WORST_CASE(?,O(n^1))
* Step 1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            fold(a,xs) -> Cons(foldl(a,xs),Cons(foldr(a,xs),Nil()))
            foldl(a,Nil()) -> a
            foldl(x,Cons(S(0()),xs)) -> foldl(S(x),xs)
            foldl(S(0()),Cons(x,xs)) -> foldl(S(x),xs)
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold,foldl,foldr,notEmpty,op} and constructors {0,Cons
            ,False,Nil,S,True}
    + 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(Cons) = {1,2},
            uargs(op) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [1]                  
                p(Cons) = [1] x1 + [1] x2 + [3]
               p(False) = [0]                  
                 p(Nil) = [0]                  
                   p(S) = [1] x1 + [0]         
                p(True) = [0]                  
                p(fold) = [5] x1 + [4] x2 + [8]
               p(foldl) = [3] x1 + [3] x2 + [0]
               p(foldr) = [2] x1 + [1] x2 + [0]
            p(notEmpty) = [0]                  
                  p(op) = [1] x1 + [1] x2 + [0]
          
          Following rules are strictly oriented:
                        fold(a,xs) = [5] a + [4] xs + [8]                     
                                   > [5] a + [4] xs + [6]                     
                                   = Cons(foldl(a,xs),Cons(foldr(a,xs),Nil()))
          
          foldl(x,Cons(S(0()),xs)) = [3] x + [3] xs + [12]                    
                                   > [3] x + [3] xs + [0]                     
                                   = foldl(S(x),xs)                           
          
          foldl(S(0()),Cons(x,xs)) = [3] x + [3] xs + [12]                    
                                   > [3] x + [3] xs + [0]                     
                                   = foldl(S(x),xs)                           
          
               foldr(a,Cons(x,xs)) = [2] a + [1] x + [1] xs + [3]             
                                   > [2] a + [1] x + [1] xs + [0]             
                                   = op(x,foldr(a,xs))                        
          
                      op(x,S(0())) = [1] x + [1]                              
                                   > [1] x + [0]                              
                                   = S(x)                                     
          
                      op(S(0()),y) = [1] y + [1]                              
                                   > [1] y + [0]                              
                                   = S(y)                                     
          
          
          Following rules are (at-least) weakly oriented:
                foldl(a,Nil()) =  [3] a + [0]
                               >= [1] a + [0]
                               =  a          
          
                foldr(a,Nil()) =  [2] a + [0]
                               >= [1] a + [0]
                               =  a          
          
          notEmpty(Cons(x,xs)) =  [0]        
                               >= [0]        
                               =  True()     
          
               notEmpty(Nil()) =  [0]        
                               >= [0]        
                               =  False()    
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            foldl(a,Nil()) -> a
            foldr(a,Nil()) -> a
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Weak TRS:
            fold(a,xs) -> Cons(foldl(a,xs),Cons(foldr(a,xs),Nil()))
            foldl(x,Cons(S(0()),xs)) -> foldl(S(x),xs)
            foldl(S(0()),Cons(x,xs)) -> foldl(S(x),xs)
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold,foldl,foldr,notEmpty,op} and constructors {0,Cons
            ,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(Cons) = {1,2},
          uargs(op) = {2}
        
        Following symbols are considered usable:
          {fold,foldl,foldr,notEmpty,op}
        TcT has computed the following interpretation:
                 p(0) = [0]                  
              p(Cons) = [1] x1 + [1] x2 + [0]
             p(False) = [0]                  
               p(Nil) = [1]                  
                 p(S) = [1] x1 + [4]         
              p(True) = [0]                  
              p(fold) = [8] x1 + [3] x2 + [2]
             p(foldl) = [2] x1 + [2] x2 + [0]
             p(foldr) = [1] x1 + [1] x2 + [1]
          p(notEmpty) = [8]                  
                p(op) = [1] x1 + [1] x2 + [0]
        
        Following rules are strictly oriented:
              foldl(a,Nil()) = [2] a + [2]
                             > [1] a + [0]
                             = a          
        
              foldr(a,Nil()) = [1] a + [2]
                             > [1] a + [0]
                             = a          
        
        notEmpty(Cons(x,xs)) = [8]        
                             > [0]        
                             = True()     
        
             notEmpty(Nil()) = [8]        
                             > [0]        
                             = False()    
        
        
        Following rules are (at-least) weakly oriented:
                      fold(a,xs) =  [8] a + [3] xs + [2]                     
                                 >= [3] a + [3] xs + [2]                     
                                 =  Cons(foldl(a,xs),Cons(foldr(a,xs),Nil()))
        
        foldl(x,Cons(S(0()),xs)) =  [2] x + [2] xs + [8]                     
                                 >= [2] x + [2] xs + [8]                     
                                 =  foldl(S(x),xs)                           
        
        foldl(S(0()),Cons(x,xs)) =  [2] x + [2] xs + [8]                     
                                 >= [2] x + [2] xs + [8]                     
                                 =  foldl(S(x),xs)                           
        
             foldr(a,Cons(x,xs)) =  [1] a + [1] x + [1] xs + [1]             
                                 >= [1] a + [1] x + [1] xs + [1]             
                                 =  op(x,foldr(a,xs))                        
        
                    op(x,S(0())) =  [1] x + [4]                              
                                 >= [1] x + [4]                              
                                 =  S(x)                                     
        
                    op(S(0()),y) =  [1] y + [4]                              
                                 >= [1] y + [4]                              
                                 =  S(y)                                     
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            fold(a,xs) -> Cons(foldl(a,xs),Cons(foldr(a,xs),Nil()))
            foldl(a,Nil()) -> a
            foldl(x,Cons(S(0()),xs)) -> foldl(S(x),xs)
            foldl(S(0()),Cons(x,xs)) -> foldl(S(x),xs)
            foldr(a,Cons(x,xs)) -> op(x,foldr(a,xs))
            foldr(a,Nil()) -> a
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            op(x,S(0())) -> S(x)
            op(S(0()),y) -> S(y)
        - Signature:
            {fold/2,foldl/2,foldr/2,notEmpty/1,op/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fold,foldl,foldr,notEmpty,op} and constructors {0,Cons
            ,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))