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)

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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(C)=  1  
  POL(g(x1))=  x1  
  POL(B)=  1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(triple(x1, x2, x3))=  x1 + x2 + x3  
  POL(foldf(x1, x2))=  x1 + x2  
  POL(f'(x1, x2))=  x1 + x2  
  POL(f''(x1))=  x1  
  POL(A)=  0  
  POL(f(x1, x2))=  x1 + x2  
was used.

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


   R
RRRPolo
       →TRS2
Removing Redundant Rules



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(C)=  0  
  POL(g(x1))=  x1  
  POL(B)=  0  
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(triple(x1, x2, x3))=  x1 + 2·x2 + 2·x3  
  POL(nil)=  0  
  POL(f'(x1, x2))=  2 + x1 + x2  
  POL(foldf(x1, x2))=  x1 + 2·x2  
  POL(f''(x1))=  x1  
  POL(A)=  0  
  POL(f(x1, x2))=  2 + x1 + x2  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
Removing Redundant Rules



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(C)=  0  
  POL(g(x1))=  x1  
  POL(B)=  0  
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(triple(x1, x2, x3))=  x1 + x2 + x3  
  POL(nil)=  0  
  POL(foldf(x1, x2))=  x1 + 2·x2  
  POL(f'(x1, x2))=  2 + x1 + x2  
  POL(f''(x1))=  2·x1  
  POL(A)=  0  
  POL(f(x1, x2))=  2 + x1 + x2  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS4
Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(C)=  1  
  POL(g(x1))=  2·x1  
  POL(B)=  1  
  POL(cons(x1, x2))=  2·x1 + x2  
  POL(triple(x1, x2, x3))=  x1 + x2 + x3  
  POL(nil)=  0  
  POL(foldf(x1, x2))=  x1 + x2  
  POL(f'(x1, x2))=  x1 + x2  
  POL(f''(x1))=  x1  
  POL(A)=  0  
  POL(f(x1, x2))=  x1 + 2·x2  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS5
Removing Redundant Rules



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(g(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(triple(x1, x2, x3))=  x1 + x2 + x3  
  POL(foldf(x1, x2))=  x1 + x2  
  POL(f'(x1, x2))=  x1 + x2  
  POL(f''(x1))=  1 + x1  
  POL(f(x1, x2))=  x1 + x2  
  POL(A)=  0  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS6
Removing Redundant Rules



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(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(nil)=  0  
  POL(foldf(x1, x2))=  x1 + x2  
  POL(f'(x1, x2))=  x1 + x2  
  POL(f(x1, x2))=  x1 + x2  
  POL(A)=  0  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS7
Removing Redundant Rules



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

foldf(x, nil) -> x

where the Polynomial interpretation:
  POL(g(x1))=  x1  
  POL(nil)=  0  
  POL(f'(x1, x2))=  x1 + x2  
  POL(foldf(x1, x2))=  1 + x1 + x2  
  POL(A)=  0  
  POL(f(x1, x2))=  x1 + x2  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS8
Removing Redundant Rules



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

g(A) -> A

where the Polynomial interpretation:
  POL(g(x1))=  1 + x1  
  POL(f'(x1, x2))=  x1 + x2  
  POL(f(x1, x2))=  1 + x1 + x2  
  POL(A)=  0  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS9
Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(g(x1))=  x1  
  POL(f'(x1, x2))=  x1 + x2  
  POL(f(x1, x2))=  1 + x1 + x2  
was used.

All Rules of R can be deleted.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS10
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes