WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + 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^3)) + 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^3)) + 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^3)) + 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: DecomposeDG WORST_CASE(?,O(n^3)) + 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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, 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(append#(bubblesort(xs),dd(x,nil())),bubblesort#(xs)) and a lower component 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))) 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)) -> append#(bubblesort(xs),dd(x,nil())) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) ** Step 5.a:1: 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(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: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)):2 2: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))):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)))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) ** Step 5.a:2: UsableRules 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: 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_10/1,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) bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) ** Step 5.a:3: Ara 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: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) bubble :: ["A"(5)] -(0)-> "A"(5) bubble' :: ["A"(0) x "A"(5) x "A"(5) x "A"(5)] -(10)-> "A"(5) bubble'' :: ["A"(5) x "A"(5)] -(5)-> "A"(5) dd :: ["A"(5) x "A"(5)] -(5)-> "A"(5) false :: [] -(0)-> "A"(0) false :: [] -(0)-> "A"(15) lt :: ["A"(0) x "A"(0)] -(0)-> "A"(15) nil :: [] -(0)-> "A"(5) nil :: [] -(0)-> "A"(15) pair :: ["A"(5) x "A"(0)] -(5)-> "A"(5) s :: ["A"(0)] -(0)-> "A"(0) true :: [] -(0)-> "A"(0) true :: [] -(0)-> "A"(15) bubblesort# :: ["A"(5)] -(14)-> "A"(1) bubblesort'# :: ["A"(5)] -(10)-> "A"(13) c_8 :: ["A"(0)] -(0)-> "A"(15) c_10 :: ["A"(0)] -(0)-> "A"(15) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "c_10_A" :: ["A"(0)] -(0)-> "A"(1) "c_8_A" :: ["A"(0)] -(0)-> "A"(1) "dd_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1) "false_A" :: [] -(0)-> "A"(1) "nil_A" :: [] -(0)-> "A"(1) "pair_A" :: ["A"(1) x "A"(0)] -(1)-> "A"(1) "s_A" :: ["A"(1)] -(1)-> "A"(1) "true_A" :: [] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: bubblesort'#(pair(xs,x)) -> c_10(bubblesort#(xs)) 2. Weak: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) ** Step 5.a:4: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(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_8/1,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: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) bubble :: ["A"(1)] -(0)-> "A"(1) bubble' :: ["A"(1) x "A"(0) x "A"(0) x "A"(1)] -(2)-> "A"(1) bubble'' :: ["A"(0) x "A"(1)] -(1)-> "A"(1) dd :: ["A"(0) x "A"(1)] -(1)-> "A"(1) false :: [] -(0)-> "A"(1) false :: [] -(0)-> "A"(15) lt :: ["A"(0) x "A"(0)] -(0)-> "A"(15) nil :: [] -(0)-> "A"(1) nil :: [] -(0)-> "A"(15) pair :: ["A"(1) x "A"(0)] -(1)-> "A"(1) s :: ["A"(0)] -(0)-> "A"(0) true :: [] -(0)-> "A"(1) true :: [] -(0)-> "A"(15) bubblesort# :: ["A"(1)] -(2)-> "A"(1) bubblesort'# :: ["A"(1)] -(1)-> "A"(13) c_8 :: ["A"(11)] -(0)-> "A"(11) c_10 :: ["A"(0)] -(0)-> "A"(15) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "c_10_A" :: ["A"(0)] -(0)-> "A"(1) "c_8_A" :: ["A"(0)] -(0)-> "A"(1) "dd_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "false_A" :: [] -(0)-> "A"(1) "nil_A" :: [] -(0)-> "A"(1) "pair_A" :: ["A"(1) x "A"(0)] -(1)-> "A"(1) "s_A" :: ["A"(1)] -(1)-> "A"(1) "true_A" :: [] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: bubblesort#(dd(x,xs)) -> c_8(bubblesort'#(bubble(dd(x,xs)))) 2. Weak: ** Step 5.b:1: DecomposeDG 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))) 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)) -> 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_8/2,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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component 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)) -> bubble#(dd(x,xs)) 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) and a lower component lt#(s(x),s(y)) -> c_14(lt#(x,y)) Further, following extension rules are added to the lower component. append#(dd(x,xs),ys) -> append#(xs,ys) bubble#(dd(x,dd(x',xs))) -> bubble'#(lt(x,x'),x,x',xs) bubble#(dd(x,dd(x',xs))) -> lt#(x,x') bubble'#(false(),x,x',xs) -> bubble#(dd(x,xs)) bubble'#(true(),x,x',xs) -> 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)) -> append#(bubblesort(xs),dd(x,nil())) bubblesort'#(pair(xs,x)) -> bubblesort#(xs) *** Step 5.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + 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))) - Weak DPs: bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) 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_8/2,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')) -->_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: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')):2 4: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')):2 5: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')):2 6:W:bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> bubblesort#(xs):8 -->_1 bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())):7 7:W:bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) -->_1 append#(dd(x,xs),ys) -> c_1(append#(xs,ys)):1 8:W:bubblesort'#(pair(xs,x)) -> bubblesort#(xs) -->_1 bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))):6 -->_1 bubblesort#(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: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) *** Step 5.b:1.a:2: Ara WORST_CASE(?,O(n^1)) + 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)) bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) - Weak DPs: bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) 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_3/1,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: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) append :: ["A"(1) x "A"(1)] -(0)-> "A"(1) bubble :: ["A"(7)] -(1)-> "A"(7) bubble' :: ["A"(1) x "A"(0) x "A"(0) x "A"(7)] -(15)-> "A"(7) bubble'' :: ["A"(0) x "A"(7)] -(7)-> "A"(7) bubblesort :: ["A"(7)] -(7)-> "A"(1) bubblesort' :: ["A"(7)] -(4)-> "A"(1) dd :: ["A"(0) x "A"(1)] -(1)-> "A"(1) dd :: ["A"(0) x "A"(7)] -(7)-> "A"(7) dd :: ["A"(0) x "A"(0)] -(0)-> "A"(0) false :: [] -(0)-> "A"(1) false :: [] -(0)-> "A"(15) lt :: ["A"(0) x "A"(0)] -(0)-> "A"(9) nil :: [] -(0)-> "A"(1) nil :: [] -(0)-> "A"(7) nil :: [] -(0)-> "A"(15) pair :: ["A"(7) x "A"(0)] -(7)-> "A"(7) pair :: ["A"(8) x "A"(0)] -(8)-> "A"(8) s :: ["A"(0)] -(0)-> "A"(0) true :: [] -(0)-> "A"(1) true :: [] -(0)-> "A"(15) append# :: ["A"(1) x "A"(0)] -(0)-> "A"(8) bubble# :: ["A"(7)] -(2)-> "A"(5) bubble'# :: ["A"(1) x "A"(0) x "A"(0) x "A"(7)] -(9)-> "A"(13) bubblesort# :: ["A"(7)] -(4)-> "A"(4) bubblesort'# :: ["A"(7)] -(0)-> "A"(4) c_1 :: ["A"(0)] -(0)-> "A"(12) c_3 :: ["A"(0)] -(0)-> "A"(12) c_5 :: ["A"(0)] -(0)-> "A"(15) c_6 :: ["A"(0)] -(0)-> "A"(15) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "c_1_A" :: ["A"(0)] -(0)-> "A"(1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1) "c_5_A" :: ["A"(0)] -(0)-> "A"(1) "c_6_A" :: ["A"(0)] -(0)-> "A"(1) "dd_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "false_A" :: [] -(0)-> "A"(1) "nil_A" :: [] -(0)-> "A"(1) "pair_A" :: ["A"(1) x "A"(0)] -(1)-> "A"(1) "s_A" :: ["A"(0)] -(0)-> "A"(1) "true_A" :: [] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) 2. Weak: 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))) *** Step 5.b:1.a:3: Ara 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))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) - Weak DPs: append#(dd(x,xs),ys) -> c_1(append#(xs,ys)) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) 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_3/1,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: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) append :: ["A"(0) x "A"(0)] -(0)-> "A"(0) bubble :: ["A"(1)] -(0)-> "A"(1) bubble' :: ["A"(1) x "A"(0) x "A"(0) x "A"(1)] -(2)-> "A"(1) bubble'' :: ["A"(0) x "A"(1)] -(1)-> "A"(1) bubblesort :: ["A"(1)] -(0)-> "A"(0) bubblesort' :: ["A"(1)] -(0)-> "A"(0) dd :: ["A"(0) x "A"(1)] -(1)-> "A"(1) dd :: ["A"(0) x "A"(0)] -(0)-> "A"(0) false :: [] -(0)-> "A"(1) false :: [] -(0)-> "A"(15) lt :: ["A"(0) x "A"(0)] -(0)-> "A"(9) nil :: [] -(0)-> "A"(0) nil :: [] -(0)-> "A"(1) nil :: [] -(0)-> "A"(15) nil :: [] -(0)-> "A"(7) pair :: ["A"(1) x "A"(0)] -(0)-> "A"(1) pair :: ["A"(8) x "A"(0)] -(0)-> "A"(8) s :: ["A"(0)] -(0)-> "A"(0) true :: [] -(0)-> "A"(1) true :: [] -(0)-> "A"(15) append# :: ["A"(0) x "A"(0)] -(0)-> "A"(0) bubble# :: ["A"(1)] -(0)-> "A"(1) bubble'# :: ["A"(1) x "A"(0) x "A"(0) x "A"(1)] -(1)-> "A"(2) bubblesort# :: ["A"(1)] -(6)-> "A"(0) bubblesort'# :: ["A"(1)] -(6)-> "A"(0) c_1 :: ["A"(0)] -(0)-> "A"(12) c_3 :: ["A"(0)] -(0)-> "A"(3) c_5 :: ["A"(0)] -(0)-> "A"(14) c_6 :: ["A"(0)] -(0)-> "A"(3) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "c_1_A" :: ["A"(0)] -(0)-> "A"(1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1) "c_5_A" :: ["A"(0)] -(0)-> "A"(1) "c_6_A" :: ["A"(0)] -(0)-> "A"(1) "dd_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "false_A" :: [] -(0)-> "A"(1) "nil_A" :: [] -(0)-> "A"(1) "pair_A" :: ["A"(1) x "A"(0)] -(0)-> "A"(1) "s_A" :: ["A"(1)] -(1)-> "A"(1) "true_A" :: [] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: bubble#(dd(x,dd(x',xs))) -> c_3(bubble'#(lt(x,x'),x,x',xs)) 2. Weak: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) *** Step 5.b:1.a:4: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) - Weak 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)) bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) 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_3/1,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: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) append :: ["A"(0) x "A"(0)] -(0)-> "A"(0) bubble :: ["A"(1)] -(1)-> "A"(1) bubble' :: ["A"(1) x "A"(0) x "A"(0) x "A"(1)] -(3)-> "A"(1) bubble'' :: ["A"(0) x "A"(1)] -(1)-> "A"(1) bubblesort :: ["A"(1)] -(15)-> "A"(0) bubblesort' :: ["A"(1)] -(14)-> "A"(0) dd :: ["A"(0) x "A"(0)] -(0)-> "A"(0) dd :: ["A"(0) x "A"(1)] -(1)-> "A"(1) false :: [] -(0)-> "A"(0) false :: [] -(0)-> "A"(1) false :: [] -(0)-> "A"(13) lt :: ["A"(0) x "A"(0)] -(0)-> "A"(11) nil :: [] -(0)-> "A"(0) nil :: [] -(0)-> "A"(1) nil :: [] -(0)-> "A"(15) nil :: [] -(0)-> "A"(7) pair :: ["A"(1) x "A"(0)] -(1)-> "A"(1) s :: ["A"(0)] -(0)-> "A"(0) true :: [] -(0)-> "A"(0) true :: [] -(0)-> "A"(1) true :: [] -(0)-> "A"(13) append# :: ["A"(0) x "A"(0)] -(0)-> "A"(1) bubble# :: ["A"(1)] -(0)-> "A"(2) bubble'# :: ["A"(0) x "A"(0) x "A"(0) x "A"(1)] -(2)-> "A"(4) bubblesort# :: ["A"(1)] -(15)-> "A"(1) bubblesort'# :: ["A"(1)] -(14)-> "A"(1) c_1 :: ["A"(0)] -(0)-> "A"(12) c_3 :: ["A"(0)] -(0)-> "A"(15) c_5 :: ["A"(0)] -(0)-> "A"(14) c_6 :: ["A"(0)] -(0)-> "A"(15) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "c_1_A" :: ["A"(0)] -(0)-> "A"(1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1) "c_5_A" :: ["A"(0)] -(0)-> "A"(1) "c_6_A" :: ["A"(0)] -(0)-> "A"(1) "dd_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "false_A" :: [] -(0)-> "A"(1) "nil_A" :: [] -(0)-> "A"(1) "pair_A" :: ["A"(1) x "A"(0)] -(1)-> "A"(1) "s_A" :: ["A"(1)] -(1)-> "A"(1) "true_A" :: [] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: bubble'#(false(),x,x',xs) -> c_5(bubble#(dd(x,xs))) 2. Weak: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) *** Step 5.b:1.a:5: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) - Weak 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)) bubble'#(false(),x,x',xs) -> c_5(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)) -> 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_3/1,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: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) append :: ["A"(0) x "A"(0)] -(0)-> "A"(0) bubble :: ["A"(4)] -(0)-> "A"(4) bubble' :: ["A"(1) x "A"(0) x "A"(0) x "A"(4)] -(8)-> "A"(4) bubble'' :: ["A"(0) x "A"(4)] -(4)-> "A"(4) bubblesort :: ["A"(4)] -(12)-> "A"(0) bubblesort' :: ["A"(4)] -(12)-> "A"(0) dd :: ["A"(0) x "A"(0)] -(0)-> "A"(0) dd :: ["A"(0) x "A"(4)] -(4)-> "A"(4) dd :: ["A"(0) x "A"(1)] -(1)-> "A"(1) false :: [] -(0)-> "A"(1) false :: [] -(0)-> "A"(0) false :: [] -(0)-> "A"(13) lt :: ["A"(0) x "A"(0)] -(0)-> "A"(11) nil :: [] -(0)-> "A"(0) nil :: [] -(0)-> "A"(4) nil :: [] -(0)-> "A"(15) nil :: [] -(0)-> "A"(7) pair :: ["A"(4) x "A"(0)] -(0)-> "A"(4) s :: ["A"(0)] -(0)-> "A"(0) true :: [] -(0)-> "A"(0) true :: [] -(0)-> "A"(1) true :: [] -(0)-> "A"(13) append# :: ["A"(0) x "A"(0)] -(0)-> "A"(1) bubble# :: ["A"(4)] -(0)-> "A"(13) bubble'# :: ["A"(0) x "A"(0) x "A"(0) x "A"(4)] -(8)-> "A"(9) bubblesort# :: ["A"(4)] -(14)-> "A"(1) bubblesort'# :: ["A"(4)] -(14)-> "A"(1) c_1 :: ["A"(0)] -(0)-> "A"(12) c_3 :: ["A"(0)] -(0)-> "A"(15) c_5 :: ["A"(13)] -(0)-> "A"(13) c_6 :: ["A"(0)] -(0)-> "A"(12) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "c_1_A" :: ["A"(0)] -(0)-> "A"(1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1) "c_5_A" :: ["A"(0)] -(0)-> "A"(1) "c_6_A" :: ["A"(0)] -(0)-> "A"(1) "dd_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "false_A" :: [] -(0)-> "A"(1) "nil_A" :: [] -(0)-> "A"(1) "pair_A" :: ["A"(1) x "A"(0)] -(0)-> "A"(1) "s_A" :: ["A"(1)] -(1)-> "A"(1) "true_A" :: [] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: bubble'#(true(),x,x',xs) -> c_6(bubble#(dd(x',xs))) 2. Weak: *** Step 5.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak DPs: append#(dd(x,xs),ys) -> append#(xs,ys) bubble#(dd(x,dd(x',xs))) -> bubble'#(lt(x,x'),x,x',xs) bubble#(dd(x,dd(x',xs))) -> lt#(x,x') bubble'#(false(),x,x',xs) -> bubble#(dd(x,xs)) bubble'#(true(),x,x',xs) -> 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)) -> 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_8/2,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:lt#(s(x),s(y)) -> c_14(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_14(lt#(x,y)):1 2:W:append#(dd(x,xs),ys) -> append#(xs,ys) -->_1 append#(dd(x,xs),ys) -> append#(xs,ys):2 3:W:bubble#(dd(x,dd(x',xs))) -> bubble'#(lt(x,x'),x,x',xs) -->_1 bubble'#(true(),x,x',xs) -> bubble#(dd(x',xs)):6 -->_1 bubble'#(false(),x,x',xs) -> bubble#(dd(x,xs)):5 4:W:bubble#(dd(x,dd(x',xs))) -> lt#(x,x') -->_1 lt#(s(x),s(y)) -> c_14(lt#(x,y)):1 5:W:bubble'#(false(),x,x',xs) -> bubble#(dd(x,xs)) -->_1 bubble#(dd(x,dd(x',xs))) -> lt#(x,x'):4 -->_1 bubble#(dd(x,dd(x',xs))) -> bubble'#(lt(x,x'),x,x',xs):3 6:W:bubble'#(true(),x,x',xs) -> bubble#(dd(x',xs)) -->_1 bubble#(dd(x,dd(x',xs))) -> lt#(x,x'):4 -->_1 bubble#(dd(x,dd(x',xs))) -> bubble'#(lt(x,x'),x,x',xs):3 7:W:bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)) -->_1 bubble#(dd(x,dd(x',xs))) -> lt#(x,x'):4 -->_1 bubble#(dd(x,dd(x',xs))) -> bubble'#(lt(x,x'),x,x',xs):3 8:W:bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))) -->_1 bubblesort'#(pair(xs,x)) -> bubblesort#(xs):10 -->_1 bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())):9 9:W:bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) -->_1 append#(dd(x,xs),ys) -> append#(xs,ys):2 10:W:bubblesort'#(pair(xs,x)) -> bubblesort#(xs) -->_1 bubblesort#(dd(x,xs)) -> bubblesort'#(bubble(dd(x,xs))):8 -->_1 bubblesort#(dd(x,xs)) -> bubble#(dd(x,xs)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: bubblesort'#(pair(xs,x)) -> append#(bubblesort(xs),dd(x,nil())) 2: append#(dd(x,xs),ys) -> append#(xs,ys) *** Step 5.b:1.b:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak DPs: bubble#(dd(x,dd(x',xs))) -> bubble'#(lt(x,x'),x,x',xs) bubble#(dd(x,dd(x',xs))) -> lt#(x,x') bubble'#(false(),x,x',xs) -> bubble#(dd(x,xs)) bubble'#(true(),x,x',xs) -> 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: 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: 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))) -> bubble'#(lt(x,x'),x,x',xs) bubble#(dd(x,dd(x',xs))) -> lt#(x,x') bubble'#(false(),x,x',xs) -> bubble#(dd(x,xs)) bubble'#(true(),x,x',xs) -> 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)) *** Step 5.b:1.b:3: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#(s(x),s(y)) -> c_14(lt#(x,y)) - Weak DPs: bubble#(dd(x,dd(x',xs))) -> bubble'#(lt(x,x'),x,x',xs) bubble#(dd(x,dd(x',xs))) -> lt#(x,x') bubble'#(false(),x,x',xs) -> bubble#(dd(x,xs)) bubble'#(true(),x,x',xs) -> 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_10/2,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: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) bubble :: ["A"(11)] -(11)-> "A"(11) bubble' :: ["A"(1) x "A"(11) x "A"(11) x "A"(11)] -(11)-> "A"(11) bubble'' :: ["A"(11) x "A"(11)] -(0)-> "A"(11) dd :: ["A"(11) x "A"(11)] -(0)-> "A"(11) false :: [] -(0)-> "A"(1) false :: [] -(0)-> "A"(15) lt :: ["A"(0) x "A"(0)] -(0)-> "A"(13) nil :: [] -(0)-> "A"(11) nil :: [] -(0)-> "A"(15) pair :: ["A"(11) x "A"(0)] -(11)-> "A"(11) s :: ["A"(11)] -(11)-> "A"(11) s :: ["A"(0)] -(0)-> "A"(0) true :: [] -(0)-> "A"(1) true :: [] -(0)-> "A"(15) bubble# :: ["A"(11)] -(12)-> "A"(1) bubble'# :: ["A"(1) x "A"(11) x "A"(11) x "A"(11)] -(12)-> "A"(1) bubblesort# :: ["A"(11)] -(12)-> "A"(0) bubblesort'# :: ["A"(11)] -(1)-> "A"(0) lt# :: ["A"(11) x "A"(11)] -(11)-> "A"(3) c_14 :: ["A"(0)] -(0)-> "A"(15) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "c_14_A" :: ["A"(0)] -(0)-> "A"(1) "dd_A" :: ["A"(1) x "A"(1)] -(0)-> "A"(1) "false_A" :: [] -(0)-> "A"(1) "nil_A" :: [] -(0)-> "A"(1) "pair_A" :: ["A"(1) x "A"(0)] -(1)-> "A"(1) "s_A" :: ["A"(1)] -(1)-> "A"(1) "true_A" :: [] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: lt#(s(x),s(y)) -> c_14(lt#(x,y)) 2. Weak: WORST_CASE(?,O(n^3))