WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            add0(x,Nil()) -> x
            add0(x',Cons(x,xs)) -> add0(Cons(Cons(Nil(),Nil()),x'),xs)
            goal(x,y) -> add0(x,y)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {add0/2,goal/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0,goal,notEmpty} and constructors {Cons,False,Nil
            ,True}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          add0#(x,Nil()) -> c_1()
          add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
          goal#(x,y) -> c_3(add0#(x,y))
          notEmpty#(Cons(x,xs)) -> c_4()
          notEmpty#(Nil()) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            add0#(x,Nil()) -> c_1()
            add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
            goal#(x,y) -> c_3(add0#(x,y))
            notEmpty#(Cons(x,xs)) -> c_4()
            notEmpty#(Nil()) -> c_5()
        - Strict TRS:
            add0(x,Nil()) -> x
            add0(x',Cons(x,xs)) -> add0(Cons(Cons(Nil(),Nil()),x'),xs)
            goal(x,y) -> add0(x,y)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {add0/2,goal/2,notEmpty/1,add0#/2,goal#/2,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/0,c_2/1,c_3/1
            ,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,notEmpty#} and constructors {Cons,False,Nil
            ,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          add0#(x,Nil()) -> c_1()
          add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
          goal#(x,y) -> c_3(add0#(x,y))
          notEmpty#(Cons(x,xs)) -> c_4()
          notEmpty#(Nil()) -> c_5()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            add0#(x,Nil()) -> c_1()
            add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
            goal#(x,y) -> c_3(add0#(x,y))
            notEmpty#(Cons(x,xs)) -> c_4()
            notEmpty#(Nil()) -> c_5()
        - Signature:
            {add0/2,goal/2,notEmpty/1,add0#/2,goal#/2,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/0,c_2/1,c_3/1
            ,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,notEmpty#} and constructors {Cons,False,Nil
            ,True}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5}
        by application of
          Pre({1,4,5}) = {2,3}.
        Here rules are labelled as follows:
          1: add0#(x,Nil()) -> c_1()
          2: add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
          3: goal#(x,y) -> c_3(add0#(x,y))
          4: notEmpty#(Cons(x,xs)) -> c_4()
          5: notEmpty#(Nil()) -> c_5()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
            goal#(x,y) -> c_3(add0#(x,y))
        - Weak DPs:
            add0#(x,Nil()) -> c_1()
            notEmpty#(Cons(x,xs)) -> c_4()
            notEmpty#(Nil()) -> c_5()
        - Signature:
            {add0/2,goal/2,notEmpty/1,add0#/2,goal#/2,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/0,c_2/1,c_3/1
            ,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,notEmpty#} and constructors {Cons,False,Nil
            ,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
             -->_1 add0#(x,Nil()) -> c_1():3
             -->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs)):1
          
          2:S:goal#(x,y) -> c_3(add0#(x,y))
             -->_1 add0#(x,Nil()) -> c_1():3
             -->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs)):1
          
          3:W:add0#(x,Nil()) -> c_1()
             
          
          4:W:notEmpty#(Cons(x,xs)) -> c_4()
             
          
          5:W:notEmpty#(Nil()) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: notEmpty#(Nil()) -> c_5()
          4: notEmpty#(Cons(x,xs)) -> c_4()
          3: add0#(x,Nil()) -> c_1()
* Step 5: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
            goal#(x,y) -> c_3(add0#(x,y))
        - Signature:
            {add0/2,goal/2,notEmpty/1,add0#/2,goal#/2,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/0,c_2/1,c_3/1
            ,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,notEmpty#} and constructors {Cons,False,Nil
            ,True}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
           -->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs)):1
        
        2:S:goal#(x,y) -> c_3(add0#(x,y))
           -->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs)):1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(2,goal#(x,y) -> c_3(add0#(x,y)))]
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
        - Signature:
            {add0/2,goal/2,notEmpty/1,add0#/2,goal#/2,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/0,c_2/1,c_3/1
            ,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,notEmpty#} and constructors {Cons,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: add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
          
        The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
        - Signature:
            {add0/2,goal/2,notEmpty/1,add0#/2,goal#/2,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/0,c_2/1,c_3/1
            ,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,notEmpty#} and constructors {Cons,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_2) = {1}
        
        Following symbols are considered usable:
          {add0#,goal#,notEmpty#}
        TcT has computed the following interpretation:
               p(Cons) = [1] x1 + [1] x2 + [8]
              p(False) = [1]                  
                p(Nil) = [0]                  
               p(True) = [0]                  
               p(add0) = [2]                  
               p(goal) = [0]                  
           p(notEmpty) = [2] x1 + [0]         
              p(add0#) = [2] x2 + [0]         
              p(goal#) = [2] x1 + [2] x2 + [0]
          p(notEmpty#) = [8] x1 + [1]         
                p(c_1) = [8]                  
                p(c_2) = [1] x1 + [8]         
                p(c_3) = [1] x1 + [1]         
                p(c_4) = [0]                  
                p(c_5) = [1]                  
        
        Following rules are strictly oriented:
        add0#(x',Cons(x,xs)) = [2] x + [2] xs + [16]                    
                             > [2] xs + [8]                             
                             = c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
        - Signature:
            {add0/2,goal/2,notEmpty/1,add0#/2,goal#/2,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/0,c_2/1,c_3/1
            ,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,notEmpty#} and constructors {Cons,False,Nil
            ,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
        - Signature:
            {add0/2,goal/2,notEmpty/1,add0#/2,goal#/2,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/0,c_2/1,c_3/1
            ,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,notEmpty#} and constructors {Cons,False,Nil
            ,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
             -->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: add0#(x',Cons(x,xs)) -> c_2(add0#(Cons(Cons(Nil(),Nil()),x'),xs))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {add0/2,goal/2,notEmpty/1,add0#/2,goal#/2,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/0,c_2/1,c_3/1
            ,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,notEmpty#} and constructors {Cons,False,Nil
            ,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))