Term Rewriting System R:
[y, x, z]
f(nil) -> nil
f(.(nil, y)) -> .(nil, f(y))
f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(nil) -> nil
g(.(x, nil)) -> .(g(x), nil)
g(.(x, .(y, z))) -> g(.(.(x, y), z))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(nil) -> nil
where the Polynomial interpretation:
| POL(g(x1)) | = x1 |
| POL(nil) | = 0 |
| POL(.(x1, x2)) | = x1 + x2 |
| POL(f(x1)) | = 1 + x1 |
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:
g(nil) -> nil
where the Polynomial interpretation:
| POL(g(x1)) | = 1 + x1 |
| POL(nil) | = 0 |
| POL(.(x1, x2)) | = x1 + x2 |
| POL(f(x1)) | = x1 |
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:
f(.(nil, y)) -> .(nil, f(y))
where the Polynomial interpretation:
| POL(g(x1)) | = x1 |
| POL(nil) | = 1 |
| POL(.(x1, x2)) | = x1 + x2 |
| POL(f(x1)) | = 2·x1 |
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
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(.(x, nil)) -> .(g(x), nil)
where the Polynomial interpretation:
| POL(g(x1)) | = 2·x1 |
| POL(nil) | = 1 |
| POL(.(x1, x2)) | = x1 + x2 |
| POL(f(x1)) | = 2 + x1 |
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
...
→TRS5
↳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
...
→TRS6
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(.(.(x, y), z)) -> F(.(x, .(y, z)))
G(.(x, .(y, z))) -> G(.(.(x, y), z))
Furthermore, R contains two SCCs.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pair:
F(.(.(x, y), z)) -> F(.(x, .(y, z)))
Rules:
f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(.(x, .(y, z))) -> g(.(.(x, y), z))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→DP Problem 3
↳Size-Change Principle
Dependency Pair:
F(.(.(x, y), z)) -> F(.(x, .(y, z)))
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- F(.(.(x, y), z)) -> F(.(x, .(y, z)))
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
.(x1, x2) -> .(x1)
We obtain no new DP problems.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→DP Problem 2
↳Usable Rules (Innermost)
Dependency Pair:
G(.(x, .(y, z))) -> G(.(.(x, y), z))
Rules:
f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(.(x, .(y, z))) -> g(.(.(x, y), z))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→DP Problem 4
↳Size-Change Principle
Dependency Pair:
G(.(x, .(y, z))) -> G(.(.(x, y), z))
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- G(.(x, .(y, z))) -> G(.(.(x, y), z))
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
.(x1, x2) -> .(x2)
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes