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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(x, x) -> F(i(x), g(g(x)))
F(x, x) -> G(g(x))
F(x, x) -> G(x)
F(x, i(x)) -> F(x, x)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rewriting Transformation`

Dependency Pairs:

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

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

On this DP problem, a Rewriting SCC transformation can be performed.
As a result of transforming the rule

F(x, x) -> F(i(x), g(g(x)))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rewriting Transformation`

Dependency Pairs:

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

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

On this DP problem, a Rewriting SCC transformation can be performed.
As a result of transforming the rule

F(x, x) -> F(i(x), i(g(x)))
one new Dependency Pair is created:

F(x, x) -> F(i(x), i(i(x)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Rw`
`           →DP Problem 2`
`             ↳Rw`
`             ...`
`               →DP Problem 3`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

F(x, x) -> F(i(x), i(i(x)))
F(x, i(x)) -> F(x, x)

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes