WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: innermost runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: innermost runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: dx(x){x -> div(x,y)} = dx(div(x,y)) ->^+ minus(div(dx(x),y),times(x,div(dx(y),exp(y,two())))) = C[dx(x) = dx(x){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: innermost runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs dx#(X) -> c_1() dx#(a()) -> c_2() dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dx#(X) -> c_1() dx#(a()) -> c_2() dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) - Weak TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2} by application of Pre({1,2}) = {3,4,5,6,7,8,9}. Here rules are labelled as follows: 1: dx#(X) -> c_1() 2: dx#(a()) -> c_2() 3: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) 4: dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) 5: dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) 6: dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) 7: dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) 8: dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) 9: dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) ** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) - Weak DPs: dx#(X) -> c_1() dx#(a()) -> c_2() - Weak TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_2 dx#(a()) -> c_2():9 -->_1 dx#(a()) -> c_2():9 -->_2 dx#(X) -> c_1():8 -->_1 dx#(X) -> c_1():8 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 2:S:dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_2 dx#(a()) -> c_2():9 -->_1 dx#(a()) -> c_2():9 -->_2 dx#(X) -> c_1():8 -->_1 dx#(X) -> c_1():8 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 3:S:dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_1 dx#(a()) -> c_2():9 -->_1 dx#(X) -> c_1():8 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 4:S:dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_2 dx#(a()) -> c_2():9 -->_1 dx#(a()) -> c_2():9 -->_2 dx#(X) -> c_1():8 -->_1 dx#(X) -> c_1():8 -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 5:S:dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_1 dx#(a()) -> c_2():9 -->_1 dx#(X) -> c_1():8 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 6:S:dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_2 dx#(a()) -> c_2():9 -->_1 dx#(a()) -> c_2():9 -->_2 dx#(X) -> c_1():8 -->_1 dx#(X) -> c_1():8 -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 7:S:dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) -->_2 dx#(a()) -> c_2():9 -->_1 dx#(a()) -> c_2():9 -->_2 dx#(X) -> c_1():8 -->_1 dx#(X) -> c_1():8 -->_2 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_1 dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)):7 -->_2 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_1 dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)):6 -->_2 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_1 dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)):5 -->_2 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_1 dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)):4 -->_2 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_1 dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)):3 -->_2 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_1 dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)):2 -->_2 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 -->_1 dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)):1 8:W:dx#(X) -> c_1() 9:W:dx#(a()) -> c_2() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: dx#(X) -> c_1() 9: dx#(a()) -> c_2() ** Step 1.b:4: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) - Weak TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) ** Step 1.b:5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) - Signature: {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + 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_3) = {1,2}, uargs(c_4) = {1,2}, uargs(c_5) = {1}, uargs(c_6) = {1,2}, uargs(c_7) = {1}, uargs(c_8) = {1,2}, uargs(c_9) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(div) = [1] x1 + [1] x2 + [0] p(dx) = [0] p(exp) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [0] p(minus) = [1] x1 + [1] x2 + [0] p(neg) = [1] x1 + [0] p(one) = [0] p(plus) = [1] x1 + [1] x2 + [0] p(times) = [1] x1 + [1] x2 + [1] p(two) = [0] p(zero) = [0] p(dx#) = [2] x1 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [1] x2 + [15] p(c_7) = [1] x1 + [5] p(c_8) = [1] x1 + [1] x2 + [15] p(c_9) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: dx#(times(ALPHA,BETA)) = [2] ALPHA + [2] BETA + [3] > [2] ALPHA + [2] BETA + [2] = c_9(dx#(ALPHA),dx#(BETA)) Following rules are (at-least) weakly oriented: dx#(div(ALPHA,BETA)) = [2] ALPHA + [2] BETA + [1] >= [2] ALPHA + [2] BETA + [2] = c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) = [2] ALPHA + [2] BETA + [1] >= [2] ALPHA + [2] BETA + [2] = c_4(dx#(ALPHA),dx#(BETA)) dx#(ln(ALPHA)) = [2] ALPHA + [1] >= [2] ALPHA + [1] = c_5(dx#(ALPHA)) dx#(minus(ALPHA,BETA)) = [2] ALPHA + [2] BETA + [1] >= [2] ALPHA + [2] BETA + [17] = c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) = [2] ALPHA + [1] >= [2] ALPHA + [6] = c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) = [2] ALPHA + [2] BETA + [1] >= [2] ALPHA + [2] BETA + [17] = c_8(dx#(ALPHA),dx#(BETA)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:6: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) - Weak DPs: dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) - Signature: {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + 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(c_3) = {1,2}, uargs(c_4) = {1,2}, uargs(c_5) = {1}, uargs(c_6) = {1,2}, uargs(c_7) = {1}, uargs(c_8) = {1,2}, uargs(c_9) = {1,2} Following symbols are considered usable: {dx#} TcT has computed the following interpretation: p(a) = 0 p(div) = 1 + x1 + x2 p(dx) = 1 p(exp) = 1 + x1 + x2 p(ln) = x1 p(minus) = 1 + x1 + x2 p(neg) = 1 + x1 p(one) = 1 p(plus) = 1 + x1 + x2 p(times) = 3 + x1 + x2 p(two) = 1 p(zero) = 1 p(dx#) = 8*x1 p(c_1) = 0 p(c_2) = 1 p(c_3) = x1 + x2 p(c_4) = 1 + x1 + x2 p(c_5) = x1 p(c_6) = 2 + x1 + x2 p(c_7) = x1 p(c_8) = x1 + x2 p(c_9) = x1 + x2 Following rules are strictly oriented: dx#(div(ALPHA,BETA)) = 8 + 8*ALPHA + 8*BETA > 8*ALPHA + 8*BETA = c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) = 8 + 8*ALPHA + 8*BETA > 1 + 8*ALPHA + 8*BETA = c_4(dx#(ALPHA),dx#(BETA)) dx#(minus(ALPHA,BETA)) = 8 + 8*ALPHA + 8*BETA > 2 + 8*ALPHA + 8*BETA = c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) = 8 + 8*ALPHA > 8*ALPHA = c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) = 8 + 8*ALPHA + 8*BETA > 8*ALPHA + 8*BETA = c_8(dx#(ALPHA),dx#(BETA)) Following rules are (at-least) weakly oriented: dx#(ln(ALPHA)) = 8*ALPHA >= 8*ALPHA = c_5(dx#(ALPHA)) dx#(times(ALPHA,BETA)) = 24 + 8*ALPHA + 8*BETA >= 8*ALPHA + 8*BETA = c_9(dx#(ALPHA),dx#(BETA)) ** Step 1.b:7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) - Weak DPs: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) - Signature: {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + 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_3) = {1,2}, uargs(c_4) = {1,2}, uargs(c_5) = {1}, uargs(c_6) = {1,2}, uargs(c_7) = {1}, uargs(c_8) = {1,2}, uargs(c_9) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(div) = [1] x1 + [1] x2 + [0] p(dx) = [0] p(exp) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [5] p(minus) = [1] x1 + [1] x2 + [0] p(neg) = [1] x1 + [0] p(one) = [0] p(plus) = [1] x1 + [1] x2 + [0] p(times) = [1] x1 + [1] x2 + [0] p(two) = [0] p(zero) = [0] p(dx#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [1] x2 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [0] p(c_9) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: dx#(ln(ALPHA)) = [1] ALPHA + [5] > [1] ALPHA + [0] = c_5(dx#(ALPHA)) Following rules are (at-least) weakly oriented: dx#(div(ALPHA,BETA)) = [1] ALPHA + [1] BETA + [0] >= [1] ALPHA + [1] BETA + [0] = c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) = [1] ALPHA + [1] BETA + [0] >= [1] ALPHA + [1] BETA + [0] = c_4(dx#(ALPHA),dx#(BETA)) dx#(minus(ALPHA,BETA)) = [1] ALPHA + [1] BETA + [0] >= [1] ALPHA + [1] BETA + [0] = c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) = [1] ALPHA + [0] >= [1] ALPHA + [0] = c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) = [1] ALPHA + [1] BETA + [0] >= [1] ALPHA + [1] BETA + [0] = c_8(dx#(ALPHA),dx#(BETA)) dx#(times(ALPHA,BETA)) = [1] ALPHA + [1] BETA + [0] >= [1] ALPHA + [1] BETA + [0] = c_9(dx#(ALPHA),dx#(BETA)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: dx#(div(ALPHA,BETA)) -> c_3(dx#(ALPHA),dx#(BETA)) dx#(exp(ALPHA,BETA)) -> c_4(dx#(ALPHA),dx#(BETA)) dx#(ln(ALPHA)) -> c_5(dx#(ALPHA)) dx#(minus(ALPHA,BETA)) -> c_6(dx#(ALPHA),dx#(BETA)) dx#(neg(ALPHA)) -> c_7(dx#(ALPHA)) dx#(plus(ALPHA,BETA)) -> c_8(dx#(ALPHA),dx#(BETA)) dx#(times(ALPHA,BETA)) -> c_9(dx#(ALPHA),dx#(BETA)) - Signature: {dx/1,dx#/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0,c_1/0,c_2/0,c_3/2,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {dx#} and constructors {a,div,exp,ln,minus,neg,one,plus ,times,two,zero} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))