WORST_CASE(?,O(n^1)) * Step 1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: sqr(x) -> *(x,x) sum(0()) -> 0() sum(s(x)) -> +(*(s(x),s(x)),sum(x)) sum(s(x)) -> +(sqr(s(x)),sum(x)) - Signature: {sqr/1,sum/1} / {*/2,+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {sqr,sum} 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(5) * :: [A(0) x A(0)] -(0)-> A(12) + :: [A(3) x A(3)] -(0)-> A(3) 0 :: [] -(0)-> A(12) 0 :: [] -(0)-> A(5) s :: [A(12)] -(12)-> A(12) s :: [A(0)] -(0)-> A(0) sqr :: [A(0)] -(2)-> A(5) sum :: [A(12)] -(8)-> A(3) 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) s :: [A_cf(0)] -(0)-> A_cf(0) sqr :: [A_cf(0)] -(0)-> A_cf(0) sum :: [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(1)] -(1)-> A(1) * Step 2: Open MAYBE - Strict TRS: sqr(x) -> *(x,x) sum(0()) -> 0() sum(s(x)) -> +(*(s(x),s(x)),sum(x)) sum(s(x)) -> +(sqr(s(x)),sum(x)) - Signature: {sqr/1,sum/1} / {*/2,+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s} Following problems could not be solved: - Strict TRS: sqr(x) -> *(x,x) sum(0()) -> 0() sum(s(x)) -> +(*(s(x),s(x)),sum(x)) sum(s(x)) -> +(sqr(s(x)),sum(x)) - Signature: {sqr/1,sum/1} / {*/2,+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s} WORST_CASE(?,O(n^1))