Term Rewriting System R:
[x, u, v, z, y]
admit(x, nil) -> nil
admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y
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:
ADMIT(x, .(u, .(v, .(w, z)))) -> COND(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)
Furthermore, R contains one SCC.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pair:
ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)
Rules:
admit(x, nil) -> nil
admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y
Strategy:
innermost
As we are in the innermost case, we can delete all 3 non-usable-rules.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
...
→DP Problem 2
↳Size-Change Principle
Dependency Pair:
ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), 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:
carry(x_{1}, x_{2}, x_{3}) -> carry(x_{1}, x_{2}, x_{3})
.(x_{1}, x_{2}) -> .(x_{1}, x_{2})
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes