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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(a, a) -> F(a, b)
F(a, b) -> F(s(a), c)
F(s(X), c) -> F(X, c)
F(c, c) -> F(a, a)

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(c, c) -> F(a, a)
F(s(X), c) -> F(X, c)
F(a, b) -> F(s(a), c)
F(a, a) -> F(a, b)

Rules:

f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)

Strategy:

innermost

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

F(s(X), c) -> F(X, c)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Forward Instantiation Transformation`

Dependency Pair:

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

Rules:

f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

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

Dependency Pair:

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

Rules:

f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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) =  0 POL(s(x1)) =  1 + x1 POL(F(x1, x2)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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