WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mtadd(x,cons(t,ts)) -> cons(tadd(x,t),mtadd(x,ts))
            mtadd(x,nil()) -> nil()
            tadd(x,leaf()) -> leaf()
            tadd(x,node(y,ts)) -> node(add(x,y),mtadd(x,ts))
        - Signature:
            {add/2,mtadd/2,tadd/2} / {0/0,cons/2,leaf/0,nil/0,node/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,mtadd,tadd} and constructors {0,cons,leaf,nil,node,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          add#(0(),y) -> c_1()
          add#(s(x),y) -> c_2(add#(x,y))
          mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
          mtadd#(x,nil()) -> c_4()
          tadd#(x,leaf()) -> c_5()
          tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(0(),y) -> c_1()
            add#(s(x),y) -> c_2(add#(x,y))
            mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
            mtadd#(x,nil()) -> c_4()
            tadd#(x,leaf()) -> c_5()
            tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mtadd(x,cons(t,ts)) -> cons(tadd(x,t),mtadd(x,ts))
            mtadd(x,nil()) -> nil()
            tadd(x,leaf()) -> leaf()
            tadd(x,node(y,ts)) -> node(add(x,y),mtadd(x,ts))
        - Signature:
            {add/2,mtadd/2,tadd/2,add#/2,mtadd#/2,tadd#/2} / {0/0,cons/2,leaf/0,nil/0,node/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mtadd#,tadd#} and constructors {0,cons,leaf,nil,node
            ,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5}
        by application of
          Pre({1,4,5}) = {2,3,6}.
        Here rules are labelled as follows:
          1: add#(0(),y) -> c_1()
          2: add#(s(x),y) -> c_2(add#(x,y))
          3: mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
          4: mtadd#(x,nil()) -> c_4()
          5: tadd#(x,leaf()) -> c_5()
          6: tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(x),y) -> c_2(add#(x,y))
            mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
            tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
        - Weak DPs:
            add#(0(),y) -> c_1()
            mtadd#(x,nil()) -> c_4()
            tadd#(x,leaf()) -> c_5()
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mtadd(x,cons(t,ts)) -> cons(tadd(x,t),mtadd(x,ts))
            mtadd(x,nil()) -> nil()
            tadd(x,leaf()) -> leaf()
            tadd(x,node(y,ts)) -> node(add(x,y),mtadd(x,ts))
        - Signature:
            {add/2,mtadd/2,tadd/2,add#/2,mtadd#/2,tadd#/2} / {0/0,cons/2,leaf/0,nil/0,node/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mtadd#,tadd#} and constructors {0,cons,leaf,nil,node
            ,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(x),y) -> c_2(add#(x,y))
             -->_1 add#(0(),y) -> c_1():4
             -->_1 add#(s(x),y) -> c_2(add#(x,y)):1
          
          2:S:mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
             -->_1 tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts)):3
             -->_1 tadd#(x,leaf()) -> c_5():6
             -->_2 mtadd#(x,nil()) -> c_4():5
             -->_2 mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts)):2
          
          3:S:tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
             -->_2 mtadd#(x,nil()) -> c_4():5
             -->_1 add#(0(),y) -> c_1():4
             -->_2 mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts)):2
             -->_1 add#(s(x),y) -> c_2(add#(x,y)):1
          
          4:W:add#(0(),y) -> c_1()
             
          
          5:W:mtadd#(x,nil()) -> c_4()
             
          
          6:W:tadd#(x,leaf()) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: tadd#(x,leaf()) -> c_5()
          5: mtadd#(x,nil()) -> c_4()
          4: add#(0(),y) -> c_1()
* Step 4: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(x),y) -> c_2(add#(x,y))
            mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
            tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mtadd(x,cons(t,ts)) -> cons(tadd(x,t),mtadd(x,ts))
            mtadd(x,nil()) -> nil()
            tadd(x,leaf()) -> leaf()
            tadd(x,node(y,ts)) -> node(add(x,y),mtadd(x,ts))
        - Signature:
            {add/2,mtadd/2,tadd/2,add#/2,mtadd#/2,tadd#/2} / {0/0,cons/2,leaf/0,nil/0,node/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mtadd#,tadd#} and constructors {0,cons,leaf,nil,node
            ,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          add#(s(x),y) -> c_2(add#(x,y))
          mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
          tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
* Step 5: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(x),y) -> c_2(add#(x,y))
            mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
            tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
        - Signature:
            {add/2,mtadd/2,tadd/2,add#/2,mtadd#/2,tadd#/2} / {0/0,cons/2,leaf/0,nil/0,node/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mtadd#,tadd#} and constructors {0,cons,leaf,nil,node
            ,s}
    + 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
          mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
          tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
        and a lower component
          add#(s(x),y) -> c_2(add#(x,y))
        Further, following extension rules are added to the lower component.
          mtadd#(x,cons(t,ts)) -> mtadd#(x,ts)
          mtadd#(x,cons(t,ts)) -> tadd#(x,t)
          tadd#(x,node(y,ts)) -> add#(x,y)
          tadd#(x,node(y,ts)) -> mtadd#(x,ts)
** Step 5.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
            tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
        - Signature:
            {add/2,mtadd/2,tadd/2,add#/2,mtadd#/2,tadd#/2} / {0/0,cons/2,leaf/0,nil/0,node/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mtadd#,tadd#} and constructors {0,cons,leaf,nil,node
            ,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
             -->_1 tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts)):2
             -->_2 mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts)):1
          
          2:S:tadd#(x,node(y,ts)) -> c_6(add#(x,y),mtadd#(x,ts))
             -->_2 mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          tadd#(x,node(y,ts)) -> c_6(mtadd#(x,ts))
** Step 5.a:2: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
            tadd#(x,node(y,ts)) -> c_6(mtadd#(x,ts))
        - Signature:
            {add/2,mtadd/2,tadd/2,add#/2,mtadd#/2,tadd#/2} / {0/0,cons/2,leaf/0,nil/0,node/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mtadd#,tadd#} and constructors {0,cons,leaf,nil,node
            ,s}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
    + Details:
        Signatures used:
        ----------------
          cons :: ["A"(8) x "A"(8)] -(8)-> "A"(8)
          node :: ["A"(0) x "A"(8)] -(0)-> "A"(8)
          mtadd# :: ["A"(0) x "A"(8)] -(0)-> "A"(0)
          tadd# :: ["A"(0) x "A"(8)] -(0)-> "A"(0)
          c_3 :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          c_6 :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
          "c_3_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1)
          "c_6_A" :: ["A"(0)] -(0)-> "A"(1)
          "cons_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1)
          "node_A" :: ["A"(0) x "A"(1)] -(0)-> "A"(1)
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
        2. Weak:
          tadd#(x,node(y,ts)) -> c_6(mtadd#(x,ts))
** Step 5.a:3: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            tadd#(x,node(y,ts)) -> c_6(mtadd#(x,ts))
        - Weak DPs:
            mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
        - Signature:
            {add/2,mtadd/2,tadd/2,add#/2,mtadd#/2,tadd#/2} / {0/0,cons/2,leaf/0,nil/0,node/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mtadd#,tadd#} and constructors {0,cons,leaf,nil,node
            ,s}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
    + Details:
        Signatures used:
        ----------------
          cons :: ["A"(14) x "A"(14)] -(14)-> "A"(14)
          node :: ["A"(0) x "A"(14)] -(14)-> "A"(14)
          mtadd# :: ["A"(0) x "A"(14)] -(0)-> "A"(14)
          tadd# :: ["A"(0) x "A"(14)] -(8)-> "A"(0)
          c_3 :: ["A"(0) x "A"(0)] -(0)-> "A"(14)
          c_6 :: ["A"(0)] -(0)-> "A"(10)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
          "c_3_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1)
          "c_6_A" :: ["A"(0)] -(0)-> "A"(1)
          "cons_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1)
          "node_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1)
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          tadd#(x,node(y,ts)) -> c_6(mtadd#(x,ts))
        2. Weak:
          

** Step 5.b:1: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            add#(s(x),y) -> c_2(add#(x,y))
        - Weak DPs:
            mtadd#(x,cons(t,ts)) -> mtadd#(x,ts)
            mtadd#(x,cons(t,ts)) -> tadd#(x,t)
            tadd#(x,node(y,ts)) -> add#(x,y)
            tadd#(x,node(y,ts)) -> mtadd#(x,ts)
        - Signature:
            {add/2,mtadd/2,tadd/2,add#/2,mtadd#/2,tadd#/2} / {0/0,cons/2,leaf/0,nil/0,node/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mtadd#,tadd#} and constructors {0,cons,leaf,nil,node
            ,s}
    + 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)
          node :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          s :: ["A"(1)] -(1)-> "A"(1)
          add# :: ["A"(1) x "A"(0)] -(0)-> "A"(0)
          mtadd# :: ["A"(1) x "A"(0)] -(0)-> "A"(0)
          tadd# :: ["A"(1) x "A"(0)] -(0)-> "A"(0)
          c_2 :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          add#(s(x),y) -> c_2(add#(x,y))
        2. Weak:
          

WORST_CASE(?,O(n^2))