WORST_CASE(?,O(n^1)) * Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(sel) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [6] p(cons) = [1] x1 + [1] x2 + [2] p(from) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(s) = [0] p(sel) = [1] x2 + [9] Following rules are strictly oriented: activate(X) = [1] X + [6] > [1] X + [0] = X activate(n__from(X)) = [1] X + [6] > [1] X + [0] = from(X) sel(0(),cons(X,Y)) = [1] X + [1] Y + [11] > [1] X + [0] = X Following rules are (at-least) weakly oriented: from(X) = [1] X + [0] >= [1] X + [2] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) sel(s(X),cons(Y,Z)) = [1] Y + [1] Z + [11] >= [1] Z + [15] = sel(X,activate(Z)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) sel(0(),cons(X,Y)) -> X - Signature: {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(sel) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [6] p(cons) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [9] p(n__from) = [1] x1 + [3] p(s) = [0] p(sel) = [1] x2 + [0] Following rules are strictly oriented: from(X) = [1] X + [9] > [1] X + [3] = cons(X,n__from(s(X))) from(X) = [1] X + [9] > [1] X + [3] = n__from(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [6] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [9] >= [1] X + [9] = from(X) sel(0(),cons(X,Y)) = [1] X + [1] Y + [0] >= [1] X + [0] = X sel(s(X),cons(Y,Z)) = [1] Y + [1] Z + [0] >= [1] Z + [6] = sel(X,activate(Z)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X - Signature: {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(11) activate :: [A(0)] -(8)-> A(0) cons :: [A(0) x A(0)] -(0)-> A(0) cons :: [A(0) x A(0)] -(0)-> A(14) from :: [A(0)] -(7)-> A(6) n__from :: [A(0)] -(0)-> A(0) n__from :: [A(0)] -(0)-> A(14) n__from :: [A(0)] -(0)-> A(6) s :: [A(11)] -(11)-> A(11) s :: [A(0)] -(0)-> A(0) sel :: [A(11) x A(0)] -(9)-> A(0) Cost-free Signatures used: -------------------------- 0 :: [] -(0)-> A_cf(0) activate :: [A_cf(0)] -(0)-> A_cf(0) cons :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) n__from :: [A_cf(0)] -(0)-> A_cf(0) s :: [A_cf(0)] -(0)-> A_cf(0) sel :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) Base Constructor Signatures used: --------------------------------- 0_A :: [] -(0)-> A(1) cons_A :: [A(0) x A(0)] -(0)-> A(1) n__from_A :: [A(0)] -(0)-> A(1) s_A :: [A(1)] -(1)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) 2. Weak: * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))