* 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_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        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: UsableRules 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()
        - 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,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))
          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()
* Step 3: 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()
        - 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 4: 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()
        - 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 5: PredecessorEstimationCP 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:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          2: ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
          
        Consider the set of all dependency pairs
          1: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
          2: ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
** Step 5.a:1: NaturalMI 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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        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:
          {append#,hd#,ifappend#,is_empty#,tl#}
        TcT has computed the following interpretation:
             p(append) = [1]                           
               p(cons) = [1] x2 + [2]                  
              p(false) = [1]                           
                 p(hd) = [0]                           
           p(ifappend) = [1] x1 + [2] x2 + [4] x3 + [8]
           p(is_empty) = [1]                           
                p(nil) = [2]                           
                 p(tl) = [0]                           
               p(true) = [2]                           
            p(append#) = [15] x1 + [4] x2 + [0]        
                p(hd#) = [1]                           
          p(ifappend#) = [4] x2 + [15] x3 + [0]        
          p(is_empty#) = [1]                           
                p(tl#) = [1] x1 + [2]                  
                p(c_1) = [1] x1 + [0]                  
                p(c_2) = [2]                           
                p(c_3) = [1] x1 + [12]                 
                p(c_4) = [0]                           
                p(c_5) = [1]                           
                p(c_6) = [1]                           
                p(c_7) = [1]                           
        
        Following rules are strictly oriented:
        ifappend#(l1,l2,cons(x,l)) = [15] l + [4] l2 + [30]
                                   > [15] l + [4] l2 + [12]
                                   = c_3(append#(l,l2))    
        
        
        Following rules are (at-least) weakly oriented:
        append#(l1,l2) =  [15] l1 + [4] l2 + [0]  
                       >= [15] l1 + [4] l2 + [0]  
                       =  c_1(ifappend#(l1,l2,l1))
        
** Step 5.a:2: Assumption WORST_CASE(?,O(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:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 5.b:1: RemoveWeakSuffixes 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:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
             -->_1 ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)):2
          
          2:W:ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
             -->_1 append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1))
          2: ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2))
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - 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))