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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  f^#(s(x)) -> c_0(f^#(p(s(x))))
    , f^#(0()) -> c_1()
    , p^#(s(x)) -> c_2()}
  
  The usable rules are:
   {p(s(x)) -> x}
  
  The estimated dependency graph contains the following edges:
   {f^#(s(x)) -> c_0(f^#(p(s(x))))}
     ==> {f^#(0()) -> c_1()}
   {f^#(s(x)) -> c_0(f^#(p(s(x))))}
     ==> {f^#(s(x)) -> c_0(f^#(p(s(x))))}
  
  We consider the following path(s):
   1) {  f^#(s(x)) -> c_0(f^#(p(s(x))))
       , f^#(0()) -> c_1()}
      
      The usable rules for this path are the following:
      {p(s(x)) -> x}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           s(x1) = [1] x1 + [0]
           p(x1) = [1] x1 + [1]
           0() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [0]
           p^#(x1) = [0] x1 + [0]
           c_2() = [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {f^#(0()) -> c_1()}
            Weak Rules:
              {  p(s(x)) -> x
               , f^#(s(x)) -> c_0(f^#(p(s(x))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {f^#(0()) -> c_1()}
            and weakly orienting the rules
            {  p(s(x)) -> x
             , f^#(s(x)) -> c_0(f^#(p(s(x))))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {f^#(0()) -> c_1()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  s(x1) = [1] x1 + [8]
                  p(x1) = [1] x1 + [0]
                  0() = [6]
                  f^#(x1) = [1] x1 + [2]
                  c_0(x1) = [1] x1 + [0]
                  c_1() = [0]
                  p^#(x1) = [0] x1 + [0]
                  c_2() = [0]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules:
                {  f^#(0()) -> c_1()
                 , p(s(x)) -> x
                 , f^#(s(x)) -> c_0(f^#(p(s(x))))}
            
            Details:         
              The given problem does not contain any strict rules
      
   2) {f^#(s(x)) -> c_0(f^#(p(s(x))))}
      
      The usable rules for this path are the following:
      {p(s(x)) -> x}
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           s(x1) = [1] x1 + [0]
           p(x1) = [1] x1 + [1]
           0() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [0]
           p^#(x1) = [0] x1 + [0]
           c_2() = [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {f^#(s(x)) -> c_0(f^#(p(s(x))))}
            Weak Rules: {p(s(x)) -> x}
          
          Details:         
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {f^#(s(x)) -> c_0(f^#(p(s(x))))}
              Weak Rules: {p(s(x)) -> x}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost DP runtime-complexity with respect to
                Strict Rules: {f^#(s(x)) -> c_0(f^#(p(s(x))))}
                Weak Rules: {p(s(x)) -> x}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  s_0(2) -> 2
                 , s_0(2) -> 7
                 , s_1(2) -> 8
                 , p_1(8) -> 7
                 , f^#_0(2) -> 5
                 , f^#_1(7) -> 6
                 , c_0_1(6) -> 5
                 , c_0_1(6) -> 6}
      
   3) {p^#(s(x)) -> c_2()}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           f(x1) = [0] x1 + [0]
           s(x1) = [0] x1 + [0]
           p(x1) = [0] x1 + [0]
           0() = [0]
           f^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           c_1() = [0]
           p^#(x1) = [0] x1 + [0]
           c_2() = [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {p^#(s(x)) -> c_2()}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {p^#(s(x)) -> c_2()}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {p^#(s(x)) -> c_2()}
              
              Details:
                 Interpretation Functions:
                  f(x1) = [0] x1 + [0]
                  s(x1) = [1] x1 + [0]
                  p(x1) = [0] x1 + [0]
                  0() = [0]
                  f^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  c_1() = [0]
                  p^#(x1) = [1] x1 + [1]
                  c_2() = [0]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules: {p^#(s(x)) -> c_2()}
            
            Details:         
              The given problem does not contain any strict rules