WORST_CASE(?,O(n^2)) * Step 1: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: concat(cons(U,V),Y) -> cons(U,concat(V,Y)) concat(leaf(),Y) -> Y lessleaves(X,leaf()) -> false() lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z)) lessleaves(leaf(),cons(W,Z)) -> true() - Signature: {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf ,true} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = MiniSMT} + Details: Signatures used: ---------------- concat :: [A(5, 0) x A(4, 1)] -(1)-> A(4, 1) cons :: [A(5, 0) x A(5, 0)] -(5)-> A(5, 0) cons :: [A(5, 0) x A(4, 1)] -(4)-> A(4, 1) false :: [] -(0)-> A(0, 1) leaf :: [] -(0)-> A(5, 0) leaf :: [] -(0)-> A(4, 1) lessleaves :: [A(4, 1) x A(4, 1)] -(1)-> A(0, 1) true :: [] -(0)-> A(4, 2) Cost-free Signatures used: -------------------------- concat :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) cons :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) false :: [] -(0)-> A_cf(2, 1) leaf :: [] -(0)-> A_cf(0, 0) lessleaves :: [A_cf(0, 0) x A_cf(0, 0)] -(4)-> A_cf(0, 0) true :: [] -(0)-> A_cf(4, 5) Base Constructor Signatures used: --------------------------------- cons_A :: [A(1, 0) x A(1, 0)] -(1)-> A(1, 0) cons_A :: [A(1, 0) x A(0, 1)] -(0)-> A(0, 1) false_A :: [] -(0)-> A(1, 0) false_A :: [] -(0)-> A(0, 1) leaf_A :: [] -(0)-> A(1, 0) leaf_A :: [] -(0)-> A(0, 1) true_A :: [] -(0)-> A(1, 0) true_A :: [] -(0)-> A(0, 1) * Step 2: Open MAYBE - Strict TRS: concat(cons(U,V),Y) -> cons(U,concat(V,Y)) concat(leaf(),Y) -> Y lessleaves(X,leaf()) -> false() lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z)) lessleaves(leaf(),cons(W,Z)) -> true() - Signature: {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf ,true} Following problems could not be solved: - Strict TRS: concat(cons(U,V),Y) -> cons(U,concat(V,Y)) concat(leaf(),Y) -> Y lessleaves(X,leaf()) -> false() lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z)) lessleaves(leaf(),cons(W,Z)) -> true() - Signature: {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf ,true} WORST_CASE(?,O(n^2))