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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

DFIB(s(s(x)), y) -> DFIB(s(x), dfib(x, y))
DFIB(s(s(x)), y) -> DFIB(x, y)

Furthermore, R contains one SCC.

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

Dependency Pairs:

DFIB(s(s(x)), y) -> DFIB(x, y)
DFIB(s(s(x)), y) -> DFIB(s(x), dfib(x, y))

Rule:

dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Strategy:

innermost

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

DFIB(s(s(x)), y) -> DFIB(s(x), dfib(x, y))
one new Dependency Pair is created:

DFIB(s(s(s(x''))), y'') -> DFIB(s(s(x'')), dfib(s(x''), y''))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

DFIB(s(s(s(x''))), y'') -> DFIB(s(s(x'')), dfib(s(x''), y''))
DFIB(s(s(x)), y) -> DFIB(x, y)

Rule:

dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Narrowing Transformation`

Dependency Pairs:

DFIB(s(s(s(s(s(x''''))))), y') -> DFIB(s(s(s(x''''))), y')
DFIB(s(s(s(s(x'')))), y'') -> DFIB(s(s(x'')), y'')
DFIB(s(s(s(x''))), y'') -> DFIB(s(s(x'')), dfib(s(x''), y''))

Rule:

dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Strategy:

innermost

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

DFIB(s(s(s(x''))), y'') -> DFIB(s(s(x'')), dfib(s(x''), y''))
one new Dependency Pair is created:

DFIB(s(s(s(s(x')))), y''') -> DFIB(s(s(s(x'))), dfib(s(x'), dfib(x', y''')))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

DFIB(s(s(s(s(x')))), y''') -> DFIB(s(s(s(x'))), dfib(s(x'), dfib(x', y''')))
DFIB(s(s(s(s(x'')))), y'') -> DFIB(s(s(x'')), y'')
DFIB(s(s(s(s(s(x''''))))), y') -> DFIB(s(s(s(x''''))), y')

Rule:

dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Strategy:

innermost

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

DFIB(s(s(s(s(x'')))), y'') -> DFIB(s(s(x'')), y'')
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

DFIB(s(s(s(s(s(s(x'''')))))), y''') -> DFIB(s(s(s(s(x'''')))), y''')
DFIB(s(s(s(s(s(s(s(x''''''))))))), y'''') -> DFIB(s(s(s(s(s(x''''''))))), y'''')
DFIB(s(s(s(s(s(s(x'''')))))), y'''') -> DFIB(s(s(s(s(x'''')))), y'''')
DFIB(s(s(s(s(s(x''''))))), y') -> DFIB(s(s(s(x''''))), y')
DFIB(s(s(s(s(x')))), y''') -> DFIB(s(s(s(x'))), dfib(s(x'), dfib(x', y''')))

Rule:

dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Strategy:

innermost

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

DFIB(s(s(s(s(s(x''''))))), y') -> DFIB(s(s(s(x''''))), y')
four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Polynomial Ordering`

Dependency Pairs:

DFIB(s(s(s(s(s(s(s(s(s(x''''''''))))))))), y'') -> DFIB(s(s(s(s(s(s(s(x''''''''))))))), y'')
DFIB(s(s(s(s(s(s(s(s(x'''''')))))))), y'') -> DFIB(s(s(s(s(s(s(x'''''')))))), y'')
DFIB(s(s(s(s(s(s(x''')))))), y'') -> DFIB(s(s(s(s(x''')))), y'')
DFIB(s(s(s(s(s(s(s(x''''''))))))), y''') -> DFIB(s(s(s(s(s(x''''''))))), y''')
DFIB(s(s(s(s(s(s(s(x''''''))))))), y'''') -> DFIB(s(s(s(s(s(x''''''))))), y'''')
DFIB(s(s(s(s(s(s(x'''')))))), y'''') -> DFIB(s(s(s(s(x'''')))), y'''')
DFIB(s(s(s(s(x')))), y''') -> DFIB(s(s(s(x'))), dfib(s(x'), dfib(x', y''')))
DFIB(s(s(s(s(s(s(x'''')))))), y''') -> DFIB(s(s(s(s(x'''')))), y''')

Rule:

dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

DFIB(s(s(s(s(s(s(s(s(s(x''''''''))))))))), y'') -> DFIB(s(s(s(s(s(s(s(x''''''''))))))), y'')
DFIB(s(s(s(s(s(s(s(s(x'''''')))))))), y'') -> DFIB(s(s(s(s(s(s(x'''''')))))), y'')
DFIB(s(s(s(s(s(s(x''')))))), y'') -> DFIB(s(s(s(s(x''')))), y'')
DFIB(s(s(s(s(s(s(s(x''''''))))))), y''') -> DFIB(s(s(s(s(s(x''''''))))), y''')
DFIB(s(s(s(s(s(s(s(x''''''))))))), y'''') -> DFIB(s(s(s(s(s(x''''''))))), y'''')
DFIB(s(s(s(s(s(s(x'''')))))), y'''') -> DFIB(s(s(s(s(x'''')))), y'''')
DFIB(s(s(s(s(x')))), y''') -> DFIB(s(s(s(x'))), dfib(s(x'), dfib(x', y''')))
DFIB(s(s(s(s(s(s(x'''')))))), y''') -> DFIB(s(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(dfib(x1, x2)) =  0 POL(DFIB(x1, x2)) =  1 + x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳Dependency Graph`

Dependency Pair:

Rule:

dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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