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 bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2} / {0/0,dd/2,false/0,nil/0,pair/2 ,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {append,bubble,bubble',bubble'',bubblesort,bubblesort' ,lt} 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() bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble#(dd(x,nil())) -> c_4() bubble'#(false(),x,x',xs) -> c_5(bubble''#(x',bubble(dd(x,xs))),bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble''#(x,bubble(dd(x',xs))),bubble#(dd(x',xs))) bubble''#(x,pair(xs,x')) -> c_7() bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort#(nil()) -> c_9() bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(0(),0()) -> c_11() lt#(0(),s(y)) -> c_12() lt#(s(x),0()) -> c_13() lt#(s(x),s(y)) -> c_14(lt#(x,y)) 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() bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble#(dd(x,nil())) -> c_4() bubble'#(false(),x,x',xs) -> c_5(bubble''#(x',bubble(dd(x,xs))),bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble''#(x,bubble(dd(x',xs))),bubble#(dd(x',xs))) bubble''#(x,pair(xs,x')) -> c_7() bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort#(nil()) -> c_9() bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(0(),0()) -> c_11() lt#(0(),s(y)) -> c_12() lt#(s(x),0()) -> c_13() lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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 {2,4,7,9,11,12,13} by application of Pre({2,4,7,9,11,12,13}) = {1,3,5,6,8,10,14}. Here rules are labelled as follows: 1: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) 2: append#(nil(),ys) -> c_2() 3: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) 4: bubble#(dd(x,nil())) -> c_4() 5: bubble'#(false(),x,x',xs) -> c_5(bubble''#(x',bubble(dd(x,xs))),bubble#(dd(x,xs))) 6: bubble'#(true(),x,x',xs) -> c_6(bubble''#(x,bubble(dd(x',xs))),bubble#(dd(x',xs))) 7: bubble''#(x,pair(xs,x')) -> c_7() 8: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) 9: bubblesort#(nil()) -> c_9() 10: bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) 11: lt#(0(),0()) -> c_11() 12: lt#(0(),s(y)) -> c_12() 13: lt#(s(x),0()) -> c_13() 14: lt#(s(x),s(y)) -> c_14(lt#(x,y)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble''#(x',bubble(dd(x,xs))),bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble''#(x,bubble(dd(x',xs))),bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak DPs: append#(nil(),ys) -> c_2() bubble#(dd(x,nil())) -> c_4() bubble''#(x,pair(xs,x')) -> c_7() bubblesort#(nil()) -> c_9() lt#(0(),0()) -> c_11() lt#(0(),s(y)) -> c_12() lt#(s(x),0()) -> c_13() - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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():8 -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 2:S:bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) -->_2 lt#(s(x),s(y)) -> c_14(lt#(x,y)):7 -->_1 bubble'#(true(),x,x',xs) -> c_6(bubble''#(x,bubble(dd(x',xs))),bubble#(dd(x',xs))):4 -->_1 bubble'#(false(),x,x',xs) -> c_5(bubble''#(x',bubble(dd(x,xs))),bubble#(dd(x,xs))):3 -->_2 lt#(s(x),0()) -> c_13():14 -->_2 lt#(0(),s(y)) -> c_12():13 -->_2 lt#(0(),0()) -> c_11():12 3:S:bubble'#(false(),x,x',xs) -> c_5(bubble''#(x',bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_1 bubble''#(x,pair(xs,x')) -> c_7():10 -->_2 bubble#(dd(x,nil())) -> c_4():9 -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):2 4:S:bubble'#(true(),x,x',xs) -> c_6(bubble''#(x,bubble(dd(x',xs))),bubble#(dd(x',xs))) -->_1 bubble''#(x,pair(xs,x')) -> c_7():10 -->_2 bubble#(dd(x,nil())) -> c_4():9 -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):2 5:S:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)):6 -->_2 bubble#(dd(x,nil())) -> c_4():9 -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):2 6:S:bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) -->_2 bubblesort#(nil()) -> c_9():11 -->_1 append#(nil(),ys) -> c_2():8 -->_2 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))):5 -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 7:S:lt#(s(x),s(y)) -> c_14(lt#(x,y)) -->_1 lt#(s(x),0()) -> c_13():14 -->_1 lt#(0(),s(y)) -> c_12():13 -->_1 lt#(0(),0()) -> c_11():12 -->_1 lt#(s(x),s(y)) -> c_14(lt#(x,y)):7 8:W:append#(nil(),ys) -> c_2() 9:W:bubble#(dd(x,nil())) -> c_4() 10:W:bubble''#(x,pair(xs,x')) -> c_7() 11:W:bubblesort#(nil()) -> c_9() 12:W:lt#(0(),0()) -> c_11() 13:W:lt#(0(),s(y)) -> c_12() 14:W:lt#(s(x),0()) -> c_13() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: bubblesort#(nil()) -> c_9() 9: bubble#(dd(x,nil())) -> c_4() 10: bubble''#(x,pair(xs,x')) -> c_7() 12: lt#(0(),0()) -> c_11() 13: lt#(0(),s(y)) -> c_12() 14: lt#(s(x),0()) -> c_13() 8: 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)) bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble''#(x',bubble(dd(x,xs))),bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble''#(x,bubble(dd(x',xs))),bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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:bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) -->_2 lt#(s(x),s(y)) -> c_14(lt#(x,y)):7 -->_1 bubble'#(true(),x,x',xs) -> c_6(bubble''#(x,bubble(dd(x',xs))),bubble#(dd(x',xs))):4 -->_1 bubble'#(false(),x,x',xs) -> c_5(bubble''#(x',bubble(dd(x,xs))),bubble#(dd(x,xs))):3 3:S:bubble'#(false(),x,x',xs) -> c_5(bubble''#(x',bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):2 4:S:bubble'#(true(),x,x',xs) -> c_6(bubble''#(x,bubble(dd(x',xs))),bubble#(dd(x',xs))) -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):2 5:S:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)):6 -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):2 6:S:bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) -->_2 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))):5 -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 7:S:lt#(s(x),s(y)) -> c_14(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_14(lt#(x,y)):7 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) * Step 5: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} Problem (S) - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - 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 bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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:bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) -->_2 lt#(s(x),s(y)) -> c_14(lt#(x,y)):7 -->_1 bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))):4 -->_1 bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))):3 3:W:bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):2 4:W:bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):2 5:W:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):2 -->_1 bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)):6 6:W:bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 -->_2 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))):5 7:W:lt#(s(x),s(y)) -> c_14(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_14(lt#(x,y)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) 4: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) 3: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) 7: lt#(s(x),s(y)) -> c_14(lt#(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: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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 5:W:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)):6 6:W:bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 -->_2 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) ** Step 5.a:3: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + 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 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) and a lower component append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) Further, following extension rules are added to the lower component. bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) *** Step 5.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) Consider the set of all dependency pairs 1: bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) 2: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) 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.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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_8) = {1}, uargs(c_10) = {1,2} Following symbols are considered usable: {bubble,bubble',bubble'',lt,append#,bubble#,bubble'#,bubble''#,bubblesort#,bubblesort'#,lt#} TcT has computed the following interpretation: p(0) = [0] p(append) = [1] x1 + [6] p(bubble) = [1] x1 + [0] p(bubble') = [1] x4 + [2] p(bubble'') = [1] x2 + [1] p(bubblesort) = [2] x1 + [0] p(bubblesort') = [4] x1 + [1] p(dd) = [1] x2 + [1] p(false) = [0] p(lt) = [0] p(nil) = [7] p(pair) = [1] x1 + [1] p(s) = [1] x1 + [1] p(true) = [0] p(append#) = [0] p(bubble#) = [1] x1 + [4] p(bubble'#) = [1] x4 + [0] p(bubble''#) = [1] x2 + [2] p(bubblesort#) = [1] x1 + [1] p(bubblesort'#) = [1] x1 + [1] p(lt#) = [1] x2 + [1] p(c_1) = [4] x1 + [0] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [4] x1 + [2] p(c_6) = [2] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [2] x1 + [1] x2 + [0] p(c_11) = [1] p(c_12) = [0] p(c_13) = [4] p(c_14) = [1] Following rules are strictly oriented: bubblesort'#(pair(xs,x)) = [1] xs + [2] > [1] xs + [1] = c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) Following rules are (at-least) weakly oriented: bubblesort#(dd(x,xs)) = [1] xs + [2] >= [1] xs + [2] = c_8(bubblesort'#(bubble(dd(x,xs)))) bubble(dd(x,dd(x',xs))) = [1] xs + [2] >= [1] xs + [2] = bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) = [8] >= [8] = pair(nil(),x) bubble'(false(),x,x',xs) = [1] xs + [2] >= [1] xs + [2] = bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) = [1] xs + [2] >= [1] xs + [2] = bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) = [1] xs + [2] >= [1] xs + [2] = pair(dd(x,xs),x') lt(0(),0()) = [0] >= [0] = false() lt(0(),s(y)) = [0] >= [0] = true() lt(s(x),0()) = [0] >= [0] = false() lt(s(x),s(y)) = [0] >= [0] = lt(x,y) **** Step 5.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) -->_1 bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)):2 2:W:bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) -->_2 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) 2: bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) **** Step 5.a:3.a:1.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 bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) The strictly oriented rules are moved into the weak component. **** Step 5.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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_1) = {1} Following symbols are considered usable: {append,bubble,bubble',bubble'',bubblesort,bubblesort',lt,append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} TcT has computed the following interpretation: p(0) = [0] p(append) = [1] x1 + [1] x2 + [0] p(bubble) = [1] x1 + [2] p(bubble') = [1] x1 + [1] x4 + [0] p(bubble'') = [1] x2 + [2] p(bubblesort) = [2] x1 + [5] p(bubblesort') = [2] x1 + [1] p(dd) = [1] x2 + [2] p(false) = [6] p(lt) = [6] p(nil) = [0] p(pair) = [1] x1 + [3] p(s) = [1] x1 + [0] p(true) = [6] p(append#) = [1] x1 + [2] x2 + [0] p(bubble#) = [1] p(bubble'#) = [1] x1 + [1] p(bubble''#) = [1] x2 + [0] p(bubblesort#) = [3] x1 + [7] p(bubblesort'#) = [3] x1 + [0] p(lt#) = [4] x1 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [2] p(c_3) = [2] x1 + [0] p(c_4) = [0] p(c_5) = [4] p(c_6) = [0] p(c_7) = [4] p(c_8) = [1] p(c_9) = [1] p(c_10) = [4] x1 + [0] p(c_11) = [4] p(c_12) = [1] p(c_13) = [0] p(c_14) = [1] Following rules are strictly oriented: append#(dd(x,xs),ys) = [1] xs + [2] ys + [2] > [1] xs + [2] ys + [0] = c_1(append#(xs,ys)) Following rules are (at-least) weakly oriented: bubblesort#(dd(x,xs)) = [3] xs + [13] >= [3] xs + [12] = bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) = [3] xs + [9] >= [2] xs + [9] = append#(bubblesort(xs),dd(x,nil())) bubblesort'#(pair(xs,x)) = [3] xs + [9] >= [3] xs + [7] = bubblesort#(xs) append(dd(x,xs),ys) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = dd(x,append(xs,ys)) append(nil(),ys) = [1] ys + [0] >= [1] ys + [0] = ys bubble(dd(x,dd(x',xs))) = [1] xs + [6] >= [1] xs + [6] = bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) = [4] >= [3] = pair(nil(),x) bubble'(false(),x,x',xs) = [1] xs + [6] >= [1] xs + [6] = bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) = [1] xs + [6] >= [1] xs + [6] = bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) = [1] xs + [5] >= [1] xs + [5] = pair(dd(x,xs),x') bubblesort(dd(x,xs)) = [2] xs + [9] >= [2] xs + [9] = bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) = [5] >= [0] = nil() bubblesort'(pair(xs,x)) = [2] xs + [7] >= [2] xs + [7] = append(bubblesort(xs),dd(x,nil())) lt(0(),0()) = [6] >= [6] = false() lt(0(),s(y)) = [6] >= [6] = true() lt(s(x),0()) = [6] >= [6] = false() lt(s(x),s(y)) = [6] >= [6] = lt(x,y) **** Step 5.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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:bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> bubblesort#(xs):4 -->_1 bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())):3 3:W:bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 4:W:bubblesort'#(pair(xs,x)) -> bubblesort#(xs) -->_1 bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) 4: bubblesort'#(pair(xs,x)) -> bubblesort#(xs) 3: bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) 1: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) **** Step 5.a:3.b:1.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 bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - 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 bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) -->_2 lt#(s(x),s(y)) -> c_14(lt#(x,y)):6 -->_1 bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))):3 -->_1 bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))):2 2:S:bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 3:S:bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 4:S:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)):5 -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 5:S:bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):7 -->_2 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))):4 6:S:lt#(s(x),s(y)) -> c_14(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_14(lt#(x,y)):6 7:W:append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/2,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) -->_2 lt#(s(x),s(y)) -> c_14(lt#(x,y)):6 -->_1 bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))):3 -->_1 bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))):2 2:S:bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 3:S:bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 4:S:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)):5 -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 5:S:bubblesort'#(pair(xs,x)) -> c_10(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) -->_2 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))):4 6:S:lt#(s(x),s(y)) -> c_14(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_14(lt#(x,y)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) ** Step 5.b:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: append(dd(x,xs),ys) -> dd(x,append(xs,ys)) append(nil(),ys) -> ys bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') bubblesort(dd(x,xs)) -> bubblesort'(bubble(dd(x,xs))) bubblesort(nil()) -> nil() bubblesort'(pair(xs,x)) -> append(bubblesort(xs),dd(x,nil())) lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) ** Step 5.b:4: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} Problem (S) - Strict DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} *** Step 5.b:4.a:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + 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 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) and a lower component bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) lt#(s(x),s(y)) -> c_14(lt#(x,y)) Further, following extension rules are added to the lower component. bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) **** Step 5.b:4.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) - Weak DPs: bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) Consider the set of all dependency pairs 1: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) 2: bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) 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.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) - Weak DPs: bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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_8) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {bubble,bubble',bubble'',lt,append#,bubble#,bubble'#,bubble''#,bubblesort#,bubblesort'#,lt#} TcT has computed the following interpretation: p(0) = [0] p(append) = [1] x1 + [2] x2 + [2] p(bubble) = [1] x1 + [6] p(bubble') = [12] x1 + [1] x4 + [0] p(bubble'') = [1] x2 + [9] p(bubblesort) = [8] p(bubblesort') = [1] x1 + [1] p(dd) = [1] x2 + [9] p(false) = [2] p(lt) = [2] p(nil) = [0] p(pair) = [1] x1 + [15] p(s) = [8] p(true) = [2] p(append#) = [1] x1 + [2] x2 + [1] p(bubble#) = [1] p(bubble'#) = [1] x2 + [2] x3 + [1] p(bubble''#) = [1] x1 + [1] x2 + [1] p(bubblesort#) = [1] x1 + [15] p(bubblesort'#) = [1] x1 + [0] p(lt#) = [1] x1 + [1] p(c_1) = [1] x1 + [1] p(c_2) = [2] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [1] p(c_7) = [2] p(c_8) = [1] x1 + [1] x2 + [6] p(c_9) = [2] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [1] p(c_13) = [2] p(c_14) = [8] Following rules are strictly oriented: bubblesort#(dd(x,xs)) = [1] xs + [24] > [1] xs + [22] = c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) Following rules are (at-least) weakly oriented: bubblesort'#(pair(xs,x)) = [1] xs + [15] >= [1] xs + [15] = c_10(bubblesort#(xs)) bubble(dd(x,dd(x',xs))) = [1] xs + [24] >= [1] xs + [24] = bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) = [15] >= [15] = pair(nil(),x) bubble'(false(),x,x',xs) = [1] xs + [24] >= [1] xs + [24] = bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) = [1] xs + [24] >= [1] xs + [24] = bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) = [1] xs + [24] >= [1] xs + [24] = pair(dd(x,xs),x') lt(0(),0()) = [2] >= [2] = false() lt(0(),s(y)) = [2] >= [2] = true() lt(s(x),0()) = [2] >= [2] = false() lt(s(x),s(y)) = [2] >= [2] = lt(x,y) ***** Step 5.b:4.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)):2 2:W:bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) -->_1 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) 2: bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) ***** Step 5.b:4.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak DPs: bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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 = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) 4: lt#(s(x),s(y)) -> c_14(lt#(x,y)) The strictly oriented rules are moved into the weak component. ***** Step 5.b:4.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak DPs: bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 2, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_3) = {1,2}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {bubble,bubble',bubble'',lt,append#,bubble#,bubble'#,bubble''#,bubblesort#,bubblesort'#,lt#} TcT has computed the following interpretation: p(0) = [0] [0] p(append) = [0 0] x1 + [0 2] x2 + [1] [0 2] [2 0] [2] p(bubble) = [0 3] x1 + [0] [2 0] [0] p(bubble') = [0 3] x1 + [0 3] x2 + [0 3] x3 + [0 3] x4 + [3] [2 0] [0 0] [0 1] [0 0] [0] p(bubble'') = [0 3] x1 + [1 0] x2 + [3] [0 0] [0 0] [0] p(bubblesort) = [1] [0] p(bubblesort') = [0 2] x1 + [0] [0 0] [0] p(dd) = [0 2] x1 + [0 3] x2 + [0] [0 1] [0 1] [1] p(false) = [0] [1] p(lt) = [0 2] x1 + [0] [0 0] [1] p(nil) = [0] [0] p(pair) = [0 3] x1 + [1] [0 0] [0] p(s) = [0 0] x1 + [0] [0 1] [3] p(true) = [0] [1] p(append#) = [1 2] x2 + [2] [2 1] [0] p(bubble#) = [1 0] x1 + [2] [0 0] [0] p(bubble'#) = [0 1] x1 + [0 2] x2 + [0 2] x3 + [0 3] x4 + [3] [0 0] [0 3] [1 1] [0 1] [0] p(bubble''#) = [1] [1] p(bubblesort#) = [0 3] x1 + [0] [0 0] [0] p(bubblesort'#) = [1 0] x1 + [0] [0 0] [0] p(lt#) = [0 0] x1 + [0 1] x2 + [1] [2 0] [0 0] [2] p(c_1) = [0 0] x1 + [0] [0 1] [0] p(c_2) = [0] [2] p(c_3) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_4) = [2] [0] p(c_5) = [1 0] x1 + [2] [0 0] [0] p(c_6) = [1 0] x1 + [0] [0 1] [0] p(c_7) = [1] [1] p(c_8) = [2 0] x1 + [0] [2 2] [0] p(c_9) = [0] [0] p(c_10) = [0 1] x1 + [0] [2 0] [0] p(c_11) = [0] [1] p(c_12) = [0] [0] p(c_13) = [1] [0] p(c_14) = [1 0] x1 + [1] [0 0] [0] Following rules are strictly oriented: bubble'#(true(),x,x',xs) = [0 2] x + [0 2] x' + [0 3] xs + [4] [0 3] [1 1] [0 1] [0] > [0 2] x' + [0 3] xs + [2] [0 0] [0 0] [0] = c_6(bubble#(dd(x',xs))) lt#(s(x),s(y)) = [0 1] y + [4] [0 0] [2] > [0 1] y + [2] [0 0] [0] = c_14(lt#(x,y)) Following rules are (at-least) weakly oriented: bubble#(dd(x,dd(x',xs))) = [0 2] x + [0 3] x' + [0 3] xs + [5] [0 0] [0 0] [0 0] [0] >= [0 2] x + [0 3] x' + [0 3] xs + [5] [0 0] [0 0] [0 0] [0] = c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) = [0 2] x + [0 2] x' + [0 3] xs + [4] [0 3] [1 1] [0 1] [0] >= [0 2] x + [0 3] xs + [4] [0 0] [0 0] [0] = c_5(bubble#(dd(x,xs))) bubblesort#(dd(x,xs)) = [0 3] x + [0 3] xs + [3] [0 0] [0 0] [0] >= [0 2] x + [0 3] xs + [2] [0 0] [0 0] [0] = bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) = [0 3] x + [0 3] xs + [3] [0 0] [0 0] [0] >= [0 3] x + [0 3] xs + [3] [0 0] [0 0] [0] = bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) = [0 3] xs + [1] [0 0] [0] >= [0 3] xs + [0] [0 0] [0] = bubblesort#(xs) bubble(dd(x,dd(x',xs))) = [0 3] x + [0 3] x' + [0 3] xs + [6] [0 4] [0 6] [0 6] [6] >= [0 3] x + [0 3] x' + [0 3] xs + [6] [0 4] [0 1] [0 0] [0] = bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) = [0 3] x + [3] [0 4] [0] >= [1] [0] = pair(nil(),x) bubble'(false(),x,x',xs) = [0 3] x + [0 3] x' + [0 3] xs + [6] [0 0] [0 1] [0 0] [0] >= [0 3] x + [0 3] x' + [0 3] xs + [6] [0 0] [0 0] [0 0] [0] = bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) = [0 3] x + [0 3] x' + [0 3] xs + [6] [0 0] [0 1] [0 0] [0] >= [0 3] x + [0 3] x' + [0 3] xs + [6] [0 0] [0 0] [0 0] [0] = bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) = [0 3] x + [0 3] xs + [4] [0 0] [0 0] [0] >= [0 3] x + [0 3] xs + [4] [0 0] [0 0] [0] = pair(dd(x,xs),x') lt(0(),0()) = [0] [1] >= [0] [1] = false() lt(0(),s(y)) = [0] [1] >= [0] [1] = true() lt(s(x),0()) = [0 2] x + [6] [0 0] [1] >= [0] [1] = false() lt(s(x),s(y)) = [0 2] x + [6] [0 0] [1] >= [0 2] x + [0] [0 0] [1] = lt(x,y) ***** Step 5.b:4.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) - Weak DPs: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) - Weak DPs: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) -->_2 lt#(s(x),s(y)) -> c_14(lt#(x,y)):7 -->_1 bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))):3 -->_1 bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))):2 2:S:bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 3:W:bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 4:W:bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 5:W:bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> bubblesort#(xs):6 6:W:bubblesort'#(pair(xs,x)) -> bubblesort#(xs) -->_1 bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))):5 -->_1 bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)):4 7:W:lt#(s(x),s(y)) -> c_14(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_14(lt#(x,y)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: lt#(s(x),s(y)) -> c_14(lt#(x,y)) ***** Step 5.b:4.a:1.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) - Weak DPs: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) -->_1 bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))):3 -->_1 bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))):2 2:S:bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 3:W:bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 4:W:bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):1 5:W:bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> bubblesort#(xs):6 6:W:bubblesort'#(pair(xs,x)) -> bubblesort#(xs) -->_1 bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))):5 -->_1 bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) ***** Step 5.b:4.a:1.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) - Weak DPs: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: 2: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) The strictly oriented rules are moved into the weak component. ****** Step 5.b:4.a:1.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) - Weak DPs: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {bubble,bubble',bubble'',lt,append#,bubble#,bubble'#,bubble''#,bubblesort#,bubblesort'#,lt#} TcT has computed the following interpretation: p(0) = [1] p(append) = [2] x1 + [4] x2 + [0] p(bubble) = [1] x1 + [0] p(bubble') = [1] x2 + [1] x3 + [1] x4 + [4] p(bubble'') = [1] x1 + [1] x2 + [2] p(bubblesort) = [1] p(bubblesort') = [1] p(dd) = [1] x1 + [1] x2 + [2] p(false) = [1] p(lt) = [1] p(nil) = [6] p(pair) = [1] x1 + [0] p(s) = [1] p(true) = [0] p(append#) = [1] x1 + [0] p(bubble#) = [2] x1 + [0] p(bubble'#) = [4] x1 + [2] x2 + [2] x3 + [2] x4 + [4] p(bubble''#) = [2] x2 + [0] p(bubblesort#) = [4] x1 + [0] p(bubblesort'#) = [4] x1 + [0] p(lt#) = [0] p(c_1) = [1] x1 + [1] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [4] p(c_5) = [1] x1 + [3] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] x2 + [0] p(c_9) = [1] p(c_10) = [4] p(c_11) = [2] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] Following rules are strictly oriented: bubble'#(false(),x,x',xs) = [2] x + [2] x' + [2] xs + [8] > [2] x + [2] xs + [7] = c_5(bubble#(dd(x,xs))) Following rules are (at-least) weakly oriented: bubble#(dd(x,dd(x',xs))) = [2] x + [2] x' + [2] xs + [8] >= [2] x + [2] x' + [2] xs + [8] = c_3(bubble'#(lt(x,x'),x,x',xs)) bubble'#(true(),x,x',xs) = [2] x + [2] x' + [2] xs + [4] >= [2] x' + [2] xs + [4] = c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) = [4] x + [4] xs + [8] >= [2] x + [2] xs + [4] = bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) = [4] x + [4] xs + [8] >= [4] x + [4] xs + [8] = bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) = [4] xs + [0] >= [4] xs + [0] = bubblesort#(xs) bubble(dd(x,dd(x',xs))) = [1] x + [1] x' + [1] xs + [4] >= [1] x + [1] x' + [1] xs + [4] = bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) = [1] x + [8] >= [6] = pair(nil(),x) bubble'(false(),x,x',xs) = [1] x + [1] x' + [1] xs + [4] >= [1] x + [1] x' + [1] xs + [4] = bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) = [1] x + [1] x' + [1] xs + [4] >= [1] x + [1] x' + [1] xs + [4] = bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) = [1] x + [1] xs + [2] >= [1] x + [1] xs + [2] = pair(dd(x,xs),x') lt(0(),0()) = [1] >= [1] = false() lt(0(),s(y)) = [1] >= [0] = true() lt(s(x),0()) = [1] >= [1] = false() lt(s(x),s(y)) = [1] >= [1] = lt(x,y) ****** Step 5.b:4.a:1.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) - Weak DPs: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) - Weak DPs: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) Consider the set of all dependency pairs 1: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) 2: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) 3: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) 4: bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) 5: bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) 6: bubblesort'#(pair(xs,x)) -> bubblesort#(xs) 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 {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 5.b:4.a:1.b:1.b:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) - Weak DPs: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {bubble,bubble',bubble'',lt,append#,bubble#,bubble'#,bubble''#,bubblesort#,bubblesort'#,lt#} TcT has computed the following interpretation: p(0) = [0] p(append) = [1] p(bubble) = [1] x1 + [0] p(bubble') = [1] x4 + [4] p(bubble'') = [1] x2 + [2] p(bubblesort) = [1] x1 + [1] p(bubblesort') = [0] p(dd) = [1] x2 + [2] p(false) = [1] p(lt) = [2] p(nil) = [2] p(pair) = [1] x1 + [0] p(s) = [1] p(true) = [1] p(append#) = [4] p(bubble#) = [3] x1 + [0] p(bubble'#) = [4] x1 + [3] x4 + [2] p(bubble''#) = [1] x2 + [0] p(bubblesort#) = [3] x1 + [0] p(bubblesort'#) = [3] x1 + [0] p(lt#) = [1] x1 + [1] x2 + [1] p(c_1) = [1] x1 + [1] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [1] x1 + [1] p(c_11) = [2] p(c_12) = [4] p(c_13) = [1] p(c_14) = [1] Following rules are strictly oriented: bubble#(dd(x,dd(x',xs))) = [3] xs + [12] > [3] xs + [10] = c_3(bubble'#(lt(x,x'),x,x',xs)) Following rules are (at-least) weakly oriented: bubble'#(false(),x,x',xs) = [3] xs + [6] >= [3] xs + [6] = c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) = [3] xs + [6] >= [3] xs + [6] = c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) = [3] xs + [6] >= [3] xs + [6] = bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) = [3] xs + [6] >= [3] xs + [6] = bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) = [3] xs + [0] >= [3] xs + [0] = bubblesort#(xs) bubble(dd(x,dd(x',xs))) = [1] xs + [4] >= [1] xs + [4] = bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) = [4] >= [2] = pair(nil(),x) bubble'(false(),x,x',xs) = [1] xs + [4] >= [1] xs + [4] = bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) = [1] xs + [4] >= [1] xs + [4] = bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) = [1] xs + [2] >= [1] xs + [2] = pair(dd(x,xs),x') lt(0(),0()) = [2] >= [1] = false() lt(0(),s(y)) = [2] >= [1] = true() lt(s(x),0()) = [2] >= [1] = false() lt(s(x),s(y)) = [2] >= [2] = lt(x,y) ******* Step 5.b:4.a:1.b:1.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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.b:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) -->_1 bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))):3 -->_1 bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))):2 2:W:bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)):1 3:W:bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)):1 4:W:bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)):1 5:W:bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> bubblesort#(xs):6 6:W:bubblesort'#(pair(xs,x)) -> bubblesort#(xs) -->_1 bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))):5 -->_1 bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) 6: bubblesort'#(pair(xs,x)) -> bubblesort#(xs) 4: bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) 1: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) 3: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) 2: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) ******* Step 5.b:4.a:1.b:1.b:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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^1)) + Considered Problem: - Strict DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak DPs: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_2 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):3 -->_1 bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)):2 2:S:bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) -->_1 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))):1 3:W:bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) -->_2 lt#(s(x),s(y)) -> c_14(lt#(x,y)):6 -->_1 bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))):5 -->_1 bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))):4 4:W:bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):3 5:W:bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) -->_1 bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')):3 6:W:lt#(s(x),s(y)) -> c_14(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_14(lt#(x,y)):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs),lt#(x,x')) 5: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) 4: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) 6: lt#(s(x),s(y)) -> c_14(lt#(x,y)) *** Step 5.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)):2 2:S:bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) -->_1 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs))),bubble#(dd(x,xs))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) *** Step 5.b:4.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) Consider the set of all dependency pairs 1: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) 2: bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) 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.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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_8) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {bubble,bubble',bubble'',lt,append#,bubble#,bubble'#,bubble''#,bubblesort#,bubblesort'#,lt#} TcT has computed the following interpretation: p(0) = [1] p(append) = [0] p(bubble) = [1] x1 + [0] p(bubble') = [8] x1 + [1] x4 + [0] p(bubble'') = [1] x2 + [8] p(bubblesort) = [2] p(bubblesort') = [1] x1 + [1] p(dd) = [1] x2 + [8] p(false) = [2] p(lt) = [2] p(nil) = [0] p(pair) = [1] x1 + [8] p(s) = [0] p(true) = [2] p(append#) = [1] x1 + [1] x2 + [1] p(bubble#) = [8] p(bubble'#) = [1] x1 + [1] x2 + [0] p(bubble''#) = [1] x2 + [0] p(bubblesort#) = [1] x1 + [8] p(bubblesort'#) = [1] x1 + [4] p(lt#) = [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [2] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [1] p(c_7) = [1] p(c_8) = [1] x1 + [0] p(c_9) = [2] p(c_10) = [1] x1 + [4] p(c_11) = [0] p(c_12) = [1] p(c_13) = [0] p(c_14) = [1] Following rules are strictly oriented: bubblesort#(dd(x,xs)) = [1] xs + [16] > [1] xs + [12] = c_8(bubblesort'#(bubble(dd(x,xs)))) Following rules are (at-least) weakly oriented: bubblesort'#(pair(xs,x)) = [1] xs + [12] >= [1] xs + [12] = c_10(bubblesort#(xs)) bubble(dd(x,dd(x',xs))) = [1] xs + [16] >= [1] xs + [16] = bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) = [8] >= [8] = pair(nil(),x) bubble'(false(),x,x',xs) = [1] xs + [16] >= [1] xs + [16] = bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) = [1] xs + [16] >= [1] xs + [16] = bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) = [1] xs + [16] >= [1] xs + [16] = pair(dd(x,xs),x') lt(0(),0()) = [2] >= [2] = false() lt(0(),s(y)) = [2] >= [2] = true() lt(s(x),0()) = [2] >= [2] = false() lt(s(x),s(y)) = [2] >= [2] = lt(x,y) **** Step 5.b:4.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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(1)) + Considered Problem: - Weak DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} and constructors {0,dd,false,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) -->_1 bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)):2 2:W:bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) -->_1 bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) 2: bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) **** Step 5.b:4.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: bubble(dd(x,dd(x',xs))) -> bubble'(lt(x,x'),x,x',xs) bubble(dd(x,nil())) -> pair(nil(),x) bubble'(false(),x,x',xs) -> bubble''(x',bubble(dd(x,xs))) bubble'(true(),x,x',xs) -> bubble''(x,bubble(dd(x',xs))) bubble''(x,pair(xs,x')) -> pair(dd(x,xs),x') lt(0(),0()) -> false() lt(0(),s(y)) -> true() lt(s(x),0()) -> false() lt(s(x),s(y)) -> lt(x,y) - Signature: {append/2,bubble/1,bubble'/4,bubble''/2,bubblesort/1,bubblesort'/1,lt/2,append#/2,bubble#/1,bubble'#/4 ,bubble''#/2,bubblesort#/1,bubblesort'#/1,lt#/2} / {0/0,dd/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0 ,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {append#,bubble#,bubble'#,bubble''#,bubblesort# ,bubblesort'#,lt#} 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))