WORST_CASE(?,O(n^1)) * Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons ,n__from,n__take,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(head) = {1}, uargs(n__take) = {2}, uargs(sel) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [2] x1 + [0] p(activate) = [1] x1 + [7] p(cons) = [1] x1 + [1] x2 + [5] p(from) = [1] x1 + [1] p(head) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__take) = [1] x2 + [0] p(nil) = [0] p(s) = [0] p(sel) = [1] x2 + [11] p(take) = [1] x2 + [0] Following rules are strictly oriented: 2nd(cons(X,XS)) = [2] X + [2] XS + [10] > [1] XS + [7] = head(activate(XS)) activate(X) = [1] X + [7] > [1] X + [0] = X activate(n__from(X)) = [1] X + [7] > [1] X + [1] = from(X) activate(n__take(X1,X2)) = [1] X2 + [7] > [1] X2 + [0] = take(X1,X2) from(X) = [1] X + [1] > [1] X + [0] = n__from(X) head(cons(X,XS)) = [1] X + [1] XS + [5] > [1] X + [0] = X sel(0(),cons(X,XS)) = [1] X + [1] XS + [16] > [1] X + [0] = X Following rules are (at-least) weakly oriented: from(X) = [1] X + [1] >= [1] X + [5] = cons(X,n__from(s(X))) sel(s(N),cons(X,XS)) = [1] X + [1] XS + [16] >= [1] XS + [18] = sel(N,activate(XS)) take(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__take(X1,X2) take(0(),XS) = [1] XS + [0] >= [0] = nil() take(s(N),cons(X,XS)) = [1] X + [1] XS + [5] >= [1] X + [1] XS + [12] = cons(X,n__take(N,activate(XS))) 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))) sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> n__from(X) head(cons(X,XS)) -> X sel(0(),cons(X,XS)) -> X - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons ,n__from,n__take,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(head) = {1}, uargs(n__take) = {2}, uargs(sel) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [4] x1 + [5] p(activate) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [4] p(head) = [1] x1 + [1] p(n__from) = [1] x1 + [4] p(n__take) = [1] x2 + [3] p(nil) = [0] p(s) = [0] p(sel) = [1] x2 + [0] p(take) = [1] x2 + [1] Following rules are strictly oriented: take(0(),XS) = [1] XS + [1] > [0] = nil() Following rules are (at-least) weakly oriented: 2nd(cons(X,XS)) = [4] X + [4] XS + [5] >= [1] XS + [1] = head(activate(XS)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [4] >= [1] X + [4] = from(X) activate(n__take(X1,X2)) = [1] X2 + [3] >= [1] X2 + [1] = take(X1,X2) from(X) = [1] X + [4] >= [1] X + [4] = cons(X,n__from(s(X))) from(X) = [1] X + [4] >= [1] X + [4] = n__from(X) head(cons(X,XS)) = [1] X + [1] XS + [1] >= [1] X + [0] = X sel(0(),cons(X,XS)) = [1] X + [1] XS + [0] >= [1] X + [0] = X sel(s(N),cons(X,XS)) = [1] X + [1] XS + [0] >= [1] XS + [0] = sel(N,activate(XS)) take(X1,X2) = [1] X2 + [1] >= [1] X2 + [3] = n__take(X1,X2) take(s(N),cons(X,XS)) = [1] X + [1] XS + [1] >= [1] X + [1] XS + [3] = cons(X,n__take(N,activate(XS))) 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: from(X) -> cons(X,n__from(s(X))) sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(X1,X2) -> n__take(X1,X2) take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> n__from(X) head(cons(X,XS)) -> X sel(0(),cons(X,XS)) -> X take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons ,n__from,n__take,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(head) = {1}, uargs(n__take) = {2}, uargs(sel) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [1] x1 + [12] p(activate) = [1] x1 + [11] p(cons) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [5] p(head) = [1] x1 + [1] p(n__from) = [1] x1 + [0] p(n__take) = [1] x2 + [1] p(nil) = [1] p(s) = [4] p(sel) = [1] x2 + [0] p(take) = [1] x2 + [1] Following rules are strictly oriented: from(X) = [1] X + [5] > [1] X + [4] = cons(X,n__from(s(X))) Following rules are (at-least) weakly oriented: 2nd(cons(X,XS)) = [1] X + [1] XS + [12] >= [1] XS + [12] = head(activate(XS)) activate(X) = [1] X + [11] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [11] >= [1] X + [5] = from(X) activate(n__take(X1,X2)) = [1] X2 + [12] >= [1] X2 + [1] = take(X1,X2) from(X) = [1] X + [5] >= [1] X + [0] = n__from(X) head(cons(X,XS)) = [1] X + [1] XS + [1] >= [1] X + [0] = X sel(0(),cons(X,XS)) = [1] X + [1] XS + [0] >= [1] X + [0] = X sel(s(N),cons(X,XS)) = [1] X + [1] XS + [0] >= [1] XS + [11] = sel(N,activate(XS)) take(X1,X2) = [1] X2 + [1] >= [1] X2 + [1] = n__take(X1,X2) take(0(),XS) = [1] XS + [1] >= [1] = nil() take(s(N),cons(X,XS)) = [1] X + [1] XS + [1] >= [1] X + [1] XS + [12] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(X1,X2) -> n__take(X1,X2) take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X sel(0(),cons(X,XS)) -> X take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons ,n__from,n__take,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(head) = {1}, uargs(n__take) = {2}, uargs(sel) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(2nd) = [1] x1 + [5] p(activate) = [1] x1 + [3] p(cons) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [0] p(head) = [1] x1 + [2] p(n__from) = [1] x1 + [0] p(n__take) = [1] x2 + [2] p(nil) = [2] p(s) = [0] p(sel) = [1] x2 + [6] p(take) = [1] x2 + [5] Following rules are strictly oriented: take(X1,X2) = [1] X2 + [5] > [1] X2 + [2] = n__take(X1,X2) Following rules are (at-least) weakly oriented: 2nd(cons(X,XS)) = [1] X + [1] XS + [5] >= [1] XS + [5] = head(activate(XS)) activate(X) = [1] X + [3] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [3] >= [1] X + [0] = from(X) activate(n__take(X1,X2)) = [1] X2 + [5] >= [1] X2 + [5] = take(X1,X2) 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) head(cons(X,XS)) = [1] X + [1] XS + [2] >= [1] X + [0] = X sel(0(),cons(X,XS)) = [1] X + [1] XS + [6] >= [1] X + [0] = X sel(s(N),cons(X,XS)) = [1] X + [1] XS + [6] >= [1] XS + [9] = sel(N,activate(XS)) take(0(),XS) = [1] XS + [5] >= [2] = nil() take(s(N),cons(X,XS)) = [1] X + [1] XS + [5] >= [1] X + [1] XS + [5] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X sel(0(),cons(X,XS)) -> X take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(15) 0 :: [] -(0)-> A(6) 2nd :: [A(8)] -(1)-> A(0) activate :: [A(6)] -(6)-> A(6) cons :: [A(0) x A(6)] -(6)-> A(6) cons :: [A(0) x A(8)] -(8)-> A(8) cons :: [A(0) x A(1)] -(1)-> A(1) from :: [A(0)] -(12)-> A(6) head :: [A(1)] -(0)-> A(0) n__from :: [A(0)] -(6)-> A(6) n__from :: [A(0)] -(9)-> A(9) n__take :: [A(6) x A(6)] -(6)-> A(6) nil :: [] -(0)-> A(8) s :: [A(15)] -(15)-> A(15) s :: [A(6)] -(6)-> A(6) s :: [A(0)] -(0)-> A(0) sel :: [A(15) x A(6)] -(13)-> A(0) take :: [A(6) x A(6)] -(11)-> A(6) 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) n__take :: [A_cf(0) x 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) take :: [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__from_A :: [A(0)] -(1)-> A(1) n__take_A :: [A(0) x A(0)] -(1)-> A(1) nil_A :: [] -(0)-> A(1) s_A :: [A(1)] -(1)-> A(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) 2. Weak: sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) * Step 6: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X sel(0(),cons(X,XS)) -> X take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(15) 0 :: [] -(0)-> A(0) 2nd :: [A(15)] -(10)-> A(0) activate :: [A(0)] -(1)-> A(0) cons :: [A(0) x A(0)] -(0)-> A(0) cons :: [A(0) x A(0)] -(15)-> A(15) from :: [A(0)] -(1)-> A(0) head :: [A(0)] -(0)-> A(0) n__from :: [A(0)] -(0)-> A(0) n__take :: [A(0) x A(0)] -(0)-> A(0) n__take :: [A(0) x A(0)] -(0)-> A(5) n__take :: [A(0) x A(0)] -(0)-> A(7) nil :: [] -(0)-> A(6) s :: [A(15)] -(15)-> A(15) s :: [A(0)] -(0)-> A(0) sel :: [A(15) x A(0)] -(9)-> A(0) take :: [A(0) x A(0)] -(1)-> 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) n__take :: [A_cf(0) x 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) take :: [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)] -(1)-> A(1) n__from_A :: [A(0)] -(1)-> A(1) n__take_A :: [A(0) x 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: sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) 2. Weak: * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))