* Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: active(c()) -> mark(a()) active(c()) -> mark(b()) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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 + [0] p(b) = [0] p(c) = [0] p(f) = [15] p(mark) = [1] x1 + [0] p(ok) = [1] x1 + [11] p(proper) = [1] x1 + [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: top(ok(X)) = [1] X + [11] > [1] X + [0] = top(active(X)) Following rules are (at-least) weakly oriented: active(c()) = [0] >= [0] = mark(a()) active(c()) = [0] >= [0] = mark(b()) f(X1,X2,mark(X3)) = [15] >= [15] = mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) = [15] >= [26] = ok(f(X1,X2,X3)) proper(a()) = [0] >= [11] = ok(a()) proper(b()) = [0] >= [11] = ok(b()) proper(c()) = [0] >= [11] = ok(c()) top(mark(X)) = [1] X + [0] >= [1] X + [0] = top(proper(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: active(c()) -> mark(a()) active(c()) -> mark(b()) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) - Weak TRS: top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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) = [4] p(active) = [1] x_1 + [0] p(b) = [2] p(c) = [5] p(f) = [8] x_1 + [5] p(mark) = [1] x_1 + [0] p(ok) = [1] x_1 + [0] p(proper) = [1] x_1 + [0] p(top) = [2] x_1 + [0] Following rules are strictly oriented: active(c()) = [5] > [4] = mark(a()) active(c()) = [5] > [2] = mark(b()) Following rules are (at-least) weakly oriented: f(X1,X2,mark(X3)) = [8] X1 + [5] >= [8] X1 + [5] = mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) = [8] X1 + [5] >= [8] X1 + [5] = ok(f(X1,X2,X3)) proper(a()) = [4] >= [4] = ok(a()) proper(b()) = [2] >= [2] = ok(b()) proper(c()) = [5] >= [5] = ok(c()) top(mark(X)) = [2] X + [0] >= [2] X + [0] = top(proper(X)) top(ok(X)) = [2] X + [0] >= [2] X + [0] = top(active(X)) * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) - Weak TRS: active(c()) -> mark(a()) active(c()) -> mark(b()) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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) = [5] p(active) = [1] x1 + [0] p(b) = [5] p(c) = [7] p(f) = [1] x2 + [9] x3 + [0] p(mark) = [1] x1 + [2] p(ok) = [1] x1 + [0] p(proper) = [1] x1 + [6] p(top) = [1] x1 + [0] Following rules are strictly oriented: f(X1,X2,mark(X3)) = [1] X2 + [9] X3 + [18] > [1] X2 + [9] X3 + [2] = mark(f(X1,X2,X3)) proper(a()) = [11] > [5] = ok(a()) proper(b()) = [11] > [5] = ok(b()) proper(c()) = [13] > [7] = ok(c()) Following rules are (at-least) weakly oriented: active(c()) = [7] >= [7] = mark(a()) active(c()) = [7] >= [7] = mark(b()) f(ok(X1),ok(X2),ok(X3)) = [1] X2 + [9] X3 + [0] >= [1] X2 + [9] X3 + [0] = ok(f(X1,X2,X3)) top(mark(X)) = [1] X + [2] >= [1] X + [6] = top(proper(X)) top(ok(X)) = [1] X + [0] >= [1] X + [0] = top(active(X)) 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: f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) - Weak TRS: active(c()) -> mark(a()) active(c()) -> mark(b()) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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) = [1] p(active) = [1] x1 + [0] p(b) = [1] p(c) = [4] p(f) = [1] x3 + [12] p(mark) = [1] x1 + [3] p(ok) = [1] x1 + [0] p(proper) = [1] x1 + [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: top(mark(X)) = [1] X + [3] > [1] X + [0] = top(proper(X)) Following rules are (at-least) weakly oriented: active(c()) = [4] >= [4] = mark(a()) active(c()) = [4] >= [4] = mark(b()) f(X1,X2,mark(X3)) = [1] X3 + [15] >= [1] X3 + [15] = mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) = [1] X3 + [12] >= [1] X3 + [12] = ok(f(X1,X2,X3)) proper(a()) = [1] >= [1] = ok(a()) proper(b()) = [1] >= [1] = ok(b()) proper(c()) = [4] >= [4] = ok(c()) top(ok(X)) = [1] X + [0] >= [1] X + [0] = top(active(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(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) - Weak TRS: active(c()) -> mark(a()) active(c()) -> mark(b()) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,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 + [1] p(b) = [6] p(c) = [8] p(f) = [2] x1 + [1] x3 + [0] p(mark) = [1] x1 + [3] p(ok) = [1] x1 + [1] p(proper) = [1] x1 + [1] p(top) = [1] x1 + [2] Following rules are strictly oriented: f(ok(X1),ok(X2),ok(X3)) = [2] X1 + [1] X3 + [3] > [2] X1 + [1] X3 + [1] = ok(f(X1,X2,X3)) Following rules are (at-least) weakly oriented: active(c()) = [9] >= [3] = mark(a()) active(c()) = [9] >= [9] = mark(b()) f(X1,X2,mark(X3)) = [2] X1 + [1] X3 + [3] >= [2] X1 + [1] X3 + [3] = mark(f(X1,X2,X3)) proper(a()) = [1] >= [1] = ok(a()) proper(b()) = [7] >= [7] = ok(b()) proper(c()) = [9] >= [9] = ok(c()) top(mark(X)) = [1] X + [5] >= [1] X + [3] = top(proper(X)) top(ok(X)) = [1] X + [3] >= [1] X + [3] = 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(c()) -> mark(a()) active(c()) -> mark(b()) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,mark,ok} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))