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