WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            main(x5,x12) -> map#2(plus_x(x12),x5)
            map#2(plus_x(x2),Nil()) -> Nil()
            map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))
            plus_x#1(0(),x8) -> x8
            plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14))
        - Signature:
            {main/2,map#2/2,plus_x#1/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main,map#2,plus_x#1} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          main#(x5,x12) -> c_1(map#2#(plus_x(x12),x5))
          map#2#(plus_x(x2),Nil()) -> c_2()
          map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
          plus_x#1#(0(),x8) -> c_4()
          plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            main#(x5,x12) -> c_1(map#2#(plus_x(x12),x5))
            map#2#(plus_x(x2),Nil()) -> c_2()
            map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
            plus_x#1#(0(),x8) -> c_4()
            plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
        - Weak TRS:
            main(x5,x12) -> map#2(plus_x(x12),x5)
            map#2(plus_x(x2),Nil()) -> Nil()
            map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))
            plus_x#1(0(),x8) -> x8
            plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14))
        - Signature:
            {main/2,map#2/2,plus_x#1/2,main#/2,map#2#/2,plus_x#1#/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1,c_1/1,c_2/0,c_3/2
            ,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#,map#2#,plus_x#1#} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4}
        by application of
          Pre({2,4}) = {1,3,5}.
        Here rules are labelled as follows:
          1: main#(x5,x12) -> c_1(map#2#(plus_x(x12),x5))
          2: map#2#(plus_x(x2),Nil()) -> c_2()
          3: map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
          4: plus_x#1#(0(),x8) -> c_4()
          5: plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            main#(x5,x12) -> c_1(map#2#(plus_x(x12),x5))
            map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
            plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
        - Weak DPs:
            map#2#(plus_x(x2),Nil()) -> c_2()
            plus_x#1#(0(),x8) -> c_4()
        - Weak TRS:
            main(x5,x12) -> map#2(plus_x(x12),x5)
            map#2(plus_x(x2),Nil()) -> Nil()
            map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))
            plus_x#1(0(),x8) -> x8
            plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14))
        - Signature:
            {main/2,map#2/2,plus_x#1/2,main#/2,map#2#/2,plus_x#1#/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1,c_1/1,c_2/0,c_3/2
            ,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#,map#2#,plus_x#1#} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:main#(x5,x12) -> c_1(map#2#(plus_x(x12),x5))
             -->_1 map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2)):2
             -->_1 map#2#(plus_x(x2),Nil()) -> c_2():4
          
          2:S:map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
             -->_1 plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14)):3
             -->_1 plus_x#1#(0(),x8) -> c_4():5
             -->_2 map#2#(plus_x(x2),Nil()) -> c_2():4
             -->_2 map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2)):2
          
          3:S:plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
             -->_1 plus_x#1#(0(),x8) -> c_4():5
             -->_1 plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14)):3
          
          4:W:map#2#(plus_x(x2),Nil()) -> c_2()
             
          
          5:W:plus_x#1#(0(),x8) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: map#2#(plus_x(x2),Nil()) -> c_2()
          5: plus_x#1#(0(),x8) -> c_4()
* Step 4: RemoveHeads WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            main#(x5,x12) -> c_1(map#2#(plus_x(x12),x5))
            map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
            plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
        - Weak TRS:
            main(x5,x12) -> map#2(plus_x(x12),x5)
            map#2(plus_x(x2),Nil()) -> Nil()
            map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))
            plus_x#1(0(),x8) -> x8
            plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14))
        - Signature:
            {main/2,map#2/2,plus_x#1/2,main#/2,map#2#/2,plus_x#1#/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1,c_1/1,c_2/0,c_3/2
            ,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#,map#2#,plus_x#1#} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:main#(x5,x12) -> c_1(map#2#(plus_x(x12),x5))
           -->_1 map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2)):2
        
        2:S:map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
           -->_1 plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14)):3
           -->_2 map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2)):2
        
        3:S:plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
           -->_1 plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14)):3
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(1,main#(x5,x12) -> c_1(map#2#(plus_x(x12),x5)))]
* Step 5: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
            plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
        - Weak TRS:
            main(x5,x12) -> map#2(plus_x(x12),x5)
            map#2(plus_x(x2),Nil()) -> Nil()
            map#2(plus_x(x6),Cons(x4,x2)) -> Cons(plus_x#1(x6,x4),map#2(plus_x(x6),x2))
            plus_x#1(0(),x8) -> x8
            plus_x#1(S(x12),x14) -> S(plus_x#1(x12,x14))
        - Signature:
            {main/2,map#2/2,plus_x#1/2,main#/2,map#2#/2,plus_x#1#/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1,c_1/1,c_2/0,c_3/2
            ,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#,map#2#,plus_x#1#} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
          plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
* Step 6: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
            plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
        - Signature:
            {main/2,map#2/2,plus_x#1/2,main#/2,map#2#/2,plus_x#1#/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1,c_1/1,c_2/0,c_3/2
            ,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#,map#2#,plus_x#1#} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
        and a lower component
          plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
        Further, following extension rules are added to the lower component.
          map#2#(plus_x(x6),Cons(x4,x2)) -> map#2#(plus_x(x6),x2)
          map#2#(plus_x(x6),Cons(x4,x2)) -> plus_x#1#(x6,x4)
** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
        - Signature:
            {main/2,map#2/2,plus_x#1/2,main#/2,map#2#/2,plus_x#1#/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1,c_1/1,c_2/0,c_3/2
            ,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#,map#2#,plus_x#1#} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2))
             -->_2 map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(plus_x#1#(x6,x4),map#2#(plus_x(x6),x2)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(map#2#(plus_x(x6),x2))
** Step 6.a:2: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(map#2#(plus_x(x6),x2))
        - Signature:
            {main/2,map#2/2,plus_x#1/2,main#/2,map#2#/2,plus_x#1#/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1,c_1/1,c_2/0,c_3/1
            ,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#,map#2#,plus_x#1#} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
    + Details:
        Signatures used:
        ----------------
          Cons :: ["A"(0) x "A"(8)] -(8)-> "A"(8)
          plus_x :: ["A"(0)] -(1)-> "A"(1)
          plus_x :: ["A"(0)] -(4)-> "A"(4)
          map#2# :: ["A"(1) x "A"(8)] -(5)-> "A"(14)
          c_3 :: ["A"(0)] -(0)-> "A"(14)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
          "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1)
          "c_3_A" :: ["A"(0)] -(0)-> "A"(1)
          "plus_x_A" :: ["A"(0)] -(1)-> "A"(1)
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          map#2#(plus_x(x6),Cons(x4,x2)) -> c_3(map#2#(plus_x(x6),x2))
        2. Weak:
          

** Step 6.b:1: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
        - Weak DPs:
            map#2#(plus_x(x6),Cons(x4,x2)) -> map#2#(plus_x(x6),x2)
            map#2#(plus_x(x6),Cons(x4,x2)) -> plus_x#1#(x6,x4)
        - Signature:
            {main/2,map#2/2,plus_x#1/2,main#/2,map#2#/2,plus_x#1#/2} / {0/0,Cons/2,Nil/0,S/1,plus_x/1,c_1/1,c_2/0,c_3/2
            ,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#,map#2#,plus_x#1#} and constructors {0,Cons,Nil,S
            ,plus_x}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
    + Details:
        Signatures used:
        ----------------
          Cons :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          S :: ["A"(1)] -(1)-> "A"(1)
          plus_x :: ["A"(1)] -(1)-> "A"(1)
          map#2# :: ["A"(1) x "A"(0)] -(0)-> "A"(0)
          plus_x#1# :: ["A"(1) x "A"(0)] -(0)-> "A"(0)
          c_5 :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          plus_x#1#(S(x12),x14) -> c_5(plus_x#1#(x12,x14))
        2. Weak:
          

WORST_CASE(?,O(n^2))