WORST_CASE(?,O(n^1)) * Step 1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: +Full(0(),y) -> y +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() - Weak TRS: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() - Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,S} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3} + Details: Signatures used: ---------------- * :: [A(2) x A(7)] -(7)-> A(2) + :: [A(0) x A(2)] -(0)-> A(2) +Full :: [A(1) x A(0)] -(14)-> A(0) 0 :: [] -(0)-> A(1) 0 :: [] -(0)-> A(7) 0 :: [] -(0)-> A(2) 0 :: [] -(0)-> A(5) 0 :: [] -(0)-> A(4) Cons :: [A(15) x A(15)] -(15)-> A(15) Cons :: [A(2) x A(2)] -(2)-> A(2) Nil :: [] -(0)-> A(15) Nil :: [] -(0)-> A(5) S :: [A(1)] -(1)-> A(1) S :: [A(7)] -(7)-> A(7) S :: [A(0)] -(0)-> A(0) f :: [A(12)] -(9)-> A(2) goal :: [A(15)] -(12)-> A(0) map :: [A(15)] -(1)-> A(2) Cost-free Signatures used: -------------------------- * :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) + :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) +Full :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) 0 :: [] -(0)-> A_cf(0) Cons :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) Nil :: [] -(0)-> A_cf(0) S :: [A_cf(0)] -(0)-> A_cf(0) f :: [A_cf(0)] -(0)-> A_cf(0) map :: [A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- +_A :: [A(0) x A(0)] -(0)-> A(0) 0_A :: [] -(0)-> A(1) Cons_A :: [A(1) x A(1)] -(1)-> A(1) Nil_A :: [] -(0)-> A(1) S_A :: [A(1)] -(1)-> A(1) * Step 2: Open MAYBE - Strict TRS: +Full(0(),y) -> y +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() - Weak TRS: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() - Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,S} Following problems could not be solved: - Strict TRS: +Full(0(),y) -> y +Full(S(x),y) -> +Full(x,S(y)) f(x) -> *(x,x) goal(xs) -> map(xs) map(Cons(x,xs)) -> Cons(f(x),map(xs)) map(Nil()) -> Nil() - Weak TRS: *(x,0()) -> 0() *(x,S(0())) -> x *(x,S(S(y))) -> +(x,*(x,S(y))) *(0(),y) -> 0() - Signature: {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,S} WORST_CASE(?,O(n^1))