Term Rewriting System R:

   [t, n, x, a, b, c]
   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)

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   g(B) -> A
   g(C) -> A
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(f''(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(B) = 1
POL(triple(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
POL(foldC(x_1, x_2)) = x_1 + 2*x_2
POL(A) = 0
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(f'(x_1, x_2)) = 1 + x_1 + x_2
POL(foldB(x_1, x_2)) = x_1 + 2*x_2
POL(0) = 0
POL(C) = 1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   g(A) -> A
   f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))

where the Polynomial interpretation:
POL(g(x_1)) = 2*x_1
POL(f''(x_1)) = x_1
POL(s(x_1)) = x_1
POL(B) = 0
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(foldC(x_1, x_2)) = x_1 + x_2
POL(A) = 1
POL(f(x_1, x_2)) = x_1 + 2*x_2
POL(f'(x_1, x_2)) = x_1 + x_2
POL(foldB(x_1, x_2)) = x_1 + x_2
POL(0) = 0
POL(C) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   g(B) -> B
   g(C) -> B
   g(C) -> C

where the Polynomial interpretation:
POL(g(x_1)) = 1 + x_1
POL(f''(x_1)) = 2*x_1
POL(s(x_1)) = 1 + x_1
POL(B) = 0
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(foldC(x_1, x_2)) = x_1 + 2*x_2
POL(f(x_1, x_2)) = 2 + x_1 + x_2
POL(f'(x_1, x_2)) = 1 + x_1 + x_2
POL(foldB(x_1, x_2)) = x_1 + 2*x_2
POL(0) = 0
POL(C) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   foldB(t, 0) -> t
   foldC(t, 0) -> t

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(f''(x_1)) = 1 + x_1
POL(s(x_1)) = x_1
POL(B) = 0
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(foldC(x_1, x_2)) = x_1 + x_2
POL(f(x_1, x_2)) = x_1 + x_2
POL(f'(x_1, x_2)) = x_1 + x_2
POL(foldB(x_1, x_2)) = x_1 + x_2
POL(0) = 1
POL(C) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   foldB(t, s(n)) -> f(foldB(t, n), B)

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(f''(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(B) = 0
POL(foldC(x_1, x_2)) = x_1 + x_2
POL(f(x_1, x_2)) = x_1 + x_2
POL(f'(x_1, x_2)) = x_1 + x_2
POL(foldB(x_1, x_2)) = x_1 + x_2
POL(0) = 0
POL(C) = 1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   f(t, x) -> f'(t, g(x))

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(f''(x_1)) = 2*x_1
POL(s(x_1)) = 1 + x_1
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(foldC(x_1, x_2)) = x_1 + 2*x_2
POL(f'(x_1, x_2)) = x_1 + x_2
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(C) = 1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   f'(triple(a, b, c), C) -> triple(a, b, s(c))

where the Polynomial interpretation:
POL(f''(x_1)) = x_1
POL(s(x_1)) = x_1
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(foldC(x_1, x_2)) = x_1 + x_2
POL(f(x_1, x_2)) = x_1 + x_2
POL(f'(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(C) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   foldC(t, s(n)) -> f(foldC(t, n), C)

where the Polynomial interpretation:
POL(f''(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(foldC(x_1, x_2)) = x_1 + x_2
POL(f(x_1, x_2)) = x_1 + x_2
POL(0) = 0
POL(C) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)

where the Polynomial interpretation:
POL(f''(x_1)) = 1 + x_1
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(foldC(x_1, x_2)) = x_1 + x_2
POL(0) = 0
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.969 seconds.