* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            times(x,0()) -> 0()
            times(x,s(y)) -> plus(times(x,y),x)
        - Signature:
            {plus/2,times/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s}
    + 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(plus) = [1] x1 + [0]          
                p(s) = [1] x1 + [1]          
            p(times) = [12] x1 + [8] x2 + [3]
          
          Following rules are strictly oriented:
           times(x,0()) = [12] x + [3]         
                        > [0]                  
                        = 0()                  
          
          times(x,s(y)) = [12] x + [8] y + [11]
                        > [12] x + [8] y + [3] 
                        = plus(times(x,y),x)   
          
          
          Following rules are (at-least) weakly oriented:
           plus(x,0()) =  [1] x + [0] 
                       >= [1] x + [0] 
                       =  x           
          
          plus(x,s(y)) =  [1] x + [0] 
                       >= [1] x + [1] 
                       =  s(plus(x,y))
          
        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:
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
        - Weak TRS:
            times(x,0()) -> 0()
            times(x,s(y)) -> plus(times(x,y),x)
        - Signature:
            {plus/2,times/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s}
    + 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) = [2]                  
             p(plus) = [1] x1 + [2]         
                p(s) = [1] x1 + [1]         
            p(times) = [1] x1 + [4] x2 + [8]
          
          Following rules are strictly oriented:
          plus(x,0()) = [1] x + [2]
                      > [1] x + [0]
                      = x          
          
          
          Following rules are (at-least) weakly oriented:
           plus(x,s(y)) =  [1] x + [2]         
                        >= [1] x + [3]         
                        =  s(plus(x,y))        
          
           times(x,0()) =  [1] x + [16]        
                        >= [2]                 
                        =  0()                 
          
          times(x,s(y)) =  [1] x + [4] y + [12]
                        >= [1] x + [4] y + [10]
                        =  plus(times(x,y),x)  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            plus(x,s(y)) -> s(plus(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            times(x,0()) -> 0()
            times(x,s(y)) -> plus(times(x,y),x)
        - Signature:
            {plus/2,times/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s}
    + 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:
          {plus,times}
        TcT has computed the following interpretation:
              p(0) = 0                    
           p(plus) = x1 + 3*x2            
              p(s) = 1 + x1               
          p(times) = 2*x1 + 3*x1*x2 + x1^2
        
        Following rules are strictly oriented:
        plus(x,s(y)) = 3 + x + 3*y 
                     > 1 + x + 3*y 
                     = s(plus(x,y))
        
        
        Following rules are (at-least) weakly oriented:
          plus(x,0()) =  x                 
                      >= x                 
                      =  x                 
        
         times(x,0()) =  2*x + x^2         
                      >= 0                 
                      =  0()               
        
        times(x,s(y)) =  5*x + 3*x*y + x^2 
                      >= 5*x + 3*x*y + x^2 
                      =  plus(times(x,y),x)
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            times(x,0()) -> 0()
            times(x,s(y)) -> plus(times(x,y),x)
        - Signature:
            {plus/2,times/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))