/export/starexec/sandbox2/solver/bin/starexec_run_tct_dc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(b()) -> mark(a()) active(f(a(),X,X)) -> mark(f(X,b(),b())) f(X1,X2,active(X3)) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(mark(X1),X2,X3) -> f(X1,X2,X3) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) - Signature: {active/1,f/3,mark/1} / {a/0,b/0} - Obligation: derivational complexity wrt. signature {a,active,b,f,mark} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [15] p(active) = [1] x1 + [0] p(b) = [8] p(f) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [1] Following rules are strictly oriented: f(X1,X2,mark(X3)) = [1] X1 + [1] X2 + [1] X3 + [1] > [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(X1,mark(X2),X3) = [1] X1 + [1] X2 + [1] X3 + [1] > [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(mark(X1),X2,X3) = [1] X1 + [1] X2 + [1] X3 + [1] > [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) mark(a()) = [16] > [15] = active(a()) mark(b()) = [9] > [8] = active(b()) Following rules are (at-least) weakly oriented: active(b()) = [8] >= [16] = mark(a()) active(f(a(),X,X)) = [2] X + [15] >= [1] X + [17] = mark(f(X,b(),b())) f(X1,X2,active(X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(X1,active(X2),X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(active(X1),X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) mark(f(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [1] = active(f(X1,mark(X2),X3)) 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: active(b()) -> mark(a()) active(f(a(),X,X)) -> mark(f(X,b(),b())) f(X1,X2,active(X3)) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) - Weak TRS: f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(mark(X1),X2,X3) -> f(X1,X2,X3) mark(a()) -> active(a()) mark(b()) -> active(b()) - Signature: {active/1,f/3,mark/1} / {a/0,b/0} - Obligation: derivational complexity wrt. signature {a,active,b,f,mark} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(active) = [1] x1 + [0] p(b) = [5] p(f) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] Following rules are strictly oriented: active(b()) = [5] > [0] = mark(a()) Following rules are (at-least) weakly oriented: active(f(a(),X,X)) = [2] X + [0] >= [1] X + [10] = mark(f(X,b(),b())) f(X1,X2,active(X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(X1,X2,mark(X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(X1,active(X2),X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(X1,mark(X2),X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(active(X1),X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(mark(X1),X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) mark(a()) = [0] >= [0] = active(a()) mark(b()) = [5] >= [5] = active(b()) mark(f(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = active(f(X1,mark(X2),X3)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(a(),X,X)) -> mark(f(X,b(),b())) f(X1,X2,active(X3)) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) - Weak TRS: active(b()) -> mark(a()) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(mark(X1),X2,X3) -> f(X1,X2,X3) mark(a()) -> active(a()) mark(b()) -> active(b()) - Signature: {active/1,f/3,mark/1} / {a/0,b/0} - Obligation: derivational complexity wrt. signature {a,active,b,f,mark} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(active) = [1] x1 + [1] p(b) = [0] p(f) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [1] Following rules are strictly oriented: f(X1,X2,active(X3)) = [1] X1 + [1] X2 + [1] X3 + [1] > [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(X1,active(X2),X3) = [1] X1 + [1] X2 + [1] X3 + [1] > [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(active(X1),X2,X3) = [1] X1 + [1] X2 + [1] X3 + [1] > [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) Following rules are (at-least) weakly oriented: active(b()) = [1] >= [1] = mark(a()) active(f(a(),X,X)) = [2] X + [1] >= [1] X + [1] = mark(f(X,b(),b())) f(X1,X2,mark(X3)) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(X1,mark(X2),X3) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) f(mark(X1),X2,X3) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) mark(a()) = [1] >= [1] = active(a()) mark(b()) = [1] >= [1] = active(b()) mark(f(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [2] = active(f(X1,mark(X2),X3)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(a(),X,X)) -> mark(f(X,b(),b())) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) - Weak TRS: active(b()) -> mark(a()) f(X1,X2,active(X3)) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(mark(X1),X2,X3) -> f(X1,X2,X3) mark(a()) -> active(a()) mark(b()) -> active(b()) - Signature: {active/1,f/3,mark/1} / {a/0,b/0} - Obligation: derivational complexity wrt. signature {a,active,b,f,mark} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] [1] [0] [0] p(active) = [1 0 0 0] [0] [0 1 1 1] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(b) = [0] [0] [0] [1] p(f) = [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] x1 + [0 0 0 0] x2 + [0 0 0 0] x3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] p(mark) = [1 0 0 0] [0] [0 1 1 1] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] Following rules are strictly oriented: active(f(a(),X,X)) = [2 1 0 0] [2] [0 0 0 0] X + [2] [0 0 0 0] [0] [0 0 0 0] [0] > [1 1 0 0] [1] [0 0 0 0] X + [2] [0 0 0 0] [0] [0 0 0 0] [0] = mark(f(X,b(),b())) Following rules are (at-least) weakly oriented: active(b()) = [0] [2] [0] [0] >= [0] [2] [0] [0] = mark(a()) f(X1,X2,active(X3)) = [1 1 0 0] [1 0 0 0] [1 1 1 1] [2] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(X1,X2,mark(X3)) = [1 1 0 0] [1 0 0 0] [1 1 1 1] [2] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(X1,active(X2),X3) = [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(X1,mark(X2),X3) = [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(active(X1),X2,X3) = [1 1 1 1] [1 0 0 0] [1 1 0 0] [2] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(mark(X1),X2,X3) = [1 1 1 1] [1 0 0 0] [1 1 0 0] [2] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) mark(a()) = [0] [2] [0] [0] >= [0] [2] [0] [0] = active(a()) mark(b()) = [0] [2] [0] [0] >= [0] [2] [0] [0] = active(b()) mark(f(X1,X2,X3)) = [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [2] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 1 0 0] [1 0 0 0] [1 1 0 0] [1] [0 0 0 0] X1 + [0 0 0 0] X2 + [0 0 0 0] X3 + [2] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = active(f(X1,mark(X2),X3)) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) - Weak TRS: active(b()) -> mark(a()) active(f(a(),X,X)) -> mark(f(X,b(),b())) f(X1,X2,active(X3)) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(mark(X1),X2,X3) -> f(X1,X2,X3) mark(a()) -> active(a()) mark(b()) -> active(b()) - Signature: {active/1,f/3,mark/1} / {a/0,b/0} - Obligation: derivational complexity wrt. signature {a,active,b,f,mark} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] [0] [1] [0] p(active) = [1 1 0 0] [0] [0 0 1 1] x1 + [1] [0 0 1 1] [0] [0 0 0 0] [0] p(b) = [0] [1] [0] [1] p(f) = [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] [0 0 1 0] x1 + [0 0 0 0] x2 + [0 0 1 0] x3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] p(mark) = [1 0 1 1] [0] [0 0 1 1] x1 + [1] [0 0 1 1] [0] [0 0 0 0] [0] Following rules are strictly oriented: mark(f(X1,X2,X3)) = [1 0 2 0] [1 0 1 1] [1 0 2 0] [1] [0 0 1 0] X1 + [0 0 1 1] X2 + [0 0 1 0] X3 + [2] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] > [1 0 2 0] [1 0 1 1] [1 0 2 0] [0] [0 0 1 0] X1 + [0 0 1 1] X2 + [0 0 1 0] X3 + [2] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = active(f(X1,mark(X2),X3)) Following rules are (at-least) weakly oriented: active(b()) = [1] [2] [1] [0] >= [1] [2] [1] [0] = mark(a()) active(f(a(),X,X)) = [2 0 2 0] [2] [0 0 2 1] X + [3] [0 0 2 1] [2] [0 0 0 0] [0] >= [1 0 2 0] [2] [0 0 1 0] X + [3] [0 0 1 0] [2] [0 0 0 0] [0] = mark(f(X,b(),b())) f(X1,X2,active(X3)) = [1 0 1 0] [1 0 0 0] [1 1 1 1] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 1] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(X1,X2,mark(X3)) = [1 0 1 0] [1 0 0 0] [1 0 2 2] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 1] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(X1,active(X2),X3) = [1 0 1 0] [1 1 0 0] [1 0 1 0] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(X1,mark(X2),X3) = [1 0 1 0] [1 0 1 1] [1 0 1 0] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(active(X1),X2,X3) = [1 1 1 1] [1 0 0 0] [1 0 1 0] [0] [0 0 1 1] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 1] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) f(mark(X1),X2,X3) = [1 0 2 2] [1 0 0 0] [1 0 1 0] [0] [0 0 1 1] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 1] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] [0 0 1 0] X1 + [0 0 0 0] X2 + [0 0 1 0] X3 + [0] [0 0 1 0] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = f(X1,X2,X3) mark(a()) = [1] [2] [1] [0] >= [0] [2] [1] [0] = active(a()) mark(b()) = [1] [2] [1] [0] >= [1] [2] [1] [0] = active(b()) * Step 6: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(b()) -> mark(a()) active(f(a(),X,X)) -> mark(f(X,b(),b())) f(X1,X2,active(X3)) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(mark(X1),X2,X3) -> f(X1,X2,X3) mark(a()) -> active(a()) mark(b()) -> active(b()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) - Signature: {active/1,f/3,mark/1} / {a/0,b/0} - Obligation: derivational complexity wrt. signature {a,active,b,f,mark} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))