WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True
            ,isEmpty[Match]}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True
            ,isEmpty[Match]}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True
            ,isEmpty[Match]}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          list(y){y -> Cons(x,y)} =
            list(Cons(x,y)) ->^+ list(y)
              = C[list(y) = list(y){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,list,notEmpty} and constructors {Cons,False,Nil,True
            ,isEmpty[Match]}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          goal#(x) -> c_1(list#(x))
          list#(Cons(x,xs)) -> c_2(list#(xs))
          list#(Nil()) -> c_3()
          list#(Nil()) -> c_4()
          notEmpty#(Cons(x,xs)) -> c_5()
          notEmpty#(Nil()) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(x) -> c_1(list#(x))
            list#(Cons(x,xs)) -> c_2(list#(xs))
            list#(Nil()) -> c_3()
            list#(Nil()) -> c_4()
            notEmpty#(Cons(x,xs)) -> c_5()
            notEmpty#(Nil()) -> c_6()
        - Weak TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1,goal#/1,list#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1,c_1/1
            ,c_2/1,c_3/0,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,list#,notEmpty#} and constructors {Cons,False,Nil
            ,True,isEmpty[Match]}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4,5,6}
        by application of
          Pre({3,4,5,6}) = {1,2}.
        Here rules are labelled as follows:
          1: goal#(x) -> c_1(list#(x))
          2: list#(Cons(x,xs)) -> c_2(list#(xs))
          3: list#(Nil()) -> c_3()
          4: list#(Nil()) -> c_4()
          5: notEmpty#(Cons(x,xs)) -> c_5()
          6: notEmpty#(Nil()) -> c_6()
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(x) -> c_1(list#(x))
            list#(Cons(x,xs)) -> c_2(list#(xs))
        - Weak DPs:
            list#(Nil()) -> c_3()
            list#(Nil()) -> c_4()
            notEmpty#(Cons(x,xs)) -> c_5()
            notEmpty#(Nil()) -> c_6()
        - Weak TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1,goal#/1,list#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1,c_1/1
            ,c_2/1,c_3/0,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,list#,notEmpty#} and constructors {Cons,False,Nil
            ,True,isEmpty[Match]}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:goal#(x) -> c_1(list#(x))
             -->_1 list#(Cons(x,xs)) -> c_2(list#(xs)):2
             -->_1 list#(Nil()) -> c_4():4
             -->_1 list#(Nil()) -> c_3():3
          
          2:S:list#(Cons(x,xs)) -> c_2(list#(xs))
             -->_1 list#(Nil()) -> c_4():4
             -->_1 list#(Nil()) -> c_3():3
             -->_1 list#(Cons(x,xs)) -> c_2(list#(xs)):2
          
          3:W:list#(Nil()) -> c_3()
             
          
          4:W:list#(Nil()) -> c_4()
             
          
          5:W:notEmpty#(Cons(x,xs)) -> c_5()
             
          
          6:W:notEmpty#(Nil()) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: notEmpty#(Nil()) -> c_6()
          5: notEmpty#(Cons(x,xs)) -> c_5()
          3: list#(Nil()) -> c_3()
          4: list#(Nil()) -> c_4()
** Step 1.b:4: RemoveHeads.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(x) -> c_1(list#(x))
            list#(Cons(x,xs)) -> c_2(list#(xs))
        - Weak TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1,goal#/1,list#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1,c_1/1
            ,c_2/1,c_3/0,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,list#,notEmpty#} and constructors {Cons,False,Nil
            ,True,isEmpty[Match]}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:goal#(x) -> c_1(list#(x))
           -->_1 list#(Cons(x,xs)) -> c_2(list#(xs)):2
        
        2:S:list#(Cons(x,xs)) -> c_2(list#(xs))
           -->_1 list#(Cons(x,xs)) -> c_2(list#(xs)):2
        
        
        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).
        
        [(1,goal#(x) -> c_1(list#(x)))]
** Step 1.b:5: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            list#(Cons(x,xs)) -> c_2(list#(xs))
        - Weak TRS:
            goal(x) -> list(x)
            list(Cons(x,xs)) -> list(xs)
            list(Nil()) -> True()
            list(Nil()) -> isEmpty[Match](Nil())
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {goal/1,list/1,notEmpty/1,goal#/1,list#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1,c_1/1
            ,c_2/1,c_3/0,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,list#,notEmpty#} and constructors {Cons,False,Nil
            ,True,isEmpty[Match]}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          list#(Cons(x,xs)) -> c_2(list#(xs))
** Step 1.b:6: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            list#(Cons(x,xs)) -> c_2(list#(xs))
        - Signature:
            {goal/1,list/1,notEmpty/1,goal#/1,list#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1,c_1/1
            ,c_2/1,c_3/0,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,list#,notEmpty#} and constructors {Cons,False,Nil
            ,True,isEmpty[Match]}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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:
          {goal#,list#,notEmpty#}
        TcT has computed the following interpretation:
                    p(Cons) = [1] x2 + [1]
                   p(False) = [1]         
                     p(Nil) = [1]         
                    p(True) = [0]         
                    p(goal) = [2]         
          p(isEmpty[Match]) = [0]         
                    p(list) = [1] x1 + [4]
                p(notEmpty) = [1]         
                   p(goal#) = [8] x1 + [0]
                   p(list#) = [2] x1 + [1]
               p(notEmpty#) = [4] x1 + [1]
                     p(c_1) = [2] x1 + [0]
                     p(c_2) = [1] x1 + [0]
                     p(c_3) = [8]         
                     p(c_4) = [0]         
                     p(c_5) = [2]         
                     p(c_6) = [0]         
        
        Following rules are strictly oriented:
        list#(Cons(x,xs)) = [2] xs + [3]  
                          > [2] xs + [1]  
                          = c_2(list#(xs))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:7: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            list#(Cons(x,xs)) -> c_2(list#(xs))
        - Signature:
            {goal/1,list/1,notEmpty/1,goal#/1,list#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,isEmpty[Match]/1,c_1/1
            ,c_2/1,c_3/0,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,list#,notEmpty#} and constructors {Cons,False,Nil
            ,True,isEmpty[Match]}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))