/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /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(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) proper(c()) -> ok(c()) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/1,g/1,proper/1,top/1} / {c/0,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,f,g,mark,ok,proper,top} + 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(active) = [1] x1 + [0] p(c) = [0] p(f) = [1] x1 + [5] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(ok) = [1] x1 + [7] p(proper) = [1] x1 + [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: active(f(g(X))) = [1] X + [5] > [1] X + [0] = mark(g(X)) top(ok(X)) = [1] X + [7] > [1] X + [0] = top(active(X)) Following rules are (at-least) weakly oriented: active(c()) = [0] >= [5] = mark(f(g(c()))) f(ok(X)) = [1] X + [12] >= [1] X + [12] = ok(f(X)) g(ok(X)) = [1] X + [7] >= [1] X + [7] = ok(g(X)) proper(c()) = [0] >= [7] = ok(c()) proper(f(X)) = [1] X + [5] >= [1] X + [5] = f(proper(X)) proper(g(X)) = [1] X + [0] >= [1] X + [0] = g(proper(X)) 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: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(c()) -> mark(f(g(c()))) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) proper(c()) -> ok(c()) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) top(mark(X)) -> top(proper(X)) - Weak TRS: active(f(g(X))) -> mark(g(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/1,g/1,proper/1,top/1} / {c/0,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,f,g,mark,ok,proper,top} + 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(active) = [1] x1 + [0] p(c) = [0] p(f) = [1] x1 + [1] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [1] 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 + [1] > [1] X + [0] = top(proper(X)) Following rules are (at-least) weakly oriented: active(c()) = [0] >= [2] = mark(f(g(c()))) active(f(g(X))) = [1] X + [1] >= [1] X + [1] = mark(g(X)) f(ok(X)) = [1] X + [1] >= [1] X + [1] = ok(f(X)) g(ok(X)) = [1] X + [0] >= [1] X + [0] = ok(g(X)) proper(c()) = [0] >= [0] = ok(c()) proper(f(X)) = [1] X + [1] >= [1] X + [1] = f(proper(X)) proper(g(X)) = [1] X + [0] >= [1] X + [0] = g(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 3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(c()) -> mark(f(g(c()))) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) proper(c()) -> ok(c()) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(f(g(X))) -> mark(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/1,g/1,proper/1,top/1} / {c/0,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,f,g,mark,ok,proper,top} + 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(active) = [1] x1 + [4] p(c) = [1] p(f) = [1] x1 + [0] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(ok) = [1] x1 + [4] p(proper) = [1] x1 + [0] p(top) = [1] x1 + [3] Following rules are strictly oriented: active(c()) = [5] > [1] = mark(f(g(c()))) Following rules are (at-least) weakly oriented: active(f(g(X))) = [1] X + [4] >= [1] X + [0] = mark(g(X)) f(ok(X)) = [1] X + [4] >= [1] X + [4] = ok(f(X)) g(ok(X)) = [1] X + [4] >= [1] X + [4] = ok(g(X)) proper(c()) = [1] >= [5] = ok(c()) proper(f(X)) = [1] X + [0] >= [1] X + [0] = f(proper(X)) proper(g(X)) = [1] X + [0] >= [1] X + [0] = g(proper(X)) top(mark(X)) = [1] X + [3] >= [1] X + [3] = top(proper(X)) top(ok(X)) = [1] X + [7] >= [1] X + [7] = top(active(X)) 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: f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) proper(c()) -> ok(c()) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/1,g/1,proper/1,top/1} / {c/0,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,f,g,mark,ok,proper,top} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, 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 1 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(active) = [1 0 2] [0] [0 0 0] x1 + [1] [0 0 0] [2] p(c) = [2] [0] [1] p(f) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] p(g) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(mark) = [1 0 0] [1] [0 0 2] x1 + [1] [0 0 0] [0] p(ok) = [1 0 0] [0] [0 0 2] x1 + [1] [0 0 0] [0] p(proper) = [1 0 0] [1] [0 0 2] x1 + [1] [0 0 0] [0] p(top) = [1 3 0] [0] [0 0 0] x1 + [2] [0 0 0] [0] Following rules are strictly oriented: proper(c()) = [3] [3] [0] > [2] [3] [0] = ok(c()) Following rules are (at-least) weakly oriented: active(c()) = [4] [1] [2] >= [4] [1] [0] = mark(f(g(c()))) active(f(g(X))) = [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [2] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [0] = mark(g(X)) f(ok(X)) = [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [0] = ok(f(X)) g(ok(X)) = [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = ok(g(X)) proper(f(X)) = [1 0 0] [2] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [2] [0 0 0] X + [1] [0 0 0] [0] = f(proper(X)) proper(g(X)) = [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [0] = g(proper(X)) top(mark(X)) = [1 0 6] [4] [0 0 0] X + [2] [0 0 0] [0] >= [1 0 6] [4] [0 0 0] X + [2] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 0 6] [3] [0 0 0] X + [2] [0 0 0] [0] >= [1 0 2] [3] [0 0 0] X + [2] [0 0 0] [0] = top(active(X)) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/1,g/1,proper/1,top/1} / {c/0,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,f,g,mark,ok,proper,top} + 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(active) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 1 1] [1] [0 0 0 0] [0] p(c) = [0] [0] [0] [1] p(f) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(g) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(mark) = [1 0 1 1] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(ok) = [1 0 0 0] [0] [0 0 0 1] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(proper) = [1 0 1 0] [0] [0 0 0 1] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(top) = [1 1 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] Following rules are strictly oriented: proper(f(X)) = [1 0 2 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] > [1 0 2 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] = f(proper(X)) Following rules are (at-least) weakly oriented: active(c()) = [1] [0] [2] [0] >= [1] [0] [2] [0] = mark(f(g(c()))) active(f(g(X))) = [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [2] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] = mark(g(X)) f(ok(X)) = [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] = ok(f(X)) g(ok(X)) = [1 0 0 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] = ok(g(X)) proper(c()) = [0] [1] [0] [0] >= [0] [1] [0] [0] = ok(c()) proper(g(X)) = [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [0] [0 0 0 0] [0] = g(proper(X)) top(mark(X)) = [1 0 1 1] [0] [0 0 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 1 1] [0] [0 0 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 0 0 1] [0] [0 0 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 1] [0] [0 0 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] = top(active(X)) * Step 6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) proper(c()) -> ok(c()) proper(f(X)) -> f(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/1,g/1,proper/1,top/1} / {c/0,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,f,g,mark,ok,proper,top} + 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(active) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c) = [0] [0] [0] [1] p(f) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(g) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(mark) = [1 0 1 0] [0] [0 0 0 1] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(ok) = [1 0 0 0] [0] [0 0 0 1] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(proper) = [1 0 1 0] [0] [0 0 0 1] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(top) = [1 1 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] Following rules are strictly oriented: f(ok(X)) = [1 0 1 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [2] [0 0 0 0] [0] > [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [2] [0 0 0 0] [0] = ok(f(X)) Following rules are (at-least) weakly oriented: active(c()) = [1] [0] [0] [0] >= [1] [0] [0] [0] = mark(f(g(c()))) active(f(g(X))) = [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = mark(g(X)) g(ok(X)) = [1 0 0 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] = ok(g(X)) proper(c()) = [0] [1] [1] [0] >= [0] [1] [1] [0] = ok(c()) proper(f(X)) = [1 0 2 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [2] [0 0 0 0] [0] >= [1 0 2 0] [1] [0 0 0 0] X + [0] [0 0 1 0] [2] [0 0 0 0] [0] = f(proper(X)) proper(g(X)) = [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 0] [1] [0 0 0 0] [0] = g(proper(X)) top(mark(X)) = [1 0 1 1] [0] [0 0 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 1] [0] [0 0 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 0 0 1] [0] [0 0 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 1] [0] [0 0 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = top(active(X)) * Step 7: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: g(ok(X)) -> ok(g(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) f(ok(X)) -> ok(f(X)) proper(c()) -> ok(c()) proper(f(X)) -> f(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/1,g/1,proper/1,top/1} / {c/0,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,f,g,mark,ok,proper,top} + Applied Processor: MI {miKind = Automaton (Just 2), miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton (Just 2): Following symbols are considered usable: all TcT has computed the following interpretation: p(active) = [1 0 0] [0] [0 0 0] x_1 + [0] [0 0 1] [0] p(c) = [0] [1] [6] p(f) = [1 0 1] [0] [0 1 0] x_1 + [2] [0 1 0] [0] p(g) = [1 0 0] [0] [0 1 0] x_1 + [1] [0 0 0] [0] p(mark) = [1 0 0] [0] [0 0 0] x_1 + [0] [0 1 1] [0] p(ok) = [1 0 0] [0] [0 1 0] x_1 + [0] [0 0 1] [0] p(proper) = [1 1 0] [0] [0 1 0] x_1 + [0] [0 0 1] [0] p(top) = [1 0 1] [6] [0 0 0] x_1 + [2] [0 0 0] [0] Following rules are strictly oriented: proper(g(X)) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] > [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = g(proper(X)) Following rules are (at-least) weakly oriented: active(c()) = [0] [0] [6] >= [0] [0] [6] = mark(f(g(c()))) active(f(g(X))) = [1 0 0] [0] [0 0 0] X + [0] [0 1 0] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 1 0] [1] = mark(g(X)) f(ok(X)) = [1 0 1] [0] [0 1 0] X + [2] [0 1 0] [0] >= [1 0 1] [0] [0 1 0] X + [2] [0 1 0] [0] = ok(f(X)) g(ok(X)) = [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(g(X)) proper(c()) = [1] [1] [6] >= [0] [1] [6] = ok(c()) proper(f(X)) = [1 1 1] [2] [0 1 0] X + [2] [0 1 0] [0] >= [1 1 1] [0] [0 1 0] X + [2] [0 1 0] [0] = f(proper(X)) top(mark(X)) = [1 1 1] [6] [0 0 0] X + [2] [0 0 0] [0] >= [1 1 1] [6] [0 0 0] X + [2] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 0 1] [6] [0 0 0] X + [2] [0 0 0] [0] >= [1 0 1] [6] [0 0 0] X + [2] [0 0 0] [0] = top(active(X)) * Step 8: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: g(ok(X)) -> ok(g(X)) - Weak TRS: active(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) f(ok(X)) -> ok(f(X)) proper(c()) -> ok(c()) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/1,g/1,proper/1,top/1} / {c/0,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,f,g,mark,ok,proper,top} + Applied Processor: MI {miKind = Automaton (Just 2), miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton (Just 2): Following symbols are considered usable: all TcT has computed the following interpretation: p(active) = [1 3 5] [0] [0 0 0] x_1 + [0] [0 0 0] [2] p(c) = [0] [0] [3] p(f) = [1 1 0] [0] [0 1 0] x_1 + [1] [0 0 1] [0] p(g) = [1 0 2] [0] [0 1 0] x_1 + [1] [0 1 0] [0] p(mark) = [1 4 5] [0] [0 0 0] x_1 + [0] [0 0 0] [2] p(ok) = [1 1 0] [3] [0 1 0] x_1 + [1] [0 0 1] [1] p(proper) = [1 2 0] [3] [0 1 0] x_1 + [1] [0 0 1] [1] p(top) = [1 2 5] [0] [0 0 0] x_1 + [3] [0 0 0] [0] Following rules are strictly oriented: g(ok(X)) = [1 1 2] [5] [0 1 0] X + [2] [0 1 0] [1] > [1 1 2] [4] [0 1 0] X + [2] [0 1 0] [1] = ok(g(X)) Following rules are (at-least) weakly oriented: active(c()) = [15] [0] [2] >= [15] [0] [2] = mark(f(g(c()))) active(f(g(X))) = [1 9 2] [7] [0 0 0] X + [0] [0 0 0] [2] >= [1 9 2] [4] [0 0 0] X + [0] [0 0 0] [2] = mark(g(X)) f(ok(X)) = [1 2 0] [4] [0 1 0] X + [2] [0 0 1] [1] >= [1 2 0] [4] [0 1 0] X + [2] [0 0 1] [1] = ok(f(X)) proper(c()) = [3] [1] [4] >= [3] [1] [4] = ok(c()) proper(f(X)) = [1 3 0] [5] [0 1 0] X + [2] [0 0 1] [1] >= [1 3 0] [4] [0 1 0] X + [2] [0 0 1] [1] = f(proper(X)) proper(g(X)) = [1 2 2] [5] [0 1 0] X + [2] [0 1 0] [1] >= [1 2 2] [5] [0 1 0] X + [2] [0 1 0] [1] = g(proper(X)) top(mark(X)) = [1 4 5] [10] [0 0 0] X + [3] [0 0 0] [0] >= [1 4 5] [10] [0 0 0] X + [3] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 3 5] [10] [0 0 0] X + [3] [0 0 0] [0] >= [1 3 5] [10] [0 0 0] X + [3] [0 0 0] [0] = top(active(X)) * Step 9: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) proper(c()) -> ok(c()) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/1,g/1,proper/1,top/1} / {c/0,mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,f,g,mark,ok,proper,top} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))