Term Rewriting System R:
[x, y, z]
:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
:(x, x) -> e
:(x, e) -> x
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
where the Polynomial interpretation:
_{ }^{ }POL(:(x_{1}, x_{2})) | = 1 + x_{1} + x_{2}_{ }^{ } |
_{ }^{ }POL(i(x_{1})) | = x_{1}_{ }^{ } |
_{ }^{ }POL(e) | = 0_{ }^{ } |
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:
I(:(x, y)) -> :'(y, x)
:'(:(x, y), z) -> :'(x, :(z, i(y)))
:'(:(x, y), z) -> :'(z, i(y))
:'(:(x, y), z) -> I(y)
Furthermore, R contains one SCC.
R
↳RRRPolo
→TRS2
↳DPs
→DP Problem 1
↳Modular Removal of Rules
Dependency Pairs:
:'(:(x, y), z) -> I(y)
:'(:(x, y), z) -> :'(z, i(y))
:'(:(x, y), z) -> :'(x, :(z, i(y)))
I(:(x, y)) -> :'(y, x)
Rules:
i(:(x, y)) -> :(y, x)
i(e) -> e
i(i(x)) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
We have the following set of usable rules:
i(:(x, y)) -> :(y, x)
i(e) -> e
i(i(x)) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
To remove rules and DPs from this DP problem we used the following monotonic and C_{E}-compatible order: Polynomial ordering.
Polynomial interpretation:
_{ }^{ }POL(:(x_{1}, x_{2})) | = 1 + x_{1} + x_{2}_{ }^{ } |
_{ }^{ }POL(I(x_{1})) | = x_{1}_{ }^{ } |
_{ }^{ }POL(i(x_{1})) | = x_{1}_{ }^{ } |
_{ }^{ }POL(e) | = 0_{ }^{ } |
_{ }^{ }POL(:'(x_{1}, x_{2})) | = x_{1} + x_{2}_{ }^{ } |
We have the following set D of usable symbols: {:, I, i, e, :'}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:
:'(:(x, y), z) -> I(y)
:'(:(x, y), z) -> :'(z, i(y))
I(:(x, y)) -> :'(y, x)
No Rules can be deleted.
The result of this processor delivers one new DP problem.
R
↳RRRPolo
→TRS2
↳DPs
→DP Problem 1
↳MRR
...
→DP Problem 2
↳Size-Change Principle
Dependency Pair:
:'(:(x, y), z) -> :'(x, :(z, i(y)))
Rules:
i(:(x, y)) -> :(y, x)
i(e) -> e
i(i(x)) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
We number the DPs as follows:
- :'(:(x, y), z) -> :'(x, :(z, i(y)))
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
D_{P}: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes