WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            add(0(),x) -> x
            add(s(x),y) -> s(add(x,y))
            p(0(),s(x),y,z) -> q(x,add(x,z))
            p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u)
            q(s(x),y) -> p(s(x),0(),s(0()),y)
        - Signature:
            {add/2,p/4,q/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,p,q} 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(q) = {2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(0) = [1]                           
            p(add) = [2] x1 + [1] x2 + [14]        
              p(p) = [2] x2 + [2] x3 + [1] x4 + [8]
              p(q) = [1] x2 + [13]                 
              p(s) = [1] x1 + [0]                  
          
          Following rules are strictly oriented:
          add(0(),x) = [1] x + [16]        
                     > [1] x + [0]         
                     = x                   
          
           q(s(x),y) = [1] y + [13]        
                     > [1] y + [12]        
                     = p(s(x),0(),s(0()),y)
          
          
          Following rules are (at-least) weakly oriented:
              add(s(x),y) =  [2] x + [1] y + [14]       
                          >= [2] x + [1] y + [14]       
                          =  s(add(x,y))                
          
          p(0(),s(x),y,z) =  [2] x + [2] y + [1] z + [8]
                          >= [2] x + [1] z + [27]       
                          =  q(x,add(x,z))              
          
            p(s(x),y,z,u) =  [1] u + [2] y + [2] z + [8]
                          >= [1] u + [2] y + [2] z + [8]
                          =  p(x,s(y),s(s(z)),u)        
          
        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))
            p(0(),s(x),y,z) -> q(x,add(x,z))
            p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u)
        - Weak TRS:
            add(0(),x) -> x
            q(s(x),y) -> p(s(x),0(),s(0()),y)
        - Signature:
            {add/2,p/4,q/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,p,q} 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(q) = {2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(0) = [3]                  
            p(add) = [3] x1 + [1] x2 + [0]
              p(p) = [4] x2 + [1] x4 + [0]
              p(q) = [1] x1 + [1] x2 + [8]
              p(s) = [1] x1 + [4]         
          
          Following rules are strictly oriented:
              add(s(x),y) = [3] x + [1] y + [12]
                          > [3] x + [1] y + [4] 
                          = s(add(x,y))         
          
          p(0(),s(x),y,z) = [4] x + [1] z + [16]
                          > [4] x + [1] z + [8] 
                          = q(x,add(x,z))       
          
          
          Following rules are (at-least) weakly oriented:
             add(0(),x) =  [1] x + [9]         
                        >= [1] x + [0]         
                        =  x                   
          
          p(s(x),y,z,u) =  [1] u + [4] y + [0] 
                        >= [1] u + [4] y + [16]
                        =  p(x,s(y),s(s(z)),u) 
          
              q(s(x),y) =  [1] x + [1] y + [12]
                        >= [1] y + [12]        
                        =  p(s(x),0(),s(0()),y)
          
        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:
            p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u)
        - Weak TRS:
            add(0(),x) -> x
            add(s(x),y) -> s(add(x,y))
            p(0(),s(x),y,z) -> q(x,add(x,z))
            q(s(x),y) -> p(s(x),0(),s(0()),y)
        - Signature:
            {add/2,p/4,q/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,p,q} 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(q) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {add,p,q}
        TcT has computed the following interpretation:
            p(0) = 0                                              
          p(add) = x1 + x2                                        
            p(p) = 5*x1 + 6*x1*x2 + 3*x1^2 + x2 + 3*x2^2 + x3 + x4
            p(q) = 4 + 5*x1 + 3*x1^2 + x2                         
            p(s) = 1 + x1                                         
        
        Following rules are strictly oriented:
        p(s(x),y,z,u) = 8 + u + 11*x + 6*x*y + 3*x^2 + 7*y + 3*y^2 + z
                      > 6 + u + 11*x + 6*x*y + 3*x^2 + 7*y + 3*y^2 + z
                      = p(x,s(y),s(s(z)),u)                           
        
        
        Following rules are (at-least) weakly oriented:
             add(0(),x) =  x                      
                        >= x                      
                        =  x                      
        
            add(s(x),y) =  1 + x + y              
                        >= 1 + x + y              
                        =  s(add(x,y))            
        
        p(0(),s(x),y,z) =  4 + 7*x + 3*x^2 + y + z
                        >= 4 + 6*x + 3*x^2 + z    
                        =  q(x,add(x,z))          
        
              q(s(x),y) =  12 + 11*x + 3*x^2 + y  
                        >= 9 + 11*x + 3*x^2 + y   
                        =  p(s(x),0(),s(0()),y)   
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add(0(),x) -> x
            add(s(x),y) -> s(add(x,y))
            p(0(),s(x),y,z) -> q(x,add(x,z))
            p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u)
            q(s(x),y) -> p(s(x),0(),s(0()),y)
        - Signature:
            {add/2,p/4,q/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,p,q} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))