Term Rewriting System R:
[x, y]
g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(x) -> x
where the Polynomial interpretation:
| POL(c(x1, x2)) | = 1 + x1 + x2 |
| POL(g(x1)) | = 1 + x1 |
| POL(d(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(f(x1)) | = 1 + x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
G(c(x, s(y))) -> G(c(s(x), y))
F(c(s(x), y)) -> F(c(x, s(y)))
F(f(x)) -> F(d(f(x)))
Furthermore, R contains two SCCs.
R
↳RRRPolo
→TRS2
↳DPs
→DP Problem 1
↳Size-Change Principle
→DP Problem 2
↳SCP
Dependency Pair:
G(c(x, s(y))) -> G(c(s(x), y))
Rules:
g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
We number the DPs as follows:
- G(c(x, s(y))) -> G(c(s(x), y))
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
c(x1, x2) -> c(x2)
s(x1) -> s(x1)
We obtain no new DP problems.
R
↳RRRPolo
→TRS2
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳Size-Change Principle
Dependency Pair:
F(c(s(x), y)) -> F(c(x, s(y)))
Rules:
g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
We number the DPs as follows:
- F(c(s(x), y)) -> F(c(x, s(y)))
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
c(x1, x2) -> c(x1)
s(x1) -> s(x1)
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes