [1] x2 + [2] p(a__from) = [1] x1 + [0] p(a__fst) = [1] x1 + [1] x2 + [0] p(a__len) = [1] x1 + [0] p(add) = [1] x1 + [1] x2 + [2] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [1] x2 + [0] p(len) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [0] Following rules are strictly oriented: a__add(0(),X) = [1] X + [2] > [1] X + [0] = mark(X) a__add(s(X),Y) = [1] Y + [2] > [0] = s(add(X,Y)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = add(X1,X2) a__from(X) = [1] X + [0] >= [1] X + [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [1] X + [0] = from(X) a__fst(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = fst(X1,X2) a__fst(0(),Z) = [1] Z + [0] >= [0] = nil() a__fst(s(X),cons(Y,Z)) = [1] Y + [0] >= [1] Y + [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1] X + [0] >= [1] X + [0] = len(X) a__len(cons(X,Z)) = [1] X + [0] >= [0] = s(len(Z)) a__len(nil()) = [0] >= [0] = 0() mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(from(X)) = [1] X + [0] >= [1] X + [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1] X + [0] >= [1] X + [0] = a__len(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) ** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [1] x2 + [0] p(a__from) = [1] x1 + [0] p(a__fst) = [1] x1 + [1] x2 + [1] p(a__len) = [1] x1 + [1] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [1] x2 + [1] p(len) = [1] x1 + [1] p(mark) = [1] x1 + [0] p(nil) = [1] p(s) = [0] Following rules are strictly oriented: a__fst(s(X),cons(Y,Z)) = [1] Y + [1] > [1] Y + [0] = cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) = [1] X + [1] > [0] = s(len(Z)) a__len(nil()) = [2] > [0] = 0() Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [1] X + [0] >= [1] X + [0] = mark(X) a__add(s(X),Y) = [1] Y + [0] >= [0] = s(add(X,Y)) a__from(X) = [1] X + [0] >= [1] X + [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [1] X + [0] = from(X) a__fst(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = fst(X1,X2) a__fst(0(),Z) = [1] Z + [1] >= [1] = nil() a__len(X) = [1] X + [1] >= [1] X + [1] = len(X) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(from(X)) = [1] X + [0] >= [1] X + [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1] X + [1] >= [1] X + [1] = a__len(mark(X)) mark(nil()) = [1] >= [1] = nil() mark(s(X)) = [0] >= [0] = s(X) ** Step 1.b:4: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__len(X) -> len(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [1] x2 + [0] p(a__from) = [1] x1 + [0] p(a__fst) = [1] x1 + [1] x2 + [2] p(a__len) = [1] x1 + [0] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [1] x2 + [2] p(len) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [0] Following rules are strictly oriented: a__fst(0(),Z) = [1] Z + [2] > [0] = nil() Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [1] X + [0] >= [1] X + [0] = mark(X) a__add(s(X),Y) = [1] Y + [0] >= [0] = s(add(X,Y)) a__from(X) = [1] X + [0] >= [1] X + [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [1] X + [0] = from(X) a__fst(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = fst(X1,X2) a__fst(s(X),cons(Y,Z)) = [1] Y + [2] >= [1] Y + [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1] X + [0] >= [1] X + [0] = len(X) a__len(cons(X,Z)) = [1] X + [0] >= [0] = s(len(Z)) a__len(nil()) = [0] >= [0] = 0() mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(from(X)) = [1] X + [0] >= [1] X + [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1] X + [0] >= [1] X + [0] = a__len(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) ** Step 1.b:5: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__len(X) -> len(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] p(a__from) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(a__fst) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [1] p(a__len) = [1 1 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(add) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(from) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(fst) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [1] p(len) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(mark) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [0] [0] [0] Following rules are strictly oriented: mark(add(X1,X2)) = [1 1 1] [1 1 1] [1] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] > [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = a__add(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] = add(X1,X2) a__add(0(),X) = [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 1 0] [0] [0 0 1] Y + [1] [0 0 1] [1] >= [0] [0] [0] = s(add(X,Y)) a__from(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = from(X) a__fst(X1,X2) = [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [1] = fst(X1,X2) a__fst(0(),Z) = [1 1 0] [0] [0 0 1] Z + [0] [0 0 1] [1] >= [0] [0] [0] = nil() a__fst(s(X),cons(Y,Z)) = [1 1 0] [0] [0 0 1] Y + [0] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] Y + [0] [0 0 1] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [1] = len(X) a__len(cons(X,Z)) = [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [0] [1] [1] >= [0] [0] [0] = 0() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [1] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 1 1] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 1] [0] [0 0 1] X + [1] [0 0 1] [1] = a__len(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) ** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__len(X) -> len(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [1] p(a__add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__fst) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__len) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(from) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(fst) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(len) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [1] p(s) = [0] [0] [0] Following rules are strictly oriented: mark(0()) = [1] [0] [1] > [0] [0] [1] = 0() mark(nil()) = [1] [0] [1] > [0] [0] [1] = nil() Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] >= [0] [0] [0] = s(add(X,Y)) a__from(X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = from(X) a__fst(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = fst(X1,X2) a__fst(0(),Z) = [1 0 1] [0] [0 0 0] Z + [0] [0 0 1] [1] >= [0] [0] [1] = nil() a__fst(s(X),cons(Y,Z)) = [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = len(X) a__len(cons(X,Z)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [0] [0] [1] >= [0] [0] [1] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__len(mark(X)) mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) ** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__len(X) -> len(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(s(X)) -> s(X) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(nil()) -> nil() - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 1] [1 1 0] [0] [0 1 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 1 0] [1] [0 0 1] x1 + [1] [0 0 1] [1] p(a__fst) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__len) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(add) = [1 0 1] [1 1 0] [0] [0 1 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(from) = [1 1 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(fst) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(len) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(mark) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [0] [0] [0] Following rules are strictly oriented: a__from(X) = [1 1 0] [1] [0 0 1] X + [1] [0 0 1] [1] > [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] = from(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 1] [1 1 0] [0] [0 1 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 1 0] [0] [0 1 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 1 0] [0] [0 0 1] Y + [0] [0 0 1] [0] >= [0] [0] [0] = s(add(X,Y)) a__from(X) = [1 1 0] [1] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__fst(X1,X2) = [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = fst(X1,X2) a__fst(0(),Z) = [1 1 0] [0] [0 0 1] Z + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__fst(s(X),cons(Y,Z)) = [1 1 0] [0] [0 0 1] Y + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] Y + [0] [0 0 1] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = len(X) a__len(cons(X,Z)) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 1 1] [1] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 1] [1] [0 0 1] X + [1] [0 0 1] [1] = a__from(mark(X)) mark(fst(X1,X2)) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = a__len(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) ** Step 1.b:8: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__fst(X1,X2) -> fst(X1,X2) a__len(X) -> len(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(s(X)) -> s(X) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(nil()) -> nil() - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(a__fst) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__len) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(add) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(from) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(fst) = [1 1 0] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [0] p(len) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(mark) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [0] [1] [1] Following rules are strictly oriented: mark(s(X)) = [1] [1] [1] > [0] [1] [1] = s(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 1 0] [0] [0 0 1] Y + [1] [0 0 1] [1] >= [0] [1] [1] = s(add(X,Y)) a__from(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [1] = from(X) a__fst(X1,X2) = [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 1 0] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = fst(X1,X2) a__fst(0(),Z) = [1 1 0] [0] [0 0 1] Z + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__fst(s(X),cons(Y,Z)) = [1 1 0] [1] [0 0 1] Y + [2] [0 0 1] [2] >= [1 1 0] [0] [0 0 1] Y + [0] [0 0 1] [1] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [0] = len(X) a__len(cons(X,Z)) = [1 1 0] [0] [0 0 1] X + [1] [0 0 1] [1] >= [0] [1] [1] = s(len(Z)) a__len(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 1 0] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 0] [0] [0 0 1] X1 + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 1] X1 + [0] [0 0 1] [1] = cons(mark(X1),X2) mark(from(X)) = [1 1 1] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [1] = a__from(mark(X)) mark(fst(X1,X2)) = [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [1 1 1] [0] [0 0 1] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 1] [0] [0 0 1] X + [0] [0 0 1] [0] = a__len(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() ** Step 1.b:9: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__fst(X1,X2) -> fst(X1,X2) a__len(X) -> len(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [1] [0] [1] p(a__add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(a__fst) = [1 0 0] [1 1 1] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__len) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(from) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(fst) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(len) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [1] [0] [1] p(s) = [1] [0] [0] Following rules are strictly oriented: mark(len(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] > [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__len(mark(X)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 0 1] [1] [0 0 0] Y + [0] [0 0 1] [0] >= [1] [0] [0] = s(add(X,Y)) a__from(X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] = from(X) a__fst(X1,X2) = [1 0 0] [1 1 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = fst(X1,X2) a__fst(0(),Z) = [1 1 1] [1] [0 0 0] Z + [0] [0 0 1] [1] >= [1] [0] [1] = nil() a__fst(s(X),cons(Y,Z)) = [1 0 1] [2] [0 0 0] Y + [0] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = len(X) a__len(cons(X,Z)) = [1 0 0] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1] [0] [0] = s(len(Z)) a__len(nil()) = [1] [0] [2] >= [1] [0] [1] = 0() mark(0()) = [2] [0] [1] >= [1] [0] [1] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [1] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__fst(mark(X1),mark(X2)) mark(nil()) = [2] [0] [1] >= [1] [0] [1] = nil() mark(s(X)) = [1] [0] [0] >= [1] [0] [0] = s(X) ** Step 1.b:10: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__fst(X1,X2) -> fst(X1,X2) a__len(X) -> len(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 1 0] [1] [0 1 1] x1 + [1] [0 0 1] [1] p(a__fst) = [1 0 0] [1 1 0] [0] [0 1 1] x1 + [0 1 1] x2 + [0] [0 0 1] [0 0 1] [0] p(a__len) = [1 1 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(add) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [1] [0 1 0] x1 + [1] [0 0 1] [0] p(from) = [1 1 0] [1] [0 1 1] x1 + [0] [0 0 1] [1] p(fst) = [1 0 0] [1 1 0] [0] [0 1 1] x1 + [0 1 1] x2 + [0] [0 0 1] [0 0 1] [0] p(len) = [1 1 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(mark) = [1 1 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [0] [0] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 1 0] [2] [0 1 1] X1 + [1] [0 0 1] [0] > [1 1 0] [1] [0 1 1] X1 + [1] [0 0 1] [0] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 1 0] [0] [0 1 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 1 1] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 1 0] [0] [0 1 1] Y + [0] [0 0 1] [0] >= [0] [0] [0] = s(add(X,Y)) a__from(X) = [1 1 0] [1] [0 1 1] X + [1] [0 0 1] [1] >= [1 1 0] [1] [0 1 1] X + [1] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [1] [0 1 1] X + [1] [0 0 1] [1] >= [1 1 0] [1] [0 1 1] X + [0] [0 0 1] [1] = from(X) a__fst(X1,X2) = [1 0 0] [1 1 0] [0] [0 1 1] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 1 0] [0] [0 1 1] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [0] = fst(X1,X2) a__fst(0(),Z) = [1 1 0] [0] [0 1 1] Z + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__fst(s(X),cons(Y,Z)) = [1 1 0] [2] [0 1 1] Y + [1] [0 0 1] [0] >= [1 1 0] [1] [0 1 1] Y + [1] [0 0 1] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 1 0] [0] [0 1 1] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 1 1] X + [0] [0 0 1] [0] = len(X) a__len(cons(X,Z)) = [1 1 0] [2] [0 1 1] X + [1] [0 0 1] [0] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 1 0] [1 2 1] [0] [0 1 1] X1 + [0 1 2] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 2 1] [0] [0 1 1] X1 + [0 1 2] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(from(X)) = [1 2 1] [1] [0 1 2] X + [1] [0 0 1] [1] >= [1 2 1] [1] [0 1 2] X + [1] [0 0 1] [1] = a__from(mark(X)) mark(fst(X1,X2)) = [1 1 1] [1 2 1] [0] [0 1 2] X1 + [0 1 2] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 2 1] [0] [0 1 2] X1 + [0 1 2] X2 + [0] [0 0 1] [0 0 1] [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 2 1] [0] [0 1 2] X + [0] [0 0 1] [0] >= [1 2 1] [0] [0 1 2] X + [0] [0 0 1] [0] = a__len(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) ** Step 1.b:11: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__fst(X1,X2) -> fst(X1,X2) a__len(X) -> len(X) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 0] [0] p(a__from) = [1 1 1] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(a__fst) = [1 1 0] [1 1 0] [1] [0 1 1] x1 + [0 1 1] x2 + [1] [0 0 0] [0 0 1] [0] p(a__len) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(add) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(cons) = [1 0 0] [0 1 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(from) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(fst) = [1 1 0] [1 1 0] [1] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(len) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(nil) = [0] [0] [0] p(s) = [0 1 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: mark(fst(X1,X2)) = [1 2 0] [1 2 0] [2] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] > [1 2 0] [1 2 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = a__fst(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 0] [0] >= [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = add(X1,X2) a__add(0(),X) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(X) a__add(s(X),Y) = [0 1 0] [1 1 0] [0] [0 0 1] X + [0 1 0] Y + [0] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 1 0] [0] [0 0 0] X + [0 0 0] Y + [0] [0 0 0] [0 0 0] [0] = s(add(X,Y)) a__from(X) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 1 1] [0] [0 1 0] X + [0] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = from(X) a__fst(X1,X2) = [1 1 0] [1 1 0] [1] [0 1 1] X1 + [0 1 1] X2 + [1] [0 0 0] [0 0 1] [0] >= [1 1 0] [1 1 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = fst(X1,X2) a__fst(0(),Z) = [1 1 0] [1] [0 1 1] Z + [1] [0 0 1] [0] >= [0] [0] [0] = nil() a__fst(s(X),cons(Y,Z)) = [0 1 1] [1 1 0] [0 1 0] [1] [0 0 1] X + [0 1 0] Y + [0 0 0] Z + [1] [0 0 0] [0 0 0] [0 0 0] [0] >= [0 1 0] [1 1 0] [0 1 0] [1] [0 0 0] X + [0 1 0] Y + [0 0 0] Z + [0] [0 0 0] [0 0 0] [0 0 0] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = len(X) a__len(cons(X,Z)) = [1 1 0] [0 1 0] [0] [0 1 0] X + [0 0 0] Z + [0] [0 0 0] [0 0 0] [0] >= [0 1 0] [0] [0 0 0] Z + [0] [0 0 0] [0] = s(len(Z)) a__len(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 0] [0 1 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [0 1 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(mark(X1),X2) mark(from(X)) = [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__from(mark(X)) mark(len(X)) = [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__len(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0 1 1] [0] [0 0 1] X + [0] [0 0 0] [0] >= [0 1 0] [0] [0 0 1] X + [0] [0 0 0] [0] = s(X) ** Step 1.b:12: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__fst(X1,X2) -> fst(X1,X2) a__len(X) -> len(X) mark(from(X)) -> a__from(mark(X)) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 1] [1 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(a__from) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(a__fst) = [1 1 1] [1 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 0] [0] p(a__len) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(add) = [1 0 1] [1 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(cons) = [1 0 0] [0 0 1] [0] [0 1 0] x1 + [0 0 1] x2 + [0] [0 0 0] [0 0 0] [0] p(from) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(fst) = [1 1 1] [1 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(len) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(nil) = [0] [0] [0] p(s) = [0] [0] [0] Following rules are strictly oriented: mark(from(X)) = [1 2 0] [1] [0 1 0] X + [1] [0 0 0] [0] > [1 2 0] [0] [0 1 0] X + [1] [0 0 0] [0] = a__from(mark(X)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 1] [1 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = add(X1,X2) a__add(0(),X) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(X) a__add(s(X),Y) = [1 1 0] [0] [0 1 0] Y + [0] [0 0 0] [0] >= [0] [0] [0] = s(add(X,Y)) a__from(X) = [1 1 0] [0] [0 1 0] X + [1] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 1 0] X + [1] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = from(X) a__fst(X1,X2) = [1 1 1] [1 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 0] [0] >= [1 1 1] [1 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = fst(X1,X2) a__fst(0(),Z) = [1 1 0] [0] [0 1 0] Z + [0] [0 0 0] [0] >= [0] [0] [0] = nil() a__fst(s(X),cons(Y,Z)) = [1 1 0] [0 0 2] [0] [0 1 0] Y + [0 0 1] Z + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] Y + [0] [0 0 0] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = len(X) a__len(cons(X,Z)) = [1 0 0] [0 0 1] [0] [0 1 0] X + [0 0 1] Z + [0] [0 0 0] [0 0 0] [0] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 1 2] [1 2 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 0] [0 0 2] [0] [0 1 0] X1 + [0 0 1] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [0 0 1] [0] [0 1 0] X1 + [0 0 1] X2 + [0] [0 0 0] [0 0 0] [0] = cons(mark(X1),X2) mark(fst(X1,X2)) = [1 2 2] [1 2 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 2 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__len(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) ** Step 1.b:13: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__fst(X1,X2) -> fst(X1,X2) a__len(X) -> len(X) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 1] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(a__from) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(a__fst) = [1 0 0] [1 1 0] [1] [0 1 1] x1 + [0 1 1] x2 + [1] [0 0 0] [0 0 0] [0] p(a__len) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(add) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(cons) = [1 0 1] [0 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 0] [0] p(from) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(fst) = [1 0 0] [1 1 0] [0] [0 1 1] x1 + [0 1 1] x2 + [1] [0 0 0] [0 0 0] [0] p(len) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(nil) = [0] [1] [0] p(s) = [0 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: a__fst(X1,X2) = [1 0 0] [1 1 0] [1] [0 1 1] X1 + [0 1 1] X2 + [1] [0 0 0] [0 0 0] [0] > [1 0 0] [1 1 0] [0] [0 1 1] X1 + [0 1 1] X2 + [1] [0 0 0] [0 0 0] [0] = fst(X1,X2) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 1] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = add(X1,X2) a__add(0(),X) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(X) a__add(s(X),Y) = [0 0 0] [1 1 0] [0] [0 0 1] X + [0 1 0] Y + [1] [0 0 0] [0 0 0] [0] >= [0] [0] [0] = s(add(X,Y)) a__from(X) = [1 1 0] [0] [0 1 0] X + [1] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 1 0] X + [1] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 1] [0] = from(X) a__fst(0(),Z) = [1 1 0] [1] [0 1 1] Z + [1] [0 0 0] [0] >= [0] [1] [0] = nil() a__fst(s(X),cons(Y,Z)) = [0 0 0] [1 1 1] [0 0 1] [2] [0 0 1] X + [0 1 1] Y + [0 0 0] Z + [2] [0 0 0] [0 0 0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] Y + [1] [0 0 0] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = len(X) a__len(cons(X,Z)) = [1 0 1] [0 0 1] [0] [0 1 0] X + [0 0 0] Z + [1] [0 0 0] [0 0 0] [0] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [0] [1] [0] >= [0] [0] [0] = 0() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 1 0] [1 2 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 1] [0 0 1] [1] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 1 0] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] = cons(mark(X1),X2) mark(from(X)) = [1 2 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] X + [1] [0 0 0] [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1 1 1] [1 2 1] [1] [0 1 1] X1 + [0 1 1] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__len(mark(X)) mark(nil()) = [1] [1] [0] >= [0] [1] [0] = nil() mark(s(X)) = [0 0 1] [0] [0 0 1] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 0 1] X + [0] [0 0 0] [0] = s(X) ** Step 1.b:14: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [1] p(a__add) = [1 0 0] [1 1 0] [1] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 0] [0] p(a__from) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__fst) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 0] [0] p(a__len) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(add) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(from) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(fst) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 0] [0] p(len) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(nil) = [0] [0] [1] p(s) = [0] [0] [0] Following rules are strictly oriented: a__add(X1,X2) = [1 0 0] [1 1 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 0] [0] > [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = add(X1,X2) Following rules are (at-least) weakly oriented: a__add(0(),X) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = mark(X) a__add(s(X),Y) = [1 1 0] [1] [0 1 0] Y + [1] [0 0 0] [0] >= [0] [0] [0] = s(add(X,Y)) a__from(X) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = from(X) a__fst(X1,X2) = [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 0] [0] >= [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 0] [0] = fst(X1,X2) a__fst(0(),Z) = [1 1 0] [0] [0 1 0] Z + [0] [0 0 0] [1] >= [0] [0] [1] = nil() a__fst(s(X),cons(Y,Z)) = [1 1 0] [0] [0 1 0] Y + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] Y + [0] [0 0 0] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = len(X) a__len(cons(X,Z)) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [0] [0] [1] >= [0] [0] [1] = 0() mark(0()) = [0] [0] [1] >= [0] [0] [1] = 0() mark(add(X1,X2)) = [1 1 0] [1 2 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [1] >= [1 1 0] [1 2 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [1] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = cons(mark(X1),X2) mark(from(X)) = [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [1] >= [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [1] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = a__len(mark(X)) mark(nil()) = [0] [0] [1] >= [0] [0] [1] = nil() mark(s(X)) = [0] [0] [1] >= [0] [0] [0] = s(X) ** Step 1.b:15: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__len(X) -> len(X) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [1] [0] [0] p(a__add) = [1 0 1] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 0] [0] p(a__from) = [1 1 0] [1] [0 1 0] x1 + [1] [0 0 0] [0] p(a__fst) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 0] [0 0 1] [0] p(a__len) = [1 1 0] [1] [0 1 0] x1 + [1] [0 0 0] [0] p(add) = [1 0 1] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(cons) = [1 0 1] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(from) = [1 1 0] [1] [0 1 0] x1 + [1] [0 0 0] [0] p(fst) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(len) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(nil) = [1] [0] [0] p(s) = [1] [1] [0] Following rules are strictly oriented: a__len(X) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] > [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = len(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 1] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 0] [0] >= [1 0 1] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = add(X1,X2) a__add(0(),X) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(X) a__add(s(X),Y) = [1 1 0] [1] [0 1 0] Y + [2] [0 0 0] [0] >= [1] [1] [0] = s(add(X,Y)) a__from(X) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] = from(X) a__fst(X1,X2) = [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 0] [0 0 1] [0] >= [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = fst(X1,X2) a__fst(0(),Z) = [1 1 0] [1] [0 1 1] Z + [0] [0 0 1] [0] >= [1] [0] [0] = nil() a__fst(s(X),cons(Y,Z)) = [1 1 1] [2] [0 1 0] Y + [2] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] Y + [1] [0 0 0] [0] = cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) = [1 1 1] [2] [0 1 0] X + [2] [0 0 0] [0] >= [1] [1] [0] = s(len(Z)) a__len(nil()) = [2] [1] [0] >= [1] [0] [0] = 0() mark(0()) = [1] [0] [0] >= [1] [0] [0] = 0() mark(add(X1,X2)) = [1 1 1] [1 2 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 1] [1] [0 1 0] X1 + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X1 + [1] [0 0 0] [0] = cons(mark(X1),X2) mark(from(X)) = [1 2 0] [2] [0 1 0] X + [1] [0 0 0] [0] >= [1 2 0] [1] [0 1 0] X + [1] [0 0 0] [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 2 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [1 2 0] [1] [0 1 0] X + [1] [0 0 0] [0] = a__len(mark(X)) mark(nil()) = [1] [0] [0] >= [1] [0] [0] = nil() mark(s(X)) = [2] [1] [0] >= [1] [1] [0] = s(X) ** Step 1.b:16: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))