Term Rewriting System R:
[X, Y]
f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(c(X, s(Y))) -> F(c(s(X), Y))
G(c(s(X), Y)) -> F(c(X, s(Y)))

Furthermore, R contains one SCC.

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

Dependency Pair:

F(c(X, s(Y))) -> F(c(s(X), Y))

Rules:

f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))

Strategy:

innermost

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

F(c(X, s(Y))) -> F(c(s(X), Y))
one new Dependency Pair is created:

F(c(s(X''), s(Y''))) -> F(c(s(s(X'')), Y''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Instantiation Transformation`

Dependency Pair:

F(c(s(X''), s(Y''))) -> F(c(s(s(X'')), Y''))

Rules:

f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))

Strategy:

innermost

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

F(c(s(X''), s(Y''))) -> F(c(s(s(X'')), Y''))
one new Dependency Pair is created:

F(c(s(s(X'''')), s(Y''''))) -> F(c(s(s(s(X''''))), Y''''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pair:

F(c(s(s(X'''')), s(Y''''))) -> F(c(s(s(s(X''''))), Y''''))

Rules:

f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(c(s(s(X'''')), s(Y''''))) -> F(c(s(s(s(X''''))), Y''''))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(c(x1, x2)) =  x2 POL(s(x1)) =  1 + x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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