Term Rewriting System R:
[y, x]
f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)

Strategy:

innermost

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

F(s(x), y) -> F(f(x, y), y)
two new Dependency Pairs are created:

F(s(0), y'') -> F(0, y'')
F(s(s(x'')), y'') -> F(f(f(x'', y''), y''), y'')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)

Strategy:

innermost

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

F(s(s(x'')), y'') -> F(f(f(x'', y''), y''), y'')
two new Dependency Pairs are created:

F(s(s(0)), y''') -> F(f(0, y'''), y''')
F(s(s(s(x'))), y''') -> F(f(f(f(x', y'''), y'''), y'''), y''')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(s(x'))), y''') -> F(f(f(f(x', y'''), y'''), y'''), y''')
F(s(s(0)), y''') -> F(f(0, y'''), y''')
F(s(x), y) -> F(x, y)

Rules:

f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)

Strategy:

innermost

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

F(s(s(0)), y''') -> F(f(0, y'''), y''')
one new Dependency Pair is created:

F(s(s(0)), y''') -> F(0, y''')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)

Strategy:

innermost

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

F(s(x), y) -> F(x, y)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)

Strategy:

innermost

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

F(s(s(s(x'))), y''') -> F(f(f(f(x', y'''), y'''), y'''), y''')
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)

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'')), y'') -> F(s(x''), y'')
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 7`
`                 ↳Polynomial Ordering`

Dependency Pairs:

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

Rules:

f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(s(s(s(s(s(x'''''))))), y'''') -> F(s(s(s(s(x''''')))), y'''')
F(s(s(s(x''''))), y'''') -> F(s(s(x'''')), y'''')
F(s(s(s(s(x''')))), y') -> F(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(s(x1)) =  1 + x1 POL(F(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 8`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

f(0, y) -> 0
f(s(x), y) -> f(f(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