Term Rewriting System R:
[x]
f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(f(x)) -> F(c(f(x)))
F(f(x)) -> F(d(f(x)))
G(c(h(0))) -> G(d(1))
G(c(1)) -> G(d(h(0)))
G(h(x)) -> G(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pair:
G(h(x)) -> G(x)
Rules:
f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)
Strategy:
innermost
The following dependency pair can be strictly oriented:
G(h(x)) -> G(x)
There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial
resulting in one new DP problem.
Used Argument Filtering System: G(x1) -> G(x1)
h(x1) -> h(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(h(0))) -> g(d(1))
g(c(1)) -> g(d(h(0)))
g(h(x)) -> g(x)
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes