WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1} / {dd/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,rev} and constructors {dd,nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
          @#(nil(),xs) -> c_2()
          rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
          rev#(nil()) -> c_4()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
            @#(nil(),xs) -> c_2()
            rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
            rev#(nil()) -> c_4()
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4}
        by application of
          Pre({2,4}) = {1,3}.
        Here rules are labelled as follows:
          1: @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
          2: @#(nil(),xs) -> c_2()
          3: rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
          4: rev#(nil()) -> c_4()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
            rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
        - Weak DPs:
            @#(nil(),xs) -> c_2()
            rev#(nil()) -> c_4()
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:@#(dd(x,xs),ys) -> c_1(@#(xs,ys))
             -->_1 @#(nil(),xs) -> c_2():3
             -->_1 @#(dd(x,xs),ys) -> c_1(@#(xs,ys)):1
          
          2:S:rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
             -->_2 rev#(nil()) -> c_4():4
             -->_1 @#(nil(),xs) -> c_2():3
             -->_2 rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs)):2
             -->_1 @#(dd(x,xs),ys) -> c_1(@#(xs,ys)):1
          
          3:W:@#(nil(),xs) -> c_2()
             
          
          4:W:rev#(nil()) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: rev#(nil()) -> c_4()
          3: @#(nil(),xs) -> c_2()
* Step 4: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
            rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,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:
              @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
          - Weak DPs:
              rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
          - Weak TRS:
              @(dd(x,xs),ys) -> dd(x,@(xs,ys))
              @(nil(),xs) -> xs
              rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
              rev(nil()) -> nil()
          - Signature:
              {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
        
        Problem (S)
          - Strict DPs:
              rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
          - Weak DPs:
              @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
          - Weak TRS:
              @(dd(x,xs),ys) -> dd(x,@(xs,ys))
              @(nil(),xs) -> xs
              rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
              rev(nil()) -> nil()
          - Signature:
              {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
** Step 4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
        - Weak DPs:
            rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
          
        The strictly oriented rules are moved into the weak component.
*** Step 4.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
        - Weak DPs:
            rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_3) = {1,2}
        
        Following symbols are considered usable:
          {@,rev,@#,rev#}
        TcT has computed the following interpretation:
             p(@) = 1 + x1 + x2 
            p(dd) = 1 + x2      
           p(nil) = 0           
           p(rev) = 2*x1        
            p(@#) = 2*x1 + x1*x2
          p(rev#) = 4*x1^2      
           p(c_1) = x1          
           p(c_2) = 4           
           p(c_3) = 4 + x1 + x2 
           p(c_4) = 1           
        
        Following rules are strictly oriented:
        @#(dd(x,xs),ys) = 2 + 2*xs + xs*ys + ys
                        > 2*xs + xs*ys         
                        = c_1(@#(xs,ys))       
        
        
        Following rules are (at-least) weakly oriented:
        rev#(dd(x,xs)) =  4 + 8*xs + 4*xs^2                    
                       >= 4 + 6*xs + 4*xs^2                    
                       =  c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
        
        @(dd(x,xs),ys) =  2 + xs + ys                          
                       >= 2 + xs + ys                          
                       =  dd(x,@(xs,ys))                       
        
           @(nil(),xs) =  1 + xs                               
                       >= xs                                   
                       =  xs                                   
        
         rev(dd(x,xs)) =  2 + 2*xs                             
                       >= 2 + 2*xs                             
                       =  @(rev(xs),dd(x,nil()))               
        
            rev(nil()) =  0                                    
                       >= 0                                    
                       =  nil()                                
        
*** Step 4.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
            rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,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:
            @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
            rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:@#(dd(x,xs),ys) -> c_1(@#(xs,ys))
             -->_1 @#(dd(x,xs),ys) -> c_1(@#(xs,ys)):1
          
          2:W:rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
             -->_2 rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs)):2
             -->_1 @#(dd(x,xs),ys) -> c_1(@#(xs,ys)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
          1: @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
*** Step 4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
        - Weak DPs:
            @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
             -->_1 @#(dd(x,xs),ys) -> c_1(@#(xs,ys)):2
             -->_2 rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs)):1
          
          2:W:@#(dd(x,xs),ys) -> c_1(@#(xs,ys))
             -->_1 @#(dd(x,xs),ys) -> c_1(@#(xs,ys)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: @#(dd(x,xs),ys) -> c_1(@#(xs,ys))
** Step 4.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs))
             -->_2 rev#(dd(x,xs)) -> c_3(@#(rev(xs),dd(x,nil())),rev#(xs)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          rev#(dd(x,xs)) -> c_3(rev#(xs))
** Step 4.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(dd(x,xs)) -> c_3(rev#(xs))
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            rev(dd(x,xs)) -> @(rev(xs),dd(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          rev#(dd(x,xs)) -> c_3(rev#(xs))
** Step 4.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(dd(x,xs)) -> c_3(rev#(xs))
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + 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: rev#(dd(x,xs)) -> c_3(rev#(xs))
          
        The strictly oriented rules are moved into the weak component.
*** Step 4.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(dd(x,xs)) -> c_3(rev#(xs))
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + 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_3) = {1}
        
        Following symbols are considered usable:
          {@#,rev#}
        TcT has computed the following interpretation:
             p(@) = [1] x1 + [8] x2 + [8]
            p(dd) = [1] x2 + [2]         
           p(nil) = [1]                  
           p(rev) = [1] x1 + [2]         
            p(@#) = [1] x2 + [2]         
          p(rev#) = [8] x1 + [0]         
           p(c_1) = [1] x1 + [0]         
           p(c_2) = [2]                  
           p(c_3) = [1] x1 + [12]        
           p(c_4) = [1]                  
        
        Following rules are strictly oriented:
        rev#(dd(x,xs)) = [8] xs + [16]
                       > [8] xs + [12]
                       = c_3(rev#(xs))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 4.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            rev#(dd(x,xs)) -> c_3(rev#(xs))
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 4.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            rev#(dd(x,xs)) -> c_3(rev#(xs))
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:rev#(dd(x,xs)) -> c_3(rev#(xs))
             -->_1 rev#(dd(x,xs)) -> c_3(rev#(xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: rev#(dd(x,xs)) -> c_3(rev#(xs))
*** Step 4.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {@/2,rev/1,@#/2,rev#/1} / {dd/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,rev#} and constructors {dd,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))