Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳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.

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 6

                 ↳Argument Filtering and 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''')

The following usable rule for innermost w.r.t. to the AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(DFIB(x₁, x₂)) = 1 + x₁ + x₂

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

DFIB(x₁, x₂) -> DFIB(x₁, x₂)
s(x₁) -> s(x₁)
dfib(x₁, x₂) -> x₂

     ↳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:
0:01 minutes

POL(DFIB(x₁, x₂))	= 1 + x₁ + x₂
POL(s(x₁))	= 1 + x₁