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