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) = [5]         
            p(activate) = [2] x1 + [1]
                 p(and) = [8] x2 + [0]
                p(plus) = [1] x1 + [0]
                   p(s) = [1] x1 + [7]
                  p(tt) = [0]         
                   p(x) = [4] x2 + [0]
          
          Following rules are strictly oriented:
          activate(X) = [2] X + [1]   
                      > [1] X + [0]   
                      = X             
          
             x(N,0()) = [20]          
                      > [5]           
                      = 0()           
          
            x(N,s(M)) = [4] M + [28]  
                      > [4] M + [0]   
                      = plus(x(N,M),N)
          
          
          Following rules are (at-least) weakly oriented:
           and(tt(),X) =  [8] X + [0] 
                       >= [2] X + [1] 
                       =  activate(X) 
          
           plus(N,0()) =  [1] N + [0] 
                       >= [1] N + [0] 
                       =  N           
          
          plus(N,s(M)) =  [1] N + [0] 
                       >= [1] N + [7] 
                       =  s(plus(N,M))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            and(tt(),X) -> activate(X)
            plus(N,0()) -> N
            plus(N,s(M)) -> s(plus(N,M))
        - Weak TRS:
            activate(X) -> X
            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) = [0]                  
            p(activate) = [1] x1 + [4]         
                 p(and) = [8] x2 + [0]         
                p(plus) = [1] x1 + [4]         
                   p(s) = [1] x1 + [4]         
                  p(tt) = [8]                  
                   p(x) = [9] x1 + [4] x2 + [6]
          
          Following rules are strictly oriented:
          plus(N,0()) = [1] N + [4]
                      > [1] N + [0]
                      = N          
          
          
          Following rules are (at-least) weakly oriented:
           activate(X) =  [1] X + [4]         
                       >= [1] X + [0]         
                       =  X                   
          
           and(tt(),X) =  [8] X + [0]         
                       >= [1] X + [4]         
                       =  activate(X)         
          
          plus(N,s(M)) =  [1] N + [4]         
                       >= [1] N + [8]         
                       =  s(plus(N,M))        
          
              x(N,0()) =  [9] N + [6]         
                       >= [0]                 
                       =  0()                 
          
             x(N,s(M)) =  [4] M + [9] N + [22]
                       >= [4] M + [9] N + [10]
                       =  plus(x(N,M),N)      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            and(tt(),X) -> activate(X)
            plus(N,s(M)) -> s(plus(N,M))
        - Weak TRS:
            activate(X) -> 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:
        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) = [0]                   
            p(activate) = [1] x1 + [0]          
                 p(and) = [9] x2 + [1]          
                p(plus) = [1] x1 + [3]          
                   p(s) = [1] x1 + [6]          
                  p(tt) = [0]                   
                   p(x) = [10] x1 + [4] x2 + [5]
          
          Following rules are strictly oriented:
          and(tt(),X) = [9] X + [1]
                      > [1] X + [0]
                      = activate(X)
          
          
          Following rules are (at-least) weakly oriented:
           activate(X) =  [1] X + [0]          
                       >= [1] X + [0]          
                       =  X                    
          
           plus(N,0()) =  [1] N + [3]          
                       >= [1] N + [0]          
                       =  N                    
          
          plus(N,s(M)) =  [1] N + [3]          
                       >= [1] N + [9]          
                       =  s(plus(N,M))         
          
              x(N,0()) =  [10] N + [5]         
                       >= [0]                  
                       =  0()                  
          
             x(N,s(M)) =  [4] M + [10] N + [29]
                       >= [4] M + [10] N + [8] 
                       =  plus(x(N,M),N)       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: 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) = 6 + 4*x1       
               p(and) = 3 + 3*x1 + 4*x2
              p(plus) = x1 + 2*x2      
                 p(s) = 2 + x1         
                p(tt) = 3              
                 p(x) = 2*x1*x2        
        
        Following rules are strictly oriented:
        plus(N,s(M)) = 4 + 2*M + N 
                     > 2 + 2*M + N 
                     = s(plus(N,M))
        
        
        Following rules are (at-least) weakly oriented:
        activate(X) =  6 + 4*X       
                    >= X             
                    =  X             
        
        and(tt(),X) =  12 + 4*X      
                    >= 6 + 4*X       
                    =  activate(X)   
        
        plus(N,0()) =  N             
                    >= N             
                    =  N             
        
           x(N,0()) =  0             
                    >= 0             
                    =  0()           
        
          x(N,s(M)) =  2*M*N + 4*N   
                    >= 2*M*N + 2*N   
                    =  plus(x(N,M),N)
        
* Step 5: 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))