/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: A(a(x1)) -> x1 A(c(b(x1))) -> b(c(A(x1))) B(C(a(x1))) -> a(C(B(x1))) B(b(x1)) -> x1 C(B(A(x1))) -> A(B(C(x1))) C(c(x1)) -> x1 a(A(x1)) -> x1 a(b(c(x1))) -> c(b(a(x1))) b(B(x1)) -> x1 b(a(C(x1))) -> C(a(b(x1))) c(A(B(x1))) -> B(A(c(x1))) c(C(x1)) -> x1 - Signature: {A/1,B/1,C/1,a/1,b/1,c/1} / {} - Obligation: derivational complexity wrt. signature {A,B,C,a,b,c} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(A) = [1] x1 + [14] p(B) = [1] x1 + [0] p(C) = [1] x1 + [5] p(a) = [1] x1 + [0] p(b) = [1] x1 + [0] p(c) = [1] x1 + [0] Following rules are strictly oriented: A(a(x1)) = [1] x1 + [14] > [1] x1 + [0] = x1 C(c(x1)) = [1] x1 + [5] > [1] x1 + [0] = x1 a(A(x1)) = [1] x1 + [14] > [1] x1 + [0] = x1 c(C(x1)) = [1] x1 + [5] > [1] x1 + [0] = x1 Following rules are (at-least) weakly oriented: A(c(b(x1))) = [1] x1 + [14] >= [1] x1 + [14] = b(c(A(x1))) B(C(a(x1))) = [1] x1 + [5] >= [1] x1 + [5] = a(C(B(x1))) B(b(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 C(B(A(x1))) = [1] x1 + [19] >= [1] x1 + [19] = A(B(C(x1))) a(b(c(x1))) = [1] x1 + [0] >= [1] x1 + [0] = c(b(a(x1))) b(B(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 b(a(C(x1))) = [1] x1 + [5] >= [1] x1 + [5] = C(a(b(x1))) c(A(B(x1))) = [1] x1 + [14] >= [1] x1 + [14] = B(A(c(x1))) 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: A(c(b(x1))) -> b(c(A(x1))) B(C(a(x1))) -> a(C(B(x1))) B(b(x1)) -> x1 C(B(A(x1))) -> A(B(C(x1))) a(b(c(x1))) -> c(b(a(x1))) b(B(x1)) -> x1 b(a(C(x1))) -> C(a(b(x1))) c(A(B(x1))) -> B(A(c(x1))) - Weak TRS: A(a(x1)) -> x1 C(c(x1)) -> x1 a(A(x1)) -> x1 c(C(x1)) -> x1 - Signature: {A/1,B/1,C/1,a/1,b/1,c/1} / {} - Obligation: derivational complexity wrt. signature {A,B,C,a,b,c} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(A) = [1] x1 + [0] p(B) = [1] x1 + [9] p(C) = [1] x1 + [0] p(a) = [1] x1 + [0] p(b) = [1] x1 + [8] p(c) = [1] x1 + [0] Following rules are strictly oriented: B(b(x1)) = [1] x1 + [17] > [1] x1 + [0] = x1 b(B(x1)) = [1] x1 + [17] > [1] x1 + [0] = x1 Following rules are (at-least) weakly oriented: A(a(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 A(c(b(x1))) = [1] x1 + [8] >= [1] x1 + [8] = b(c(A(x1))) B(C(a(x1))) = [1] x1 + [9] >= [1] x1 + [9] = a(C(B(x1))) C(B(A(x1))) = [1] x1 + [9] >= [1] x1 + [9] = A(B(C(x1))) C(c(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 a(A(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 a(b(c(x1))) = [1] x1 + [8] >= [1] x1 + [8] = c(b(a(x1))) b(a(C(x1))) = [1] x1 + [8] >= [1] x1 + [8] = C(a(b(x1))) c(A(B(x1))) = [1] x1 + [9] >= [1] x1 + [9] = B(A(c(x1))) c(C(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 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: A(c(b(x1))) -> b(c(A(x1))) B(C(a(x1))) -> a(C(B(x1))) C(B(A(x1))) -> A(B(C(x1))) a(b(c(x1))) -> c(b(a(x1))) b(a(C(x1))) -> C(a(b(x1))) c(A(B(x1))) -> B(A(c(x1))) - Weak TRS: A(a(x1)) -> x1 B(b(x1)) -> x1 C(c(x1)) -> x1 a(A(x1)) -> x1 b(B(x1)) -> x1 c(C(x1)) -> x1 - Signature: {A/1,B/1,C/1,a/1,b/1,c/1} / {} - Obligation: derivational complexity wrt. signature {A,B,C,a,b,c} + 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(A) = [1 0] x1 + [0] [0 1] [0] p(B) = [1 0] x1 + [0] [0 1] [6] p(C) = [1 0] x1 + [0] [0 1] [2] p(a) = [1 0] x1 + [3] [0 1] [0] p(b) = [1 0] x1 + [1] [0 1] [0] p(c) = [1 2] x1 + [0] [0 1] [0] Following rules are strictly oriented: c(A(B(x1))) = [1 2] x1 + [12] [0 1] [6] > [1 2] x1 + [0] [0 1] [6] = B(A(c(x1))) Following rules are (at-least) weakly oriented: A(a(x1)) = [1 0] x1 + [3] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 A(c(b(x1))) = [1 2] x1 + [1] [0 1] [0] >= [1 2] x1 + [1] [0 1] [0] = b(c(A(x1))) B(C(a(x1))) = [1 0] x1 + [3] [0 1] [8] >= [1 0] x1 + [3] [0 1] [8] = a(C(B(x1))) B(b(x1)) = [1 0] x1 + [1] [0 1] [6] >= [1 0] x1 + [0] [0 1] [0] = x1 C(B(A(x1))) = [1 0] x1 + [0] [0 1] [8] >= [1 0] x1 + [0] [0 1] [8] = A(B(C(x1))) C(c(x1)) = [1 2] x1 + [0] [0 1] [2] >= [1 0] x1 + [0] [0 1] [0] = x1 a(A(x1)) = [1 0] x1 + [3] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 a(b(c(x1))) = [1 2] x1 + [4] [0 1] [0] >= [1 2] x1 + [4] [0 1] [0] = c(b(a(x1))) b(B(x1)) = [1 0] x1 + [1] [0 1] [6] >= [1 0] x1 + [0] [0 1] [0] = x1 b(a(C(x1))) = [1 0] x1 + [4] [0 1] [2] >= [1 0] x1 + [4] [0 1] [2] = C(a(b(x1))) c(C(x1)) = [1 2] x1 + [4] [0 1] [2] >= [1 0] x1 + [0] [0 1] [0] = x1 * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: A(c(b(x1))) -> b(c(A(x1))) B(C(a(x1))) -> a(C(B(x1))) C(B(A(x1))) -> A(B(C(x1))) a(b(c(x1))) -> c(b(a(x1))) b(a(C(x1))) -> C(a(b(x1))) - Weak TRS: A(a(x1)) -> x1 B(b(x1)) -> x1 C(c(x1)) -> x1 a(A(x1)) -> x1 b(B(x1)) -> x1 c(A(B(x1))) -> B(A(c(x1))) c(C(x1)) -> x1 - Signature: {A/1,B/1,C/1,a/1,b/1,c/1} / {} - Obligation: derivational complexity wrt. signature {A,B,C,a,b,c} + 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(A) = [1 1] x1 + [0] [0 1] [1] p(B) = [1 1] x1 + [0] [0 1] [1] p(C) = [1 0] x1 + [3] [0 1] [0] p(a) = [1 0] x1 + [0] [0 1] [1] p(b) = [1 0] x1 + [0] [0 1] [0] p(c) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: B(C(a(x1))) = [1 1] x1 + [4] [0 1] [2] > [1 1] x1 + [3] [0 1] [2] = a(C(B(x1))) Following rules are (at-least) weakly oriented: A(a(x1)) = [1 1] x1 + [1] [0 1] [2] >= [1 0] x1 + [0] [0 1] [0] = x1 A(c(b(x1))) = [1 1] x1 + [0] [0 1] [1] >= [1 1] x1 + [0] [0 1] [1] = b(c(A(x1))) B(b(x1)) = [1 1] x1 + [0] [0 1] [1] >= [1 0] x1 + [0] [0 1] [0] = x1 C(B(A(x1))) = [1 2] x1 + [4] [0 1] [2] >= [1 2] x1 + [4] [0 1] [2] = A(B(C(x1))) C(c(x1)) = [1 0] x1 + [3] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 a(A(x1)) = [1 1] x1 + [0] [0 1] [2] >= [1 0] x1 + [0] [0 1] [0] = x1 a(b(c(x1))) = [1 0] x1 + [0] [0 1] [1] >= [1 0] x1 + [0] [0 1] [1] = c(b(a(x1))) b(B(x1)) = [1 1] x1 + [0] [0 1] [1] >= [1 0] x1 + [0] [0 1] [0] = x1 b(a(C(x1))) = [1 0] x1 + [3] [0 1] [1] >= [1 0] x1 + [3] [0 1] [1] = C(a(b(x1))) c(A(B(x1))) = [1 2] x1 + [1] [0 1] [2] >= [1 2] x1 + [1] [0 1] [2] = B(A(c(x1))) c(C(x1)) = [1 0] x1 + [3] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: A(c(b(x1))) -> b(c(A(x1))) C(B(A(x1))) -> A(B(C(x1))) a(b(c(x1))) -> c(b(a(x1))) b(a(C(x1))) -> C(a(b(x1))) - Weak TRS: A(a(x1)) -> x1 B(C(a(x1))) -> a(C(B(x1))) B(b(x1)) -> x1 C(c(x1)) -> x1 a(A(x1)) -> x1 b(B(x1)) -> x1 c(A(B(x1))) -> B(A(c(x1))) c(C(x1)) -> x1 - Signature: {A/1,B/1,C/1,a/1,b/1,c/1} / {} - Obligation: derivational complexity wrt. signature {A,B,C,a,b,c} + 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(A) = [1 0] x1 + [0] [0 1] [0] p(B) = [1 0] x1 + [0] [0 1] [0] p(C) = [1 2] x1 + [0] [0 1] [4] p(a) = [1 0] x1 + [4] [0 1] [0] p(b) = [1 2] x1 + [0] [0 1] [2] p(c) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: b(a(C(x1))) = [1 4] x1 + [12] [0 1] [6] > [1 4] x1 + [8] [0 1] [6] = C(a(b(x1))) Following rules are (at-least) weakly oriented: A(a(x1)) = [1 0] x1 + [4] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 A(c(b(x1))) = [1 2] x1 + [0] [0 1] [2] >= [1 2] x1 + [0] [0 1] [2] = b(c(A(x1))) B(C(a(x1))) = [1 2] x1 + [4] [0 1] [4] >= [1 2] x1 + [4] [0 1] [4] = a(C(B(x1))) B(b(x1)) = [1 2] x1 + [0] [0 1] [2] >= [1 0] x1 + [0] [0 1] [0] = x1 C(B(A(x1))) = [1 2] x1 + [0] [0 1] [4] >= [1 2] x1 + [0] [0 1] [4] = A(B(C(x1))) C(c(x1)) = [1 2] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [0] = x1 a(A(x1)) = [1 0] x1 + [4] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 a(b(c(x1))) = [1 2] x1 + [4] [0 1] [2] >= [1 2] x1 + [4] [0 1] [2] = c(b(a(x1))) b(B(x1)) = [1 2] x1 + [0] [0 1] [2] >= [1 0] x1 + [0] [0 1] [0] = x1 c(A(B(x1))) = [1 0] x1 + [0] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = B(A(c(x1))) c(C(x1)) = [1 2] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [0] = x1 * Step 6: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: A(c(b(x1))) -> b(c(A(x1))) C(B(A(x1))) -> A(B(C(x1))) a(b(c(x1))) -> c(b(a(x1))) - Weak TRS: A(a(x1)) -> x1 B(C(a(x1))) -> a(C(B(x1))) B(b(x1)) -> x1 C(c(x1)) -> x1 a(A(x1)) -> x1 b(B(x1)) -> x1 b(a(C(x1))) -> C(a(b(x1))) c(A(B(x1))) -> B(A(c(x1))) c(C(x1)) -> x1 - Signature: {A/1,B/1,C/1,a/1,b/1,c/1} / {} - Obligation: derivational complexity wrt. signature {A,B,C,a,b,c} + 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(A) = [1 0] x_1 + [0] [0 1] [0] p(B) = [1 0] x_1 + [0] [0 1] [0] p(C) = [1 2] x_1 + [0] [0 1] [4] p(a) = [1 2] x_1 + [0] [0 1] [4] p(b) = [1 0] x_1 + [0] [0 1] [0] p(c) = [1 0] x_1 + [0] [0 1] [4] Following rules are strictly oriented: a(b(c(x1))) = [1 2] x1 + [8] [0 1] [8] > [1 2] x1 + [0] [0 1] [8] = c(b(a(x1))) Following rules are (at-least) weakly oriented: A(a(x1)) = [1 2] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [0] = x1 A(c(b(x1))) = [1 0] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [4] = b(c(A(x1))) B(C(a(x1))) = [1 4] x1 + [8] [0 1] [8] >= [1 4] x1 + [8] [0 1] [8] = a(C(B(x1))) B(b(x1)) = [1 0] x1 + [0] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 C(B(A(x1))) = [1 2] x1 + [0] [0 1] [4] >= [1 2] x1 + [0] [0 1] [4] = A(B(C(x1))) C(c(x1)) = [1 2] x1 + [8] [0 1] [8] >= [1 0] x1 + [0] [0 1] [0] = x1 a(A(x1)) = [1 2] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [0] = x1 b(B(x1)) = [1 0] x1 + [0] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 b(a(C(x1))) = [1 4] x1 + [8] [0 1] [8] >= [1 4] x1 + [8] [0 1] [8] = C(a(b(x1))) c(A(B(x1))) = [1 0] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [4] = B(A(c(x1))) c(C(x1)) = [1 2] x1 + [0] [0 1] [8] >= [1 0] x1 + [0] [0 1] [0] = x1 * Step 7: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: A(c(b(x1))) -> b(c(A(x1))) C(B(A(x1))) -> A(B(C(x1))) - Weak TRS: A(a(x1)) -> x1 B(C(a(x1))) -> a(C(B(x1))) B(b(x1)) -> x1 C(c(x1)) -> x1 a(A(x1)) -> x1 a(b(c(x1))) -> c(b(a(x1))) b(B(x1)) -> x1 b(a(C(x1))) -> C(a(b(x1))) c(A(B(x1))) -> B(A(c(x1))) c(C(x1)) -> x1 - Signature: {A/1,B/1,C/1,a/1,b/1,c/1} / {} - Obligation: derivational complexity wrt. signature {A,B,C,a,b,c} + 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(A) = [1 0] x1 + [0] [0 1] [3] p(B) = [1 0] x1 + [4] [0 1] [0] p(C) = [1 2] x1 + [5] [0 1] [0] p(a) = [1 0] x1 + [4] [0 1] [0] p(b) = [1 0] x1 + [2] [0 1] [0] p(c) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: C(B(A(x1))) = [1 2] x1 + [15] [0 1] [3] > [1 2] x1 + [9] [0 1] [3] = A(B(C(x1))) Following rules are (at-least) weakly oriented: A(a(x1)) = [1 0] x1 + [4] [0 1] [3] >= [1 0] x1 + [0] [0 1] [0] = x1 A(c(b(x1))) = [1 0] x1 + [2] [0 1] [3] >= [1 0] x1 + [2] [0 1] [3] = b(c(A(x1))) B(C(a(x1))) = [1 2] x1 + [13] [0 1] [0] >= [1 2] x1 + [13] [0 1] [0] = a(C(B(x1))) B(b(x1)) = [1 0] x1 + [6] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 C(c(x1)) = [1 2] x1 + [5] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 a(A(x1)) = [1 0] x1 + [4] [0 1] [3] >= [1 0] x1 + [0] [0 1] [0] = x1 a(b(c(x1))) = [1 0] x1 + [6] [0 1] [0] >= [1 0] x1 + [6] [0 1] [0] = c(b(a(x1))) b(B(x1)) = [1 0] x1 + [6] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 b(a(C(x1))) = [1 2] x1 + [11] [0 1] [0] >= [1 2] x1 + [11] [0 1] [0] = C(a(b(x1))) c(A(B(x1))) = [1 0] x1 + [4] [0 1] [3] >= [1 0] x1 + [4] [0 1] [3] = B(A(c(x1))) c(C(x1)) = [1 2] x1 + [5] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 * Step 8: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: A(c(b(x1))) -> b(c(A(x1))) - Weak TRS: A(a(x1)) -> x1 B(C(a(x1))) -> a(C(B(x1))) B(b(x1)) -> x1 C(B(A(x1))) -> A(B(C(x1))) C(c(x1)) -> x1 a(A(x1)) -> x1 a(b(c(x1))) -> c(b(a(x1))) b(B(x1)) -> x1 b(a(C(x1))) -> C(a(b(x1))) c(A(B(x1))) -> B(A(c(x1))) c(C(x1)) -> x1 - Signature: {A/1,B/1,C/1,a/1,b/1,c/1} / {} - Obligation: derivational complexity wrt. signature {A,B,C,a,b,c} + 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(A) = [1 1] x1 + [0] [0 1] [0] p(B) = [1 0] x1 + [0] [0 1] [0] p(C) = [1 0] x1 + [0] [0 1] [0] p(a) = [1 0] x1 + [0] [0 1] [0] p(b) = [1 0] x1 + [0] [0 1] [4] p(c) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: A(c(b(x1))) = [1 1] x1 + [4] [0 1] [4] > [1 1] x1 + [0] [0 1] [4] = b(c(A(x1))) Following rules are (at-least) weakly oriented: A(a(x1)) = [1 1] x1 + [0] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 B(C(a(x1))) = [1 0] x1 + [0] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = a(C(B(x1))) B(b(x1)) = [1 0] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [0] = x1 C(B(A(x1))) = [1 1] x1 + [0] [0 1] [0] >= [1 1] x1 + [0] [0 1] [0] = A(B(C(x1))) C(c(x1)) = [1 0] x1 + [0] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 a(A(x1)) = [1 1] x1 + [0] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 a(b(c(x1))) = [1 0] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [4] = c(b(a(x1))) b(B(x1)) = [1 0] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [0] = x1 b(a(C(x1))) = [1 0] x1 + [0] [0 1] [4] >= [1 0] x1 + [0] [0 1] [4] = C(a(b(x1))) c(A(B(x1))) = [1 1] x1 + [0] [0 1] [0] >= [1 1] x1 + [0] [0 1] [0] = B(A(c(x1))) c(C(x1)) = [1 0] x1 + [0] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = x1 * Step 9: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: A(a(x1)) -> x1 A(c(b(x1))) -> b(c(A(x1))) B(C(a(x1))) -> a(C(B(x1))) B(b(x1)) -> x1 C(B(A(x1))) -> A(B(C(x1))) C(c(x1)) -> x1 a(A(x1)) -> x1 a(b(c(x1))) -> c(b(a(x1))) b(B(x1)) -> x1 b(a(C(x1))) -> C(a(b(x1))) c(A(B(x1))) -> B(A(c(x1))) c(C(x1)) -> x1 - Signature: {A/1,B/1,C/1,a/1,b/1,c/1} / {} - Obligation: derivational complexity wrt. signature {A,B,C,a,b,c} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))