WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() eqNatList(Nil(),Cons(y,ys)) -> False() eqNatList(Nil(),Nil()) -> True() goal(a1,b1,a2,b2,a3,b3) -> nolexicord(a1,b1,a2,b2,a3,b3) nolexicord(Cons(x,xs),b1,a2,b2,a3,b3) -> nolexicord[Ite][False][Ite](eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) nolexicord(Nil(),b1,a2,b2,a3,b3) -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) number42() -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) nolexicord[Ite][False][Ite](False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> nolexicord(xs',xs',xs',xs',xs',xs) nolexicord[Ite][False][Ite](True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> nolexicord(xs',xs',xs',xs',xs',xs) - Signature: {!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0} / {0/0,Cons/2,False/0,Nil/0 ,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ,eqNatList,goal,nolexicord,nolexicord[Ite][False][Ite] ,number42} and constructors {0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y)) eqNatList#(Cons(x,xs),Nil()) -> c_2() eqNatList#(Nil(),Cons(y,ys)) -> c_3() eqNatList#(Nil(),Nil()) -> c_4() goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)) nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() number42#() -> c_8() Weak DPs !EQ#(0(),0()) -> c_9() !EQ#(0(),S(y)) -> c_10() !EQ#(S(x),0()) -> c_11() !EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)) nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y)) eqNatList#(Cons(x,xs),Nil()) -> c_2() eqNatList#(Nil(),Cons(y,ys)) -> c_3() eqNatList#(Nil(),Nil()) -> c_4() goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)) nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() number42#() -> c_8() - Weak DPs: !EQ#(0(),0()) -> c_9() !EQ#(0(),S(y)) -> c_10() !EQ#(S(x),0()) -> c_11() !EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)) nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() eqNatList(Nil(),Cons(y,ys)) -> False() eqNatList(Nil(),Nil()) -> True() goal(a1,b1,a2,b2,a3,b3) -> nolexicord(a1,b1,a2,b2,a3,b3) nolexicord(Cons(x,xs),b1,a2,b2,a3,b3) -> nolexicord[Ite][False][Ite](eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) nolexicord(Nil(),b1,a2,b2,a3,b3) -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) nolexicord[Ite][False][Ite](False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> nolexicord(xs',xs',xs',xs',xs',xs) nolexicord[Ite][False][Ite](True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> nolexicord(xs',xs',xs',xs',xs',xs) number42() -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) - Signature: {!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6 ,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/0,c_12/1,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,eqNatList#,goal#,nolexicord# ,nolexicord[Ite][False][Ite]#,number42#} and constructors {0,Cons,False,Nil,S,True ,eqNatList[Match][Cons][Match][Cons][Ite]} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,8} by application of Pre({1,2,3,4,8}) = {6}. Here rules are labelled as follows: 1: eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y)) 2: eqNatList#(Cons(x,xs),Nil()) -> c_2() 3: eqNatList#(Nil(),Cons(y,ys)) -> c_3() 4: eqNatList#(Nil(),Nil()) -> c_4() 5: goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) 6: nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)) 7: nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() 8: number42#() -> c_8() 9: !EQ#(0(),0()) -> c_9() 10: !EQ#(0(),S(y)) -> c_10() 11: !EQ#(S(x),0()) -> c_11() 12: !EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)) 13: nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) 14: nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)) nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() - Weak DPs: !EQ#(0(),0()) -> c_9() !EQ#(0(),S(y)) -> c_10() !EQ#(S(x),0()) -> c_11() !EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)) eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y)) eqNatList#(Cons(x,xs),Nil()) -> c_2() eqNatList#(Nil(),Cons(y,ys)) -> c_3() eqNatList#(Nil(),Nil()) -> c_4() nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) number42#() -> c_8() - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() eqNatList(Nil(),Cons(y,ys)) -> False() eqNatList(Nil(),Nil()) -> True() goal(a1,b1,a2,b2,a3,b3) -> nolexicord(a1,b1,a2,b2,a3,b3) nolexicord(Cons(x,xs),b1,a2,b2,a3,b3) -> nolexicord[Ite][False][Ite](eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) nolexicord(Nil(),b1,a2,b2,a3,b3) -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) nolexicord[Ite][False][Ite](False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> nolexicord(xs',xs',xs',xs',xs',xs) nolexicord[Ite][False][Ite](True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> nolexicord(xs',xs',xs',xs',xs',xs) number42() -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) - Signature: {!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6 ,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/0,c_12/1,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,eqNatList#,goal#,nolexicord# ,nolexicord[Ite][False][Ite]#,number42#} and constructors {0,Cons,False,Nil,S,True ,eqNatList[Match][Cons][Match][Cons][Ite]} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) -->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)):2 -->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3 2:S:nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)) -->_1 nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)):13 -->_1 nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)):12 -->_2 eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y)):8 -->_2 eqNatList#(Cons(x,xs),Nil()) -> c_2():9 3:S:nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() 4:W:!EQ#(0(),0()) -> c_9() 5:W:!EQ#(0(),S(y)) -> c_10() 6:W:!EQ#(S(x),0()) -> c_11() 7:W:!EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)) -->_1 !EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)):7 -->_1 !EQ#(S(x),0()) -> c_11():6 -->_1 !EQ#(0(),S(y)) -> c_10():5 -->_1 !EQ#(0(),0()) -> c_9():4 8:W:eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y)) -->_1 !EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)):7 -->_1 !EQ#(S(x),0()) -> c_11():6 -->_1 !EQ#(0(),S(y)) -> c_10():5 -->_1 !EQ#(0(),0()) -> c_9():4 9:W:eqNatList#(Cons(x,xs),Nil()) -> c_2() 10:W:eqNatList#(Nil(),Cons(y,ys)) -> c_3() 11:W:eqNatList#(Nil(),Nil()) -> c_4() 12:W:nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) -->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3 -->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)):2 13:W:nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) -->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3 -->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)):2 14:W:number42#() -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: number42#() -> c_8() 11: eqNatList#(Nil(),Nil()) -> c_4() 10: eqNatList#(Nil(),Cons(y,ys)) -> c_3() 9: eqNatList#(Cons(x,xs),Nil()) -> c_2() 8: eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y)) 7: !EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)) 4: !EQ#(0(),0()) -> c_9() 5: !EQ#(0(),S(y)) -> c_10() 6: !EQ#(S(x),0()) -> c_11() * Step 4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)) nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() - Weak DPs: nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() eqNatList(Nil(),Cons(y,ys)) -> False() eqNatList(Nil(),Nil()) -> True() goal(a1,b1,a2,b2,a3,b3) -> nolexicord(a1,b1,a2,b2,a3,b3) nolexicord(Cons(x,xs),b1,a2,b2,a3,b3) -> nolexicord[Ite][False][Ite](eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) nolexicord(Nil(),b1,a2,b2,a3,b3) -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) nolexicord[Ite][False][Ite](False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> nolexicord(xs',xs',xs',xs',xs',xs) nolexicord[Ite][False][Ite](True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> nolexicord(xs',xs',xs',xs',xs',xs) number42() -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) - Signature: {!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6 ,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/0,c_12/1,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,eqNatList#,goal#,nolexicord# ,nolexicord[Ite][False][Ite]#,number42#} and constructors {0,Cons,False,Nil,S,True ,eqNatList[Match][Cons][Match][Cons][Ite]} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) -->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)):2 -->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3 2:S:nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)) -->_1 nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)):13 -->_1 nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)):12 3:S:nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() 12:W:nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) -->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3 -->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)):2 13:W:nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) -->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3 -->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) ,eqNatList#(Cons(x,xs),b1)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)) * Step 5: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)) nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() - Weak DPs: nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() eqNatList(Nil(),Cons(y,ys)) -> False() eqNatList(Nil(),Nil()) -> True() goal(a1,b1,a2,b2,a3,b3) -> nolexicord(a1,b1,a2,b2,a3,b3) nolexicord(Cons(x,xs),b1,a2,b2,a3,b3) -> nolexicord[Ite][False][Ite](eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3) nolexicord(Nil(),b1,a2,b2,a3,b3) -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) nolexicord[Ite][False][Ite](False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> nolexicord(xs',xs',xs',xs',xs',xs) nolexicord[Ite][False][Ite](True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> nolexicord(xs',xs',xs',xs',xs',xs) number42() -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) - Signature: {!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6 ,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/0,c_12/1,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,eqNatList#,goal#,nolexicord# ,nolexicord[Ite][False][Ite]#,number42#} and constructors {0,Cons,False,Nil,S,True ,eqNatList[Match][Cons][Match][Cons][Ite]} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)) nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) * Step 6: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)) nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() - Weak DPs: nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() - Signature: {!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6 ,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/0,c_12/1,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,eqNatList#,goal#,nolexicord# ,nolexicord[Ite][False][Ite]#,number42#} and constructors {0,Cons,False,Nil,S,True ,eqNatList[Match][Cons][Match][Cons][Ite]} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)) -->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)):2 -->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3 2:S:nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)) -->_1 nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)):5 -->_1 nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)):4 3:S:nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() 4:W:nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) -->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3 -->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)):2 5:W:nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) -->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3 -->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)):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#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)))] * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)) nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() - Weak DPs: nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() - Signature: {!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6 ,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/0,c_12/1,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,eqNatList#,goal#,nolexicord# ,nolexicord[Ite][False][Ite]#,number42#} and constructors {0,Cons,False,Nil,S,True ,eqNatList[Match][Cons][Match][Cons][Ite]} + 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(eqNatList[Match][Cons][Match][Cons][Ite]) = {1}, uargs(nolexicord[Ite][False][Ite]#) = {1}, uargs(c_6) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [2] p(0) = [0] p(Cons) = [4] p(False) = [0] p(Nil) = [1] p(S) = [1] x1 + [0] p(True) = [2] p(eqNatList) = [2] x1 + [1] p(eqNatList[Match][Cons][Match][Cons][Ite]) = [1] x1 + [4] p(goal) = [0] p(nolexicord) = [4] x1 + [1] x3 + [4] x4 + [1] x6 + [2] p(nolexicord[Ite][False][Ite]) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [2] x6 + [4] x7 + [2] p(number42) = [2] p(!EQ#) = [1] x2 + [0] p(eqNatList#) = [1] p(goal#) = [2] x2 + [1] x3 + [1] x4 + [1] x6 + [0] p(nolexicord#) = [3] p(nolexicord[Ite][False][Ite]#) = [1] x1 + [5] p(number42#) = [0] p(c_1) = [1] x1 + [1] p(c_2) = [2] p(c_3) = [0] p(c_4) = [1] p(c_5) = [4] p(c_6) = [1] x1 + [1] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [3] Following rules are strictly oriented: nolexicord#(Nil(),b1,a2,b2,a3,b3) = [3] > [1] = c_7() Following rules are (at-least) weakly oriented: nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) = [3] >= [15] = c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)) nolexicord[Ite][False][Ite]#(False() = [5] ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) >= [3] = c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() = [7] ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) >= [6] = c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) !EQ(0(),0()) = [2] >= [2] = True() !EQ(0(),S(y)) = [2] >= [0] = False() !EQ(S(x),0()) = [2] >= [0] = False() !EQ(S(x),S(y)) = [2] >= [2] = !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) = [9] >= [6] = eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) = [9] >= [0] = False() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)) - Weak DPs: nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() - Signature: {!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6 ,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/0,c_12/1,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,eqNatList#,goal#,nolexicord# ,nolexicord[Ite][False][Ite]#,number42#} and constructors {0,Cons,False,Nil,S,True ,eqNatList[Match][Cons][Match][Cons][Ite]} + 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(eqNatList[Match][Cons][Match][Cons][Ite]) = {1}, uargs(nolexicord[Ite][False][Ite]#) = {1}, uargs(c_6) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(!EQ) = [0] p(0) = [0] p(Cons) = [1] x2 + [1] p(False) = [0] p(Nil) = [1] p(S) = [1] p(True) = [0] p(eqNatList) = [5] p(eqNatList[Match][Cons][Match][Cons][Ite]) = [1] x1 + [5] p(goal) = [1] x3 + [4] p(nolexicord) = [1] x5 + [1] p(nolexicord[Ite][False][Ite]) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [4] x5 + [0] p(number42) = [1] p(!EQ#) = [2] x1 + [1] x2 + [0] p(eqNatList#) = [1] x1 + [2] x2 + [0] p(goal#) = [2] x2 + [4] x3 + [1] x4 + [1] x6 + [0] p(nolexicord#) = [4] x1 + [1] x3 + [6] x4 + [6] p(nolexicord[Ite][False][Ite]#) = [1] x1 + [4] x2 + [1] x4 + [6] x5 + [0] p(number42#) = [0] p(c_1) = [1] x1 + [1] p(c_2) = [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [0] p(c_7) = [2] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] x1 + [4] p(c_13) = [1] x1 + [5] p(c_14) = [1] x1 + [2] Following rules are strictly oriented: nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) = [1] a2 + [6] b2 + [4] xs + [10] > [1] a2 + [6] b2 + [4] xs + [9] = c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)) Following rules are (at-least) weakly oriented: nolexicord#(Nil(),b1,a2,b2,a3,b3) = [1] a2 + [6] b2 + [10] >= [2] = c_7() nolexicord[Ite][False][Ite]#(False() = [11] xs' + [11] ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) >= [11] xs' + [11] = c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() = [11] xs' + [11] ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) >= [11] xs' + [8] = c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) !EQ(0(),0()) = [0] >= [0] = True() !EQ(0(),S(y)) = [0] >= [0] = False() !EQ(S(x),0()) = [0] >= [0] = False() !EQ(S(x),S(y)) = [0] >= [0] = !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) = [5] >= [5] = eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) = [5] >= [0] = False() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1) ,Cons(x,xs) ,b1 ,a2 ,b2 ,a3 ,b3)) nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7() nolexicord[Ite][False][Ite]#(False() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)) nolexicord[Ite][False][Ite]#(True() ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x',xs') ,Cons(x,xs) ,Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)) - Weak TRS: !EQ(0(),0()) -> True() !EQ(0(),S(y)) -> False() !EQ(S(x),0()) -> False() !EQ(S(x),S(y)) -> !EQ(x,y) eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs) eqNatList(Cons(x,xs),Nil()) -> False() - Signature: {!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6 ,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/0,c_12/1,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {!EQ#,eqNatList#,goal#,nolexicord# ,nolexicord[Ite][False][Ite]#,number42#} and constructors {0,Cons,False,Nil,S,True ,eqNatList[Match][Cons][Match][Cons][Ite]} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))