Term Rewriting System R:
[x, y, z]
f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
f(s(x), y, z) -> f(x, s(c(y)), c(z))
f(c(x), x, y) -> c(y)
g(x, y) -> x
g(x, y) -> y

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(x, c(x), c(y)) -> F(y, y, f(y, x, y))
F(x, c(x), c(y)) -> F(y, x, y)
F(s(x), y, z) -> F(x, s(c(y)), c(z))

Furthermore, R contains two SCCs.

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

Dependency Pair:

F(s(x), y, z) -> F(x, s(c(y)), c(z))

Rules:

f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
f(s(x), y, z) -> f(x, s(c(y)), c(z))
f(c(x), x, y) -> c(y)
g(x, y) -> x
g(x, y) -> y

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(s(x), y, z) -> F(x, s(c(y)), c(z))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2, x3) -> x1
s(x1) -> s(x1)

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

Dependency Pair:

Rules:

f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
f(s(x), y, z) -> f(x, s(c(y)), c(z))
f(c(x), x, y) -> c(y)
g(x, y) -> x
g(x, y) -> y

Strategy:

innermost

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:

F(x, c(x), c(y)) -> F(y, x, y)

Rules:

f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
f(s(x), y, z) -> f(x, s(c(y)), c(z))
f(c(x), x, y) -> c(y)
g(x, y) -> x
g(x, y) -> y

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(x, c(x), c(y)) -> F(y, x, y)

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2, x3) -> x2
c(x1) -> c(x1)

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

Dependency Pair:

Rules:

f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
f(s(x), y, z) -> f(x, s(c(y)), c(z))
f(c(x), x, y) -> c(y)
g(x, y) -> x
g(x, y) -> y

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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