* Step 1: ToInnermost WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            car(.(x,y)) -> x
            cdr(.(x,y)) -> y
            null(.(x,y)) -> false()
            null(nil()) -> true()
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1} / {./2,false/0,nil/0,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {++,car,cdr,null,rev} and constructors {.,false,nil,true}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            car(.(x,y)) -> x
            cdr(.(x,y)) -> y
            null(.(x,y)) -> false()
            null(nil()) -> true()
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1} / {./2,false/0,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++,car,cdr,null,rev} and constructors {.,false,nil,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          ++#(.(x,y),z) -> c_1(++#(y,z))
          ++#(nil(),y) -> c_2()
          car#(.(x,y)) -> c_3()
          cdr#(.(x,y)) -> c_4()
          null#(.(x,y)) -> c_5()
          null#(nil()) -> c_6()
          rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
          rev#(nil()) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(.(x,y),z) -> c_1(++#(y,z))
            ++#(nil(),y) -> c_2()
            car#(.(x,y)) -> c_3()
            cdr#(.(x,y)) -> c_4()
            null#(.(x,y)) -> c_5()
            null#(nil()) -> c_6()
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
            rev#(nil()) -> c_8()
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            car(.(x,y)) -> x
            cdr(.(x,y)) -> y
            null(.(x,y)) -> false()
            null(nil()) -> true()
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          ++(.(x,y),z) -> .(x,++(y,z))
          ++(nil(),y) -> y
          rev(.(x,y)) -> ++(rev(y),.(x,nil()))
          rev(nil()) -> nil()
          ++#(.(x,y),z) -> c_1(++#(y,z))
          ++#(nil(),y) -> c_2()
          car#(.(x,y)) -> c_3()
          cdr#(.(x,y)) -> c_4()
          null#(.(x,y)) -> c_5()
          null#(nil()) -> c_6()
          rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
          rev#(nil()) -> c_8()
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(.(x,y),z) -> c_1(++#(y,z))
            ++#(nil(),y) -> c_2()
            car#(.(x,y)) -> c_3()
            cdr#(.(x,y)) -> c_4()
            null#(.(x,y)) -> c_5()
            null#(nil()) -> c_6()
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
            rev#(nil()) -> c_8()
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4,5,6,8}
        by application of
          Pre({2,3,4,5,6,8}) = {1,7}.
        Here rules are labelled as follows:
          1: ++#(.(x,y),z) -> c_1(++#(y,z))
          2: ++#(nil(),y) -> c_2()
          3: car#(.(x,y)) -> c_3()
          4: cdr#(.(x,y)) -> c_4()
          5: null#(.(x,y)) -> c_5()
          6: null#(nil()) -> c_6()
          7: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
          8: rev#(nil()) -> c_8()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(.(x,y),z) -> c_1(++#(y,z))
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
        - Weak DPs:
            ++#(nil(),y) -> c_2()
            car#(.(x,y)) -> c_3()
            cdr#(.(x,y)) -> c_4()
            null#(.(x,y)) -> c_5()
            null#(nil()) -> c_6()
            rev#(nil()) -> c_8()
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:++#(.(x,y),z) -> c_1(++#(y,z))
             -->_1 ++#(nil(),y) -> c_2():3
             -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):1
          
          2:S:rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
             -->_2 rev#(nil()) -> c_8():8
             -->_1 ++#(nil(),y) -> c_2():3
             -->_2 rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)):2
             -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):1
          
          3:W:++#(nil(),y) -> c_2()
             
          
          4:W:car#(.(x,y)) -> c_3()
             
          
          5:W:cdr#(.(x,y)) -> c_4()
             
          
          6:W:null#(.(x,y)) -> c_5()
             
          
          7:W:null#(nil()) -> c_6()
             
          
          8:W:rev#(nil()) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: null#(nil()) -> c_6()
          6: null#(.(x,y)) -> c_5()
          5: cdr#(.(x,y)) -> c_4()
          4: car#(.(x,y)) -> c_3()
          8: rev#(nil()) -> c_8()
          3: ++#(nil(),y) -> c_2()
* Step 6: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(.(x,y),z) -> c_1(++#(y,z))
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + 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:
              ++#(.(x,y),z) -> c_1(++#(y,z))
          - Weak DPs:
              rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
          - Weak TRS:
              ++(.(x,y),z) -> .(x,++(y,z))
              ++(nil(),y) -> y
              rev(.(x,y)) -> ++(rev(y),.(x,nil()))
              rev(nil()) -> nil()
          - Signature:
              {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
              ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
              ,true}
        
        Problem (S)
          - Strict DPs:
              rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
          - Weak DPs:
              ++#(.(x,y),z) -> c_1(++#(y,z))
          - Weak TRS:
              ++(.(x,y),z) -> .(x,++(y,z))
              ++(nil(),y) -> y
              rev(.(x,y)) -> ++(rev(y),.(x,nil()))
              rev(nil()) -> nil()
          - Signature:
              {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
              ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
              ,true}
** Step 6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(.(x,y),z) -> c_1(++#(y,z))
        - Weak DPs:
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + 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: ++#(.(x,y),z) -> c_1(++#(y,z))
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(.(x,y),z) -> c_1(++#(y,z))
        - Weak DPs:
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + 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_7) = {1,2}
        
        Following symbols are considered usable:
          {++,rev,++#,car#,cdr#,null#,rev#}
        TcT has computed the following interpretation:
             p(++) = x1 + 2*x2      
              p(.) = 1 + x2         
            p(car) = 1 + x1 + 2*x1^2
            p(cdr) = 1              
          p(false) = 0              
            p(nil) = 0              
           p(null) = 4*x1 + x1^2    
            p(rev) = 2 + 2*x1       
           p(true) = 0              
            p(++#) = 4*x1           
           p(car#) = 0              
           p(cdr#) = 4 + x1         
          p(null#) = 1 + 2*x1^2     
           p(rev#) = 4*x1 + 4*x1^2  
            p(c_1) = 1 + x1         
            p(c_2) = 0              
            p(c_3) = 1              
            p(c_4) = 0              
            p(c_5) = 0              
            p(c_6) = 0              
            p(c_7) = x1 + x2        
            p(c_8) = 0              
        
        Following rules are strictly oriented:
        ++#(.(x,y),z) = 4 + 4*y      
                      > 1 + 4*y      
                      = c_1(++#(y,z))
        
        
        Following rules are (at-least) weakly oriented:
        rev#(.(x,y)) =  8 + 12*y + 4*y^2                   
                     >= 8 + 12*y + 4*y^2                   
                     =  c_7(++#(rev(y),.(x,nil())),rev#(y))
        
        ++(.(x,y),z) =  1 + y + 2*z                        
                     >= 1 + y + 2*z                        
                     =  .(x,++(y,z))                       
        
         ++(nil(),y) =  2*y                                
                     >= y                                  
                     =  y                                  
        
         rev(.(x,y)) =  4 + 2*y                            
                     >= 4 + 2*y                            
                     =  ++(rev(y),.(x,nil()))              
        
          rev(nil()) =  2                                  
                     >= 0                                  
                     =  nil()                              
        
*** Step 6.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ++#(.(x,y),z) -> c_1(++#(y,z))
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ++#(.(x,y),z) -> c_1(++#(y,z))
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:++#(.(x,y),z) -> c_1(++#(y,z))
             -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):1
          
          2:W:rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
             -->_2 rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)):2
             -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
          1: ++#(.(x,y),z) -> c_1(++#(y,z))
*** Step 6.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
        - Weak DPs:
            ++#(.(x,y),z) -> c_1(++#(y,z))
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
             -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):2
             -->_2 rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)):1
          
          2:W:++#(.(x,y),z) -> c_1(++#(y,z))
             -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: ++#(.(x,y),z) -> c_1(++#(y,z))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y))
             -->_2 rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          rev#(.(x,y)) -> c_7(rev#(y))
** Step 6.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(.(x,y)) -> c_7(rev#(y))
        - Weak TRS:
            ++(.(x,y),z) -> .(x,++(y,z))
            ++(nil(),y) -> y
            rev(.(x,y)) -> ++(rev(y),.(x,nil()))
            rev(nil()) -> nil()
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          rev#(.(x,y)) -> c_7(rev#(y))
** Step 6.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(.(x,y)) -> c_7(rev#(y))
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + 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#(.(x,y)) -> c_7(rev#(y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            rev#(.(x,y)) -> c_7(rev#(y))
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + 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_7) = {1}
        
        Following symbols are considered usable:
          {++#,car#,cdr#,null#,rev#}
        TcT has computed the following interpretation:
             p(++) = [2] x1 + [1] x2 + [0]
              p(.) = [1] x2 + [4]         
            p(car) = [2] x1 + [1]         
            p(cdr) = [0]                  
          p(false) = [0]                  
            p(nil) = [1]                  
           p(null) = [0]                  
            p(rev) = [4] x1 + [1]         
           p(true) = [0]                  
            p(++#) = [1] x1 + [1]         
           p(car#) = [1] x1 + [1]         
           p(cdr#) = [1] x1 + [2]         
          p(null#) = [1] x1 + [4]         
           p(rev#) = [4] x1 + [8]         
            p(c_1) = [1] x1 + [0]         
            p(c_2) = [1]                  
            p(c_3) = [1]                  
            p(c_4) = [1]                  
            p(c_5) = [0]                  
            p(c_6) = [0]                  
            p(c_7) = [1] x1 + [12]        
            p(c_8) = [0]                  
        
        Following rules are strictly oriented:
        rev#(.(x,y)) = [4] y + [24]
                     > [4] y + [20]
                     = c_7(rev#(y))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 6.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            rev#(.(x,y)) -> c_7(rev#(y))
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            rev#(.(x,y)) -> c_7(rev#(y))
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:rev#(.(x,y)) -> c_7(rev#(y))
             -->_1 rev#(.(x,y)) -> c_7(rev#(y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: rev#(.(x,y)) -> c_7(rev#(y))
*** Step 6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))