YES

Problem:
 f(0()) -> true()
 f(1()) -> false()
 f(s(x)) -> f(x)
 if(true(),s(x),s(y)) -> s(x)
 if(false(),s(x),s(y)) -> s(y)
 g(x,c(y)) -> c(g(x,y))
 g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))

Proof:
 DP Processor:
  DPs:
   f#(s(x)) -> f#(x)
   g#(x,c(y)) -> g#(x,y)
   g#(x,c(y)) -> g#(s(x),y)
   g#(x,c(y)) -> f#(x)
   g#(x,c(y)) -> if#(f(x),c(g(s(x),y)),c(y))
   g#(x,c(y)) -> g#(x,if(f(x),c(g(s(x),y)),c(y)))
  TRS:
   f(0()) -> true()
   f(1()) -> false()
   f(s(x)) -> f(x)
   if(true(),s(x),s(y)) -> s(x)
   if(false(),s(x),s(y)) -> s(y)
   g(x,c(y)) -> c(g(x,y))
   g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
  Usable Rule Processor:
   DPs:
    f#(s(x)) -> f#(x)
    g#(x,c(y)) -> g#(x,y)
    g#(x,c(y)) -> g#(s(x),y)
    g#(x,c(y)) -> f#(x)
    g#(x,c(y)) -> if#(f(x),c(g(s(x),y)),c(y))
    g#(x,c(y)) -> g#(x,if(f(x),c(g(s(x),y)),c(y)))
   TRS:
    f12(x,y) -> x
    f12(x,y) -> y
    g(x,c(y)) -> c(g(x,y))
    g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
    f(0()) -> true()
    f(1()) -> false()
    f(s(x)) -> f(x)
   CDG Processor:
    DPs:
     f#(s(x)) -> f#(x)
     g#(x,c(y)) -> g#(x,y)
     g#(x,c(y)) -> g#(s(x),y)
     g#(x,c(y)) -> f#(x)
     g#(x,c(y)) -> if#(f(x),c(g(s(x),y)),c(y))
     g#(x,c(y)) -> g#(x,if(f(x),c(g(s(x),y)),c(y)))
    TRS:
     f12(x,y) -> x
     f12(x,y) -> y
     g(x,c(y)) -> c(g(x,y))
     g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
     f(0()) -> true()
     f(1()) -> false()
     f(s(x)) -> f(x)
    graph:
     g#(x,c(y)) -> g#(s(x),y) -> g#(x,c(y)) -> g#(x,y)
     g#(x,c(y)) -> g#(s(x),y) -> g#(x,c(y)) -> g#(s(x),y)
     g#(x,c(y)) -> g#(s(x),y) -> g#(x,c(y)) -> f#(x)
     g#(x,c(y)) -> g#(s(x),y) -> g#(x,c(y)) -> if#(f(x),c(g(s(x),y)),c(y))
     g#(x,c(y)) -> g#(s(x),y) ->
     g#(x,c(y)) -> g#(x,if(f(x),c(g(s(x),y)),c(y)))
     g#(x,c(y)) -> g#(x,y) -> g#(x,c(y)) -> g#(x,y)
     g#(x,c(y)) -> g#(x,y) -> g#(x,c(y)) -> g#(s(x),y)
     g#(x,c(y)) -> g#(x,y) -> g#(x,c(y)) -> f#(x)
     g#(x,c(y)) -> g#(x,y) -> g#(x,c(y)) -> if#(f(x),c(g(s(x),y)),c(y))
     g#(x,c(y)) -> g#(x,y) -> g#(x,c(y)) -> g#(x,if(f(x),c(g(s(x),y)),c(y)))
     g#(x,c(y)) -> f#(x) -> f#(s(x)) -> f#(x)
     f#(s(x)) -> f#(x) -> f#(s(x)) -> f#(x)
    Restore Modifier:
     DPs:
      f#(s(x)) -> f#(x)
      g#(x,c(y)) -> g#(x,y)
      g#(x,c(y)) -> g#(s(x),y)
      g#(x,c(y)) -> f#(x)
      g#(x,c(y)) -> if#(f(x),c(g(s(x),y)),c(y))
      g#(x,c(y)) -> g#(x,if(f(x),c(g(s(x),y)),c(y)))
     TRS:
      f(0()) -> true()
      f(1()) -> false()
      f(s(x)) -> f(x)
      if(true(),s(x),s(y)) -> s(x)
      if(false(),s(x),s(y)) -> s(y)
      g(x,c(y)) -> c(g(x,y))
      g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
     SCC Processor:
      #sccs: 2
      #rules: 3
      #arcs: 12/36
      DPs:
       g#(x,c(y)) -> g#(s(x),y)
       g#(x,c(y)) -> g#(x,y)
      TRS:
       f(0()) -> true()
       f(1()) -> false()
       f(s(x)) -> f(x)
       if(true(),s(x),s(y)) -> s(x)
       if(false(),s(x),s(y)) -> s(y)
       g(x,c(y)) -> c(g(x,y))
       g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [g#](x0, x1) = x0 + x1,
        
        [g](x0, x1) = x0 + x1 + 1,
        
        [c](x0) = x0 + 1,
        
        [if](x0, x1, x2) = x2,
        
        [s](x0) = 1,
        
        [false] = 0,
        
        [1] = 0,
        
        [true] = 0,
        
        [f](x0) = 0,
        
        [0] = 0
       orientation:
        g#(x,c(y)) = x + y + 1 >= y + 1 = g#(s(x),y)
        
        g#(x,c(y)) = x + y + 1 >= x + y = g#(x,y)
        
        f(0()) = 0 >= 0 = true()
        
        f(1()) = 0 >= 0 = false()
        
        f(s(x)) = 0 >= 0 = f(x)
        
        if(true(),s(x),s(y)) = 1 >= 1 = s(x)
        
        if(false(),s(x),s(y)) = 1 >= 1 = s(y)
        
        g(x,c(y)) = x + y + 2 >= x + y + 2 = c(g(x,y))
        
        g(x,c(y)) = x + y + 2 >= x + y + 2 = g(x,if(f(x),c(g(s(x),y)),c(y)))
       problem:
        DPs:
         g#(x,c(y)) -> g#(s(x),y)
        TRS:
         f(0()) -> true()
         f(1()) -> false()
         f(s(x)) -> f(x)
         if(true(),s(x),s(y)) -> s(x)
         if(false(),s(x),s(y)) -> s(y)
         g(x,c(y)) -> c(g(x,y))
         g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
       Matrix Interpretation Processor:
        dimension: 1
        interpretation:
         [g#](x0, x1) = x1,
         
         [g](x0, x1) = x1 + 1,
         
         [c](x0) = x0 + 1,
         
         [if](x0, x1, x2) = 0,
         
         [s](x0) = 0,
         
         [false] = 0,
         
         [1] = 0,
         
         [true] = 0,
         
         [f](x0) = 0,
         
         [0] = 0
        orientation:
         g#(x,c(y)) = y + 1 >= y = g#(s(x),y)
         
         f(0()) = 0 >= 0 = true()
         
         f(1()) = 0 >= 0 = false()
         
         f(s(x)) = 0 >= 0 = f(x)
         
         if(true(),s(x),s(y)) = 0 >= 0 = s(x)
         
         if(false(),s(x),s(y)) = 0 >= 0 = s(y)
         
         g(x,c(y)) = y + 2 >= y + 2 = c(g(x,y))
         
         g(x,c(y)) = y + 2 >= 1 = g(x,if(f(x),c(g(s(x),y)),c(y)))
        problem:
         DPs:
          
         TRS:
          f(0()) -> true()
          f(1()) -> false()
          f(s(x)) -> f(x)
          if(true(),s(x),s(y)) -> s(x)
          if(false(),s(x),s(y)) -> s(y)
          g(x,c(y)) -> c(g(x,y))
          g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
        Qed
      
      DPs:
       f#(s(x)) -> f#(x)
      TRS:
       f(0()) -> true()
       f(1()) -> false()
       f(s(x)) -> f(x)
       if(true(),s(x),s(y)) -> s(x)
       if(false(),s(x),s(y)) -> s(y)
       g(x,c(y)) -> c(g(x,y))
       g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0) = x0 + 1,
        
        [g](x0, x1) = 0,
        
        [c](x0) = 0,
        
        [if](x0, x1, x2) = x0 + x1 + x2,
        
        [s](x0) = x0 + 1,
        
        [false] = 1,
        
        [1] = 1,
        
        [true] = 1,
        
        [f](x0) = x0,
        
        [0] = 1
       orientation:
        f#(s(x)) = x + 2 >= x + 1 = f#(x)
        
        f(0()) = 1 >= 1 = true()
        
        f(1()) = 1 >= 1 = false()
        
        f(s(x)) = x + 1 >= x = f(x)
        
        if(true(),s(x),s(y)) = x + y + 3 >= x + 1 = s(x)
        
        if(false(),s(x),s(y)) = x + y + 3 >= y + 1 = s(y)
        
        g(x,c(y)) = 0 >= 0 = c(g(x,y))
        
        g(x,c(y)) = 0 >= 0 = g(x,if(f(x),c(g(s(x),y)),c(y)))
       problem:
        DPs:
         
        TRS:
         f(0()) -> true()
         f(1()) -> false()
         f(s(x)) -> f(x)
         if(true(),s(x),s(y)) -> s(x)
         if(false(),s(x),s(y)) -> s(y)
         g(x,c(y)) -> c(g(x,y))
         g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
       Qed