WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(x,mult(x,y))
        - Signature:
            {add/2,mult/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,mult} 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(add) = {2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [0]         
             p(add) = [1] x2 + [8]
            p(mult) = [13]        
               p(s) = [1] x1 + [2]
          
          Following rules are strictly oriented:
           add(0(),y) = [1] y + [8]
                      > [1] y + [0]
                      = y          
          
          mult(0(),y) = [13]       
                      > [0]        
                      = 0()        
          
          
          Following rules are (at-least) weakly oriented:
           add(s(x),y) =  [1] y + [8]     
                       >= [1] y + [10]    
                       =  s(add(x,y))     
          
          mult(s(x),y) =  [13]            
                       >= [21]            
                       =  add(x,mult(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:
            add(s(x),y) -> s(add(x,y))
            mult(s(x),y) -> add(x,mult(x,y))
        - Weak TRS:
            add(0(),y) -> y
            mult(0(),y) -> 0()
        - Signature:
            {add/2,mult/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,mult} 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(add) = {2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [0]                  
             p(add) = [1] x2 + [0]         
            p(mult) = [1] x1 + [4] x2 + [0]
               p(s) = [1] x1 + [8]         
          
          Following rules are strictly oriented:
          mult(s(x),y) = [1] x + [4] y + [8]
                       > [1] x + [4] y + [0]
                       = add(x,mult(x,y))   
          
          
          Following rules are (at-least) weakly oriented:
           add(0(),y) =  [1] y + [0]
                      >= [1] y + [0]
                      =  y          
          
          add(s(x),y) =  [1] y + [0]
                      >= [1] y + [8]
                      =  s(add(x,y))
          
          mult(0(),y) =  [4] y + [0]
                      >= [0]        
                      =  0()        
          
        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:
            add(s(x),y) -> s(add(x,y))
        - Weak TRS:
            add(0(),y) -> y
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(x,mult(x,y))
        - Signature:
            {add/2,mult/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,mult} 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(add) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {add,mult}
        TcT has computed the following interpretation:
             p(0) = 0                   
           p(add) = 6 + 4*x1 + x2       
          p(mult) = 3 + 2*x1 + x1^2 + x2
             p(s) = 2 + x1              
        
        Following rules are strictly oriented:
        add(s(x),y) = 14 + 4*x + y
                    > 8 + 4*x + y 
                    = s(add(x,y)) 
        
        
        Following rules are (at-least) weakly oriented:
          add(0(),y) =  6 + y             
                     >= y                 
                     =  y                 
        
         mult(0(),y) =  3 + y             
                     >= 0                 
                     =  0()               
        
        mult(s(x),y) =  11 + 6*x + x^2 + y
                     >= 9 + 6*x + x^2 + y 
                     =  add(x,mult(x,y))  
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(x,mult(x,y))
        - Signature:
            {add/2,mult/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,mult} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))