WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(x){x -> cons(x,y)} = mark(cons(x,y)) ->^+ cons(mark(x),y) = C[mark(x) = mark(x){}] ** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = 0 p(a__from) = 4 + x1 p(a__length) = 0 p(a__length1) = 0 p(cons) = x1 p(from) = 4 + x1 p(length) = 0 p(length1) = 0 p(mark) = x1 p(nil) = 0 p(s) = x1 Following rules are strictly oriented: a__from(X) = 4 + X > X = cons(mark(X),from(s(X))) Following rules are (at-least) weakly oriented: a__from(X) = 4 + X >= 4 + X = from(X) a__length(X) = 0 >= 0 = length(X) a__length(cons(X,Y)) = 0 >= 0 = s(a__length1(Y)) a__length(nil()) = 0 >= 0 = 0() a__length1(X) = 0 >= 0 = a__length(X) a__length1(X) = 0 >= 0 = length1(X) mark(0()) = 0 >= 0 = 0() mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(from(X)) = 4 + X >= 4 + X = a__from(mark(X)) mark(length(X)) = 0 >= 0 = a__length(X) mark(length1(X)) = 0 >= 0 = a__length1(X) mark(nil()) = 0 >= 0 = nil() mark(s(X)) = X >= X = s(mark(X)) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = 0 p(a__from) = 4 + x1 p(a__length) = 0 p(a__length1) = 0 p(cons) = x1 p(from) = 4 + x1 p(length) = 0 p(length1) = 0 p(mark) = 2 + x1 p(nil) = 0 p(s) = x1 Following rules are strictly oriented: mark(0()) = 2 > 0 = 0() mark(length(X)) = 2 > 0 = a__length(X) mark(length1(X)) = 2 > 0 = a__length1(X) mark(nil()) = 2 > 0 = nil() Following rules are (at-least) weakly oriented: a__from(X) = 4 + X >= 2 + X = cons(mark(X),from(s(X))) a__from(X) = 4 + X >= 4 + X = from(X) a__length(X) = 0 >= 0 = length(X) a__length(cons(X,Y)) = 0 >= 0 = s(a__length1(Y)) a__length(nil()) = 0 >= 0 = 0() a__length1(X) = 0 >= 0 = a__length(X) a__length1(X) = 0 >= 0 = length1(X) mark(cons(X1,X2)) = 2 + X1 >= 2 + X1 = cons(mark(X1),X2) mark(from(X)) = 6 + X >= 6 + X = a__from(mark(X)) mark(s(X)) = 2 + X >= 2 + X = s(mark(X)) ** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) mark(0()) -> 0() mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,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(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [0] p(a__from) = [1] x1 + [0] p(a__length) = [1] p(a__length1) = [1] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(length) = [1] p(length1) = [1] p(mark) = [1] x1 + [0] p(nil) = [13] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__length(nil()) = [1] > [0] = 0() Following rules are (at-least) weakly oriented: a__from(X) = [1] X + [0] >= [1] X + [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [1] X + [0] = from(X) a__length(X) = [1] >= [1] = length(X) a__length(cons(X,Y)) = [1] >= [1] = s(a__length1(Y)) a__length1(X) = [1] >= [1] = a__length(X) a__length1(X) = [1] >= [1] = length1(X) mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(from(X)) = [1] X + [0] >= [1] X + [0] = a__from(mark(X)) mark(length(X)) = [1] >= [1] = a__length(X) mark(length1(X)) = [1] >= [1] = a__length1(X) mark(nil()) = [13] >= [13] = nil() mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) ** Step 1.b:4: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__length(nil()) -> 0() mark(0()) -> 0() mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,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(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [0] p(a__from) = [1] x1 + [2] p(a__length) = [1] p(a__length1) = [1] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [2] p(length) = [0] p(length1) = [0] p(mark) = [1] x1 + [1] p(nil) = [7] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__length(X) = [1] > [0] = length(X) a__length1(X) = [1] > [0] = length1(X) Following rules are (at-least) weakly oriented: a__from(X) = [1] X + [2] >= [1] X + [1] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [2] >= [1] X + [2] = from(X) a__length(cons(X,Y)) = [1] >= [1] = s(a__length1(Y)) a__length(nil()) = [1] >= [0] = 0() a__length1(X) = [1] >= [1] = a__length(X) mark(0()) = [1] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] >= [1] X1 + [1] = cons(mark(X1),X2) mark(from(X)) = [1] X + [3] >= [1] X + [3] = a__from(mark(X)) mark(length(X)) = [1] >= [1] = a__length(X) mark(length1(X)) = [1] >= [1] = a__length1(X) mark(nil()) = [8] >= [7] = nil() mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(mark(X)) ** Step 1.b:5: Ara WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__length(X) -> length(X) a__length(nil()) -> 0() a__length1(X) -> length1(X) mark(0()) -> 0() mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0, 12) 0 :: [] -(0)-> "A"(13, 15) 0 :: [] -(0)-> "A"(15, 13) a__from :: ["A"(0, 12)] -(12)-> "A"(2, 12) a__length :: ["A"(12, 0)] -(7)-> "A"(0, 12) a__length1 :: ["A"(12, 0)] -(7)-> "A"(0, 12) cons :: ["A"(0, 0) x "A"(12, 0)] -(12)-> "A"(12, 0) cons :: ["A"(0, 12) x "A"(0, 0)] -(0)-> "A"(0, 12) cons :: ["A"(0, 12) x "A"(2, 0)] -(2)-> "A"(2, 12) from :: ["A"(0, 12)] -(12)-> "A"(0, 12) from :: ["A"(0, 12)] -(12)-> "A"(8, 12) from :: ["A"(0, 0)] -(0)-> "A"(8, 0) length :: ["A"(12, 0)] -(0)-> "A"(0, 12) length :: ["A"(12, 0)] -(0)-> "A"(15, 12) length1 :: ["A"(12, 0)] -(0)-> "A"(0, 12) length1 :: ["A"(12, 0)] -(0)-> "A"(4, 12) mark :: ["A"(0, 12)] -(10)-> "A"(0, 12) nil :: [] -(0)-> "A"(12, 0) nil :: [] -(0)-> "A"(0, 12) nil :: [] -(0)-> "A"(8, 14) s :: ["A"(0, 12)] -(0)-> "A"(0, 12) s :: ["A"(0, 12)] -(0)-> "A"(7, 12) s :: ["A"(0, 12)] -(0)-> "A"(6, 12) s :: ["A"(0, 0)] -(0)-> "A"(14, 0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1, 0) "0_A" :: [] -(0)-> "A"(0, 1) "cons_A" :: ["A"(0, 0) x "A"(1, 0)] -(1)-> "A"(1, 0) "cons_A" :: ["A"(0, 1) x "A"(0, 0)] -(0)-> "A"(0, 1) "from_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "from_A" :: ["A"(0, 1)] -(1)-> "A"(0, 1) "length1_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "length1_A" :: ["A"(1, 0)] -(0)-> "A"(0, 1) "length_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "length_A" :: ["A"(1, 0)] -(0)-> "A"(0, 1) "nil_A" :: [] -(0)-> "A"(1, 0) "nil_A" :: [] -(0)-> "A"(0, 1) "s_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "s_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: a__length(cons(X,Y)) -> s(a__length1(Y)) 2. Weak: a__from(X) -> from(X) a__length1(X) -> a__length(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) ** Step 1.b:6: Ara WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) a__length1(X) -> a__length(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> length1(X) mark(0()) -> 0() mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0, 12) 0 :: [] -(0)-> "A"(15, 13) 0 :: [] -(0)-> "A"(15, 12) a__from :: ["A"(0, 12)] -(12)-> "A"(4, 12) a__length :: ["A"(12, 0)] -(12)-> "A"(7, 12) a__length1 :: ["A"(12, 0)] -(15)-> "A"(0, 12) cons :: ["A"(0, 12) x "A"(0, 0)] -(0)-> "A"(0, 12) cons :: ["A"(0, 0) x "A"(12, 0)] -(12)-> "A"(12, 0) cons :: ["A"(0, 12) x "A"(4, 0)] -(4)-> "A"(4, 12) from :: ["A"(0, 12)] -(12)-> "A"(0, 12) from :: ["A"(0, 12)] -(12)-> "A"(12, 12) from :: ["A"(0, 0)] -(0)-> "A"(4, 0) length :: ["A"(12, 0)] -(12)-> "A"(0, 12) length :: ["A"(12, 0)] -(12)-> "A"(7, 12) length1 :: ["A"(12, 0)] -(12)-> "A"(0, 12) mark :: ["A"(0, 12)] -(8)-> "A"(0, 12) nil :: [] -(0)-> "A"(12, 0) nil :: [] -(0)-> "A"(0, 12) nil :: [] -(0)-> "A"(6, 14) s :: ["A"(0, 12)] -(0)-> "A"(0, 12) s :: ["A"(0, 12)] -(0)-> "A"(15, 12) s :: ["A"(0, 0)] -(0)-> "A"(0, 0) s :: ["A"(0, 12)] -(0)-> "A"(7, 12) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1, 0) "0_A" :: [] -(0)-> "A"(0, 1) "cons_A" :: ["A"(0, 0) x "A"(1, 0)] -(1)-> "A"(1, 0) "cons_A" :: ["A"(0, 1) x "A"(0, 0)] -(0)-> "A"(0, 1) "from_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "from_A" :: ["A"(0, 1)] -(1)-> "A"(0, 1) "length1_A" :: ["A"(0, 0)] -(1)-> "A"(1, 0) "length1_A" :: ["A"(1, 0)] -(1)-> "A"(0, 1) "length_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "length_A" :: ["A"(1, 0)] -(1)-> "A"(0, 1) "nil_A" :: [] -(0)-> "A"(1, 0) "nil_A" :: [] -(0)-> "A"(0, 1) "s_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "s_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: a__length1(X) -> a__length(X) 2. Weak: a__from(X) -> from(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) ** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__from) = [1 1 2] [0] [0 0 2] x1 + [0] [0 0 1] [1] p(a__length) = [0] [0] [0] p(a__length1) = [0] [0] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(from) = [1 1 2] [0] [0 0 2] x1 + [0] [0 0 1] [1] p(length) = [0] [0] [0] p(length1) = [0] [0] [0] p(mark) = [1 1 2] [0] [0 0 2] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 2] [0] [0 1 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: mark(from(X)) = [1 1 6] [2] [0 0 2] X + [2] [0 0 1] [1] > [1 1 6] [0] [0 0 2] X + [0] [0 0 1] [1] = a__from(mark(X)) Following rules are (at-least) weakly oriented: a__from(X) = [1 1 2] [0] [0 0 2] X + [0] [0 0 1] [1] >= [1 1 2] [0] [0 0 2] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 2] [0] [0 0 2] X + [0] [0 0 1] [1] >= [1 1 2] [0] [0 0 2] X + [0] [0 0 1] [1] = from(X) a__length(X) = [0] [0] [0] >= [0] [0] [0] = length(X) a__length(cons(X,Y)) = [0] [0] [0] >= [0] [0] [0] = s(a__length1(Y)) a__length(nil()) = [0] [0] [0] >= [0] [0] [0] = 0() a__length1(X) = [0] [0] [0] >= [0] [0] [0] = a__length(X) a__length1(X) = [0] [0] [0] >= [0] [0] [0] = length1(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 1 2] [0] [0 0 2] X1 + [0] [0 0 1] [0] >= [1 1 2] [0] [0 0 2] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(length(X)) = [0] [0] [0] >= [0] [0] [0] = a__length(X) mark(length1(X)) = [0] [0] [0] >= [0] [0] [0] = a__length1(X) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 1 4] [0] [0 0 2] X + [0] [0 0 1] [0] >= [1 1 4] [0] [0 0 2] X + [0] [0 0 1] [0] = s(mark(X)) ** Step 1.b:8: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__from) = [1 1 0] [0] [0 0 1] x1 + [2] [0 0 1] [2] p(a__length) = [2] [0] [0] p(a__length1) = [2] [0] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [2] [0 0 1] [2] p(from) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [2] p(length) = [2] [0] [0] p(length1) = [2] [0] [0] p(mark) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(nil) = [3] [0] [2] p(s) = [1 1 3] [0] [0 0 1] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 1 0] [2] [0 0 1] X1 + [2] [0 0 1] [2] > [1 1 0] [0] [0 0 1] X1 + [2] [0 0 1] [2] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__from(X) = [1 1 0] [0] [0 0 1] X + [2] [0 0 1] [2] >= [1 1 0] [0] [0 0 1] X + [2] [0 0 1] [2] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 0 1] X + [2] [0 0 1] [2] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 1] [2] = from(X) a__length(X) = [2] [0] [0] >= [2] [0] [0] = length(X) a__length(cons(X,Y)) = [2] [0] [0] >= [2] [0] [0] = s(a__length1(Y)) a__length(nil()) = [2] [0] [0] >= [0] [0] [0] = 0() a__length1(X) = [2] [0] [0] >= [2] [0] [0] = a__length(X) a__length1(X) = [2] [0] [0] >= [2] [0] [0] = length1(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(from(X)) = [1 1 1] [0] [0 0 1] X + [2] [0 0 1] [2] >= [1 1 1] [0] [0 0 1] X + [2] [0 0 1] [2] = a__from(mark(X)) mark(length(X)) = [2] [0] [0] >= [2] [0] [0] = a__length(X) mark(length1(X)) = [2] [0] [0] >= [2] [0] [0] = a__length1(X) mark(nil()) = [3] [2] [2] >= [3] [0] [2] = nil() mark(s(X)) = [1 1 4] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 1 4] [0] [0 0 1] X + [0] [0 0 1] [0] = s(mark(X)) ** Step 1.b:9: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__from(X) -> from(X) mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [0] [2] [0] p(a__from) = [1 0 1] [1] [0 0 0] x1 + [2] [0 0 1] [1] p(a__length) = [0] [2] [0] p(a__length1) = [0] [2] [0] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(from) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [1] p(length) = [0] [1] [0] p(length1) = [0] [2] [0] p(mark) = [1 0 1] [0] [0 2 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: a__from(X) = [1 0 1] [1] [0 0 0] X + [2] [0 0 1] [1] > [1 0 1] [0] [0 0 0] X + [1] [0 0 1] [1] = from(X) Following rules are (at-least) weakly oriented: a__from(X) = [1 0 1] [1] [0 0 0] X + [2] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__length(X) = [0] [2] [0] >= [0] [1] [0] = length(X) a__length(cons(X,Y)) = [0] [2] [0] >= [0] [0] [0] = s(a__length1(Y)) a__length(nil()) = [0] [2] [0] >= [0] [2] [0] = 0() a__length1(X) = [0] [2] [0] >= [0] [2] [0] = a__length(X) a__length1(X) = [0] [2] [0] >= [0] [2] [0] = length1(X) mark(0()) = [0] [4] [0] >= [0] [2] [0] = 0() mark(cons(X1,X2)) = [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 0 2] [1] [0 0 0] X + [2] [0 0 1] [1] >= [1 0 2] [1] [0 0 0] X + [2] [0 0 1] [1] = a__from(mark(X)) mark(length(X)) = [0] [2] [0] >= [0] [2] [0] = a__length(X) mark(length1(X)) = [0] [4] [0] >= [0] [2] [0] = a__length1(X) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = s(mark(X)) ** Step 1.b:10: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(s(X)) -> s(mark(X)) - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(a__from) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__from,a__length,a__length1,mark} TcT has computed the following interpretation: p(0) = [1] [0] [0] p(a__from) = [1 2 0] [3] [0 1 0] x1 + [0] [1 0 0] [3] p(a__length) = [0 0 1] [2] [0 0 1] x1 + [0] [0 0 0] [0] p(a__length1) = [0 0 1] [2] [0 0 1] x1 + [0] [0 0 0] [0] p(cons) = [1 0 0] [0 0 2] [2] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 1] [2] p(from) = [1 2 0] [3] [0 1 0] x1 + [0] [0 0 0] [0] p(length) = [0 0 0] [2] [0 0 1] x1 + [0] [0 0 0] [0] p(length1) = [0 0 0] [1] [0 0 1] x1 + [0] [0 0 0] [0] p(mark) = [1 2 0] [1] [0 1 0] x1 + [0] [2 0 0] [0] p(nil) = [0] [1] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [2] [0 0 0] [0] Following rules are strictly oriented: mark(s(X)) = [1 2 0] [5] [0 1 0] X + [2] [2 0 0] [0] > [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] = s(mark(X)) Following rules are (at-least) weakly oriented: a__from(X) = [1 2 0] [3] [0 1 0] X + [0] [1 0 0] [3] >= [1 2 0] [3] [0 1 0] X + [0] [0 0 0] [2] = cons(mark(X),from(s(X))) a__from(X) = [1 2 0] [3] [0 1 0] X + [0] [1 0 0] [3] >= [1 2 0] [3] [0 1 0] X + [0] [0 0 0] [0] = from(X) a__length(X) = [0 0 1] [2] [0 0 1] X + [0] [0 0 0] [0] >= [0 0 0] [2] [0 0 1] X + [0] [0 0 0] [0] = length(X) a__length(cons(X,Y)) = [0 0 1] [4] [0 0 1] Y + [2] [0 0 0] [0] >= [0 0 1] [2] [0 0 1] Y + [2] [0 0 0] [0] = s(a__length1(Y)) a__length(nil()) = [2] [0] [0] >= [1] [0] [0] = 0() a__length1(X) = [0 0 1] [2] [0 0 1] X + [0] [0 0 0] [0] >= [0 0 1] [2] [0 0 1] X + [0] [0 0 0] [0] = a__length(X) a__length1(X) = [0 0 1] [2] [0 0 1] X + [0] [0 0 0] [0] >= [0 0 0] [1] [0 0 1] X + [0] [0 0 0] [0] = length1(X) mark(0()) = [2] [0] [2] >= [1] [0] [0] = 0() mark(cons(X1,X2)) = [1 2 0] [0 0 2] [3] [0 1 0] X1 + [0 0 0] X2 + [0] [2 0 0] [0 0 4] [4] >= [1 2 0] [0 0 2] [3] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 1] [2] = cons(mark(X1),X2) mark(from(X)) = [1 4 0] [4] [0 1 0] X + [0] [2 4 0] [6] >= [1 4 0] [4] [0 1 0] X + [0] [1 2 0] [4] = a__from(mark(X)) mark(length(X)) = [0 0 2] [3] [0 0 1] X + [0] [0 0 0] [4] >= [0 0 1] [2] [0 0 1] X + [0] [0 0 0] [0] = a__length(X) mark(length1(X)) = [0 0 2] [2] [0 0 1] X + [0] [0 0 0] [2] >= [0 0 1] [2] [0 0 1] X + [0] [0 0 0] [0] = a__length1(X) mark(nil()) = [3] [1] [0] >= [0] [1] [0] = nil() ** Step 1.b:11: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__length(X) -> length(X) a__length(cons(X,Y)) -> s(a__length1(Y)) a__length(nil()) -> 0() a__length1(X) -> a__length(X) a__length1(X) -> length1(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) - Signature: {a__from/1,a__length/1,a__length1/1,mark/1} / {0/0,cons/2,from/1,length/1,length1/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__from,a__length,a__length1,mark} and constructors {0 ,cons,from,length,length1,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))