/export/starexec/sandbox/solver/bin/starexec_run_tct_dc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(c()) -> mark(d()) active(g(X)) -> mark(h(X)) active(h(d())) -> mark(g(c())) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) proper(c()) -> ok(c()) proper(d()) -> ok(d()) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,g/1,h/1,proper/1,top/1} / {c/0,d/0,mark/1,ok/1} - Obligation: derivational complexity wrt. signature {active,c,d,g,h,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(d) = [0] p(g) = [1] x1 + [0] p(h) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(ok) = [1] x1 + [9] p(proper) = [1] x1 + [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: top(ok(X)) = [1] X + [9] > [1] X + [0] = top(active(X)) Following rules are (at-least) weakly oriented: active(c()) = [0] >= [0] = mark(d()) active(g(X)) = [1] X + [0] >= [1] X + [0] = mark(h(X)) active(h(d())) = [0] >= [0] = mark(g(c())) g(ok(X)) = [1] X + [9] >= [1] X + [9] = ok(g(X)) h(ok(X)) = [1] X + [9] >= [1] X + [9] = ok(h(X)) proper(c()) = [0] >= [9] = ok(c()) proper(d()) = [0] >= [9] = ok(d()) proper(g(X)) = [1] X + [0] >= [1] X + [0] = g(proper(X)) proper(h(X)) = [1] X + [0] >= [1] X + [0] = h(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(d()) active(g(X)) -> mark(h(X)) active(h(d())) -> mark(g(c())) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) proper(c()) -> ok(c()) proper(d()) -> ok(d()) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) - Weak TRS: top(ok(X)) -> top(active(X)) - Signature: {active/1,g/1,h/1,proper/1,top/1} / {c/0,d/0,mark/1,ok/1} - Obligation: derivational complexity wrt. signature {active,c,d,g,h,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) = [9] p(d) = [0] p(g) = [1] x1 + [0] p(h) = [1] x1 + [0] p(mark) = [1] x1 + [5] p(ok) = [1] x1 + [0] p(proper) = [1] x1 + [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: active(c()) = [9] > [5] = mark(d()) top(mark(X)) = [1] X + [5] > [1] X + [0] = top(proper(X)) Following rules are (at-least) weakly oriented: active(g(X)) = [1] X + [0] >= [1] X + [5] = mark(h(X)) active(h(d())) = [0] >= [14] = mark(g(c())) g(ok(X)) = [1] X + [0] >= [1] X + [0] = ok(g(X)) h(ok(X)) = [1] X + [0] >= [1] X + [0] = ok(h(X)) proper(c()) = [9] >= [9] = ok(c()) proper(d()) = [0] >= [0] = ok(d()) proper(g(X)) = [1] X + [0] >= [1] X + [0] = g(proper(X)) proper(h(X)) = [1] X + [0] >= [1] X + [0] = h(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(g(X)) -> mark(h(X)) active(h(d())) -> mark(g(c())) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) proper(c()) -> ok(c()) proper(d()) -> ok(d()) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c()) -> mark(d()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,g/1,h/1,proper/1,top/1} / {c/0,d/0,mark/1,ok/1} - Obligation: derivational complexity wrt. signature {active,c,d,g,h,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 + [6] p(c) = [0] p(d) = [0] p(g) = [1] x1 + [0] p(h) = [1] x1 + [2] p(mark) = [1] x1 + [2] p(ok) = [1] x1 + [12] p(proper) = [1] x1 + [0] p(top) = [1] x1 + [1] Following rules are strictly oriented: active(g(X)) = [1] X + [6] > [1] X + [4] = mark(h(X)) active(h(d())) = [8] > [2] = mark(g(c())) Following rules are (at-least) weakly oriented: active(c()) = [6] >= [2] = mark(d()) g(ok(X)) = [1] X + [12] >= [1] X + [12] = ok(g(X)) h(ok(X)) = [1] X + [14] >= [1] X + [14] = ok(h(X)) proper(c()) = [0] >= [12] = ok(c()) proper(d()) = [0] >= [12] = ok(d()) proper(g(X)) = [1] X + [0] >= [1] X + [0] = g(proper(X)) proper(h(X)) = [1] X + [2] >= [1] X + [2] = h(proper(X)) top(mark(X)) = [1] X + [3] >= [1] X + [1] = top(proper(X)) top(ok(X)) = [1] X + [13] >= [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: g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) proper(c()) -> ok(c()) proper(d()) -> ok(d()) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c()) -> mark(d()) active(g(X)) -> mark(h(X)) active(h(d())) -> mark(g(c())) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,g/1,h/1,proper/1,top/1} / {c/0,d/0,mark/1,ok/1} - Obligation: derivational complexity wrt. signature {active,c,d,g,h,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 + [2] [0 0 0] [1] p(c) = [0] [0] [3] p(d) = [2] [0] [1] p(g) = [1 0 0] [1] [0 0 2] x1 + [0] [0 0 0] [2] p(h) = [1 0 0] [0] [0 0 2] x1 + [0] [0 0 0] [2] p(mark) = [1 0 2] [1] [0 0 0] x1 + [2] [0 0 0] [1] p(ok) = [1 0 0] [0] [0 0 2] x1 + [0] [0 0 0] [2] p(proper) = [1 0 0] [1] [0 0 2] x1 + [0] [0 0 0] [2] p(top) = [1 1 2] [0] [0 0 0] x1 + [0] [0 0 0] [2] Following rules are strictly oriented: proper(c()) = [1] [6] [2] > [0] [6] [2] = ok(c()) proper(d()) = [3] [2] [2] > [2] [2] [2] = ok(d()) Following rules are (at-least) weakly oriented: active(c()) = [6] [2] [1] >= [5] [2] [1] = mark(d()) active(g(X)) = [1 0 0] [5] [0 0 0] X + [2] [0 0 0] [1] >= [1 0 0] [5] [0 0 0] X + [2] [0 0 0] [1] = mark(h(X)) active(h(d())) = [6] [2] [1] >= [6] [2] [1] = mark(g(c())) g(ok(X)) = [1 0 0] [1] [0 0 0] X + [4] [0 0 0] [2] >= [1 0 0] [1] [0 0 0] X + [4] [0 0 0] [2] = ok(g(X)) h(ok(X)) = [1 0 0] [0] [0 0 0] X + [4] [0 0 0] [2] >= [1 0 0] [0] [0 0 0] X + [4] [0 0 0] [2] = ok(h(X)) proper(g(X)) = [1 0 0] [2] [0 0 0] X + [4] [0 0 0] [2] >= [1 0 0] [2] [0 0 0] X + [4] [0 0 0] [2] = g(proper(X)) proper(h(X)) = [1 0 0] [1] [0 0 0] X + [4] [0 0 0] [2] >= [1 0 0] [1] [0 0 0] X + [4] [0 0 0] [2] = h(proper(X)) top(mark(X)) = [1 0 2] [5] [0 0 0] X + [0] [0 0 0] [2] >= [1 0 2] [5] [0 0 0] X + [0] [0 0 0] [2] = top(proper(X)) top(ok(X)) = [1 0 2] [4] [0 0 0] X + [0] [0 0 0] [2] >= [1 0 2] [4] [0 0 0] X + [0] [0 0 0] [2] = top(active(X)) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c()) -> mark(d()) active(g(X)) -> mark(h(X)) active(h(d())) -> mark(g(c())) proper(c()) -> ok(c()) proper(d()) -> ok(d()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,g/1,h/1,proper/1,top/1} / {c/0,d/0,mark/1,ok/1} - Obligation: derivational complexity wrt. signature {active,c,d,g,h,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] [1] p(c) = [0] [0] [0] [1] p(d) = [1] [0] [0] [0] p(g) = [1 1 0 0] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(h) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(mark) = [1 1 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(ok) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(proper) = [1 1 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(top) = [1 0 1 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] Following rules are strictly oriented: proper(g(X)) = [1 2 0 0] [1] [0 1 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] > [1 2 0 0] [0] [0 1 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] = g(proper(X)) Following rules are (at-least) weakly oriented: active(c()) = [1] [0] [0] [1] >= [1] [0] [0] [0] = mark(d()) active(g(X)) = [1 1 0 0] [0] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [0] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = mark(h(X)) active(h(d())) = [1] [0] [0] [1] >= [1] [0] [0] [0] = mark(g(c())) g(ok(X)) = [1 1 0 0] [0] [0 1 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] = ok(g(X)) h(ok(X)) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = ok(h(X)) proper(c()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = ok(c()) proper(d()) = [1] [0] [0] [0] >= [1] [0] [0] [0] = ok(d()) proper(h(X)) = [1 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 1 0 1] [1] [0 0 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 1] [1] [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] [1] [0 0 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 1] [1] [0 0 0 0] X + [1] [0 0 0 0] [1] [0 0 0 0] [0] = top(active(X)) * Step 6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c()) -> mark(d()) active(g(X)) -> mark(h(X)) active(h(d())) -> mark(g(c())) proper(c()) -> ok(c()) proper(d()) -> ok(d()) proper(g(X)) -> g(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,g/1,h/1,proper/1,top/1} / {c/0,d/0,mark/1,ok/1} - Obligation: derivational complexity wrt. signature {active,c,d,g,h,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(d) = [1] [0] [0] [0] p(g) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(h) = [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 0 0] [0] [0 0 0 0] [0] p(ok) = [1 1 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 + [0] [0 0 0 0] [1] [0 0 0 0] [1] Following rules are strictly oriented: g(ok(X)) = [1 1 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(g(X)) Following rules are (at-least) weakly oriented: active(c()) = [1] [0] [0] [0] >= [1] [0] [0] [0] = mark(d()) active(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(h(X)) active(h(d())) = [1] [0] [0] [0] >= [1] [0] [0] [0] = mark(g(c())) h(ok(X)) = [1 1 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(h(X)) proper(c()) = [0] [1] [1] [0] >= [0] [1] [1] [0] = ok(c()) proper(d()) = [1] [0] [1] [0] >= [1] [0] [1] [0] = ok(d()) proper(g(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] = g(proper(X)) proper(h(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] = h(proper(X)) top(mark(X)) = [1 0 1 1] [0] [0 0 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [1 0 1 1] [0] [0 0 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] = top(proper(X)) top(ok(X)) = [1 1 0 1] [0] [0 0 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [1 0 0 1] [0] [0 0 0 0] X + [0] [0 0 0 0] [1] [0 0 0 0] [1] = top(active(X)) * Step 7: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: h(ok(X)) -> ok(h(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c()) -> mark(d()) active(g(X)) -> mark(h(X)) active(h(d())) -> mark(g(c())) g(ok(X)) -> ok(g(X)) proper(c()) -> ok(c()) proper(d()) -> ok(d()) proper(g(X)) -> g(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,g/1,h/1,proper/1,top/1} / {c/0,d/0,mark/1,ok/1} - Obligation: derivational complexity wrt. signature {active,c,d,g,h,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 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c) = [0] [0] [0] [0] p(d) = [0] [0] [0] [1] p(g) = [1 1 1 1] [1] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(h) = [1 0 1 1] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(mark) = [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(ok) = [1 0 1 0] [0] [0 1 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(proper) = [1 1 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(top) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] Following rules are strictly oriented: proper(h(X)) = [1 1 1 1] [1] [0 1 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] > [1 1 0 1] [0] [0 1 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] = h(proper(X)) Following rules are (at-least) weakly oriented: active(c()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = mark(d()) active(g(X)) = [1 1 1 1] [1] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 1 1] [1] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = mark(h(X)) active(h(d())) = [1] [0] [0] [0] >= [1] [0] [0] [0] = mark(g(c())) g(ok(X)) = [1 1 1 1] [1] [0 1 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 1 1] [1] [0 1 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = ok(g(X)) h(ok(X)) = [1 0 1 1] [0] [0 1 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 1 1] [0] [0 1 0 0] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] = ok(h(X)) proper(c()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = ok(c()) proper(d()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = ok(d()) proper(g(X)) = [1 2 1 1] [1] [0 1 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 2 0 1] [1] [0 1 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = g(proper(X)) top(mark(X)) = [1 1 0 0] [0] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [0] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [1] = top(proper(X)) top(ok(X)) = [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 0 0 0] [0] [0 0 0 0] X + [0] [0 0 0 0] [0] [0 0 0 0] [1] = top(active(X)) * Step 8: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: h(ok(X)) -> ok(h(X)) - Weak TRS: active(c()) -> mark(d()) active(g(X)) -> mark(h(X)) active(h(d())) -> mark(g(c())) g(ok(X)) -> ok(g(X)) proper(c()) -> ok(c()) proper(d()) -> ok(d()) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,g/1,h/1,proper/1,top/1} / {c/0,d/0,mark/1,ok/1} - Obligation: derivational complexity wrt. signature {active,c,d,g,h,mark,ok,proper,top} + Applied Processor: NaturalMI {miDimension = 3, 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 2] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(c) = [3] [0] [0] p(d) = [0] [3] [0] p(g) = [1 3 0] [0] [0 1 0] x1 + [3] [0 0 0] [1] p(h) = [1 2 0] [0] [0 1 0] x1 + [2] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(ok) = [1 0 2] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(proper) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [2] p(top) = [1 0 0] [0] [0 0 0] x1 + [2] [0 0 0] [2] Following rules are strictly oriented: h(ok(X)) = [1 2 2] [2] [0 1 0] X + [3] [0 0 0] [0] > [1 2 0] [0] [0 1 0] X + [3] [0 0 0] [0] = ok(h(X)) Following rules are (at-least) weakly oriented: active(c()) = [3] [0] [0] >= [3] [0] [0] = mark(d()) active(g(X)) = [1 3 0] [2] [0 1 0] X + [3] [0 0 0] [0] >= [1 3 0] [2] [0 0 0] X + [0] [0 0 0] [0] = mark(h(X)) active(h(d())) = [6] [5] [0] >= [6] [0] [0] = mark(g(c())) g(ok(X)) = [1 3 2] [3] [0 1 0] X + [4] [0 0 0] [1] >= [1 3 0] [2] [0 1 0] X + [4] [0 0 0] [0] = ok(g(X)) proper(c()) = [3] [1] [2] >= [3] [1] [0] = ok(c()) proper(d()) = [3] [4] [2] >= [0] [4] [0] = ok(d()) proper(g(X)) = [1 4 0] [3] [0 1 0] X + [4] [0 0 0] [2] >= [1 4 0] [3] [0 1 0] X + [4] [0 0 0] [1] = g(proper(X)) proper(h(X)) = [1 3 0] [2] [0 1 0] X + [3] [0 0 0] [2] >= [1 3 0] [2] [0 1 0] X + [3] [0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 1 0] [0] [0 0 0] X + [2] [0 0 0] [2] >= [1 1 0] [0] [0 0 0] X + [2] [0 0 0] [2] = top(proper(X)) top(ok(X)) = [1 0 2] [0] [0 0 0] X + [2] [0 0 0] [2] >= [1 0 2] [0] [0 0 0] X + [2] [0 0 0] [2] = top(active(X)) * Step 9: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(c()) -> mark(d()) active(g(X)) -> mark(h(X)) active(h(d())) -> mark(g(c())) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) proper(c()) -> ok(c()) proper(d()) -> ok(d()) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,g/1,h/1,proper/1,top/1} / {c/0,d/0,mark/1,ok/1} - Obligation: derivational complexity wrt. signature {active,c,d,g,h,mark,ok,proper,top} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))