Term Rewriting System R:
[x, y]
g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
G(f(x), y) -> H(x, y)
H(x, y) -> G(x, f(y))
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pairs:
H(x, y) -> G(x, f(y))
G(f(x), y) -> H(x, y)
Rules:
g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))
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:
H(x, y) -> G(x, f(y))
G(f(x), y) -> H(x, y)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- H(x, y) -> G(x, f(y))
- G(f(x), y) -> H(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:
f(x_{1}) -> f(x_{1})
We obtain no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes