Term Rewriting System R:
[x]
f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(h(x)) -> F(i(x))
F(h(x)) -> I(x)
G(i(x)) -> G(h(x))
G(i(x)) -> H(x)

Furthermore, R contains two SCCs.

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

Dependency Pair:

F(h(x)) -> F(i(x))

Rules:

f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b

The following dependency pair can be strictly oriented:

F(h(x)) -> F(i(x))

The following rules can be oriented:

i(a) -> b
f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{a, b}
h > i

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

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

Dependency Pair:

Rules:

f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

G(i(x)) -> G(h(x))

Rules:

f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b

The following dependency pair can be strictly oriented:

G(i(x)) -> G(h(x))

The following rules can be oriented:

h(a) -> b
f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
i(a) -> b

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
i > h
a > b

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b

Using the Dependency Graph resulted in no new DP problems.

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