WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            and(tt(),X) -> activate(X)
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          and#(tt(),X) -> c_2(activate#(X))
          plus#(N,0()) -> c_3()
          plus#(N,s(M)) -> c_4(plus#(N,M))
          x#(N,0()) -> c_5()
          x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            and#(tt(),X) -> c_2(activate#(X))
            plus#(N,0()) -> c_3()
            plus#(N,s(M)) -> c_4(plus#(N,M))
            x#(N,0()) -> c_5()
            x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
        - Weak TRS:
            activate(X) -> X
            and(tt(),X) -> activate(X)
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,5}
        by application of
          Pre({1,3,5}) = {2,4,6}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: and#(tt(),X) -> c_2(activate#(X))
          3: plus#(N,0()) -> c_3()
          4: plus#(N,s(M)) -> c_4(plus#(N,M))
          5: x#(N,0()) -> c_5()
          6: x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            and#(tt(),X) -> c_2(activate#(X))
            plus#(N,s(M)) -> c_4(plus#(N,M))
            x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
        - Weak DPs:
            activate#(X) -> c_1()
            plus#(N,0()) -> c_3()
            x#(N,0()) -> c_5()
        - Weak TRS:
            activate(X) -> X
            and(tt(),X) -> activate(X)
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: and#(tt(),X) -> c_2(activate#(X))
          2: plus#(N,s(M)) -> c_4(plus#(N,M))
          3: x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
          4: activate#(X) -> c_1()
          5: plus#(N,0()) -> c_3()
          6: x#(N,0()) -> c_5()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(N,s(M)) -> c_4(plus#(N,M))
            x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
        - Weak DPs:
            activate#(X) -> c_1()
            and#(tt(),X) -> c_2(activate#(X))
            plus#(N,0()) -> c_3()
            x#(N,0()) -> c_5()
        - Weak TRS:
            activate(X) -> X
            and(tt(),X) -> activate(X)
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:plus#(N,s(M)) -> c_4(plus#(N,M))
             -->_1 plus#(N,0()) -> c_3():5
             -->_1 plus#(N,s(M)) -> c_4(plus#(N,M)):1
          
          2:S:x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
             -->_2 x#(N,0()) -> c_5():6
             -->_1 plus#(N,0()) -> c_3():5
             -->_2 x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M)):2
             -->_1 plus#(N,s(M)) -> c_4(plus#(N,M)):1
          
          3:W:activate#(X) -> c_1()
             
          
          4:W:and#(tt(),X) -> c_2(activate#(X))
             -->_1 activate#(X) -> c_1():3
          
          5:W:plus#(N,0()) -> c_3()
             
          
          6:W:x#(N,0()) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: and#(tt(),X) -> c_2(activate#(X))
          3: activate#(X) -> c_1()
          6: x#(N,0()) -> c_5()
          5: plus#(N,0()) -> c_3()
* Step 5: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(N,s(M)) -> c_4(plus#(N,M))
            x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
        - Weak TRS:
            activate(X) -> X
            and(tt(),X) -> activate(X)
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          plus(N,0()) -> N
          plus(N,s(M)) -> s(plus(N,M))
          x(N,0()) -> 0()
          x(N,s(M)) -> plus(x(N,M),N)
          plus#(N,s(M)) -> c_4(plus#(N,M))
          x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
* Step 6: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(N,s(M)) -> c_4(plus#(N,M))
            x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
        - Weak TRS:
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
        and a lower component
          plus#(N,s(M)) -> c_4(plus#(N,M))
        Further, following extension rules are added to the lower component.
          x#(N,s(M)) -> plus#(x(N,M),N)
          x#(N,s(M)) -> x#(N,M)
** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
        - Weak TRS:
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
             -->_2 x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          x#(N,s(M)) -> c_6(x#(N,M))
** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            x#(N,s(M)) -> c_6(x#(N,M))
        - Weak TRS:
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          x#(N,s(M)) -> c_6(x#(N,M))
** Step 6.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            x#(N,s(M)) -> c_6(x#(N,M))
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + 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_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]         
             p(activate) = [0]         
                  p(and) = [0]         
                 p(plus) = [0]         
                    p(s) = [1] x1 + [5]
                   p(tt) = [0]         
                    p(x) = [0]         
            p(activate#) = [0]         
                 p(and#) = [0]         
                p(plus#) = [0]         
                   p(x#) = [3] x2 + [0]
                  p(c_1) = [0]         
                  p(c_2) = [0]         
                  p(c_3) = [0]         
                  p(c_4) = [0]         
                  p(c_5) = [0]         
                  p(c_6) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          x#(N,s(M)) = [3] M + [15]
                     > [3] M + [0] 
                     = c_6(x#(N,M))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            x#(N,s(M)) -> c_6(x#(N,M))
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            plus#(N,s(M)) -> c_4(plus#(N,M))
        - Weak DPs:
            x#(N,s(M)) -> plus#(x(N,M),N)
            x#(N,s(M)) -> x#(N,M)
        - Weak TRS:
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + 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_4) = {1}
        
        Following symbols are considered usable:
          {activate#,and#,plus#,x#}
        TcT has computed the following interpretation:
                  p(0) = [1]                  
           p(activate) = [0]                  
                p(and) = [1] x2 + [0]         
               p(plus) = [3] x1 + [7] x2 + [0]
                  p(s) = [1] x1 + [2]         
                 p(tt) = [0]                  
                  p(x) = [6] x1 + [0]         
          p(activate#) = [0]                  
               p(and#) = [2] x1 + [0]         
              p(plus#) = [1] x2 + [4]         
                 p(x#) = [8] x1 + [4]         
                p(c_1) = [0]                  
                p(c_2) = [0]                  
                p(c_3) = [0]                  
                p(c_4) = [1] x1 + [0]         
                p(c_5) = [0]                  
                p(c_6) = [8] x2 + [0]         
        
        Following rules are strictly oriented:
        plus#(N,s(M)) = [1] M + [6]    
                      > [1] M + [4]    
                      = c_4(plus#(N,M))
        
        
        Following rules are (at-least) weakly oriented:
        x#(N,s(M)) =  [8] N + [4]    
                   >= [1] N + [4]    
                   =  plus#(x(N,M),N)
        
        x#(N,s(M)) =  [8] N + [4]    
                   >= [8] N + [4]    
                   =  x#(N,M)        
        
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            plus#(N,s(M)) -> c_4(plus#(N,M))
            x#(N,s(M)) -> plus#(x(N,M),N)
            x#(N,s(M)) -> x#(N,M)
        - Weak TRS:
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
            x(N,0()) -> 0()
            x(N,s(M)) -> plus(x(N,M),N)
        - Signature:
            {activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))