WORST_CASE(?,O(n^2))
* Step 1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a__add(X1,X2) -> add(X1,X2)
            a__add(0(),X) -> mark(X)
            a__add(s(X),Y) -> s(add(X,Y))
            a__dbl(X) -> dbl(X)
            a__dbl(0()) -> 0()
            a__dbl(s(X)) -> s(s(dbl(X)))
            a__first(X1,X2) -> first(X1,X2)
            a__first(0(),X) -> nil()
            a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
            a__sqr(X) -> sqr(X)
            a__sqr(0()) -> 0()
            a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
            a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
            a__terms(X) -> terms(X)
            mark(0()) -> 0()
            mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(dbl(X)) -> a__dbl(mark(X))
            mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
            mark(nil()) -> nil()
            mark(recip(X)) -> recip(mark(X))
            mark(s(X)) -> s(X)
            mark(sqr(X)) -> a__sqr(mark(X))
            mark(terms(X)) -> a__terms(mark(X))
        - Signature:
            {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1
            ,s/1,sqr/1,terms/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms
            ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(a__add) = {1,2},
          uargs(a__dbl) = {1},
          uargs(a__first) = {1,2},
          uargs(a__sqr) = {1},
          uargs(a__terms) = {1},
          uargs(cons) = {1},
          uargs(recip) = {1}
        
        Following symbols are considered usable:
          {a__add,a__dbl,a__first,a__sqr,a__terms,mark}
        TcT has computed the following interpretation:
                 p(0) = [4]                      
                        [1]                      
            p(a__add) = [1 0] x1 + [1 4] x2 + [4]
                        [0 1]      [0 1]      [1]
            p(a__dbl) = [1 4] x1 + [1]           
                        [0 1]      [1]           
          p(a__first) = [1 2] x1 + [1 4] x2 + [4]
                        [0 1]      [0 1]      [3]
            p(a__sqr) = [1 0] x1 + [4]           
                        [0 1]      [1]           
          p(a__terms) = [1 4] x1 + [5]           
                        [0 1]      [3]           
               p(add) = [1 0] x1 + [1 4] x2 + [1]
                        [0 1]      [0 1]      [1]
              p(cons) = [1 0] x1 + [0]           
                        [0 1]      [1]           
               p(dbl) = [1 4] x1 + [0]           
                        [0 1]      [1]           
             p(first) = [1 2] x1 + [1 4] x2 + [0]
                        [0 1]      [0 1]      [3]
              p(mark) = [1 4] x1 + [0]           
                        [0 1]      [0]           
               p(nil) = [4]                      
                        [1]                      
             p(recip) = [1 0] x1 + [0]           
                        [0 1]      [1]           
                 p(s) = [0]                      
                        [2]                      
               p(sqr) = [1 0] x1 + [2]           
                        [0 1]      [1]           
             p(terms) = [1 4] x1 + [0]           
                        [0 1]      [3]           
        
        Following rules are strictly oriented:
                   a__add(X1,X2) = [1 0] X1 + [1 4] X2 + [4]               
                                   [0 1]      [0 1]      [1]               
                                 > [1 0] X1 + [1 4] X2 + [1]               
                                   [0 1]      [0 1]      [1]               
                                 = add(X1,X2)                              
        
                   a__add(0(),X) = [1 4] X + [8]                           
                                   [0 1]     [2]                           
                                 > [1 4] X + [0]                           
                                   [0 1]     [0]                           
                                 = mark(X)                                 
        
                  a__add(s(X),Y) = [1 4] Y + [4]                           
                                   [0 1]     [3]                           
                                 > [0]                                     
                                   [2]                                     
                                 = s(add(X,Y))                             
        
                       a__dbl(X) = [1 4] X + [1]                           
                                   [0 1]     [1]                           
                                 > [1 4] X + [0]                           
                                   [0 1]     [1]                           
                                 = dbl(X)                                  
        
                     a__dbl(0()) = [9]                                     
                                   [2]                                     
                                 > [4]                                     
                                   [1]                                     
                                 = 0()                                     
        
                    a__dbl(s(X)) = [9]                                     
                                   [3]                                     
                                 > [0]                                     
                                   [2]                                     
                                 = s(s(dbl(X)))                            
        
                 a__first(X1,X2) = [1 2] X1 + [1 4] X2 + [4]               
                                   [0 1]      [0 1]      [3]               
                                 > [1 2] X1 + [1 4] X2 + [0]               
                                   [0 1]      [0 1]      [3]               
                                 = first(X1,X2)                            
        
                 a__first(0(),X) = [1 4] X + [10]                          
                                   [0 1]     [4]                           
                                 > [4]                                     
                                   [1]                                     
                                 = nil()                                   
        
        a__first(s(X),cons(Y,Z)) = [1 4] Y + [12]                          
                                   [0 1]     [6]                           
                                 > [1 4] Y + [0]                           
                                   [0 1]     [1]                           
                                 = cons(mark(Y),first(X,Z))                
        
                       a__sqr(X) = [1 0] X + [4]                           
                                   [0 1]     [1]                           
                                 > [1 0] X + [2]                           
                                   [0 1]     [1]                           
                                 = sqr(X)                                  
        
                     a__sqr(0()) = [8]                                     
                                   [2]                                     
                                 > [4]                                     
                                   [1]                                     
                                 = 0()                                     
        
                    a__sqr(s(X)) = [4]                                     
                                   [3]                                     
                                 > [0]                                     
                                   [2]                                     
                                 = s(add(sqr(X),dbl(X)))                   
        
                     a__terms(N) = [1 4] N + [5]                           
                                   [0 1]     [3]                           
                                 > [1 4] N + [4]                           
                                   [0 1]     [3]                           
                                 = cons(recip(a__sqr(mark(N))),terms(s(N)))
        
                     a__terms(X) = [1 4] X + [5]                           
                                   [0 1]     [3]                           
                                 > [1 4] X + [0]                           
                                   [0 1]     [3]                           
                                 = terms(X)                                
        
                       mark(0()) = [8]                                     
                                   [1]                                     
                                 > [4]                                     
                                   [1]                                     
                                 = 0()                                     
        
                mark(add(X1,X2)) = [1 4] X1 + [1 8] X2 + [5]               
                                   [0 1]      [0 1]      [1]               
                                 > [1 4] X1 + [1 8] X2 + [4]               
                                   [0 1]      [0 1]      [1]               
                                 = a__add(mark(X1),mark(X2))               
        
               mark(cons(X1,X2)) = [1 4] X1 + [4]                          
                                   [0 1]      [1]                          
                                 > [1 4] X1 + [0]                          
                                   [0 1]      [1]                          
                                 = cons(mark(X1),X2)                       
        
                    mark(dbl(X)) = [1 8] X + [4]                           
                                   [0 1]     [1]                           
                                 > [1 8] X + [1]                           
                                   [0 1]     [1]                           
                                 = a__dbl(mark(X))                         
        
              mark(first(X1,X2)) = [1 6] X1 + [1 8] X2 + [12]              
                                   [0 1]      [0 1]      [3]               
                                 > [1 6] X1 + [1 8] X2 + [4]               
                                   [0 1]      [0 1]      [3]               
                                 = a__first(mark(X1),mark(X2))             
        
                     mark(nil()) = [8]                                     
                                   [1]                                     
                                 > [4]                                     
                                   [1]                                     
                                 = nil()                                   
        
                  mark(recip(X)) = [1 4] X + [4]                           
                                   [0 1]     [1]                           
                                 > [1 4] X + [0]                           
                                   [0 1]     [1]                           
                                 = recip(mark(X))                          
        
                      mark(s(X)) = [8]                                     
                                   [2]                                     
                                 > [0]                                     
                                   [2]                                     
                                 = s(X)                                    
        
                    mark(sqr(X)) = [1 4] X + [6]                           
                                   [0 1]     [1]                           
                                 > [1 4] X + [4]                           
                                   [0 1]     [1]                           
                                 = a__sqr(mark(X))                         
        
                  mark(terms(X)) = [1 8] X + [12]                          
                                   [0 1]     [3]                           
                                 > [1 8] X + [5]                           
                                   [0 1]     [3]                           
                                 = a__terms(mark(X))                       
        
        
        Following rules are (at-least) weakly oriented:
        

WORST_CASE(?,O(n^2))