/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^3)) * Step 1: WeightGap. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: active(f(0())) -> mark(cons(0(),f(s(0())))) active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) - Signature: {active/1,cons/2,f/1,mark/1,p/1,s/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {0,active,cons,f,mark,p,s} + 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(0) = [3] p(active) = [1] x1 + [1] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [0] p(mark) = [1] x1 + [7] p(p) = [1] x1 + [5] p(s) = [1] x1 + [0] Following rules are strictly oriented: cons(X1,active(X2)) = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = cons(X1,X2) cons(X1,mark(X2)) = [1] X1 + [1] X2 + [7] > [1] X1 + [1] X2 + [0] = cons(X1,X2) cons(active(X1),X2) = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = cons(X1,X2) cons(mark(X1),X2) = [1] X1 + [1] X2 + [7] > [1] X1 + [1] X2 + [0] = cons(X1,X2) f(active(X)) = [1] X + [1] > [1] X + [0] = f(X) f(mark(X)) = [1] X + [7] > [1] X + [0] = f(X) mark(0()) = [10] > [4] = active(0()) p(active(X)) = [1] X + [6] > [1] X + [5] = p(X) p(mark(X)) = [1] X + [12] > [1] X + [5] = p(X) s(active(X)) = [1] X + [1] > [1] X + [0] = s(X) s(mark(X)) = [1] X + [7] > [1] X + [0] = s(X) Following rules are (at-least) weakly oriented: active(f(0())) = [4] >= [13] = mark(cons(0(),f(s(0())))) active(f(s(0()))) = [4] >= [15] = mark(f(p(s(0())))) active(p(s(0()))) = [9] >= [10] = mark(0()) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [8] = active(cons(mark(X1),X2)) mark(f(X)) = [1] X + [7] >= [1] X + [8] = active(f(mark(X))) mark(p(X)) = [1] X + [12] >= [1] X + [13] = active(p(mark(X))) mark(s(X)) = [1] X + [7] >= [1] X + [8] = active(s(mark(X))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: active(f(0())) -> mark(cons(0(),f(s(0())))) active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) - Weak TRS: cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(0()) -> active(0()) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) - Signature: {active/1,cons/2,f/1,mark/1,p/1,s/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {0,active,cons,f,mark,p,s} + 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(0) = [4] p(active) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [1] Following rules are strictly oriented: active(p(s(0()))) = [5] > [4] = mark(0()) Following rules are (at-least) weakly oriented: active(f(0())) = [4] >= [9] = mark(cons(0(),f(s(0())))) active(f(s(0()))) = [5] >= [5] = mark(f(p(s(0())))) cons(X1,active(X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(X1,X2) cons(X1,mark(X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(X1,X2) cons(active(X1),X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(X1,X2) cons(mark(X1),X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(X1,X2) f(active(X)) = [1] X + [0] >= [1] X + [0] = f(X) f(mark(X)) = [1] X + [0] >= [1] X + [0] = f(X) mark(0()) = [4] >= [4] = active(0()) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = active(cons(mark(X1),X2)) mark(f(X)) = [1] X + [0] >= [1] X + [0] = active(f(mark(X))) mark(p(X)) = [1] X + [0] >= [1] X + [0] = active(p(mark(X))) mark(s(X)) = [1] X + [1] >= [1] X + [1] = active(s(mark(X))) p(active(X)) = [1] X + [0] >= [1] X + [0] = p(X) p(mark(X)) = [1] X + [0] >= [1] X + [0] = p(X) s(active(X)) = [1] X + [1] >= [1] X + [1] = s(X) s(mark(X)) = [1] X + [1] >= [1] X + [1] = s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: active(f(0())) -> mark(cons(0(),f(s(0())))) active(f(s(0()))) -> mark(f(p(s(0())))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) - Weak TRS: active(p(s(0()))) -> mark(0()) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(0()) -> active(0()) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) - Signature: {active/1,cons/2,f/1,mark/1,p/1,s/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {0,active,cons,f,mark,p,s} + 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(0) = [0] [0] [0] p(active) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(cons) = [1 0 1] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(f) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(p) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(s) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: active(f(0())) = [1] [0] [0] > [0] [0] [0] = mark(cons(0(),f(s(0())))) Following rules are (at-least) weakly oriented: active(f(s(0()))) = [1] [0] [0] >= [1] [0] [0] = mark(f(p(s(0())))) active(p(s(0()))) = [0] [0] [0] >= [0] [0] [0] = mark(0()) cons(X1,active(X2)) = [1 0 1] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(X1,mark(X2)) = [1 0 1] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(active(X1),X2) = [1 0 1] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(mark(X1),X2) = [1 0 1] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) f(active(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [1] = f(X) f(mark(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [1] = f(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = active(0()) mark(cons(X1,X2)) = [1 0 1] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = active(cons(mark(X1),X2)) mark(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] = active(f(mark(X))) mark(p(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] = active(p(mark(X))) mark(s(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] = active(s(mark(X))) p(active(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] = p(X) p(mark(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] = p(X) s(active(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] = s(X) s(mark(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] = s(X) * Step 4: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: active(f(s(0()))) -> mark(f(p(s(0())))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) - Weak TRS: active(f(0())) -> mark(cons(0(),f(s(0())))) active(p(s(0()))) -> mark(0()) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(0()) -> active(0()) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) - Signature: {active/1,cons/2,f/1,mark/1,p/1,s/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {0,active,cons,f,mark,p,s} + 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(0) = [0] [0] [0] p(active) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(cons) = [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [1] [0 0 0] [0 0 0] [0] p(f) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(p) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 1 0] [1 0 0] [1] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] > [1 1 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] = active(cons(mark(X1),X2)) Following rules are (at-least) weakly oriented: active(f(0())) = [1] [1] [0] >= [1] [1] [0] = mark(cons(0(),f(s(0())))) active(f(s(0()))) = [1] [1] [0] >= [1] [1] [0] = mark(f(p(s(0())))) active(p(s(0()))) = [0] [0] [0] >= [0] [0] [0] = mark(0()) cons(X1,active(X2)) = [1 0 0] [1 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(X1,mark(X2)) = [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(active(X1),X2) = [1 0 1] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(mark(X1),X2) = [1 1 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [1] [0 0 0] [0 0 0] [0] = cons(X1,X2) f(active(X)) = [1 0 1] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [1] = f(X) f(mark(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [1] = f(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = active(0()) mark(f(X)) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] = active(f(mark(X))) mark(p(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] = active(p(mark(X))) mark(s(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] = active(s(mark(X))) p(active(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] = p(X) p(mark(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] = p(X) s(active(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] = s(X) s(mark(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] = s(X) * Step 5: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: active(f(s(0()))) -> mark(f(p(s(0())))) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) - Weak TRS: active(f(0())) -> mark(cons(0(),f(s(0())))) active(p(s(0()))) -> mark(0()) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) - Signature: {active/1,cons/2,f/1,mark/1,p/1,s/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {0,active,cons,f,mark,p,s} + 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(0) = [0] [0] [0] p(active) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(cons) = [1 1 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(f) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(p) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(s) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] Following rules are strictly oriented: active(f(s(0()))) = [1] [0] [0] > [0] [0] [0] = mark(f(p(s(0())))) Following rules are (at-least) weakly oriented: active(f(0())) = [0] [0] [0] >= [0] [0] [0] = mark(cons(0(),f(s(0())))) active(p(s(0()))) = [0] [0] [0] >= [0] [0] [0] = mark(0()) cons(X1,active(X2)) = [1 1 0] [1 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(X1,mark(X2)) = [1 1 0] [1 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(active(X1),X2) = [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(mark(X1),X2) = [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) f(active(X)) = [1 1 0] [0] [0 0 1] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 0] [0] = f(X) f(mark(X)) = [1 1 0] [0] [0 0 1] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 1] X + [0] [0 0 0] [0] = f(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = active(0()) mark(cons(X1,X2)) = [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 0 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = active(cons(mark(X1),X2)) mark(f(X)) = [1 1 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 1] [0] [0 0 0] X + [0] [0 0 0] [0] = active(f(mark(X))) mark(p(X)) = [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = active(p(mark(X))) mark(s(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] [1] = active(s(mark(X))) p(active(X)) = [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = p(X) p(mark(X)) = [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = p(X) s(active(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] [1] = s(X) s(mark(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] [1] = s(X) * Step 6: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) - Weak TRS: active(f(0())) -> mark(cons(0(),f(s(0())))) active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) - Signature: {active/1,cons/2,f/1,mark/1,p/1,s/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {0,active,cons,f,mark,p,s} + 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(0) = [0] [0] [0] [0] p(active) = [1 0 1 0] [0] [0 1 1 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(cons) = [1 0 0 0] [1 0 0 0] [0] [0 1 1 1] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] p(f) = [1 0 1 1] [0] [0 1 1 1] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(mark) = [1 1 1 1] [0] [0 1 1 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(p) = [1 0 0 0] [0] [0 1 1 1] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(s) = [1 0 0 0] [0] [0 1 1 1] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] Following rules are strictly oriented: mark(s(X)) = [1 1 1 1] [1] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] > [1 1 1 1] [0] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = active(s(mark(X))) Following rules are (at-least) weakly oriented: active(f(0())) = [1] [1] [0] [0] >= [1] [0] [0] [0] = mark(cons(0(),f(s(0())))) active(f(s(0()))) = [2] [2] [0] [0] >= [2] [2] [0] [0] = mark(f(p(s(0())))) active(p(s(0()))) = [0] [1] [0] [0] >= [0] [0] [0] [0] = mark(0()) cons(X1,active(X2)) = [1 0 0 0] [1 0 1 0] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 0 0] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = cons(X1,X2) cons(X1,mark(X2)) = [1 0 0 0] [1 1 1 1] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 0 0] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = cons(X1,X2) cons(active(X1),X2) = [1 0 1 0] [1 0 0 0] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 0 0] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = cons(X1,X2) cons(mark(X1),X2) = [1 1 1 1] [1 0 0 0] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 0 0] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = cons(X1,X2) f(active(X)) = [1 0 1 1] [0] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 1] [0] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = f(X) f(mark(X)) = [1 1 1 2] [0] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 1 1] [0] [0 1 1 1] X + [0] [0 0 0 0] [1] [0 0 0 0] [0] = f(X) mark(0()) = [0] [0] [0] [0] >= [0] [0] [0] [0] = active(0()) mark(cons(X1,X2)) = [1 1 1 1] [1 0 0 0] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 1 1 1] [1 0 0 0] [0] [0 1 1 1] X1 + [0 0 0 0] X2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = active(cons(mark(X1),X2)) mark(f(X)) = [1 1 2 2] [1] [0 1 1 1] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 1 2] [1] [0 1 1 1] X + [1] [0 0 0 0] [0] [0 0 0 0] [0] = active(f(mark(X))) mark(p(X)) = [1 1 1 1] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 1 1] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = active(p(mark(X))) p(active(X)) = [1 0 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = p(X) p(mark(X)) = [1 1 1 1] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [0] = p(X) s(active(X)) = [1 0 1 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 0 0 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [1] = s(X) s(mark(X)) = [1 1 1 1] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 0 0 0] [0] [0 1 1 1] X + [0] [0 0 0 0] [0] [0 0 0 0] [1] = s(X) * Step 7: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) - Weak TRS: active(f(0())) -> mark(cons(0(),f(s(0())))) active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) - Signature: {active/1,cons/2,f/1,mark/1,p/1,s/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {0,active,cons,f,mark,p,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(0) = [0] [0] [0] p(active) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(cons) = [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(f) = [1 0 1] [0] [0 1 1] x1 + [1] [0 0 1] [1] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(p) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] Following rules are strictly oriented: mark(p(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] = active(p(mark(X))) Following rules are (at-least) weakly oriented: active(f(0())) = [1] [1] [1] >= [1] [0] [0] = mark(cons(0(),f(s(0())))) active(f(s(0()))) = [3] [3] [2] >= [3] [3] [1] = mark(f(p(s(0())))) active(p(s(0()))) = [0] [2] [0] >= [0] [0] [0] = mark(0()) cons(X1,active(X2)) = [1 0 0] [1 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(X1,mark(X2)) = [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(active(X1),X2) = [1 0 1] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(mark(X1),X2) = [1 1 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) f(active(X)) = [1 0 2] [0] [0 1 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 1 1] X + [1] [0 0 1] [1] = f(X) f(mark(X)) = [1 1 1] [0] [0 1 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 1 1] X + [1] [0 0 1] [1] = f(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = active(0()) mark(cons(X1,X2)) = [1 1 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = active(cons(mark(X1),X2)) mark(f(X)) = [1 1 2] [1] [0 1 1] X + [1] [0 0 1] [1] >= [1 1 2] [1] [0 1 1] X + [1] [0 0 1] [1] = active(f(mark(X))) mark(s(X)) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [1] >= [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [1] = active(s(mark(X))) p(active(X)) = [1 0 1] [0] [0 1 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] = p(X) p(mark(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] = p(X) s(active(X)) = [1 0 1] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [1] = s(X) s(mark(X)) = [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [1] = s(X) * Step 8: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: mark(f(X)) -> active(f(mark(X))) - Weak TRS: active(f(0())) -> mark(cons(0(),f(s(0())))) active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) - Signature: {active/1,cons/2,f/1,mark/1,p/1,s/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {0,active,cons,f,mark,p,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(0) = [0] [0] [0] p(active) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(cons) = [1 0 0] [1 0 0] [0] [0 1 1] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(f) = [1 0 1] [0] [0 1 1] x1 + [1] [0 0 1] [1] p(mark) = [1 1 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(p) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(s) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [1] Following rules are strictly oriented: mark(f(X)) = [1 1 3] [2] [0 1 1] X + [1] [0 0 1] [1] > [1 1 3] [1] [0 1 1] X + [1] [0 0 1] [1] = active(f(mark(X))) Following rules are (at-least) weakly oriented: active(f(0())) = [1] [1] [1] >= [1] [0] [0] = mark(cons(0(),f(s(0())))) active(f(s(0()))) = [3] [2] [2] >= [3] [2] [1] = mark(f(p(s(0())))) active(p(s(0()))) = [0] [1] [0] >= [0] [0] [0] = mark(0()) cons(X1,active(X2)) = [1 0 0] [1 0 1] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(X1,mark(X2)) = [1 0 0] [1 1 1] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(active(X1),X2) = [1 0 1] [1 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) cons(mark(X1),X2) = [1 1 1] [1 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = cons(X1,X2) f(active(X)) = [1 0 2] [0] [0 1 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 1 1] X + [1] [0 0 1] [1] = f(X) f(mark(X)) = [1 1 2] [0] [0 1 1] X + [1] [0 0 1] [1] >= [1 0 1] [0] [0 1 1] X + [1] [0 0 1] [1] = f(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = active(0()) mark(cons(X1,X2)) = [1 1 1] [1 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 1] [1 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [0] [0 0 0] [0 0 0] [0] = active(cons(mark(X1),X2)) mark(p(X)) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] = active(p(mark(X))) mark(s(X)) = [1 1 1] [1] [0 1 1] X + [0] [0 0 0] [1] >= [1 1 1] [1] [0 1 1] X + [0] [0 0 0] [1] = active(s(mark(X))) p(active(X)) = [1 0 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 1] X + [0] [0 0 0] [0] = p(X) p(mark(X)) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 1] X + [0] [0 0 0] [0] = p(X) s(active(X)) = [1 0 1] [0] [0 1 1] X + [0] [0 0 0] [1] >= [1 0 0] [0] [0 1 1] X + [0] [0 0 0] [1] = s(X) s(mark(X)) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [1] >= [1 0 0] [0] [0 1 1] X + [0] [0 0 0] [1] = s(X) * Step 9: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(f(0())) -> mark(cons(0(),f(s(0())))) active(f(s(0()))) -> mark(f(p(s(0())))) active(p(s(0()))) -> mark(0()) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) f(active(X)) -> f(X) f(mark(X)) -> f(X) mark(0()) -> active(0()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(f(X)) -> active(f(mark(X))) mark(p(X)) -> active(p(mark(X))) mark(s(X)) -> active(s(mark(X))) p(active(X)) -> p(X) p(mark(X)) -> p(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) - Signature: {active/1,cons/2,f/1,mark/1,p/1,s/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {0,active,cons,f,mark,p,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))