WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(+(x,y)) -> +(D(x),D(y)) D(-(x,y)) -> -(D(x),D(y)) D(constant()) -> 0() D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(minus(x)) -> minus(D(x)) D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y))) D(t()) -> 1() - Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,2,constant,div,ln,minus ,pow,t} + 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(*) = {2}, uargs(+) = {1,2}, uargs(-) = {1,2}, uargs(div) = {1}, uargs(minus) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [1] x2 + [1] p(+) = [1] x1 + [1] x2 + [4] p(-) = [1] x1 + [1] x2 + [5] p(0) = [2] p(1) = [3] p(2) = [0] p(D) = [4] p(constant) = [1] p(div) = [1] x1 + [0] p(ln) = [7] p(minus) = [1] x1 + [0] p(pow) = [7] p(t) = [4] Following rules are strictly oriented: D(constant()) = [4] > [2] = 0() D(t()) = [4] > [3] = 1() Following rules are (at-least) weakly oriented: D(*(x,y)) = [4] >= [14] = +(*(y,D(x)),*(x,D(y))) D(+(x,y)) = [4] >= [12] = +(D(x),D(y)) D(-(x,y)) = [4] >= [13] = -(D(x),D(y)) D(div(x,y)) = [4] >= [14] = -(div(D(x),y),div(*(x,D(y)),pow(y,2()))) D(ln(x)) = [4] >= [4] = div(D(x),x) D(minus(x)) = [4] >= [4] = minus(D(x)) D(pow(x,y)) = [4] >= [14] = +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(+(x,y)) -> +(D(x),D(y)) D(-(x,y)) -> -(D(x),D(y)) D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(minus(x)) -> minus(D(x)) D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y))) - Weak TRS: D(constant()) -> 0() D(t()) -> 1() - Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,2,constant,div,ln,minus ,pow,t} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(*) = {2}, uargs(+) = {1,2}, uargs(-) = {1,2}, uargs(div) = {1}, uargs(minus) = {1} Following symbols are considered usable: {D} TcT has computed the following interpretation: p(*) = 2 + x1 + x2 p(+) = x1 + x2 p(-) = x1 + x2 p(0) = 4 p(1) = 0 p(2) = 4 p(D) = 3*x1 + 2*x1^2 p(constant) = 2 p(div) = 2 + x1 + x2 p(ln) = 1 + x1 p(minus) = 1 + x1 p(pow) = 2 + x1 + x2 p(t) = 0 Following rules are strictly oriented: D(*(x,y)) = 14 + 11*x + 4*x*y + 2*x^2 + 11*y + 2*y^2 > 4 + 4*x + 2*x^2 + 4*y + 2*y^2 = +(*(y,D(x)),*(x,D(y))) D(div(x,y)) = 14 + 11*x + 4*x*y + 2*x^2 + 11*y + 2*y^2 > 12 + 4*x + 2*x^2 + 5*y + 2*y^2 = -(div(D(x),y),div(*(x,D(y)),pow(y,2()))) D(ln(x)) = 5 + 7*x + 2*x^2 > 2 + 4*x + 2*x^2 = div(D(x),x) D(minus(x)) = 5 + 7*x + 2*x^2 > 1 + 3*x + 2*x^2 = minus(D(x)) D(pow(x,y)) = 14 + 11*x + 4*x*y + 2*x^2 + 11*y + 2*y^2 > 13 + 6*x + 2*x^2 + 6*y + 2*y^2 = +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y))) Following rules are (at-least) weakly oriented: D(+(x,y)) = 3*x + 4*x*y + 2*x^2 + 3*y + 2*y^2 >= 3*x + 2*x^2 + 3*y + 2*y^2 = +(D(x),D(y)) D(-(x,y)) = 3*x + 4*x*y + 2*x^2 + 3*y + 2*y^2 >= 3*x + 2*x^2 + 3*y + 2*y^2 = -(D(x),D(y)) D(constant()) = 14 >= 4 = 0() D(t()) = 0 >= 0 = 1() * Step 3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: D(+(x,y)) -> +(D(x),D(y)) D(-(x,y)) -> -(D(x),D(y)) - Weak TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(constant()) -> 0() D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(minus(x)) -> minus(D(x)) D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y))) D(t()) -> 1() - Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,2,constant,div,ln,minus ,pow,t} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(*) = {2}, uargs(+) = {1,2}, uargs(-) = {1,2}, uargs(div) = {1}, uargs(minus) = {1} Following symbols are considered usable: {D} TcT has computed the following interpretation: p(*) = 1 + x1 + x2 p(+) = x1 + x2 p(-) = 1 + x1 + x2 p(0) = 0 p(1) = 0 p(2) = 2 p(D) = 4*x1 + 4*x1^2 p(constant) = 1 p(div) = 1 + x1 + x2 p(ln) = 1 + x1 p(minus) = x1 p(pow) = 1 + x1 + x2 p(t) = 0 Following rules are strictly oriented: D(-(x,y)) = 8 + 12*x + 8*x*y + 4*x^2 + 12*y + 4*y^2 > 1 + 4*x + 4*x^2 + 4*y + 4*y^2 = -(D(x),D(y)) Following rules are (at-least) weakly oriented: D(*(x,y)) = 8 + 12*x + 8*x*y + 4*x^2 + 12*y + 4*y^2 >= 2 + 5*x + 4*x^2 + 5*y + 4*y^2 = +(*(y,D(x)),*(x,D(y))) D(+(x,y)) = 4*x + 8*x*y + 4*x^2 + 4*y + 4*y^2 >= 4*x + 4*x^2 + 4*y + 4*y^2 = +(D(x),D(y)) D(constant()) = 8 >= 0 = 0() D(div(x,y)) = 8 + 12*x + 8*x*y + 4*x^2 + 12*y + 4*y^2 >= 7 + 5*x + 4*x^2 + 6*y + 4*y^2 = -(div(D(x),y),div(*(x,D(y)),pow(y,2()))) D(ln(x)) = 8 + 12*x + 4*x^2 >= 1 + 5*x + 4*x^2 = div(D(x),x) D(minus(x)) = 4*x + 4*x^2 >= 4*x + 4*x^2 = minus(D(x)) D(pow(x,y)) = 8 + 12*x + 8*x*y + 4*x^2 + 12*y + 4*y^2 >= 8 + 7*x + 4*x^2 + 7*y + 4*y^2 = +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y))) D(t()) = 0 >= 0 = 1() * Step 4: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: D(+(x,y)) -> +(D(x),D(y)) - Weak TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(-(x,y)) -> -(D(x),D(y)) D(constant()) -> 0() D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(minus(x)) -> minus(D(x)) D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y))) D(t()) -> 1() - Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,2,constant,div,ln,minus ,pow,t} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(*) = {2}, uargs(+) = {1,2}, uargs(-) = {1,2}, uargs(div) = {1}, uargs(minus) = {1} Following symbols are considered usable: {D} TcT has computed the following interpretation: p(*) = 2 + x1 + x2 p(+) = 1 + x1 + x2 p(-) = x1 + x2 p(0) = 0 p(1) = 0 p(2) = 0 p(D) = 5*x1 + x1^2 p(constant) = 0 p(div) = 1 + x1 + x2 p(ln) = 1 + x1 p(minus) = x1 p(pow) = 2 + x1 + x2 p(t) = 0 Following rules are strictly oriented: D(+(x,y)) = 6 + 7*x + 2*x*y + x^2 + 7*y + y^2 > 1 + 5*x + x^2 + 5*y + y^2 = +(D(x),D(y)) Following rules are (at-least) weakly oriented: D(*(x,y)) = 14 + 9*x + 2*x*y + x^2 + 9*y + y^2 >= 5 + 6*x + x^2 + 6*y + y^2 = +(*(y,D(x)),*(x,D(y))) D(-(x,y)) = 5*x + 2*x*y + x^2 + 5*y + y^2 >= 5*x + x^2 + 5*y + y^2 = -(D(x),D(y)) D(constant()) = 0 >= 0 = 0() D(div(x,y)) = 6 + 7*x + 2*x*y + x^2 + 7*y + y^2 >= 6 + 6*x + x^2 + 7*y + y^2 = -(div(D(x),y),div(*(x,D(y)),pow(y,2()))) D(ln(x)) = 6 + 7*x + x^2 >= 1 + 6*x + x^2 = div(D(x),x) D(minus(x)) = 5*x + x^2 >= 5*x + x^2 = minus(D(x)) D(pow(x,y)) = 14 + 9*x + 2*x*y + x^2 + 9*y + y^2 >= 14 + 8*x + x^2 + 8*y + y^2 = +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y))) D(t()) = 0 >= 0 = 1() * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(+(x,y)) -> +(D(x),D(y)) D(-(x,y)) -> -(D(x),D(y)) D(constant()) -> 0() D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(minus(x)) -> minus(D(x)) D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y))) D(t()) -> 1() - Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,2,constant,div,ln,minus ,pow,t} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))