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_ = DT}
    + Details:
        We add the following dependency tuples:
        
        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: 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()
        - Weak 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:
        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 3: 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()
        - Weak 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:
        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 4: 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)))
        - Weak 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:
        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 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
        - Weak 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:
          revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest)))
* Step 6: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          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:
            all
          TcT has computed the following interpretation:
               p(Cons) = [1] x2 + [1]
                p(Nil) = [0]         
               p(goal) = [0]         
             p(revapp) = [0]         
              p(goal#) = [0]         
            p(revapp#) = [1] x1 + [0]
                p(c_1) = [0]         
                p(c_2) = [1] x1 + [0]
                p(c_3) = [0]         
          
          Following rules are strictly oriented:
          revapp#(Cons(x,xs),rest) = [1] xs + [1]                 
                                   > [1] xs + [0]                 
                                   = c_2(revapp#(xs,Cons(x,rest)))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor 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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))