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