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