WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {append,gt,quicksort,quicksort',split ,split'} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) append#(nil(),ys) -> c_2() gt#(0(),0()) -> c_3() gt#(0(),s(y)) -> c_4() gt#(s(x),0()) -> c_5() gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort#(nil()) -> c_8() quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)),gt#(x,pivot),split#(pivot,xs)) split#(pivot,nil()) -> c_11() split'#(false(),x,pair(ls,rs)) -> c_12() split'#(true(),x,pair(ls,rs)) -> c_13() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) append#(nil(),ys) -> c_2() gt#(0(),0()) -> c_3() gt#(0(),s(y)) -> c_4() gt#(s(x),0()) -> c_5() gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort#(nil()) -> c_8() quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)),gt#(x,pivot),split#(pivot,xs)) split#(pivot,nil()) -> c_11() split'#(false(),x,pair(ls,rs)) -> c_12() split'#(true(),x,pair(ls,rs)) -> c_13() - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/3,c_10/3,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,4,5,8,11,12,13} by application of Pre({2,3,4,5,8,11,12,13}) = {1,6,7,9,10}. Here rules are labelled as follows: 1: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) 2: append#(nil(),ys) -> c_2() 3: gt#(0(),0()) -> c_3() 4: gt#(0(),s(y)) -> c_4() 5: gt#(s(x),0()) -> c_5() 6: gt#(s(x),s(y)) -> c_6(gt#(x,y)) 7: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) 8: quicksort#(nil()) -> c_8() 9: quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) 10: split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)),gt#(x,pivot),split#(pivot,xs)) 11: split#(pivot,nil()) -> c_11() 12: split'#(false(),x,pair(ls,rs)) -> c_12() 13: split'#(true(),x,pair(ls,rs)) -> c_13() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)),gt#(x,pivot),split#(pivot,xs)) - Weak DPs: append#(nil(),ys) -> c_2() gt#(0(),0()) -> c_3() gt#(0(),s(y)) -> c_4() gt#(s(x),0()) -> c_5() quicksort#(nil()) -> c_8() split#(pivot,nil()) -> c_11() split'#(false(),x,pair(ls,rs)) -> c_12() split'#(true(),x,pair(ls,rs)) -> c_13() - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/3,c_10/3,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(nil(),ys) -> c_2():6 -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 2:S:gt#(s(x),s(y)) -> c_6(gt#(x,y)) -->_1 gt#(s(x),0()) -> c_5():9 -->_1 gt#(0(),s(y)) -> c_4():8 -->_1 gt#(0(),0()) -> c_3():7 -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):2 3:S:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)) ,gt#(x,pivot) ,split#(pivot,xs)):5 -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)):4 -->_2 split#(pivot,nil()) -> c_11():11 4:S:quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) -->_3 quicksort#(nil()) -> c_8():10 -->_2 quicksort#(nil()) -> c_8():10 -->_1 append#(nil(),ys) -> c_2():6 -->_3 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):3 -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):3 -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 5:S:split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)),gt#(x,pivot),split#(pivot,xs)) -->_1 split'#(true(),x,pair(ls,rs)) -> c_13():13 -->_1 split'#(false(),x,pair(ls,rs)) -> c_12():12 -->_3 split#(pivot,nil()) -> c_11():11 -->_2 gt#(s(x),0()) -> c_5():9 -->_2 gt#(0(),s(y)) -> c_4():8 -->_2 gt#(0(),0()) -> c_3():7 -->_3 split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)) ,gt#(x,pivot) ,split#(pivot,xs)):5 -->_2 gt#(s(x),s(y)) -> c_6(gt#(x,y)):2 6:W:append#(nil(),ys) -> c_2() 7:W:gt#(0(),0()) -> c_3() 8:W:gt#(0(),s(y)) -> c_4() 9:W:gt#(s(x),0()) -> c_5() 10:W:quicksort#(nil()) -> c_8() 11:W:split#(pivot,nil()) -> c_11() 12:W:split'#(false(),x,pair(ls,rs)) -> c_12() 13:W:split'#(true(),x,pair(ls,rs)) -> c_13() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: quicksort#(nil()) -> c_8() 11: split#(pivot,nil()) -> c_11() 12: split'#(false(),x,pair(ls,rs)) -> c_12() 13: split'#(true(),x,pair(ls,rs)) -> c_13() 7: gt#(0(),0()) -> c_3() 8: gt#(0(),s(y)) -> c_4() 9: gt#(s(x),0()) -> c_5() 6: append#(nil(),ys) -> c_2() * Step 4: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)),gt#(x,pivot),split#(pivot,xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/3,c_10/3,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 2:S:gt#(s(x),s(y)) -> c_6(gt#(x,y)) -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):2 3:S:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)) ,gt#(x,pivot) ,split#(pivot,xs)):5 -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)):4 4:S:quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) -->_3 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):3 -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):3 -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 5:S:split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)),gt#(x,pivot),split#(pivot,xs)) -->_3 split#(pivot,dd(x,xs)) -> c_10(split'#(gt(x,pivot),x,split(pivot,xs)) ,gt#(x,pivot) ,split#(pivot,xs)):5 -->_2 gt#(s(x),s(y)) -> c_6(gt#(x,y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) * Step 5: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2 ,split#/2,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2 ,c_8/0,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} Problem (S) - Strict DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2 ,split#/2,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2 ,c_8/0,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} ** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 2:W:gt#(s(x),s(y)) -> c_6(gt#(x,y)) -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):2 3:W:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):5 -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)):4 4:W:quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 -->_3 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):3 -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):3 5:W:split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):2 -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) 2: gt#(s(x),s(y)) -> c_6(gt#(x,y)) ** Step 5.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 3:W:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)):4 4:W:quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 -->_3 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):3 -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) ** Step 5.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) The strictly oriented rules are moved into the weak component. *** Step 5.a:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1,2,3} Following symbols are considered usable: {append,quicksort,quicksort',split,split',append#,gt#,quicksort#,quicksort'#,split#,split'#} TcT has computed the following interpretation: p(0) = 0 p(append) = x1 + x2 p(dd) = 1 + x2 p(false) = 0 p(gt) = 2*x2 p(nil) = 0 p(pair) = x1 + x2 p(quicksort) = x1 p(quicksort') = 1 + x2 p(s) = 2 p(split) = x2 p(split') = 1 + x3 p(true) = 0 p(append#) = x1 p(gt#) = 2 + x1*x2 + 2*x2 p(quicksort#) = 2*x1^2 p(quicksort'#) = 1 + x2 + 2*x2^2 p(split#) = 1 p(split'#) = 2*x2*x3 + x2^2 + 2*x3 p(c_1) = x1 p(c_2) = 1 p(c_3) = 0 p(c_4) = 0 p(c_5) = 2 p(c_6) = 0 p(c_7) = x1 p(c_8) = 0 p(c_9) = x1 + x2 + x3 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = 2 Following rules are strictly oriented: append#(dd(x,xs),ys) = 1 + xs > xs = c_1(append#(xs,ys)) Following rules are (at-least) weakly oriented: quicksort#(dd(z,zs)) = 2 + 4*zs + 2*zs^2 >= 1 + zs + 2*zs^2 = c_7(quicksort'#(z,split(z,zs))) quicksort'#(z,pair(xs,ys)) = 1 + xs + 4*xs*ys + 2*xs^2 + ys + 2*ys^2 >= xs + 2*xs^2 + 2*ys^2 = c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) append(dd(x,xs),ys) = 1 + xs + ys >= 1 + xs + ys = dd(x,append(xs,ys)) append(nil(),ys) = ys >= ys = ys quicksort(dd(z,zs)) = 1 + zs >= 1 + zs = quicksort'(z,split(z,zs)) quicksort(nil()) = 0 >= 0 = nil() quicksort'(z,pair(xs,ys)) = 1 + xs + ys >= 1 + xs + ys = append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) = 1 + xs >= 1 + xs = split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) = 0 >= 0 = pair(nil(),nil()) split'(false(),x,pair(ls,rs)) = 1 + ls + rs >= 1 + ls + rs = pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) = 1 + ls + rs >= 1 + ls + rs = pair(ls,dd(x,rs)) *** Step 5.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 2:W:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)):3 3:W:quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) -->_3 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))):2 -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))):2 -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) 3: quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) 1: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) *** Step 5.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:gt#(s(x),s(y)) -> c_6(gt#(x,y)) -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):1 2:S:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):4 -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)):3 3:S:quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):5 -->_3 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):2 -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):2 4:S:split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):4 -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):1 5:W:append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))),quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/3,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:gt#(s(x),s(y)) -> c_6(gt#(x,y)) -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):1 2:S:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):4 -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)):3 3:S:quicksort'#(z,pair(xs,ys)) -> c_9(append#(quicksort(xs),dd(z,quicksort(ys))) ,quicksort#(xs) ,quicksort#(ys)) -->_3 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):2 -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):2 4:S:split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):4 -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) ** Step 5.b:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) quicksort(dd(z,zs)) -> quicksort'(z,split(z,zs)) quicksort(nil()) -> nil() quicksort'(z,pair(xs,ys)) -> append(quicksort(xs),dd(z,quicksort(ys))) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) ** Step 5.b:4: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) - Weak DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2 ,split#/2,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2 ,c_8/0,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} Problem (S) - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2 ,split#/2,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2 ,c_8/0,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} *** Step 5.b:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) - Weak DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: gt#(s(x),s(y)) -> c_6(gt#(x,y)) The strictly oriented rules are moved into the weak component. **** Step 5.b:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) - Weak DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_6) = {1}, uargs(c_7) = {1,2}, uargs(c_9) = {1,2}, uargs(c_10) = {1,2} Following symbols are considered usable: {split,split',append#,gt#,quicksort#,quicksort'#,split#,split'#} TcT has computed the following interpretation: p(0) = [1] [0] [1] p(append) = [0] [0] [0] p(dd) = [0 1 0] [1 1 0] [1] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(false) = [0] [0] [0] p(gt) = [0 0 1] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [1 0 1] [0 0 0] [0] p(nil) = [0] [0] [0] p(pair) = [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [1] p(quicksort) = [0] [0] [0] p(quicksort') = [0] [0] [0] p(s) = [0 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(split) = [0 1 0] [1 0 0] [0] [0 0 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [1] p(split') = [0 1 0] [1 1 0] [0] [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 0] [1] p(true) = [0] [0] [0] p(append#) = [0] [0] [0] p(gt#) = [0 1 0] [0 0 0] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 0] [0 1 0] [0] p(quicksort#) = [1 0 0] [0] [0 0 0] x1 + [0] [1 1 0] [1] p(quicksort'#) = [0 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 1 0] [0 0 0] [0] p(split#) = [0 0 0] [0 1 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [1] p(split'#) = [0] [0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [0] [0] [0] p(c_4) = [0] [0] [0] p(c_5) = [0] [0] [0] p(c_6) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(c_7) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 1 0] [0] p(c_8) = [0] [0] [0] p(c_9) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 0] [0 0 0] [0] p(c_10) = [1 0 0] [1 0 0] [0] [0 0 1] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 1] [0] p(c_11) = [0] [0] [0] p(c_12) = [0] [0] [0] p(c_13) = [0] [0] [0] Following rules are strictly oriented: gt#(s(x),s(y)) = [0 1 0] [0 0 0] [1] [0 0 0] x + [0 0 0] y + [1] [0 0 0] [0 1 0] [1] > [0 1 0] [0] [0 0 0] x + [0] [0 0 0] [1] = c_6(gt#(x,y)) Following rules are (at-least) weakly oriented: quicksort#(dd(z,zs)) = [0 1 0] [1 1 0] [1] [0 0 0] z + [0 0 0] zs + [0] [0 2 0] [1 2 0] [2] >= [0 1 0] [1 1 0] [1] [0 0 0] z + [0 0 0] zs + [0] [0 2 0] [0 0 0] [0] = c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) = [1 0 0] [1 0 0] [0 0 0] [0] [0 0 0] xs + [0 0 0] ys + [0 0 0] z + [1] [0 0 0] [0 0 0] [0 1 0] [0] >= [1 0 0] [1 0 0] [0] [0 0 0] xs + [0 0 0] ys + [1] [0 0 0] [0 0 0] [0] = c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) = [0 0 0] [0 1 0] [0 1 0] [0] [0 1 0] pivot + [0 0 0] x + [0 0 0] xs + [0] [0 0 0] [0 0 0] [0 0 0] [1] >= [0 0 0] [0 1 0] [0 1 0] [0] [0 1 0] pivot + [0 0 0] x + [0 0 0] xs + [0] [0 0 0] [0 0 0] [0 0 0] [1] = c_10(gt#(x,pivot),split#(pivot,xs)) split(pivot,dd(x,xs)) = [0 1 0] [0 1 0] [1 1 0] [1] [0 0 0] pivot + [0 1 0] x + [0 1 0] xs + [1] [0 0 0] [0 0 0] [0 0 0] [1] >= [0 1 0] [0 1 0] [1 1 0] [1] [0 0 0] pivot + [0 1 0] x + [0 1 0] xs + [1] [0 0 0] [0 0 0] [0 0 0] [1] = split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) = [0 1 0] [0] [0 0 0] pivot + [1] [0 0 0] [1] >= [0] [1] [1] = pair(nil(),nil()) split'(false(),x,pair(ls,rs)) = [1 1 0] [1 1 0] [0 1 0] [1] [0 1 0] ls + [0 1 0] rs + [0 1 0] x + [1] [0 0 0] [0 0 0] [0 0 0] [1] >= [1 1 0] [1 0 0] [0 1 0] [1] [0 1 0] ls + [0 1 0] rs + [0 1 0] x + [1] [0 0 0] [0 0 0] [0 0 0] [1] = pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) = [1 1 0] [1 1 0] [0 1 0] [1] [0 1 0] ls + [0 1 0] rs + [0 1 0] x + [1] [0 0 0] [0 0 0] [0 0 0] [1] >= [1 0 0] [1 1 0] [0 1 0] [1] [0 1 0] ls + [0 1 0] rs + [0 1 0] x + [1] [0 0 0] [0 0 0] [0 0 0] [1] = pair(ls,dd(x,rs)) **** Step 5.b:4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.b:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:gt#(s(x),s(y)) -> c_6(gt#(x,y)) -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):1 2:W:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):4 -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)):3 3:W:quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):2 -->_1 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):2 4:W:split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):4 -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) 3: quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) 4: split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) 1: gt#(s(x),s(y)) -> c_6(gt#(x,y)) **** Step 5.b:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak DPs: gt#(s(x),s(y)) -> c_6(gt#(x,y)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):3 -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)):2 2:S:quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):1 -->_1 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):1 3:S:split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):4 -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):3 4:W:gt#(s(x),s(y)) -> c_6(gt#(x,y)) -->_1 gt#(s(x),s(y)) -> c_6(gt#(x,y)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: gt#(s(x),s(y)) -> c_6(gt#(x,y)) *** Step 5.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/2,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):3 -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)):2 2:S:quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):1 -->_1 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):1 3:S:split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(gt#(x,pivot),split#(pivot,xs)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)) *** Step 5.b:4.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)) The strictly oriented rules are moved into the weak component. **** Step 5.b:4.b:3.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_7) = {1,2}, uargs(c_9) = {1,2}, uargs(c_10) = {1} Following symbols are considered usable: {gt,split,split',append#,gt#,quicksort#,quicksort'#,split#,split'#} TcT has computed the following interpretation: p(0) = [0] [1] [1] p(append) = [0] [0] [0] p(dd) = [0 1 1] [0 0 0] [1] [0 1 0] x1 + [0 1 1] x2 + [0] [0 0 0] [0 0 1] [1] p(false) = [0] [0] [0] p(gt) = [0 1 0] [0 0 0] [1] [0 0 0] x1 + [0 1 0] x2 + [1] [0 1 1] [0 0 0] [1] p(nil) = [0] [0] [0] p(pair) = [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(quicksort) = [0] [0] [0] p(quicksort') = [0] [0] [0] p(s) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(split) = [1 1 0] [1 0 1] [1] [0 0 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 1] [0] p(split') = [0 1 1] [0 0 0] [0 0 0] [1] [0 0 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 0] [0 0 1] [1] p(true) = [1] [1] [1] p(append#) = [0] [0] [0] p(gt#) = [0] [0] [0] p(quicksort#) = [0 1 0] [0] [1 1 0] x1 + [0] [1 0 0] [1] p(quicksort'#) = [0 1 0] [0 1 0] [0] [1 1 0] x1 + [1 1 0] x2 + [1] [1 0 1] [0 0 0] [1] p(split#) = [0 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 1 0] [0 0 0] [1] p(split'#) = [0] [0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [0] [0] [0] p(c_4) = [0] [0] [0] p(c_5) = [0] [0] [0] p(c_6) = [0] [0] [0] p(c_7) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [1 0 1] x2 + [0] [0 0 0] [0 0 1] [0] p(c_8) = [0] [0] [0] p(c_9) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [1 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_10) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_11) = [0] [0] [0] p(c_12) = [0] [0] [0] p(c_13) = [0] [0] [0] Following rules are strictly oriented: split#(pivot,dd(x,xs)) = [0 0 0] [0 0 1] [1] [0 0 0] pivot + [0 0 0] xs + [0] [0 1 0] [0 0 0] [1] > [0 0 1] [0] [0 0 0] xs + [0] [0 0 0] [0] = c_10(split#(pivot,xs)) Following rules are (at-least) weakly oriented: quicksort#(dd(z,zs)) = [0 1 0] [0 1 1] [0] [0 2 1] z + [0 1 1] zs + [1] [0 1 1] [0 0 0] [2] >= [0 1 0] [0 1 1] [0] [0 1 0] z + [0 0 1] zs + [1] [0 1 0] [0 0 0] [1] = c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) = [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] xs + [0 1 0] ys + [1 1 0] z + [1] [0 0 0] [0 0 0] [1 0 1] [1] >= [0 1 0] [0 1 0] [0] [0 0 0] xs + [0 1 0] ys + [0] [0 0 0] [0 0 0] [0] = c_9(quicksort#(xs),quicksort#(ys)) gt(0(),0()) = [2] [2] [3] >= [0] [0] [0] = false() gt(0(),s(y)) = [0 0 0] [2] [0 1 0] y + [1] [0 0 0] [3] >= [0] [0] [0] = false() gt(s(x),0()) = [0 1 0] [1] [0 0 0] x + [2] [0 1 1] [1] >= [1] [1] [1] = true() gt(s(x),s(y)) = [0 1 0] [0 0 0] [1] [0 0 0] x + [0 1 0] y + [1] [0 1 1] [0 0 0] [1] >= [0 1 0] [0 0 0] [1] [0 0 0] x + [0 1 0] y + [1] [0 1 1] [0 0 0] [1] = gt(x,y) split(pivot,dd(x,xs)) = [1 1 0] [0 1 1] [0 0 1] [3] [0 0 0] pivot + [0 1 0] x + [0 1 1] xs + [0] [0 0 0] [0 0 0] [0 0 1] [1] >= [0 1 0] [0 1 1] [0 0 0] [3] [0 0 0] pivot + [0 1 0] x + [0 1 1] xs + [0] [0 0 0] [0 0 0] [0 0 1] [1] = split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) = [1 1 0] [1] [0 0 0] pivot + [0] [0 0 0] [0] >= [0] [0] [0] = pair(nil(),nil()) split'(false(),x,pair(ls,rs)) = [0 0 0] [0 0 0] [0 0 0] [1] [0 1 1] ls + [0 1 1] rs + [0 1 0] x + [0] [0 0 1] [0 0 1] [0 0 0] [1] >= [0 0 0] [0 0 0] [0 0 0] [0] [0 1 1] ls + [0 1 0] rs + [0 1 0] x + [0] [0 0 1] [0 0 1] [0 0 0] [1] = pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) = [0 0 0] [0 0 0] [0 0 0] [3] [0 1 1] ls + [0 1 1] rs + [0 1 0] x + [0] [0 0 1] [0 0 1] [0 0 0] [1] >= [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] ls + [0 1 1] rs + [0 1 0] x + [0] [0 0 1] [0 0 1] [0 0 0] [1] = pair(ls,dd(x,rs)) **** Step 5.b:4.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) - Weak DPs: split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.b:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) - Weak DPs: split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_2 split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)):3 -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)):2 2:S:quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):1 -->_1 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):1 3:W:split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)) -->_1 split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: split#(pivot,dd(x,xs)) -> c_10(split#(pivot,xs)) **** Step 5.b:4.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/2,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)) -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)):2 2:S:quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):1 -->_1 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs)),split#(z,zs)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) **** Step 5.b:4.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + 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: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) Consider the set of all dependency pairs 1: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) 2: quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 5.b:4.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + 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_7) = {1}, uargs(c_9) = {1,2} Following symbols are considered usable: {split,split',append#,gt#,quicksort#,quicksort'#,split#,split'#} TcT has computed the following interpretation: p(0) = [0] p(append) = [0] p(dd) = [1] x2 + [3] p(false) = [2] p(gt) = [11] x1 + [10] x2 + [9] p(nil) = [2] p(pair) = [1] x1 + [1] x2 + [0] p(quicksort) = [1] p(quicksort') = [8] x1 + [2] x2 + [1] p(s) = [1] x1 + [1] p(split) = [1] x2 + [2] p(split') = [1] x3 + [3] p(true) = [8] p(append#) = [8] x1 + [1] x2 + [4] p(gt#) = [2] x1 + [0] p(quicksort#) = [8] x1 + [0] p(quicksort'#) = [8] x2 + [4] p(split#) = [2] x1 + [2] x2 + [0] p(split'#) = [2] x1 + [4] p(c_1) = [4] x1 + [1] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] p(c_6) = [2] x1 + [0] p(c_7) = [1] x1 + [2] p(c_8) = [2] p(c_9) = [1] x1 + [1] x2 + [4] p(c_10) = [8] x1 + [2] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] Following rules are strictly oriented: quicksort#(dd(z,zs)) = [8] zs + [24] > [8] zs + [22] = c_7(quicksort'#(z,split(z,zs))) Following rules are (at-least) weakly oriented: quicksort'#(z,pair(xs,ys)) = [8] xs + [8] ys + [4] >= [8] xs + [8] ys + [4] = c_9(quicksort#(xs),quicksort#(ys)) split(pivot,dd(x,xs)) = [1] xs + [5] >= [1] xs + [5] = split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) = [4] >= [4] = pair(nil(),nil()) split'(false(),x,pair(ls,rs)) = [1] ls + [1] rs + [3] >= [1] ls + [1] rs + [3] = pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) = [1] ls + [1] rs + [3] >= [1] ls + [1] rs + [3] = pair(ls,dd(x,rs)) ***** Step 5.b:4.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) - Weak DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.b:4.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) -->_1 quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)):2 2:W:quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) -->_2 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))):1 -->_1 quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: quicksort#(dd(z,zs)) -> c_7(quicksort'#(z,split(z,zs))) 2: quicksort'#(z,pair(xs,ys)) -> c_9(quicksort#(xs),quicksort#(ys)) ***** Step 5.b:4.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: gt(0(),0()) -> false() gt(0(),s(y)) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) split(pivot,dd(x,xs)) -> split'(gt(x,pivot),x,split(pivot,xs)) split(pivot,nil()) -> pair(nil(),nil()) split'(false(),x,pair(ls,rs)) -> pair(dd(x,ls),rs) split'(true(),x,pair(ls,rs)) -> pair(ls,dd(x,rs)) - Signature: {append/2,gt/2,quicksort/1,quicksort'/2,split/2,split'/3,append#/2,gt#/2,quicksort#/1,quicksort'#/2,split#/2 ,split'#/3} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0 ,c_9/2,c_10/1,c_11/0,c_12/0,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,gt#,quicksort#,quicksort'#,split# ,split'#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))