Term Rewriting System R:
[x]
f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(s(x)) -> G(f(x))
F(s(x)) -> F(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pair:
F(s(x)) -> F(x)
Rules:
f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))
Strategy:
innermost
As we are in the innermost case, we can delete all 4 non-usable-rules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳Size-Change Principle
Dependency Pair:
F(s(x)) -> F(x)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- F(s(x)) -> F(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.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes