Term Rewriting System R:
[x]
a(f, a(f, x)) -> a(x, x)
a(h, x) -> a(f, a(g, a(f, x)))
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
A(f, a(f, x)) -> A(x, x)
A(h, x) -> A(f, a(g, a(f, x)))
A(h, x) -> A(g, a(f, x))
A(h, x) -> A(f, x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pairs:
A(h, x) -> A(f, x)
A(f, a(f, x)) -> A(x, x)
Rules:
a(f, a(f, x)) -> a(x, x)
a(h, x) -> a(f, a(g, a(f, x)))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳Size-Change Principle
Dependency Pairs:
A(h, x) -> A(f, x)
A(f, a(f, x)) -> A(x, x)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- A(h, x) -> A(f, x)
- A(f, a(f, x)) -> A(x, x)
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:
a(x1, x2) -> a(x1, x2)
We obtain no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes