WORST_CASE(?,O(n^2)) * Step 1: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(1, 6) 0 :: [] -(0)-> A(7, 6) 0 :: [] -(0)-> A(7, 12) a__first :: [A(1, 6) x A(7, 6)] -(2)-> A(1, 6) a__from :: [A(7, 6)] -(6)-> A(1, 6) cons :: [A(7, 6) x A(0, 0)] -(7)-> A(7, 6) cons :: [A(1, 6) x A(0, 0)] -(1)-> A(1, 6) first :: [A(7, 6) x A(13, 6)] -(7)-> A(7, 6) first :: [A(1, 6) x A(7, 6)] -(1)-> A(1, 6) first :: [A(0, 0) x A(0, 0)] -(0)-> A(0, 0) from :: [A(13, 6)] -(7)-> A(7, 6) from :: [A(0, 0)] -(0)-> A(0, 0) from :: [A(7, 6)] -(1)-> A(1, 6) mark :: [A(7, 6)] -(4)-> A(1, 6) nil :: [] -(0)-> A(7, 6) nil :: [] -(0)-> A(7, 12) s :: [A(1, 6)] -(1)-> A(1, 6) s :: [A(7, 6)] -(7)-> A(7, 6) s :: [A(0, 0)] -(0)-> A(0, 0) Cost-free Signatures used: -------------------------- 0 :: [] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(6, 0) 0 :: [] -(0)-> A_cf(10, 2) a__first :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) a__first :: [A_cf(6, 0) x A_cf(6, 0)] -(6)-> A_cf(6, 0) a__from :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) a__from :: [A_cf(6, 0)] -(6)-> A_cf(6, 0) cons :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) cons :: [A_cf(6, 0) x A_cf(0, 0)] -(6)-> A_cf(6, 0) first :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) first :: [A_cf(6, 0) x A_cf(6, 0)] -(6)-> A_cf(6, 0) from :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) from :: [A_cf(6, 0)] -(6)-> A_cf(6, 0) mark :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) mark :: [A_cf(6, 0)] -(0)-> A_cf(6, 0) nil :: [] -(0)-> A_cf(0, 0) nil :: [] -(0)-> A_cf(6, 0) nil :: [] -(0)-> A_cf(10, 2) s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(6, 0)] -(6)-> A_cf(6, 0) Base Constructor Signatures used: --------------------------------- 0_A :: [] -(0)-> A(1, 0) 0_A :: [] -(0)-> A(0, 1) cons_A :: [A(1, 0) x A(0, 0)] -(1)-> A(1, 0) cons_A :: [A(0, 1) x A(0, 0)] -(0)-> A(0, 1) first_A :: [A(1, 0) x A(1, 0)] -(1)-> A(1, 0) first_A :: [A(0, 1) x A(1, 1)] -(0)-> A(0, 1) from_A :: [A(1, 0)] -(1)-> A(1, 0) from_A :: [A(1, 1)] -(0)-> A(0, 1) nil_A :: [] -(0)-> A(1, 0) nil_A :: [] -(0)-> A(0, 1) s_A :: [A(1, 0)] -(1)-> A(1, 0) s_A :: [A(0, 1)] -(0)-> A(0, 1) * Step 2: Open MAYBE - Strict TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,s} Following problems could not be solved: - Strict TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__first/2,a__from/1,mark/1} / {0/0,cons/2,first/2,from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__first,a__from,mark} and constructors {0,cons,first ,from,nil,s} WORST_CASE(?,O(n^2))