Term Rewriting System R:
[x]
p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))
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:
FAC(s(x)) -> FAC(p(s(x)))
FAC(s(x)) -> P(s(x))
Furthermore, R contains one SCC.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pair:
FAC(s(x)) -> FAC(p(s(x)))
Rules:
p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
...
→DP Problem 2
↳Rewriting Transformation
Dependency Pair:
FAC(s(x)) -> FAC(p(s(x)))
Rule:
p(s(x)) -> x
Strategy:
innermost
On this DP problem, a Rewriting SCC transformation can be performed.
As a result of transforming the rule
FAC(s(x)) -> FAC(p(s(x)))
one new Dependency Pair
is created:
FAC(s(x)) -> FAC(x)
The transformation is resulting in one new DP problem:
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
...
→DP Problem 3
↳Usable Rules (Innermost)
Dependency Pair:
FAC(s(x)) -> FAC(x)
Rule:
p(s(x)) -> x
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 4
↳Size-Change Principle
Dependency Pair:
FAC(s(x)) -> FAC(x)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- FAC(s(x)) -> FAC(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