YES

Problem:
 g(A()) -> A()
 g(B()) -> A()
 g(B()) -> B()
 g(C()) -> A()
 g(C()) -> B()
 g(C()) -> C()
 foldB(t,0()) -> t
 foldB(t,s(n)) -> f(foldB(t,n),B())
 foldC(t,0()) -> t
 foldC(t,s(n)) -> f(foldC(t,n),C())
 f(t,x) -> f'(t,g(x))
 f'(triple(a,b,c),C()) -> triple(a,b,s(c))
 f'(triple(a,b,c),B()) -> f(triple(a,b,c),A())
 f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b))
 f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c)
 fold(t,x,0()) -> t
 fold(t,x,s(n)) -> f(fold(t,x,n),x)

Proof:
 DP Processor:
  DPs:
   foldB#(t,s(n)) -> foldB#(t,n)
   foldB#(t,s(n)) -> f#(foldB(t,n),B())
   foldC#(t,s(n)) -> foldC#(t,n)
   foldC#(t,s(n)) -> f#(foldC(t,n),C())
   f#(t,x) -> g#(x)
   f#(t,x) -> f'#(t,g(x))
   f'#(triple(a,b,c),B()) -> f#(triple(a,b,c),A())
   f'#(triple(a,b,c),A()) -> foldB#(triple(s(a),0(),c),b)
   f'#(triple(a,b,c),A()) -> f''#(foldB(triple(s(a),0(),c),b))
   f''#(triple(a,b,c)) -> foldC#(triple(a,b,0()),c)
   fold#(t,x,s(n)) -> fold#(t,x,n)
   fold#(t,x,s(n)) -> f#(fold(t,x,n),x)
  TRS:
   g(A()) -> A()
   g(B()) -> A()
   g(B()) -> B()
   g(C()) -> A()
   g(C()) -> B()
   g(C()) -> C()
   foldB(t,0()) -> t
   foldB(t,s(n)) -> f(foldB(t,n),B())
   foldC(t,0()) -> t
   foldC(t,s(n)) -> f(foldC(t,n),C())
   f(t,x) -> f'(t,g(x))
   f'(triple(a,b,c),C()) -> triple(a,b,s(c))
   f'(triple(a,b,c),B()) -> f(triple(a,b,c),A())
   f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b))
   f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c)
   fold(t,x,0()) -> t
   fold(t,x,s(n)) -> f(fold(t,x,n),x)
  TDG Processor:
   DPs:
    foldB#(t,s(n)) -> foldB#(t,n)
    foldB#(t,s(n)) -> f#(foldB(t,n),B())
    foldC#(t,s(n)) -> foldC#(t,n)
    foldC#(t,s(n)) -> f#(foldC(t,n),C())
    f#(t,x) -> g#(x)
    f#(t,x) -> f'#(t,g(x))
    f'#(triple(a,b,c),B()) -> f#(triple(a,b,c),A())
    f'#(triple(a,b,c),A()) -> foldB#(triple(s(a),0(),c),b)
    f'#(triple(a,b,c),A()) -> f''#(foldB(triple(s(a),0(),c),b))
    f''#(triple(a,b,c)) -> foldC#(triple(a,b,0()),c)
    fold#(t,x,s(n)) -> fold#(t,x,n)
    fold#(t,x,s(n)) -> f#(fold(t,x,n),x)
   TRS:
    g(A()) -> A()
    g(B()) -> A()
    g(B()) -> B()
    g(C()) -> A()
    g(C()) -> B()
    g(C()) -> C()
    foldB(t,0()) -> t
    foldB(t,s(n)) -> f(foldB(t,n),B())
    foldC(t,0()) -> t
    foldC(t,s(n)) -> f(foldC(t,n),C())
    f(t,x) -> f'(t,g(x))
    f'(triple(a,b,c),C()) -> triple(a,b,s(c))
    f'(triple(a,b,c),B()) -> f(triple(a,b,c),A())
    f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b))
    f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c)
    fold(t,x,0()) -> t
    fold(t,x,s(n)) -> f(fold(t,x,n),x)
   graph:
    fold#(t,x,s(n)) -> fold#(t,x,n) ->
    fold#(t,x,s(n)) -> f#(fold(t,x,n),x)
    fold#(t,x,s(n)) -> fold#(t,x,n) ->
    fold#(t,x,s(n)) -> fold#(t,x,n)
    fold#(t,x,s(n)) -> f#(fold(t,x,n),x) ->
    f#(t,x) -> f'#(t,g(x))
    fold#(t,x,s(n)) -> f#(fold(t,x,n),x) ->
    f#(t,x) -> g#(x)
    f''#(triple(a,b,c)) -> foldC#(triple(a,b,0()),c) ->
    foldC#(t,s(n)) -> f#(foldC(t,n),C())
    f''#(triple(a,b,c)) -> foldC#(triple(a,b,0()),c) ->
    foldC#(t,s(n)) -> foldC#(t,n)
    f'#(triple(a,b,c),B()) -> f#(triple(a,b,c),A()) ->
    f#(t,x) -> f'#(t,g(x))
    f'#(triple(a,b,c),B()) -> f#(triple(a,b,c),A()) ->
    f#(t,x) -> g#(x)
    f'#(triple(a,b,c),A()) -> f''#(foldB(triple(s(a),0(),c),b)) ->
    f''#(triple(a,b,c)) -> foldC#(triple(a,b,0()),c)
    f'#(triple(a,b,c),A()) -> foldB#(triple(s(a),0(),c),b) ->
    foldB#(t,s(n)) -> f#(foldB(t,n),B())
    f'#(triple(a,b,c),A()) -> foldB#(triple(s(a),0(),c),b) ->
    foldB#(t,s(n)) -> foldB#(t,n)
    foldC#(t,s(n)) -> foldC#(t,n) ->
    foldC#(t,s(n)) -> f#(foldC(t,n),C())
    foldC#(t,s(n)) -> foldC#(t,n) ->
    foldC#(t,s(n)) -> foldC#(t,n)
    foldC#(t,s(n)) -> f#(foldC(t,n),C()) ->
    f#(t,x) -> f'#(t,g(x))
    foldC#(t,s(n)) -> f#(foldC(t,n),C()) -> f#(t,x) -> g#(x)
    f#(t,x) -> f'#(t,g(x)) ->
    f'#(triple(a,b,c),A()) -> f''#(foldB(triple(s(a),0(),c),b))
    f#(t,x) -> f'#(t,g(x)) ->
    f'#(triple(a,b,c),A()) -> foldB#(triple(s(a),0(),c),b)
    f#(t,x) -> f'#(t,g(x)) ->
    f'#(triple(a,b,c),B()) -> f#(triple(a,b,c),A())
    foldB#(t,s(n)) -> f#(foldB(t,n),B()) ->
    f#(t,x) -> f'#(t,g(x))
    foldB#(t,s(n)) -> f#(foldB(t,n),B()) -> f#(t,x) -> g#(x)
    foldB#(t,s(n)) -> foldB#(t,n) ->
    foldB#(t,s(n)) -> f#(foldB(t,n),B())
    foldB#(t,s(n)) -> foldB#(t,n) -> foldB#(t,s(n)) -> foldB#(t,n)
   SCC Processor:
    #sccs: 2
    #rules: 10
    #arcs: 22/144
    DPs:
     fold#(t,x,s(n)) -> fold#(t,x,n)
    TRS:
     g(A()) -> A()
     g(B()) -> A()
     g(B()) -> B()
     g(C()) -> A()
     g(C()) -> B()
     g(C()) -> C()
     foldB(t,0()) -> t
     foldB(t,s(n)) -> f(foldB(t,n),B())
     foldC(t,0()) -> t
     foldC(t,s(n)) -> f(foldC(t,n),C())
     f(t,x) -> f'(t,g(x))
     f'(triple(a,b,c),C()) -> triple(a,b,s(c))
     f'(triple(a,b,c),B()) -> f(triple(a,b,c),A())
     f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b))
     f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c)
     fold(t,x,0()) -> t
     fold(t,x,s(n)) -> f(fold(t,x,n),x)
    KBO Processor:
     argument filtering:
      pi(A) = []
      pi(g) = 0
      pi(B) = []
      pi(C) = []
      pi(0) = []
      pi(foldB) = 0
      pi(s) = [0]
      pi(f) = 0
      pi(foldC) = 0
      pi(f') = 0
      pi(triple) = []
      pi(f'') = 0
      pi(fold) = [0,2]
      pi(fold#) = 2
     weight function:
      w0 = 1
      w(fold#) = w(triple) = w(foldC) = w(0) = w(C) = w(B) = w(A) = 1
      w(fold) = w(f'') = w(f') = w(f) = w(s) = w(foldB) = w(g) = 0
     precedence:
      fold# ~ fold ~ s ~ 0 ~ C > B > f'' ~ triple ~ f' ~ foldC ~ f ~ foldB ~ g ~ A
     problem:
      DPs:
       
      TRS:
       g(A()) -> A()
       g(B()) -> A()
       g(B()) -> B()
       g(C()) -> A()
       g(C()) -> B()
       g(C()) -> C()
       foldB(t,0()) -> t
       foldB(t,s(n)) -> f(foldB(t,n),B())
       foldC(t,0()) -> t
       foldC(t,s(n)) -> f(foldC(t,n),C())
       f(t,x) -> f'(t,g(x))
       f'(triple(a,b,c),C()) -> triple(a,b,s(c))
       f'(triple(a,b,c),B()) -> f(triple(a,b,c),A())
       f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b))
       f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c)
       fold(t,x,0()) -> t
       fold(t,x,s(n)) -> f(fold(t,x,n),x)
     Qed
    
    DPs:
     f#(t,x) -> f'#(t,g(x))
     f'#(triple(a,b,c),B()) -> f#(triple(a,b,c),A())
     f'#(triple(a,b,c),A()) -> foldB#(triple(s(a),0(),c),b)
     foldB#(t,s(n)) -> foldB#(t,n)
     foldB#(t,s(n)) -> f#(foldB(t,n),B())
     f'#(triple(a,b,c),A()) -> f''#(foldB(triple(s(a),0(),c),b))
     f''#(triple(a,b,c)) -> foldC#(triple(a,b,0()),c)
     foldC#(t,s(n)) -> foldC#(t,n)
     foldC#(t,s(n)) -> f#(foldC(t,n),C())
    TRS:
     g(A()) -> A()
     g(B()) -> A()
     g(B()) -> B()
     g(C()) -> A()
     g(C()) -> B()
     g(C()) -> C()
     foldB(t,0()) -> t
     foldB(t,s(n)) -> f(foldB(t,n),B())
     foldC(t,0()) -> t
     foldC(t,s(n)) -> f(foldC(t,n),C())
     f(t,x) -> f'(t,g(x))
     f'(triple(a,b,c),C()) -> triple(a,b,s(c))
     f'(triple(a,b,c),B()) -> f(triple(a,b,c),A())
     f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b))
     f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c)
     fold(t,x,0()) -> t
     fold(t,x,s(n)) -> f(fold(t,x,n),x)
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [f''#](x0) = x0 + 7,
      
      [f'#](x0, x1) = x0 + 2x1 + 1,
      
      [foldC#](x0, x1) = x0 + x1 + 6,
      
      [f#](x0, x1) = x0 + 2x1 + 2,
      
      [foldB#](x0, x1) = x0 + 3x1,
      
      [fold](x0, x1, x2) = 6x0 + 2x1 + x2,
      
      [f''](x0) = x0,
      
      [triple](x0, x1, x2) = 4x1 + x2 + 2,
      
      [f'](x0, x1) = x0 + 7,
      
      [foldC](x0, x1) = x0 + x1,
      
      [f](x0, x1) = x0 + 7,
      
      [s](x0) = x0 + 7,
      
      [foldB](x0, x1) = x0 + 3x1,
      
      [0] = 0,
      
      [C] = 5,
      
      [B] = 5,
      
      [g](x0) = x0,
      
      [A] = 4
     orientation:
      f#(t,x) = t + 2x + 2 >= t + 2x + 1 = f'#(t,g(x))
      
      f'#(triple(a,b,c),B()) = 4b + c + 13 >= 4b + c + 12 = f#(triple(a,b,c),A())
      
      f'#(triple(a,b,c),A()) = 4b + c + 11 >= 3b + c + 2 = foldB#(triple(s(a),0(),c),b)
      
      foldB#(t,s(n)) = 3n + t + 21 >= 3n + t = foldB#(t,n)
      
      foldB#(t,s(n)) = 3n + t + 21 >= 3n + t + 12 = f#(foldB(t,n),B())
      
      f'#(triple(a,b,c),A()) = 4b + c + 11 >= 3b + c + 9 = f''#(foldB(triple(s(a),0(),c),b))
      
      f''#(triple(a,b,c)) = 4b + c + 9 >= 4b + c + 8 = foldC#(triple(a,b,0()),c)
      
      foldC#(t,s(n)) = n + t + 13 >= n + t + 6 = foldC#(t,n)
      
      foldC#(t,s(n)) = n + t + 13 >= n + t + 12 = f#(foldC(t,n),C())
      
      g(A()) = 4 >= 4 = A()
      
      g(B()) = 5 >= 4 = A()
      
      g(B()) = 5 >= 5 = B()
      
      g(C()) = 5 >= 4 = A()
      
      g(C()) = 5 >= 5 = B()
      
      g(C()) = 5 >= 5 = C()
      
      foldB(t,0()) = t >= t = t
      
      foldB(t,s(n)) = 3n + t + 21 >= 3n + t + 7 = f(foldB(t,n),B())
      
      foldC(t,0()) = t >= t = t
      
      foldC(t,s(n)) = n + t + 7 >= n + t + 7 = f(foldC(t,n),C())
      
      f(t,x) = t + 7 >= t + 7 = f'(t,g(x))
      
      f'(triple(a,b,c),C()) = 4b + c + 9 >= 4b + c + 9 = triple(a,b,s(c))
      
      f'(triple(a,b,c),B()) = 4b + c + 9 >= 4b + c + 9 = f(triple(a,b,c),A())
      
      f'(triple(a,b,c),A()) = 4b + c + 9 >= 3b + c + 2 = f''(foldB(triple(s(a),0(),c),b))
      
      f''(triple(a,b,c)) = 4b + c + 2 >= 4b + c + 2 = foldC(triple(a,b,0()),c)
      
      fold(t,x,0()) = 6t + 2x >= t = t
      
      fold(t,x,s(n)) = n + 6t + 2x + 7 >= n + 6t + 2x + 7 = f(fold(t,x,n),x)
     problem:
      DPs:
       
      TRS:
       g(A()) -> A()
       g(B()) -> A()
       g(B()) -> B()
       g(C()) -> A()
       g(C()) -> B()
       g(C()) -> C()
       foldB(t,0()) -> t
       foldB(t,s(n)) -> f(foldB(t,n),B())
       foldC(t,0()) -> t
       foldC(t,s(n)) -> f(foldC(t,n),C())
       f(t,x) -> f'(t,g(x))
       f'(triple(a,b,c),C()) -> triple(a,b,s(c))
       f'(triple(a,b,c),B()) -> f(triple(a,b,c),A())
       f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b))
       f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c)
       fold(t,x,0()) -> t
       fold(t,x,s(n)) -> f(fold(t,x,n),x)
     Qed