MAYBE * Step 1: DependencyPairs MAYBE + 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 2: UsableRules MAYBE + 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: 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)) <=#(0(),y) -> c_12() <=#(S(x),0()) -> c_13() <=#(S(x),S(y)) -> c_14(<=#(x,y)) 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() 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#(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)) * Step 3: PredecessorEstimation MAYBE + 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) 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_3/2,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 4: RemoveWeakSuffixes MAYBE + 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) 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_3/2,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 5: SimplifyRHS MAYBE + 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) 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_3/2,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 6: RemoveHeads MAYBE + 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 7: Decompose MAYBE + 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: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: merge#(Cons(x,xs),Nil()) -> c_2() - Weak DPs: 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)) - 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} Problem (S) - Strict DPs: 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#(Cons(x,xs),Nil()) -> c_2() 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} ** Step 7.a:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: merge#(Cons(x,xs),Nil()) -> c_2() - Weak DPs: 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)) - 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: RemoveWeakSuffixes + Details: Consider the dependency graph 2:S:merge#(Cons(x,xs),Nil()) -> c_2() 3:W: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:W:merge#(Nil(),xs2) -> c_4() 5:W: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:W: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:W:splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)) -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):3 -->_1 merge#(Nil(),xs2) -> c_4():4 -->_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 8:W:merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):3 -->_1 merge#(Nil(),xs2) -> c_4():4 9:W:merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) -->_1 merge#(Cons(x,xs),Nil()) -> c_2():2 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):3 -->_1 merge#(Nil(),xs2) -> c_4():4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: merge#(Nil(),xs2) -> c_4() ** Step 7.a:2: Failure MAYBE + Considered Problem: - Strict DPs: merge#(Cons(x,xs),Nil()) -> c_2() - Weak DPs: merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) 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)) - 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 still open. ** Step 7.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: 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#(Cons(x,xs),Nil()) -> c_2() 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1: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)):8 -->_1 merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)):7 2:S:merge#(Nil(),xs2) -> c_4() 3: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)):4 4: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)):5 -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):4 5:S:splitmerge#(Nil(),xs1,xs2) -> c_11(merge#(mergesort(xs1),mergesort(xs2)) ,mergesort#(xs1) ,mergesort#(xs2)) -->_1 merge#(Cons(x,xs),Nil()) -> c_2():6 -->_3 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):3 -->_2 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):3 -->_1 merge#(Nil(),xs2) -> c_4():2 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):1 6:W:merge#(Cons(x,xs),Nil()) -> c_2() 7:W:merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) -->_1 merge#(Cons(x,xs),Nil()) -> c_2():6 -->_1 merge#(Nil(),xs2) -> c_4():2 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):1 8:W:merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) -->_1 merge#(Cons(x,xs),Nil()) -> c_2():6 -->_1 merge#(Nil(),xs2) -> c_4():2 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: merge#(Cons(x,xs),Nil()) -> c_2() ** Step 7.b:2: Decompose MAYBE + Considered Problem: - Strict DPs: 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: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: 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))) -> 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} Problem (S) - 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 DPs: 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)) - 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} *** Step 7.b:2.a:1: Failure MAYBE + Considered Problem: - Strict DPs: 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))) -> 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: EmptyProcessor + Details: The problem is still open. *** Step 7.b:2.b:1: RemoveWeakSuffixes 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 DPs: 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)) - 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: RemoveWeakSuffixes + 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)) -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):4 -->_1 merge#(Nil(),xs2) -> c_4():5 -->_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 4:W: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)):7 -->_1 merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)):6 5:W:merge#(Nil(),xs2) -> c_4() 6:W:merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) -->_1 merge#(Nil(),xs2) -> c_4():5 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):4 7:W:merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) -->_1 merge#(Nil(),xs2) -> c_4():5 -->_1 merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: merge#(Cons(x',xs'),Cons(x,xs)) -> c_3(merge[Ite]#(<=(x',x),Cons(x',xs'),Cons(x,xs))) 7: merge[Ite]#(True(),Cons(x,xs),xs2) -> c_16(merge#(xs,xs2)) 6: merge[Ite]#(False(),xs1,Cons(x,xs)) -> c_15(merge#(xs1,xs)) 5: merge#(Nil(),xs2) -> c_4() *** Step 7.b:2.b:2: 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 7.b:2.b:3: 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 7.b:2.b:4: PredecessorEstimationCP 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) Consider the set of all dependency pairs 1: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) 2: splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) 3: splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 7.b:2.b:4.a:1: NaturalPI 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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_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) = 0 p(<=) = 0 p(Cons) = 1 + x2 p(False) = 0 p(Nil) = 0 p(S) = x1 p(True) = 0 p(goal) = 0 p(merge) = 0 p(merge[Ite]) = 0 p(mergesort) = 0 p(notEmpty) = 0 p(splitmerge) = 0 p(<=#) = x1 + 4*x1^2 p(goal#) = 2 p(merge#) = 2 + 4*x1 + 2*x1^2 + x2^2 p(merge[Ite]#) = x1*x2 + 2*x1^2 + x2 + x2*x3 + 2*x2^2 + 4*x3 p(mergesort#) = 3*x1 + 2*x1^2 p(notEmpty#) = 1 + x1 p(splitmerge#) = 5*x1 + 2*x1*x2 + 2*x1*x3 + x1^2 + 3*x2 + 2*x2^2 + 5*x3 + 2*x3^2 p(c_1) = 1 p(c_2) = 0 p(c_3) = 1 p(c_4) = 0 p(c_5) = 0 p(c_6) = x1 p(c_7) = 4 p(c_8) = 0 p(c_9) = 1 p(c_10) = x1 p(c_11) = x1 + x2 p(c_12) = 1 p(c_13) = 2 p(c_14) = 1 p(c_15) = 1 + x1 p(c_16) = 2 + x1 Following rules are strictly oriented: splitmerge#(Cons(x,xs),xs1,xs2) = 6 + 7*xs + 2*xs*xs1 + 2*xs*xs2 + xs^2 + 5*xs1 + 2*xs1^2 + 7*xs2 + 2*xs2^2 > 5 + 7*xs + 2*xs*xs1 + 2*xs*xs2 + xs^2 + 5*xs1 + 2*xs1^2 + 7*xs2 + 2*xs2^2 = c_10(splitmerge#(xs,Cons(x,xs2),xs1)) Following rules are (at-least) weakly oriented: mergesort#(Cons(x',Cons(x,xs))) = 14 + 11*xs + 2*xs^2 >= 14 + 9*xs + xs^2 = c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) splitmerge#(Nil(),xs1,xs2) = 3*xs1 + 2*xs1^2 + 5*xs2 + 2*xs2^2 >= 3*xs1 + 2*xs1^2 + 3*xs2 + 2*xs2^2 = c_11(mergesort#(xs1),mergesort#(xs2)) **** Step 7.b:2.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict 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)) - Weak DPs: splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.b:2.b:4.b:1: RemoveWeakSuffixes 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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:W:splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) -->_1 splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)):3 -->_1 splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)):2 3:W:splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) -->_2 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):1 -->_1 mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: mergesort#(Cons(x',Cons(x,xs))) -> c_6(splitmerge#(Cons(x',Cons(x,xs)),Nil(),Nil())) 3: splitmerge#(Nil(),xs1,xs2) -> c_11(mergesort#(xs1),mergesort#(xs2)) 2: splitmerge#(Cons(x,xs),xs1,xs2) -> c_10(splitmerge#(xs,Cons(x,xs2),xs1)) **** Step 7.b:2.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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). MAYBE