WORST_CASE(?,O(n^2)) * Step 1: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(3, 3) 0 :: [] -(0)-> A(0, 0) 0 :: [] -(0)-> A(6, 3) 0 :: [] -(0)-> A(7, 7) 0 :: [] -(0)-> A(7, 13) 0 :: [] -(0)-> A(5, 5) log :: [A(0, 3)] -(3)-> A(0, 0) min :: [A(9, 3) x A(0, 0)] -(2)-> A(9, 3) quot :: [A(6, 3) x A(0, 0)] -(1)-> A(3, 3) s :: [A(3, 3)] -(0)-> A(0, 3) s :: [A(6, 3)] -(3)-> A(3, 3) s :: [A(12, 3)] -(9)-> A(9, 3) s :: [A(0, 0)] -(0)-> A(0, 0) s :: [A(9, 3)] -(6)-> A(6, 3) Cost-free Signatures used: -------------------------- 0 :: [] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(3, 3) 0 :: [] -(0)-> A_cf(3, 11) 0 :: [] -(0)-> A_cf(3, 0) 0 :: [] -(0)-> A_cf(3, 2) log :: [A_cf(0, 0)] -(1)-> A_cf(0, 0) min :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) min :: [A_cf(3, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) min :: [A_cf(3, 0) x A_cf(0, 0)] -(0)-> A_cf(3, 0) quot :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) quot :: [A_cf(3, 0) x A_cf(0, 0)] -(0)-> A_cf(3, 0) s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(3, 0)] -(3)-> A_cf(3, 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: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} Following problems could not be solved: - Strict TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} WORST_CASE(?,O(n^2))