WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + 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: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(constant()) -> c_4() D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) D#(t()) -> c_9() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(constant()) -> c_4() D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) D#(t()) -> c_9() - 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,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2 ,c_6/1,c_7/1,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus ,pow,t} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,9} by application of Pre({4,9}) = {1,2,3,5,6,7,8}. Here rules are labelled as follows: 1: D#(*(x,y)) -> c_1(D#(x),D#(y)) 2: D#(+(x,y)) -> c_2(D#(x),D#(y)) 3: D#(-(x,y)) -> c_3(D#(x),D#(y)) 4: D#(constant()) -> c_4() 5: D#(div(x,y)) -> c_5(D#(x),D#(y)) 6: D#(ln(x)) -> c_6(D#(x)) 7: D#(minus(x)) -> c_7(D#(x)) 8: D#(pow(x,y)) -> c_8(D#(x),D#(y)) 9: D#(t()) -> c_9() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak DPs: D#(constant()) -> c_4() D#(t()) -> c_9() - 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,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2 ,c_6/1,c_7/1,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus ,pow,t} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:D#(*(x,y)) -> c_1(D#(x),D#(y)) -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_2 D#(minus(x)) -> c_7(D#(x)):6 -->_1 D#(minus(x)) -> c_7(D#(x)):6 -->_2 D#(ln(x)) -> c_6(D#(x)):5 -->_1 D#(ln(x)) -> c_6(D#(x)):5 -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(t()) -> c_9():9 -->_1 D#(t()) -> c_9():9 -->_2 D#(constant()) -> c_4():8 -->_1 D#(constant()) -> c_4():8 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 2:S:D#(+(x,y)) -> c_2(D#(x),D#(y)) -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_2 D#(minus(x)) -> c_7(D#(x)):6 -->_1 D#(minus(x)) -> c_7(D#(x)):6 -->_2 D#(ln(x)) -> c_6(D#(x)):5 -->_1 D#(ln(x)) -> c_6(D#(x)):5 -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(t()) -> c_9():9 -->_1 D#(t()) -> c_9():9 -->_2 D#(constant()) -> c_4():8 -->_1 D#(constant()) -> c_4():8 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 3:S:D#(-(x,y)) -> c_3(D#(x),D#(y)) -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_2 D#(minus(x)) -> c_7(D#(x)):6 -->_1 D#(minus(x)) -> c_7(D#(x)):6 -->_2 D#(ln(x)) -> c_6(D#(x)):5 -->_1 D#(ln(x)) -> c_6(D#(x)):5 -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_2 D#(t()) -> c_9():9 -->_1 D#(t()) -> c_9():9 -->_2 D#(constant()) -> c_4():8 -->_1 D#(constant()) -> c_4():8 -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 4:S:D#(div(x,y)) -> c_5(D#(x),D#(y)) -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_2 D#(minus(x)) -> c_7(D#(x)):6 -->_1 D#(minus(x)) -> c_7(D#(x)):6 -->_2 D#(ln(x)) -> c_6(D#(x)):5 -->_1 D#(ln(x)) -> c_6(D#(x)):5 -->_2 D#(t()) -> c_9():9 -->_1 D#(t()) -> c_9():9 -->_2 D#(constant()) -> c_4():8 -->_1 D#(constant()) -> c_4():8 -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 5:S:D#(ln(x)) -> c_6(D#(x)) -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_1 D#(minus(x)) -> c_7(D#(x)):6 -->_1 D#(t()) -> c_9():9 -->_1 D#(constant()) -> c_4():8 -->_1 D#(ln(x)) -> c_6(D#(x)):5 -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 6:S:D#(minus(x)) -> c_7(D#(x)) -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_1 D#(t()) -> c_9():9 -->_1 D#(constant()) -> c_4():8 -->_1 D#(minus(x)) -> c_7(D#(x)):6 -->_1 D#(ln(x)) -> c_6(D#(x)):5 -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 7:S:D#(pow(x,y)) -> c_8(D#(x),D#(y)) -->_2 D#(t()) -> c_9():9 -->_1 D#(t()) -> c_9():9 -->_2 D#(constant()) -> c_4():8 -->_1 D#(constant()) -> c_4():8 -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_2 D#(minus(x)) -> c_7(D#(x)):6 -->_1 D#(minus(x)) -> c_7(D#(x)):6 -->_2 D#(ln(x)) -> c_6(D#(x)):5 -->_1 D#(ln(x)) -> c_6(D#(x)):5 -->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4 -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3 -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1 8:W:D#(constant()) -> c_4() 9:W:D#(t()) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: D#(constant()) -> c_4() 9: D#(t()) -> c_9() * Step 4: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - 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,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2 ,c_6/1,c_7/1,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,2,constant,div,ln,minus ,pow,t} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) * Step 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2 ,c_6/1,c_7/1,c_8/2,c_9/0} - 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 constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_2) = {1,2}, uargs(c_3) = {1,2}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [1] x1 + [1] x2 + [1] p(+) = [1] x1 + [1] x2 + [0] p(-) = [1] x1 + [1] x2 + [0] p(0) = [0] p(1) = [0] p(2) = [0] p(D) = [1] x1 + [8] p(constant) = [0] p(div) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(pow) = [1] x1 + [1] x2 + [0] p(t) = [1] p(D#) = [8] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [5] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] p(c_5) = [1] x1 + [1] x2 + [3] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [2] p(c_9) = [0] Following rules are strictly oriented: D#(*(x,y)) = [8] x + [8] y + [8] > [8] x + [8] y + [5] = c_1(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(+(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = c_2(D#(x),D#(y)) D#(-(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = c_3(D#(x),D#(y)) D#(div(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [3] = c_5(D#(x),D#(y)) D#(ln(x)) = [8] x + [0] >= [8] x + [0] = c_6(D#(x)) D#(minus(x)) = [8] x + [0] >= [8] x + [0] = c_7(D#(x)) D#(pow(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [2] = c_8(D#(x),D#(y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2 ,c_6/1,c_7/1,c_8/2,c_9/0} - 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 constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_2) = {1,2}, uargs(c_3) = {1,2}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [1] x1 + [1] x2 + [4] p(+) = [1] x1 + [1] x2 + [3] p(-) = [1] x1 + [1] x2 + [0] p(0) = [0] p(1) = [0] p(2) = [0] p(D) = [0] p(constant) = [0] p(div) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(pow) = [1] x1 + [1] x2 + [0] p(t) = [1] p(D#) = [2] x1 + [1] p(c_1) = [1] x1 + [1] x2 + [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [1] p(c_9) = [0] Following rules are strictly oriented: D#(+(x,y)) = [2] x + [2] y + [7] > [2] x + [2] y + [2] = c_2(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(*(x,y)) = [2] x + [2] y + [9] >= [2] x + [2] y + [2] = c_1(D#(x),D#(y)) D#(-(x,y)) = [2] x + [2] y + [1] >= [2] x + [2] y + [2] = c_3(D#(x),D#(y)) D#(div(x,y)) = [2] x + [2] y + [1] >= [2] x + [2] y + [2] = c_5(D#(x),D#(y)) D#(ln(x)) = [2] x + [1] >= [2] x + [1] = c_6(D#(x)) D#(minus(x)) = [2] x + [1] >= [2] x + [1] = c_7(D#(x)) D#(pow(x,y)) = [2] x + [2] y + [1] >= [2] x + [2] y + [3] = c_8(D#(x),D#(y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2 ,c_6/1,c_7/1,c_8/2,c_9/0} - 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 constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_2) = {1,2}, uargs(c_3) = {1,2}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [1] x1 + [1] x2 + [5] p(+) = [1] x1 + [1] x2 + [3] p(-) = [1] x1 + [1] x2 + [1] p(0) = [1] p(1) = [0] p(2) = [1] p(D) = [1] x1 + [2] p(constant) = [1] p(div) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [13] p(minus) = [1] x1 + [1] p(pow) = [1] x1 + [1] x2 + [12] p(t) = [0] p(D#) = [1] x1 + [3] p(c_1) = [1] x1 + [1] x2 + [2] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [0] p(c_9) = [0] Following rules are strictly oriented: D#(ln(x)) = [1] x + [16] > [1] x + [3] = c_6(D#(x)) D#(minus(x)) = [1] x + [4] > [1] x + [3] = c_7(D#(x)) D#(pow(x,y)) = [1] x + [1] y + [15] > [1] x + [1] y + [6] = c_8(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(*(x,y)) = [1] x + [1] y + [8] >= [1] x + [1] y + [8] = c_1(D#(x),D#(y)) D#(+(x,y)) = [1] x + [1] y + [6] >= [1] x + [1] y + [6] = c_2(D#(x),D#(y)) D#(-(x,y)) = [1] x + [1] y + [4] >= [1] x + [1] y + [6] = c_3(D#(x),D#(y)) D#(div(x,y)) = [1] x + [1] y + [3] >= [1] x + [1] y + [6] = c_5(D#(x),D#(y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(div(x,y)) -> c_5(D#(x),D#(y)) - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2 ,c_6/1,c_7/1,c_8/2,c_9/0} - 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 constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_2) = {1,2}, uargs(c_3) = {1,2}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [1] x1 + [1] x2 + [2] p(+) = [1] x1 + [1] x2 + [1] p(-) = [1] x1 + [1] x2 + [1] p(0) = [0] p(1) = [0] p(2) = [0] p(D) = [8] p(constant) = [8] p(div) = [1] x1 + [1] x2 + [10] p(ln) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(pow) = [1] x1 + [1] x2 + [1] p(t) = [0] p(D#) = [1] x1 + [1] p(c_1) = [1] x1 + [1] x2 + [1] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [0] p(c_9) = [0] Following rules are strictly oriented: D#(div(x,y)) = [1] x + [1] y + [11] > [1] x + [1] y + [2] = c_5(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(*(x,y)) = [1] x + [1] y + [3] >= [1] x + [1] y + [3] = c_1(D#(x),D#(y)) D#(+(x,y)) = [1] x + [1] y + [2] >= [1] x + [1] y + [2] = c_2(D#(x),D#(y)) D#(-(x,y)) = [1] x + [1] y + [2] >= [1] x + [1] y + [2] = c_3(D#(x),D#(y)) D#(ln(x)) = [1] x + [1] >= [1] x + [1] = c_6(D#(x)) D#(minus(x)) = [1] x + [1] >= [1] x + [1] = c_7(D#(x)) D#(pow(x,y)) = [1] x + [1] y + [2] >= [1] x + [1] y + [2] = c_8(D#(x),D#(y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(-(x,y)) -> c_3(D#(x),D#(y)) - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2 ,c_6/1,c_7/1,c_8/2,c_9/0} - 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 constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_2) = {1,2}, uargs(c_3) = {1,2}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [1] x1 + [1] x2 + [1] p(+) = [1] x1 + [1] x2 + [0] p(-) = [1] x1 + [1] x2 + [1] p(0) = [0] p(1) = [0] p(2) = [1] p(D) = [1] p(constant) = [0] p(div) = [1] x1 + [1] x2 + [1] p(ln) = [1] x1 + [1] p(minus) = [1] x1 + [1] p(pow) = [1] x1 + [1] x2 + [1] p(t) = [0] p(D#) = [4] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [2] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [1] x2 + [3] p(c_4) = [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [4] p(c_7) = [1] x1 + [2] p(c_8) = [1] x1 + [1] x2 + [1] p(c_9) = [2] Following rules are strictly oriented: D#(-(x,y)) = [4] x + [4] y + [4] > [4] x + [4] y + [3] = c_3(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(*(x,y)) = [4] x + [4] y + [4] >= [4] x + [4] y + [2] = c_1(D#(x),D#(y)) D#(+(x,y)) = [4] x + [4] y + [0] >= [4] x + [4] y + [0] = c_2(D#(x),D#(y)) D#(div(x,y)) = [4] x + [4] y + [4] >= [4] x + [4] y + [0] = c_5(D#(x),D#(y)) D#(ln(x)) = [4] x + [4] >= [4] x + [4] = c_6(D#(x)) D#(minus(x)) = [4] x + [4] >= [4] x + [2] = c_7(D#(x)) D#(pow(x,y)) = [4] x + [4] y + [4] >= [4] x + [4] y + [1] = c_8(D#(x),D#(y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 10: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: D#(*(x,y)) -> c_1(D#(x),D#(y)) D#(+(x,y)) -> c_2(D#(x),D#(y)) D#(-(x,y)) -> c_3(D#(x),D#(y)) D#(div(x,y)) -> c_5(D#(x),D#(y)) D#(ln(x)) -> c_6(D#(x)) D#(minus(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Signature: {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2 ,c_6/1,c_7/1,c_8/2,c_9/0} - 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^1))