'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  *(i(x), x) -> 1()
     , *(1(), y) -> y
     , *(x, 0()) -> 0()
     , *(*(x, y), z) -> *(x, *(y, z))}

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  *^#(i(x), x) -> c_0()
    , *^#(1(), y) -> c_1()
    , *^#(x, 0()) -> c_2()
    , *^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))}
  
  The usable rules are:
   {  *(i(x), x) -> 1()
    , *(1(), y) -> y
    , *(x, 0()) -> 0()
    , *(*(x, y), z) -> *(x, *(y, z))}
  
  The estimated dependency graph contains the following edges:
   {*^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))}
     ==> {*^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))}
   {*^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))}
     ==> {*^#(x, 0()) -> c_2()}
   {*^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))}
     ==> {*^#(1(), y) -> c_1()}
   {*^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))}
     ==> {*^#(i(x), x) -> c_0()}
  
  We consider the following path(s):
   1) {  *^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))
       , *^#(x, 0()) -> c_2()}
      
      The usable rules for this path are the following:
      {  *(i(x), x) -> 1()
       , *(1(), y) -> y
       , *(x, 0()) -> 0()
       , *(*(x, y), z) -> *(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:
              {  *(i(x), x) -> 1()
               , *(1(), y) -> y
               , *(x, 0()) -> 0()
               , *(*(x, y), z) -> *(x, *(y, z))
               , *^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))
               , *^#(x, 0()) -> c_2()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  *(i(x), x) -> 1()
             , *(1(), y) -> y
             , *(x, 0()) -> 0()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  *(i(x), x) -> 1()
               , *(1(), y) -> y
               , *(x, 0()) -> 0()}
              
              Details:
                 Interpretation Functions:
                  *(x1, x2) = [1] x1 + [1] x2 + [1]
                  i(x1) = [1] x1 + [0]
                  1() = [0]
                  0() = [0]
                  *^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_0() = [0]
                  c_1() = [0]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {*^#(x, 0()) -> c_2()}
            and weakly orienting the rules
            {  *(i(x), x) -> 1()
             , *(1(), y) -> y
             , *(x, 0()) -> 0()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {*^#(x, 0()) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  *(x1, x2) = [1] x1 + [1] x2 + [0]
                  i(x1) = [1] x1 + [0]
                  1() = [0]
                  0() = [0]
                  *^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  c_2() = [0]
                  c_3(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_3(*^#(x, *(y, z)))}
              Weak Rules:
                {  *^#(x, 0()) -> c_2()
                 , *(i(x), x) -> 1()
                 , *(1(), y) -> y
                 , *(x, 0()) -> 0()}
            
            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_3(*^#(x, *(y, z)))}
                Weak Rules:
                  {  *^#(x, 0()) -> c_2()
                   , *(i(x), x) -> 1()
                   , *(1(), y) -> y
                   , *(x, 0()) -> 0()}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  i_0(2) -> 2
                 , i_0(3) -> 2
                 , i_0(4) -> 2
                 , 1_0() -> 3
                 , 0_0() -> 4
                 , *^#_0(2, 2) -> 5
                 , *^#_0(2, 3) -> 5
                 , *^#_0(2, 4) -> 5
                 , *^#_0(3, 2) -> 5
                 , *^#_0(3, 3) -> 5
                 , *^#_0(3, 4) -> 5
                 , *^#_0(4, 2) -> 5
                 , *^#_0(4, 3) -> 5
                 , *^#_0(4, 4) -> 5
                 , c_2_0() -> 5}
      
   2) {  *^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))
       , *^#(i(x), x) -> c_0()}
      
      The usable rules for this path are the following:
      {  *(i(x), x) -> 1()
       , *(1(), y) -> y
       , *(x, 0()) -> 0()
       , *(*(x, y), z) -> *(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:
              {  *(i(x), x) -> 1()
               , *(1(), y) -> y
               , *(x, 0()) -> 0()
               , *(*(x, y), z) -> *(x, *(y, z))
               , *^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))
               , *^#(i(x), x) -> c_0()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  *(i(x), x) -> 1()
             , *(1(), y) -> y
             , *(x, 0()) -> 0()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  *(i(x), x) -> 1()
               , *(1(), y) -> y
               , *(x, 0()) -> 0()}
              
              Details:
                 Interpretation Functions:
                  *(x1, x2) = [1] x1 + [1] x2 + [1]
                  i(x1) = [1] x1 + [0]
                  1() = [0]
                  0() = [0]
                  *^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_0() = [0]
                  c_1() = [0]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {*^#(i(x), x) -> c_0()}
            and weakly orienting the rules
            {  *(i(x), x) -> 1()
             , *(1(), y) -> y
             , *(x, 0()) -> 0()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {*^#(i(x), x) -> c_0()}
              
              Details:
                 Interpretation Functions:
                  *(x1, x2) = [1] x1 + [1] x2 + [0]
                  i(x1) = [1] x1 + [2]
                  1() = [0]
                  0() = [6]
                  *^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  c_2() = [0]
                  c_3(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_3(*^#(x, *(y, z)))}
              Weak Rules:
                {  *^#(i(x), x) -> c_0()
                 , *(i(x), x) -> 1()
                 , *(1(), y) -> y
                 , *(x, 0()) -> 0()}
            
            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_3(*^#(x, *(y, z)))}
                Weak Rules:
                  {  *^#(i(x), x) -> c_0()
                   , *(i(x), x) -> 1()
                   , *(1(), y) -> y
                   , *(x, 0()) -> 0()}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  i_0(2) -> 2
                 , 1_0() -> 2
                 , 0_0() -> 2
                 , *^#_0(2, 2) -> 1
                 , c_0_0() -> 1}
      
   3) {  *^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))
       , *^#(1(), y) -> c_1()}
      
      The usable rules for this path are the following:
      {  *(i(x), x) -> 1()
       , *(1(), y) -> y
       , *(x, 0()) -> 0()
       , *(*(x, y), z) -> *(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:
              {  *(i(x), x) -> 1()
               , *(1(), y) -> y
               , *(x, 0()) -> 0()
               , *(*(x, y), z) -> *(x, *(y, z))
               , *^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))
               , *^#(1(), y) -> c_1()}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  *(i(x), x) -> 1()
             , *(1(), y) -> y
             , *(x, 0()) -> 0()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  *(i(x), x) -> 1()
               , *(1(), y) -> y
               , *(x, 0()) -> 0()}
              
              Details:
                 Interpretation Functions:
                  *(x1, x2) = [1] x1 + [1] x2 + [1]
                  i(x1) = [1] x1 + [0]
                  1() = [0]
                  0() = [0]
                  *^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_0() = [0]
                  c_1() = [0]
                  c_2() = [0]
                  c_3(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {*^#(1(), y) -> c_1()}
            and weakly orienting the rules
            {  *(i(x), x) -> 1()
             , *(1(), y) -> y
             , *(x, 0()) -> 0()}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {*^#(1(), y) -> c_1()}
              
              Details:
                 Interpretation Functions:
                  *(x1, x2) = [1] x1 + [1] x2 + [0]
                  i(x1) = [1] x1 + [0]
                  1() = [0]
                  0() = [0]
                  *^#(x1, x2) = [1] x1 + [1] x2 + [1]
                  c_0() = [0]
                  c_1() = [0]
                  c_2() = [0]
                  c_3(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_3(*^#(x, *(y, z)))}
              Weak Rules:
                {  *^#(1(), y) -> c_1()
                 , *(i(x), x) -> 1()
                 , *(1(), y) -> y
                 , *(x, 0()) -> 0()}
            
            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_3(*^#(x, *(y, z)))}
                Weak Rules:
                  {  *^#(1(), y) -> c_1()
                   , *(i(x), x) -> 1()
                   , *(1(), y) -> y
                   , *(x, 0()) -> 0()}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  i_0(2) -> 2
                 , i_0(3) -> 2
                 , i_0(4) -> 2
                 , 1_0() -> 3
                 , 0_0() -> 4
                 , *^#_0(2, 2) -> 5
                 , *^#_0(2, 3) -> 5
                 , *^#_0(2, 4) -> 5
                 , *^#_0(3, 2) -> 5
                 , *^#_0(3, 3) -> 5
                 , *^#_0(3, 4) -> 5
                 , *^#_0(4, 2) -> 5
                 , *^#_0(4, 3) -> 5
                 , *^#_0(4, 4) -> 5
                 , c_1_0() -> 5}
      
   4) {*^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))}
      
      The usable rules for this path are the following:
      {  *(i(x), x) -> 1()
       , *(1(), y) -> y
       , *(x, 0()) -> 0()
       , *(*(x, y), z) -> *(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:
              {  *(i(x), x) -> 1()
               , *(1(), y) -> y
               , *(x, 0()) -> 0()
               , *(*(x, y), z) -> *(x, *(y, z))
               , *^#(*(x, y), z) -> c_3(*^#(x, *(y, z)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {  *(i(x), x) -> 1()
             , *(1(), y) -> y
             , *(x, 0()) -> 0()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  *(i(x), x) -> 1()
               , *(1(), y) -> y
               , *(x, 0()) -> 0()}
              
              Details:
                 Interpretation Functions:
                  *(x1, x2) = [1] x1 + [1] x2 + [1]
                  i(x1) = [1] x1 + [0]
                  1() = [0]
                  0() = [0]
                  *^#(x1, x2) = [1] x1 + [1] x2 + [0]
                  c_0() = [0]
                  c_1() = [0]
                  c_2() = [0]
                  c_3(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_3(*^#(x, *(y, z)))}
              Weak Rules:
                {  *(i(x), x) -> 1()
                 , *(1(), y) -> y
                 , *(x, 0()) -> 0()}
            
            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_3(*^#(x, *(y, z)))}
                Weak Rules:
                  {  *(i(x), x) -> 1()
                   , *(1(), y) -> y
                   , *(x, 0()) -> 0()}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  i_0(2) -> 2
                 , i_0(3) -> 2
                 , i_0(4) -> 2
                 , 1_0() -> 3
                 , 0_0() -> 4
                 , *^#_0(2, 2) -> 5
                 , *^#_0(2, 3) -> 5
                 , *^#_0(2, 4) -> 5
                 , *^#_0(3, 2) -> 5
                 , *^#_0(3, 3) -> 5
                 , *^#_0(3, 4) -> 5
                 , *^#_0(4, 2) -> 5
                 , *^#_0(4, 3) -> 5
                 , *^#_0(4, 4) -> 5}