WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {2}, uargs(n__first) = {2}, uargs(sel) = {2} Following symbols are considered usable: {activate,first,from,sel} TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(first) = [1] x2 + [0] p(from) = [1] x1 + [0] p(n__first) = [1] x2 + [0] p(n__from) = [1] x1 + [0] p(nil) = [0] p(s) = [0] p(sel) = [4] x2 + [3] Following rules are strictly oriented: sel(0(),cons(X,Z)) = [4] X + [4] Z + [3] > [1] X + [0] = X Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = first(X1,X2) activate(n__from(X)) = [1] X + [0] >= [1] X + [0] = from(X) first(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__first(X1,X2) first(0(),Z) = [1] Z + [0] >= [0] = nil() first(s(X),cons(Y,Z)) = [1] Y + [1] Z + [0] >= [1] Y + [1] Z + [0] = cons(Y,n__first(X,activate(Z))) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) sel(s(X),cons(Y,Z)) = [4] Y + [4] Z + [3] >= [4] Z + [3] = sel(X,activate(Z)) * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) 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: sel(0(),cons(X,Z)) -> X - Signature: {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons ,n__first,n__from,nil,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(cons) = {2}, uargs(n__first) = {2}, uargs(sel) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [9] p(cons) = [1] x1 + [1] x2 + [9] p(first) = [1] x2 + [0] p(from) = [1] x1 + [0] p(n__first) = [1] x2 + [1] p(n__from) = [1] x1 + [0] p(nil) = [0] p(s) = [0] p(sel) = [1] x2 + [0] Following rules are strictly oriented: activate(X) = [1] X + [9] > [1] X + [0] = X activate(n__first(X1,X2)) = [1] X2 + [10] > [1] X2 + [0] = first(X1,X2) activate(n__from(X)) = [1] X + [9] > [1] X + [0] = from(X) Following rules are (at-least) weakly oriented: first(X1,X2) = [1] X2 + [0] >= [1] X2 + [1] = n__first(X1,X2) first(0(),Z) = [1] Z + [0] >= [0] = nil() first(s(X),cons(Y,Z)) = [1] Y + [1] Z + [9] >= [1] Y + [1] Z + [19] = cons(Y,n__first(X,activate(Z))) from(X) = [1] X + [0] >= [1] X + [9] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) sel(0(),cons(X,Z)) = [1] X + [1] Z + [9] >= [1] X + [0] = X sel(s(X),cons(Y,Z)) = [1] Y + [1] Z + [9] >= [1] Z + [9] = sel(X,activate(Z)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) 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__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) sel(0(),cons(X,Z)) -> X - Signature: {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons ,n__first,n__from,nil,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(cons) = {2}, uargs(n__first) = {2}, uargs(sel) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [1] p(cons) = [1] x1 + [1] x2 + [4] p(first) = [1] x2 + [1] p(from) = [1] x1 + [1] p(n__first) = [1] x2 + [0] p(n__from) = [1] x1 + [0] p(nil) = [0] p(s) = [0] p(sel) = [1] x2 + [7] Following rules are strictly oriented: first(X1,X2) = [1] X2 + [1] > [1] X2 + [0] = n__first(X1,X2) first(0(),Z) = [1] Z + [1] > [0] = nil() from(X) = [1] X + [1] > [1] X + [0] = n__from(X) sel(s(X),cons(Y,Z)) = [1] Y + [1] Z + [11] > [1] Z + [8] = sel(X,activate(Z)) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X2 + [1] >= [1] X2 + [1] = first(X1,X2) activate(n__from(X)) = [1] X + [1] >= [1] X + [1] = from(X) first(s(X),cons(Y,Z)) = [1] Y + [1] Z + [5] >= [1] Y + [1] Z + [5] = cons(Y,n__first(X,activate(Z))) from(X) = [1] X + [1] >= [1] X + [4] = cons(X,n__from(s(X))) sel(0(),cons(X,Z)) = [1] X + [1] Z + [11] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() from(X) -> n__from(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(4) 0 :: [] -(0)-> A(15) activate :: [A(4)] -(5)-> A(4) cons :: [A(0) x A(4)] -(4)-> A(4) first :: [A(4) x A(4)] -(1)-> A(4) from :: [A(0)] -(5)-> A(4) n__first :: [A(4) x A(4)] -(0)-> A(4) n__from :: [A(0)] -(0)-> A(4) n__from :: [A(0)] -(0)-> A(6) n__from :: [A(0)] -(0)-> A(8) nil :: [] -(0)-> A(10) s :: [A(4)] -(4)-> A(4) s :: [A(15)] -(15)-> A(15) s :: [A(0)] -(0)-> A(0) sel :: [A(15) x A(4)] -(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) first :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) n__first :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) n__from :: [A_cf(0)] -(0)-> A_cf(0) nil :: [] -(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(1)] -(1)-> A(1) n__first_A :: [A(0) x A(0)] -(0)-> A(1) n__from_A :: [A(0)] -(0)-> A(1) nil_A :: [] -(0)-> A(1) s_A :: [A(1)] -(1)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: from(X) -> cons(X,n__from(s(X))) 2. Weak: first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) * Step 5: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(3) 0 :: [] -(0)-> A(11) activate :: [A(3)] -(11)-> A(3) cons :: [A(0) x A(3)] -(0)-> A(3) cons :: [A(0) x A(6)] -(0)-> A(6) first :: [A(3) x A(3)] -(13)-> A(3) from :: [A(0)] -(11)-> A(3) n__first :: [A(3) x A(3)] -(3)-> A(3) n__from :: [A(0)] -(0)-> A(3) n__from :: [A(0)] -(0)-> A(12) n__from :: [A(0)] -(0)-> A(7) nil :: [] -(0)-> A(5) s :: [A(3)] -(3)-> A(3) s :: [A(11)] -(11)-> A(11) s :: [A(0)] -(0)-> A(0) sel :: [A(11) x A(3)] -(0)-> 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) first :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) from :: [A_cf(0)] -(0)-> A_cf(0) n__first :: [A_cf(0) x A_cf(0)] -(0)-> A_cf(0) n__from :: [A_cf(0)] -(0)-> A_cf(0) nil :: [] -(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(1)] -(0)-> A(1) n__first_A :: [A(0) x A(0)] -(1)-> A(1) n__from_A :: [A(0)] -(0)-> A(1) nil_A :: [] -(0)-> A(1) s_A :: [A(1)] -(1)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) 2. Weak: * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons ,n__first,n__from,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))