WORST_CASE(?,O(n^2)) * Step 1: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) f(0()) -> 1() f(s(x)) -> g(x,s(x)) g(0(),y) -> y g(s(x),y) -> g(x,+(y,s(x))) g(s(x),y) -> g(x,s(+(y,x))) - Signature: {+/2,f/1,g/2} / {0/0,1/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = Z3} + Details: Signatures used: ---------------- + :: [A(0, 0) x A(1, 0)] -(1)-> A(0, 0) 0 :: [] -(0)-> A(1, 0) 0 :: [] -(0)-> A(0, 14) 0 :: [] -(0)-> A(0, 13) 1 :: [] -(0)-> A(6, 6) f :: [A(0, 14)] -(10)-> A(0, 0) g :: [A(0, 13) x A(0, 0)] -(6)-> A(0, 0) s :: [A(1, 0)] -(1)-> A(1, 0) s :: [A(14, 14)] -(14)-> A(0, 14) s :: [A(13, 13)] -(13)-> A(0, 13) s :: [A(0, 0)] -(0)-> A(0, 0) s :: [A(7, 0)] -(7)-> A(7, 0) Cost-free Signatures used: -------------------------- + :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(1, 0) g :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) g :: [A_cf(1, 0) x A_cf(0, 0)] -(4)-> A_cf(0, 0) s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(1, 0)] -(1)-> A_cf(1, 0) Base Constructor Signatures used: --------------------------------- 0_A :: [] -(0)-> A(1, 0) 0_A :: [] -(0)-> A(0, 1) 1_A :: [] -(0)-> A(1, 0) 1_A :: [] -(0)-> A(0, 1) s_A :: [A(1, 0)] -(1)-> A(1, 0) s_A :: [A(1, 1)] -(1)-> A(0, 1) * Step 2: Open MAYBE - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) f(0()) -> 1() f(s(x)) -> g(x,s(x)) g(0(),y) -> y g(s(x),y) -> g(x,+(y,s(x))) g(s(x),y) -> g(x,s(+(y,x))) - Signature: {+/2,f/1,g/2} / {0/0,1/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s} Following problems could not be solved: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) f(0()) -> 1() f(s(x)) -> g(x,s(x)) g(0(),y) -> y g(s(x),y) -> g(x,+(y,s(x))) g(s(x),y) -> g(x,s(+(y,x))) - Signature: {+/2,f/1,g/2} / {0/0,1/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s} WORST_CASE(?,O(n^2))