YES
Time: 0.006332
TRS:
 {  if(true(), X, Y) -> X,
   if(false(), X, Y) -> activate Y,
                 f X -> if(X, c(), n__f n__true()),
                 f X -> n__f X,
              true() -> n__true(),
          activate X -> X,
     activate n__f X -> f activate X,
  activate n__true() -> true()}
 DP:
  DP:
   { if#(false(), X, Y) -> activate# Y,
                   f# X -> if#(X, c(), n__f n__true()),
       activate# n__f X -> f# activate X,
       activate# n__f X -> activate# X,
    activate# n__true() -> true#()}
  TRS:
  {  if(true(), X, Y) -> X,
    if(false(), X, Y) -> activate Y,
                  f X -> if(X, c(), n__f n__true()),
                  f X -> n__f X,
               true() -> n__true(),
           activate X -> X,
      activate n__f X -> f activate X,
   activate n__true() -> true()}
  UR:
   {  if(true(), X, Y) -> X,
     if(false(), X, Y) -> activate Y,
                   f X -> if(X, c(), n__f n__true()),
                   f X -> n__f X,
                true() -> n__true(),
            activate X -> X,
       activate n__f X -> f activate X,
    activate n__true() -> true()}
   EDG:
    {(activate# n__f X -> activate# X, activate# n__f X -> f# activate X)
     (activate# n__f X -> activate# X, activate# n__f X -> activate# X)
     (activate# n__f X -> activate# X, activate# n__true() -> true#())
     (activate# n__f X -> f# activate X, f# X -> if#(X, c(), n__f n__true()))
     (if#(false(), X, Y) -> activate# Y, activate# n__f X -> f# activate X)
     (if#(false(), X, Y) -> activate# Y, activate# n__f X -> activate# X)
     (if#(false(), X, Y) -> activate# Y, activate# n__true() -> true#())}
    STATUS:
     arrows: 0.720000
     SCCS (1):
      Scc:
       {activate# n__f X -> activate# X}
      
      
      SCC (1):
       Strict:
        {activate# n__f X -> activate# X}
       Weak:
       {  if(true(), X, Y) -> X,
         if(false(), X, Y) -> activate Y,
                       f X -> if(X, c(), n__f n__true()),
                       f X -> n__f X,
                    true() -> n__true(),
                activate X -> X,
           activate n__f X -> f activate X,
        activate n__true() -> true()}
       POLY:
        Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true
        Interpretation:
         [if](x0, x1, x2) = x0 + x1 + x2 + 1,
         
         [n__f](x0) = x0 + 1,
         
         [f](x0) = 0,
         
         [activate](x0) = 0,
         
         [c] = 1,
         
         [n__true] = 1,
         
         [true] = 1,
         
         [false] = 1,
         
         [activate#](x0) = x0
        Strict:
         activate# n__f X -> activate# X
         1 + 1X >= 0 + 1X
        Weak:
         activate n__true() -> true()
         0 >= 1
         activate n__f X -> f activate X
         0 + 0X >= 0 + 0X
         activate X -> X
         0 + 0X >= 1X
         true() -> n__true()
         1 >= 1
         f X -> n__f X
         0 + 0X >= 1 + 1X
         f X -> if(X, c(), n__f n__true())
         0 + 0X >= 4 + 1X
         if(false(), X, Y) -> activate Y
         2 + 1X + 1Y >= 0 + 0Y
         if(true(), X, Y) -> X
         2 + 1X + 1Y >= 1X
       Qed