Term Rewriting System R:
[x, y, z]
*(x, +(y, z)) -> +(*(x, y), *(x, z))
Termination of R to be shown.
R
↳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
↳OC
→TRS2
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
*'(x, +(y, z)) -> *'(x, y)
*'(x, +(y, z)) -> *'(x, z)
Furthermore, R contains one SCC.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pairs:
*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)
Rule:
*(x, +(y, z)) -> +(*(x, y), *(x, z))
Strategy:
innermost
As we are in the innermost case, we can delete all 1 non-usable-rules.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
...
→DP Problem 2
↳Size-Change Principle
Dependency Pairs:
*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- *'(x, +(y, z)) -> *'(x, z)
- *'(x, +(y, z)) -> *'(x, y)
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.
Termination of R successfully shown.
Duration:
0:00 minutes