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: WeightGap 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:
        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_3) = {1,2},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]                  
               p(add) = [0]                  
              p(cons) = [1] x1 + [1] x2 + [0]
              p(leaf) = [0]                  
             p(mtadd) = [0]                  
               p(nil) = [0]                  
              p(node) = [1] x1 + [1] x2 + [0]
                 p(s) = [1] x1 + [0]         
              p(tadd) = [0]                  
              p(add#) = [0]                  
            p(mtadd#) = [0]                  
             p(tadd#) = [13]                 
               p(c_1) = [0]                  
               p(c_2) = [0]                  
               p(c_3) = [1] x1 + [1] x2 + [0]
               p(c_4) = [0]                  
               p(c_5) = [0]                  
               p(c_6) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          tadd#(x,node(y,ts)) = [13]             
                              > [0]              
                              = c_6(mtadd#(x,ts))
          
          
          Following rules are (at-least) weakly oriented:
          mtadd#(x,cons(t,ts)) =  [0]                         
                               >= [13]                        
                               =  c_3(tadd#(x,t),mtadd#(x,ts))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 5.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mtadd#(x,cons(t,ts)) -> c_3(tadd#(x,t),mtadd#(x,ts))
        - Weak DPs:
            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:
        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_3) = {1,2},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [1]                  
               p(add) = [4] x1 + [1] x2 + [0]
              p(cons) = [1] x1 + [1] x2 + [4]
              p(leaf) = [0]                  
             p(mtadd) = [4]                  
               p(nil) = [4]                  
              p(node) = [1] x2 + [10]        
                 p(s) = [1]                  
              p(tadd) = [1] x1 + [1] x2 + [0]
              p(add#) = [8] x1 + [2] x2 + [1]
            p(mtadd#) = [1] x2 + [11]        
             p(tadd#) = [1] x2 + [1]         
               p(c_1) = [0]                  
               p(c_2) = [0]                  
               p(c_3) = [1] x1 + [1] x2 + [0]
               p(c_4) = [0]                  
               p(c_5) = [0]                  
               p(c_6) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          mtadd#(x,cons(t,ts)) = [1] t + [1] ts + [15]       
                               > [1] t + [1] ts + [12]       
                               = c_3(tadd#(x,t),mtadd#(x,ts))
          
          
          Following rules are (at-least) weakly oriented:
          tadd#(x,node(y,ts)) =  [1] ts + [11]    
                              >= [1] ts + [11]    
                              =  c_6(mtadd#(x,ts))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 5.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak 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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: WeightGap 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:
        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_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]                  
               p(add) = [0]                  
              p(cons) = [1] x1 + [1] x2 + [0]
              p(leaf) = [0]                  
             p(mtadd) = [0]                  
               p(nil) = [0]                  
              p(node) = [1] x1 + [1] x2 + [0]
                 p(s) = [1] x1 + [13]        
              p(tadd) = [0]                  
              p(add#) = [1] x1 + [0]         
            p(mtadd#) = [1] x1 + [0]         
             p(tadd#) = [1] x1 + [0]         
               p(c_1) = [0]                  
               p(c_2) = [1] x1 + [0]         
               p(c_3) = [0]                  
               p(c_4) = [0]                  
               p(c_5) = [0]                  
               p(c_6) = [0]                  
          
          Following rules are strictly oriented:
          add#(s(x),y) = [1] x + [13]  
                       > [1] x + [0]   
                       = c_2(add#(x,y))
          
          
          Following rules are (at-least) weakly oriented:
          mtadd#(x,cons(t,ts)) =  [1] x + [0] 
                               >= [1] x + [0] 
                               =  mtadd#(x,ts)
          
          mtadd#(x,cons(t,ts)) =  [1] x + [0] 
                               >= [1] x + [0] 
                               =  tadd#(x,t)  
          
           tadd#(x,node(y,ts)) =  [1] x + [0] 
                               >= [1] x + [0] 
                               =  add#(x,y)   
          
           tadd#(x,node(y,ts)) =  [1] x + [0] 
                               >= [1] x + [0] 
                               =  mtadd#(x,ts)
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add#(s(x),y) -> c_2(add#(x,y))
            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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))