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: innermost 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: 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: innermost 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: innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs check#(cons(x,y)) -> c_1() check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(rest#(check(x)),check#(x)) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6() rest#(nil()) -> c_7() top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),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() check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(rest#(check(x)),check#(x)) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6() rest#(nil()) -> c_7() top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),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()) 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_1/0,c_2/1,c_3/1,c_4/2,c_5/1,c_6/0 ,c_7/0,c_8/3} - Obligation: innermost 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() check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(rest#(check(x)),check#(x)) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6() rest#(nil()) -> c_7() top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x)) ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_1() check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(rest#(check(x)),check#(x)) check#(sent(x)) -> c_5(check#(x)) rest#(cons(x,y)) -> c_6() rest#(nil()) -> c_7() top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),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_1/0,c_2/1,c_3/1,c_4/2,c_5/1,c_6/0 ,c_7/0,c_8/3} - Obligation: innermost 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 {1,6,7} by application of Pre({1,6,7}) = {2,3,4,5,8}. Here rules are labelled as follows: 1: check#(cons(x,y)) -> c_1() 2: check#(cons(x,y)) -> c_2(check#(y)) 3: check#(cons(x,y)) -> c_3(check#(x)) 4: check#(rest(x)) -> c_4(rest#(check(x)),check#(x)) 5: check#(sent(x)) -> c_5(check#(x)) 6: rest#(cons(x,y)) -> c_6() 7: rest#(nil()) -> c_7() 8: top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x)) ** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(rest#(check(x)),check#(x)) check#(sent(x)) -> c_5(check#(x)) top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x)) - Weak DPs: check#(cons(x,y)) -> c_1() rest#(cons(x,y)) -> c_6() 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_1/0,c_2/1,c_3/1,c_4/2,c_5/1,c_6/0 ,c_7/0,c_8/3} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:check#(cons(x,y)) -> c_2(check#(y)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_1():6 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 2:S:check#(cons(x,y)) -> c_3(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_1 check#(cons(x,y)) -> c_1():6 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 3:S:check#(rest(x)) -> c_4(rest#(check(x)),check#(x)) -->_2 check#(sent(x)) -> c_5(check#(x)):4 -->_1 rest#(nil()) -> c_7():8 -->_1 rest#(cons(x,y)) -> c_6():7 -->_2 check#(cons(x,y)) -> c_1():6 -->_2 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_2 check#(cons(x,y)) -> c_3(check#(x)):2 -->_2 check#(cons(x,y)) -> c_2(check#(y)):1 4:S:check#(sent(x)) -> c_5(check#(x)) -->_1 check#(cons(x,y)) -> c_1():6 -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 5:S:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x)) -->_3 rest#(nil()) -> c_7():8 -->_3 rest#(cons(x,y)) -> c_6():7 -->_2 check#(cons(x,y)) -> c_1():6 -->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x)):5 -->_2 check#(sent(x)) -> c_5(check#(x)):4 -->_2 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_2 check#(cons(x,y)) -> c_3(check#(x)):2 -->_2 check#(cons(x,y)) -> c_2(check#(y)):1 6:W:check#(cons(x,y)) -> c_1() 7:W:rest#(cons(x,y)) -> c_6() 8:W:rest#(nil()) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: rest#(cons(x,y)) -> c_6() 8: rest#(nil()) -> c_7() 6: check#(cons(x,y)) -> c_1() ** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(rest#(check(x)),check#(x)) check#(sent(x)) -> c_5(check#(x)) top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),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_1/0,c_2/1,c_3/1,c_4/2,c_5/1,c_6/0 ,c_7/0,c_8/3} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:check#(cons(x,y)) -> c_2(check#(y)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 2:S:check#(cons(x,y)) -> c_3(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 3:S:check#(rest(x)) -> c_4(rest#(check(x)),check#(x)) -->_2 check#(sent(x)) -> c_5(check#(x)):4 -->_2 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_2 check#(cons(x,y)) -> c_3(check#(x)):2 -->_2 check#(cons(x,y)) -> c_2(check#(y)):1 4:S:check#(sent(x)) -> c_5(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 5:S:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x)) -->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x)):5 -->_2 check#(sent(x)) -> c_5(check#(x)):4 -->_2 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3 -->_2 check#(cons(x,y)) -> c_3(check#(x)):2 -->_2 check#(cons(x,y)) -> c_2(check#(y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: check#(rest(x)) -> c_4(check#(x)) top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))) ** Step 1.b:6: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} Problem (S) - Strict DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))) - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} *** Step 1.b:6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: check#(cons(x,y)) -> c_3(check#(x)) The strictly oriented rules are moved into the weak component. **** Step 1.b:6.a:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost 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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-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(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_8) = {1,2} Following symbols are considered usable: {check,rest,check#,rest#,top#} TcT has computed the following interpretation: p(check) = [1 0 0] [0] [0 0 1] x1 + [1] [0 0 1] [0] p(cons) = [0 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(nil) = [0] [0] [0] p(rest) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(sent) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(top) = [0] [0] [0] p(check#) = [0 0 1] [0] [0 0 0] x1 + [1] [1 0 0] [0] p(rest#) = [0] [0] [0] p(top#) = [1 0 0] [0] [1 1 0] x1 + [0] [1 0 1] [0] p(c_1) = [0] [0] [0] p(c_2) = [1 0 0] [1] [0 0 0] x1 + [0] [1 0 0] [0] p(c_3) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(c_4) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(c_5) = [1 0 0] [0] [0 0 0] x1 + [0] [1 0 0] [0] p(c_6) = [0] [0] [0] p(c_7) = [0] [0] [0] p(c_8) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [1 0 0] [0 0 0] [0] Following rules are strictly oriented: check#(cons(x,y)) = [0 0 1] [0 0 1] [1] [0 0 0] x + [0 0 0] y + [1] [0 0 0] [1 0 1] [0] > [0 0 1] [0] [0 0 0] x + [1] [0 0 0] [0] = c_3(check#(x)) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [0 0 1] [0 0 1] [1] [0 0 0] x + [0 0 0] y + [1] [0 0 0] [1 0 1] [0] >= [0 0 1] [1] [0 0 0] y + [0] [0 0 1] [0] = c_2(check#(y)) check#(rest(x)) = [0 0 1] [0] [0 0 0] x + [1] [1 0 0] [0] >= [0 0 1] [0] [0 0 0] x + [1] [1 0 0] [0] = c_4(check#(x)) check#(sent(x)) = [0 0 1] [0] [0 0 0] x + [1] [1 0 1] [0] >= [0 0 1] [0] [0 0 0] x + [0] [0 0 1] [0] = c_5(check#(x)) top#(sent(x)) = [1 0 1] [0] [1 0 1] x + [0] [1 0 2] [0] >= [1 0 1] [0] [0 0 0] x + [0] [1 0 0] [0] = c_8(top#(check(rest(x))),check#(rest(x))) check(cons(x,y)) = [0 0 0] [1 0 1] [0] [0 0 1] x + [0 0 1] y + [2] [0 0 1] [0 0 1] [1] >= [0 0 0] [1 0 1] [0] [0 0 0] x + [0 0 0] y + [0] [0 0 1] [0 0 1] [1] = cons(x,y) check(cons(x,y)) = [0 0 0] [1 0 1] [0] [0 0 1] x + [0 0 1] y + [2] [0 0 1] [0 0 1] [1] >= [0 0 0] [1 0 1] [0] [0 0 0] x + [0 0 0] y + [0] [0 0 1] [0 0 1] [1] = cons(x,check(y)) check(cons(x,y)) = [0 0 0] [1 0 1] [0] [0 0 1] x + [0 0 1] y + [2] [0 0 1] [0 0 1] [1] >= [0 0 0] [1 0 1] [0] [0 0 0] x + [0 0 0] y + [0] [0 0 1] [0 0 1] [1] = cons(check(x),y) check(rest(x)) = [1 0 0] [0] [0 0 1] x + [1] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] x + [0] [0 0 1] [0] = rest(check(x)) check(sent(x)) = [1 0 1] [0] [0 0 1] x + [1] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] x + [0] [0 0 1] [0] = sent(check(x)) rest(cons(x,y)) = [0 0 0] [1 0 1] [0] [0 0 0] x + [0 0 0] y + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] y + [0] [0 0 1] [0] = sent(y) rest(nil()) = [0] [0] [0] >= [0] [0] [0] = sent(nil()) **** Step 1.b:6.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_3(check#(x)) top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 1.b:6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_3(check#(x)) top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: check#(cons(x,y)) -> c_2(check#(y)) The strictly oriented rules are moved into the weak component. ***** Step 1.b:6.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_3(check#(x)) top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost 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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-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(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_8) = {1,2} Following symbols are considered usable: {check,rest,check#,rest#,top#} TcT has computed the following interpretation: p(check) = [0 0 0] [1] [0 1 0] x1 + [0] [0 0 1] [0] p(cons) = [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 1 1] x2 + [0] [0 0 1] [0 0 1] [1] p(nil) = [0] [1] [0] p(rest) = [0 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(sent) = [0 0 0] [0] [0 1 1] x1 + [1] [0 0 1] [0] p(top) = [0] [0] [0] p(check#) = [0 0 1] [0] [0 0 0] x1 + [1] [0 0 0] [1] p(rest#) = [0] [0] [0] p(top#) = [0 1 0] [0] [0 0 1] x1 + [1] [0 0 0] [1] p(c_1) = [0] [0] [0] p(c_2) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(c_3) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_4) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(c_5) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(c_6) = [0] [0] [0] p(c_7) = [0] [0] [0] p(c_8) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [1 1 0] x2 + [0] [0 0 0] [0 0 0] [0] Following rules are strictly oriented: check#(cons(x,y)) = [0 0 1] [0 0 1] [1] [0 0 0] x + [0 0 0] y + [1] [0 0 0] [0 0 0] [1] > [0 0 1] [0] [0 0 0] y + [0] [0 0 0] [1] = c_2(check#(y)) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [0 0 1] [0 0 1] [1] [0 0 0] x + [0 0 0] y + [1] [0 0 0] [0 0 0] [1] >= [0 0 1] [0] [0 0 0] x + [0] [0 0 0] [0] = c_3(check#(x)) check#(rest(x)) = [0 0 1] [0] [0 0 0] x + [1] [0 0 0] [1] >= [0 0 1] [0] [0 0 0] x + [0] [0 0 0] [1] = c_4(check#(x)) check#(sent(x)) = [0 0 1] [0] [0 0 0] x + [1] [0 0 0] [1] >= [0 0 1] [0] [0 0 0] x + [1] [0 0 0] [0] = c_5(check#(x)) top#(sent(x)) = [0 1 1] [1] [0 0 1] x + [1] [0 0 0] [1] >= [0 1 1] [1] [0 0 1] x + [1] [0 0 0] [0] = c_8(top#(check(rest(x))),check#(rest(x))) check(cons(x,y)) = [0 0 0] [0 0 0] [1] [0 0 0] x + [0 1 1] y + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 0] x + [0 1 1] y + [0] [0 0 1] [0 0 1] [1] = cons(x,y) check(cons(x,y)) = [0 0 0] [0 0 0] [1] [0 0 0] x + [0 1 1] y + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 0] x + [0 1 1] y + [0] [0 0 1] [0 0 1] [1] = cons(x,check(y)) check(cons(x,y)) = [0 0 0] [0 0 0] [1] [0 0 0] x + [0 1 1] y + [0] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 0 0] x + [0 1 1] y + [0] [0 0 1] [0 0 1] [1] = cons(check(x),y) check(rest(x)) = [0 0 0] [1] [0 1 0] x + [1] [0 0 1] [0] >= [0 0 0] [0] [0 1 0] x + [1] [0 0 1] [0] = rest(check(x)) check(sent(x)) = [0 0 0] [1] [0 1 1] x + [1] [0 0 1] [0] >= [0 0 0] [0] [0 1 1] x + [1] [0 0 1] [0] = sent(check(x)) rest(cons(x,y)) = [0 0 0] [0 0 0] [0] [0 0 0] x + [0 1 1] y + [1] [0 0 1] [0 0 1] [1] >= [0 0 0] [0] [0 1 1] y + [1] [0 0 1] [0] = sent(y) rest(nil()) = [0] [2] [0] >= [0] [2] [0] = sent(nil()) ***** Step 1.b:6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 1.b:6.a:1.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))) and a lower component check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) Further, following extension rules are added to the lower component. top#(sent(x)) -> check#(rest(x)) top#(sent(x)) -> top#(check(rest(x))) ****** Step 1.b:6.a:1.b:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))) The strictly oriented rules are moved into the weak component. ******* Step 1.b:6.a:1.b:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_8) = {1,2} Following symbols are considered usable: {check,rest,check#,rest#,top#} TcT has computed the following interpretation: p(check) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(cons) = [0 1 1] [2] [0 1 2] x2 + [2] [0 0 0] [0] p(nil) = [3] [2] [2] p(rest) = [1 0 1] [1] [0 1 1] x1 + [1] [0 0 0] [0] p(sent) = [0 1 1] [2] [0 1 0] x1 + [2] [0 0 0] [0] p(top) = [0 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(check#) = [0 0 0] [0] [0 2 0] x1 + [3] [0 0 1] [1] p(rest#) = [0 0 0] [2] [0 1 0] x1 + [2] [0 0 2] [0] p(top#) = [2 0 0] [0] [1 1 0] x1 + [1] [2 0 0] [0] p(c_1) = [0] [2] [0] p(c_2) = [2] [1] [0] p(c_3) = [2] [2] [1] p(c_4) = [0 0 0] [0] [1 0 0] x1 + [0] [2 2 0] [2] p(c_5) = [2] [0] [1] p(c_6) = [2] [1] [2] p(c_7) = [2] [0] [0] p(c_8) = [1 0 0] [2 0 0] [1] [0 0 0] x1 + [2 0 2] x2 + [1] [0 0 0] [0 0 3] [1] Following rules are strictly oriented: top#(sent(x)) = [0 2 2] [4] [0 2 1] x + [5] [0 2 2] [4] > [0 2 2] [3] [0 0 0] x + [3] [0 0 0] [4] = c_8(top#(check(rest(x))),check#(rest(x))) Following rules are (at-least) weakly oriented: check(cons(x,y)) = [0 1 2] [2] [0 1 2] y + [2] [0 0 0] [0] >= [0 1 1] [2] [0 1 2] y + [2] [0 0 0] [0] = cons(x,y) check(cons(x,y)) = [0 1 2] [2] [0 1 2] y + [2] [0 0 0] [0] >= [0 1 0] [2] [0 1 0] y + [2] [0 0 0] [0] = cons(x,check(y)) check(cons(x,y)) = [0 1 2] [2] [0 1 2] y + [2] [0 0 0] [0] >= [0 1 1] [2] [0 1 2] y + [2] [0 0 0] [0] = cons(check(x),y) check(rest(x)) = [0 1 1] [1] [0 1 1] x + [1] [0 0 0] [0] >= [0 1 0] [1] [0 1 0] x + [1] [0 0 0] [0] = rest(check(x)) check(sent(x)) = [0 1 0] [2] [0 1 0] x + [2] [0 0 0] [0] >= [0 1 0] [2] [0 1 0] x + [2] [0 0 0] [0] = sent(check(x)) rest(cons(x,y)) = [0 1 1] [3] [0 1 2] y + [3] [0 0 0] [0] >= [0 1 1] [2] [0 1 0] y + [2] [0 0 0] [0] = sent(y) rest(nil()) = [6] [5] [0] >= [6] [4] [0] = sent(nil()) ******* Step 1.b:6.a:1.b:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 1.b:6.a:1.b:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))) -->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))) ******* Step 1.b:6.a:1.b:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost 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:6.a:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) top#(sent(x)) -> check#(rest(x)) top#(sent(x)) -> 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: check#(sent(x)) -> c_5(check#(x)) The strictly oriented rules are moved into the weak component. ******* Step 1.b:6.a:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) top#(sent(x)) -> check#(rest(x)) top#(sent(x)) -> 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {check,rest,check#,rest#,top#} TcT has computed the following interpretation: p(check) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(nil) = [0] p(rest) = [1] x1 + [1] p(sent) = [1] x1 + [1] p(top) = [8] x1 + [0] p(check#) = [8] x1 + [0] p(rest#) = [8] p(top#) = [8] x1 + [2] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [8] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [8] x1 + [2] Following rules are strictly oriented: check#(sent(x)) = [8] x + [8] > [8] x + [0] = c_5(check#(x)) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [8] x + [8] y + [0] >= [8] y + [0] = c_2(check#(y)) check#(cons(x,y)) = [8] x + [8] y + [0] >= [8] x + [0] = c_3(check#(x)) check#(rest(x)) = [8] x + [8] >= [8] x + [8] = c_4(check#(x)) top#(sent(x)) = [8] x + [10] >= [8] x + [8] = check#(rest(x)) top#(sent(x)) = [8] x + [10] >= [8] x + [10] = top#(check(rest(x))) 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 + [1] >= [1] x + [1] = rest(check(x)) check(sent(x)) = [1] x + [1] >= [1] x + [1] = sent(check(x)) rest(cons(x,y)) = [1] x + [1] y + [1] >= [1] y + [1] = sent(y) rest(nil()) = [1] >= [1] = sent(nil()) ******* Step 1.b:6.a:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: check#(rest(x)) -> c_4(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(sent(x)) -> c_5(check#(x)) top#(sent(x)) -> check#(rest(x)) top#(sent(x)) -> 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 1.b:6.a:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: check#(rest(x)) -> c_4(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(sent(x)) -> c_5(check#(x)) top#(sent(x)) -> check#(rest(x)) top#(sent(x)) -> 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: check#(rest(x)) -> c_4(check#(x)) The strictly oriented rules are moved into the weak component. ******** Step 1.b:6.a:1.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: check#(rest(x)) -> c_4(check#(x)) - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(sent(x)) -> c_5(check#(x)) top#(sent(x)) -> check#(rest(x)) top#(sent(x)) -> 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {check,rest,check#,rest#,top#} TcT has computed the following interpretation: p(check) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(nil) = [0] p(rest) = [1] x1 + [2] p(sent) = [1] x1 + [2] p(top) = [0] p(check#) = [8] x1 + [0] p(rest#) = [0] p(top#) = [8] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [8] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] Following rules are strictly oriented: check#(rest(x)) = [8] x + [16] > [8] x + [8] = c_4(check#(x)) Following rules are (at-least) weakly oriented: check#(cons(x,y)) = [8] x + [8] y + [0] >= [8] y + [0] = c_2(check#(y)) check#(cons(x,y)) = [8] x + [8] y + [0] >= [8] x + [0] = c_3(check#(x)) check#(sent(x)) = [8] x + [16] >= [8] x + [0] = c_5(check#(x)) top#(sent(x)) = [8] x + [16] >= [8] x + [16] = check#(rest(x)) top#(sent(x)) = [8] x + [16] >= [8] x + [16] = top#(check(rest(x))) 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()) = [2] >= [2] = sent(nil()) ******** Step 1.b:6.a:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) top#(sent(x)) -> check#(rest(x)) top#(sent(x)) -> 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 1.b:6.a:1.b:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(x)) top#(sent(x)) -> check#(rest(x)) top#(sent(x)) -> 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:check#(cons(x,y)) -> c_2(check#(y)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 2:W:check#(cons(x,y)) -> c_3(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 3:W:check#(rest(x)) -> c_4(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 4:W:check#(sent(x)) -> c_5(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 5:W:top#(sent(x)) -> check#(rest(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):4 -->_1 check#(rest(x)) -> c_4(check#(x)):3 -->_1 check#(cons(x,y)) -> c_3(check#(x)):2 -->_1 check#(cons(x,y)) -> c_2(check#(y)):1 6:W:top#(sent(x)) -> top#(check(rest(x))) -->_1 top#(sent(x)) -> top#(check(rest(x))):6 -->_1 top#(sent(x)) -> check#(rest(x)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: top#(sent(x)) -> top#(check(rest(x))) 5: top#(sent(x)) -> check#(rest(x)) 1: check#(cons(x,y)) -> c_2(check#(y)) 4: check#(sent(x)) -> c_5(check#(x)) 3: check#(rest(x)) -> c_4(check#(x)) 2: check#(cons(x,y)) -> c_3(check#(x)) ******** Step 1.b:6.a:1.b:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost 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:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))) - Weak DPs: check#(cons(x,y)) -> c_2(check#(y)) check#(cons(x,y)) -> c_3(check#(x)) check#(rest(x)) -> c_4(check#(x)) check#(sent(x)) -> c_5(check#(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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))) -->_2 check#(sent(x)) -> c_5(check#(x)):5 -->_2 check#(rest(x)) -> c_4(check#(x)):4 -->_2 check#(cons(x,y)) -> c_3(check#(x)):3 -->_2 check#(cons(x,y)) -> c_2(check#(y)):2 -->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))):1 2:W:check#(cons(x,y)) -> c_2(check#(y)) -->_1 check#(sent(x)) -> c_5(check#(x)):5 -->_1 check#(rest(x)) -> c_4(check#(x)):4 -->_1 check#(cons(x,y)) -> c_3(check#(x)):3 -->_1 check#(cons(x,y)) -> c_2(check#(y)):2 3:W:check#(cons(x,y)) -> c_3(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):5 -->_1 check#(rest(x)) -> c_4(check#(x)):4 -->_1 check#(cons(x,y)) -> c_3(check#(x)):3 -->_1 check#(cons(x,y)) -> c_2(check#(y)):2 4:W:check#(rest(x)) -> c_4(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):5 -->_1 check#(rest(x)) -> c_4(check#(x)):4 -->_1 check#(cons(x,y)) -> c_3(check#(x)):3 -->_1 check#(cons(x,y)) -> c_2(check#(y)):2 5:W:check#(sent(x)) -> c_5(check#(x)) -->_1 check#(sent(x)) -> c_5(check#(x)):5 -->_1 check#(rest(x)) -> c_4(check#(x)):4 -->_1 check#(cons(x,y)) -> c_3(check#(x)):3 -->_1 check#(cons(x,y)) -> c_2(check#(y)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: check#(sent(x)) -> c_5(check#(x)) 4: check#(rest(x)) -> c_4(check#(x)) 3: check#(cons(x,y)) -> c_3(check#(x)) 2: check#(cons(x,y)) -> c_2(check#(y)) *** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: top#(sent(x)) -> c_8(top#(check(rest(x))),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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))) -->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: top#(sent(x)) -> c_8(top#(check(rest(x)))) *** Step 1.b:6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: top#(sent(x)) -> c_8(top#(check(rest(x)))) The strictly oriented rules are moved into the weak component. **** Step 1.b:6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_8) = {1} Following symbols are considered usable: {check,rest,check#,rest#,top#} TcT has computed the following interpretation: p(check) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(cons) = [0 0 2] [0] [0 1 2] x2 + [2] [0 0 0] [0] p(nil) = [0] [0] [2] p(rest) = [0 1 2] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(sent) = [0 1 1] [2] [0 1 0] x1 + [2] [0 0 0] [0] p(top) = [0 0 1] [2] [2 2 0] x1 + [1] [1 0 0] [1] p(check#) = [0] [0] [2] p(rest#) = [2 2 0] [1] [2 0 2] x1 + [2] [1 1 0] [2] p(top#) = [1 0 0] [0] [0 0 2] x1 + [1] [0 0 0] [0] p(c_1) = [0] [2] [1] p(c_2) = [2] [0] [0] p(c_3) = [0 0 0] [0] [0 0 0] x1 + [0] [2 1 2] [2] p(c_4) = [0 0 0] [0] [0 0 0] x1 + [2] [2 0 0] [0] p(c_5) = [0 0 0] [0] [0 0 1] x1 + [1] [0 0 2] [0] p(c_6) = [1] [1] [0] p(c_7) = [0] [0] [0] p(c_8) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] Following rules are strictly oriented: top#(sent(x)) = [0 1 1] [2] [0 0 0] x + [1] [0 0 0] [0] > [0 1 1] [1] [0 0 0] x + [1] [0 0 0] [0] = c_8(top#(check(rest(x)))) Following rules are (at-least) weakly oriented: check(cons(x,y)) = [0 1 2] [2] [0 1 2] y + [2] [0 0 0] [0] >= [0 0 2] [0] [0 1 2] y + [2] [0 0 0] [0] = cons(x,y) check(cons(x,y)) = [0 1 2] [2] [0 1 2] y + [2] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] y + [2] [0 0 0] [0] = cons(x,check(y)) check(cons(x,y)) = [0 1 2] [2] [0 1 2] y + [2] [0 0 0] [0] >= [0 0 2] [0] [0 1 2] y + [2] [0 0 0] [0] = cons(check(x),y) check(rest(x)) = [0 1 1] [0] [0 1 1] x + [0] [0 0 0] [0] >= [0 1 0] [0] [0 1 0] x + [0] [0 0 0] [0] = rest(check(x)) check(sent(x)) = [0 1 0] [2] [0 1 0] x + [2] [0 0 0] [0] >= [0 1 0] [2] [0 1 0] x + [2] [0 0 0] [0] = sent(check(x)) rest(cons(x,y)) = [0 1 2] [2] [0 1 2] y + [2] [0 0 0] [0] >= [0 1 1] [2] [0 1 0] y + [2] [0 0 0] [0] = sent(y) rest(nil()) = [4] [2] [0] >= [4] [2] [0] = sent(nil()) **** Step 1.b:6.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {check#,rest#,top#} and constructors {cons,nil,sent} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:top#(sent(x)) -> c_8(top#(check(rest(x)))) -->_1 top#(sent(x)) -> c_8(top#(check(rest(x)))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: top#(sent(x)) -> c_8(top#(check(rest(x)))) **** Step 1.b:6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0 ,c_7/0,c_8/1} - Obligation: innermost 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))