Term Rewriting System R:
[x]
f(a, h(x)) -> f(g(x), h(x))
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(a, h(x)) -> F(g(x), h(x))
F(a, h(x)) -> G(x)
H(g(x)) -> H(a)
G(h(x)) -> G(x)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Size-Change Principle
→DP Problem 2
↳Neg POLO
Dependency Pair:
G(h(x)) -> G(x)
Rules:
f(a, h(x)) -> f(g(x), h(x))
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)
We number the DPs as follows:
- G(h(x)) -> G(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:
h(x_{1}) -> h(x_{1})
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳Negative Polynomial Order
Dependency Pair:
F(a, h(x)) -> F(g(x), h(x))
Rules:
f(a, h(x)) -> f(g(x), h(x))
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)
The following Dependency Pair can be strictly oriented using the given order.
F(a, h(x)) -> F(g(x), h(x))
Moreover, the following usable rules (regarding the implicit AFS) are oriented.
g(h(x)) -> g(x)
Used ordering:
Polynomial Order with Interpretation:
POL( F(x_{1}, x_{2}) ) = x_{1}
POL( a ) = 1
POL( g(x_{1}) ) = 0
This results in one new DP problem.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳Neg POLO
→DP Problem 3
↳Dependency Graph
Dependency Pair:
Rules:
f(a, h(x)) -> f(g(x), h(x))
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:06 minutes