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)
  fold(t,x,|0|()) -> t
  fold(t,x,s(n)) -> f(fold(t,x,n),x)


-- 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)
p11: fold#(t,x,s(n)) -> f#(fold(t,x,n),x)
p12: fold#(t,x,s(n)) -> fold#(t,x,n)

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)
r16: fold(t,x,|0|()) -> t
r17: fold(t,x,s(n)) -> f(fold(t,x,n),x)

The estimated dependency graph contains the following SCCs:

  {p12}
  {p1, p2, p3, p4, p5, p7, p8, p9, p10}


-- Reduction pair.

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

p1: fold#(t,x,s(n)) -> fold#(t,x,n)

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)
r16: fold(t,x,|0|()) -> t
r17: fold(t,x,s(n)) -> f(fold(t,x,n),x)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          fold#_A(x1,x2,x3) = x1 + x2 + x3 + 7
          s_A(x1) = x1 + 6
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          fold#_A(x1,x2,x3) = x3
          s_A(x1) = max{11, x1}
          fold#!_A(x1,x2,x3) = max{6, x1 - 1, x2 - 1, x3 - 12}
          s!_A(x1) = max{5, x1 - 187}
  
    partial status:
  
      pi(fold#) = [1]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.

-- 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)
r16: fold(t,x,|0|()) -> t
r17: fold(t,x,s(n)) -> f(fold(t,x,n),x)

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:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          foldB#_A(x1,x2) = x1 + x2 + 10
          s_A(x1) = x1 + 9
          f#_A(x1,x2) = x1 + x2 + 8
          foldB_A(x1,x2) = x1 + x2
          B_A() = 7
          |f'|#_A(x1,x2) = x1 + x2 + 7
          g_A(x1) = x1
          triple_A(x1,x2,x3) = x2 + x3 + 4
          A_A() = 5
          |0|_A() = 0
          |f''|#_A(x1) = x1 + 11
          foldC#_A(x1,x2) = x1 + x2 + 10
          foldC_A(x1,x2) = x1 + x2 + 1
          C_A() = 8
          f_A(x1,x2) = x1 + x2 + 1
          |f'|_A(x1,x2) = x1 + x2 + 1
          |f''|_A(x1) = x1 + 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          foldB#_A(x1,x2) = max{1, x1 - 72, x2 - 71}
          s_A(x1) = max{63, x1}
          f#_A(x1,x2) = max{1, x1 - 72, x2 - 13}
          foldB_A(x1,x2) = max{17, x1, x2 + 1}
          B_A = 6
          |f'|#_A(x1,x2) = max{1, x1 - 72, x2 - 136}
          g_A(x1) = max{12, x1 - 7}
          triple_A(x1,x2,x3) = max{x1 - 126, x2 + 41, x3 - 64}
          A_A = 0
          |0|_A = 0
          |f''|#_A(x1) = max{1, x1 - 72}
          foldC#_A(x1,x2) = max{1, x1 - 72}
          foldC_A(x1,x2) = max{41, x1}
          C_A = 12
          f_A(x1,x2) = max{x1, x2 + 29}
          |f'|_A(x1,x2) = max{x1, x2 + 17}
          |f''|_A(x1) = max{41, x1}
          foldB#!_A(x1,x2) = max{0, x2 - 67}
          s!_A(x1) = 3
          f#!_A(x1,x2) = max{2, x2 - 9}
          foldB!_A(x1,x2) = max{0, x2 - 68}
          B!_A = 308
          |f'|#!_A(x1,x2) = max{x1 + 265, x2 - 1}
          g!_A(x1) = max{310, x1 - 13}
          triple!_A(x1,x2,x3) = max{x1 - 140, x2 - 10, x3 + 31}
          A!_A = 307
          |0|!_A = 138
          |f''|#!_A(x1) = 308
          foldC#!_A(x1,x2) = max{1, x1 - 77, x2 - 77}
          foldC!_A(x1,x2) = max{x1 + 310, x2 - 64}
          C!_A = 0
          f!_A(x1,x2) = max{309, x1 + 125, x2 - 17}
          |f'|!_A(x1,x2) = max{x1 + 96, x2 - 11}
          |f''|!_A(x1) = max{110, x1 - 12}
  
    partial status:
  
      pi(foldB#) = [1, 2]
      pi(s) = [1]
      pi(f#) = []
      pi(foldB) = [1]
      pi(B) = []
      pi(|f'|#) = [1, 2]
      pi(g) = []
      pi(triple) = [2]
      pi(A) = []
      pi(|0|) = []
      pi(|f''|#) = [1]
      pi(foldC#) = [2]
      pi(foldC) = []
      pi(C) = []
      pi(f) = []
      pi(|f'|) = []
      pi(|f''|) = [1]

The following rules are removed:

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

Then no dependency pair remains.