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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.

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

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(c) =  0 POL(s(x1)) =  1 + x1 POL(F(x1, x2)) =  x1 + x2

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

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

Dependency Pair:

Rules:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(c(x1)) =  1 + x1

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> 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(y)) -> f(x, s(f(y, y)))
f(s(x), s(y)) -> f(x, s(c(s(y))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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