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.
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(0) | = 0 |
| POL(g(x1)) | = x1 |
| POL(B) | = 1 |
| POL(foldB(x1, x2)) | = x1 + 2·x2 |
| POL(triple(x1, x2, x3)) | = x1 + 2·x2 + 2·x3 |
| POL(s(x1)) | = 1 + x1 |
| POL(f'(x1, x2)) | = 1 + x1 + x2 |
| POL(f''(x1)) | = x1 |
| POL(A) | = 0 |
| POL(f(x1, x2)) | = 1 + x1 + x2 |
| POL(foldC(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
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
foldC(t, 0) -> t
foldB(t, 0) -> t
where the Polynomial interpretation:
| POL(C) | = 0 |
| POL(0) | = 1 |
| POL(g(x1)) | = x1 |
| POL(B) | = 0 |
| POL(foldB(x1, x2)) | = x1 + x2 |
| POL(triple(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(s(x1)) | = x1 |
| POL(f'(x1, x2)) | = x1 + x2 |
| POL(f''(x1)) | = 1 + x1 |
| POL(foldC(x1, x2)) | = x1 + x2 |
| POL(f(x1, x2)) | = x1 + x2 |
| POL(A) | = 2 |
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:
g(A) -> A
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
where the Polynomial interpretation:
| POL(C) | = 0 |
| POL(0) | = 0 |
| POL(g(x1)) | = 2·x1 |
| POL(B) | = 0 |
| POL(foldB(x1, x2)) | = x1 + x2 |
| POL(triple(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(s(x1)) | = x1 |
| POL(f'(x1, x2)) | = x1 + x2 |
| POL(f''(x1)) | = x1 |
| POL(foldC(x1, x2)) | = x1 + x2 |
| POL(f(x1, x2)) | = x1 + 2·x2 |
| POL(A) | = 1 |
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:
foldC(t, s(n)) -> f(foldC(t, n), C)
where the Polynomial interpretation:
| POL(C) | = 0 |
| POL(0) | = 0 |
| POL(g(x1)) | = x1 |
| POL(B) | = 0 |
| POL(foldB(x1, x2)) | = x1 + x2 |
| POL(triple(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(f'(x1, x2)) | = 1 + x1 + x2 |
| POL(s(x1)) | = 1 + x1 |
| POL(f''(x1)) | = 2·x1 |
| POL(f(x1, x2)) | = 1 + x1 + x2 |
| POL(foldC(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:
g(B) -> B
g(C) -> C
g(C) -> B
where the Polynomial interpretation:
| POL(C) | = 0 |
| POL(0) | = 0 |
| POL(g(x1)) | = 1 + x1 |
| POL(B) | = 0 |
| POL(foldB(x1, x2)) | = x1 + 2·x2 |
| POL(triple(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(f'(x1, x2)) | = 1 + x1 + x2 |
| POL(s(x1)) | = 1 + x1 |
| POL(f''(x1)) | = x1 |
| POL(f(x1, x2)) | = 2 + x1 + x2 |
| POL(foldC(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
...
→TRS6
↳Removing Redundant Rules
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(0) | = 0 |
| POL(C) | = 1 |
| POL(g(x1)) | = x1 |
| POL(foldB(x1, x2)) | = x1 + x2 |
| POL(B) | = 0 |
| POL(triple(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(f'(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(f''(x1)) | = x1 |
| POL(f(x1, x2)) | = x1 + x2 |
| POL(foldC(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
...
→TRS7
↳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(0) | = 0 |
| POL(foldB(x1, x2)) | = x1 + x2 |
| POL(B) | = 0 |
| POL(g(x1)) | = x1 |
| POL(triple(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(s(x1)) | = 1 + x1 |
| POL(f'(x1, x2)) | = x1 + x2 |
| POL(f''(x1)) | = x1 |
| POL(foldC(x1, x2)) | = x1 + x2 |
| POL(f(x1, x2)) | = 1 + 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:
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(foldB(x1, x2)) | = x1 + x2 |
| POL(B) | = 0 |
| POL(triple(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(s(x1)) | = x1 |
| POL(f''(x1)) | = 1 + x1 |
| POL(f(x1, x2)) | = x1 + x2 |
| POL(foldC(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
...
→TRS9
↳Removing Redundant Rules
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(foldB(x1, x2)) | = x1 + x2 |
| POL(B) | = 0 |
| POL(s(x1)) | = 1 + x1 |
| POL(f(x1, x2)) | = x1 + x2 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS10
↳Overlay and local confluence Check
The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS11
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:04 minutes