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

  r1, r2, r3, r4, r5, r6, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          fold#_A(x1,x2,x3) = x1 + x3
          s_A(x1) = x1 + 6
          g_A(x1) = 5
          A_A() = 2
          B_A() = 3
          C_A() = 4
          |f'|_A(x1,x2) = x1 + 22
          triple_A(x1,x2,x3) = x1 + x2 + x3 + 7
  
    base order 2:
  
      precedence:
      
        |f'| > triple > s = g > fold# = C > B > A
      
  
    partial status:
  
      pi(fold#) = [1]
      pi(s) = [1]
      pi(g) = []
      pi(A) = []
      pi(B) = []
      pi(C) = []
      pi(|f'|) = [1]
      pi(triple) = [3]

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, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          foldB#_A(x1,x2) = x1 + x2 + 5
          s_A(x1) = x1 + 14
          f#_A(x1,x2) = x1 + x2 + 7
          foldB_A(x1,x2) = x1 + x2 + 1
          B_A() = 5
          |f'|#_A(x1,x2) = x1 + x2 + 5
          g_A(x1) = x1 + 1
          triple_A(x1,x2,x3) = x2 + x3 + 15
          A_A() = 2
          |0|_A() = 1
          |f''|#_A(x1) = x1 + 4
          foldC#_A(x1,x2) = x1 + x2 + 2
          foldC_A(x1,x2) = x1 + x2 + 1
          C_A() = 7
          f_A(x1,x2) = x1 + 14
          |f'|_A(x1,x2) = x1 + 14
          |f''|_A(x1) = x1 + 3
  
    base order 2:
  
      precedence:
      
        foldB# > foldB = foldC# > s = g > C > foldC > |f''| > f = |f'| > f# = |f'|# = triple > B = |f''|# > A > |0|
      
  
    partial status:
  
      pi(foldB#) = []
      pi(s) = []
      pi(f#) = []
      pi(foldB) = []
      pi(B) = []
      pi(|f'|#) = [1, 2]
      pi(g) = []
      pi(triple) = [3]
      pi(A) = []
      pi(|0|) = []
      pi(|f''|#) = [1]
      pi(foldC#) = [2]
      pi(foldC) = [2]
      pi(C) = []
      pi(f) = [1]
      pi(|f'|) = [1]
      pi(|f''|) = [1]

The following rules are removed:

  p7



-- SCC decomposition.

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)) -> f#(foldC(t,n),C())
p8: |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, p3, p4, p5, p6, p7, p8}


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

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

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          foldB#_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 30
          f#_A(x1,x2) = x1 + x2 + 2
          foldB_A(x1,x2) = x1 + x2 + 2
          B_A() = 25
          |f'|#_A(x1,x2) = x1 + x2
          g_A(x1) = x1 + 1
          triple_A(x1,x2,x3) = x2 + x3 + 10
          A_A() = 12
          |f''|#_A(x1) = x1 + 8
          |0|_A() = 1
          foldC#_A(x1,x2) = x1 + x2 + 6
          foldC_A(x1,x2) = x1 + x2 + 2
          C_A() = 31
          f_A(x1,x2) = x1 + 30
          |f'|_A(x1,x2) = x1 + 30
          |f''|_A(x1) = x1 + 27
  
    base order 2:
  
      precedence:
      
        foldC > C > foldB# = s = foldB = |f'|# > f# > B > f = |f'| > |f''| > triple = foldC# > A > g = |f''|# > |0|
      
  
    partial status:
  
      pi(foldB#) = [2]
      pi(s) = [1]
      pi(f#) = []
      pi(foldB) = []
      pi(B) = []
      pi(|f'|#) = [1, 2]
      pi(g) = []
      pi(triple) = [2]
      pi(A) = []
      pi(|f''|#) = []
      pi(|0|) = []
      pi(foldC#) = []
      pi(foldC) = [2]
      pi(C) = []
      pi(f) = []
      pi(|f'|) = []
      pi(|f''|) = [1]

The following rules are removed:

  p8



-- SCC decomposition.

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),B()) -> f#(triple(a,b,c),A())
p4: |f'|#(triple(a,b,c),A()) -> |f''|#(foldB(triple(s(a),|0|(),c),b))
p5: |f''|#(triple(a,b,c)) -> foldC#(triple(a,b,|0|()),c)
p6: foldC#(t,s(n)) -> f#(foldC(t,n),C())
p7: |f'|#(triple(a,b,c),A()) -> foldB#(triple(s(a),|0|(),c),b)

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, p3, p4, p5, p6, p7}


-- 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: |f'|#(triple(a,b,c),A()) -> |f''|#(foldB(triple(s(a),|0|(),c),b))
p5: |f''|#(triple(a,b,c)) -> foldC#(triple(a,b,|0|()),c)
p6: foldC#(t,s(n)) -> f#(foldC(t,n),C())
p7: |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, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          foldB#_A(x1,x2) = x1 + x2 + 12
          s_A(x1) = x1 + 28
          f#_A(x1,x2) = x1 + x2 + 2
          foldB_A(x1,x2) = x1 + x2 + 5
          B_A() = 32
          |f'|#_A(x1,x2) = x1 + x2
          g_A(x1) = x1 + 1
          triple_A(x1,x2,x3) = x2 + x3 + 12
          A_A() = 17
          |0|_A() = 1
          |f''|#_A(x1) = x1 + 10
          foldC#_A(x1,x2) = x1 + x2 + 8
          foldC_A(x1,x2) = x1 + x2
          C_A() = 33
          f_A(x1,x2) = x1 + 28
          |f'|_A(x1,x2) = x1 + 28
          |f''|_A(x1) = x1 + 2
          fold_A(x1,x2,x3) = x1 + 29
  
    base order 2:
  
      precedence:
      
        foldB = foldC = fold > |0| = f = |f'| > triple > |f'|# > foldB# = |f''|# > f# > B = foldC# > g > C > s > A > |f''|
      
  
    partial status:
  
      pi(foldB#) = []
      pi(s) = []
      pi(f#) = [1]
      pi(foldB) = []
      pi(B) = []
      pi(|f'|#) = [1]
      pi(g) = []
      pi(triple) = []
      pi(A) = []
      pi(|0|) = []
      pi(|f''|#) = []
      pi(foldC#) = []
      pi(foldC) = []
      pi(C) = []
      pi(f) = []
      pi(|f'|) = []
      pi(|f''|) = []
      pi(fold) = [1]

The following rules are removed:

  p5



-- SCC decomposition.

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: |f'|#(triple(a,b,c),A()) -> |f''|#(foldB(triple(s(a),|0|(),c),b))
p5: foldC#(t,s(n)) -> f#(foldC(t,n),C())
p6: |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, p3, p6}


-- 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),B()) -> f#(triple(a,b,c),A())
p4: |f'|#(triple(a,b,c),A()) -> foldB#(triple(s(a),|0|(),c),b)

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, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          foldB#_A(x1,x2) = x1 + x2 + 2
          s_A(x1) = x1 + 9
          f#_A(x1,x2) = x1 + x2 + 4
          foldB_A(x1,x2) = x1 + x2 + 1
          B_A() = 5
          |f'|#_A(x1,x2) = x1 + x2 + 2
          g_A(x1) = x1 + 1
          triple_A(x1,x2,x3) = x2 + x3 + 10
          A_A() = 2
          |0|_A() = 1
          C_A() = 6
          f_A(x1,x2) = x1 + 9
          foldC_A(x1,x2) = x1 + x2 + 1
          |f'|_A(x1,x2) = x1 + 9
          |f''|_A(x1) = x1 + 3
  
    base order 2:
  
      precedence:
      
        |f'|# > foldB# = |f''| > foldB = foldC > f# = f = |f'| > triple > s > A > g = |0| > C > B
      
  
    partial status:
  
      pi(foldB#) = [2]
      pi(s) = [1]
      pi(f#) = []
      pi(foldB) = [2]
      pi(B) = []
      pi(|f'|#) = []
      pi(g) = []
      pi(triple) = []
      pi(A) = []
      pi(|0|) = []
      pi(C) = []
      pi(f) = []
      pi(foldC) = [2]
      pi(|f'|) = []
      pi(|f''|) = [1]

The following rules are removed:

  p1



-- 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())
p3: |f'|#(triple(a,b,c),A()) -> foldB#(triple(s(a),|0|(),c),b)

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, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          f#_A(x1,x2) = x1 + x2 + 5
          |f'|#_A(x1,x2) = x2
          g_A(x1) = x1 + 4
          triple_A(x1,x2,x3) = 4
          B_A() = 20
          A_A() = 10
          C_A() = 21
          |f'|_A(x1,x2) = 8
          s_A(x1) = x1 + 5
  
    base order 2:
  
      precedence:
      
        C > B > |f'|# = s > triple = A = |f'| > f# > g
      
  
    partial status:
  
      pi(f#) = []
      pi(|f'|#) = []
      pi(g) = [1]
      pi(triple) = []
      pi(B) = []
      pi(A) = []
      pi(C) = []
      pi(|f'|) = []
      pi(s) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: |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:

  (no SCCs)