WORST_CASE(?,O(n^2))
* Step 1: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(plus) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [3]                  
            p(activate) = [2] x1 + [4]         
                 p(and) = [7] x1 + [4] x2 + [0]
                p(plus) = [1] x1 + [1]         
                   p(s) = [1] x1 + [7]         
                  p(tt) = [1]                  
                   p(x) = [1] x1 + [3] x2 + [0]
          
          Following rules are strictly oriented:
          activate(X) = [2] X + [4]         
                      > [1] X + [0]         
                      = X                   
          
          and(tt(),X) = [4] X + [7]         
                      > [2] X + [4]         
                      = activate(X)         
          
          plus(N,0()) = [1] N + [1]         
                      > [1] N + [0]         
                      = N                   
          
             x(N,0()) = [1] N + [9]         
                      > [3]                 
                      = 0()                 
          
            x(N,s(M)) = [3] M + [1] N + [21]
                      > [3] M + [1] N + [1] 
                      = plus(x(N,M),N)      
          
          
          Following rules are (at-least) weakly oriented:
          plus(N,s(M)) =  [1] N + [1] 
                       >= [1] N + [8] 
                       =  s(plus(N,M))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            plus(N,s(M)) -> s(plus(N,M))
        - Weak TRS:
            activate(X) -> X
            and(tt(),X) -> activate(X)
            plus(N,0()) -> N
            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:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(plus) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {activate,and,plus,x}
        TcT has computed the following interpretation:
                 p(0) = 0                          
          p(activate) = 3 + 6*x1                   
               p(and) = 3 + 4*x1 + 4*x1^2 + 6*x2   
              p(plus) = 2 + x1 + 4*x2              
                 p(s) = 2 + x1                     
                p(tt) = 0                          
                 p(x) = 2 + 2*x1 + 2*x1*x2 + 2*x2^2
        
        Following rules are strictly oriented:
        plus(N,s(M)) = 10 + 4*M + N
                     > 4 + 4*M + N 
                     = s(plus(N,M))
        
        
        Following rules are (at-least) weakly oriented:
        activate(X) =  3 + 6*X                       
                    >= X                             
                    =  X                             
        
        and(tt(),X) =  3 + 6*X                       
                    >= 3 + 6*X                       
                    =  activate(X)                   
        
        plus(N,0()) =  2 + N                         
                    >= N                             
                    =  N                             
        
           x(N,0()) =  2 + 2*N                       
                    >= 0                             
                    =  0()                           
        
          x(N,s(M)) =  10 + 8*M + 2*M*N + 2*M^2 + 6*N
                    >= 4 + 2*M*N + 2*M^2 + 6*N       
                    =  plus(x(N,M),N)                
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - 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} / {0/0,s/1,tt/0}
        - 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))