WORST_CASE(?,O(n^1)) * Step 1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(+(x,y)) -> *(f(x),f(y)) f(+(x,s(0()))) -> +(s(s(0())),f(x)) f(0()) -> s(0()) f(s(0())) -> *(s(s(0())),f(0())) f(s(0())) -> s(s(0())) - Signature: {f/1} / {*/2,+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {*,+,0,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3} + Details: Signatures used: ---------------- * :: [A(0) x A(0)] -(0)-> A(7) * :: [A(0) x A(0)] -(0)-> A(4) + :: [A(6) x A(6)] -(6)-> A(6) + :: [A(0) x A(0)] -(0)-> A(0) 0 :: [] -(0)-> A(0) 0 :: [] -(0)-> A(6) 0 :: [] -(0)-> A(12) 0 :: [] -(0)-> A(14) f :: [A(6)] -(2)-> A(0) s :: [A(0)] -(6)-> A(6) s :: [A(0)] -(2)-> A(2) s :: [A(0)] -(0)-> A(0) s :: [A(0)] -(1)-> A(1) 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) 0 :: [] -(0)-> A_cf(0) f :: [A_cf(0)] -(0)-> A_cf(0) s :: [A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- *_A :: [A(0) x A(0)] -(0)-> A(0) +_A :: [A(0) x A(0)] -(0)-> A(0) 0_A :: [] -(0)-> A(1) s_A :: [A(0)] -(1)-> A(1) * Step 2: Open MAYBE - Strict TRS: f(+(x,y)) -> *(f(x),f(y)) f(+(x,s(0()))) -> +(s(s(0())),f(x)) f(0()) -> s(0()) f(s(0())) -> *(s(s(0())),f(0())) f(s(0())) -> s(s(0())) - Signature: {f/1} / {*/2,+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {*,+,0,s} Following problems could not be solved: - Strict TRS: f(+(x,y)) -> *(f(x),f(y)) f(+(x,s(0()))) -> +(s(s(0())),f(x)) f(0()) -> s(0()) f(s(0())) -> *(s(s(0())),f(0())) f(s(0())) -> s(s(0())) - Signature: {f/1} / {*/2,+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {*,+,0,s} WORST_CASE(?,O(n^1))