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