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