WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: DX(z0) -> c() DX(a()) -> c1() DX(div(z0,z1)) -> c10(DX(z1)) DX(div(z0,z1)) -> c9(DX(z0)) DX(exp(z0,z1)) -> c12(DX(z0)) DX(exp(z0,z1)) -> c13(DX(z1)) DX(ln(z0)) -> c11(DX(z0)) DX(minus(z0,z1)) -> c6(DX(z0)) DX(minus(z0,z1)) -> c7(DX(z1)) DX(neg(z0)) -> c8(DX(z0)) DX(plus(z0,z1)) -> c2(DX(z0)) DX(plus(z0,z1)) -> c3(DX(z1)) DX(times(z0,z1)) -> c4(DX(z0)) DX(times(z0,z1)) -> c5(DX(z1)) - Weak TRS: dx(z0) -> one() dx(a()) -> zero() dx(div(z0,z1)) -> minus(div(dx(z0),z1),times(z0,div(dx(z1),exp(z1,two())))) dx(exp(z0,z1)) -> plus(times(z1,times(exp(z0,minus(z1,one())),dx(z0))) ,times(exp(z0,z1),times(ln(z0),dx(z1)))) dx(ln(z0)) -> div(dx(z0),z0) dx(minus(z0,z1)) -> minus(dx(z0),dx(z1)) dx(neg(z0)) -> neg(dx(z0)) dx(plus(z0,z1)) -> plus(dx(z0),dx(z1)) dx(times(z0,z1)) -> plus(times(z1,dx(z0)),times(z0,dx(z1))) - Signature: {DX/1,dx/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1,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,dx} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3,c4 ,c5,c6,c7,c8,c9,div,exp,ln,minus,neg,one,plus,times,two,zero} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: DX(z0) -> c() DX(a()) -> c1() DX(div(z0,z1)) -> c10(DX(z1)) DX(div(z0,z1)) -> c9(DX(z0)) DX(exp(z0,z1)) -> c12(DX(z0)) DX(exp(z0,z1)) -> c13(DX(z1)) DX(ln(z0)) -> c11(DX(z0)) DX(minus(z0,z1)) -> c6(DX(z0)) DX(minus(z0,z1)) -> c7(DX(z1)) DX(neg(z0)) -> c8(DX(z0)) DX(plus(z0,z1)) -> c2(DX(z0)) DX(plus(z0,z1)) -> c3(DX(z1)) DX(times(z0,z1)) -> c4(DX(z0)) DX(times(z0,z1)) -> c5(DX(z1)) - Weak TRS: dx(z0) -> one() dx(a()) -> zero() dx(div(z0,z1)) -> minus(div(dx(z0),z1),times(z0,div(dx(z1),exp(z1,two())))) dx(exp(z0,z1)) -> plus(times(z1,times(exp(z0,minus(z1,one())),dx(z0))) ,times(exp(z0,z1),times(ln(z0),dx(z1)))) dx(ln(z0)) -> div(dx(z0),z0) dx(minus(z0,z1)) -> minus(dx(z0),dx(z1)) dx(neg(z0)) -> neg(dx(z0)) dx(plus(z0,z1)) -> plus(dx(z0),dx(z1)) dx(times(z0,z1)) -> plus(times(z1,dx(z0)),times(z0,dx(z1))) - Signature: {DX/1,dx/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1,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,dx} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3,c4 ,c5,c6,c7,c8,c9,div,exp,ln,minus,neg,one,plus,times,two,zero} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: DX(z0) -> c() DX(a()) -> c1() DX(div(z0,z1)) -> c10(DX(z1)) DX(div(z0,z1)) -> c9(DX(z0)) DX(exp(z0,z1)) -> c12(DX(z0)) DX(exp(z0,z1)) -> c13(DX(z1)) DX(ln(z0)) -> c11(DX(z0)) DX(minus(z0,z1)) -> c6(DX(z0)) DX(minus(z0,z1)) -> c7(DX(z1)) DX(neg(z0)) -> c8(DX(z0)) DX(plus(z0,z1)) -> c2(DX(z0)) DX(plus(z0,z1)) -> c3(DX(z1)) DX(times(z0,z1)) -> c4(DX(z0)) DX(times(z0,z1)) -> c5(DX(z1)) - Weak TRS: dx(z0) -> one() dx(a()) -> zero() dx(div(z0,z1)) -> minus(div(dx(z0),z1),times(z0,div(dx(z1),exp(z1,two())))) dx(exp(z0,z1)) -> plus(times(z1,times(exp(z0,minus(z1,one())),dx(z0))) ,times(exp(z0,z1),times(ln(z0),dx(z1)))) dx(ln(z0)) -> div(dx(z0),z0) dx(minus(z0,z1)) -> minus(dx(z0),dx(z1)) dx(neg(z0)) -> neg(dx(z0)) dx(plus(z0,z1)) -> plus(dx(z0),dx(z1)) dx(times(z0,z1)) -> plus(times(z1,dx(z0)),times(z0,dx(z1))) - Signature: {DX/1,dx/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1,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,dx} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3,c4 ,c5,c6,c7,c8,c9,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(y){y -> div(x,y)} = DX(div(x,y)) ->^+ c10(DX(y)) = C[DX(y) = DX(y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: DX(z0) -> c() DX(a()) -> c1() DX(div(z0,z1)) -> c10(DX(z1)) DX(div(z0,z1)) -> c9(DX(z0)) DX(exp(z0,z1)) -> c12(DX(z0)) DX(exp(z0,z1)) -> c13(DX(z1)) DX(ln(z0)) -> c11(DX(z0)) DX(minus(z0,z1)) -> c6(DX(z0)) DX(minus(z0,z1)) -> c7(DX(z1)) DX(neg(z0)) -> c8(DX(z0)) DX(plus(z0,z1)) -> c2(DX(z0)) DX(plus(z0,z1)) -> c3(DX(z1)) DX(times(z0,z1)) -> c4(DX(z0)) DX(times(z0,z1)) -> c5(DX(z1)) - Weak TRS: dx(z0) -> one() dx(a()) -> zero() dx(div(z0,z1)) -> minus(div(dx(z0),z1),times(z0,div(dx(z1),exp(z1,two())))) dx(exp(z0,z1)) -> plus(times(z1,times(exp(z0,minus(z1,one())),dx(z0))) ,times(exp(z0,z1),times(ln(z0),dx(z1)))) dx(ln(z0)) -> div(dx(z0),z0) dx(minus(z0,z1)) -> minus(dx(z0),dx(z1)) dx(neg(z0)) -> neg(dx(z0)) dx(plus(z0,z1)) -> plus(dx(z0),dx(z1)) dx(times(z0,z1)) -> plus(times(z1,dx(z0)),times(z0,dx(z1))) - Signature: {DX/1,dx/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1,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,dx} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3,c4 ,c5,c6,c7,c8,c9,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#(z0) -> c_1() DX#(a()) -> c_2() DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) Weak DPs dx#(z0) -> c_15() dx#(a()) -> c_16() dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)) dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)) dx#(ln(z0)) -> c_19(dx#(z0)) dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)) dx#(neg(z0)) -> c_21(dx#(z0)) dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)) dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)) and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(z0) -> c_1() DX#(a()) -> c_2() DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Weak DPs: dx#(z0) -> c_15() dx#(a()) -> c_16() dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)) dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)) dx#(ln(z0)) -> c_19(dx#(z0)) dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)) dx#(neg(z0)) -> c_21(dx#(z0)) dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)) dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)) - Weak TRS: DX(z0) -> c() DX(a()) -> c1() DX(div(z0,z1)) -> c10(DX(z1)) DX(div(z0,z1)) -> c9(DX(z0)) DX(exp(z0,z1)) -> c12(DX(z0)) DX(exp(z0,z1)) -> c13(DX(z1)) DX(ln(z0)) -> c11(DX(z0)) DX(minus(z0,z1)) -> c6(DX(z0)) DX(minus(z0,z1)) -> c7(DX(z1)) DX(neg(z0)) -> c8(DX(z0)) DX(plus(z0,z1)) -> c2(DX(z0)) DX(plus(z0,z1)) -> c3(DX(z1)) DX(times(z0,z1)) -> c4(DX(z0)) DX(times(z0,z1)) -> c5(DX(z1)) dx(z0) -> one() dx(a()) -> zero() dx(div(z0,z1)) -> minus(div(dx(z0),z1),times(z0,div(dx(z1),exp(z1,two())))) dx(exp(z0,z1)) -> plus(times(z1,times(exp(z0,minus(z1,one())),dx(z0))) ,times(exp(z0,z1),times(ln(z0),dx(z1)))) dx(ln(z0)) -> div(dx(z0),z0) dx(minus(z0,z1)) -> minus(dx(z0),dx(z1)) dx(neg(z0)) -> neg(dx(z0)) dx(plus(z0,z1)) -> plus(dx(z0),dx(z1)) dx(times(z0,z1)) -> plus(times(z1,dx(z0)),times(z0,dx(z1))) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,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}) = {3,4,5,6,7,8,9,10,11,12,13,14}. Here rules are labelled as follows: 1: DX#(z0) -> c_1() 2: DX#(a()) -> c_2() 3: DX#(div(z0,z1)) -> c_3(DX#(z1)) 4: DX#(div(z0,z1)) -> c_4(DX#(z0)) 5: DX#(exp(z0,z1)) -> c_5(DX#(z0)) 6: DX#(exp(z0,z1)) -> c_6(DX#(z1)) 7: DX#(ln(z0)) -> c_7(DX#(z0)) 8: DX#(minus(z0,z1)) -> c_8(DX#(z0)) 9: DX#(minus(z0,z1)) -> c_9(DX#(z1)) 10: DX#(neg(z0)) -> c_10(DX#(z0)) 11: DX#(plus(z0,z1)) -> c_11(DX#(z0)) 12: DX#(plus(z0,z1)) -> c_12(DX#(z1)) 13: DX#(times(z0,z1)) -> c_13(DX#(z0)) 14: DX#(times(z0,z1)) -> c_14(DX#(z1)) 15: dx#(z0) -> c_15() 16: dx#(a()) -> c_16() 17: dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)) 18: dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)) 19: dx#(ln(z0)) -> c_19(dx#(z0)) 20: dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)) 21: dx#(neg(z0)) -> c_21(dx#(z0)) 22: dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)) 23: dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Weak DPs: DX#(z0) -> c_1() DX#(a()) -> c_2() dx#(z0) -> c_15() dx#(a()) -> c_16() dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)) dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)) dx#(ln(z0)) -> c_19(dx#(z0)) dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)) dx#(neg(z0)) -> c_21(dx#(z0)) dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)) dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)) - Weak TRS: DX(z0) -> c() DX(a()) -> c1() DX(div(z0,z1)) -> c10(DX(z1)) DX(div(z0,z1)) -> c9(DX(z0)) DX(exp(z0,z1)) -> c12(DX(z0)) DX(exp(z0,z1)) -> c13(DX(z1)) DX(ln(z0)) -> c11(DX(z0)) DX(minus(z0,z1)) -> c6(DX(z0)) DX(minus(z0,z1)) -> c7(DX(z1)) DX(neg(z0)) -> c8(DX(z0)) DX(plus(z0,z1)) -> c2(DX(z0)) DX(plus(z0,z1)) -> c3(DX(z1)) DX(times(z0,z1)) -> c4(DX(z0)) DX(times(z0,z1)) -> c5(DX(z1)) dx(z0) -> one() dx(a()) -> zero() dx(div(z0,z1)) -> minus(div(dx(z0),z1),times(z0,div(dx(z1),exp(z1,two())))) dx(exp(z0,z1)) -> plus(times(z1,times(exp(z0,minus(z1,one())),dx(z0))) ,times(exp(z0,z1),times(ln(z0),dx(z1)))) dx(ln(z0)) -> div(dx(z0),z0) dx(minus(z0,z1)) -> minus(dx(z0),dx(z1)) dx(neg(z0)) -> neg(dx(z0)) dx(plus(z0,z1)) -> plus(dx(z0),dx(z1)) dx(times(z0,z1)) -> plus(times(z1,dx(z0)),times(z0,dx(z1))) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,div,exp,ln,minus,neg,one,plus,times,two,zero} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:DX#(div(z0,z1)) -> c_3(DX#(z1)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 2:S:DX#(div(z0,z1)) -> c_4(DX#(z0)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 3:S:DX#(exp(z0,z1)) -> c_5(DX#(z0)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 4:S:DX#(exp(z0,z1)) -> c_6(DX#(z1)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 5:S:DX#(ln(z0)) -> c_7(DX#(z0)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 6:S:DX#(minus(z0,z1)) -> c_8(DX#(z0)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 7:S:DX#(minus(z0,z1)) -> c_9(DX#(z1)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 8:S:DX#(neg(z0)) -> c_10(DX#(z0)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 9:S:DX#(plus(z0,z1)) -> c_11(DX#(z0)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 10:S:DX#(plus(z0,z1)) -> c_12(DX#(z1)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 11:S:DX#(times(z0,z1)) -> c_13(DX#(z0)) -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 12:S:DX#(times(z0,z1)) -> c_14(DX#(z1)) -->_1 DX#(a()) -> c_2():14 -->_1 DX#(z0) -> c_1():13 -->_1 DX#(times(z0,z1)) -> c_14(DX#(z1)):12 -->_1 DX#(times(z0,z1)) -> c_13(DX#(z0)):11 -->_1 DX#(plus(z0,z1)) -> c_12(DX#(z1)):10 -->_1 DX#(plus(z0,z1)) -> c_11(DX#(z0)):9 -->_1 DX#(neg(z0)) -> c_10(DX#(z0)):8 -->_1 DX#(minus(z0,z1)) -> c_9(DX#(z1)):7 -->_1 DX#(minus(z0,z1)) -> c_8(DX#(z0)):6 -->_1 DX#(ln(z0)) -> c_7(DX#(z0)):5 -->_1 DX#(exp(z0,z1)) -> c_6(DX#(z1)):4 -->_1 DX#(exp(z0,z1)) -> c_5(DX#(z0)):3 -->_1 DX#(div(z0,z1)) -> c_4(DX#(z0)):2 -->_1 DX#(div(z0,z1)) -> c_3(DX#(z1)):1 13:W:DX#(z0) -> c_1() 14:W:DX#(a()) -> c_2() 15:W:dx#(z0) -> c_15() 16:W:dx#(a()) -> c_16() 17:W:dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)) -->_2 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_1 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_2 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_1 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_2 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_1 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_2 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_1 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_2 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_1 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_2 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_1 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_2 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_1 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_2 dx#(a()) -> c_16():16 -->_1 dx#(a()) -> c_16():16 -->_2 dx#(z0) -> c_15():15 -->_1 dx#(z0) -> c_15():15 18:W:dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)) -->_2 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_1 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_2 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_1 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_2 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_1 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_2 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_1 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_2 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_1 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_2 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_1 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_2 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_1 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_2 dx#(a()) -> c_16():16 -->_1 dx#(a()) -> c_16():16 -->_2 dx#(z0) -> c_15():15 -->_1 dx#(z0) -> c_15():15 19:W:dx#(ln(z0)) -> c_19(dx#(z0)) -->_1 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_1 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_1 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_1 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_1 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_1 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_1 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_1 dx#(a()) -> c_16():16 -->_1 dx#(z0) -> c_15():15 20:W:dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)) -->_2 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_1 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_2 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_1 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_2 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_1 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_2 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_1 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_2 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_1 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_2 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_1 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_2 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_1 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_2 dx#(a()) -> c_16():16 -->_1 dx#(a()) -> c_16():16 -->_2 dx#(z0) -> c_15():15 -->_1 dx#(z0) -> c_15():15 21:W:dx#(neg(z0)) -> c_21(dx#(z0)) -->_1 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_1 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_1 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_1 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_1 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_1 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_1 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_1 dx#(a()) -> c_16():16 -->_1 dx#(z0) -> c_15():15 22:W:dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)) -->_2 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_1 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_2 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_1 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_2 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_1 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_2 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_1 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_2 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_1 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_2 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_1 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_2 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_1 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_2 dx#(a()) -> c_16():16 -->_1 dx#(a()) -> c_16():16 -->_2 dx#(z0) -> c_15():15 -->_1 dx#(z0) -> c_15():15 23:W:dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)) -->_2 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_1 dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)):23 -->_2 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_1 dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)):22 -->_2 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_1 dx#(neg(z0)) -> c_21(dx#(z0)):21 -->_2 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_1 dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)):20 -->_2 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_1 dx#(ln(z0)) -> c_19(dx#(z0)):19 -->_2 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_1 dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)):18 -->_2 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_1 dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)):17 -->_2 dx#(a()) -> c_16():16 -->_1 dx#(a()) -> c_16():16 -->_2 dx#(z0) -> c_15():15 -->_1 dx#(z0) -> c_15():15 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 17: dx#(div(z0,z1)) -> c_17(dx#(z0),dx#(z1)) 23: dx#(times(z0,z1)) -> c_23(dx#(z0),dx#(z1)) 22: dx#(plus(z0,z1)) -> c_22(dx#(z0),dx#(z1)) 21: dx#(neg(z0)) -> c_21(dx#(z0)) 20: dx#(minus(z0,z1)) -> c_20(dx#(z0),dx#(z1)) 19: dx#(ln(z0)) -> c_19(dx#(z0)) 18: dx#(exp(z0,z1)) -> c_18(dx#(z0),dx#(z1)) 16: dx#(a()) -> c_16() 15: dx#(z0) -> c_15() 13: DX#(z0) -> c_1() 14: DX#(a()) -> c_2() ** Step 1.b:4: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Weak TRS: DX(z0) -> c() DX(a()) -> c1() DX(div(z0,z1)) -> c10(DX(z1)) DX(div(z0,z1)) -> c9(DX(z0)) DX(exp(z0,z1)) -> c12(DX(z0)) DX(exp(z0,z1)) -> c13(DX(z1)) DX(ln(z0)) -> c11(DX(z0)) DX(minus(z0,z1)) -> c6(DX(z0)) DX(minus(z0,z1)) -> c7(DX(z1)) DX(neg(z0)) -> c8(DX(z0)) DX(plus(z0,z1)) -> c2(DX(z0)) DX(plus(z0,z1)) -> c3(DX(z1)) DX(times(z0,z1)) -> c4(DX(z0)) DX(times(z0,z1)) -> c5(DX(z1)) dx(z0) -> one() dx(a()) -> zero() dx(div(z0,z1)) -> minus(div(dx(z0),z1),times(z0,div(dx(z1),exp(z1,two())))) dx(exp(z0,z1)) -> plus(times(z1,times(exp(z0,minus(z1,one())),dx(z0))) ,times(exp(z0,z1),times(ln(z0),dx(z1)))) dx(ln(z0)) -> div(dx(z0),z0) dx(minus(z0,z1)) -> minus(dx(z0),dx(z1)) dx(neg(z0)) -> neg(dx(z0)) dx(plus(z0,z1)) -> plus(dx(z0),dx(z1)) dx(times(z0,z1)) -> plus(times(z1,dx(z0)),times(z0,dx(z1))) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,div,exp,ln,minus,neg,one,plus,times,two,zero} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) ** Step 1.b:5: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,div,exp,ln,minus,neg,one,plus,times,two,zero} + 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(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {DX#,dx#} TcT has computed the following interpretation: p(DX) = [0] p(a) = [2] p(c) = [1] p(c1) = [2] p(c10) = [0] p(c11) = [1] x1 + [8] p(c12) = [0] p(c13) = [1] x1 + [0] p(c2) = [1] x1 + [0] p(c3) = [1] p(c4) = [1] p(c5) = [8] p(c6) = [4] p(c7) = [4] p(c8) = [1] p(c9) = [0] p(div) = [1] x1 + [1] x2 + [0] p(dx) = [1] x1 + [2] p(exp) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [0] p(minus) = [1] x1 + [1] x2 + [0] p(neg) = [1] x1 + [1] p(one) = [1] p(plus) = [1] x1 + [1] x2 + [0] p(times) = [1] x1 + [1] x2 + [0] p(two) = [1] p(zero) = [0] p(DX#) = [4] x1 + [0] p(dx#) = [2] x1 + [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [3] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] p(c_18) = [2] x2 + [2] p(c_19) = [1] p(c_20) = [2] x1 + [1] x2 + [0] p(c_21) = [1] p(c_22) = [1] x1 + [2] x2 + [0] p(c_23) = [2] x2 + [1] Following rules are strictly oriented: DX#(neg(z0)) = [4] z0 + [4] > [4] z0 + [3] = c_10(DX#(z0)) Following rules are (at-least) weakly oriented: DX#(div(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_3(DX#(z1)) DX#(div(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_4(DX#(z0)) DX#(exp(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_5(DX#(z0)) DX#(exp(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_6(DX#(z1)) DX#(ln(z0)) = [4] z0 + [0] >= [4] z0 + [0] = c_7(DX#(z0)) DX#(minus(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_8(DX#(z0)) DX#(minus(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_9(DX#(z1)) DX#(plus(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_11(DX#(z0)) DX#(plus(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_12(DX#(z1)) DX#(times(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_13(DX#(z0)) DX#(times(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_14(DX#(z1)) ** Step 1.b:6: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Weak DPs: DX#(neg(z0)) -> c_10(DX#(z0)) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,div,exp,ln,minus,neg,one,plus,times,two,zero} + 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(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {DX#,dx#} TcT has computed the following interpretation: p(DX) = [1] x1 + [1] p(a) = [0] p(c) = [0] p(c1) = [1] p(c10) = [4] p(c11) = [1] x1 + [1] p(c12) = [1] x1 + [1] p(c13) = [1] p(c2) = [1] p(c3) = [0] p(c4) = [1] x1 + [1] p(c5) = [1] p(c6) = [1] p(c7) = [1] x1 + [0] p(c8) = [1] x1 + [4] p(c9) = [2] p(div) = [1] x1 + [1] x2 + [0] p(dx) = [2] x1 + [0] p(exp) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [0] p(minus) = [1] x1 + [1] x2 + [2] p(neg) = [1] x1 + [3] p(one) = [0] p(plus) = [1] x1 + [1] x2 + [0] p(times) = [1] x1 + [1] x2 + [0] p(two) = [0] p(zero) = [8] p(DX#) = [8] x1 + [0] p(dx#) = [1] p(c_1) = [1] p(c_2) = [8] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [8] p(c_9) = [1] x1 + [8] p(c_10) = [1] x1 + [15] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [2] p(c_17) = [8] x1 + [8] p(c_18) = [1] x1 + [4] p(c_19) = [1] x1 + [1] p(c_20) = [2] x2 + [2] p(c_21) = [2] x1 + [1] p(c_22) = [2] x2 + [0] p(c_23) = [1] x1 + [1] x2 + [2] Following rules are strictly oriented: DX#(minus(z0,z1)) = [8] z0 + [8] z1 + [16] > [8] z0 + [8] = c_8(DX#(z0)) DX#(minus(z0,z1)) = [8] z0 + [8] z1 + [16] > [8] z1 + [8] = c_9(DX#(z1)) Following rules are (at-least) weakly oriented: DX#(div(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z1 + [0] = c_3(DX#(z1)) DX#(div(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z0 + [0] = c_4(DX#(z0)) DX#(exp(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z0 + [0] = c_5(DX#(z0)) DX#(exp(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z1 + [0] = c_6(DX#(z1)) DX#(ln(z0)) = [8] z0 + [0] >= [8] z0 + [0] = c_7(DX#(z0)) DX#(neg(z0)) = [8] z0 + [24] >= [8] z0 + [15] = c_10(DX#(z0)) DX#(plus(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z0 + [0] = c_11(DX#(z0)) DX#(plus(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z1 + [0] = c_12(DX#(z1)) DX#(times(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z0 + [0] = c_13(DX#(z0)) DX#(times(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z1 + [0] = c_14(DX#(z1)) ** Step 1.b:7: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Weak DPs: DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,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}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(DX) = [0] p(a) = [0] p(c) = [0] p(c1) = [0] p(c10) = [1] x1 + [0] p(c11) = [1] x1 + [0] p(c12) = [1] x1 + [0] p(c13) = [1] x1 + [0] p(c2) = [1] x1 + [0] p(c3) = [1] x1 + [0] p(c4) = [1] x1 + [0] p(c5) = [1] x1 + [0] p(c6) = [1] x1 + [0] p(c7) = [1] x1 + [0] p(c8) = [1] x1 + [0] p(c9) = [1] x1 + [0] p(div) = [1] x1 + [1] x2 + [5] p(dx) = [0] p(exp) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [3] 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 + [5] p(dx#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [1] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [8] x1 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [1] x1 + [1] x2 + [0] p(c_23) = [2] x1 + [1] Following rules are strictly oriented: DX#(div(z0,z1)) = [2] z0 + [2] z1 + [15] > [2] z1 + [5] = c_3(DX#(z1)) DX#(div(z0,z1)) = [2] z0 + [2] z1 + [15] > [2] z0 + [5] = c_4(DX#(z0)) DX#(ln(z0)) = [2] z0 + [11] > [2] z0 + [5] = c_7(DX#(z0)) DX#(times(z0,z1)) = [2] z0 + [2] z1 + [7] > [2] z0 + [5] = c_13(DX#(z0)) DX#(times(z0,z1)) = [2] z0 + [2] z1 + [7] > [2] z1 + [6] = c_14(DX#(z1)) Following rules are (at-least) weakly oriented: DX#(exp(z0,z1)) = [2] z0 + [2] z1 + [5] >= [2] z0 + [5] = c_5(DX#(z0)) DX#(exp(z0,z1)) = [2] z0 + [2] z1 + [5] >= [2] z1 + [5] = c_6(DX#(z1)) DX#(minus(z0,z1)) = [2] z0 + [2] z1 + [5] >= [2] z0 + [5] = c_8(DX#(z0)) DX#(minus(z0,z1)) = [2] z0 + [2] z1 + [5] >= [2] z1 + [5] = c_9(DX#(z1)) DX#(neg(z0)) = [2] z0 + [5] >= [2] z0 + [5] = c_10(DX#(z0)) DX#(plus(z0,z1)) = [2] z0 + [2] z1 + [5] >= [2] z0 + [5] = c_11(DX#(z0)) DX#(plus(z0,z1)) = [2] z0 + [2] z1 + [5] >= [2] z1 + [5] = c_12(DX#(z1)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:8: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) - Weak DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,div,exp,ln,minus,neg,one,plus,times,two,zero} + 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(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {DX#,dx#} TcT has computed the following interpretation: p(DX) = [2] x1 + [0] p(a) = [0] p(c) = [2] p(c1) = [0] p(c10) = [2] p(c11) = [1] p(c12) = [2] p(c13) = [1] p(c2) = [1] p(c3) = [1] p(c4) = [1] x1 + [1] p(c5) = [0] p(c6) = [4] p(c7) = [1] p(c8) = [1] x1 + [0] p(c9) = [1] p(div) = [1] x1 + [1] x2 + [0] p(dx) = [0] p(exp) = [1] x1 + [1] x2 + [1] p(ln) = [1] x1 + [0] p(minus) = [1] x1 + [1] x2 + [0] p(neg) = [1] x1 + [2] p(one) = [1] p(plus) = [1] x1 + [1] x2 + [0] p(times) = [1] x1 + [1] x2 + [0] p(two) = [1] p(zero) = [1] p(DX#) = [8] x1 + [0] p(dx#) = [2] x1 + [1] p(c_1) = [2] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [8] p(c_6) = [1] x1 + [7] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [12] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [2] p(c_16) = [0] p(c_17) = [1] x1 + [2] x2 + [1] p(c_18) = [2] x1 + [4] x2 + [2] p(c_19) = [0] p(c_20) = [1] x2 + [1] p(c_21) = [1] x1 + [2] p(c_22) = [2] x1 + [1] x2 + [0] p(c_23) = [2] x2 + [4] Following rules are strictly oriented: DX#(exp(z0,z1)) = [8] z0 + [8] z1 + [8] > [8] z1 + [7] = c_6(DX#(z1)) Following rules are (at-least) weakly oriented: DX#(div(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z1 + [0] = c_3(DX#(z1)) DX#(div(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z0 + [0] = c_4(DX#(z0)) DX#(exp(z0,z1)) = [8] z0 + [8] z1 + [8] >= [8] z0 + [8] = c_5(DX#(z0)) DX#(ln(z0)) = [8] z0 + [0] >= [8] z0 + [0] = c_7(DX#(z0)) DX#(minus(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z0 + [0] = c_8(DX#(z0)) DX#(minus(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z1 + [0] = c_9(DX#(z1)) DX#(neg(z0)) = [8] z0 + [16] >= [8] z0 + [12] = c_10(DX#(z0)) DX#(plus(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z0 + [0] = c_11(DX#(z0)) DX#(plus(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z1 + [0] = c_12(DX#(z1)) DX#(times(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z0 + [0] = c_13(DX#(z0)) DX#(times(z0,z1)) = [8] z0 + [8] z1 + [0] >= [8] z1 + [0] = c_14(DX#(z1)) ** Step 1.b:9: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) - Weak DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,div,exp,ln,minus,neg,one,plus,times,two,zero} + 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(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {DX#,dx#} TcT has computed the following interpretation: p(DX) = [1] x1 + [1] p(a) = [1] p(c) = [1] p(c1) = [1] p(c10) = [1] x1 + [0] p(c11) = [1] x1 + [0] p(c12) = [4] p(c13) = [1] x1 + [0] p(c2) = [1] x1 + [1] p(c3) = [0] p(c4) = [1] x1 + [1] p(c5) = [0] p(c6) = [1] x1 + [1] p(c7) = [1] x1 + [0] p(c8) = [2] p(c9) = [1] p(div) = [1] x1 + [1] x2 + [5] p(dx) = [1] x1 + [0] p(exp) = [1] x1 + [1] x2 + [3] p(ln) = [1] x1 + [4] p(minus) = [1] x1 + [1] x2 + [10] p(neg) = [1] x1 + [14] p(one) = [0] p(plus) = [1] x1 + [1] x2 + [0] p(times) = [1] x1 + [1] x2 + [0] p(two) = [1] p(zero) = [0] p(DX#) = [2] x1 + [0] p(dx#) = [1] x1 + [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [10] p(c_4) = [1] x1 + [10] p(c_5) = [1] x1 + [2] p(c_6) = [1] x1 + [6] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [8] p(c_9) = [1] x1 + [1] p(c_10) = [1] x1 + [8] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [8] p(c_16) = [2] p(c_17) = [2] x1 + [0] p(c_18) = [1] x1 + [1] p(c_19) = [2] x1 + [1] p(c_20) = [1] x1 + [0] p(c_21) = [2] x1 + [8] p(c_22) = [2] x1 + [4] x2 + [8] p(c_23) = [2] x1 + [1] Following rules are strictly oriented: DX#(exp(z0,z1)) = [2] z0 + [2] z1 + [6] > [2] z0 + [2] = c_5(DX#(z0)) Following rules are (at-least) weakly oriented: DX#(div(z0,z1)) = [2] z0 + [2] z1 + [10] >= [2] z1 + [10] = c_3(DX#(z1)) DX#(div(z0,z1)) = [2] z0 + [2] z1 + [10] >= [2] z0 + [10] = c_4(DX#(z0)) DX#(exp(z0,z1)) = [2] z0 + [2] z1 + [6] >= [2] z1 + [6] = c_6(DX#(z1)) DX#(ln(z0)) = [2] z0 + [8] >= [2] z0 + [0] = c_7(DX#(z0)) DX#(minus(z0,z1)) = [2] z0 + [2] z1 + [20] >= [2] z0 + [8] = c_8(DX#(z0)) DX#(minus(z0,z1)) = [2] z0 + [2] z1 + [20] >= [2] z1 + [1] = c_9(DX#(z1)) DX#(neg(z0)) = [2] z0 + [28] >= [2] z0 + [8] = c_10(DX#(z0)) DX#(plus(z0,z1)) = [2] z0 + [2] z1 + [0] >= [2] z0 + [0] = c_11(DX#(z0)) DX#(plus(z0,z1)) = [2] z0 + [2] z1 + [0] >= [2] z1 + [0] = c_12(DX#(z1)) DX#(times(z0,z1)) = [2] z0 + [2] z1 + [0] >= [2] z0 + [0] = c_13(DX#(z0)) DX#(times(z0,z1)) = [2] z0 + [2] z1 + [0] >= [2] z1 + [0] = c_14(DX#(z1)) ** Step 1.b:10: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) - Weak DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,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}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(DX) = [0] p(a) = [0] p(c) = [0] p(c1) = [0] p(c10) = [1] x1 + [0] p(c11) = [1] x1 + [0] p(c12) = [1] x1 + [0] p(c13) = [1] x1 + [0] p(c2) = [1] x1 + [0] p(c3) = [1] x1 + [0] p(c4) = [1] x1 + [0] p(c5) = [1] x1 + [0] p(c6) = [1] x1 + [0] p(c7) = [1] x1 + [0] p(c8) = [1] x1 + [0] p(c9) = [1] x1 + [0] p(div) = [1] x1 + [1] x2 + [1] p(dx) = [0] p(exp) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [0] p(minus) = [1] x1 + [1] x2 + [1] p(neg) = [1] x1 + [0] p(one) = [0] p(plus) = [1] x1 + [1] x2 + [1] p(times) = [1] x1 + [1] x2 + [1] p(two) = [0] p(zero) = [0] p(DX#) = [4] x1 + [0] p(dx#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [4] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [4] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [4] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] Following rules are strictly oriented: DX#(plus(z0,z1)) = [4] z0 + [4] z1 + [4] > [4] z0 + [1] = c_11(DX#(z0)) Following rules are (at-least) weakly oriented: DX#(div(z0,z1)) = [4] z0 + [4] z1 + [4] >= [4] z1 + [0] = c_3(DX#(z1)) DX#(div(z0,z1)) = [4] z0 + [4] z1 + [4] >= [4] z0 + [4] = c_4(DX#(z0)) DX#(exp(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_5(DX#(z0)) DX#(exp(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_6(DX#(z1)) DX#(ln(z0)) = [4] z0 + [0] >= [4] z0 + [0] = c_7(DX#(z0)) DX#(minus(z0,z1)) = [4] z0 + [4] z1 + [4] >= [4] z0 + [0] = c_8(DX#(z0)) DX#(minus(z0,z1)) = [4] z0 + [4] z1 + [4] >= [4] z1 + [4] = c_9(DX#(z1)) DX#(neg(z0)) = [4] z0 + [0] >= [4] z0 + [0] = c_10(DX#(z0)) DX#(plus(z0,z1)) = [4] z0 + [4] z1 + [4] >= [4] z1 + [4] = c_12(DX#(z1)) DX#(times(z0,z1)) = [4] z0 + [4] z1 + [4] >= [4] z0 + [0] = c_13(DX#(z0)) DX#(times(z0,z1)) = [4] z0 + [4] z1 + [4] >= [4] z1 + [0] = c_14(DX#(z1)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:11: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DX#(plus(z0,z1)) -> c_12(DX#(z1)) - Weak DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,div,exp,ln,minus,neg,one,plus,times,two,zero} + 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(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {DX#,dx#} TcT has computed the following interpretation: p(DX) = [1] x1 + [1] p(a) = [1] p(c) = [1] p(c1) = [0] p(c10) = [0] p(c11) = [0] p(c12) = [4] p(c13) = [1] p(c2) = [1] x1 + [2] p(c3) = [0] p(c4) = [1] x1 + [2] p(c5) = [1] x1 + [0] p(c6) = [0] p(c7) = [1] p(c8) = [1] p(c9) = [0] p(div) = [1] x1 + [1] x2 + [0] p(dx) = [2] p(exp) = [1] x1 + [1] x2 + [0] p(ln) = [1] x1 + [1] p(minus) = [1] x1 + [1] x2 + [0] p(neg) = [1] x1 + [2] p(one) = [1] p(plus) = [1] x1 + [1] x2 + [2] p(times) = [1] x1 + [1] x2 + [0] p(two) = [1] p(zero) = [0] p(DX#) = [4] x1 + [0] p(dx#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [4] p(c_11) = [1] x1 + [8] p(c_12) = [1] x1 + [4] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] p(c_17) = [2] x2 + [2] p(c_18) = [1] x2 + [1] p(c_19) = [2] p(c_20) = [1] x1 + [1] x2 + [0] p(c_21) = [8] p(c_22) = [1] x2 + [0] p(c_23) = [1] x1 + [8] x2 + [0] Following rules are strictly oriented: DX#(plus(z0,z1)) = [4] z0 + [4] z1 + [8] > [4] z1 + [4] = c_12(DX#(z1)) Following rules are (at-least) weakly oriented: DX#(div(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_3(DX#(z1)) DX#(div(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_4(DX#(z0)) DX#(exp(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_5(DX#(z0)) DX#(exp(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_6(DX#(z1)) DX#(ln(z0)) = [4] z0 + [4] >= [4] z0 + [0] = c_7(DX#(z0)) DX#(minus(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_8(DX#(z0)) DX#(minus(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_9(DX#(z1)) DX#(neg(z0)) = [4] z0 + [8] >= [4] z0 + [4] = c_10(DX#(z0)) DX#(plus(z0,z1)) = [4] z0 + [4] z1 + [8] >= [4] z0 + [8] = c_11(DX#(z0)) DX#(times(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z0 + [0] = c_13(DX#(z0)) DX#(times(z0,z1)) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_14(DX#(z1)) ** Step 1.b:12: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: DX#(div(z0,z1)) -> c_3(DX#(z1)) DX#(div(z0,z1)) -> c_4(DX#(z0)) DX#(exp(z0,z1)) -> c_5(DX#(z0)) DX#(exp(z0,z1)) -> c_6(DX#(z1)) DX#(ln(z0)) -> c_7(DX#(z0)) DX#(minus(z0,z1)) -> c_8(DX#(z0)) DX#(minus(z0,z1)) -> c_9(DX#(z1)) DX#(neg(z0)) -> c_10(DX#(z0)) DX#(plus(z0,z1)) -> c_11(DX#(z0)) DX#(plus(z0,z1)) -> c_12(DX#(z1)) DX#(times(z0,z1)) -> c_13(DX#(z0)) DX#(times(z0,z1)) -> c_14(DX#(z1)) - Signature: {DX/1,dx/1,DX#/1,dx#/1} / {a/0,c/0,c1/0,c10/1,c11/1,c12/1,c13/1,c2/1,c3/1,c4/1,c5/1,c6/1,c7/1,c8/1,c9/1 ,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_3/1,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/1,c_22/2 ,c_23/2} - Obligation: innermost runtime complexity wrt. defined symbols {DX#,dx#} and constructors {a,c,c1,c10,c11,c12,c13,c2,c3 ,c4,c5,c6,c7,c8,c9,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))