WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs 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:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
          hd#(cons(x,l)) -> c_2()
          ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
          ifappend#(l1,l2,nil()) -> c_4()
          is_empty#(cons(x,l)) -> c_5()
          is_empty#(nil()) -> c_6()
          tl#(cons(x,l)) -> c_7()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
            hd#(cons(x,l)) -> c_2()
            ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
            ifappend#(l1,l2,nil()) -> c_4()
            is_empty#(cons(x,l)) -> c_5()
            is_empty#(nil()) -> c_6()
            tl#(cons(x,l)) -> c_7()
        - 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,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0
            ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty#
            ,tl#} and constructors {cons,false,nil,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,5,6,7}
        by application of
          Pre({2,4,5,6,7}) = {1}.
        Here rules are labelled as follows:
          1: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
          2: hd#(cons(x,l)) -> c_2()
          3: ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
          4: ifappend#(l1,l2,nil()) -> c_4()
          5: is_empty#(cons(x,l)) -> c_5()
          6: is_empty#(nil()) -> c_6()
          7: tl#(cons(x,l)) -> c_7()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
            ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
        - Weak DPs:
            hd#(cons(x,l)) -> c_2()
            ifappend#(l1,l2,nil()) -> c_4()
            is_empty#(cons(x,l)) -> c_5()
            is_empty#(nil()) -> c_6()
            tl#(cons(x,l)) -> c_7()
        - 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,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0
            ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty#
            ,tl#} and constructors {cons,false,nil,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
             -->_1 ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)):2
             -->_1 ifappend#(l1,l2,nil()) -> c_4():4
          
          2:S:ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
             -->_1 append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)):1
          
          3:W:hd#(cons(x,l)) -> c_2()
             
          
          4:W:ifappend#(l1,l2,nil()) -> c_4()
             
          
          5:W:is_empty#(cons(x,l)) -> c_5()
             
          
          6:W:is_empty#(nil()) -> c_6()
             
          
          7:W:tl#(cons(x,l)) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: tl#(cons(x,l)) -> c_7()
          6: is_empty#(nil()) -> c_6()
          5: is_empty#(cons(x,l)) -> c_5()
          3: hd#(cons(x,l)) -> c_2()
          4: ifappend#(l1,l2,nil()) -> c_4()
* Step 4: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
            ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
        - 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,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0
            ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty#
            ,tl#} and constructors {cons,false,nil,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
          ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
* Step 5: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
            ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0
            ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - 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 constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(append) = [0]                  
                 p(cons) = [1] x1 + [1] x2 + [0]
                p(false) = [0]                  
                   p(hd) = [0]                  
             p(ifappend) = [0]                  
             p(is_empty) = [0]                  
                  p(nil) = [0]                  
                   p(tl) = [0]                  
                 p(true) = [0]                  
              p(append#) = [0]                  
                  p(hd#) = [0]                  
            p(ifappend#) = [5]                  
            p(is_empty#) = [0]                  
                  p(tl#) = [0]                  
                  p(c_1) = [1] x1 + [0]         
                  p(c_2) = [0]                  
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [0]                  
                  p(c_5) = [0]                  
                  p(c_6) = [0]                  
                  p(c_7) = [0]                  
          
          Following rules are strictly oriented:
          ifappend#(l1,l2,cons(x,l)) = [5]               
                                     > [0]               
                                     = c_3(append#(l,l2))
          
          
          Following rules are (at-least) weakly oriented:
          append#(l1,l2) =  [0]                     
                         >= [5]                     
                         =  c_1(ifappend#(l1,l2,l1))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
        - Weak DPs:
            ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0
            ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - 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 constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(append) = [0]                  
                 p(cons) = [1] x1 + [1] x2 + [1]
                p(false) = [0]                  
                   p(hd) = [0]                  
             p(ifappend) = [0]                  
             p(is_empty) = [0]                  
                  p(nil) = [0]                  
                   p(tl) = [0]                  
                 p(true) = [0]                  
              p(append#) = [1] x1 + [1]         
                  p(hd#) = [0]                  
            p(ifappend#) = [1] x3 + [0]         
            p(is_empty#) = [0]                  
                  p(tl#) = [0]                  
                  p(c_1) = [1] x1 + [0]         
                  p(c_2) = [0]                  
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [0]                  
                  p(c_5) = [0]                  
                  p(c_6) = [0]                  
                  p(c_7) = [0]                  
          
          Following rules are strictly oriented:
          append#(l1,l2) = [1] l1 + [1]            
                         > [1] l1 + [0]            
                         = c_1(ifappend#(l1,l2,l1))
          
          
          Following rules are (at-least) weakly oriented:
          ifappend#(l1,l2,cons(x,l)) =  [1] l + [1] x + [1]
                                     >= [1] l + [1]        
                                     =  c_3(append#(l,l2)) 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
            ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
        - Signature:
            {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0
            ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0}
        - 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))