MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D,b} and constructors {1,2,c,constant,div,h,ln,m,opp,pow ,s,t} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)) D#(constant()) -> c_3() D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)) D#(t()) -> c_9() b#(x,h()) -> c_10() b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) b#(h(),x) -> c_12() b#(s(x),s(y)) -> c_13(b#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)) D#(constant()) -> c_3() D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)) D#(t()) -> c_9() b#(x,h()) -> c_10() b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) b#(h(),x) -> c_12() b#(s(x),s(y)) -> c_13(b#(x,y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/3,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/3,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3,9,10,12} by application of Pre({3,9,10,12}) = {1,2,4,5,6,7,8,11,13}. Here rules are labelled as follows: 1: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) 2: D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)) 3: D#(constant()) -> c_3() 4: D#(div(x,y)) -> c_4(D#(x),D#(y)) 5: D#(ln(x)) -> c_5(D#(x)) 6: D#(m(x,y)) -> c_6(D#(x),D#(y)) 7: D#(opp(x)) -> c_7(D#(x)) 8: D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)) 9: D#(t()) -> c_9() 10: b#(x,h()) -> c_10() 11: b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) 12: b#(h(),x) -> c_12() 13: b#(s(x),s(y)) -> c_13(b#(x,y)) * Step 3: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)) b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) b#(s(x),s(y)) -> c_13(b#(x,y)) - Weak DPs: D#(constant()) -> c_3() D#(t()) -> c_9() b#(x,h()) -> c_10() b#(h(),x) -> c_12() - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/3,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/3,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) -->_1 b#(s(x),s(y)) -> c_13(b#(x,y)):9 -->_1 b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)):8 -->_3 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_3 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_3 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_3 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_3 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_3 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_1 b#(h(),x) -> c_12():13 -->_1 b#(x,h()) -> c_10():12 -->_3 D#(t()) -> c_9():11 -->_2 D#(t()) -> c_9():11 -->_3 D#(constant()) -> c_3():10 -->_2 D#(constant()) -> c_3():10 -->_3 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 2:S:D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)) -->_3 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_3 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_3 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_3 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_3 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_3 D#(t()) -> c_9():11 -->_2 D#(t()) -> c_9():11 -->_3 D#(constant()) -> c_3():10 -->_2 D#(constant()) -> c_3():10 -->_3 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_3 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 3:S:D#(div(x,y)) -> c_4(D#(x),D#(y)) -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_1 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(t()) -> c_9():11 -->_1 D#(t()) -> c_9():11 -->_2 D#(constant()) -> c_3():10 -->_1 D#(constant()) -> c_3():10 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_1 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 4:S:D#(ln(x)) -> c_5(D#(x)) -->_1 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(t()) -> c_9():11 -->_1 D#(constant()) -> c_3():10 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 5:S:D#(m(x,y)) -> c_6(D#(x),D#(y)) -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_1 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(t()) -> c_9():11 -->_1 D#(t()) -> c_9():11 -->_2 D#(constant()) -> c_3():10 -->_1 D#(constant()) -> c_3():10 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_1 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 6:S:D#(opp(x)) -> c_7(D#(x)) -->_1 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_1 D#(t()) -> c_9():11 -->_1 D#(constant()) -> c_3():10 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 7:S:D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)) -->_3 D#(t()) -> c_9():11 -->_2 D#(t()) -> c_9():11 -->_3 D#(constant()) -> c_3():10 -->_2 D#(constant()) -> c_3():10 -->_3 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_3 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_3 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_3 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_3 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_3 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_3 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 8:S:b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) -->_2 b#(s(x),s(y)) -> c_13(b#(x,y)):9 -->_1 b#(s(x),s(y)) -> c_13(b#(x,y)):9 -->_2 b#(h(),x) -> c_12():13 -->_1 b#(h(),x) -> c_12():13 -->_2 b#(x,h()) -> c_10():12 -->_1 b#(x,h()) -> c_10():12 -->_2 b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)):8 -->_1 b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)):8 9:S:b#(s(x),s(y)) -> c_13(b#(x,y)) -->_1 b#(h(),x) -> c_12():13 -->_1 b#(x,h()) -> c_10():12 -->_1 b#(s(x),s(y)) -> c_13(b#(x,y)):9 -->_1 b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)):8 10:W:D#(constant()) -> c_3() 11:W:D#(t()) -> c_9() 12:W:b#(x,h()) -> c_10() 13:W:b#(h(),x) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: D#(constant()) -> c_3() 11: D#(t()) -> c_9() 12: b#(x,h()) -> c_10() 13: b#(h(),x) -> c_12() * Step 4: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)) b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) b#(s(x),s(y)) -> c_13(b#(x,y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/3,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/3,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) -->_1 b#(s(x),s(y)) -> c_13(b#(x,y)):9 -->_1 b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)):8 -->_3 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_3 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_3 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_3 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_3 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_3 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_3 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 2:S:D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)) -->_3 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_3 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_3 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_3 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_3 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_3 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_3 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 3:S:D#(div(x,y)) -> c_4(D#(x),D#(y)) -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_1 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_1 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 4:S:D#(ln(x)) -> c_5(D#(x)) -->_1 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 5:S:D#(m(x,y)) -> c_6(D#(x),D#(y)) -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_1 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_1 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 6:S:D#(opp(x)) -> c_7(D#(x)) -->_1 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 7:S:D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)) -->_3 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_2 D#(pow(x,y)) -> c_8(b#(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))),D#(x),D#(y)):7 -->_3 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_3 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_3 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_3 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_3 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_2 D#(c(x,y)) -> c_2(b#(c(y,D(x)),c(x,D(y))),D#(x),D#(y)):2 -->_3 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 8:S:b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) -->_2 b#(s(x),s(y)) -> c_13(b#(x,y)):9 -->_1 b#(s(x),s(y)) -> c_13(b#(x,y)):9 -->_2 b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)):8 -->_1 b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)):8 9:S:b#(s(x),s(y)) -> c_13(b#(x,y)) -->_1 b#(s(x),s(y)) -> c_13(b#(x,y)):9 -->_1 b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)):8 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) * Step 5: DecomposeDG MAYBE + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) b#(s(x),s(y)) -> c_13(b#(x,y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) and a lower component b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) b#(s(x),s(y)) -> c_13(b#(x,y)) Further, following extension rules are added to the lower component. D#(b(x,y)) -> D#(x) D#(b(x,y)) -> D#(y) D#(b(x,y)) -> b#(D(x),D(y)) D#(c(x,y)) -> D#(x) D#(c(x,y)) -> D#(y) D#(div(x,y)) -> D#(x) D#(div(x,y)) -> D#(y) D#(ln(x)) -> D#(x) D#(m(x,y)) -> D#(x) D#(m(x,y)) -> D#(y) D#(opp(x)) -> D#(x) D#(pow(x,y)) -> D#(x) D#(pow(x,y)) -> D#(y) ** Step 5.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 4: D#(ln(x)) -> c_5(D#(x)) 5: D#(m(x,y)) -> c_6(D#(x),D#(y)) 7: D#(pow(x,y)) -> c_8(D#(x),D#(y)) The strictly oriented rules are moved into the weak component. *** Step 5.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2,3}, uargs(c_2) = {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} Following symbols are considered usable: {D#,b#} TcT has computed the following interpretation: p(1) = [0] p(2) = [7] p(D) = [0] p(b) = [1] x1 + [2] x2 + [0] p(c) = [1] x1 + [1] x2 + [0] p(constant) = [1] p(div) = [1] x1 + [1] x2 + [0] p(h) = [0] p(ln) = [1] x1 + [3] p(m) = [1] x1 + [1] x2 + [3] p(opp) = [1] x1 + [0] p(pow) = [1] x1 + [1] x2 + [2] p(s) = [2] p(t) = [0] p(D#) = [4] x1 + [0] p(b#) = [0] p(c_1) = [4] x1 + [1] x2 + [2] x3 + [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [4] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] x1 + [7] p(c_6) = [1] x1 + [1] x2 + [7] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [4] p(c_9) = [0] p(c_10) = [1] p(c_11) = [1] x2 + [1] p(c_12) = [1] p(c_13) = [0] Following rules are strictly oriented: D#(ln(x)) = [4] x + [12] > [4] x + [7] = c_5(D#(x)) D#(m(x,y)) = [4] x + [4] y + [12] > [4] x + [4] y + [7] = c_6(D#(x),D#(y)) D#(pow(x,y)) = [4] x + [4] y + [8] > [4] x + [4] y + [4] = c_8(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(b(x,y)) = [4] x + [8] y + [0] >= [4] x + [8] y + [0] = c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(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 + [0] >= [4] x + [4] y + [0] = c_4(D#(x),D#(y)) D#(opp(x)) = [4] x + [0] >= [4] x + [0] = c_7(D#(x)) *** Step 5.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) - Weak DPs: D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) - Weak DPs: D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 4: D#(opp(x)) -> c_7(D#(x)) The strictly oriented rules are moved into the weak component. **** Step 5.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) - Weak DPs: D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2,3}, uargs(c_2) = {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} Following symbols are considered usable: {D#,b#} TcT has computed the following interpretation: p(1) = [2] p(2) = [7] p(D) = [1] p(b) = [2] x1 + [2] x2 + [0] p(c) = [1] x1 + [1] x2 + [0] p(constant) = [0] p(div) = [1] x1 + [1] x2 + [0] p(h) = [0] p(ln) = [1] x1 + [0] p(m) = [1] x1 + [1] x2 + [0] p(opp) = [1] x1 + [3] p(pow) = [1] x1 + [1] x2 + [1] p(s) = [1] p(t) = [0] p(D#) = [4] x1 + [0] p(b#) = [0] p(c_1) = [1] x1 + [2] x2 + [2] x3 + [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] 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 + [6] p(c_8) = [1] x1 + [1] x2 + [1] p(c_9) = [1] p(c_10) = [2] p(c_11) = [1] x1 + [1] x2 + [0] p(c_12) = [1] p(c_13) = [4] Following rules are strictly oriented: D#(opp(x)) = [4] x + [12] > [4] x + [6] = c_7(D#(x)) Following rules are (at-least) weakly oriented: D#(b(x,y)) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(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 + [0] >= [4] x + [4] y + [0] = c_4(D#(x),D#(y)) D#(ln(x)) = [4] x + [0] >= [4] x + [0] = c_5(D#(x)) D#(m(x,y)) = [4] x + [4] y + [0] >= [4] x + [4] y + [0] = c_6(D#(x),D#(y)) D#(pow(x,y)) = [4] x + [4] y + [4] >= [4] x + [4] y + [1] = c_8(D#(x),D#(y)) **** Step 5.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) - Weak DPs: D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) - Weak DPs: D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: D#(div(x,y)) -> c_4(D#(x),D#(y)) The strictly oriented rules are moved into the weak component. ***** Step 5.a:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) - Weak DPs: D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2,3}, uargs(c_2) = {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} Following symbols are considered usable: {D#,b#} TcT has computed the following interpretation: p(1) = [7] p(2) = [3] p(D) = [0] p(b) = [1] x1 + [2] x2 + [0] p(c) = [1] x1 + [1] x2 + [0] p(constant) = [0] p(div) = [1] x1 + [1] x2 + [1] p(h) = [0] p(ln) = [1] x1 + [0] p(m) = [1] x1 + [1] x2 + [1] p(opp) = [1] x1 + [0] p(pow) = [1] x1 + [1] x2 + [0] p(s) = [3] p(t) = [0] p(D#) = [4] x1 + [0] p(b#) = [0] p(c_1) = [1] x1 + [1] x2 + [2] x3 + [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] p(c_4) = [1] x1 + [1] x2 + [2] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [1] x2 + [4] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x2 + [1] p(c_12) = [1] p(c_13) = [0] Following rules are strictly oriented: D#(div(x,y)) = [4] x + [4] y + [4] > [4] x + [4] y + [2] = c_4(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(b(x,y)) = [4] x + [8] y + [0] >= [4] x + [8] y + [0] = c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) = [4] x + [4] y + [0] >= [4] x + [4] y + [0] = c_2(D#(x),D#(y)) D#(ln(x)) = [4] x + [0] >= [4] x + [0] = c_5(D#(x)) D#(m(x,y)) = [4] x + [4] y + [4] >= [4] x + [4] y + [4] = c_6(D#(x),D#(y)) D#(opp(x)) = [4] x + [0] >= [4] x + [0] = c_7(D#(x)) D#(pow(x,y)) = [4] x + [4] y + [0] >= [4] x + [4] y + [0] = c_8(D#(x),D#(y)) ***** Step 5.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.a:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) The strictly oriented rules are moved into the weak component. ****** Step 5.a:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2,3}, uargs(c_2) = {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} Following symbols are considered usable: {D#,b#} TcT has computed the following interpretation: p(1) = [0] p(2) = [3] p(D) = [3] p(b) = [2] x1 + [2] x2 + [1] p(c) = [1] x1 + [1] x2 + [0] p(constant) = [0] p(div) = [1] x1 + [1] x2 + [2] p(h) = [0] p(ln) = [1] x1 + [1] p(m) = [1] x1 + [1] x2 + [0] p(opp) = [1] x1 + [1] p(pow) = [1] x1 + [1] x2 + [0] p(s) = [0] p(t) = [0] p(D#) = [4] x1 + [0] p(b#) = [0] p(c_1) = [1] x1 + [2] x2 + [2] x3 + [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] x1 + [4] p(c_6) = [1] x1 + [1] x2 + [0] p(c_7) = [1] x1 + [1] p(c_8) = [1] x1 + [1] x2 + [0] p(c_9) = [1] p(c_10) = [4] p(c_11) = [4] x1 + [2] x2 + [4] p(c_12) = [2] p(c_13) = [1] x1 + [1] Following rules are strictly oriented: D#(b(x,y)) = [8] x + [8] y + [4] > [8] x + [8] y + [0] = c_1(b#(D(x),D(y)),D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(c(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 + [8] >= [4] x + [4] y + [0] = c_4(D#(x),D#(y)) D#(ln(x)) = [4] x + [4] >= [4] x + [4] = c_5(D#(x)) D#(m(x,y)) = [4] x + [4] y + [0] >= [4] x + [4] y + [0] = c_6(D#(x),D#(y)) D#(opp(x)) = [4] x + [4] >= [4] x + [1] = c_7(D#(x)) D#(pow(x,y)) = [4] x + [4] y + [0] >= [4] x + [4] y + [0] = c_8(D#(x),D#(y)) ****** Step 5.a:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: D#(c(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 5.a:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(c(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: D#(c(x,y)) -> c_2(D#(x),D#(y)) The strictly oriented rules are moved into the weak component. ******* Step 5.a:1.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: D#(c(x,y)) -> c_2(D#(x),D#(y)) - Weak DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2,3}, uargs(c_2) = {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} Following symbols are considered usable: {D#,b#} TcT has computed the following interpretation: p(1) = [2] p(2) = [4] p(D) = [2] x1 + [0] p(b) = [1] x1 + [2] x2 + [0] p(c) = [1] x1 + [1] x2 + [2] p(constant) = [0] p(div) = [1] x1 + [1] x2 + [2] p(h) = [4] p(ln) = [1] x1 + [0] p(m) = [1] x1 + [1] x2 + [1] p(opp) = [1] x1 + [0] p(pow) = [1] x1 + [1] x2 + [0] p(s) = [2] p(t) = [0] p(D#) = [4] x1 + [0] p(b#) = [0] p(c_1) = [1] x1 + [1] x2 + [1] x3 + [0] p(c_2) = [1] x1 + [1] x2 + [4] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [1] x2 + [1] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [0] p(c_9) = [1] p(c_10) = [1] p(c_11) = [4] x2 + [1] p(c_12) = [0] p(c_13) = [1] Following rules are strictly oriented: D#(c(x,y)) = [4] x + [4] y + [8] > [4] x + [4] y + [4] = c_2(D#(x),D#(y)) Following rules are (at-least) weakly oriented: D#(b(x,y)) = [4] x + [8] y + [0] >= [4] x + [4] y + [0] = c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(div(x,y)) = [4] x + [4] y + [8] >= [4] x + [4] y + [0] = c_4(D#(x),D#(y)) D#(ln(x)) = [4] x + [0] >= [4] x + [0] = c_5(D#(x)) D#(m(x,y)) = [4] x + [4] y + [4] >= [4] x + [4] y + [1] = c_6(D#(x),D#(y)) D#(opp(x)) = [4] x + [0] >= [4] x + [0] = c_7(D#(x)) D#(pow(x,y)) = [4] x + [4] y + [0] >= [4] x + [4] y + [0] = c_8(D#(x),D#(y)) ******* Step 5.a:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 5.a:1.b:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) D#(c(x,y)) -> c_2(D#(x),D#(y)) D#(div(x,y)) -> c_4(D#(x),D#(y)) D#(ln(x)) -> c_5(D#(x)) D#(m(x,y)) -> c_6(D#(x),D#(y)) D#(opp(x)) -> c_7(D#(x)) D#(pow(x,y)) -> c_8(D#(x),D#(y)) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) -->_3 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_3 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(opp(x)) -> c_7(D#(x)):6 -->_3 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_3 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_3 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_3 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_3 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 2:W:D#(c(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#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 3:W:D#(div(x,y)) -> c_4(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#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 4:W:D#(ln(x)) -> c_5(D#(x)) -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 5:W:D#(m(x,y)) -> c_6(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#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 6:W:D#(opp(x)) -> c_7(D#(x)) -->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 7:W:D#(pow(x,y)) -> c_8(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#(opp(x)) -> c_7(D#(x)):6 -->_1 D#(opp(x)) -> c_7(D#(x)):6 -->_2 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_1 D#(m(x,y)) -> c_6(D#(x),D#(y)):5 -->_2 D#(ln(x)) -> c_5(D#(x)):4 -->_1 D#(ln(x)) -> c_5(D#(x)):4 -->_2 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_1 D#(div(x,y)) -> c_4(D#(x),D#(y)):3 -->_2 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_1 D#(c(x,y)) -> c_2(D#(x),D#(y)):2 -->_2 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 -->_1 D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: D#(b(x,y)) -> c_1(b#(D(x),D(y)),D#(x),D#(y)) 7: D#(pow(x,y)) -> c_8(D#(x),D#(y)) 6: D#(opp(x)) -> c_7(D#(x)) 5: D#(m(x,y)) -> c_6(D#(x),D#(y)) 4: D#(ln(x)) -> c_5(D#(x)) 3: D#(div(x,y)) -> c_4(D#(x),D#(y)) 2: D#(c(x,y)) -> c_2(D#(x),D#(y)) ******* Step 5.a:1.b:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: Failure MAYBE + Considered Problem: - Strict DPs: b#(b(x,y),z) -> c_11(b#(x,b(y,z)),b#(y,z)) b#(s(x),s(y)) -> c_13(b#(x,y)) - Weak DPs: D#(b(x,y)) -> D#(x) D#(b(x,y)) -> D#(y) D#(b(x,y)) -> b#(D(x),D(y)) D#(c(x,y)) -> D#(x) D#(c(x,y)) -> D#(y) D#(div(x,y)) -> D#(x) D#(div(x,y)) -> D#(y) D#(ln(x)) -> D#(x) D#(m(x,y)) -> D#(x) D#(m(x,y)) -> D#(y) D#(opp(x)) -> D#(x) D#(pow(x,y)) -> D#(x) D#(pow(x,y)) -> D#(y) - Weak TRS: D(b(x,y)) -> b(D(x),D(y)) D(c(x,y)) -> b(c(y,D(x)),c(x,D(y))) D(constant()) -> h() D(div(x,y)) -> m(div(D(x),y),div(c(x,D(y)),pow(y,2()))) D(ln(x)) -> div(D(x),x) D(m(x,y)) -> m(D(x),D(y)) D(opp(x)) -> opp(D(x)) D(pow(x,y)) -> b(c(c(y,pow(x,m(y,1()))),D(x)),c(c(pow(x,y),ln(x)),D(y))) D(t()) -> s(h()) b(x,h()) -> x b(b(x,y),z) -> b(x,b(y,z)) b(h(),x) -> x b(s(x),s(y)) -> s(s(b(x,y))) - Signature: {D/1,b/2,D#/1,b#/2} / {1/0,2/0,c/2,constant/0,div/2,h/0,ln/1,m/2,opp/1,pow/2,s/1,t/0,c_1/3,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/2,c_7/1,c_8/2,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {D#,b#} and constructors {1,2,c,constant,div,h,ln,m,opp ,pow,s,t} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE