/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: check(y){y -> cons(x,y)} = check(cons(x,y)) ->^+ cons(x,check(y)) = C[check(y) = check(y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak dependency pairs: Strict DPs check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) rest#(nil()) -> c_7() top#(sent(x)) -> c_8(top#(check(rest(x)))) Weak DPs and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) rest#(nil()) -> c_7() top#(sent(x)) -> c_8(top#(check(rest(x)))) - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) rest#(nil()) -> c_7() top#(sent(x)) -> c_8(top#(check(rest(x)))) ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) rest#(nil()) -> c_7() top#(sent(x)) -> c_8(top#(check(rest(x)))) - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {7} by application of Pre({7}) = {1,2,3,4,6}. Here rules are labelled as follows: 1: check#(cons(x,y)) -> c_1(x,y) 2: check#(cons(x,y)) -> c_2(x,check#(y)) 3: check#(cons(x,y)) -> c_3(check#(x),y) 4: check#(rest(x)) -> c_4(rest#(check(x))) 5: check#(sent(x)) -> c_5(check#(x)) 6: rest#(cons(x,y)) -> c_6(y) 7: rest#(nil()) -> c_7() 8: top#(sent(x)) -> c_8(top#(check(rest(x)))) ** Step 1.b:4: PathAnalysis. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) top#(sent(x)) -> c_8(top#(check(rest(x)))) - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Weak DPs: rest#(nil()) -> c_7() - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: PathAnalysis {onlyLinear = True} + Details: We employ 'linear path analysis' using the following approximated dependency graph: 1:S:check#(cons(x,y)) -> c_1(x,y) -->_2 top#(sent(x)) -> c_8(top#(check(rest(x)))):7 -->_1 top#(sent(x)) -> c_8(top#(check(rest(x)))):7 -->_2 rest#(cons(x,y)) -> c_6(y):6 -->_1 rest#(cons(x,y)) -> c_6(y):6 -->_2 check#(sent(x)) -> c_5(check#(x)):5 -->_1 check#(sent(x)) -> c_5(check#(x)):5 -->_2 check#(rest(x)) -> c_4(rest#(check(x))):4 -->_1 check#(rest(x)) -> c_4(rest#(check(x))):4 -->_2 check#(cons(x,y)) -> c_3(check#(x),y):3 -->_1 check#(cons(x,y)) -> c_3(check#(x),y):3 -->_2 check#(cons(x,y)) -> c_2(x,check#(y)):2 -->_1 check#(cons(x,y)) -> c_2(x,check#(y)):2 -->_2 rest#(nil()) -> c_7():8 -->_1 rest#(nil()) -> c_7():8 -->_2 check#(cons(x,y)) -> c_1(x,y):1 -->_1 check#(cons(x,y)) -> c_1(x,y):1 2:S:check#(cons(x,y)) -> c_2(x,check#(y)) -->_1 top#(sent(x)) -> c_8(top#(check(rest(x)))):7 -->_1 rest#(cons(x,y)) -> c_6(y):6 -->_2 check#(sent(x)) -> c_5(check#(x)):5 -->_1 check#(sent(x)) -> c_5(check#(x)):5 -->_2 check#(rest(x)) -> c_4(rest#(check(x))):4 -->_1 check#(rest(x)) -> c_4(rest#(check(x))):4 -->_2 check#(cons(x,y)) -> c_3(check#(x),y):3 -->_1 check#(cons(x,y)) -> c_3(check#(x),y):3 -->_1 rest#(nil()) -> c_7():8 -->_2 check#(cons(x,y)) -> c_2(x,check#(y)):2 -->_1 check#(cons(x,y)) -> c_2(x,check#(y)):2 -->_2 check#(cons(x,y)) -> c_1(x,y):1 -->_1 check#(cons(x,y)) -> c_1(x,y):1 3:S:check#(cons(x,y)) -> c_3(check#(x),y) -->_2 top#(sent(x)) -> c_8(top#(check(rest(x)))):7 -->_2 rest#(cons(x,y)) -> c_6(y):6 -->_2 check#(sent(x)) -> c_5(check#(x)):5 -->_1 check#(sent(x)) -> c_5(check#(x)):5 -->_2 check#(rest(x)) -> c_4(rest#(check(x))):4 -->_1 check#(rest(x)) -> c_4(rest#(check(x))):4 -->_2 rest#(nil()) -> c_7():8 -->_2 check#(cons(x,y)) -> c_3(check#(x),y):3 -->_1 check#(cons(x,y)) -> c_3(check#(x),y):3 -->_2 check#(cons(x,y)) -> c_2(x,check#(y)):2 -->_1 check#(cons(x,y)) -> c_2(x,check#(y)):2 -->_2 check#(cons(x,y)) -> c_1(x,y):1 -->_1 check#(cons(x,y)) -> c_1(x,y):1 4:S:check#(rest(x)) -> c_4(rest#(check(x))) -->_1 rest#(cons(x,y)) -> c_6(y):6 -->_1 rest#(nil()) -> c_7():8 5:S:check#(sent(x)) -> c_5(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):5 -->_1 check#(rest(x)) -> c_4(rest#(check(x))):4 -->_1 check#(cons(x,y)) -> c_3(check#(x),y):3 -->_1 check#(cons(x,y)) -> c_2(x,check#(y)):2 -->_1 check#(cons(x,y)) -> c_1(x,y):1 6:S:rest#(cons(x,y)) -> c_6(y) -->_1 top#(sent(x)) -> c_8(top#(check(rest(x)))):7 -->_1 rest#(nil()) -> c_7():8 -->_1 rest#(cons(x,y)) -> c_6(y):6 -->_1 check#(sent(x)) -> c_5(check#(x)):5 -->_1 check#(rest(x)) -> c_4(rest#(check(x))):4 -->_1 check#(cons(x,y)) -> c_3(check#(x),y):3 -->_1 check#(cons(x,y)) -> c_2(x,check#(y)):2 -->_1 check#(cons(x,y)) -> c_1(x,y):1 7:S:top#(sent(x)) -> c_8(top#(check(rest(x)))) -->_1 top#(sent(x)) -> c_8(top#(check(rest(x)))):7 8:W:rest#(nil()) -> c_7() Obtaining following paths: 1.) Path: [1,3] 2.) Path: [1,2] *** Step 1.b:4.a:1: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) top#(sent(x)) -> c_8(top#(check(rest(x)))) - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(check) = {1}, uargs(cons) = {1,2}, uargs(rest) = {1}, uargs(sent) = {1}, uargs(rest#) = {1}, uargs(top#) = {1}, uargs(c_2) = {2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [3] p(nil) = [4] p(rest) = [1] x1 + [1] p(sent) = [1] x1 + [2] p(top) = [4] x1 + [4] p(check#) = [4] x1 + [2] p(rest#) = [1] x1 + [4] p(top#) = [1] x1 + [5] p(c_1) = [4] x1 + [4] x2 + [0] p(c_2) = [4] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [4] x2 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] Following rules are strictly oriented: check#(cons(x,y)) = [4] x + [4] y + [14] > [4] x + [4] y + [0] = c_1(x,y) check#(cons(x,y)) = [4] x + [4] y + [14] > [4] x + [4] y + [2] = c_2(x,check#(y)) check#(cons(x,y)) = [4] x + [4] y + [14] > [4] x + [4] y + [2] = c_3(check#(x),y) check#(rest(x)) = [4] x + [6] > [1] x + [4] = c_4(rest#(check(x))) check#(sent(x)) = [4] x + [10] > [4] x + [2] = c_5(check#(x)) rest#(cons(x,y)) = [1] x + [1] y + [7] > [1] y + [0] = c_6(y) top#(sent(x)) = [1] x + [7] > [1] x + [6] = c_8(top#(check(rest(x)))) rest(cons(x,y)) = [1] x + [1] y + [4] > [1] y + [2] = sent(y) Following rules are (at-least) weakly oriented: check(cons(x,y)) = [1] x + [1] y + [3] >= [1] x + [1] y + [3] = cons(x,y) check(cons(x,y)) = [1] x + [1] y + [3] >= [1] x + [1] y + [3] = cons(x,check(y)) check(cons(x,y)) = [1] x + [1] y + [3] >= [1] x + [1] y + [3] = cons(check(x),y) check(rest(x)) = [1] x + [1] >= [1] x + [1] = rest(check(x)) check(sent(x)) = [1] x + [2] >= [1] x + [2] = sent(check(x)) rest(nil()) = [5] >= [6] = sent(nil()) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:4.a:2: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(nil()) -> sent(nil()) - Weak DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) top#(sent(x)) -> c_8(top#(check(rest(x)))) - Weak TRS: rest(cons(x,y)) -> sent(y) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(check) = {1}, uargs(cons) = {1,2}, uargs(rest) = {1}, uargs(sent) = {1}, uargs(rest#) = {1}, uargs(top#) = {1}, uargs(c_2) = {2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [1] x1 + [1] p(cons) = [1] x1 + [1] x2 + [2] p(nil) = [0] p(rest) = [1] x1 + [0] p(sent) = [1] x1 + [1] p(top) = [1] x1 + [1] p(check#) = [1] x1 + [2] p(rest#) = [1] x1 + [0] p(top#) = [1] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [1] p(c_2) = [1] x2 + [2] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [1] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] x1 + [0] Following rules are strictly oriented: check(cons(x,y)) = [1] x + [1] y + [3] > [1] x + [1] y + [2] = cons(x,y) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [1] x + [1] y + [4] >= [1] x + [1] y + [1] = c_1(x,y) check#(cons(x,y)) = [1] x + [1] y + [4] >= [1] y + [4] = c_2(x,check#(y)) check#(cons(x,y)) = [1] x + [1] y + [4] >= [1] x + [3] = c_3(check#(x),y) check#(rest(x)) = [1] x + [2] >= [1] x + [2] = c_4(rest#(check(x))) check#(sent(x)) = [1] x + [3] >= [1] x + [3] = c_5(check#(x)) rest#(cons(x,y)) = [1] x + [1] y + [2] >= [1] y + [0] = c_6(y) top#(sent(x)) = [1] x + [1] >= [1] x + [1] = c_8(top#(check(rest(x)))) check(cons(x,y)) = [1] x + [1] y + [3] >= [1] x + [1] y + [3] = cons(x,check(y)) check(cons(x,y)) = [1] x + [1] y + [3] >= [1] x + [1] y + [3] = cons(check(x),y) check(rest(x)) = [1] x + [1] >= [1] x + [1] = rest(check(x)) check(sent(x)) = [1] x + [2] >= [1] x + [2] = sent(check(x)) rest(cons(x,y)) = [1] x + [1] y + [2] >= [1] y + [1] = sent(y) rest(nil()) = [0] >= [1] = sent(nil()) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:4.a:3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(nil()) -> sent(nil()) - Weak DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) top#(sent(x)) -> c_8(top#(check(rest(x)))) - Weak TRS: check(cons(x,y)) -> cons(x,y) rest(cons(x,y)) -> sent(y) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(check) = {1}, uargs(cons) = {1,2}, uargs(rest) = {1}, uargs(sent) = {1}, uargs(rest#) = {1}, uargs(top#) = {1}, uargs(c_2) = {2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(cons) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 1] x2 + [1] [0 0 0] [0 0 0] [0] p(nil) = [0] [0] [0] p(rest) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(sent) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(top) = [0] [0] [0] p(check#) = [1 1 0] [0] [1 0 0] x1 + [0] [0 1 0] [1] p(rest#) = [1 0 0] [0] [0 0 0] x1 + [1] [1 0 0] [1] p(top#) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(c_1) = [0 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_2) = [0 0 0] [1 0 1] [0] [1 0 0] x1 + [0 0 0] x2 + [0] [0 1 0] [0 0 0] [0] p(c_3) = [1 0 0] [0 0 0] [0] [0 0 0] x1 + [1 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(c_4) = [1 0 0] [0] [0 0 0] x1 + [0] [0 1 0] [0] p(c_5) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_6) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(c_7) = [0] [0] [0] p(c_8) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: check(cons(x,y)) = [1 1 0] [1 2 1] [1] [0 1 0] x + [0 1 1] y + [1] [0 0 0] [0 0 0] [0] > [1 0 0] [1 2 0] [0] [0 1 0] x + [0 1 0] y + [1] [0 0 0] [0 0 0] [0] = cons(x,check(y)) check(cons(x,y)) = [1 1 0] [1 2 1] [1] [0 1 0] x + [0 1 1] y + [1] [0 0 0] [0 0 0] [0] > [1 1 0] [1 1 0] [0] [0 1 0] x + [0 1 1] y + [1] [0 0 0] [0 0 0] [0] = cons(check(x),y) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [1 1 0] [1 2 1] [1] [1 0 0] x + [1 1 0] y + [0] [0 1 0] [0 1 1] [2] >= [0 1 0] [0] [0 0 0] x + [0] [0 0 0] [0] = c_1(x,y) check#(cons(x,y)) = [1 1 0] [1 2 1] [1] [1 0 0] x + [1 1 0] y + [0] [0 1 0] [0 1 1] [2] >= [0 0 0] [1 2 0] [1] [1 0 0] x + [0 0 0] y + [0] [0 1 0] [0 0 0] [0] = c_2(x,check#(y)) check#(cons(x,y)) = [1 1 0] [1 2 1] [1] [1 0 0] x + [1 1 0] y + [0] [0 1 0] [0 1 1] [2] >= [1 1 0] [0 0 0] [0] [0 0 0] x + [1 0 0] y + [0] [0 1 0] [0 0 0] [1] = c_3(check#(x),y) check#(rest(x)) = [1 1 0] [0] [1 0 0] x + [0] [0 1 0] [1] >= [1 1 0] [0] [0 0 0] x + [0] [0 0 0] [1] = c_4(rest#(check(x))) check#(sent(x)) = [1 2 0] [0] [1 1 0] x + [0] [0 1 0] [1] >= [1 1 0] [0] [0 0 0] x + [0] [0 0 0] [0] = c_5(check#(x)) rest#(cons(x,y)) = [1 0 0] [1 1 0] [0] [0 0 0] x + [0 0 0] y + [1] [1 0 0] [1 1 0] [1] >= [1 0 0] [0] [0 0 0] y + [1] [0 0 0] [0] = c_6(y) top#(sent(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] = c_8(top#(check(rest(x)))) check(cons(x,y)) = [1 1 0] [1 2 1] [1] [0 1 0] x + [0 1 1] y + [1] [0 0 0] [0 0 0] [0] >= [1 0 0] [1 1 0] [0] [0 1 0] x + [0 1 1] y + [1] [0 0 0] [0 0 0] [0] = cons(x,y) check(rest(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] = rest(check(x)) check(sent(x)) = [1 2 0] [0] [0 1 0] x + [0] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] x + [0] [0 0 0] [0] = sent(check(x)) rest(cons(x,y)) = [1 0 0] [1 1 0] [0] [0 1 0] x + [0 1 1] y + [1] [0 0 0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] y + [0] [0 0 0] [0] = sent(y) rest(nil()) = [0] [0] [0] >= [0] [0] [0] = sent(nil()) *** Step 1.b:4.a:4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(nil()) -> sent(nil()) - Weak DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) top#(sent(x)) -> c_8(top#(check(rest(x)))) - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) rest(cons(x,y)) -> sent(y) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(check) = {1}, uargs(cons) = {1,2}, uargs(rest) = {1}, uargs(sent) = {1}, uargs(rest#) = {1}, uargs(top#) = {1}, uargs(c_2) = {2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [1 0 0 0] [0] [0 0 1 0] x1 + [0] [0 1 0 0] [0] [0 0 0 0] [0] p(cons) = [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x1 + [0 0 0 1] x2 + [0] [0 0 0 0] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] p(nil) = [0] [0] [0] [1] p(rest) = [1 0 0 1] [0] [0 0 1 0] x1 + [0] [0 1 0 1] [0] [0 0 0 0] [0] p(sent) = [1 1 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 0] [0] p(top) = [0] [0] [0] [0] p(check#) = [1 0 0 0] [0] [0 0 1 0] x1 + [0] [0 1 0 0] [0] [1 0 0 0] [1] p(rest#) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [1 1 0 0] [1] p(top#) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [1 1 1 0] [1] [0 0 0 0] [0] p(c_1) = [1 0 0 0] [0 1 0 0] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [1 0 0 0] [0] p(c_2) = [1 1 1 0] [0] [0 0 0 0] x2 + [0] [0 0 0 0] [0] [1 0 0 0] [0] p(c_3) = [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] x1 + [0 0 0 1] x2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [1 1 0 0] [0] p(c_4) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] p(c_5) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c_6) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [1 1 1 1] [0] p(c_7) = [0] [0] [0] [0] p(c_8) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [1 0 0 0] [1] [0 0 0 0] [0] Following rules are strictly oriented: rest(nil()) = [1] [0] [1] [0] > [0] [0] [1] [0] = sent(nil()) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 1] [0] [1 0 0 0] [1 1 1 0] [1] >= [1 0 0 0] [0 1 0 0] [0] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [1 0 0 0] [0] = c_1(x,y) check#(cons(x,y)) = [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 1] [0] [1 0 0 0] [1 1 1 0] [1] >= [1 1 1 0] [0] [0 0 0 0] y + [0] [0 0 0 0] [0] [1 0 0 0] [0] = c_2(x,check#(y)) check#(cons(x,y)) = [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 1] [0] [1 0 0 0] [1 1 1 0] [1] >= [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [1 1 0 0] [0] = c_3(check#(x),y) check#(rest(x)) = [1 0 0 1] [0] [0 1 0 1] x + [0] [0 0 1 0] [0] [1 0 0 1] [1] >= [1 0 0 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [1] = c_4(rest#(check(x))) check#(sent(x)) = [1 1 1 0] [0] [0 0 0 1] x + [0] [0 0 0 0] [0] [1 1 1 0] [1] >= [1 1 0 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] = c_5(check#(x)) rest#(cons(x,y)) = [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 0 0] [0 0 0 0] [0] [1 0 0 0] [1 1 1 1] [1] >= [1 0 1 0] [0] [0 0 0 0] y + [0] [0 0 0 0] [0] [1 1 1 1] [0] = c_6(y) top#(sent(x)) = [1 1 1 1] [0] [0 0 0 0] x + [0] [1 1 1 1] [1] [0 0 0 0] [0] >= [1 0 1 1] [0] [0 0 0 0] x + [0] [1 0 1 1] [1] [0 0 0 0] [0] = c_8(top#(check(rest(x)))) check(cons(x,y)) = [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] = cons(x,y) check(cons(x,y)) = [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = cons(x,check(y)) check(cons(x,y)) = [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] = cons(check(x),y) check(rest(x)) = [1 0 0 1] [0] [0 1 0 1] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 1 0 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] = rest(check(x)) check(sent(x)) = [1 1 1 0] [0] [0 0 0 1] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] = sent(check(x)) rest(cons(x,y)) = [1 0 0 0] [1 1 1 0] [0] [0 0 0 0] x + [0 0 0 1] y + [0] [0 0 0 0] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 1 1 0] [0] [0 0 0 0] y + [0] [0 0 0 1] [0] [0 0 0 0] [0] = sent(y) *** Step 1.b:4.a:5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) - Weak DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) top#(sent(x)) -> c_8(top#(check(rest(x)))) - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(check) = {1}, uargs(cons) = {1,2}, uargs(rest) = {1}, uargs(sent) = {1}, uargs(rest#) = {1}, uargs(top#) = {1}, uargs(c_2) = {2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [1 0 1 0] [0] [0 0 1 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(cons) = [1 0 0 0] [1 0 1 1] [0] [0 0 0 0] x1 + [0 0 1 1] x2 + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] p(nil) = [0] [1] [0] [1] p(rest) = [1 0 0 0] [1] [0 1 0 1] x1 + [0] [0 0 1 1] [0] [0 0 0 0] [0] p(sent) = [1 0 1 1] [0] [0 0 1 1] x1 + [1] [0 0 1 0] [1] [0 0 0 0] [0] p(top) = [0] [0] [0] [0] p(check#) = [1 0 1 0] [0] [0 0 1 0] x1 + [1] [0 0 0 0] [0] [0 0 0 1] [0] p(rest#) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [1 1 0 0] [0] [0 0 0 0] [0] p(top#) = [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 1 0] [0] [0 1 1 0] [0] p(c_1) = [1 0 0 0] [0] [0 0 0 0] x2 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c_2) = [0 0 0 0] [1 0 0 0] [1] [0 0 1 0] x1 + [0 0 0 1] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] p(c_3) = [1 0 0 0] [1 0 0 0] [0] [0 1 0 0] x1 + [0 0 0 1] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] p(c_4) = [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c_5) = [1 1 0 1] [0] [0 1 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(c_6) = [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(c_7) = [0] [0] [0] [0] p(c_8) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 1 0 0] [1] Following rules are strictly oriented: check(sent(x)) = [1 0 2 1] [1] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] > [1 0 2 0] [0] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] = sent(check(x)) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [1 0 1 0] [1 0 2 2] [1] [0 0 1 0] x + [0 0 1 1] y + [2] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 0 0 0] y + [0] [0 0 0 0] [0] [0 0 0 0] [0] = c_1(x,y) check#(cons(x,y)) = [1 0 1 0] [1 0 2 2] [1] [0 0 1 0] x + [0 0 1 1] y + [2] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [1 0 1 0] [1] [0 0 1 0] x + [0 0 0 1] y + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = c_2(x,check#(y)) check#(cons(x,y)) = [1 0 1 0] [1 0 2 2] [1] [0 0 1 0] x + [0 0 1 1] y + [2] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 0 0] [0] [0 0 1 0] x + [0 0 0 1] y + [2] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = c_3(check#(x),y) check#(rest(x)) = [1 0 1 1] [1] [0 0 1 1] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 1 0] [1] [0 0 0 0] x + [0] [0 0 0 0] [0] [0 0 0 0] [0] = c_4(rest#(check(x))) check#(sent(x)) = [1 0 2 1] [1] [0 0 1 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 2 1] [1] [0 0 1 0] x + [2] [0 0 0 0] [0] [0 0 0 0] [0] = c_5(check#(x)) rest#(cons(x,y)) = [1 0 0 0] [1 0 1 1] [0] [0 0 0 0] x + [0 0 0 0] y + [1] [1 0 0 0] [1 0 2 2] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 1] [0] [0 0 0 0] y + [0] [0 0 0 0] [1] [0 0 0 0] [0] = c_6(y) top#(sent(x)) = [1 0 2 2] [2] [0 0 0 0] x + [1] [0 0 1 0] [1] [0 0 2 1] [2] >= [1 0 2 2] [2] [0 0 0 0] x + [0] [0 0 0 0] [1] [0 0 0 0] [2] = c_8(top#(check(rest(x)))) check(cons(x,y)) = [1 0 1 0] [1 0 2 2] [1] [0 0 1 0] x + [0 0 1 1] y + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 1 1] [0] [0 0 0 0] x + [0 0 1 1] y + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] = cons(x,y) check(cons(x,y)) = [1 0 1 0] [1 0 2 2] [1] [0 0 1 0] x + [0 0 1 1] y + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 2 0] [0] [0 0 0 0] x + [0 0 1 0] y + [1] [0 0 1 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] = cons(x,check(y)) check(cons(x,y)) = [1 0 1 0] [1 0 2 2] [1] [0 0 1 0] x + [0 0 1 1] y + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 1 1] [0] [0 0 0 0] x + [0 0 1 1] y + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] = cons(check(x),y) check(rest(x)) = [1 0 1 1] [1] [0 0 1 1] x + [0] [0 0 1 1] [0] [0 0 0 0] [0] >= [1 0 1 0] [1] [0 0 1 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] = rest(check(x)) rest(cons(x,y)) = [1 0 0 0] [1 0 1 1] [1] [0 0 0 0] x + [0 0 1 1] y + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 1] [0] [0 0 1 1] y + [1] [0 0 1 0] [1] [0 0 0 0] [0] = sent(y) rest(nil()) = [1] [2] [1] [0] >= [1] [2] [1] [0] = sent(nil()) *** Step 1.b:4.a:6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(rest(x)) -> rest(check(x)) - Weak DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) top#(sent(x)) -> c_8(top#(check(rest(x)))) - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(check) = {1}, uargs(cons) = {1,2}, uargs(rest) = {1}, uargs(sent) = {1}, uargs(rest#) = {1}, uargs(top#) = {1}, uargs(c_2) = {2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [1 0 1 0] [0] [0 0 1 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(cons) = [1 0 0 1] [1 0 1 1] [1] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 1 1] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [0] p(nil) = [0] [0] [0] [1] p(rest) = [1 0 0 1] [0] [0 0 1 1] x1 + [1] [0 0 1 0] [1] [0 0 0 0] [0] p(sent) = [1 0 1 0] [1] [0 0 1 1] x1 + [1] [0 0 1 0] [1] [0 0 0 0] [0] p(top) = [0] [0] [0] [0] p(check#) = [1 0 1 0] [0] [1 0 0 0] x1 + [0] [1 1 0 0] [1] [1 1 1 0] [0] p(rest#) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [1 0 1 0] [0] [0 0 0 0] [1] p(top#) = [1 1 0 0] [1] [0 1 1 0] x1 + [0] [0 1 0 0] [0] [0 0 0 0] [0] p(c_1) = [1 0 1 0] [1 0 0 0] [0] [0 0 0 1] x1 + [0 0 1 0] x2 + [1] [0 0 0 1] [0 0 1 0] [0] [1 0 0 0] [0 0 0 0] [0] p(c_2) = [0 0 1 0] [1 0 0 0] [0] [1 0 0 1] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 1 0 0] [0] [0 0 0 0] [0 0 0 0] [0] p(c_3) = [1 0 0 0] [0 0 0 0] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 1 0 0] [0 0 0 1] [0] [1 0 0 0] [0 0 0 0] [0] p(c_4) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [1 0 0 1] [1] [0 0 1 0] [0] p(c_5) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 1 0 0] [1] p(c_6) = [1 0 1 1] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(c_7) = [0] [0] [0] [0] p(c_8) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] Following rules are strictly oriented: check(rest(x)) = [1 0 1 1] [1] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] > [1 0 1 0] [0] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] = rest(check(x)) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [1 0 1 2] [1 0 2 2] [1] [1 0 0 1] x + [1 0 1 1] y + [1] [1 0 0 1] [1 0 1 1] [2] [1 0 1 2] [1 0 2 2] [1] >= [1 0 1 0] [1 0 0 0] [0] [0 0 0 1] x + [0 0 1 0] y + [1] [0 0 0 1] [0 0 1 0] [0] [1 0 0 0] [0 0 0 0] [0] = c_1(x,y) check#(cons(x,y)) = [1 0 1 2] [1 0 2 2] [1] [1 0 0 1] x + [1 0 1 1] y + [1] [1 0 0 1] [1 0 1 1] [2] [1 0 1 2] [1 0 2 2] [1] >= [0 0 1 0] [1 0 1 0] [0] [1 0 0 1] x + [0 0 0 0] y + [0] [0 0 0 0] [1 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = c_2(x,check#(y)) check#(cons(x,y)) = [1 0 1 2] [1 0 2 2] [1] [1 0 0 1] x + [1 0 1 1] y + [1] [1 0 0 1] [1 0 1 1] [2] [1 0 1 2] [1 0 2 2] [1] >= [1 0 1 0] [0 0 0 0] [0] [0 0 0 0] x + [0 0 0 0] y + [0] [1 0 0 0] [0 0 0 1] [0] [1 0 1 0] [0 0 0 0] [0] = c_3(check#(x),y) check#(rest(x)) = [1 0 1 1] [1] [1 0 0 1] x + [0] [1 0 1 2] [2] [1 0 2 2] [2] >= [1 0 1 0] [0] [0 0 0 0] x + [0] [1 0 1 0] [2] [1 0 2 0] [0] = c_4(rest#(check(x))) check#(sent(x)) = [1 0 2 0] [2] [1 0 1 0] x + [1] [1 0 2 1] [3] [1 0 3 1] [3] >= [1 0 1 0] [0] [0 0 0 0] x + [0] [0 0 0 0] [0] [1 0 0 0] [1] = c_5(check#(x)) rest#(cons(x,y)) = [1 0 0 1] [1 0 1 1] [1] [0 0 0 0] x + [0 0 0 0] y + [1] [1 0 1 2] [1 0 2 2] [1] [0 0 0 0] [0 0 0 0] [1] >= [1 0 1 1] [0] [0 0 0 0] y + [1] [0 0 0 0] [0] [0 0 0 0] [0] = c_6(y) top#(sent(x)) = [1 0 2 1] [3] [0 0 2 1] x + [2] [0 0 1 1] [1] [0 0 0 0] [0] >= [1 0 2 1] [3] [0 0 0 0] x + [1] [0 0 0 0] [0] [0 0 0 0] [0] = c_8(top#(check(rest(x)))) check(cons(x,y)) = [1 0 1 2] [1 0 2 2] [1] [0 0 1 1] x + [0 0 1 1] y + [0] [0 0 1 1] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 1] [1 0 1 1] [1] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 1 1] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [0] = cons(x,y) check(cons(x,y)) = [1 0 1 2] [1 0 2 2] [1] [0 0 1 1] x + [0 0 1 1] y + [0] [0 0 1 1] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 1] [1 0 2 0] [1] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 1 1] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] = cons(x,check(y)) check(cons(x,y)) = [1 0 1 2] [1 0 2 2] [1] [0 0 1 1] x + [0 0 1 1] y + [0] [0 0 1 1] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 1 1] [1] [0 0 0 0] x + [0 0 0 0] y + [0] [0 0 1 0] [0 0 1 1] [0] [0 0 0 0] [0 0 0 0] [0] = cons(check(x),y) check(sent(x)) = [1 0 2 0] [2] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] >= [1 0 2 0] [1] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] = sent(check(x)) rest(cons(x,y)) = [1 0 0 1] [1 0 1 1] [1] [0 0 1 1] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1] [0 0 1 1] y + [1] [0 0 1 0] [1] [0 0 0 0] [0] = sent(y) rest(nil()) = [1] [2] [1] [0] >= [1] [2] [1] [0] = sent(nil()) *** Step 1.b:4.a:7: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) top#(sent(x)) -> c_8(top#(check(rest(x)))) - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:4.b:1: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Weak DPs: rest#(nil()) -> c_7() - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {1,2}, uargs(rest) = {1}, uargs(sent) = {1}, uargs(rest#) = {1}, uargs(c_2) = {2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [3] x1 + [0] p(cons) = [1] x1 + [1] x2 + [1] p(nil) = [0] p(rest) = [1] x1 + [5] p(sent) = [1] x1 + [0] p(top) = [0] p(check#) = [3] x1 + [12] p(rest#) = [1] x1 + [0] p(top#) = [0] p(c_1) = [3] x1 + [3] x2 + [0] p(c_2) = [3] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [3] x2 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] Following rules are strictly oriented: check#(cons(x,y)) = [3] x + [3] y + [15] > [3] x + [3] y + [0] = c_1(x,y) check#(cons(x,y)) = [3] x + [3] y + [15] > [3] x + [3] y + [12] = c_2(x,check#(y)) check#(cons(x,y)) = [3] x + [3] y + [15] > [3] x + [3] y + [12] = c_3(check#(x),y) check#(rest(x)) = [3] x + [27] > [3] x + [0] = c_4(rest#(check(x))) rest#(cons(x,y)) = [1] x + [1] y + [1] > [1] y + [0] = c_6(y) check(cons(x,y)) = [3] x + [3] y + [3] > [1] x + [1] y + [1] = cons(x,y) check(cons(x,y)) = [3] x + [3] y + [3] > [1] x + [3] y + [1] = cons(x,check(y)) check(cons(x,y)) = [3] x + [3] y + [3] > [3] x + [1] y + [1] = cons(check(x),y) check(rest(x)) = [3] x + [15] > [3] x + [5] = rest(check(x)) rest(cons(x,y)) = [1] x + [1] y + [6] > [1] y + [0] = sent(y) rest(nil()) = [5] > [0] = sent(nil()) Following rules are (at-least) weakly oriented: check#(sent(x)) = [3] x + [12] >= [3] x + [12] = c_5(check#(x)) rest#(nil()) = [0] >= [0] = c_7() check(sent(x)) = [3] x + [0] >= [3] x + [0] = sent(check(x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:4.b:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: check#(sent(x)) -> c_5(check#(x)) - Strict TRS: check(sent(x)) -> sent(check(x)) - Weak DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) rest#(cons(x,y)) -> c_6(y) rest#(nil()) -> c_7() - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {1,2}, uargs(rest) = {1}, uargs(sent) = {1}, uargs(rest#) = {1}, uargs(c_2) = {2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(nil) = [6] p(rest) = [1] x1 + [2] p(sent) = [1] x1 + [2] p(top) = [0] p(check#) = [1] x1 + [1] p(rest#) = [1] x1 + [1] p(top#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [2] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [0] p(c_7) = [4] p(c_8) = [0] Following rules are strictly oriented: check#(sent(x)) = [1] x + [3] > [1] x + [2] = c_5(check#(x)) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [1] x + [1] y + [1] >= [0] = c_1(x,y) check#(cons(x,y)) = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = c_2(x,check#(y)) check#(cons(x,y)) = [1] x + [1] y + [1] >= [1] x + [1] = c_3(check#(x),y) check#(rest(x)) = [1] x + [3] >= [1] x + [3] = c_4(rest#(check(x))) rest#(cons(x,y)) = [1] x + [1] y + [1] >= [1] y + [0] = c_6(y) rest#(nil()) = [7] >= [4] = c_7() check(cons(x,y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = cons(x,y) check(cons(x,y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = cons(x,check(y)) check(cons(x,y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = cons(check(x),y) check(rest(x)) = [1] x + [2] >= [1] x + [2] = rest(check(x)) check(sent(x)) = [1] x + [2] >= [1] x + [2] = sent(check(x)) rest(cons(x,y)) = [1] x + [1] y + [2] >= [1] y + [2] = sent(y) rest(nil()) = [8] >= [8] = sent(nil()) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:4.b:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: check(sent(x)) -> sent(check(x)) - Weak DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) rest#(nil()) -> c_7() - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {1,2}, uargs(rest) = {1}, uargs(sent) = {1}, uargs(rest#) = {1}, uargs(c_2) = {2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [2] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(nil) = [0] p(rest) = [1] x1 + [5] p(sent) = [1] x1 + [5] p(top) = [0] p(check#) = [2] x1 + [2] p(rest#) = [1] x1 + [5] p(top#) = [0] p(c_1) = [2] x1 + [2] x2 + [2] p(c_2) = [2] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [4] p(c_8) = [1] Following rules are strictly oriented: check(sent(x)) = [2] x + [10] > [2] x + [5] = sent(check(x)) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [2] x + [2] y + [2] >= [2] x + [2] y + [2] = c_1(x,y) check#(cons(x,y)) = [2] x + [2] y + [2] >= [2] x + [2] y + [2] = c_2(x,check#(y)) check#(cons(x,y)) = [2] x + [2] y + [2] >= [2] x + [1] y + [2] = c_3(check#(x),y) check#(rest(x)) = [2] x + [12] >= [2] x + [5] = c_4(rest#(check(x))) check#(sent(x)) = [2] x + [12] >= [2] x + [2] = c_5(check#(x)) rest#(cons(x,y)) = [1] x + [1] y + [5] >= [1] y + [0] = c_6(y) rest#(nil()) = [5] >= [4] = c_7() check(cons(x,y)) = [2] x + [2] y + [0] >= [1] x + [1] y + [0] = cons(x,y) check(cons(x,y)) = [2] x + [2] y + [0] >= [1] x + [2] y + [0] = cons(x,check(y)) check(cons(x,y)) = [2] x + [2] y + [0] >= [2] x + [1] y + [0] = cons(check(x),y) check(rest(x)) = [2] x + [10] >= [2] x + [5] = rest(check(x)) rest(cons(x,y)) = [1] x + [1] y + [5] >= [1] y + [5] = sent(y) rest(nil()) = [5] >= [5] = sent(nil()) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:4.b:4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: check#(cons(x,y)) -> c_1(x,y) check#(cons(x,y)) -> c_2(x,check#(y)) check#(cons(x,y)) -> c_3(check#(x),y) check#(rest(x)) -> c_4(rest#(check(x))) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6(y) rest#(nil()) -> c_7() - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Signature: {check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/2,c_2/2,c_3/2,c_4/1,c_5/1,c_6/1 ,c_7/0,c_8/1} - Obligation: runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))