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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
    , +^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))
    , +^#(*(x, y), +(*(x, z), u())) -> c_2(+^#(*(x, +(y, z)), u()))}
  
  The usable rules are:
   {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
    , +(+(x, y), z) -> +(x, +(y, z))
    , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())}
  
  The estimated dependency graph contains the following edges:
   {+^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))}
     ==> {+^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))}
   {+^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))}
     ==> {+^#(*(x, y), +(*(x, z), u())) -> c_2(+^#(*(x, +(y, z)), u()))}
   {+^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))}
     ==> {+^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))}
   {+^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))}
     ==> {+^#(*(x, y), +(*(x, z), u())) -> c_2(+^#(*(x, +(y, z)), u()))}
   {+^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))}
     ==> {+^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))}
   {+^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))}
     ==> {+^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))}
  
  We consider the following path(s):
   1) {  +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
       , +^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))}
      
      The usable rules for this path are the following:
      {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
       , +(+(x, y), z) -> +(x, +(y, z))
       , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())}
      
        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), *(x, z)) -> *(x, +(y, z))
               , +(+(x, y), z) -> +(x, +(y, z))
               , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())
               , +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
               , +^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
             , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
               , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())}
              
              Details:
                 Interpretation Functions:
                  +(x1, x2) = [1] x1 + [1] x2 + [0]
                  *(x1, x2) = [1] x1 + [1] x2 + [3]
                  u() = [0]
                  +^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_0(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [1]
                  c_2(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {+^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))}
            and weakly orienting the rules
            {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
             , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {+^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))}
              
              Details:
                 Interpretation Functions:
                  +(x1, x2) = [1] x1 + [1] x2 + [0]
                  *(x1, x2) = [1] x1 + [1] x2 + [8]
                  u() = [9]
                  +^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_0(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
              
            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_1(+^#(x, +(y, z)))}
              Weak Rules:
                {  +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
                 , +(*(x, y), *(x, z)) -> *(x, +(y, z))
                 , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())}
            
            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_1(+^#(x, +(y, z)))}
                Weak Rules:
                  {  +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
                   , +(*(x, y), *(x, z)) -> *(x, +(y, z))
                   , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  *_0(2, 2) -> 2
                 , *_0(2, 3) -> 2
                 , *_0(3, 2) -> 2
                 , *_0(3, 3) -> 2
                 , u_0() -> 3
                 , +^#_0(2, 2) -> 4
                 , +^#_0(2, 3) -> 4
                 , +^#_0(3, 2) -> 4
                 , +^#_0(3, 3) -> 4
                 , c_0_0(4) -> 4}
      
   2) {  +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
       , +^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))
       , +^#(*(x, y), +(*(x, z), u())) -> c_2(+^#(*(x, +(y, z)), u()))}
      
      The usable rules for this path are the following:
      {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
       , +(+(x, y), z) -> +(x, +(y, z))
       , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())}
      
        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), *(x, z)) -> *(x, +(y, z))
               , +(+(x, y), z) -> +(x, +(y, z))
               , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())
               , +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
               , +^#(+(x, y), z) -> c_1(+^#(x, +(y, z)))
               , +^#(*(x, y), +(*(x, z), u())) -> c_2(+^#(*(x, +(y, z)), u()))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
             , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())
             , +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
             , +^#(*(x, y), +(*(x, z), u())) -> c_2(+^#(*(x, +(y, z)), u()))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
               , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())
               , +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
               , +^#(*(x, y), +(*(x, z), u())) -> c_2(+^#(*(x, +(y, z)), u()))}
              
              Details:
                 Interpretation Functions:
                  +(x1, x2) = [1] x1 + [1] x2 + [4]
                  *(x1, x2) = [1] x1 + [1] x2 + [5]
                  u() = [0]
                  +^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_0(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [13]
                  c_2(x1) = [1] x1 + [0]
              
            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_1(+^#(x, +(y, z)))}
              Weak Rules:
                {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
                 , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())
                 , +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
                 , +^#(*(x, y), +(*(x, z), u())) -> c_2(+^#(*(x, +(y, z)), u()))}
            
            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_1(+^#(x, +(y, z)))}
                Weak Rules:
                  {  +(*(x, y), *(x, z)) -> *(x, +(y, z))
                   , +(*(x, y), +(*(x, z), u())) -> +(*(x, +(y, z)), u())
                   , +^#(*(x, y), *(x, z)) -> c_0(+^#(y, z))
                   , +^#(*(x, y), +(*(x, z), u())) -> c_2(+^#(*(x, +(y, z)), u()))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  *_0(2, 2) -> 2
                 , *_0(2, 3) -> 2
                 , *_0(3, 2) -> 2
                 , *_0(3, 3) -> 2
                 , u_0() -> 3
                 , +^#_0(2, 2) -> 4
                 , +^#_0(2, 3) -> 4
                 , +^#_0(3, 2) -> 4
                 , +^#_0(3, 3) -> 4
                 , c_0_0(4) -> 4}