WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 main(x4,x2) -> app_xs#1(x4,app_xs#1(x4,x2)) - Signature: {app_xs#1/2,main/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1,main} and constructors {Cons,Nil} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) app_xs#1#(Nil(),x6) -> c_2() main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) app_xs#1#(Nil(),x6) -> c_2() main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Strict TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 main(x4,x2) -> app_xs#1(x4,app_xs#1(x4,x2)) - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) app_xs#1#(Nil(),x6) -> c_2() main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) app_xs#1#(Nil(),x6) -> c_2() main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Strict TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(app_xs#1#) = {2}, uargs(c_1) = {1}, uargs(c_3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [1] x2 + [1] p(Nil) = [0] p(app_xs#1) = [4] x1 + [1] x2 + [3] p(main) = [0] p(app_xs#1#) = [9] x1 + [1] x2 + [0] p(main#) = [14] x1 + [1] x2 + [2] p(c_1) = [1] x1 + [2] p(c_2) = [0] p(c_3) = [1] x1 + [0] Following rules are strictly oriented: app_xs#1#(Cons(x7,x8),x10) = [1] x10 + [9] x8 + [9] > [1] x10 + [9] x8 + [2] = c_1(app_xs#1#(x8,x10)) app_xs#1(Cons(x7,x8),x10) = [1] x10 + [4] x8 + [7] > [1] x10 + [4] x8 + [4] = Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) = [1] x6 + [3] > [1] x6 + [0] = x6 Following rules are (at-least) weakly oriented: app_xs#1#(Nil(),x6) = [1] x6 + [0] >= [0] = c_2() main#(x4,x2) = [1] x2 + [14] x4 + [2] >= [1] x2 + [13] x4 + [3] = c_3(app_xs#1#(x4,app_xs#1(x4,x2))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: Decompose WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: app_xs#1#(Nil(),x6) -> c_2() main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Weak DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + 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: app_xs#1#(Nil(),x6) -> c_2() - Weak DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} Problem (S) - Strict DPs: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Weak DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) app_xs#1#(Nil(),x6) -> c_2() - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} ** Step 4.a:1: PredecessorEstimationCP WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: app_xs#1#(Nil(),x6) -> c_2() - Weak DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: app_xs#1#(Nil(),x6) -> c_2() Consider the set of all dependency pairs 1: app_xs#1#(Nil(),x6) -> c_2() 2: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) 3: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) Processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 4.a:1.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: app_xs#1#(Nil(),x6) -> c_2() - Weak DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, 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 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_3) = {1} Following symbols are considered usable: {app_xs#1,app_xs#1#,main#} TcT has computed the following interpretation: p(Cons) = [2] p(Nil) = [0] p(app_xs#1) = [2] x1 + [1] x2 + [0] p(main) = [1] x1 + [1] p(app_xs#1#) = [1] x2 + [1] p(main#) = [8] x1 + [4] x2 + [4] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [4] x1 + [0] Following rules are strictly oriented: app_xs#1#(Nil(),x6) = [1] x6 + [1] > [0] = c_2() Following rules are (at-least) weakly oriented: app_xs#1#(Cons(x7,x8),x10) = [1] x10 + [1] >= [1] x10 + [1] = c_1(app_xs#1#(x8,x10)) main#(x4,x2) = [4] x2 + [8] x4 + [4] >= [4] x2 + [8] x4 + [4] = c_3(app_xs#1#(x4,app_xs#1(x4,x2))) app_xs#1(Cons(x7,x8),x10) = [1] x10 + [4] >= [2] = Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) = [1] x6 + [0] >= [1] x6 + [0] = x6 *** Step 4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) app_xs#1#(Nil(),x6) -> c_2() main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) app_xs#1#(Nil(),x6) -> c_2() main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) -->_1 app_xs#1#(Nil(),x6) -> c_2():2 -->_1 app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)):1 2:W:app_xs#1#(Nil(),x6) -> c_2() 3:W:main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) -->_1 app_xs#1#(Nil(),x6) -> c_2():2 -->_1 app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) 1: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) 2: app_xs#1#(Nil(),x6) -> c_2() *** Step 4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 4.b:1: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Weak DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) app_xs#1#(Nil(),x6) -> c_2() - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) 2: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) 3: app_xs#1#(Nil(),x6) -> c_2() ** Step 4.b:2: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) app_xs#1#(Nil(),x6) -> c_2() main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) -->_1 app_xs#1#(Nil(),x6) -> c_2():2 -->_1 app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)):1 2:W:app_xs#1#(Nil(),x6) -> c_2() 3:W:main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) -->_1 app_xs#1#(Nil(),x6) -> c_2():2 -->_1 app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: main#(x4,x2) -> c_3(app_xs#1#(x4,app_xs#1(x4,x2))) 1: app_xs#1#(Cons(x7,x8),x10) -> c_1(app_xs#1#(x8,x10)) 2: app_xs#1#(Nil(),x6) -> c_2() ** Step 4.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: app_xs#1(Cons(x7,x8),x10) -> Cons(x7,app_xs#1(x8,x10)) app_xs#1(Nil(),x6) -> x6 - Signature: {app_xs#1/2,main/2,app_xs#1#/2,main#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {app_xs#1#,main#} and constructors {Cons,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))