WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs,ys) -> revapp(xs,ys)
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
        - Signature:
            {goal/2,revapp/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          goal#(xs,ys) -> c_1(revapp#(xs,ys))
          revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
          revapp#(Nil(),rest) -> c_3()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(xs,ys) -> c_1(revapp#(xs,ys))
            revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
            revapp#(Nil(),rest) -> c_3()
        - Strict TRS:
            goal(xs,ys) -> revapp(xs,ys)
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
        - Signature:
            {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          goal#(xs,ys) -> c_1(revapp#(xs,ys))
          revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
          revapp#(Nil(),rest) -> c_3()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(xs,ys) -> c_1(revapp#(xs,ys))
            revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
            revapp#(Nil(),rest) -> c_3()
        - Signature:
            {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3}
        by application of
          Pre({3}) = {1,2}.
        Here rules are labelled as follows:
          1: goal#(xs,ys) -> c_1(revapp#(xs,ys))
          2: revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
          3: revapp#(Nil(),rest) -> c_3()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(xs,ys) -> c_1(revapp#(xs,ys))
            revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
        - Weak DPs:
            revapp#(Nil(),rest) -> c_3()
        - Signature:
            {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:goal#(xs,ys) -> c_1(revapp#(xs,ys))
             -->_1 revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))):2
             -->_1 revapp#(Nil(),rest) -> c_3():3
          
          2:S:revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
             -->_1 revapp#(Nil(),rest) -> c_3():3
             -->_1 revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))):2
          
          3:W:revapp#(Nil(),rest) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: revapp#(Nil(),rest) -> c_3()
* Step 5: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(xs,ys) -> c_1(revapp#(xs,ys))
            revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
        - Signature:
            {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:goal#(xs,ys) -> c_1(revapp#(xs,ys))
           -->_1 revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))):2
        
        2:S:revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
           -->_1 revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))):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#(xs,ys) -> c_1(revapp#(xs,ys)))]
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
        - Signature:
            {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,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:
          2: revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
          
        The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
        - Signature:
            {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,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_2) = {1}
        
        Following symbols are considered usable:
          {goal#,revapp#}
        TcT has computed the following interpretation:
             p(Cons) = [1] x2 + [4]         
              p(Nil) = [2]                  
             p(goal) = [2] x2 + [1]         
           p(revapp) = [4] x1 + [2] x2 + [2]
            p(goal#) = [1] x1 + [4] x2 + [0]
          p(revapp#) = [4] x1 + [4]         
              p(c_1) = [0]                  
              p(c_2) = [1] x1 + [15]        
              p(c_3) = [0]                  
        
        Following rules are strictly oriented:
        revapp#(Cons(x,xs),rest) = [4] xs + [20]                
                                 > [4] xs + [19]                
                                 = c_2(revapp#(xs,Cons(x,rest)))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
        - Signature:
            {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil}
    + 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:
            revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
        - Signature:
            {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
             -->_1 revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))