WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) mark(g(X)) -> g(mark(X)) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + 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(a__f) = {1}, uargs(g) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a__f) = [1] x1 + [0] p(f) = [1] x1 + [15] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [8] Following rules are strictly oriented: mark(f(X1,X2)) = [1] X1 + [23] > [1] X1 + [8] = a__f(mark(X1),X2) Following rules are (at-least) weakly oriented: a__f(X1,X2) = [1] X1 + [0] >= [1] X1 + [15] = f(X1,X2) a__f(g(X),Y) = [1] X + [0] >= [1] X + [8] = a__f(mark(X),f(g(X),Y)) mark(g(X)) = [1] X + [8] >= [1] X + [8] = g(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(g(X)) -> g(mark(X)) - Weak TRS: mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + 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(a__f) = {1}, uargs(g) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a__f) = [1] x1 + [8] p(f) = [1] x1 + [9] p(g) = [1] x1 + [1] p(mark) = [1] x1 + [0] Following rules are strictly oriented: a__f(g(X),Y) = [1] X + [9] > [1] X + [8] = a__f(mark(X),f(g(X),Y)) Following rules are (at-least) weakly oriented: a__f(X1,X2) = [1] X1 + [8] >= [1] X1 + [9] = f(X1,X2) mark(f(X1,X2)) = [1] X1 + [9] >= [1] X1 + [8] = a__f(mark(X1),X2) mark(g(X)) = [1] X + [1] >= [1] X + [1] = g(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) mark(g(X)) -> g(mark(X)) - Weak TRS: a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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__f) = {1}, uargs(g) = {1} Following symbols are considered usable: {a__f,mark} TcT has computed the following interpretation: p(a__f) = [1 1] x1 + [3] [0 1] [4] p(f) = [1 1] x1 + [0] [0 1] [4] p(g) = [1 2] x1 + [4] [0 1] [0] p(mark) = [1 1] x1 + [1] [0 1] [0] Following rules are strictly oriented: a__f(X1,X2) = [1 1] X1 + [3] [0 1] [4] > [1 1] X1 + [0] [0 1] [4] = f(X1,X2) Following rules are (at-least) weakly oriented: a__f(g(X),Y) = [1 3] X + [7] [0 1] [4] >= [1 2] X + [4] [0 1] [4] = a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) = [1 2] X1 + [5] [0 1] [4] >= [1 2] X1 + [4] [0 1] [4] = a__f(mark(X1),X2) mark(g(X)) = [1 3] X + [5] [0 1] [0] >= [1 3] X + [5] [0 1] [0] = g(mark(X)) * Step 4: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(g(X)) -> g(mark(X)) - Weak TRS: a__f(X1,X2) -> f(X1,X2) a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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__f) = {1}, uargs(g) = {1} Following symbols are considered usable: {a__f,mark} TcT has computed the following interpretation: p(a__f) = [1 0] x1 + [0 1] x2 + [0] [0 1] [0 0] [2] p(f) = [1 0] x1 + [0 1] x2 + [0] [0 1] [0 0] [2] p(g) = [1 4] x1 + [6] [0 1] [4] p(mark) = [1 2] x1 + [0] [0 1] [0] Following rules are strictly oriented: mark(g(X)) = [1 6] X + [14] [0 1] [4] > [1 6] X + [6] [0 1] [4] = g(mark(X)) Following rules are (at-least) weakly oriented: a__f(X1,X2) = [1 0] X1 + [0 1] X2 + [0] [0 1] [0 0] [2] >= [1 0] X1 + [0 1] X2 + [0] [0 1] [0 0] [2] = f(X1,X2) a__f(g(X),Y) = [1 4] X + [0 1] Y + [6] [0 1] [0 0] [6] >= [1 3] X + [6] [0 1] [2] = a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) = [1 2] X1 + [0 1] X2 + [4] [0 1] [0 0] [2] >= [1 2] X1 + [0 1] X2 + [0] [0 1] [0 0] [2] = a__f(mark(X1),X2) * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__f(X1,X2) -> f(X1,X2) a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) mark(g(X)) -> g(mark(X)) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))