WORST_CASE(?,O(n^1))
* Step 1: Sum.  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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: NaturalPI.  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:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        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) = 1 + x1 + 2*x2
              p(cons) = 1 + x1 + x2  
             p(false) = 0            
                p(hd) = 6 + 2*x1     
          p(ifappend) = 1 + 2*x2 + x3
          p(is_empty) = 15           
               p(nil) = 0            
                p(tl) = 12 + 2*x1    
              p(true) = 11           
        
        Following rules are strictly oriented:
                hd(cons(x,l)) = 8 + 2*l + 2*x 
                              > x             
                              = x             
        
        ifappend(l1,l2,nil()) = 1 + 2*l2      
                              > l2            
                              = l2            
        
          is_empty(cons(x,l)) = 15            
                              > 0             
                              = false()       
        
              is_empty(nil()) = 15            
                              > 11            
                              = true()        
        
                tl(cons(x,l)) = 14 + 2*l + 2*x
                              > l             
                              = l             
        
        
        Following rules are (at-least) weakly oriented:
                    append(l1,l2) =  1 + l1 + 2*l2       
                                  >= 1 + l1 + 2*l2       
                                  =  ifappend(l1,l2,l1)  
        
        ifappend(l1,l2,cons(x,l)) =  2 + l + 2*l2 + x    
                                  >= 2 + l + 2*l2 + x    
                                  =  cons(x,append(l,l2))
        
* Step 3: 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))
        - Weak TRS:
            hd(cons(x,l)) -> x
            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:
        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) = [4] x1 + [1] x2 + [4]
              p(cons) = [1] x1 + [1] x2 + [4]
             p(false) = [5]                  
                p(hd) = [4] x1 + [0]         
          p(ifappend) = [1] x2 + [4] x3 + [0]
          p(is_empty) = [1] x1 + [1]         
               p(nil) = [0]                  
                p(tl) = [1] x1 + [1]         
              p(true) = [0]                  
        
        Following rules are strictly oriented:
                    append(l1,l2) = [4] l1 + [1] l2 + [4]        
                                  > [4] l1 + [1] l2 + [0]        
                                  = ifappend(l1,l2,l1)           
        
        ifappend(l1,l2,cons(x,l)) = [4] l + [1] l2 + [4] x + [16]
                                  > [4] l + [1] l2 + [1] x + [8] 
                                  = cons(x,append(l,l2))         
        
        
        Following rules are (at-least) weakly oriented:
                hd(cons(x,l)) =  [4] l + [4] x + [16]
                              >= [1] x + [0]         
                              =  x                   
        
        ifappend(l1,l2,nil()) =  [1] l2 + [0]        
                              >= [1] l2 + [0]        
                              =  l2                  
        
          is_empty(cons(x,l)) =  [1] l + [1] x + [5] 
                              >= [5]                 
                              =  false()             
        
              is_empty(nil()) =  [1]                 
                              >= [0]                 
                              =  true()              
        
                tl(cons(x,l)) =  [1] l + [1] x + [5] 
                              >= [1] l + [0]         
                              =  l                   
        
* Step 4: 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))