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

  r12

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              fold#_A(x1,x2,x3) = x3
              s_A(x1) = x1 + 3
              |f'|_A(x1,x2) = x1 + 2
              triple_A(x1,x2,x3) = x1 + x2 + 1
              C_A() = 2
              fold#!_A(x1,x2,x3) = x3
              s!_A(x1) = 11
              |f'|!_A(x1,x2) = x1 + 1
              triple!_A(x1,x2,x3) = x1 + 1
              C!_A() = 10
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              fold#_A(x1,x2,x3) = x3
              s_A(x1) = x1 + 5
              |f'|_A(x1,x2) = x1 + 5
              triple_A(x1,x2,x3) = x1 + x2 + 1
              C_A() = 2
              fold#!_A(x1,x2,x3) = x3 + 6
              s!_A(x1) = x1 + 5
              |f'|!_A(x1,x2) = x1 + 3
              triple!_A(x1,x2,x3) = 2
              C!_A() = 17
        
  
    partial status:
  
      pi(fold#) = [3]
      pi(s) = [1]
      pi(|f'|) = [1, 2]
      pi(triple) = []
      pi(C) = []

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

  r6, r7, r8, r9, r10, r11, r12, r13, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              foldB#_A(x1,x2) = x1 + x2
              s_A(x1) = x1 + 5
              f#_A(x1,x2) = x1
              foldB_A(x1,x2) = x1 + x2
              B_A() = 2
              |f'|#_A(x1,x2) = x1
              g_A(x1) = 1
              triple_A(x1,x2,x3) = x2 + x3 + 13
              A_A() = 2
              |0|_A() = 0
              |f''|#_A(x1) = x1
              foldC#_A(x1,x2) = x1 + x2
              foldC_A(x1,x2) = x1 + x2 + 1
              C_A() = 1
              f_A(x1,x2) = x1 + 5
              |f'|_A(x1,x2) = x1 + 5
              |f''|_A(x1) = x1 + 1
              foldB#!_A(x1,x2) = x1 + x2
              s!_A(x1) = 13
              f#!_A(x1,x2) = x1 + 4
              foldB!_A(x1,x2) = x1 + x2 + 3
              B!_A() = 4
              |f'|#!_A(x1,x2) = x1 + 4
              g!_A(x1) = x1 + 8
              triple!_A(x1,x2,x3) = x2 + x3 + 14
              A!_A() = 3
              |0|!_A() = 21
              |f''|#!_A(x1) = x1
              foldC#!_A(x1,x2) = x1 + x2
              foldC!_A(x1,x2) = x1 + x2 + 7
              C!_A() = 13
              f!_A(x1,x2) = x1 + 7
              |f'|!_A(x1,x2) = x1 + 7
              |f''|!_A(x1) = x1 + 7
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              foldB#_A(x1,x2) = x1
              s_A(x1) = x1 + 6
              f#_A(x1,x2) = x1
              foldB_A(x1,x2) = x1
              B_A() = 4
              |f'|#_A(x1,x2) = x1
              g_A(x1) = 3
              triple_A(x1,x2,x3) = 13
              A_A() = 4
              |0|_A() = 1
              |f''|#_A(x1) = x1
              foldC#_A(x1,x2) = x1
              foldC_A(x1,x2) = x1
              C_A() = 3
              f_A(x1,x2) = x1
              |f'|_A(x1,x2) = x1
              |f''|_A(x1) = 13
              foldB#!_A(x1,x2) = x1 + x2 + 6
              s!_A(x1) = 14
              f#!_A(x1,x2) = x1
              foldB!_A(x1,x2) = x2 + 12
              B!_A() = 16
              |f'|#!_A(x1,x2) = x1
              g!_A(x1) = x1 + 16
              triple!_A(x1,x2,x3) = 15
              A!_A() = 18
              |0|!_A() = 20
              |f''|#!_A(x1) = 19
              foldC#!_A(x1,x2) = x1 + 1
              foldC!_A(x1,x2) = 16
              C!_A() = 14
              f!_A(x1,x2) = 17
              |f'|!_A(x1,x2) = 17
              |f''|!_A(x1) = 16
        
  
    partial status:
  
      pi(foldB#) = []
      pi(s) = []
      pi(f#) = []
      pi(foldB) = [1]
      pi(B) = []
      pi(|f'|#) = []
      pi(g) = [1]
      pi(triple) = [1]
      pi(A) = []
      pi(|0|) = []
      pi(|f''|#) = [1]
      pi(foldC#) = []
      pi(foldC) = [1]
      pi(C) = []
      pi(f) = [1]
      pi(|f'|) = [1]
      pi(|f''|) = [1]

The following rules are removed:

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



-- SCC decomposition.

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

p1: f#(t,x) -> |f'|#(t,g(x))
p2: |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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: f#(t,x) -> |f'|#(t,g(x))
p2: |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

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              f#_A(x1,x2) = x1
              |f'|#_A(x1,x2) = x1
              g_A(x1) = 5
              triple_A(x1,x2,x3) = 1
              B_A() = 5
              A_A() = 1
              C_A() = 5
              f#!_A(x1,x2) = x1 + x2 + 4
              |f'|#!_A(x1,x2) = x1 + 3
              g!_A(x1) = x1 + 3
              triple!_A(x1,x2,x3) = x2 + 8
              B!_A() = 7
              A!_A() = 3
              C!_A() = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              f#_A(x1,x2) = x1
              |f'|#_A(x1,x2) = x1
              g_A(x1) = 18
              triple_A(x1,x2,x3) = x1 + x2 + x3 + 1
              B_A() = 18
              A_A() = 7
              C_A() = 18
              f#!_A(x1,x2) = x2 + 8
              |f'|#!_A(x1,x2) = 7
              g!_A(x1) = x1
              triple!_A(x1,x2,x3) = x2 + x3 + 16
              B!_A() = 17
              A!_A() = 6
              C!_A() = 0
        
  
    partial status:
  
      pi(f#) = []
      pi(|f'|#) = [2]
      pi(g) = []
      pi(triple) = [2]
      pi(B) = []
      pi(A) = []
      pi(C) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.