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