WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: main(x5,x12) -> map#2(plus_x(x12),x5) map#2(plus_x(x2),Nil()) -> Nil() map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2)) plus_x#1(0(),x8) -> x8 plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14)) - Signature: {main/2,map#2/2,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {main,map#2,plus_x#1} and constructors {0,Cons,Nil,S ,plus_x} + 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) = {1,2}, uargs(S) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [10] p(Nil) = [1] p(S) = [1] x1 + [7] p(main) = [1] x1 + [1] x2 + [13] p(map#2) = [4] x1 + [1] x2 + [8] p(plus_x) = [1] p(plus_x#1) = [1] x2 + [2] Following rules are strictly oriented: main(x5,x12) = [1] x12 + [1] x5 + [13] > [1] x5 + [12] = map#2(plus_x(x12),x5) map#2(plus_x(x2),Nil()) = [13] > [1] = Nil() plus_x#1(0(),x8) = [1] x8 + [2] > [1] x8 + [0] = x8 Following rules are (at-least) weakly oriented: map#2(plus_x(x6),Cons(x4,x2)) = [1] x2 + [1] x4 + [22] >= [1] x2 + [1] x4 + [24] = Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2)) plus_x#1(S(x12),x14) = [1] x14 + [2] >= [1] x14 + [9] = S(plus_x#1(x12,x14)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2)) plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14)) - Weak TRS: main(x5,x12) -> map#2(plus_x(x12),x5) map#2(plus_x(x2),Nil()) -> Nil() plus_x#1(0(),x8) -> x8 - Signature: {main/2,map#2/2,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {main,map#2,plus_x#1} and constructors {0,Cons,Nil,S ,plus_x} + 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) = {1,2}, uargs(S) = {1} Following symbols are considered usable: {main,map#2,plus_x#1} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [2] p(Nil) = [1] p(S) = [1] x1 + [0] p(main) = [6] x1 + [3] x2 + [2] p(map#2) = [2] x1 + [5] x2 + [0] p(plus_x) = [1] x1 + [0] p(plus_x#1) = [4] x2 + [2] Following rules are strictly oriented: map#2(plus_x(x6),Cons(x4,x2)) = [5] x2 + [5] x4 + [2] x6 + [10] > [5] x2 + [4] x4 + [2] x6 + [4] = Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2)) Following rules are (at-least) weakly oriented: main(x5,x12) = [3] x12 + [6] x5 + [2] >= [2] x12 + [5] x5 + [0] = map#2(plus_x(x12),x5) map#2(plus_x(x2),Nil()) = [2] x2 + [5] >= [1] = Nil() plus_x#1(0(),x8) = [4] x8 + [2] >= [1] x8 + [0] = x8 plus_x#1(S(x12),x14) = [4] x14 + [2] >= [4] x14 + [2] = S(plus_x#1(x12,x14)) * Step 3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14)) - Weak TRS: main(x5,x12) -> map#2(plus_x(x12),x5) map#2(plus_x(x2),Nil()) -> Nil() map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2)) plus_x#1(0(),x8) -> x8 - Signature: {main/2,map#2/2,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {main,map#2,plus_x#1} and constructors {0,Cons,Nil,S ,plus_x} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(Cons) = {1,2}, uargs(S) = {1} Following symbols are considered usable: {main,map#2,plus_x#1} TcT has computed the following interpretation: p(0) = 1 p(Cons) = 2 + x1 + x2 p(Nil) = 0 p(S) = 1 + x1 p(main) = 1 + 4*x1 + 4*x1*x2 + 2*x1^2 + 4*x2 + 4*x2^2 p(map#2) = 1 + 4*x1 + 4*x1*x2 + 2*x1^2 + x2 + 2*x2^2 p(plus_x) = x1 p(plus_x#1) = 3*x1 + 4*x1*x2 Following rules are strictly oriented: plus_x#1(S(x12),x14) = 3 + 3*x12 + 4*x12*x14 + 4*x14 > 1 + 3*x12 + 4*x12*x14 = S(plus_x#1(x12,x14)) Following rules are (at-least) weakly oriented: main(x5,x12) = 1 + 4*x12 + 4*x12*x5 + 4*x12^2 + 4*x5 + 2*x5^2 >= 1 + 4*x12 + 4*x12*x5 + 2*x12^2 + x5 + 2*x5^2 = map#2(plus_x(x12),x5) map#2(plus_x(x2),Nil()) = 1 + 4*x2 + 2*x2^2 >= 0 = Nil() map#2(plus_x(x6),Cons(x4,x2)) = 11 + 9*x2 + 4*x2*x4 + 4*x2*x6 + 2*x2^2 + 9*x4 + 4*x4*x6 + 2*x4^2 + 12*x6 + 2*x6^2 >= 3 + x2 + 4*x2*x6 + 2*x2^2 + 4*x4*x6 + 7*x6 + 2*x6^2 = Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2)) plus_x#1(0(),x8) = 3 + 4*x8 >= x8 = x8 * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: main(x5,x12) -> map#2(plus_x(x12),x5) map#2(plus_x(x2),Nil()) -> Nil() map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2)) plus_x#1(0(),x8) -> x8 plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14)) - Signature: {main/2,map#2/2,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {main,map#2,plus_x#1} and constructors {0,Cons,Nil,S ,plus_x} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))