Term Rewriting System R:
[x, y]
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, 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:
SUM(s(x)) -> +'(sum(x), s(x))
SUM(s(x)) -> SUM(x)
+'(x, s(y)) -> +'(x, y)
Furthermore, R contains two SCCs.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
→DP Problem 2
↳UsableRules
Dependency Pair:
+'(x, s(y)) -> +'(x, y)
Rules:
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
Strategy:
innermost
As we are in the innermost case, we can delete all 4 non-usable-rules.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
...
→DP Problem 3
↳Size-Change Principle
→DP Problem 2
↳UsableRules
Dependency Pair:
+'(x, s(y)) -> +'(x, y)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- +'(x, s(y)) -> +'(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:
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳Usable Rules (Innermost)
Dependency Pair:
SUM(s(x)) -> SUM(x)
Rules:
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
Strategy:
innermost
As we are in the innermost case, we can delete all 4 non-usable-rules.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳UsableRules
...
→DP Problem 4
↳Size-Change Principle
Dependency Pair:
SUM(s(x)) -> SUM(x)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- SUM(s(x)) -> SUM(x)
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:
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes