Term Rewriting System R:

   [x, y, z, t, a, b, c]
   g(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldf(x, nil) -> x
   foldf(x, cons(y, z)) -> f(foldf(x, z), y)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, cons(C, c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b))
   f''(triple(a, b, c)) -> foldf(triple(a, b, nil), 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(nil) = 0
POL(f''(x_1)) = x_1
POL(foldf(x_1, x_2)) = x_1 + x_2
POL(B) = 1
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(A) = 0
POL(f(x_1, x_2)) = x_1 + x_2
POL(f'(x_1, x_2)) = x_1 + x_2
POL(C) = 1
POL(cons(x_1, x_2)) = x_1 + x_2
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

where the Polynomial interpretation:
POL(g(x_1)) = 2*x_1
POL(nil) = 0
POL(f''(x_1)) = x_1
POL(foldf(x_1, x_2)) = x_1 + 2*x_2
POL(B) = 0
POL(triple(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
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(C) = 0
POL(cons(x_1, x_2)) = x_1 + x_2
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(C) -> B

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(nil) = 0
POL(f''(x_1)) = x_1
POL(foldf(x_1, x_2)) = x_1 + x_2
POL(B) = 0
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(A) = 0
POL(f(x_1, x_2)) = x_1 + x_2
POL(f'(x_1, x_2)) = x_1 + x_2
POL(C) = 1
POL(cons(x_1, x_2)) = x_1 + x_2
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

where the Polynomial interpretation:
POL(g(x_1)) = 2*x_1
POL(nil) = 0
POL(f''(x_1)) = x_1
POL(foldf(x_1, x_2)) = x_1 + x_2
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(B) = 1
POL(A) = 0
POL(f(x_1, x_2)) = x_1 + 2*x_2
POL(f'(x_1, x_2)) = x_1 + x_2
POL(C) = 0
POL(cons(x_1, x_2)) = 2*x_1 + x_2
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: 

   foldf(x, nil) -> x

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(nil) = 0
POL(f''(x_1)) = 1 + x_1
POL(foldf(x_1, x_2)) = 1 + x_1 + 2*x_2
POL(triple(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
POL(A) = 2
POL(f(x_1, x_2)) = x_1 + 2*x_2
POL(f'(x_1, x_2)) = x_1 + 2*x_2
POL(C) = 0
POL(cons(x_1, x_2)) = x_1 + x_2
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), A) -> f''(foldf(triple(cons(A, a), nil, c), b))

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(nil) = 0
POL(f''(x_1)) = x_1
POL(foldf(x_1, x_2)) = x_1 + 2*x_2
POL(triple(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
POL(A) = 0
POL(f(x_1, x_2)) = 2 + x_1 + x_2
POL(f'(x_1, x_2)) = 2 + x_1 + x_2
POL(C) = 0
POL(cons(x_1, x_2)) = 1 + x_1 + x_2
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(C) -> C

where the Polynomial interpretation:
POL(g(x_1)) = 1 + x_1
POL(nil) = 0
POL(f''(x_1)) = 2*x_1
POL(foldf(x_1, x_2)) = x_1 + 2*x_2
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(f(x_1, x_2)) = 2 + x_1 + x_2
POL(f'(x_1, x_2)) = 1 + x_1 + x_2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2
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: 

   foldf(x, cons(y, z)) -> f(foldf(x, z), y)

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(nil) = 0
POL(f''(x_1)) = 2*x_1
POL(foldf(x_1, x_2)) = x_1 + 2*x_2
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(f'(x_1, x_2)) = 1 + x_1 + x_2
POL(C) = 0
POL(cons(x_1, x_2)) = 1 + x_1 + x_2
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(nil) = 0
POL(g(x_1)) = x_1
POL(f''(x_1)) = x_1
POL(foldf(x_1, x_2)) = x_1 + x_2
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(f'(x_1, x_2)) = x_1 + x_2
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(C) = 0
POL(cons(x_1, x_2)) = x_1 + x_2
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, cons(C, c))

where the Polynomial interpretation:
POL(nil) = 0
POL(f''(x_1)) = x_1
POL(foldf(x_1, x_2)) = x_1 + x_2
POL(triple(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(f'(x_1, x_2)) = 1 + x_1 + x_2
POL(C) = 0
POL(cons(x_1, x_2)) = x_1 + x_2
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)) -> foldf(triple(a, b, nil), c)

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 2.430 seconds.