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:
0:01 minutes