WORST_CASE(?,O(n^1))
* Step 1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            append(l1,l2) -> ifappend(l1,l2,l1)
            hd(cons(x,l)) -> x
            ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
            ifappend(l1,l2,nil()) -> l2
            is_empty(cons(x,l)) -> false()
            is_empty(nil()) -> true()
            tl(cons(x,l)) -> l
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
            ,false,nil,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) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(append) = [1] x1 + [2] x2 + [0]
                p(cons) = [1] x1 + [1] x2 + [3]
               p(false) = [0]                  
                  p(hd) = [2] x1 + [0]         
            p(ifappend) = [2] x2 + [1] x3 + [0]
            p(is_empty) = [0]                  
                 p(nil) = [0]                  
                  p(tl) = [7] x1 + [0]         
                p(true) = [0]                  
          
          Following rules are strictly oriented:
          hd(cons(x,l)) = [2] l + [2] x + [6] 
                        > [1] x + [0]         
                        = x                   
          
          tl(cons(x,l)) = [7] l + [7] x + [21]
                        > [1] l + [0]         
                        = l                   
          
          
          Following rules are (at-least) weakly oriented:
                      append(l1,l2) =  [1] l1 + [2] l2 + [0]       
                                    >= [1] l1 + [2] l2 + [0]       
                                    =  ifappend(l1,l2,l1)          
          
          ifappend(l1,l2,cons(x,l)) =  [1] l + [2] l2 + [1] x + [3]
                                    >= [1] l + [2] l2 + [1] x + [3]
                                    =  cons(x,append(l,l2))        
          
              ifappend(l1,l2,nil()) =  [2] l2 + [0]                
                                    >= [1] l2 + [0]                
                                    =  l2                          
          
                is_empty(cons(x,l)) =  [0]                         
                                    >= [0]                         
                                    =  false()                     
          
                    is_empty(nil()) =  [0]                         
                                    >= [0]                         
                                    =  true()                      
          
        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:
            append(l1,l2) -> ifappend(l1,l2,l1)
            ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
            ifappend(l1,l2,nil()) -> l2
            is_empty(cons(x,l)) -> false()
            is_empty(nil()) -> true()
        - Weak TRS:
            hd(cons(x,l)) -> x
            tl(cons(x,l)) -> l
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
            ,false,nil,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) = {2}
        
        Following symbols are considered usable:
          {append,hd,ifappend,is_empty,tl}
        TcT has computed the following interpretation:
            p(append) = [7] x1 + [4] x2 + [11]
              p(cons) = [1] x1 + [1] x2 + [2] 
             p(false) = [2]                   
                p(hd) = [6] x1 + [4]          
          p(ifappend) = [4] x2 + [7] x3 + [0] 
          p(is_empty) = [4]                   
               p(nil) = [1]                   
                p(tl) = [1] x1 + [0]          
              p(true) = [0]                   
        
        Following rules are strictly oriented:
                    append(l1,l2) = [7] l1 + [4] l2 + [11]       
                                  > [7] l1 + [4] l2 + [0]        
                                  = ifappend(l1,l2,l1)           
        
        ifappend(l1,l2,cons(x,l)) = [7] l + [4] l2 + [7] x + [14]
                                  > [7] l + [4] l2 + [1] x + [13]
                                  = cons(x,append(l,l2))         
        
            ifappend(l1,l2,nil()) = [4] l2 + [7]                 
                                  > [1] l2 + [0]                 
                                  = l2                           
        
              is_empty(cons(x,l)) = [4]                          
                                  > [2]                          
                                  = false()                      
        
                  is_empty(nil()) = [4]                          
                                  > [0]                          
                                  = true()                       
        
        
        Following rules are (at-least) weakly oriented:
        hd(cons(x,l)) =  [6] l + [6] x + [16]
                      >= [1] x + [0]         
                      =  x                   
        
        tl(cons(x,l)) =  [1] l + [1] x + [2] 
                      >= [1] l + [0]         
                      =  l                   
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            append(l1,l2) -> ifappend(l1,l2,l1)
            hd(cons(x,l)) -> x
            ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
            ifappend(l1,l2,nil()) -> l2
            is_empty(cons(x,l)) -> false()
            is_empty(nil()) -> true()
            tl(cons(x,l)) -> l
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
            ,false,nil,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))