WORST_CASE(?,O(n^2)) * Step 1: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: app(Cons(x,xs),ys) -> Cons(x,app(xs,ys)) app(Nil(),ys) -> ys goal(xs) -> naiverev(xs) naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil())) naiverev(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {app/2,goal/1,naiverev/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,goal,naiverev,notEmpty} and constructors {Cons,False ,Nil,True} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3} + Details: Signatures used: ---------------- Cons :: [A(0, 0) x A(2, 0)] -(2)-> A(2, 0) Cons :: [A(9, 0) x A(9, 9)] -(9)-> A(0, 9) Cons :: [A(8, 0) x A(8, 8)] -(8)-> A(0, 8) Cons :: [A(0, 0) x A(0, 0)] -(0)-> A(0, 0) False :: [] -(0)-> A(14, 14) Nil :: [] -(0)-> A(2, 0) Nil :: [] -(0)-> A(0, 9) Nil :: [] -(0)-> A(0, 8) Nil :: [] -(0)-> A(7, 13) Nil :: [] -(0)-> A(7, 7) True :: [] -(0)-> A(14, 14) app :: [A(2, 0) x A(0, 0)] -(4)-> A(0, 0) goal :: [A(14, 15)] -(14)-> A(0, 0) naiverev :: [A(0, 9)] -(11)-> A(0, 0) notEmpty :: [A(0, 8)] -(10)-> A(0, 0) Cost-free Signatures used: -------------------------- Cons :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) Cons :: [A_cf(0, 0) x A_cf(9, 0)] -(9)-> A_cf(9, 0) Cons :: [A_cf(0, 0) x A_cf(2, 0)] -(2)-> A_cf(2, 0) Nil :: [] -(0)-> A_cf(0, 0) Nil :: [] -(0)-> A_cf(9, 0) Nil :: [] -(0)-> A_cf(2, 0) Nil :: [] -(0)-> A_cf(5, 11) Nil :: [] -(0)-> A_cf(3, 2) app :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) app :: [A_cf(2, 0) x A_cf(2, 0)] -(0)-> A_cf(2, 0) naiverev :: [A_cf(9, 0)] -(1)-> A_cf(2, 0) Base Constructor Signatures used: --------------------------------- Cons_A :: [A(0, 0) x A(1, 0)] -(1)-> A(1, 0) Cons_A :: [A(1, 0) x A(1, 1)] -(1)-> A(0, 1) False_A :: [] -(0)-> A(1, 0) False_A :: [] -(0)-> A(0, 1) Nil_A :: [] -(0)-> A(1, 0) Nil_A :: [] -(0)-> A(0, 1) True_A :: [] -(0)-> A(1, 0) True_A :: [] -(0)-> A(0, 1) * Step 2: Open MAYBE - Strict TRS: app(Cons(x,xs),ys) -> Cons(x,app(xs,ys)) app(Nil(),ys) -> ys goal(xs) -> naiverev(xs) naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil())) naiverev(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {app/2,goal/1,naiverev/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,goal,naiverev,notEmpty} and constructors {Cons,False ,Nil,True} Following problems could not be solved: - Strict TRS: app(Cons(x,xs),ys) -> Cons(x,app(xs,ys)) app(Nil(),ys) -> ys goal(xs) -> naiverev(xs) naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil())) naiverev(Nil()) -> Nil() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Signature: {app/2,goal/1,naiverev/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,goal,naiverev,notEmpty} and constructors {Cons,False ,Nil,True} WORST_CASE(?,O(n^2))