Term Rewriting System R:
[x, y]
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(f(x)) -> F(g(f(x), x))
F(f(x)) -> G(f(x), x)
F(f(x)) -> F(h(f(x), f(x)))
F(f(x)) -> H(f(x), f(x))
H(x, x) -> G(x, 0)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`

Dependency Pairs:

F(f(x)) -> F(h(f(x), f(x)))
F(f(x)) -> F(g(f(x), x))

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

The following dependency pair can be strictly oriented:

F(f(x)) -> F(g(f(x), x))

The following usable rules using the Ce-refinement can be oriented:

g(x, y) -> y
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
h(x, x) -> g(x, 0)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
F > 0
h > 0
f > 0
g > 0

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
f(x1) -> f(x1)
g(x1, x2) -> x2
h(x1, x2) -> x1

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Argument Filtering and Ordering`

Dependency Pair:

F(f(x)) -> F(h(f(x), f(x)))

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

The following dependency pair can be strictly oriented:

F(f(x)) -> F(h(f(x), f(x)))

The following usable rules using the Ce-refinement can be oriented:

h(x, x) -> g(x, 0)
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
f > {h, 0}

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
f(x1) -> f(x1)
h(x1, x2) -> h
g(x1, x2) -> x2

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳AFS`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes