/export/starexec/sandbox/solver/bin/starexec_run_tct_dci /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(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) 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,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 + [4] p(c) = [1] x1 + [0] p(d) = [1] x1 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [0] p(h) = [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(c(X)) = [1] X + [4] > [1] X + [0] = mark(d(X)) active(f(f(X))) = [1] X + [4] > [1] X + [0] = mark(c(f(g(f(X))))) active(h(X)) = [1] X + [4] > [1] X + [0] = mark(c(d(X))) top(ok(X)) = [1] X + [7] > [1] X + [4] = top(active(X)) Following rules are (at-least) weakly oriented: active(f(X)) = [1] X + [4] >= [1] X + [4] = f(active(X)) active(h(X)) = [1] X + [4] >= [1] X + [4] = h(active(X)) c(ok(X)) = [1] X + [7] >= [1] X + [7] = ok(c(X)) d(ok(X)) = [1] X + [7] >= [1] X + [7] = ok(d(X)) f(mark(X)) = [1] X + [0] >= [1] X + [0] = mark(f(X)) f(ok(X)) = [1] X + [7] >= [1] X + [7] = ok(f(X)) g(ok(X)) = [1] X + [7] >= [1] X + [7] = ok(g(X)) h(mark(X)) = [1] X + [0] >= [1] X + [0] = mark(h(X)) h(ok(X)) = [1] X + [7] >= [1] X + [7] = ok(h(X)) proper(c(X)) = [1] X + [0] >= [1] X + [0] = c(proper(X)) proper(d(X)) = [1] X + [0] >= [1] X + [0] = d(proper(X)) proper(f(X)) = [1] X + [0] >= [1] X + [0] = f(proper(X)) 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(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> mark(c(d(X))) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 + [5] p(c) = [1] x1 + [0] p(d) = [1] x1 + [2] p(f) = [1] x1 + [2] p(g) = [1] x1 + [4] p(h) = [1] x1 + [0] p(mark) = [1] x1 + [1] p(ok) = [1] x1 + [5] 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(X)) = [1] X + [5] >= [1] X + [3] = mark(d(X)) active(f(X)) = [1] X + [7] >= [1] X + [7] = f(active(X)) active(f(f(X))) = [1] X + [9] >= [1] X + [9] = mark(c(f(g(f(X))))) active(h(X)) = [1] X + [5] >= [1] X + [5] = h(active(X)) active(h(X)) = [1] X + [5] >= [1] X + [3] = mark(c(d(X))) c(ok(X)) = [1] X + [5] >= [1] X + [5] = ok(c(X)) d(ok(X)) = [1] X + [7] >= [1] X + [7] = ok(d(X)) f(mark(X)) = [1] X + [3] >= [1] X + [3] = mark(f(X)) f(ok(X)) = [1] X + [7] >= [1] X + [7] = ok(f(X)) g(ok(X)) = [1] X + [9] >= [1] X + [9] = ok(g(X)) h(mark(X)) = [1] X + [1] >= [1] X + [1] = mark(h(X)) h(ok(X)) = [1] X + [5] >= [1] X + [5] = ok(h(X)) proper(c(X)) = [1] X + [0] >= [1] X + [0] = c(proper(X)) proper(d(X)) = [1] X + [2] >= [1] X + [2] = d(proper(X)) proper(f(X)) = [1] X + [2] >= [1] X + [2] = f(proper(X)) proper(g(X)) = [1] X + [4] >= [1] X + [4] = g(proper(X)) proper(h(X)) = [1] X + [0] >= [1] X + [0] = h(proper(X)) top(ok(X)) = [1] X + [5] >= [1] X + [5] = top(active(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> mark(c(d(X))) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(c) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(d) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(f) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(g) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(h) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(mark) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(ok) = [1 1 1] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(proper) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(top) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] Following rules are strictly oriented: h(mark(X)) = [1 1 0] [1] [0 1 0] X + [2] [0 0 0] [0] > [1 1 0] [0] [0 1 0] X + [2] [0 0 0] [0] = mark(h(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] = mark(d(X)) active(f(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = f(active(X)) active(f(f(X))) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] = mark(c(f(g(f(X))))) active(h(X)) = [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] >= [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] = h(active(X)) active(h(X)) = [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] = mark(c(d(X))) c(ok(X)) = [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(c(X)) d(ok(X)) = [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(d(X)) f(mark(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] = mark(f(X)) f(ok(X)) = [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(f(X)) g(ok(X)) = [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(g(X)) h(ok(X)) = [1 2 1] [1] [0 1 0] X + [2] [0 0 0] [0] >= [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] = ok(h(X)) proper(c(X)) = [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = c(proper(X)) proper(d(X)) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = d(proper(X)) proper(f(X)) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = f(proper(X)) proper(g(X)) = [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = g(proper(X)) proper(h(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 0 0] [0] [0 1 0] X + [2] [0 0 0] [1] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [1] = top(proper(X)) top(ok(X)) = [1 1 1] [0] [0 1 0] X + [2] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [2] [0 0 0] [1] = top(active(X)) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> mark(c(d(X))) h(mark(X)) -> mark(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 1] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(c) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(d) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(f) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(g) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(h) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(mark) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(ok) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(proper) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(top) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: active(f(X)) = [1 0 2] [1] [0 0 1] X + [1] [0 0 1] [1] > [1 0 2] [0] [0 0 1] X + [0] [0 0 1] [1] = f(active(X)) g(ok(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] > [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(g(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(d(X)) active(f(f(X))) = [1 0 3] [3] [0 0 1] X + [2] [0 0 1] [2] >= [1 0 3] [2] [0 0 0] X + [0] [0 0 1] [2] = mark(c(f(g(f(X))))) active(h(X)) = [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] = h(active(X)) active(h(X)) = [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(c(d(X))) c(ok(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(c(X)) d(ok(X)) = [1 0 1] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(d(X)) f(mark(X)) = [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = mark(f(X)) f(ok(X)) = [1 0 2] [1] [0 0 1] X + [1] [0 0 1] [2] >= [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [2] = ok(f(X)) h(mark(X)) = [1 0 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(h(X)) h(ok(X)) = [1 0 1] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(h(X)) proper(c(X)) = [1 0 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = c(proper(X)) proper(d(X)) = [1 0 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 1] X + [0] [0 0 1] [0] = d(proper(X)) proper(f(X)) = [1 0 1] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [1] = f(proper(X)) proper(g(X)) = [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = g(proper(X)) proper(h(X)) = [1 0 0] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 1] X + [0] [0 0 1] [0] = h(proper(X)) top(mark(X)) = [1 0 0] [0] [0 0 1] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 1] X + [0] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 0 1] [0] [0 0 1] X + [1] [0 0 0] [0] >= [1 0 1] [0] [0 0 1] X + [0] [0 0 0] [0] = top(active(X)) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(h(X)) -> h(active(X)) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) h(ok(X)) -> ok(h(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> mark(c(d(X))) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(c) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(d) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(f) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(g) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(h) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(ok) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(proper) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(top) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: h(ok(X)) = [1 1 1] [1] [0 0 0] X + [0] [0 0 1] [1] > [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(h(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(d(X)) active(f(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = f(active(X)) active(f(f(X))) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(c(f(g(f(X))))) active(h(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = h(active(X)) active(h(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(c(d(X))) c(ok(X)) = [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(c(X)) d(ok(X)) = [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(d(X)) f(mark(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(f(X)) f(ok(X)) = [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(f(X)) g(ok(X)) = [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(g(X)) h(mark(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(h(X)) proper(c(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = c(proper(X)) proper(d(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = d(proper(X)) proper(f(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = f(proper(X)) proper(g(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = g(proper(X)) proper(h(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 1 1] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = top(active(X)) * Step 6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(h(X)) -> h(active(X)) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> mark(c(d(X))) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(c) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(d) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(f) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(g) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(h) = [1 1 0] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(mark) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(ok) = [1 1 0] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(proper) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(top) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] Following rules are strictly oriented: active(h(X)) = [1 2 1] [1] [0 1 1] X + [1] [0 0 0] [0] > [1 2 0] [0] [0 1 0] X + [1] [0 0 0] [0] = h(active(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(d(X)) active(f(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = f(active(X)) active(f(f(X))) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(c(f(g(f(X))))) active(h(X)) = [1 2 1] [1] [0 1 1] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(c(d(X))) c(ok(X)) = [1 1 0] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(c(X)) d(ok(X)) = [1 1 0] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(d(X)) f(mark(X)) = [1 0 0] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(f(X)) f(ok(X)) = [1 1 0] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(f(X)) g(ok(X)) = [1 2 1] [1] [0 1 1] X + [1] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(g(X)) h(mark(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 1] X + [1] [0 0 0] [0] = mark(h(X)) h(ok(X)) = [1 2 1] [1] [0 1 1] X + [2] [0 0 0] [0] >= [1 2 1] [1] [0 1 1] X + [2] [0 0 0] [0] = ok(h(X)) proper(c(X)) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = c(proper(X)) proper(d(X)) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = d(proper(X)) proper(f(X)) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = f(proper(X)) proper(g(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = g(proper(X)) proper(h(X)) = [1 1 0] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] X + [1] [0 0 0] [1] = top(proper(X)) top(ok(X)) = [1 1 0] [1] [0 0 0] X + [1] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] X + [1] [0 0 0] [1] = top(active(X)) * Step 7: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(c) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(d) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(f) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(g) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(h) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(ok) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(proper) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(top) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: d(ok(X)) = [1 0 2] [2] [0 0 0] X + [0] [0 0 1] [1] > [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] = ok(d(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] = mark(d(X)) active(f(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] = f(active(X)) active(f(f(X))) = [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] = mark(c(f(g(f(X))))) active(h(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] = h(active(X)) active(h(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] = mark(c(d(X))) c(ok(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(c(X)) f(mark(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = mark(f(X)) f(ok(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(f(X)) g(ok(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] = ok(g(X)) h(mark(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = mark(h(X)) h(ok(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(h(X)) proper(c(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = c(proper(X)) proper(d(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] = d(proper(X)) proper(f(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = f(proper(X)) proper(g(X)) = [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] = g(proper(X)) proper(h(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] = top(active(X)) * Step 8: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: c(ok(X)) -> ok(c(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) d(ok(X)) -> ok(d(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(c) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(d) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(f) = [1 1 0] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(g) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(h) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(mark) = [1 1 1] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(ok) = [1 1 1] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(proper) = [1 1 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(top) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: proper(f(X)) = [1 2 1] [1] [0 1 1] X + [1] [0 0 0] [0] > [1 2 1] [0] [0 1 1] X + [1] [0 0 0] [0] = f(proper(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] = mark(d(X)) active(f(X)) = [1 2 1] [1] [0 1 1] X + [2] [0 0 0] [0] >= [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] = f(active(X)) active(f(f(X))) = [1 3 2] [3] [0 1 1] X + [3] [0 0 0] [0] >= [1 3 2] [3] [0 1 1] X + [3] [0 0 0] [0] = mark(c(f(g(f(X))))) active(h(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = h(active(X)) active(h(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] = mark(c(d(X))) c(ok(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] = ok(c(X)) d(ok(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] = ok(d(X)) f(mark(X)) = [1 2 2] [1] [0 1 1] X + [2] [0 0 0] [0] >= [1 2 1] [1] [0 1 1] X + [2] [0 0 0] [0] = mark(f(X)) f(ok(X)) = [1 2 2] [1] [0 1 1] X + [2] [0 0 0] [0] >= [1 2 1] [1] [0 1 1] X + [2] [0 0 0] [0] = ok(f(X)) g(ok(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] = ok(g(X)) h(mark(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] = mark(h(X)) h(ok(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] = ok(h(X)) proper(c(X)) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 1] X + [0] [0 0 0] [0] = c(proper(X)) proper(d(X)) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 1] X + [0] [0 0 0] [0] = d(proper(X)) proper(g(X)) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 1] X + [0] [0 0 0] [0] = g(proper(X)) proper(h(X)) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 1] X + [0] [0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 1 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 1 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = top(active(X)) * Step 9: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: c(ok(X)) -> ok(c(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) d(ok(X)) -> ok(d(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(f(X)) -> f(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(c) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(d) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(f) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(g) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(h) = [1 1 0] [1] [0 1 0] x1 + [1] [0 0 0] [1] p(mark) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(ok) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(proper) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(top) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: proper(h(X)) = [1 2 0] [2] [0 1 0] X + [1] [0 0 0] [1] > [1 2 0] [1] [0 1 0] X + [1] [0 0 0] [1] = h(proper(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = mark(d(X)) active(f(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = f(active(X)) active(f(f(X))) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = mark(c(f(g(f(X))))) active(h(X)) = [1 2 0] [2] [0 1 0] X + [2] [0 0 0] [1] >= [1 2 0] [2] [0 1 0] X + [2] [0 0 0] [1] = h(active(X)) active(h(X)) = [1 2 0] [2] [0 1 0] X + [2] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = mark(c(d(X))) c(ok(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = ok(c(X)) d(ok(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = ok(d(X)) f(mark(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = mark(f(X)) f(ok(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = ok(f(X)) g(ok(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = ok(g(X)) h(mark(X)) = [1 2 0] [2] [0 1 0] X + [2] [0 0 0] [1] >= [1 2 0] [2] [0 1 0] X + [2] [0 0 0] [1] = mark(h(X)) h(ok(X)) = [1 2 0] [2] [0 1 0] X + [2] [0 0 0] [1] >= [1 2 0] [2] [0 1 0] X + [2] [0 0 0] [1] = ok(h(X)) proper(c(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = c(proper(X)) proper(d(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = d(proper(X)) proper(f(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = f(proper(X)) proper(g(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = g(proper(X)) top(mark(X)) = [1 1 0] [2] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [2] [0 0 0] X + [0] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 1 0] [2] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [2] [0 0 0] X + [0] [0 0 0] [0] = top(active(X)) * Step 10: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: c(ok(X)) -> ok(c(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) d(ok(X)) -> ok(d(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(f(X)) -> f(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(c) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(d) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(f) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(g) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(h) = [1 0 0] [1] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(ok) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(proper) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(top) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: c(ok(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] = ok(c(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 2 0] [0] [0 0 0] X + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = mark(d(X)) active(f(X)) = [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = f(active(X)) active(f(f(X))) = [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = mark(c(f(g(f(X))))) active(h(X)) = [1 1 0] [1] [0 0 0] X + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] X + [0] [0 0 0] [0] = h(active(X)) active(h(X)) = [1 1 0] [1] [0 0 0] X + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = mark(c(d(X))) d(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(d(X)) f(mark(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = mark(f(X)) f(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(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)) h(mark(X)) = [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] = mark(h(X)) h(ok(X)) = [1 0 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [1 0 0] [1] [0 1 0] X + [1] [0 0 0] [0] = ok(h(X)) proper(c(X)) = [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = c(proper(X)) proper(d(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = d(proper(X)) proper(f(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = f(proper(X)) proper(g(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = g(proper(X)) proper(h(X)) = [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 1 0] [2] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] X + [0] [0 0 0] [0] = top(active(X)) * Step 11: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(f(X)) -> f(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(c) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(d) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(f) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(g) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(h) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(ok) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(proper) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(top) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: f(ok(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] > [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(f(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(d(X)) active(f(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = f(active(X)) active(f(f(X))) = [1 0 3] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 3] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(c(f(g(f(X))))) active(h(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = h(active(X)) active(h(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(c(d(X))) c(ok(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(c(X)) d(ok(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(d(X)) f(mark(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(f(X)) g(ok(X)) = [1 0 1] [0] [0 0 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(g(X)) h(mark(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(h(X)) h(ok(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = ok(h(X)) proper(c(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = c(proper(X)) proper(d(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = d(proper(X)) proper(f(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = f(proper(X)) proper(g(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = g(proper(X)) proper(h(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 0 1] [0] [0 0 1] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 0 2] [1] [0 0 1] X + [1] [0 0 1] [1] >= [1 0 2] [0] [0 0 1] X + [0] [0 0 1] [0] = top(active(X)) * Step 12: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(mark(X)) -> mark(f(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(f(X)) -> f(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(c) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(d) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(f) = [1 1 0] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(g) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(h) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(mark) = [1 0 1] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(ok) = [1 1 1] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(proper) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(top) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: f(mark(X)) = [1 1 2] [1] [0 1 1] X + [2] [0 0 0] [0] > [1 1 0] [0] [0 1 1] X + [2] [0 0 0] [0] = mark(f(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 0 1] [0] [0 1 0] X + [1] [0 0 0] [0] = mark(d(X)) active(f(X)) = [1 2 1] [1] [0 1 1] X + [2] [0 0 0] [0] >= [1 2 0] [1] [0 1 0] X + [2] [0 0 0] [0] = f(active(X)) active(f(f(X))) = [1 3 2] [3] [0 1 1] X + [3] [0 0 0] [0] >= [1 2 1] [1] [0 1 1] X + [3] [0 0 0] [0] = mark(c(f(g(f(X))))) active(h(X)) = [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = h(active(X)) active(h(X)) = [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 0 1] [0] [0 1 0] X + [1] [0 0 0] [0] = mark(c(d(X))) c(ok(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(c(X)) d(ok(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(d(X)) f(ok(X)) = [1 2 2] [1] [0 1 1] X + [2] [0 0 0] [0] >= [1 2 1] [1] [0 1 1] X + [2] [0 0 0] [0] = ok(f(X)) g(ok(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(g(X)) h(mark(X)) = [1 0 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 0 1] [0] [0 1 0] X + [1] [0 0 0] [0] = mark(h(X)) h(ok(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 1] [0] [0 1 0] X + [1] [0 0 0] [0] = ok(h(X)) proper(c(X)) = [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = c(proper(X)) proper(d(X)) = [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = d(proper(X)) proper(f(X)) = [1 1 0] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = f(proper(X)) proper(g(X)) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = g(proper(X)) proper(h(X)) = [1 0 1] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = h(proper(X)) top(mark(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [0] = top(proper(X)) top(ok(X)) = [1 1 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = top(active(X)) * Step 13: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(f(X)) -> f(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,g,h,mark,ok,proper,top} + Applied Processor: MI {miKind = Automaton Nothing, miDimension = 2, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton Nothing: Following symbols are considered usable: all TcT has computed the following interpretation: p(active) = [1 3] x_1 + [2] [0 0] [5] p(c) = [1 0] x_1 + [0] [0 1] [0] p(d) = [1 1] x_1 + [0] [0 1] [2] p(f) = [1 0] x_1 + [0] [0 1] [0] p(g) = [1 2] x_1 + [0] [0 1] [0] p(h) = [1 0] x_1 + [0] [0 1] [0] p(mark) = [1 1] x_1 + [0] [0 0] [5] p(ok) = [1 3] x_1 + [2] [0 1] [6] p(proper) = [1 1] x_1 + [0] [0 1] [0] p(top) = [1 0] x_1 + [0] [0 0] [7] Following rules are strictly oriented: proper(d(X)) = [1 2] X + [2] [0 1] [2] > [1 2] X + [0] [0 1] [2] = d(proper(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 3] X + [2] [0 0] [5] >= [1 2] X + [2] [0 0] [5] = mark(d(X)) active(f(X)) = [1 3] X + [2] [0 0] [5] >= [1 3] X + [2] [0 0] [5] = f(active(X)) active(f(f(X))) = [1 3] X + [2] [0 0] [5] >= [1 3] X + [0] [0 0] [5] = mark(c(f(g(f(X))))) active(h(X)) = [1 3] X + [2] [0 0] [5] >= [1 3] X + [2] [0 0] [5] = h(active(X)) active(h(X)) = [1 3] X + [2] [0 0] [5] >= [1 2] X + [2] [0 0] [5] = mark(c(d(X))) c(ok(X)) = [1 3] X + [2] [0 1] [6] >= [1 3] X + [2] [0 1] [6] = ok(c(X)) d(ok(X)) = [1 4] X + [8] [0 1] [8] >= [1 4] X + [8] [0 1] [8] = ok(d(X)) f(mark(X)) = [1 1] X + [0] [0 0] [5] >= [1 1] X + [0] [0 0] [5] = mark(f(X)) f(ok(X)) = [1 3] X + [2] [0 1] [6] >= [1 3] X + [2] [0 1] [6] = ok(f(X)) g(ok(X)) = [1 5] X + [14] [0 1] [6] >= [1 5] X + [2] [0 1] [6] = ok(g(X)) h(mark(X)) = [1 1] X + [0] [0 0] [5] >= [1 1] X + [0] [0 0] [5] = mark(h(X)) h(ok(X)) = [1 3] X + [2] [0 1] [6] >= [1 3] X + [2] [0 1] [6] = ok(h(X)) proper(c(X)) = [1 1] X + [0] [0 1] [0] >= [1 1] X + [0] [0 1] [0] = c(proper(X)) proper(f(X)) = [1 1] X + [0] [0 1] [0] >= [1 1] X + [0] [0 1] [0] = f(proper(X)) proper(g(X)) = [1 3] X + [0] [0 1] [0] >= [1 3] X + [0] [0 1] [0] = g(proper(X)) proper(h(X)) = [1 1] X + [0] [0 1] [0] >= [1 1] X + [0] [0 1] [0] = h(proper(X)) top(mark(X)) = [1 1] X + [0] [0 0] [7] >= [1 1] X + [0] [0 0] [7] = top(proper(X)) top(ok(X)) = [1 3] X + [2] [0 0] [7] >= [1 3] X + [2] [0 0] [7] = top(active(X)) * Step 14: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,g,h,mark,ok,proper,top} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: 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 5] x1 + [5] [0 0] [0] p(c) = [1 1] x1 + [0] [0 1] [1] p(d) = [1 0] x1 + [0] [0 1] [0] p(f) = [1 0] x1 + [0] [0 1] [0] p(g) = [1 0] x1 + [0] [0 1] [0] p(h) = [1 0] x1 + [0] [0 1] [0] p(mark) = [1 4] x1 + [1] [0 0] [0] p(ok) = [1 3] x1 + [0] [0 1] [3] p(proper) = [1 2] x1 + [1] [0 1] [0] p(top) = [1 2] x1 + [0] [0 0] [0] Following rules are strictly oriented: proper(c(X)) = [1 3] X + [3] [0 1] [1] > [1 3] X + [1] [0 1] [1] = c(proper(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 6] X + [10] [0 0] [0] >= [1 4] X + [1] [0 0] [0] = mark(d(X)) active(f(X)) = [1 5] X + [5] [0 0] [0] >= [1 5] X + [5] [0 0] [0] = f(active(X)) active(f(f(X))) = [1 5] X + [5] [0 0] [0] >= [1 5] X + [5] [0 0] [0] = mark(c(f(g(f(X))))) active(h(X)) = [1 5] X + [5] [0 0] [0] >= [1 5] X + [5] [0 0] [0] = h(active(X)) active(h(X)) = [1 5] X + [5] [0 0] [0] >= [1 5] X + [5] [0 0] [0] = mark(c(d(X))) c(ok(X)) = [1 4] X + [3] [0 1] [4] >= [1 4] X + [3] [0 1] [4] = ok(c(X)) d(ok(X)) = [1 3] X + [0] [0 1] [3] >= [1 3] X + [0] [0 1] [3] = ok(d(X)) f(mark(X)) = [1 4] X + [1] [0 0] [0] >= [1 4] X + [1] [0 0] [0] = mark(f(X)) f(ok(X)) = [1 3] X + [0] [0 1] [3] >= [1 3] X + [0] [0 1] [3] = ok(f(X)) g(ok(X)) = [1 3] X + [0] [0 1] [3] >= [1 3] X + [0] [0 1] [3] = ok(g(X)) h(mark(X)) = [1 4] X + [1] [0 0] [0] >= [1 4] X + [1] [0 0] [0] = mark(h(X)) h(ok(X)) = [1 3] X + [0] [0 1] [3] >= [1 3] X + [0] [0 1] [3] = ok(h(X)) proper(d(X)) = [1 2] X + [1] [0 1] [0] >= [1 2] X + [1] [0 1] [0] = d(proper(X)) proper(f(X)) = [1 2] X + [1] [0 1] [0] >= [1 2] X + [1] [0 1] [0] = f(proper(X)) proper(g(X)) = [1 2] X + [1] [0 1] [0] >= [1 2] X + [1] [0 1] [0] = g(proper(X)) proper(h(X)) = [1 2] X + [1] [0 1] [0] >= [1 2] X + [1] [0 1] [0] = h(proper(X)) top(mark(X)) = [1 4] X + [1] [0 0] [0] >= [1 4] X + [1] [0 0] [0] = top(proper(X)) top(ok(X)) = [1 5] X + [6] [0 0] [0] >= [1 5] X + [5] [0 0] [0] = top(active(X)) * Step 15: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: proper(g(X)) -> g(proper(X)) - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) proper(h(X)) -> h(proper(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,g,h,mark,ok,proper,top} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: 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 6] x1 + [4] [0 0] [1] p(c) = [1 0] x1 + [0] [0 1] [0] p(d) = [1 2] x1 + [2] [0 1] [0] p(f) = [1 0] x1 + [4] [0 1] [0] p(g) = [1 1] x1 + [0] [0 1] [1] p(h) = [1 0] x1 + [0] [0 1] [0] p(mark) = [1 4] x1 + [0] [0 0] [1] p(ok) = [1 4] x1 + [2] [0 1] [4] p(proper) = [1 2] x1 + [1] [0 1] [0] p(top) = [1 2] x1 + [0] [0 0] [2] Following rules are strictly oriented: proper(g(X)) = [1 3] X + [3] [0 1] [1] > [1 3] X + [1] [0 1] [1] = g(proper(X)) Following rules are (at-least) weakly oriented: active(c(X)) = [1 6] X + [4] [0 0] [1] >= [1 6] X + [2] [0 0] [1] = mark(d(X)) active(f(X)) = [1 6] X + [8] [0 0] [1] >= [1 6] X + [8] [0 0] [1] = f(active(X)) active(f(f(X))) = [1 6] X + [12] [0 0] [1] >= [1 5] X + [12] [0 0] [1] = mark(c(f(g(f(X))))) active(h(X)) = [1 6] X + [4] [0 0] [1] >= [1 6] X + [4] [0 0] [1] = h(active(X)) active(h(X)) = [1 6] X + [4] [0 0] [1] >= [1 6] X + [2] [0 0] [1] = mark(c(d(X))) c(ok(X)) = [1 4] X + [2] [0 1] [4] >= [1 4] X + [2] [0 1] [4] = ok(c(X)) d(ok(X)) = [1 6] X + [12] [0 1] [4] >= [1 6] X + [4] [0 1] [4] = ok(d(X)) f(mark(X)) = [1 4] X + [4] [0 0] [1] >= [1 4] X + [4] [0 0] [1] = mark(f(X)) f(ok(X)) = [1 4] X + [6] [0 1] [4] >= [1 4] X + [6] [0 1] [4] = ok(f(X)) g(ok(X)) = [1 5] X + [6] [0 1] [5] >= [1 5] X + [6] [0 1] [5] = ok(g(X)) h(mark(X)) = [1 4] X + [0] [0 0] [1] >= [1 4] X + [0] [0 0] [1] = mark(h(X)) h(ok(X)) = [1 4] X + [2] [0 1] [4] >= [1 4] X + [2] [0 1] [4] = ok(h(X)) proper(c(X)) = [1 2] X + [1] [0 1] [0] >= [1 2] X + [1] [0 1] [0] = c(proper(X)) proper(d(X)) = [1 4] X + [3] [0 1] [0] >= [1 4] X + [3] [0 1] [0] = d(proper(X)) proper(f(X)) = [1 2] X + [5] [0 1] [0] >= [1 2] X + [5] [0 1] [0] = f(proper(X)) proper(h(X)) = [1 2] X + [1] [0 1] [0] >= [1 2] X + [1] [0 1] [0] = h(proper(X)) top(mark(X)) = [1 4] X + [2] [0 0] [2] >= [1 4] X + [1] [0 0] [2] = top(proper(X)) top(ok(X)) = [1 6] X + [10] [0 0] [2] >= [1 6] X + [6] [0 0] [2] = top(active(X)) * Step 16: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(c(X)) -> mark(d(X)) active(f(X)) -> f(active(X)) active(f(f(X))) -> mark(c(f(g(f(X))))) active(h(X)) -> h(active(X)) active(h(X)) -> mark(c(d(X))) c(ok(X)) -> ok(c(X)) d(ok(X)) -> ok(d(X)) f(mark(X)) -> mark(f(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(mark(X)) -> mark(h(X)) h(ok(X)) -> ok(h(X)) proper(c(X)) -> c(proper(X)) proper(d(X)) -> d(proper(X)) proper(f(X)) -> f(proper(X)) 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,c/1,d/1,f/1,g/1,h/1,proper/1,top/1} / {mark/1,ok/1} - Obligation: innermost derivational complexity wrt. signature {active,c,d,f,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))