WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: +Full(0(),y) -> y +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() - Weak TRS: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() - Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,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(+) = {2}, uargs(Cons) = {1,2} Following symbols are considered usable: {*,+Full,f,goal,map} TcT has computed the following interpretation: p(*) = [2] x1 + [3] x2 + [0] p(+) = [1] x2 + [0] p(+Full) = [8] x2 + [0] p(0) = [1] p(Cons) = [1] x1 + [1] x2 + [2] p(Nil) = [0] p(S) = [0] p(f) = [5] x1 + [0] p(goal) = [8] x1 + [2] p(map) = [5] x1 + [2] Following rules are strictly oriented: map(Cons(x,xs)) = [5] x + [5] xs + [12] > [5] x + [5] xs + [4] = Cons(f(x),map(xs)) map(Nil()) = [2] > [0] = Nil() Following rules are (at-least) weakly oriented: *(x,0()) = [2] x + [3] >= [1] = 0() *(x,S(0())) = [2] x + [0] >= [1] x + [0] = x *(x,S(S(y))) = [2] x + [0] >= [2] x + [0] = +(x,*(x,S(y))) *(0(),y) = [3] y + [2] >= [1] = 0() +Full(0(),y) = [8] y + [0] >= [1] y + [0] = y +Full(S(x),y) = [8] y + [0] >= [0] = +Full(x,S(y)) f(x) = [5] x + [0] >= [5] x + [0] = *(x,x) goal(xs) = [8] xs + [2] >= [5] xs + [2] = map(xs) * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: +Full(0(),y) -> y +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) - Weak TRS: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() - Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,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(+) = {2}, uargs(Cons) = {1,2} Following symbols are considered usable: {*,+Full,f,goal,map} TcT has computed the following interpretation: p(*) = [2] x1 + [8] p(+) = [1] x2 + [0] p(+Full) = [1] x1 + [2] x2 + [3] p(0) = [1] p(Cons) = [1] x1 + [1] x2 + [2] p(Nil) = [2] p(S) = [1] x1 + [0] p(f) = [5] x1 + [8] p(goal) = [8] x1 + [1] p(map) = [5] x1 + [0] Following rules are strictly oriented: +Full(0(),y) = [2] y + [4] > [1] y + [0] = y goal(xs) = [8] xs + [1] > [5] xs + [0] = map(xs) Following rules are (at-least) weakly oriented: *(x,0()) = [2] x + [8] >= [1] = 0() *(x,S(0())) = [2] x + [8] >= [1] x + [0] = x *(x,S(S(y))) = [2] x + [8] >= [2] x + [8] = +(x,*(x,S(y))) *(0(),y) = [10] >= [1] = 0() +Full(S(x),y) = [1] x + [2] y + [3] >= [1] x + [2] y + [3] = +Full(x,S(y)) f(x) = [5] x + [8] >= [2] x + [8] = *(x,x) map(Cons(x,xs)) = [5] x + [5] xs + [10] >= [5] x + [5] xs + [10] = Cons(f(x),map(xs)) map(Nil()) = [10] >= [2] = Nil() * Step 3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) - Weak TRS: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() +Full(0(),y) -> y goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() - Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,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(+) = {2}, uargs(Cons) = {1,2} Following symbols are considered usable: {*,+Full,f,goal,map} TcT has computed the following interpretation: p(*) = [2] x1 + [4] x2 + [0] p(+) = [1] x2 + [0] p(+Full) = [2] x2 + [2] p(0) = [4] p(Cons) = [1] x1 + [1] x2 + [4] p(Nil) = [0] p(S) = [0] p(f) = [7] x1 + [1] p(goal) = [7] x1 + [2] p(map) = [7] x1 + [2] Following rules are strictly oriented: f(x) = [7] x + [1] > [6] x + [0] = *(x,x) Following rules are (at-least) weakly oriented: *(x,0()) = [2] x + [16] >= [4] = 0() *(x,S(0())) = [2] x + [0] >= [1] x + [0] = x *(x,S(S(y))) = [2] x + [0] >= [2] x + [0] = +(x,*(x,S(y))) *(0(),y) = [4] y + [8] >= [4] = 0() +Full(0(),y) = [2] y + [2] >= [1] y + [0] = y +Full(S(x),y) = [2] y + [2] >= [2] = +Full(x,S(y)) goal(xs) = [7] xs + [2] >= [7] xs + [2] = map(xs) map(Cons(x,xs)) = [7] x + [7] xs + [30] >= [7] x + [7] xs + [7] = Cons(f(x),map(xs)) map(Nil()) = [2] >= [0] = Nil() * Step 4: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: +Full(S(x),y) -> +Full(x,S(y)) - Weak TRS: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() +Full(0(),y) -> y f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() - Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,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(+) = {2}, uargs(Cons) = {1,2} Following symbols are considered usable: {*,+Full,f,goal,map} TcT has computed the following interpretation: p(*) = [1] x1 + [0] p(+) = [1] x2 + [0] p(+Full) = [8] x1 + [4] x2 + [0] p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(Nil) = [1] p(S) = [1] x1 + [2] p(f) = [2] x1 + [0] p(goal) = [3] x1 + [2] p(map) = [2] x1 + [2] Following rules are strictly oriented: +Full(S(x),y) = [8] x + [4] y + [16] > [8] x + [4] y + [8] = +Full(x,S(y)) Following rules are (at-least) weakly oriented: *(x,0()) = [1] x + [0] >= [0] = 0() *(x,S(0())) = [1] x + [0] >= [1] x + [0] = x *(x,S(S(y))) = [1] x + [0] >= [1] x + [0] = +(x,*(x,S(y))) *(0(),y) = [0] >= [0] = 0() +Full(0(),y) = [4] y + [0] >= [1] y + [0] = y f(x) = [2] x + [0] >= [1] x + [0] = *(x,x) goal(xs) = [3] xs + [2] >= [2] xs + [2] = map(xs) map(Cons(x,xs)) = [2] x + [2] xs + [2] >= [2] x + [2] xs + [2] = Cons(f(x),map(xs)) map(Nil()) = [4] >= [1] = Nil() * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() +Full(0(),y) -> y +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() - Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,S} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))