Term Rewriting System R:
[x, y, z]
a(lambda(x), y) -> lambda(a(x, 1))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
a(x, y) -> x
a(x, y) -> y
lambda(x) -> x
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
lambda(x) -> x
where the Polynomial interpretation:
_{ }^{ }POL(t) | = 0_{ }^{ } |
_{ }^{ }POL(1) | = 0_{ }^{ } |
_{ }^{ }POL(lambda(x_{1})) | = 1 + x_{1}_{ }^{ } |
_{ }^{ }POL(a(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
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
A(lambda(x), y) -> A(x, a(y, t))
A(lambda(x), y) -> A(y, t)
A(lambda(x), y) -> A(x, 1)
A(a(x, y), z) -> A(x, a(y, z))
A(a(x, y), z) -> A(y, z)
Furthermore, R contains one SCC.
R
↳RRRPolo
→TRS2
↳DPs
→DP Problem 1
↳Modular Removal of Rules
Dependency Pairs:
A(a(x, y), z) -> A(y, z)
A(a(x, y), z) -> A(x, a(y, z))
A(lambda(x), y) -> A(x, 1)
A(lambda(x), y) -> A(y, t)
A(lambda(x), y) -> A(x, a(y, t))
Rules:
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(x, y) -> y
a(lambda(x), y) -> lambda(a(x, 1))
a(x, y) -> x
a(a(x, y), z) -> a(x, a(y, z))
We have the following set of usable rules:
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(x, y) -> y
a(lambda(x), y) -> lambda(a(x, 1))
a(x, y) -> x
a(a(x, y), z) -> a(x, a(y, z))
To remove rules and DPs from this DP problem we used the following monotonic and C_{E}-compatible order: Polynomial ordering.
Polynomial interpretation:
_{ }^{ }POL(t) | = 0_{ }^{ } |
_{ }^{ }POL(1) | = 0_{ }^{ } |
_{ }^{ }POL(lambda(x_{1})) | = 1 + x_{1}_{ }^{ } |
_{ }^{ }POL(a(x_{1}, x_{2})) | = x_{1} + x_{2}_{ }^{ } |
_{ }^{ }POL(A(x_{1}, x_{2})) | = 1 + x_{1} + x_{2}_{ }^{ } |
We have the following set D of usable symbols: {t, 1, lambda, a, A}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:
A(lambda(x), y) -> A(x, 1)
A(lambda(x), y) -> A(y, t)
A(lambda(x), y) -> A(x, a(y, t))
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 Pairs:
A(a(x, y), z) -> A(y, z)
A(a(x, y), z) -> A(x, a(y, z))
Rules:
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(x, y) -> y
a(lambda(x), y) -> lambda(a(x, 1))
a(x, y) -> x
a(a(x, y), z) -> a(x, a(y, z))
We number the DPs as follows:
- A(a(x, y), z) -> A(y, z)
- A(a(x, y), z) -> A(x, a(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
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes