WORST_CASE(?,O(n^3)) * Step 1: WeightGap 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__adx) = [1] x1 + [0] p(a__hd) = [1] x1 + [0] p(a__incr) = [1] x1 + [0] p(a__nats) = [0] p(a__tl) = [1] x1 + [0] p(a__zeros) = [0] p(adx) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(hd) = [1] x1 + [0] p(incr) = [1] x1 + [0] p(mark) = [1] x1 + [5] p(nats) = [0] p(s) = [1] x1 + [0] p(tl) = [1] x1 + [0] p(zeros) = [0] Following rules are strictly oriented: mark(0()) = [5] > [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [0] = cons(X1,X2) mark(nats()) = [5] > [0] = a__nats() mark(s(X)) = [1] X + [5] > [1] X + [0] = s(X) mark(zeros()) = [5] > [0] = a__zeros() Following rules are (at-least) weakly oriented: a__adx(X) = [1] X + [0] >= [1] X + [0] = adx(X) a__adx(cons(X,Y)) = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = a__incr(cons(X,adx(Y))) a__hd(X) = [1] X + [0] >= [1] X + [0] = hd(X) a__hd(cons(X,Y)) = [1] X + [1] Y + [0] >= [1] X + [5] = mark(X) a__incr(X) = [1] X + [0] >= [1] X + [0] = incr(X) a__incr(cons(X,Y)) = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = cons(s(X),incr(Y)) a__nats() = [0] >= [0] = a__adx(a__zeros()) a__nats() = [0] >= [0] = nats() a__tl(X) = [1] X + [0] >= [1] X + [0] = tl(X) a__tl(cons(X,Y)) = [1] X + [1] Y + [0] >= [1] Y + [5] = mark(Y) a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(adx(X)) = [1] X + [5] >= [1] X + [5] = a__adx(mark(X)) mark(hd(X)) = [1] X + [5] >= [1] X + [5] = a__hd(mark(X)) mark(incr(X)) = [1] X + [5] >= [1] X + [5] = a__incr(mark(X)) mark(tl(X)) = [1] X + [5] >= [1] X + [5] = a__tl(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap 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(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: 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__adx) = [1] x1 + [0] p(a__hd) = [1] x1 + [12] p(a__incr) = [1] x1 + [0] p(a__nats) = [0] p(a__tl) = [1] x1 + [8] p(a__zeros) = [0] p(adx) = [0] p(cons) = [0] p(hd) = [0] p(incr) = [1] x1 + [0] p(mark) = [0] p(nats) = [0] p(s) = [0] p(tl) = [1] p(zeros) = [0] Following rules are strictly oriented: a__hd(X) = [1] X + [12] > [0] = hd(X) a__hd(cons(X,Y)) = [12] > [0] = mark(X) a__tl(X) = [1] X + [8] > [1] = tl(X) a__tl(cons(X,Y)) = [8] > [0] = mark(Y) Following rules are (at-least) weakly oriented: a__adx(X) = [1] X + [0] >= [0] = adx(X) a__adx(cons(X,Y)) = [0] >= [0] = a__incr(cons(X,adx(Y))) a__incr(X) = [1] X + [0] >= [1] X + [0] = incr(X) a__incr(cons(X,Y)) = [0] >= [0] = cons(s(X),incr(Y)) a__nats() = [0] >= [0] = a__adx(a__zeros()) a__nats() = [0] >= [0] = nats() a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [0] >= [0] = 0() mark(adx(X)) = [0] >= [0] = a__adx(mark(X)) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(hd(X)) = [0] >= [12] = a__hd(mark(X)) mark(incr(X)) = [0] >= [0] = a__incr(mark(X)) mark(nats()) = [0] >= [0] = a__nats() mark(s(X)) = [0] >= [0] = s(X) mark(tl(X)) = [0] >= [8] = a__tl(mark(X)) mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap 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__nats() -> a__adx(a__zeros()) a__nats() -> nats() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() 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(X) -> hd(X) a__hd(cons(X,Y)) -> mark(X) a__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(a__adx) = [1] x1 + [3] p(a__hd) = [1] x1 + [0] p(a__incr) = [1] x1 + [0] p(a__nats) = [0] p(a__tl) = [1] x1 + [0] p(a__zeros) = [0] p(adx) = [1] x1 + [2] p(cons) = [1] x1 + [1] x2 + [0] p(hd) = [1] x1 + [0] p(incr) = [1] x1 + [2] p(mark) = [1] x1 + [0] p(nats) = [0] p(s) = [1] x1 + [1] p(tl) = [1] x1 + [0] p(zeros) = [0] Following rules are strictly oriented: a__adx(X) = [1] X + [3] > [1] X + [2] = adx(X) a__adx(cons(X,Y)) = [1] X + [1] Y + [3] > [1] X + [1] Y + [2] = a__incr(cons(X,adx(Y))) mark(incr(X)) = [1] X + [2] > [1] X + [0] = a__incr(mark(X)) Following rules are (at-least) weakly oriented: a__hd(X) = [1] X + [0] >= [1] X + [0] = hd(X) a__hd(cons(X,Y)) = [1] X + [1] Y + [0] >= [1] X + [0] = mark(X) a__incr(X) = [1] X + [0] >= [1] X + [2] = incr(X) a__incr(cons(X,Y)) = [1] X + [1] Y + [0] >= [1] X + [1] Y + [3] = cons(s(X),incr(Y)) a__nats() = [0] >= [3] = a__adx(a__zeros()) a__nats() = [0] >= [0] = nats() a__tl(X) = [1] X + [0] >= [1] X + [0] = tl(X) a__tl(cons(X,Y)) = [1] X + [1] Y + [0] >= [1] Y + [0] = mark(Y) a__zeros() = [0] >= [4] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [4] >= [4] = 0() mark(adx(X)) = [1] X + [2] >= [1] X + [3] = a__adx(mark(X)) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(X1,X2) mark(hd(X)) = [1] X + [0] >= [1] X + [0] = a__hd(mark(X)) mark(nats()) = [0] >= [0] = a__nats() mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(X) mark(tl(X)) = [1] X + [0] >= [1] X + [0] = a__tl(mark(X)) mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: 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__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(adx(X)) -> a__adx(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(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__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(incr(X)) -> a__incr(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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(a__adx) = [1] x1 + [2] p(a__hd) = [1] x1 + [0] p(a__incr) = [1] x1 + [0] p(a__nats) = [4] p(a__tl) = [1] x1 + [0] p(a__zeros) = [0] p(adx) = [1] x1 + [2] p(cons) = [1] x1 + [1] x2 + [6] p(hd) = [1] x1 + [0] p(incr) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nats) = [4] p(s) = [1] x1 + [4] p(tl) = [1] x1 + [0] p(zeros) = [0] Following rules are strictly oriented: a__nats() = [4] > [2] = a__adx(a__zeros()) Following rules are (at-least) weakly oriented: a__adx(X) = [1] X + [2] >= [1] X + [2] = adx(X) a__adx(cons(X,Y)) = [1] X + [1] Y + [8] >= [1] X + [1] Y + [8] = a__incr(cons(X,adx(Y))) a__hd(X) = [1] X + [0] >= [1] X + [0] = hd(X) a__hd(cons(X,Y)) = [1] X + [1] Y + [6] >= [1] X + [0] = mark(X) a__incr(X) = [1] X + [0] >= [1] X + [0] = incr(X) a__incr(cons(X,Y)) = [1] X + [1] Y + [6] >= [1] X + [1] Y + [10] = cons(s(X),incr(Y)) a__nats() = [4] >= [4] = nats() a__tl(X) = [1] X + [0] >= [1] X + [0] = tl(X) a__tl(cons(X,Y)) = [1] X + [1] Y + [6] >= [1] Y + [0] = mark(Y) a__zeros() = [0] >= [10] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [4] >= [4] = 0() mark(adx(X)) = [1] X + [2] >= [1] X + [2] = a__adx(mark(X)) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [6] = cons(X1,X2) mark(hd(X)) = [1] X + [0] >= [1] X + [0] = a__hd(mark(X)) mark(incr(X)) = [1] X + [0] >= [1] X + [0] = a__incr(mark(X)) mark(nats()) = [4] >= [4] = a__nats() mark(s(X)) = [1] X + [4] >= [1] X + [4] = s(X) mark(tl(X)) = [1] X + [0] >= [1] X + [0] = a__tl(mark(X)) mark(zeros()) = [0] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__nats() -> nats() a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(adx(X)) -> a__adx(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(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__tl(X) -> tl(X) a__tl(cons(X,Y)) -> mark(Y) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(incr(X)) -> a__incr(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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__adx) = [1] x1 + [1] p(a__hd) = [1] x1 + [5] p(a__incr) = [1] x1 + [0] p(a__nats) = [1] p(a__tl) = [1] x1 + [1] p(a__zeros) = [0] p(adx) = [0] p(cons) = [0] p(hd) = [0] p(incr) = [0] p(mark) = [1] p(nats) = [0] p(s) = [0] p(tl) = [1] x1 + [0] p(zeros) = [0] Following rules are strictly oriented: a__nats() = [1] > [0] = nats() Following rules are (at-least) weakly oriented: a__adx(X) = [1] X + [1] >= [0] = adx(X) a__adx(cons(X,Y)) = [1] >= [0] = a__incr(cons(X,adx(Y))) a__hd(X) = [1] X + [5] >= [0] = hd(X) a__hd(cons(X,Y)) = [5] >= [1] = mark(X) a__incr(X) = [1] X + [0] >= [0] = incr(X) a__incr(cons(X,Y)) = [0] >= [0] = cons(s(X),incr(Y)) a__nats() = [1] >= [1] = a__adx(a__zeros()) a__tl(X) = [1] X + [1] >= [1] X + [0] = tl(X) a__tl(cons(X,Y)) = [1] >= [1] = mark(Y) a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [1] >= [0] = 0() mark(adx(X)) = [1] >= [2] = a__adx(mark(X)) mark(cons(X1,X2)) = [1] >= [0] = cons(X1,X2) mark(hd(X)) = [1] >= [6] = a__hd(mark(X)) mark(incr(X)) = [1] >= [1] = a__incr(mark(X)) mark(nats()) = [1] >= [1] = a__nats() mark(s(X)) = [1] >= [0] = s(X) mark(tl(X)) = [1] >= [2] = a__tl(mark(X)) mark(zeros()) = [1] >= [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__incr(X) -> incr(X) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(adx(X)) -> a__adx(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(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) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(incr(X)) -> a__incr(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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__adx) = {1}, uargs(a__hd) = {1}, uargs(a__incr) = {1}, uargs(a__tl) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__adx) = [1] x1 + [1] p(a__hd) = [1] x1 + [5] p(a__incr) = [1] x1 + [0] p(a__nats) = [5] p(a__tl) = [1] x1 + [6] p(a__zeros) = [4] p(adx) = [0] p(cons) = [0] p(hd) = [1] x1 + [5] p(incr) = [1] x1 + [0] p(mark) = [5] p(nats) = [5] p(s) = [0] p(tl) = [1] x1 + [0] p(zeros) = [0] Following rules are strictly oriented: a__zeros() = [4] > [0] = cons(0(),zeros()) a__zeros() = [4] > [0] = zeros() Following rules are (at-least) weakly oriented: a__adx(X) = [1] X + [1] >= [0] = adx(X) a__adx(cons(X,Y)) = [1] >= [0] = a__incr(cons(X,adx(Y))) a__hd(X) = [1] X + [5] >= [1] X + [5] = hd(X) a__hd(cons(X,Y)) = [5] >= [5] = mark(X) a__incr(X) = [1] X + [0] >= [1] X + [0] = incr(X) a__incr(cons(X,Y)) = [0] >= [0] = cons(s(X),incr(Y)) a__nats() = [5] >= [5] = a__adx(a__zeros()) a__nats() = [5] >= [5] = nats() a__tl(X) = [1] X + [6] >= [1] X + [0] = tl(X) a__tl(cons(X,Y)) = [6] >= [5] = mark(Y) mark(0()) = [5] >= [0] = 0() mark(adx(X)) = [5] >= [6] = a__adx(mark(X)) mark(cons(X1,X2)) = [5] >= [0] = cons(X1,X2) mark(hd(X)) = [5] >= [10] = a__hd(mark(X)) mark(incr(X)) = [5] >= [5] = a__incr(mark(X)) mark(nats()) = [5] >= [5] = a__nats() mark(s(X)) = [5] >= [0] = s(X) mark(tl(X)) = [5] >= [11] = a__tl(mark(X)) mark(zeros()) = [5] >= [4] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: 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(adx(X)) -> a__adx(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(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(cons(X1,X2)) -> cons(X1,X2) mark(incr(X)) -> a__incr(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 = 2, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): 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] p(a__adx) = [1 0] x1 + [0] [0 1] [0] p(a__hd) = [1 0] x1 + [0] [0 1] [0] p(a__incr) = [1 0] x1 + [0] [0 1] [0] p(a__nats) = [0] [0] p(a__tl) = [1 0] x1 + [4] [0 1] [4] p(a__zeros) = [0] [0] p(adx) = [0 0] x1 + [0] [0 1] [0] p(cons) = [0 2] x1 + [0 2] x2 + [0] [0 1] [0 1] [0] p(hd) = [0 0] x1 + [0] [0 1] [0] p(incr) = [0 0] x1 + [0] [0 1] [0] p(mark) = [0 2] x1 + [0] [0 1] [0] p(nats) = [0] [0] p(s) = [0] [0] p(tl) = [0 0] x1 + [0] [0 1] [4] p(zeros) = [0] [0] Following rules are strictly oriented: mark(tl(X)) = [0 2] X + [8] [0 1] [4] > [0 2] X + [4] [0 1] [4] = a__tl(mark(X)) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0] X + [0] [0 1] [0] >= [0 0] X + [0] [0 1] [0] = adx(X) a__adx(cons(X,Y)) = [0 2] X + [0 2] Y + [0] [0 1] [0 1] [0] >= [0 2] X + [0 2] Y + [0] [0 1] [0 1] [0] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0] X + [0] [0 1] [0] >= [0 0] X + [0] [0 1] [0] = hd(X) a__hd(cons(X,Y)) = [0 2] X + [0 2] Y + [0] [0 1] [0 1] [0] >= [0 2] X + [0] [0 1] [0] = mark(X) a__incr(X) = [1 0] X + [0] [0 1] [0] >= [0 0] X + [0] [0 1] [0] = incr(X) a__incr(cons(X,Y)) = [0 2] X + [0 2] Y + [0] [0 1] [0 1] [0] >= [0 2] Y + [0] [0 1] [0] = cons(s(X),incr(Y)) a__nats() = [0] [0] >= [0] [0] = a__adx(a__zeros()) a__nats() = [0] [0] >= [0] [0] = nats() a__tl(X) = [1 0] X + [4] [0 1] [4] >= [0 0] X + [0] [0 1] [4] = tl(X) a__tl(cons(X,Y)) = [0 2] X + [0 2] Y + [4] [0 1] [0 1] [4] >= [0 2] Y + [0] [0 1] [0] = mark(Y) a__zeros() = [0] [0] >= [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] >= [0] [0] = zeros() mark(0()) = [0] [0] >= [0] [0] = 0() mark(adx(X)) = [0 2] X + [0] [0 1] [0] >= [0 2] X + [0] [0 1] [0] = a__adx(mark(X)) mark(cons(X1,X2)) = [0 2] X1 + [0 2] X2 + [0] [0 1] [0 1] [0] >= [0 2] X1 + [0 2] X2 + [0] [0 1] [0 1] [0] = cons(X1,X2) mark(hd(X)) = [0 2] X + [0] [0 1] [0] >= [0 2] X + [0] [0 1] [0] = a__hd(mark(X)) mark(incr(X)) = [0 2] X + [0] [0 1] [0] >= [0 2] X + [0] [0 1] [0] = a__incr(mark(X)) mark(nats()) = [0] [0] >= [0] [0] = a__nats() mark(s(X)) = [0] [0] >= [0] [0] = s(X) mark(zeros()) = [0] [0] >= [0] [0] = a__zeros() * Step 8: 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(adx(X)) -> a__adx(mark(X)) mark(hd(X)) -> a__hd(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(cons(X1,X2)) -> cons(X1,X2) 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 = 2, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): 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] p(a__adx) = [1 0] x1 + [0] [0 1] [0] p(a__hd) = [1 0] x1 + [7] [0 1] [2] p(a__incr) = [1 0] x1 + [0] [0 1] [0] p(a__nats) = [0] [0] p(a__tl) = [1 0] x1 + [0] [0 1] [3] p(a__zeros) = [0] [0] p(adx) = [0 0] x1 + [0] [0 1] [0] p(cons) = [0 4] x1 + [0 4] x2 + [0] [0 1] [0 1] [0] p(hd) = [0 0] x1 + [0] [0 1] [2] p(incr) = [0 0] x1 + [0] [0 1] [0] p(mark) = [0 4] x1 + [0] [0 1] [0] p(nats) = [0] [0] p(s) = [0 1] x1 + [0] [0 1] [0] p(tl) = [0 0] x1 + [0] [0 1] [3] p(zeros) = [0] [0] Following rules are strictly oriented: mark(hd(X)) = [0 4] X + [8] [0 1] [2] > [0 4] X + [7] [0 1] [2] = a__hd(mark(X)) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0] X + [0] [0 1] [0] >= [0 0] X + [0] [0 1] [0] = adx(X) a__adx(cons(X,Y)) = [0 4] X + [0 4] Y + [0] [0 1] [0 1] [0] >= [0 4] X + [0 4] Y + [0] [0 1] [0 1] [0] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0] X + [7] [0 1] [2] >= [0 0] X + [0] [0 1] [2] = hd(X) a__hd(cons(X,Y)) = [0 4] X + [0 4] Y + [7] [0 1] [0 1] [2] >= [0 4] X + [0] [0 1] [0] = mark(X) a__incr(X) = [1 0] X + [0] [0 1] [0] >= [0 0] X + [0] [0 1] [0] = incr(X) a__incr(cons(X,Y)) = [0 4] X + [0 4] Y + [0] [0 1] [0 1] [0] >= [0 4] X + [0 4] Y + [0] [0 1] [0 1] [0] = cons(s(X),incr(Y)) a__nats() = [0] [0] >= [0] [0] = a__adx(a__zeros()) a__nats() = [0] [0] >= [0] [0] = nats() a__tl(X) = [1 0] X + [0] [0 1] [3] >= [0 0] X + [0] [0 1] [3] = tl(X) a__tl(cons(X,Y)) = [0 4] X + [0 4] Y + [0] [0 1] [0 1] [3] >= [0 4] Y + [0] [0 1] [0] = mark(Y) a__zeros() = [0] [0] >= [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] >= [0] [0] = zeros() mark(0()) = [0] [0] >= [0] [0] = 0() mark(adx(X)) = [0 4] X + [0] [0 1] [0] >= [0 4] X + [0] [0 1] [0] = a__adx(mark(X)) mark(cons(X1,X2)) = [0 4] X1 + [0 4] X2 + [0] [0 1] [0 1] [0] >= [0 4] X1 + [0 4] X2 + [0] [0 1] [0 1] [0] = cons(X1,X2) mark(incr(X)) = [0 4] X + [0] [0 1] [0] >= [0 4] X + [0] [0 1] [0] = a__incr(mark(X)) mark(nats()) = [0] [0] >= [0] [0] = a__nats() mark(s(X)) = [0 4] X + [0] [0 1] [0] >= [0 1] X + [0] [0 1] [0] = s(X) mark(tl(X)) = [0 4] X + [12] [0 1] [3] >= [0 4] X + [0] [0 1] [3] = a__tl(mark(X)) mark(zeros()) = [0] [0] >= [0] [0] = a__zeros() * Step 9: WeightGap 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(adx(X)) -> a__adx(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(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: WeightGap {wgDimension = 2, wgDegree = 2, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): 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: all TcT has computed the following interpretation: p(0) = [0] [0] p(a__adx) = [1 0] x1 + [1] [0 1] [4] p(a__hd) = [1 0] x1 + [0] [0 1] [5] p(a__incr) = [1 0] x1 + [1] [0 1] [4] p(a__nats) = [1] [4] p(a__tl) = [1 0] x1 + [0] [0 1] [0] p(a__zeros) = [0] [0] p(adx) = [0 0] x1 + [0] [0 1] [0] p(cons) = [0 2] x1 + [0 2] x2 + [0] [0 1] [0 1] [0] p(hd) = [0 0] x1 + [0] [0 1] [5] p(incr) = [0 0] x1 + [0] [0 1] [4] p(mark) = [0 2] x1 + [0] [0 1] [0] p(nats) = [0] [4] p(s) = [0 2] x1 + [0] [0 1] [0] p(tl) = [0 0] x1 + [0] [0 1] [0] p(zeros) = [0] [0] Following rules are strictly oriented: a__incr(X) = [1 0] X + [1] [0 1] [4] > [0 0] X + [0] [0 1] [4] = incr(X) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0] X + [1] [0 1] [4] >= [0 0] X + [0] [0 1] [0] = adx(X) a__adx(cons(X,Y)) = [0 2] X + [0 2] Y + [1] [0 1] [0 1] [4] >= [0 2] X + [0 2] Y + [1] [0 1] [0 1] [4] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0] X + [0] [0 1] [5] >= [0 0] X + [0] [0 1] [5] = hd(X) a__hd(cons(X,Y)) = [0 2] X + [0 2] Y + [0] [0 1] [0 1] [5] >= [0 2] X + [0] [0 1] [0] = mark(X) a__incr(cons(X,Y)) = [0 2] X + [0 2] Y + [1] [0 1] [0 1] [4] >= [0 2] X + [0 2] Y + [8] [0 1] [0 1] [4] = cons(s(X),incr(Y)) a__nats() = [1] [4] >= [1] [4] = a__adx(a__zeros()) a__nats() = [1] [4] >= [0] [4] = nats() a__tl(X) = [1 0] X + [0] [0 1] [0] >= [0 0] X + [0] [0 1] [0] = tl(X) a__tl(cons(X,Y)) = [0 2] X + [0 2] Y + [0] [0 1] [0 1] [0] >= [0 2] Y + [0] [0 1] [0] = mark(Y) a__zeros() = [0] [0] >= [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] >= [0] [0] = zeros() mark(0()) = [0] [0] >= [0] [0] = 0() mark(adx(X)) = [0 2] X + [0] [0 1] [0] >= [0 2] X + [1] [0 1] [4] = a__adx(mark(X)) mark(cons(X1,X2)) = [0 2] X1 + [0 2] X2 + [0] [0 1] [0 1] [0] >= [0 2] X1 + [0 2] X2 + [0] [0 1] [0 1] [0] = cons(X1,X2) mark(hd(X)) = [0 2] X + [10] [0 1] [5] >= [0 2] X + [0] [0 1] [5] = a__hd(mark(X)) mark(incr(X)) = [0 2] X + [8] [0 1] [4] >= [0 2] X + [1] [0 1] [4] = a__incr(mark(X)) mark(nats()) = [8] [4] >= [1] [4] = a__nats() mark(s(X)) = [0 2] X + [0] [0 1] [0] >= [0 2] X + [0] [0 1] [0] = s(X) mark(tl(X)) = [0 2] X + [0] [0 1] [0] >= [0 2] X + [0] [0 1] [0] = a__tl(mark(X)) mark(zeros()) = [0] [0] >= [0] [0] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 10: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) mark(adx(X)) -> a__adx(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(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 = 2, miDegree = 2, 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] p(a__adx) = [1 0] x1 + [0] [0 1] [1] p(a__hd) = [1 4] x1 + [5] [0 1] [1] p(a__incr) = [1 0] x1 + [0] [0 1] [0] p(a__nats) = [1] [3] p(a__tl) = [1 4] x1 + [0] [0 1] [1] p(a__zeros) = [0] [2] p(adx) = [1 0] x1 + [0] [0 1] [1] p(cons) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [1] p(hd) = [1 4] x1 + [5] [0 1] [1] p(incr) = [1 0] x1 + [0] [0 1] [0] p(mark) = [1 4] x1 + [2] [0 1] [1] p(nats) = [0] [2] p(s) = [0] [0] p(tl) = [1 4] x1 + [0] [0 1] [1] p(zeros) = [0] [1] Following rules are strictly oriented: mark(adx(X)) = [1 4] X + [6] [0 1] [2] > [1 4] X + [2] [0 1] [2] = a__adx(mark(X)) Following rules are (at-least) weakly oriented: a__adx(X) = [1 0] X + [0] [0 1] [1] >= [1 0] X + [0] [0 1] [1] = adx(X) a__adx(cons(X,Y)) = [1 0] X + [1 0] Y + [0] [0 1] [0 1] [2] >= [1 0] X + [1 0] Y + [0] [0 1] [0 1] [2] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 4] X + [5] [0 1] [1] >= [1 4] X + [5] [0 1] [1] = hd(X) a__hd(cons(X,Y)) = [1 4] X + [1 4] Y + [9] [0 1] [0 1] [2] >= [1 4] X + [2] [0 1] [1] = mark(X) a__incr(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = incr(X) a__incr(cons(X,Y)) = [1 0] X + [1 0] Y + [0] [0 1] [0 1] [1] >= [1 0] Y + [0] [0 1] [1] = cons(s(X),incr(Y)) a__nats() = [1] [3] >= [0] [3] = a__adx(a__zeros()) a__nats() = [1] [3] >= [0] [2] = nats() a__tl(X) = [1 4] X + [0] [0 1] [1] >= [1 4] X + [0] [0 1] [1] = tl(X) a__tl(cons(X,Y)) = [1 4] X + [1 4] Y + [4] [0 1] [0 1] [2] >= [1 4] Y + [2] [0 1] [1] = mark(Y) a__zeros() = [0] [2] >= [0] [2] = cons(0(),zeros()) a__zeros() = [0] [2] >= [0] [1] = zeros() mark(0()) = [2] [1] >= [0] [0] = 0() mark(cons(X1,X2)) = [1 4] X1 + [1 4] X2 + [6] [0 1] [0 1] [2] >= [1 0] X1 + [1 0] X2 + [0] [0 1] [0 1] [1] = cons(X1,X2) mark(hd(X)) = [1 8] X + [11] [0 1] [2] >= [1 8] X + [11] [0 1] [2] = a__hd(mark(X)) mark(incr(X)) = [1 4] X + [2] [0 1] [1] >= [1 4] X + [2] [0 1] [1] = a__incr(mark(X)) mark(nats()) = [10] [3] >= [1] [3] = a__nats() mark(s(X)) = [2] [1] >= [0] [0] = s(X) mark(tl(X)) = [1 8] X + [6] [0 1] [2] >= [1 8] X + [6] [0 1] [2] = a__tl(mark(X)) mark(zeros()) = [6] [2] >= [0] [2] = a__zeros() * Step 11: 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 = 4, 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 (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): 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] [0] p(a__adx) = [1 1 0 0] [0] [0 0 0 0] x1 + [1] [0 0 1 0] [1] [0 0 0 1] [1] p(a__hd) = [1 0 0 1] [1] [0 0 0 0] x1 + [1] [0 0 1 1] [0] [0 0 0 1] [0] p(a__incr) = [1 1 0 0] [0] [0 0 0 0] x1 + [1] [0 0 1 0] [1] [0 0 0 1] [0] p(a__nats) = [1] [1] [1] [1] p(a__tl) = [1 0 1 1] [0] [0 1 0 0] x1 + [0] [0 0 1 1] [1] [0 0 0 1] [1] p(a__zeros) = [0] [1] [0] [0] p(adx) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [1] p(cons) = [1 0 1 0] [1 0 0 0] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [1] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] p(hd) = [1 0 0 1] [1] [0 0 0 0] x1 + [0] [0 0 1 1] [0] [0 0 0 1] [0] p(incr) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 1] [0] p(mark) = [1 0 1 1] [0] [0 0 0 0] x1 + [1] [0 0 1 1] [0] [0 0 0 1] [0] p(nats) = [0] [0] [0] [1] p(s) = [0 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(tl) = [1 0 1 1] [0] [0 0 0 0] x1 + [0] [0 0 1 1] [1] [0 0 0 1] [1] p(zeros) = [0] [0] [0] [0] Following rules are strictly oriented: a__incr(cons(X,Y)) = [1 0 1 0] [1 0 0 0] [1] [0 0 0 0] X + [0 0 0 0] Y + [1] [0 0 1 0] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [0] > [0 0 1 0] [1 0 0 0] [0] [0 0 0 0] X + [0 0 0 0] Y + [1] [0 0 1 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 1] [0] = cons(s(X),incr(Y)) Following rules are (at-least) weakly oriented: a__adx(X) = [1 1 0 0] [0] [0 0 0 0] X + [1] [0 0 1 0] [1] [0 0 0 1] [1] >= [1 0 0 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 1] [1] = adx(X) a__adx(cons(X,Y)) = [1 0 1 0] [1 0 0 0] [1] [0 0 0 0] X + [0 0 0 0] Y + [1] [0 0 1 0] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [1] >= [1 0 1 0] [1 0 0 0] [1] [0 0 0 0] X + [0 0 0 0] Y + [1] [0 0 1 0] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [1] = a__incr(cons(X,adx(Y))) a__hd(X) = [1 0 0 1] [1] [0 0 0 0] X + [1] [0 0 1 1] [0] [0 0 0 1] [0] >= [1 0 0 1] [1] [0 0 0 0] X + [0] [0 0 1 1] [0] [0 0 0 1] [0] = hd(X) a__hd(cons(X,Y)) = [1 0 1 1] [1 0 0 1] [1] [0 0 0 0] X + [0 0 0 0] Y + [1] [0 0 1 1] [0 0 1 1] [0] [0 0 0 1] [0 0 0 1] [0] >= [1 0 1 1] [0] [0 0 0 0] X + [1] [0 0 1 1] [0] [0 0 0 1] [0] = mark(X) a__incr(X) = [1 1 0 0] [0] [0 0 0 0] X + [1] [0 0 1 0] [1] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 1] [0] = incr(X) a__nats() = [1] [1] [1] [1] >= [1] [1] [1] [1] = a__adx(a__zeros()) a__nats() = [1] [1] [1] [1] >= [0] [0] [0] [1] = nats() a__tl(X) = [1 0 1 1] [0] [0 1 0 0] X + [0] [0 0 1 1] [1] [0 0 0 1] [1] >= [1 0 1 1] [0] [0 0 0 0] X + [0] [0 0 1 1] [1] [0 0 0 1] [1] = tl(X) a__tl(cons(X,Y)) = [1 0 2 1] [1 0 1 1] [0] [0 0 0 0] X + [0 0 0 0] Y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 1] [0 0 0 1] [1] >= [1 0 1 1] [0] [0 0 0 0] Y + [1] [0 0 1 1] [0] [0 0 0 1] [0] = mark(Y) a__zeros() = [0] [1] [0] [0] >= [0] [1] [0] [0] = cons(0(),zeros()) a__zeros() = [0] [1] [0] [0] >= [0] [0] [0] [0] = zeros() mark(0()) = [0] [1] [0] [0] >= [0] [0] [0] [0] = 0() mark(adx(X)) = [1 0 1 1] [1] [0 0 0 0] X + [1] [0 0 1 1] [1] [0 0 0 1] [1] >= [1 0 1 1] [1] [0 0 0 0] X + [1] [0 0 1 1] [1] [0 0 0 1] [1] = a__adx(mark(X)) mark(cons(X1,X2)) = [1 0 2 1] [1 0 1 1] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [1] [0 0 1 1] [0 0 1 1] [0] [0 0 0 1] [0 0 0 1] [0] >= [1 0 1 0] [1 0 0 0] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [1] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] = cons(X1,X2) mark(hd(X)) = [1 0 1 3] [1] [0 0 0 0] X + [1] [0 0 1 2] [0] [0 0 0 1] [0] >= [1 0 1 2] [1] [0 0 0 0] X + [1] [0 0 1 2] [0] [0 0 0 1] [0] = a__hd(mark(X)) mark(incr(X)) = [1 0 1 1] [1] [0 0 0 0] X + [1] [0 0 1 1] [1] [0 0 0 1] [0] >= [1 0 1 1] [1] [0 0 0 0] X + [1] [0 0 1 1] [1] [0 0 0 1] [0] = a__incr(mark(X)) mark(nats()) = [1] [1] [1] [1] >= [1] [1] [1] [1] = a__nats() mark(s(X)) = [0 0 1 0] [0] [0 0 0 0] X + [1] [0 0 1 0] [0] [0 0 0 0] [0] >= [0 0 0 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] = s(X) mark(tl(X)) = [1 0 2 3] [2] [0 0 0 0] X + [1] [0 0 1 2] [2] [0 0 0 1] [1] >= [1 0 2 3] [0] [0 0 0 0] X + [1] [0 0 1 2] [1] [0 0 0 1] [1] = a__tl(mark(X)) mark(zeros()) = [0] [1] [0] [0] >= [0] [1] [0] [0] = a__zeros() * Step 12: 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))