Term Rewriting System R:
[x, y]
f(x, a(b(y))) -> a(f(a(b(x)), y))
f(a(x), y) -> f(x, a(y))
f(b(x), y) -> f(x, b(y))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(x, a(b(y))) -> F(a(b(x)), y)
F(a(x), y) -> F(x, a(y))
F(b(x), y) -> F(x, b(y))
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Modular Removal of Rules
Dependency Pairs:
F(b(x), y) -> F(x, b(y))
F(a(x), y) -> F(x, a(y))
F(x, a(b(y))) -> F(a(b(x)), y)
Rules:
f(x, a(b(y))) -> a(f(a(b(x)), y))
f(a(x), y) -> f(x, a(y))
f(b(x), y) -> f(x, b(y))
We have the following set of usable rules:
none
To remove rules and DPs from this DP problem we used the following monotonic and C_{E}-compatible order: Polynomial ordering.
Polynomial interpretation:
_{ }^{ }POL(b(x_{1})) | = x_{1}_{ }^{ } |
_{ }^{ }POL(a(x_{1})) | = x_{1}_{ }^{ } |
_{ }^{ }POL(F(x_{1}, x_{2})) | = 1 + x_{1} + x_{2}_{ }^{ } |
We have the following set D of usable symbols: {b, a, F}
No Dependency Pairs can be deleted.
3 non usable rules have been deleted.
The result of this processor delivers one new DP problem.
R
↳DPs
→DP Problem 1
↳MRR
→DP Problem 2
↳Non-Overlappingness Check
Dependency Pairs:
F(b(x), y) -> F(x, b(y))
F(a(x), y) -> F(x, a(y))
F(x, a(b(y))) -> F(a(b(x)), y)
Rule:
none
R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle:
R
↳DPs
→DP Problem 1
↳MRR
→DP Problem 2
↳NOC
...
→DP Problem 3
↳Scc To SRS
Dependency Pairs:
F(b(x), y) -> F(x, b(y))
F(a(x), y) -> F(x, a(y))
F(x, a(b(y))) -> F(a(b(x)), y)
Rule:
none
Strategy:
innermost
It has been determined that showing finiteness of this DP problem is equivalent to showing termination of a string rewrite system.
(Re)applying the dependency pair method (incl. the dependency graph) for the following SRS:
a(b(x)) -> b(a(x))
There is only one SCC in the graph and, thus, we obtain one new DP problem.
R
↳DPs
→DP Problem 1
↳MRR
→DP Problem 2
↳NOC
...
→DP Problem 4
↳Non-Overlappingness Check
Dependency Pair:
A(b(x)) -> A(x)
Rule:
a(b(x)) -> b(a(x))
R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle:
R
↳DPs
→DP Problem 1
↳MRR
→DP Problem 2
↳NOC
...
→DP Problem 5
↳Usable Rules (Innermost)
Dependency Pair:
A(b(x)) -> A(x)
Rule:
a(b(x)) -> b(a(x))
Strategy:
innermost
As we are in the innermost case, we can delete all 1 non-usable-rules.
R
↳DPs
→DP Problem 1
↳MRR
→DP Problem 2
↳NOC
...
→DP Problem 6
↳Size-Change Principle
Dependency Pair:
A(b(x)) -> A(x)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- A(b(x)) -> A(x)
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:
b(x_{1}) -> b(x_{1})
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes