'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  +(+(x, y), z) -> +(x, +(y, z))
     , +(f(x), f(y)) -> f(+(x, y))
     , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)}

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  +^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))
    , +^#(f(x), f(y)) -> c_1(+^#(x, y))
    , +^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
  
  The usable rules are:
   {  +(+(x, y), z) -> +(x, +(y, z))
    , +(f(x), f(y)) -> f(+(x, y))
    , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)}
  
  The estimated dependency graph contains the following edges:
   {+^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))}
     ==> {+^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
   {+^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))}
     ==> {+^#(f(x), f(y)) -> c_1(+^#(x, y))}
   {+^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))}
     ==> {+^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))}
   {+^#(f(x), f(y)) -> c_1(+^#(x, y))}
     ==> {+^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
   {+^#(f(x), f(y)) -> c_1(+^#(x, y))}
     ==> {+^#(f(x), f(y)) -> c_1(+^#(x, y))}
   {+^#(f(x), f(y)) -> c_1(+^#(x, y))}
     ==> {+^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))}
   {+^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
     ==> {+^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
   {+^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
     ==> {+^#(f(x), f(y)) -> c_1(+^#(x, y))}
  
  We consider the following path(s):
   1) {  +^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))
       , +^#(f(x), f(y)) -> c_1(+^#(x, y))
       , +^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
      
      The usable rules for this path are the following:
      {  +(+(x, y), z) -> +(x, +(y, z))
       , +(f(x), f(y)) -> f(+(x, y))
       , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  +(+(x, y), z) -> +(x, +(y, z))
               , +(f(x), f(y)) -> f(+(x, y))
               , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
               , +^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))
               , +^#(f(x), f(y)) -> c_1(+^#(x, y))
               , +^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  +(f(x), f(y)) -> f(+(x, y))
             , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
             , +^#(f(x), f(y)) -> c_1(+^#(x, y))
             , +^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  +(f(x), f(y)) -> f(+(x, y))
               , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
               , +^#(f(x), f(y)) -> c_1(+^#(x, y))
               , +^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
              
              Details:
                 Interpretation Functions:
                  +(x1, x2) = [1] x1 + [1] x2 + [0]
                  f(x1) = [1] x1 + [8]
                  +^#(x1, x2) = [1] x1 + [1] x2 + [8]
                  c_0(x1) = [1] x1 + [2]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [4]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules:
                {  +(+(x, y), z) -> +(x, +(y, z))
                 , +^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))}
              Weak Rules:
                {  +(f(x), f(y)) -> f(+(x, y))
                 , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
                 , +^#(f(x), f(y)) -> c_1(+^#(x, y))
                 , +^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules:
                  {  +(+(x, y), z) -> +(x, +(y, z))
                   , +^#(+(x, y), z) -> c_0(+^#(x, +(y, z)))}
                Weak Rules:
                  {  +(f(x), f(y)) -> f(+(x, y))
                   , +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
                   , +^#(f(x), f(y)) -> c_1(+^#(x, y))
                   , +^#(f(x), +(f(y), z)) -> c_2(+^#(f(+(x, y)), z))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  f_0(2) -> 2
                 , +^#_0(2, 2) -> 3
                 , c_1_0(3) -> 3}