* Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: active(b()) -> mark(a()) f(mark(X1),X2) -> mark(f(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok} + 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(mark) = {1}, uargs(ok) = {1}, uargs(top) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(active) = [1] x1 + [9] p(b) = [0] p(f) = [0] p(mark) = [1] x1 + [0] p(ok) = [1] x1 + [0] p(proper) = [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: active(b()) = [9] > [0] = mark(a()) Following rules are (at-least) weakly oriented: f(mark(X1),X2) = [0] >= [0] = mark(f(X1,X2)) f(ok(X1),ok(X2)) = [0] >= [0] = ok(f(X1,X2)) proper(a()) = [0] >= [0] = ok(a()) proper(b()) = [0] >= [0] = ok(b()) top(mark(X)) = [1] X + [0] >= [0] = top(proper(X)) top(ok(X)) = [1] X + [0] >= [1] X + [9] = top(active(X)) 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: f(mark(X1),X2) -> mark(f(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Weak TRS: active(b()) -> mark(a()) - Signature: {active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok} + 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(mark) = {1}, uargs(ok) = {1}, uargs(top) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [8] p(active) = [8] p(b) = [0] p(f) = [9] x1 + [10] p(mark) = [1] x1 + [0] p(ok) = [1] x1 + [1] p(proper) = [1] x1 + [9] p(top) = [1] x1 + [0] Following rules are strictly oriented: f(ok(X1),ok(X2)) = [9] X1 + [19] > [9] X1 + [11] = ok(f(X1,X2)) proper(a()) = [17] > [9] = ok(a()) proper(b()) = [9] > [1] = ok(b()) Following rules are (at-least) weakly oriented: active(b()) = [8] >= [8] = mark(a()) f(mark(X1),X2) = [9] X1 + [10] >= [9] X1 + [10] = mark(f(X1,X2)) top(mark(X)) = [1] X + [0] >= [1] X + [9] = top(proper(X)) top(ok(X)) = [1] X + [1] >= [8] = top(active(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(mark(X1),X2) -> mark(f(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Weak TRS: active(b()) -> mark(a()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) - Signature: {active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)): The following argument positions are considered usable: uargs(mark) = {1}, uargs(ok) = {1}, uargs(top) = {1} Following symbols are considered usable: {active,f,proper,top} TcT has computed the following interpretation: p(a) = [0] p(active) = [1] x_1 + [0] p(b) = [8] p(f) = [1] x_1 + [1] x_2 + [8] p(mark) = [1] x_1 + [8] p(ok) = [1] x_1 + [0] p(proper) = [1] x_1 + [0] p(top) = [1] x_1 + [4] Following rules are strictly oriented: top(mark(X)) = [1] X + [12] > [1] X + [4] = top(proper(X)) Following rules are (at-least) weakly oriented: active(b()) = [8] >= [8] = mark(a()) f(mark(X1),X2) = [1] X1 + [1] X2 + [16] >= [1] X1 + [1] X2 + [16] = mark(f(X1,X2)) f(ok(X1),ok(X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = ok(f(X1,X2)) proper(a()) = [0] >= [0] = ok(a()) proper(b()) = [8] >= [8] = ok(b()) top(ok(X)) = [1] X + [4] >= [1] X + [4] = top(active(X)) * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(mark(X1),X2) -> mark(f(X1,X2)) top(ok(X)) -> top(active(X)) - Weak TRS: active(b()) -> mark(a()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) top(mark(X)) -> top(proper(X)) - Signature: {active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok} + 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(mark) = {1}, uargs(ok) = {1}, uargs(top) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [9] p(active) = [1] x1 + [0] p(b) = [12] p(f) = [1] x1 + [14] p(mark) = [1] x1 + [3] p(ok) = [1] x1 + [3] p(proper) = [1] x1 + [3] p(top) = [1] x1 + [0] Following rules are strictly oriented: top(ok(X)) = [1] X + [3] > [1] X + [0] = top(active(X)) Following rules are (at-least) weakly oriented: active(b()) = [12] >= [12] = mark(a()) f(mark(X1),X2) = [1] X1 + [17] >= [1] X1 + [17] = mark(f(X1,X2)) f(ok(X1),ok(X2)) = [1] X1 + [17] >= [1] X1 + [17] = ok(f(X1,X2)) proper(a()) = [12] >= [12] = ok(a()) proper(b()) = [15] >= [15] = ok(b()) top(mark(X)) = [1] X + [3] >= [1] X + [3] = top(proper(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(mark(X1),X2) -> mark(f(X1,X2)) - Weak TRS: active(b()) -> mark(a()) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok} + 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(mark) = {1}, uargs(ok) = {1}, uargs(top) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [2] p(active) = [1] x1 + [1] p(b) = [2] p(f) = [12] x1 + [14] x2 + [3] p(mark) = [1] x1 + [1] p(ok) = [1] x1 + [1] p(proper) = [1] x1 + [1] p(top) = [1] x1 + [0] Following rules are strictly oriented: f(mark(X1),X2) = [12] X1 + [14] X2 + [15] > [12] X1 + [14] X2 + [4] = mark(f(X1,X2)) Following rules are (at-least) weakly oriented: active(b()) = [3] >= [3] = mark(a()) f(ok(X1),ok(X2)) = [12] X1 + [14] X2 + [29] >= [12] X1 + [14] X2 + [4] = ok(f(X1,X2)) proper(a()) = [3] >= [3] = ok(a()) proper(b()) = [3] >= [3] = ok(b()) top(mark(X)) = [1] X + [1] >= [1] X + [1] = top(proper(X)) top(ok(X)) = [1] X + [1] >= [1] X + [1] = top(active(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(b()) -> mark(a()) f(mark(X1),X2) -> mark(f(X1,X2)) f(ok(X1),ok(X2)) -> ok(f(X1,X2)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/2,proper/1,top/1} / {a/0,b/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,mark,ok} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))