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