Term Rewriting System R:
[x, y, z]
minus(0) -> 0
minus(minus(x)) -> x
+(x, 0) -> x
+(0, y) -> y
+(minus(1), 1) -> 0
+(x, minus(y)) -> minus(+(minus(x), y))
+(x, +(y, z)) -> +(+(x, y), z)
+(minus(+(x, 1)), 1) -> minus(x)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
minus(0) -> 0
minus(minus(x)) -> x
+(minus(1), 1) -> 0
where the Polynomial interpretation:
_{ }^{ }POL(0) | = 0_{ }^{ } |
_{ }^{ }POL(1) | = 0_{ }^{ } |
_{ }^{ }POL(minus(x_{1})) | = 1 + x_{1}_{ }^{ } |
_{ }^{ }POL(+(x_{1}, x_{2})) | = x_{1} + 2·x_{2}_{ }^{ } |
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:
+(x, 0) -> x
+(0, y) -> y
where the Polynomial interpretation:
_{ }^{ }POL(0) | = 1_{ }^{ } |
_{ }^{ }POL(1) | = 0_{ }^{ } |
_{ }^{ }POL(minus(x_{1})) | = x_{1}_{ }^{ } |
_{ }^{ }POL(+(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.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(x, +(y, z)) -> +(+(x, y), z)
+(minus(+(x, 1)), 1) -> minus(x)
where the Polynomial interpretation:
_{ }^{ }POL(1) | = 0_{ }^{ } |
_{ }^{ }POL(minus(x_{1})) | = x_{1}_{ }^{ } |
_{ }^{ }POL(+(x_{1}, x_{2})) | = 1 + x_{1} + 2·x_{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
↳RRRPolo
...
→TRS4
↳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
...
→TRS5
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
+'(x, minus(y)) -> +'(minus(x), y)
Furthermore, R contains one SCC.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pair:
+'(x, minus(y)) -> +'(minus(x), y)
Rule:
+(x, minus(y)) -> minus(+(minus(x), y))
Strategy:
innermost
As we are in the innermost case, we can delete all 1 non-usable-rules.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→DP Problem 2
↳Size-Change Principle
Dependency Pair:
+'(x, minus(y)) -> +'(minus(x), y)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- +'(x, minus(y)) -> +'(minus(x), 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
with Argument Filtering System:
minus(x_{1}) -> minus(x_{1})
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes