WORST_CASE(?,O(n^1)) * Step 1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> c(f(g(f(a())))) mark(a()) -> a() mark(c(X)) -> c(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,c/1,f/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,c,f,g} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3} + Details: Signatures used: ---------------- a :: [] -(0)-> A(3) a :: [] -(0)-> A(15) a :: [] -(0)-> A(13) a :: [] -(0)-> A(5) a__f :: [A(3)] -(4)-> A(3) c :: [A(0)] -(0)-> A(15) c :: [A(0)] -(0)-> A(9) f :: [A(3)] -(3)-> A(3) f :: [A(15)] -(15)-> A(15) f :: [A(0)] -(0)-> A(0) g :: [A(15)] -(15)-> A(15) g :: [A(0)] -(0)-> A(0) g :: [A(3)] -(3)-> A(3) mark :: [A(15)] -(9)-> A(3) Cost-free Signatures used: -------------------------- a :: [] -(0)-> A_cf(0) a__f :: [A_cf(0)] -(0)-> A_cf(0) c :: [A_cf(0)] -(0)-> A_cf(0) f :: [A_cf(0)] -(0)-> A_cf(0) g :: [A_cf(0)] -(0)-> A_cf(0) mark :: [A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- a_A :: [] -(0)-> A(1) c_A :: [A(0)] -(0)-> A(1) f_A :: [A(1)] -(1)-> A(1) g_A :: [A(1)] -(1)-> A(1) * Step 2: Open MAYBE - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> c(f(g(f(a())))) mark(a()) -> a() mark(c(X)) -> c(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,c/1,f/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,c,f,g} Following problems could not be solved: - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> c(f(g(f(a())))) mark(a()) -> a() mark(c(X)) -> c(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,c/1,f/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,c,f,g} WORST_CASE(?,O(n^1))