YES

We show the termination of the TRS R:

  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)

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: foldB#(t,s(n)) -> f#(foldB(t,n),B())
p2: foldB#(t,s(n)) -> foldB#(t,n)
p3: foldC#(t,s(n)) -> f#(foldC(t,n),C())
p4: foldC#(t,s(n)) -> foldC#(t,n)
p5: f#(t,x) -> |f'|#(t,g(x))
p6: f#(t,x) -> g#(x)
p7: |f'|#(triple(a,b,c),B()) -> f#(triple(a,b,c),A())
p8: |f'|#(triple(a,b,c),A()) -> |f''|#(foldB(triple(s(a),|0|(),c),b))
p9: |f'|#(triple(a,b,c),A()) -> foldB#(triple(s(a),|0|(),c),b)
p10: |f''|#(triple(a,b,c)) -> foldC#(triple(a,b,|0|()),c)

and R consists of:

r1: g(A()) -> A()
r2: g(B()) -> A()
r3: g(B()) -> B()
r4: g(C()) -> A()
r5: g(C()) -> B()
r6: g(C()) -> C()
r7: foldB(t,|0|()) -> t
r8: foldB(t,s(n)) -> f(foldB(t,n),B())
r9: foldC(t,|0|()) -> t
r10: foldC(t,s(n)) -> f(foldC(t,n),C())
r11: f(t,x) -> |f'|(t,g(x))
r12: |f'|(triple(a,b,c),C()) -> triple(a,b,s(c))
r13: |f'|(triple(a,b,c),B()) -> f(triple(a,b,c),A())
r14: |f'|(triple(a,b,c),A()) -> |f''|(foldB(triple(s(a),|0|(),c),b))
r15: |f''|(triple(a,b,c)) -> foldC(triple(a,b,|0|()),c)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p7, p8, p9, p10}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: foldB#(t,s(n)) -> f#(foldB(t,n),B())
p2: f#(t,x) -> |f'|#(t,g(x))
p3: |f'|#(triple(a,b,c),A()) -> foldB#(triple(s(a),|0|(),c),b)
p4: foldB#(t,s(n)) -> foldB#(t,n)
p5: |f'|#(triple(a,b,c),A()) -> |f''|#(foldB(triple(s(a),|0|(),c),b))
p6: |f''|#(triple(a,b,c)) -> foldC#(triple(a,b,|0|()),c)
p7: foldC#(t,s(n)) -> foldC#(t,n)
p8: foldC#(t,s(n)) -> f#(foldC(t,n),C())
p9: |f'|#(triple(a,b,c),B()) -> f#(triple(a,b,c),A())

and R consists of:

r1: g(A()) -> A()
r2: g(B()) -> A()
r3: g(B()) -> B()
r4: g(C()) -> A()
r5: g(C()) -> B()
r6: g(C()) -> C()
r7: foldB(t,|0|()) -> t
r8: foldB(t,s(n)) -> f(foldB(t,n),B())
r9: foldC(t,|0|()) -> t
r10: foldC(t,s(n)) -> f(foldC(t,n),C())
r11: f(t,x) -> |f'|(t,g(x))
r12: |f'|(triple(a,b,c),C()) -> triple(a,b,s(c))
r13: |f'|(triple(a,b,c),B()) -> f(triple(a,b,c),A())
r14: |f'|(triple(a,b,c),A()) -> |f''|(foldB(triple(s(a),|0|(),c),b))
r15: |f''|(triple(a,b,c)) -> foldC(triple(a,b,|0|()),c)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^4
    order: lexicographic order
    interpretations:
      foldB#_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(1,0,0,0),(0,0,0,0)) x1 + ((1,0,0,0),(1,1,0,0),(1,0,0,0),(1,0,0,0)) x2 + (5,7,4,0)
      s_A(x1) = ((1,0,0,0),(0,1,0,0),(1,0,0,0),(1,1,1,0)) x1 + (25,1,1,1)
      f#_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(0,0,0,0),(0,1,0,0)) x1 + ((0,0,0,0),(1,0,0,0),(1,1,0,0),(0,0,0,0)) x2 + (7,15,1,25)
      foldB_A(x1,x2) = x1 + ((1,0,0,0),(1,1,0,0),(0,1,0,0),(0,0,0,0)) x2 + (3,1,8,1)
      B_A() = (16,1,1,0)
      |f'|#_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,1,0)) x1 + ((0,0,0,0),(1,0,0,0),(0,0,0,0),(1,0,0,0)) x2 + (7,6,0,16)
      g_A(x1) = ((1,0,0,0),(0,0,0,0),(0,0,0,0),(0,1,0,0)) x1 + (8,3,2,0)
      triple_A(x1,x2,x3) = x2 + ((1,0,0,0),(0,0,0,0),(0,1,0,0),(0,0,0,0)) x3 + (2,6,0,3)
      A_A() = (1,4,1,3)
      |0|_A() = (1,1,1,1)
      |f''|#_A(x1) = ((1,0,0,0),(1,0,0,0),(0,0,0,0),(1,0,0,0)) x1 + (2,2,5,12)
      foldC#_A(x1,x2) = ((1,0,0,0),(1,0,0,0),(1,1,0,0),(0,0,0,0)) x1 + ((1,0,0,0),(1,0,0,0),(1,0,0,0),(1,0,0,0)) x2
      foldC_A(x1,x2) = x1 + ((1,0,0,0),(1,1,0,0),(1,0,0,0),(0,0,0,0)) x2 + (1,1,1,1)
      C_A() = (9,2,1,1)
      |f''|_A(x1) = ((1,0,0,0),(1,0,0,0),(1,0,0,0),(0,0,0,0)) x1 + (3,0,0,3)
      |f'|_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(1,0,0,0),(0,0,0,0)) x1 + ((0,0,0,0),(1,0,0,0),(0,0,0,0),(0,0,0,0)) x2 + (25,0,5,2)
      f_A(x1,x2) = ((1,0,0,0),(0,1,0,0),(1,0,0,0),(0,0,0,0)) x1 + ((0,0,0,0),(1,0,0,0),(0,0,0,0),(0,0,0,0)) x2 + (25,9,5,2)

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9

We remove them from the problem.  Then no dependency pair remains.