WORST_CASE(?,O(n^3)) * Step 1: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + 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__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = 0 p(a__adx) = x1 p(a__hd) = x1 p(a__incr) = x1 p(a__nats) = 1 p(a__tl) = x1 p(a__zeros) = 1 p(adx) = x1 p(cons) = 1 p(hd) = x1 p(incr) = x1 p(mark) = 1 p(nats) = 0 p(s) = 0 p(tl) = x1 p(zeros) = 1 Following rules are strictly oriented: a__nats() = 1 > 0 = nats() mark(0()) = 1 > 0 = 0() mark(s(X)) = 1 > 0 = s(X) Following rules are (at-least) weakly oriented: a__adx(X) = X >= X = adx(X) a__adx(cons(X,Y)) = 1 >= 1 = a__incr(cons(X,adx(Y))) a__hd(X) = X >= X = hd(X) a__hd(cons(X,Y)) = 1 >= 1 = mark(X) a__incr(X) = X >= X = incr(X) a__incr(cons(X,Y)) = 1 >= 1 = cons(s(X),incr(Y)) a__nats() = 1 >= 1 = a__adx(a__zeros()) a__tl(X) = X >= X = tl(X) a__tl(cons(X,Y)) = 1 >= 1 = mark(Y) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__zeros() = 1 >= 1 = zeros() mark(adx(X)) = 1 >= 1 = a__adx(mark(X)) mark(cons(X1,X2)) = 1 >= 1 = cons(X1,X2) mark(hd(X)) = 1 >= 1 = a__hd(mark(X)) mark(incr(X)) = 1 >= 1 = a__incr(mark(X)) mark(nats()) = 1 >= 1 = a__nats() mark(tl(X)) = 1 >= 1 = a__tl(mark(X)) mark(zeros()) = 1 >= 1 = a__zeros() * Step 2: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__nats() -> a__adx(a__zeros()) a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__nats() -> nats() mark(0()) -> 0() mark(s(X)) -> s(X) - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + 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__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = 0 p(a__adx) = x1 p(a__hd) = x1 p(a__incr) = x1 p(a__nats) = 1 p(a__tl) = x1 p(a__zeros) = 1 p(adx) = 0 p(cons) = 1 p(hd) = 0 p(incr) = x1 p(mark) = 1 p(nats) = 0 p(s) = 1 p(tl) = 0 p(zeros) = 0 Following rules are strictly oriented: a__zeros() = 1 > 0 = zeros() Following rules are (at-least) weakly oriented: a__adx(X) = X >= 0 = adx(X) a__adx(cons(X,Y)) = 1 >= 1 = a__incr(cons(X,adx(Y))) a__hd(X) = X >= 0 = hd(X) a__hd(cons(X,Y)) = 1 >= 1 = mark(X) a__incr(X) = X >= X = incr(X) a__incr(cons(X,Y)) = 1 >= 1 = cons(s(X),incr(Y)) a__nats() = 1 >= 1 = a__adx(a__zeros()) a__nats() = 1 >= 0 = nats() a__tl(X) = X >= 0 = tl(X) a__tl(cons(X,Y)) = 1 >= 1 = mark(Y) a__zeros() = 1 >= 1 = cons(0(),zeros()) mark(0()) = 1 >= 0 = 0() mark(adx(X)) = 1 >= 1 = a__adx(mark(X)) mark(cons(X1,X2)) = 1 >= 1 = cons(X1,X2) mark(hd(X)) = 1 >= 1 = a__hd(mark(X)) mark(incr(X)) = 1 >= 1 = a__incr(mark(X)) mark(nats()) = 1 >= 1 = a__nats() mark(s(X)) = 1 >= 1 = s(X) mark(tl(X)) = 1 >= 1 = a__tl(mark(X)) mark(zeros()) = 1 >= 1 = a__zeros() * Step 3: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__nats() -> a__adx(a__zeros()) a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__nats() -> nats() a__zeros() -> zeros() mark(0()) -> 0() mark(s(X)) -> s(X) - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + 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__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = 0 p(a__adx) = x1 p(a__hd) = x1 p(a__incr) = x1 p(a__nats) = 2 p(a__tl) = 4 + x1 p(a__zeros) = 2 p(adx) = x1 p(cons) = x1 + x2 p(hd) = x1 p(incr) = x1 p(mark) = x1 p(nats) = 2 p(s) = x1 p(tl) = 4 + x1 p(zeros) = 2 Following rules are strictly oriented: a__tl(cons(X,Y)) = 4 + X + Y > Y = mark(Y) Following rules are (at-least) weakly oriented: a__adx(X) = X >= X = adx(X) a__adx(cons(X,Y)) = X + Y >= X + Y = a__incr(cons(X,adx(Y))) a__hd(X) = X >= X = hd(X) a__hd(cons(X,Y)) = X + Y >= X = mark(X) a__incr(X) = X >= X = incr(X) a__incr(cons(X,Y)) = X + Y >= X + Y = cons(s(X),incr(Y)) a__nats() = 2 >= 2 = a__adx(a__zeros()) a__nats() = 2 >= 2 = nats() a__tl(X) = 4 + X >= 4 + X = tl(X) a__zeros() = 2 >= 2 = cons(0(),zeros()) a__zeros() = 2 >= 2 = zeros() mark(0()) = 0 >= 0 = 0() mark(adx(X)) = X >= X = a__adx(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) mark(hd(X)) = X >= X = a__hd(mark(X)) mark(incr(X)) = X >= X = a__incr(mark(X)) mark(nats()) = 2 >= 2 = a__nats() mark(s(X)) = X >= X = s(X) mark(tl(X)) = 4 + X >= 4 + X = a__tl(mark(X)) mark(zeros()) = 2 >= 2 = a__zeros() * Step 4: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__nats() -> a__adx(a__zeros()) a__tl(X) -> tl(X) a__zeros() -> cons(0(),zeros()) mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__nats() -> nats() a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> zeros() mark(0()) -> 0() mark(s(X)) -> s(X) - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + 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__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = 0 p(a__adx) = x1 p(a__hd) = 4 + x1 p(a__incr) = x1 p(a__nats) = 2 p(a__tl) = 4 + x1 p(a__zeros) = 0 p(adx) = x1 p(cons) = x1 + x2 p(hd) = 4 + x1 p(incr) = x1 p(mark) = x1 p(nats) = 2 p(s) = x1 p(tl) = 4 + x1 p(zeros) = 0 Following rules are strictly oriented: a__hd(cons(X,Y)) = 4 + X + Y > X = mark(X) a__nats() = 2 > 0 = a__adx(a__zeros()) Following rules are (at-least) weakly oriented: a__adx(X) = X >= X = adx(X) a__adx(cons(X,Y)) = X + Y >= X + Y = a__incr(cons(X,adx(Y))) a__hd(X) = 4 + X >= 4 + X = hd(X) a__incr(X) = X >= X = incr(X) a__incr(cons(X,Y)) = X + Y >= X + Y = cons(s(X),incr(Y)) a__nats() = 2 >= 2 = nats() a__tl(X) = 4 + X >= 4 + X = tl(X) a__tl(cons(X,Y)) = 4 + X + Y >= Y = mark(Y) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__zeros() = 0 >= 0 = zeros() mark(0()) = 0 >= 0 = 0() mark(adx(X)) = X >= X = a__adx(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) mark(hd(X)) = 4 + X >= 4 + X = a__hd(mark(X)) mark(incr(X)) = X >= X = a__incr(mark(X)) mark(nats()) = 2 >= 2 = a__nats() mark(s(X)) = X >= X = s(X) mark(tl(X)) = 4 + X >= 4 + X = a__tl(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() * Step 5: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__tl(X) -> tl(X) a__zeros() -> cons(0(),zeros()) mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__hd(cons(X,Y)) -> mark(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> zeros() mark(0()) -> 0() mark(s(X)) -> s(X) - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + 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__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] p(a__adx) = [1] x1 + [7] p(a__hd) = [1] x1 + [4] p(a__incr) = [1] x1 + [0] p(a__nats) = [7] p(a__tl) = [1] x1 + [4] p(a__zeros) = [0] p(adx) = [1] x1 + [7] p(cons) = [1] x1 + [1] x2 + [0] p(hd) = [1] x1 + [4] p(incr) = [1] x1 + [0] p(mark) = [1] x1 + [1] p(nats) = [6] p(s) = [0] p(tl) = [1] x1 + [4] p(zeros) = [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = cons(X1,X2) mark(zeros()) = [1] > [0] = a__zeros() Following rules are (at-least) weakly oriented: a__adx(X) = [1] X + [7] >= [1] X + [7] = adx(X) a__adx(cons(X,Y)) = [1] X + [1] Y + [7] >= [1] X + [1] Y + [7] = a__incr(cons(X,adx(Y))) a__hd(X) = [1] X + [4] >= [1] X + [4] = hd(X) a__hd(cons(X,Y)) = [1] X + [1] Y + [4] >= [1] X + [1] = mark(X) a__incr(X) = [1] X + [0] >= [1] X + [0] = incr(X) a__incr(cons(X,Y)) = [1] X + [1] Y + [0] >= [1] Y + [0] = cons(s(X),incr(Y)) a__nats() = [7] >= [7] = a__adx(a__zeros()) a__nats() = [7] >= [6] = nats() a__tl(X) = [1] X + [4] >= [1] X + [4] = tl(X) a__tl(cons(X,Y)) = [1] X + [1] Y + [4] >= [1] Y + [1] = mark(Y) a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [1] >= [0] = 0() mark(adx(X)) = [1] X + [8] >= [1] X + [8] = a__adx(mark(X)) mark(hd(X)) = [1] X + [5] >= [1] X + [5] = a__hd(mark(X)) mark(incr(X)) = [1] X + [1] >= [1] X + [1] = a__incr(mark(X)) mark(nats()) = [7] >= [7] = a__nats() mark(s(X)) = [1] >= [0] = s(X) mark(tl(X)) = [1] X + [5] >= [1] X + [5] = a__tl(mark(X)) * Step 6: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__tl(X) -> tl(X) a__zeros() -> cons(0(),zeros()) mark(adx(X)) -> a__adx(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(tl(X)) -> a__tl(mark(X)) - Weak TRS: a__hd(cons(X,Y)) -> mark(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> zeros() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(s(X)) -> s(X) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + 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__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = 0 p(a__adx) = x1 p(a__hd) = 1 + x1 p(a__incr) = x1 p(a__nats) = 3 p(a__tl) = 7 + x1 p(a__zeros) = 1 p(adx) = x1 p(cons) = x1 + x2 p(hd) = 1 + x1 p(incr) = x1 p(mark) = 1 + x1 p(nats) = 3 p(s) = 0 p(tl) = 7 + x1 p(zeros) = 0 Following rules are strictly oriented: a__zeros() = 1 > 0 = cons(0(),zeros()) mark(nats()) = 4 > 3 = a__nats() Following rules are (at-least) weakly oriented: a__adx(X) = X >= X = adx(X) a__adx(cons(X,Y)) = X + Y >= X + Y = a__incr(cons(X,adx(Y))) a__hd(X) = 1 + X >= 1 + X = hd(X) a__hd(cons(X,Y)) = 1 + X + Y >= 1 + X = mark(X) a__incr(X) = X >= X = incr(X) a__incr(cons(X,Y)) = X + Y >= Y = cons(s(X),incr(Y)) a__nats() = 3 >= 1 = a__adx(a__zeros()) a__nats() = 3 >= 3 = nats() a__tl(X) = 7 + X >= 7 + X = tl(X) a__tl(cons(X,Y)) = 7 + X + Y >= 1 + Y = mark(Y) a__zeros() = 1 >= 0 = zeros() mark(0()) = 1 >= 0 = 0() mark(adx(X)) = 1 + X >= 1 + X = a__adx(mark(X)) mark(cons(X1,X2)) = 1 + X1 + X2 >= X1 + X2 = cons(X1,X2) mark(hd(X)) = 2 + X >= 2 + X = a__hd(mark(X)) mark(incr(X)) = 1 + X >= 1 + X = a__incr(mark(X)) mark(s(X)) = 1 >= 0 = s(X) mark(tl(X)) = 8 + X >= 8 + X = a__tl(mark(X)) mark(zeros()) = 1 >= 1 = a__zeros() * Step 7: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__tl(X) -> tl(X) mark(adx(X)) -> a__adx(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(tl(X)) -> a__tl(mark(X)) - Weak TRS: a__hd(cons(X,Y)) -> mark(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__adx) = [1 0 0] [0] [0 1 0] x1 + [0] [0 1 0] [0] p(a__hd) = [1 0 0] [3] [0 1 0] x1 + [2] [0 2 0] [1] p(a__incr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 1 0] [0] p(a__nats) = [0] [0] [1] p(a__tl) = [1 0 0] [0] [0 1 0] x1 + [0] [0 2 0] [1] p(a__zeros) = [0] [0] [0] p(adx) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(cons) = [0 2 0] [0 2 0] [0] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 0] [0 0 0] [0] p(hd) = [0 0 0] [1] [0 1 0] x1 + [2] [0 0 0] [0] p(incr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [0 2 0] [0] [0 1 0] x1 + [0] [0 2 0] [1] p(nats) = [0] [0] [1] p(s) = [0 2 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(tl) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(zeros) = [0] [0] [0] Following rules are strictly oriented: a__hd(X) = [1 0 0] [3] [0 1 0] X + [2] [0 2 0] [1] > [0 0 0] [1] [0 1 0] X + [2] [0 0 0] [0] = hd(X) mark(hd(X)) = [0 2 0] [4] [0 1 0] X + [2] [0 2 0] [5] > [0 2 0] [3] [0 1 0] X + [2] [0 2 0] [1] = a__hd(mark(X)) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0 0] [0] [0 1 0] X + [0] [0 1 0] [0] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = adx(X) a__adx(cons(X,Y)) = [0 2 0] [0 2 0] [0] [0 1 0] X + [0 1 1] Y + [0] [0 1 0] [0 1 1] [0] >= [0 2 0] [0 2 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 1 0] [0 1 0] [0] = a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) = [0 2 0] [0 2 0] [3] [0 1 0] X + [0 1 1] Y + [2] [0 2 0] [0 2 2] [1] >= [0 2 0] [0] [0 1 0] X + [0] [0 2 0] [1] = mark(X) a__incr(X) = [1 0 0] [0] [0 1 0] X + [0] [0 1 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = incr(X) a__incr(cons(X,Y)) = [0 2 0] [0 2 0] [0] [0 1 0] X + [0 1 1] Y + [0] [0 1 0] [0 1 1] [0] >= [0 2 0] [0 2 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 0] [0 0 0] [0] = cons(s(X),incr(Y)) a__nats() = [0] [0] [1] >= [0] [0] [0] = a__adx(a__zeros()) a__nats() = [0] [0] [1] >= [0] [0] [1] = nats() a__tl(X) = [1 0 0] [0] [0 1 0] X + [0] [0 2 0] [1] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [1] = tl(X) a__tl(cons(X,Y)) = [0 2 0] [0 2 0] [0] [0 1 0] X + [0 1 1] Y + [0] [0 2 0] [0 2 2] [1] >= [0 2 0] [0] [0 1 0] Y + [0] [0 2 0] [1] = mark(Y) a__zeros() = [0] [0] [0] >= [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [0] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [1] >= [0] [0] [0] = 0() mark(adx(X)) = [0 2 0] [0] [0 1 0] X + [0] [0 2 0] [1] >= [0 2 0] [0] [0 1 0] X + [0] [0 1 0] [0] = a__adx(mark(X)) mark(cons(X1,X2)) = [0 2 0] [0 2 2] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [0 2 0] [0 2 2] [1] >= [0 2 0] [0 2 0] [0] [0 1 0] X1 + [0 1 1] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) mark(incr(X)) = [0 2 0] [0] [0 1 0] X + [0] [0 2 0] [1] >= [0 2 0] [0] [0 1 0] X + [0] [0 1 0] [0] = a__incr(mark(X)) mark(nats()) = [0] [0] [1] >= [0] [0] [1] = a__nats() mark(s(X)) = [0 2 0] [0] [0 1 0] X + [0] [0 2 0] [1] >= [0 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] = s(X) mark(tl(X)) = [0 2 0] [0] [0 1 0] X + [0] [0 2 0] [1] >= [0 2 0] [0] [0 1 0] X + [0] [0 2 0] [1] = a__tl(mark(X)) mark(zeros()) = [0] [0] [1] >= [0] [0] [0] = a__zeros() * Step 8: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__tl(X) -> tl(X) mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(tl(X)) -> a__tl(mark(X)) - Weak TRS: a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__adx) = [1 0 0] [0] [0 2 0] x1 + [0] [0 0 1] [0] p(a__hd) = [1 0 0] [2] [0 0 0] x1 + [0] [0 0 1] [2] p(a__incr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__nats) = [0] [0] [0] p(a__tl) = [1 0 0] [2] [0 0 0] x1 + [0] [0 0 1] [2] p(a__zeros) = [0] [0] [0] p(adx) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(cons) = [0 0 2] [0 0 2] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(hd) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [2] p(incr) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [0 0 2] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nats) = [0] [0] [0] p(s) = [0] [0] [0] p(tl) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [2] p(zeros) = [0] [0] [0] Following rules are strictly oriented: a__tl(X) = [1 0 0] [2] [0 0 0] X + [0] [0 0 1] [2] > [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [2] = tl(X) mark(tl(X)) = [0 0 2] [4] [0 0 0] X + [0] [0 0 1] [2] > [0 0 2] [2] [0 0 0] X + [0] [0 0 1] [2] = a__tl(mark(X)) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0 0] [0] [0 2 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = adx(X) a__adx(cons(X,Y)) = [0 0 2] [0 0 2] [0] [0 0 0] X + [0 0 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 0 2] [0 0 2] [0] [0 0 0] X + [0 0 0] Y + [0] [0 0 1] [0 0 1] [0] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0 0] [2] [0 0 0] X + [0] [0 0 1] [2] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [2] = hd(X) a__hd(cons(X,Y)) = [0 0 2] [0 0 2] [2] [0 0 0] X + [0 0 0] Y + [0] [0 0 1] [0 0 1] [2] >= [0 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__incr(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = incr(X) a__incr(cons(X,Y)) = [0 0 2] [0 0 2] [0] [0 0 0] X + [0 0 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 0 2] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(s(X),incr(Y)) a__nats() = [0] [0] [0] >= [0] [0] [0] = a__adx(a__zeros()) a__nats() = [0] [0] [0] >= [0] [0] [0] = nats() a__tl(cons(X,Y)) = [0 0 2] [0 0 2] [2] [0 0 0] X + [0 0 0] Y + [0] [0 0 1] [0 0 1] [2] >= [0 0 2] [0] [0 0 0] Y + [0] [0 0 1] [0] = mark(Y) a__zeros() = [0] [0] [0] >= [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [0] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(adx(X)) = [0 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = a__adx(mark(X)) mark(cons(X1,X2)) = [0 0 2] [0 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 0 2] [0 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = cons(X1,X2) mark(hd(X)) = [0 0 2] [4] [0 0 0] X + [0] [0 0 1] [2] >= [0 0 2] [2] [0 0 0] X + [0] [0 0 1] [2] = a__hd(mark(X)) mark(incr(X)) = [0 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = a__incr(mark(X)) mark(nats()) = [0] [0] [0] >= [0] [0] [0] = a__nats() mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) mark(zeros()) = [0] [0] [0] >= [0] [0] [0] = a__zeros() * Step 9: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) - Weak TRS: a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = [2] [0] [0] p(a__adx) = [1 0 0] [2] [0 1 0] x1 + [2] [0 0 0] [0] p(a__hd) = [1 0 0] [3] [0 1 1] x1 + [3] [0 0 0] [0] p(a__incr) = [1 0 0] [0] [0 1 2] x1 + [0] [0 0 0] [0] p(a__nats) = [2] [2] [0] p(a__tl) = [1 0 0] [2] [0 1 0] x1 + [3] [0 0 0] [0] p(a__zeros) = [0] [0] [0] p(adx) = [1 0 0] [0] [0 1 0] x1 + [2] [0 0 0] [0] p(cons) = [0 1 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(hd) = [0 0 0] [2] [0 1 0] x1 + [3] [0 0 0] [0] p(incr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [0 1 0] [2] [0 1 0] x1 + [2] [0 0 0] [0] p(nats) = [0] [0] [0] p(s) = [0] [0] [0] p(tl) = [1 0 0] [2] [0 1 0] x1 + [3] [0 0 0] [0] p(zeros) = [0] [0] [0] Following rules are strictly oriented: a__adx(X) = [1 0 0] [2] [0 1 0] X + [2] [0 0 0] [0] > [1 0 0] [0] [0 1 0] X + [2] [0 0 0] [0] = adx(X) Following rules are (at-least) weakly oriented: a__adx(cons(X,Y)) = [0 1 0] [0 1 0] [2] [0 1 0] X + [0 1 0] Y + [2] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 1 0] [2] [0 1 0] X + [0 1 0] Y + [2] [0 0 0] [0 0 0] [0] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0 0] [3] [0 1 1] X + [3] [0 0 0] [0] >= [0 0 0] [2] [0 1 0] X + [3] [0 0 0] [0] = hd(X) a__hd(cons(X,Y)) = [0 1 0] [0 1 0] [3] [0 1 0] X + [0 1 0] Y + [3] [0 0 0] [0 0 0] [0] >= [0 1 0] [2] [0 1 0] X + [2] [0 0 0] [0] = mark(X) a__incr(X) = [1 0 0] [0] [0 1 2] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = incr(X) a__incr(cons(X,Y)) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 0] [0 0 0] [0] >= [0 1 0] [0] [0 1 0] Y + [0] [0 0 0] [0] = cons(s(X),incr(Y)) a__nats() = [2] [2] [0] >= [2] [2] [0] = a__adx(a__zeros()) a__nats() = [2] [2] [0] >= [0] [0] [0] = nats() a__tl(X) = [1 0 0] [2] [0 1 0] X + [3] [0 0 0] [0] >= [1 0 0] [2] [0 1 0] X + [3] [0 0 0] [0] = tl(X) a__tl(cons(X,Y)) = [0 1 0] [0 1 0] [2] [0 1 0] X + [0 1 0] Y + [3] [0 0 0] [0 0 0] [0] >= [0 1 0] [2] [0 1 0] Y + [2] [0 0 0] [0] = mark(Y) a__zeros() = [0] [0] [0] >= [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [0] >= [0] [0] [0] = zeros() mark(0()) = [2] [2] [0] >= [2] [0] [0] = 0() mark(adx(X)) = [0 1 0] [4] [0 1 0] X + [4] [0 0 0] [0] >= [0 1 0] [4] [0 1 0] X + [4] [0 0 0] [0] = a__adx(mark(X)) mark(cons(X1,X2)) = [0 1 0] [0 1 0] [2] [0 1 0] X1 + [0 1 0] X2 + [2] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) mark(hd(X)) = [0 1 0] [5] [0 1 0] X + [5] [0 0 0] [0] >= [0 1 0] [5] [0 1 0] X + [5] [0 0 0] [0] = a__hd(mark(X)) mark(incr(X)) = [0 1 0] [2] [0 1 0] X + [2] [0 0 0] [0] >= [0 1 0] [2] [0 1 0] X + [2] [0 0 0] [0] = a__incr(mark(X)) mark(nats()) = [2] [2] [0] >= [2] [2] [0] = a__nats() mark(s(X)) = [2] [2] [0] >= [0] [0] [0] = s(X) mark(tl(X)) = [0 1 0] [5] [0 1 0] X + [5] [0 0 0] [0] >= [0 1 0] [4] [0 1 0] X + [5] [0 0 0] [0] = a__tl(mark(X)) mark(zeros()) = [2] [2] [0] >= [0] [0] [0] = a__zeros() * Step 10: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) - Weak TRS: a__adx(X) -> adx(X) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__adx) = [1 0 2] [0] [0 1 2] x1 + [0] [0 0 1] [2] p(a__hd) = [1 0 1] [0] [0 1 2] x1 + [0] [0 0 1] [2] p(a__incr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__nats) = [2] [1] [3] p(a__tl) = [1 0 2] [2] [0 1 3] x1 + [0] [0 0 1] [2] p(a__zeros) = [0] [1] [0] p(adx) = [1 0 0] [0] [0 1 2] x1 + [0] [0 0 1] [2] p(cons) = [0 1 0] [0 1 0] [0] [0 1 2] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 1] [0] p(hd) = [1 0 1] [0] [0 1 2] x1 + [0] [0 0 1] [2] p(incr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(mark) = [0 1 1] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(nats) = [0] [0] [3] p(s) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(tl) = [1 0 0] [0] [0 1 3] x1 + [0] [0 0 1] [2] p(zeros) = [0] [0] [0] Following rules are strictly oriented: mark(adx(X)) = [0 1 3] [2] [0 1 2] X + [1] [0 0 1] [2] > [0 1 3] [0] [0 1 2] X + [1] [0 0 1] [2] = a__adx(mark(X)) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0 2] [0] [0 1 2] X + [0] [0 0 1] [2] >= [1 0 0] [0] [0 1 2] X + [0] [0 0 1] [2] = adx(X) a__adx(cons(X,Y)) = [0 1 2] [0 1 2] [0] [0 1 4] X + [0 1 2] Y + [1] [0 0 1] [0 0 1] [2] >= [0 1 0] [0 1 2] [0] [0 1 2] X + [0 1 2] Y + [1] [0 0 1] [0 0 1] [2] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0 1] [0] [0 1 2] X + [0] [0 0 1] [2] >= [1 0 1] [0] [0 1 2] X + [0] [0 0 1] [2] = hd(X) a__hd(cons(X,Y)) = [0 1 1] [0 1 1] [0] [0 1 4] X + [0 1 2] Y + [1] [0 0 1] [0 0 1] [2] >= [0 1 1] [0] [0 1 0] X + [1] [0 0 1] [0] = mark(X) a__incr(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] = incr(X) a__incr(cons(X,Y)) = [0 1 0] [0 1 0] [0] [0 1 2] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [1] [0 0 0] [0 0 1] [0] = cons(s(X),incr(Y)) a__nats() = [2] [1] [3] >= [0] [1] [2] = a__adx(a__zeros()) a__nats() = [2] [1] [3] >= [0] [0] [3] = nats() a__tl(X) = [1 0 2] [2] [0 1 3] X + [0] [0 0 1] [2] >= [1 0 0] [0] [0 1 3] X + [0] [0 0 1] [2] = tl(X) a__tl(cons(X,Y)) = [0 1 2] [0 1 2] [2] [0 1 5] X + [0 1 3] Y + [1] [0 0 1] [0 0 1] [2] >= [0 1 1] [0] [0 1 0] Y + [1] [0 0 1] [0] = mark(Y) a__zeros() = [0] [1] [0] >= [0] [1] [0] = cons(0(),zeros()) a__zeros() = [0] [1] [0] >= [0] [0] [0] = zeros() mark(0()) = [0] [1] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [0 1 3] [0 1 1] [1] [0 1 2] X1 + [0 1 0] X2 + [2] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0] [0 1 2] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 1] [0] = cons(X1,X2) mark(hd(X)) = [0 1 3] [2] [0 1 2] X + [1] [0 0 1] [2] >= [0 1 2] [0] [0 1 2] X + [1] [0 0 1] [2] = a__hd(mark(X)) mark(incr(X)) = [0 1 1] [0] [0 1 0] X + [1] [0 0 1] [0] >= [0 1 1] [0] [0 1 0] X + [1] [0 0 1] [0] = a__incr(mark(X)) mark(nats()) = [3] [1] [3] >= [2] [1] [3] = a__nats() mark(s(X)) = [0 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = s(X) mark(tl(X)) = [0 1 4] [2] [0 1 3] X + [1] [0 0 1] [2] >= [0 1 3] [2] [0 1 3] X + [1] [0 0 1] [2] = a__tl(mark(X)) mark(zeros()) = [0] [1] [0] >= [0] [1] [0] = a__zeros() * Step 11: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) mark(incr(X)) -> a__incr(mark(X)) - Weak TRS: a__adx(X) -> adx(X) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__adx) = [1 1 1] [0] [0 0 0] x1 + [2] [0 0 1] [1] p(a__hd) = [1 1 2] [0] [0 0 0] x1 + [2] [0 0 1] [2] p(a__incr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__nats) = [3] [2] [3] p(a__tl) = [1 1 2] [0] [0 0 0] x1 + [2] [0 0 1] [3] p(a__zeros) = [0] [1] [2] p(adx) = [1 1 1] [0] [0 0 0] x1 + [2] [0 0 1] [1] p(cons) = [1 2 2] [1 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [2] p(hd) = [1 1 2] [0] [0 0 0] x1 + [2] [0 0 1] [2] p(incr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(mark) = [1 1 2] [0] [0 0 0] x1 + [2] [0 0 1] [2] p(nats) = [0] [2] [2] p(s) = [0 2 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tl) = [1 1 2] [0] [0 0 0] x1 + [0] [0 0 1] [3] p(zeros) = [0] [0] [0] Following rules are strictly oriented: a__adx(cons(X,Y)) = [1 2 3] [1 1 1] [3] [0 0 0] X + [0 0 0] Y + [2] [0 0 1] [0 0 1] [3] > [1 2 2] [1 1 1] [2] [0 0 0] X + [0 0 0] Y + [1] [0 0 1] [0 0 1] [3] = a__incr(cons(X,adx(Y))) Following rules are (at-least) weakly oriented: a__adx(X) = [1 1 1] [0] [0 0 0] X + [2] [0 0 1] [1] >= [1 1 1] [0] [0 0 0] X + [2] [0 0 1] [1] = adx(X) a__hd(X) = [1 1 2] [0] [0 0 0] X + [2] [0 0 1] [2] >= [1 1 2] [0] [0 0 0] X + [2] [0 0 1] [2] = hd(X) a__hd(cons(X,Y)) = [1 2 4] [1 1 2] [5] [0 0 0] X + [0 0 0] Y + [2] [0 0 1] [0 0 1] [4] >= [1 1 2] [0] [0 0 0] X + [2] [0 0 1] [2] = mark(X) a__incr(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] = incr(X) a__incr(cons(X,Y)) = [1 2 2] [1 1 0] [0] [0 0 0] X + [0 0 0] Y + [1] [0 0 1] [0 0 1] [2] >= [0 2 2] [1 1 0] [0] [0 0 0] X + [0 0 0] Y + [1] [0 0 1] [0 0 1] [2] = cons(s(X),incr(Y)) a__nats() = [3] [2] [3] >= [3] [2] [3] = a__adx(a__zeros()) a__nats() = [3] [2] [3] >= [0] [2] [2] = nats() a__tl(X) = [1 1 2] [0] [0 0 0] X + [2] [0 0 1] [3] >= [1 1 2] [0] [0 0 0] X + [0] [0 0 1] [3] = tl(X) a__tl(cons(X,Y)) = [1 2 4] [1 1 2] [5] [0 0 0] X + [0 0 0] Y + [2] [0 0 1] [0 0 1] [5] >= [1 1 2] [0] [0 0 0] Y + [2] [0 0 1] [2] = mark(Y) a__zeros() = [0] [1] [2] >= [0] [1] [2] = cons(0(),zeros()) a__zeros() = [0] [1] [2] >= [0] [0] [0] = zeros() mark(0()) = [0] [2] [2] >= [0] [0] [0] = 0() mark(adx(X)) = [1 1 3] [4] [0 0 0] X + [2] [0 0 1] [3] >= [1 1 3] [4] [0 0 0] X + [2] [0 0 1] [3] = a__adx(mark(X)) mark(cons(X1,X2)) = [1 2 4] [1 1 2] [5] [0 0 0] X1 + [0 0 0] X2 + [2] [0 0 1] [0 0 1] [4] >= [1 2 2] [1 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [2] = cons(X1,X2) mark(hd(X)) = [1 1 4] [6] [0 0 0] X + [2] [0 0 1] [4] >= [1 1 4] [6] [0 0 0] X + [2] [0 0 1] [4] = a__hd(mark(X)) mark(incr(X)) = [1 1 2] [0] [0 0 0] X + [2] [0 0 1] [2] >= [1 1 2] [0] [0 0 0] X + [2] [0 0 1] [2] = a__incr(mark(X)) mark(nats()) = [6] [2] [4] >= [3] [2] [3] = a__nats() mark(s(X)) = [0 2 2] [0] [0 0 0] X + [2] [0 0 1] [2] >= [0 2 0] [0] [0 0 0] X + [0] [0 0 1] [0] = s(X) mark(tl(X)) = [1 1 4] [6] [0 0 0] X + [2] [0 0 1] [5] >= [1 1 4] [6] [0 0 0] X + [2] [0 0 1] [5] = a__tl(mark(X)) mark(zeros()) = [0] [2] [2] >= [0] [1] [2] = a__zeros() * Step 12: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) mark(incr(X)) -> a__incr(mark(X)) - Weak TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__adx) = [1 0 0] [1] [0 1 0] x1 + [1] [0 0 1] [2] p(a__hd) = [1 0 0] [0] [0 1 2] x1 + [0] [0 0 1] [0] p(a__incr) = [1 0 0] [1] [0 1 0] x1 + [1] [0 0 1] [0] p(a__nats) = [2] [2] [3] p(a__tl) = [1 0 2] [2] [0 1 2] x1 + [0] [0 0 1] [1] p(a__zeros) = [0] [0] [0] p(adx) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [2] p(cons) = [0 1 2] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(hd) = [1 0 0] [0] [0 1 2] x1 + [0] [0 0 1] [0] p(incr) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(mark) = [0 1 2] [0] [0 1 2] x1 + [0] [0 0 1] [0] p(nats) = [0] [0] [3] p(s) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(tl) = [1 0 2] [0] [0 1 2] x1 + [0] [0 0 1] [1] p(zeros) = [0] [0] [0] Following rules are strictly oriented: a__incr(X) = [1 0 0] [1] [0 1 0] X + [1] [0 0 1] [0] > [1 0 0] [0] [0 1 0] X + [1] [0 0 1] [0] = incr(X) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0 0] [1] [0 1 0] X + [1] [0 0 1] [2] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [2] = adx(X) a__adx(cons(X,Y)) = [0 1 2] [0 1 0] [1] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [2] >= [0 1 2] [0 1 0] [1] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [2] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0 0] [0] [0 1 2] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 2] X + [0] [0 0 1] [0] = hd(X) a__hd(cons(X,Y)) = [0 1 2] [0 1 0] [0] [0 1 2] X + [0 1 2] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 2] [0] [0 1 2] X + [0] [0 0 1] [0] = mark(X) a__incr(cons(X,Y)) = [0 1 2] [0 1 0] [1] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [1] [0 1 0] X + [0 1 0] Y + [1] [0 0 0] [0 0 1] [0] = cons(s(X),incr(Y)) a__nats() = [2] [2] [3] >= [1] [1] [2] = a__adx(a__zeros()) a__nats() = [2] [2] [3] >= [0] [0] [3] = nats() a__tl(X) = [1 0 2] [2] [0 1 2] X + [0] [0 0 1] [1] >= [1 0 2] [0] [0 1 2] X + [0] [0 0 1] [1] = tl(X) a__tl(cons(X,Y)) = [0 1 4] [0 1 2] [2] [0 1 2] X + [0 1 2] Y + [0] [0 0 1] [0 0 1] [1] >= [0 1 2] [0] [0 1 2] Y + [0] [0 0 1] [0] = mark(Y) a__zeros() = [0] [0] [0] >= [0] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] [0] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(adx(X)) = [0 1 2] [4] [0 1 2] X + [4] [0 0 1] [2] >= [0 1 2] [1] [0 1 2] X + [1] [0 0 1] [2] = a__adx(mark(X)) mark(cons(X1,X2)) = [0 1 2] [0 1 2] [0] [0 1 2] X1 + [0 1 2] X2 + [0] [0 0 1] [0 0 1] [0] >= [0 1 2] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = cons(X1,X2) mark(hd(X)) = [0 1 4] [0] [0 1 4] X + [0] [0 0 1] [0] >= [0 1 2] [0] [0 1 4] X + [0] [0 0 1] [0] = a__hd(mark(X)) mark(incr(X)) = [0 1 2] [1] [0 1 2] X + [1] [0 0 1] [0] >= [0 1 2] [1] [0 1 2] X + [1] [0 0 1] [0] = a__incr(mark(X)) mark(nats()) = [6] [6] [3] >= [2] [2] [3] = a__nats() mark(s(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = s(X) mark(tl(X)) = [0 1 4] [2] [0 1 4] X + [2] [0 0 1] [1] >= [0 1 4] [2] [0 1 4] X + [0] [0 0 1] [1] = a__tl(mark(X)) mark(zeros()) = [0] [0] [0] >= [0] [0] [0] = a__zeros() * Step 13: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) mark(incr(X)) -> a__incr(mark(X)) - Weak TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__incr(X) -> incr(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__adx) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [1] p(a__hd) = [1 0 0] [3] [0 1 0] x1 + [1] [0 0 1] [0] p(a__incr) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(a__nats) = [1] [3] [1] p(a__tl) = [1 2 0] [1] [0 1 1] x1 + [0] [0 0 1] [3] p(a__zeros) = [1] [2] [0] p(adx) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(cons) = [1 2 0] [1 0 0] [1] [0 1 2] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(hd) = [1 0 0] [2] [0 1 0] x1 + [1] [0 0 1] [0] p(incr) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(mark) = [1 2 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(nats) = [0] [3] [1] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tl) = [1 2 0] [1] [0 1 1] x1 + [0] [0 0 1] [3] p(zeros) = [0] [2] [0] Following rules are strictly oriented: mark(incr(X)) = [1 2 0] [2] [0 1 1] X + [1] [0 0 1] [0] > [1 2 0] [0] [0 1 1] X + [1] [0 0 1] [0] = a__incr(mark(X)) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0 0] [0] [0 1 0] X + [1] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] = adx(X) a__adx(cons(X,Y)) = [1 2 0] [1 0 0] [1] [0 1 2] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [1] >= [1 2 0] [1 0 0] [1] [0 1 2] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [1] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0 0] [3] [0 1 0] X + [1] [0 0 1] [0] >= [1 0 0] [2] [0 1 0] X + [1] [0 0 1] [0] = hd(X) a__hd(cons(X,Y)) = [1 2 0] [1 0 0] [4] [0 1 2] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [1 2 0] [0] [0 1 1] X + [0] [0 0 1] [0] = mark(X) a__incr(X) = [1 0 0] [0] [0 1 0] X + [1] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 1] [0] = incr(X) a__incr(cons(X,Y)) = [1 2 0] [1 0 0] [1] [0 1 2] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [1] [0 0 2] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] = cons(s(X),incr(Y)) a__nats() = [1] [3] [1] >= [1] [3] [1] = a__adx(a__zeros()) a__nats() = [1] [3] [1] >= [0] [3] [1] = nats() a__tl(X) = [1 2 0] [1] [0 1 1] X + [0] [0 0 1] [3] >= [1 2 0] [1] [0 1 1] X + [0] [0 0 1] [3] = tl(X) a__tl(cons(X,Y)) = [1 4 4] [1 2 0] [2] [0 1 3] X + [0 1 1] Y + [0] [0 0 1] [0 0 1] [3] >= [1 2 0] [0] [0 1 1] Y + [0] [0 0 1] [0] = mark(Y) a__zeros() = [1] [2] [0] >= [1] [2] [0] = cons(0(),zeros()) a__zeros() = [1] [2] [0] >= [0] [2] [0] = zeros() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(adx(X)) = [1 2 0] [0] [0 1 1] X + [1] [0 0 1] [1] >= [1 2 0] [0] [0 1 1] X + [1] [0 0 1] [1] = a__adx(mark(X)) mark(cons(X1,X2)) = [1 4 4] [1 2 0] [1] [0 1 3] X1 + [0 1 1] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 2 0] [1 0 0] [1] [0 1 2] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = cons(X1,X2) mark(hd(X)) = [1 2 0] [4] [0 1 1] X + [1] [0 0 1] [0] >= [1 2 0] [3] [0 1 1] X + [1] [0 0 1] [0] = a__hd(mark(X)) mark(nats()) = [6] [4] [1] >= [1] [3] [1] = a__nats() mark(s(X)) = [1 0 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = s(X) mark(tl(X)) = [1 4 2] [1] [0 1 2] X + [3] [0 0 1] [3] >= [1 4 2] [1] [0 1 2] X + [0] [0 0 1] [3] = a__tl(mark(X)) mark(zeros()) = [4] [2] [0] >= [1] [2] [0] = a__zeros() * Step 14: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) - Weak TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__incr(X) -> incr(X) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__adx) = [1 0 1] [2] [0 1 1] x1 + [2] [0 0 1] [2] p(a__hd) = [1 0 0] [2] [0 1 0] x1 + [0] [0 0 1] [0] p(a__incr) = [1 0 1] [0] [0 1 0] x1 + [2] [0 0 1] [0] p(a__nats) = [3] [3] [3] p(a__tl) = [1 3 1] [0] [0 1 2] x1 + [2] [0 0 1] [3] p(a__zeros) = [0] [0] [1] p(adx) = [1 0 0] [0] [0 1 1] x1 + [1] [0 0 1] [2] p(cons) = [1 2 2] [1 0 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(hd) = [1 0 0] [2] [0 1 0] x1 + [0] [0 0 1] [0] p(incr) = [1 0 1] [0] [0 1 0] x1 + [2] [0 0 1] [0] p(mark) = [1 2 1] [0] [0 1 1] x1 + [0] [0 0 1] [1] p(nats) = [1] [0] [3] p(s) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(tl) = [1 3 0] [0] [0 1 2] x1 + [1] [0 0 1] [3] p(zeros) = [0] [0] [0] Following rules are strictly oriented: a__incr(cons(X,Y)) = [1 2 3] [1 0 1] [1] [0 1 1] X + [0 1 0] Y + [2] [0 0 1] [0 0 1] [1] > [0 0 2] [1 0 1] [0] [0 0 1] X + [0 1 0] Y + [2] [0 0 1] [0 0 1] [1] = cons(s(X),incr(Y)) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0 1] [2] [0 1 1] X + [2] [0 0 1] [2] >= [1 0 0] [0] [0 1 1] X + [1] [0 0 1] [2] = adx(X) a__adx(cons(X,Y)) = [1 2 3] [1 0 1] [3] [0 1 2] X + [0 1 1] Y + [3] [0 0 1] [0 0 1] [3] >= [1 2 3] [1 0 1] [3] [0 1 1] X + [0 1 1] Y + [3] [0 0 1] [0 0 1] [3] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0 0] [2] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [2] [0 1 0] X + [0] [0 0 1] [0] = hd(X) a__hd(cons(X,Y)) = [1 2 2] [1 0 0] [2] [0 1 1] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [1] >= [1 2 1] [0] [0 1 1] X + [0] [0 0 1] [1] = mark(X) a__incr(X) = [1 0 1] [0] [0 1 0] X + [2] [0 0 1] [0] >= [1 0 1] [0] [0 1 0] X + [2] [0 0 1] [0] = incr(X) a__nats() = [3] [3] [3] >= [3] [3] [3] = a__adx(a__zeros()) a__nats() = [3] [3] [3] >= [1] [0] [3] = nats() a__tl(X) = [1 3 1] [0] [0 1 2] X + [2] [0 0 1] [3] >= [1 3 0] [0] [0 1 2] X + [1] [0 0 1] [3] = tl(X) a__tl(cons(X,Y)) = [1 5 6] [1 3 1] [1] [0 1 3] X + [0 1 2] Y + [4] [0 0 1] [0 0 1] [4] >= [1 2 1] [0] [0 1 1] Y + [0] [0 0 1] [1] = mark(Y) a__zeros() = [0] [0] [1] >= [0] [0] [1] = cons(0(),zeros()) a__zeros() = [0] [0] [1] >= [0] [0] [0] = zeros() mark(0()) = [0] [0] [1] >= [0] [0] [0] = 0() mark(adx(X)) = [1 2 3] [4] [0 1 2] X + [3] [0 0 1] [3] >= [1 2 2] [3] [0 1 2] X + [3] [0 0 1] [3] = a__adx(mark(X)) mark(cons(X1,X2)) = [1 4 5] [1 2 1] [1] [0 1 2] X1 + [0 1 1] X2 + [1] [0 0 1] [0 0 1] [2] >= [1 2 2] [1 0 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] = cons(X1,X2) mark(hd(X)) = [1 2 1] [2] [0 1 1] X + [0] [0 0 1] [1] >= [1 2 1] [2] [0 1 1] X + [0] [0 0 1] [1] = a__hd(mark(X)) mark(incr(X)) = [1 2 2] [4] [0 1 1] X + [2] [0 0 1] [1] >= [1 2 2] [1] [0 1 1] X + [2] [0 0 1] [1] = a__incr(mark(X)) mark(nats()) = [4] [3] [4] >= [3] [3] [3] = a__nats() mark(s(X)) = [0 0 1] [0] [0 0 1] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = s(X) mark(tl(X)) = [1 5 5] [5] [0 1 3] X + [4] [0 0 1] [4] >= [1 5 5] [1] [0 1 3] X + [4] [0 0 1] [4] = a__tl(mark(X)) mark(zeros()) = [0] [0] [1] >= [0] [0] [1] = a__zeros() * Step 15: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__adx(X) -> adx(X) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(adx(X)) -> a__adx(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(hd(X)) -> a__hd(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(s(X)) -> s(X) mark(tl(X)) -> a__tl(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__adx/1,a__hd/1,a__incr/1,a__nats/0,a__tl/1,a__zeros/0,mark/1} / {0/0,adx/1,cons/2,hd/1,incr/1,nats/0,s/1 ,tl/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros ,mark} and constructors {0,adx,cons,hd,incr,nats,s,tl,zeros} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))