WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: goal(xs) -> mergesort(xs) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3} / {0/0,Cons/2,False/0,Nil/0,S/1 ,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {<=,goal,merge,merge[Ite],mergesort,notEmpty ,splitmerge} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: goal(xs) -> mergesort(xs) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3} / {0/0,Cons/2,False/0,Nil/0,S/1 ,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {<=,goal,merge,merge[Ite],mergesort,notEmpty ,splitmerge} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: splitmerge(y,z,u){y -> Cons(x,y)} = splitmerge(Cons(x,y),z,u) ->^+ splitmerge(y,Cons(x,u),z) = C[splitmerge(y,Cons(x,u),z) = splitmerge(y,z,u){z -> Cons(x,u),u -> z}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: goal(xs) -> mergesort(xs) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3} / {0/0,Cons/2,False/0,Nil/0,S/1 ,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {<=,goal,merge,merge[Ite],mergesort,notEmpty ,splitmerge} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs goal#(xs) -> c_1(mergesort#(xs)) merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)) merge#(Nil(),xs2) -> c_4() mergesort#(Cons(x,Nil())) -> c_5() mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) mergesort#(Nil()) -> c_7() notEmpty#(Cons(x,xs)) -> c_8() notEmpty#(Nil()) -> c_9() splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) Weak DPs <=#(0(),y) -> c_12() <=#(S(x),0()) -> c_13() <=#(S(x),S(y)) -> c_14(<=#(x,y)) merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: goal#(xs) -> c_1(mergesort#(xs)) merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)) merge#(Nil(),xs2) -> c_4() mergesort#(Cons(x,Nil())) -> c_5() mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) mergesort#(Nil()) -> c_7() notEmpty#(Cons(x,xs)) -> c_8() notEmpty#(Nil()) -> c_9() splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) - Weak DPs: <=#(0(),y) -> c_12() <=#(S(x),0()) -> c_13() <=#(S(x),S(y)) -> c_14(<=#(x,y)) merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) goal(xs) -> mergesort(xs) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5,7,8,9} by application of Pre({5,7,8,9}) = {1,11}. Here rules are labelled as follows: 1: goal#(xs) -> c_1(mergesort#(xs)) 2: merge#(Cons(x,xs),Nil()) -> c_2() 3: merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)) 4: merge#(Nil(),xs2) -> c_4() 5: mergesort#(Cons(x,Nil())) -> c_5() 6: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) 7: mergesort#(Nil()) -> c_7() 8: notEmpty#(Cons(x,xs)) -> c_8() 9: notEmpty#(Nil()) -> c_9() 10: splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) 11: splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)) 12: <=#(0(),y) -> c_12() 13: <=#(S(x),0()) -> c_13() 14: <=#(S(x),S(y)) -> c_14(<=#(x,y)) 15: merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) 16: merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) ** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: goal#(xs) -> c_1(mergesort#(xs)) merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)) merge#(Nil(),xs2) -> c_4() mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) - Weak DPs: <=#(0(),y) -> c_12() <=#(S(x),0()) -> c_13() <=#(S(x),S(y)) -> c_14(<=#(x,y)) merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) mergesort#(Cons(x,Nil())) -> c_5() mergesort#(Nil()) -> c_7() notEmpty#(Cons(x,xs)) -> c_8() notEmpty#(Nil()) -> c_9() - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) goal(xs) -> mergesort(xs) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:goal#(xs) -> c_1(mergesort#(xs)) -->_1 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):5 -->_1 mergesort#(Nil()) -> c_7():14 -->_1 mergesort#(Cons(x,Nil())) -> c_5():13 2:S:merge#(Cons(x,xs),Nil()) -> c_2() 3:S:merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)) -->_1 merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)):12 -->_1 merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)):11 -->_2 <=#(S(x),S(y)) -> c_14(<=#(x,y)):10 -->_2 <=#(S(x),0()) -> c_13():9 -->_2 <=#(0(),y) -> c_12():8 4:S:merge#(Nil(),xs2) -> c_4() 5:S:mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):6 6:S:splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) -->_1 splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)):7 -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):6 7:S:splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)) -->_3 mergesort#(Nil()) -> c_7():14 -->_2 mergesort#(Nil()) -> c_7():14 -->_3 mergesort#(Cons(x,Nil())) -> c_5():13 -->_2 mergesort#(Cons(x,Nil())) -> c_5():13 -->_3 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):5 -->_2 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):5 -->_1 merge#(Nil(),xs2) -> c_4():4 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)):3 -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 8:W:<=#(0(),y) -> c_12() 9:W:<=#(S(x),0()) -> c_13() 10:W:<=#(S(x),S(y)) -> c_14(<=#(x,y)) -->_1 <=#(S(x),S(y)) -> c_14(<=#(x,y)):10 -->_1 <=#(S(x),0()) -> c_13():9 -->_1 <=#(0(),y) -> c_12():8 11:W:merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) -->_1 merge#(Nil(),xs2) -> c_4():4 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)):3 -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 12:W:merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) -->_1 merge#(Nil(),xs2) -> c_4():4 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)):3 -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 13:W:mergesort#(Cons(x,Nil())) -> c_5() 14:W:mergesort#(Nil()) -> c_7() 15:W:notEmpty#(Cons(x,xs)) -> c_8() 16:W:notEmpty#(Nil()) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 16: notEmpty#(Nil()) -> c_9() 15: notEmpty#(Cons(x,xs)) -> c_8() 10: <=#(S(x),S(y)) -> c_14(<=#(x,y)) 8: <=#(0(),y) -> c_12() 9: <=#(S(x),0()) -> c_13() 13: mergesort#(Cons(x,Nil())) -> c_5() 14: mergesort#(Nil()) -> c_7() ** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: goal#(xs) -> c_1(mergesort#(xs)) merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)) merge#(Nil(),xs2) -> c_4() mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) - Weak DPs: merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) goal(xs) -> mergesort(xs) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/2,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:goal#(xs) -> c_1(mergesort#(xs)) -->_1 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):5 2:S:merge#(Cons(x,xs),Nil()) -> c_2() 3:S:merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)) -->_1 merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)):12 -->_1 merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)):11 4:S:merge#(Nil(),xs2) -> c_4() 5:S:mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):6 6:S:splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) -->_1 splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)):7 -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):6 7:S:splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)) -->_3 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):5 -->_2 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):5 -->_1 merge#(Nil(),xs2) -> c_4():4 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)):3 -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 11:W:merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) -->_1 merge#(Nil(),xs2) -> c_4():4 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)):3 -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 12:W:merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) -->_1 merge#(Nil(),xs2) -> c_4():4 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs)),<=#(x',x)):3 -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) ** Step 1.b:5: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: goal#(xs) -> c_1(mergesort#(xs)) merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) merge#(Nil(),xs2) -> c_4() mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) - Weak DPs: merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) goal(xs) -> mergesort(xs) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) goal#(xs) -> c_1(mergesort#(xs)) merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) merge#(Nil(),xs2) -> c_4() merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) ** Step 1.b:6: RemoveHeads WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: goal#(xs) -> c_1(mergesort#(xs)) merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) merge#(Nil(),xs2) -> c_4() mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) - Weak DPs: merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:goal#(xs) -> c_1(mergesort#(xs)) -->_1 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):5 2:S:merge#(Cons(x,xs),Nil()) -> c_2() 3:S:merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) -->_1 merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)):9 -->_1 merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)):8 4:S:merge#(Nil(),xs2) -> c_4() 5:S:mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):6 6:S:splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) -->_1 splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)):7 -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):6 7:S:splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)) -->_3 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):5 -->_2 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):5 -->_1 merge#(Nil(),xs2) -> c_4():4 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):3 -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 8:W:merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) -->_1 merge#(Nil(),xs2) -> c_4():4 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):3 -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 9:W:merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) -->_1 merge#(Nil(),xs2) -> c_4():4 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):3 -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(1,goal#(xs) -> c_1(mergesort#(xs)))] ** Step 1.b:7: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) merge#(Nil(),xs2) -> c_4() mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) - Weak DPs: merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,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 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) and a lower component merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) merge#(Nil(),xs2) -> c_4() merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) Further, following extension rules are added to the lower component. mergesort#(Cons(x',Cons(x,xs))) -> splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil()) splitmerge#(Cons(x,xs),xs1,xs2) -> splitmerge#(xs,Cons(x,xs2),xs1) splitmerge#(Nil(),xs1,xs2) -> merge#(mergesort(xs1),mergesort(xs2)) splitmerge#(Nil(),xs1,xs2) -> mergesort#(xs1) splitmerge#(Nil(),xs1,xs2) -> mergesort#(xs2) *** Step 1.b:7.a:1: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)),mergesort#(xs1),mergesort#(xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):2 2:S:splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) -->_1 splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)):3 -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):2 3:S:splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)) -->_3 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):1 -->_2 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) *** Step 1.b:7.a:2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) *** Step 1.b:7.a:3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_6) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(<=) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(False) = [0] p(Nil) = [1] p(S) = [0] p(True) = [0] p(goal) = [0] p(merge) = [0] p(merge[Ite]) = [0] p(mergesort) = [0] p(notEmpty) = [0] p(splitmerge) = [0] p(<=#) = [0] p(goal#) = [0] p(merge#) = [0] p(merge[Ite]#) = [0] p(mergesort#) = [1] p(notEmpty#) = [0] p(splitmerge#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [1] x2 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [2] x1 + [0] p(c_16) = [4] x1 + [1] Following rules are strictly oriented: mergesort#(Cons(x',Cons(x,xs))) = [1] > [0] = c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) Following rules are (at-least) weakly oriented: splitmerge#(Cons(x,xs),xs1,xs2) = [0] >= [0] = c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) = [0] >= [2] = c_11(mergesort#(xs1),mergesort#(xs2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:7.a:4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) - Weak DPs: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_6) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(<=) = [1] x2 + [1] p(Cons) = [8] p(False) = [0] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [0] p(goal) = [0] p(merge) = [0] p(merge[Ite]) = [0] p(mergesort) = [0] p(notEmpty) = [0] p(splitmerge) = [0] p(<=#) = [0] p(goal#) = [0] p(merge#) = [0] p(merge[Ite]#) = [2] x1 + [1] x3 + [2] p(mergesort#) = [2] x1 + [0] p(notEmpty#) = [4] x1 + [0] p(splitmerge#) = [3] x2 + [3] x3 + [5] p(c_1) = [2] x1 + [0] p(c_2) = [2] p(c_3) = [1] x1 + [8] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [1] p(c_7) = [0] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [1] x2 + [0] p(c_12) = [1] p(c_13) = [8] p(c_14) = [1] x1 + [2] p(c_15) = [1] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: splitmerge#(Nil(),xs1,xs2) = [3] xs1 + [3] xs2 + [5] > [2] xs1 + [2] xs2 + [0] = c_11(mergesort#(xs1),mergesort#(xs2)) Following rules are (at-least) weakly oriented: mergesort#(Cons(x',Cons(x,xs))) = [16] >= [6] = c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) = [3] xs1 + [3] xs2 + [5] >= [3] xs1 + [30] = c_10(splitmerge#(xs,Cons(x,xs2),xs1)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:7.a:5: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) - Weak DPs: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_6) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1,2} Following symbols are considered usable: {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty#,splitmerge#} TcT has computed the following interpretation: p(0) = 1 p(<=) = 1 + x1^2 p(Cons) = 1 + x2 p(False) = 2 p(Nil) = 0 p(S) = 1 p(True) = 1 p(goal) = 4*x1 + 2*x1^2 p(merge) = 4 + x1 + x2^2 p(merge[Ite]) = 2*x1*x2 + x1^2 + 2*x2 + x3 + x3^2 p(mergesort) = 1 + x1 + 4*x1^2 p(notEmpty) = 1 + x1 p(splitmerge) = x1 + 4*x1*x3 + 2*x1^2 + x2 + 2*x2^2 + x3 + 4*x3^2 p(<=#) = 1 + 4*x1 + x2 p(goal#) = 0 p(merge#) = 4*x1 + 2*x1*x2 + x1^2 p(merge[Ite]#) = 1 + 4*x2^2 + x3 + x3^2 p(mergesort#) = 3*x1^2 p(notEmpty#) = 0 p(splitmerge#) = 2*x1 + 3*x1*x2 + 3*x1*x3 + 2*x1^2 + 3*x2^2 + 3*x3 + 3*x3^2 p(c_1) = 1 p(c_2) = 2 p(c_3) = 1 p(c_4) = 0 p(c_5) = 0 p(c_6) = x1 p(c_7) = 1 p(c_8) = 0 p(c_9) = 1 p(c_10) = x1 p(c_11) = x1 + x2 p(c_12) = 0 p(c_13) = 1 p(c_14) = 1 + x1 p(c_15) = 1 p(c_16) = 1 Following rules are strictly oriented: splitmerge#(Cons(x,xs),xs1,xs2) = 4 + 6*xs + 3*xs*xs1 + 3*xs*xs2 + 2*xs^2 + 3*xs1 + 3*xs1^2 + 6*xs2 + 3*xs2^2 > 3 + 5*xs + 3*xs*xs1 + 3*xs*xs2 + 2*xs^2 + 3*xs1 + 3*xs1^2 + 6*xs2 + 3*xs2^2 = c_10(splitmerge#(xs,Cons(x,xs2),xs1)) Following rules are (at-least) weakly oriented: mergesort#(Cons(x',Cons(x,xs))) = 12 + 12*xs + 3*xs^2 >= 12 + 10*xs + 2*xs^2 = c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Nil(),xs1,xs2) = 3*xs1^2 + 3*xs2 + 3*xs2^2 >= 3*xs1^2 + 3*xs2^2 = c_11(mergesort#(xs1),mergesort#(xs2)) *** Step 1.b:7.a:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:7.b:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) merge#(Nil(),xs2) -> c_4() - Weak DPs: merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) mergesort#(Cons(x',Cons(x,xs))) -> splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil()) splitmerge#(Cons(x,xs),xs1,xs2) -> splitmerge#(xs,Cons(x,xs2),xs1) splitmerge#(Nil(),xs1,xs2) -> merge#(mergesort(xs1),mergesort(xs2)) splitmerge#(Nil(),xs1,xs2) -> mergesort#(xs1) splitmerge#(Nil(),xs1,xs2) -> mergesort#(xs2) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {<=,<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty#,splitmerge#} TcT has computed the following interpretation: p(0) = [1] p(<=) = [1] p(Cons) = [0] p(False) = [1] p(Nil) = [0] p(S) = [1] p(True) = [1] p(goal) = [1] p(merge) = [4] p(merge[Ite]) = [2] x1 + [2] p(mergesort) = [4] x1 + [0] p(notEmpty) = [1] p(splitmerge) = [4] x1 + [1] x3 + [0] p(<=#) = [1] x2 + [0] p(goal#) = [4] p(merge#) = [4] p(merge[Ite]#) = [4] x1 + [0] p(mergesort#) = [7] p(notEmpty#) = [0] p(splitmerge#) = [1] x2 + [1] x3 + [7] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [2] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [2] p(c_11) = [2] x1 + [1] x2 + [0] p(c_12) = [0] p(c_13) = [4] p(c_14) = [1] x1 + [2] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: merge#(Cons(x,xs),Nil()) = [4] > [1] = c_2() merge#(Nil(),xs2) = [4] > [2] = c_4() Following rules are (at-least) weakly oriented: merge#(Cons(x',xs'),Cons(x,xs)) = [4] >= [4] = c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) merge[Ite]#(False(),xs1,Cons(x,xs)) = [4] >= [4] = c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) = [4] >= [4] = c_16(merge#(xs,xs2)) mergesort#(Cons(x',Cons(x,xs))) = [7] >= [7] = splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil()) splitmerge#(Cons(x,xs),xs1,xs2) = [1] xs1 + [1] xs2 + [7] >= [1] xs1 + [7] = splitmerge#(xs,Cons(x,xs2),xs1) splitmerge#(Nil(),xs1,xs2) = [1] xs1 + [1] xs2 + [7] >= [4] = merge#(mergesort(xs1),mergesort(xs2)) splitmerge#(Nil(),xs1,xs2) = [1] xs1 + [1] xs2 + [7] >= [7] = mergesort#(xs1) splitmerge#(Nil(),xs1,xs2) = [1] xs1 + [1] xs2 + [7] >= [7] = mergesort#(xs2) <=(0(),y) = [1] >= [1] = True() <=(S(x),0()) = [1] >= [1] = False() <=(S(x),S(y)) = [1] >= [1] = <=(x,y) *** Step 1.b:7.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) - Weak DPs: merge#(Cons(x,xs),Nil()) -> c_2() merge#(Nil(),xs2) -> c_4() merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) mergesort#(Cons(x',Cons(x,xs))) -> splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil()) splitmerge#(Cons(x,xs),xs1,xs2) -> splitmerge#(xs,Cons(x,xs2),xs1) splitmerge#(Nil(),xs1,xs2) -> merge#(mergesort(xs1),mergesort(xs2)) splitmerge#(Nil(),xs1,xs2) -> mergesort#(xs1) splitmerge#(Nil(),xs1,xs2) -> mergesort#(xs2) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(merge) = {1,2}, uargs(merge[Ite]) = {1}, uargs(merge#) = {1,2}, uargs(merge[Ite]#) = {1}, uargs(c_3) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(<=) = [0] p(Cons) = [1] x2 + [6] p(False) = [0] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [0] p(goal) = [4] x1 + [1] p(merge) = [1] x1 + [1] x2 + [0] p(merge[Ite]) = [1] x1 + [1] x2 + [1] x3 + [0] p(mergesort) = [1] x1 + [0] p(notEmpty) = [2] x1 + [1] p(splitmerge) = [1] x1 + [1] x2 + [1] x3 + [0] p(<=#) = [1] x2 + [0] p(goal#) = [1] x1 + [2] p(merge#) = [1] x1 + [1] x2 + [1] p(merge[Ite]#) = [1] x1 + [1] x2 + [1] x3 + [0] p(mergesort#) = [1] x1 + [1] p(notEmpty#) = [1] p(splitmerge#) = [1] x1 + [1] x2 + [1] x3 + [1] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] x1 + [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [0] p(c_10) = [4] p(c_11) = [1] x2 + [4] p(c_12) = [2] p(c_13) = [1] p(c_14) = [1] x1 + [4] p(c_15) = [1] x1 + [1] p(c_16) = [1] x1 + [5] Following rules are strictly oriented: merge#(Cons(x',xs'),Cons(x,xs)) = [1] xs + [1] xs' + [13] > [1] xs + [1] xs' + [12] = c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) Following rules are (at-least) weakly oriented: merge#(Cons(x,xs),Nil()) = [1] xs + [7] >= [0] = c_2() merge#(Nil(),xs2) = [1] xs2 + [1] >= [1] = c_4() merge[Ite]#(False(),xs1,Cons(x,xs)) = [1] xs + [1] xs1 + [6] >= [1] xs + [1] xs1 + [2] = c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) = [1] xs + [1] xs2 + [6] >= [1] xs + [1] xs2 + [6] = c_16(merge#(xs,xs2)) mergesort#(Cons(x',Cons(x,xs))) = [1] xs + [13] >= [1] xs + [13] = splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil()) splitmerge#(Cons(x,xs),xs1,xs2) = [1] xs + [1] xs1 + [1] xs2 + [7] >= [1] xs + [1] xs1 + [1] xs2 + [7] = splitmerge#(xs,Cons(x,xs2),xs1) splitmerge#(Nil(),xs1,xs2) = [1] xs1 + [1] xs2 + [1] >= [1] xs1 + [1] xs2 + [1] = merge#(mergesort(xs1),mergesort(xs2)) splitmerge#(Nil(),xs1,xs2) = [1] xs1 + [1] xs2 + [1] >= [1] xs1 + [1] = mergesort#(xs1) splitmerge#(Nil(),xs1,xs2) = [1] xs1 + [1] xs2 + [1] >= [1] xs2 + [1] = mergesort#(xs2) <=(0(),y) = [0] >= [0] = True() <=(S(x),0()) = [0] >= [0] = False() <=(S(x),S(y)) = [0] >= [0] = <=(x,y) merge(Cons(x,xs),Nil()) = [1] xs + [6] >= [1] xs + [6] = Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) = [1] xs + [1] xs' + [12] >= [1] xs + [1] xs' + [12] = merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) = [1] xs2 + [0] >= [1] xs2 + [0] = xs2 merge[Ite](False(),xs1,Cons(x,xs)) = [1] xs + [1] xs1 + [6] >= [1] xs + [1] xs1 + [6] = Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) = [1] xs + [1] xs2 + [6] >= [1] xs + [1] xs2 + [6] = Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) = [6] >= [6] = Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) = [1] xs + [12] >= [1] xs + [12] = splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) = [0] >= [0] = Nil() splitmerge(Cons(x,xs),xs1,xs2) = [1] xs + [1] xs1 + [1] xs2 + [6] >= [1] xs + [1] xs1 + [1] xs2 + [6] = splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) = [1] xs1 + [1] xs2 + [0] >= [1] xs1 + [1] xs2 + [0] = merge(mergesort(xs1),mergesort(xs2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:7.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: merge#(Cons(x,xs),Nil()) -> c_2() merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) merge#(Nil(),xs2) -> c_4() merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) mergesort#(Cons(x',Cons(x,xs))) -> splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil()) splitmerge#(Cons(x,xs),xs1,xs2) -> splitmerge#(xs,Cons(x,xs2),xs1) splitmerge#(Nil(),xs1,xs2) -> merge#(mergesort(xs1),mergesort(xs2)) splitmerge#(Nil(),xs1,xs2) -> mergesort#(xs1) splitmerge#(Nil(),xs1,xs2) -> mergesort#(xs2) - Weak TRS: <=(0(),y) -> True() <=(S(x),0()) -> False() <=(S(x),S(y)) -> <=(x,y) merge(Cons(x,xs),Nil()) -> Cons(x,xs) merge(Cons(x',xs'),Cons(x,xs)) -> merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs)) merge(Nil(),xs2) -> xs2 merge[Ite](False(),xs1,Cons(x,xs)) -> Cons(x,merge(xs1,xs)) merge[Ite](True(),Cons(x,xs),xs2) -> Cons(x,merge(xs,xs2)) mergesort(Cons(x,Nil())) -> Cons(x,Nil()) mergesort(Cons(x',Cons(x,xs))) -> splitmerge(Cons(x',Cons(x,xs)),Nil(),Nil()) mergesort(Nil()) -> Nil() splitmerge(Cons(x,xs),xs1,xs2) -> splitmerge(xs,Cons(x,xs2),xs1) splitmerge(Nil(),xs1,xs2) -> merge(mergesort(xs1),mergesort(xs2)) - Signature: {<=/2,goal/1,merge/2,merge[Ite]/3,mergesort/1,notEmpty/1,splitmerge/3,<=#/2,goal#/1,merge#/2,merge[Ite]#/3 ,mergesort#/1,notEmpty#/1,splitmerge#/3} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {<=#,goal#,merge#,merge[Ite]#,mergesort#,notEmpty# ,splitmerge#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))