Term Rewriting System R:
[x, y]
f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(x, y) -> G(x, y)
G(h(x), y) -> F(x, y)
G(h(x), y) -> G(x, y)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation


Dependency Pairs:

G(h(x), y) -> G(x, y)
G(h(x), y) -> F(x, y)
F(x, y) -> G(x, y)


Rules:


f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))


Strategy:

innermost




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

F(x, y) -> G(x, y)
one new Dependency Pair is created:

F(h(x''), y'') -> G(h(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:

F(h(x''), y'') -> G(h(x''), y'')
G(h(x), y) -> F(x, y)
G(h(x), y) -> G(x, y)


Rules:


f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))


Strategy:

innermost




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

G(h(x), y) -> F(x, y)
one new Dependency Pair is created:

G(h(h(x'''')), y') -> F(h(x''''), y')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 3
Forward Instantiation Transformation


Dependency Pairs:

G(h(h(x'''')), y') -> F(h(x''''), y')
G(h(x), y) -> G(x, y)
F(h(x''), y'') -> G(h(x''), y'')


Rules:


f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))


Strategy:

innermost




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

G(h(x), y) -> G(x, y)
two new Dependency Pairs are created:

G(h(h(x'')), y'') -> G(h(x''), y'')
G(h(h(h(x''''''))), y') -> G(h(h(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:

G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')
G(h(h(x'')), y'') -> G(h(x''), y'')
F(h(x''), y'') -> G(h(x''), y'')
G(h(h(x'''')), y') -> F(h(x''''), y')


Rules:


f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))


Strategy:

innermost




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

F(h(x''), y'') -> G(h(x''), y'')
three new Dependency Pairs are created:

F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(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:

F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
G(h(h(x'')), y'') -> G(h(x''), y'')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(x'''')), y') -> F(h(x''''), y')
G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')


Rules:


f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))


Strategy:

innermost




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

G(h(h(x'''')), y') -> F(h(x''''), y')
three new Dependency Pairs are created:

G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 6
Forward Instantiation Transformation


Dependency Pairs:

G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')
G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')
G(h(h(x'')), y'') -> G(h(x''), y'')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')


Rules:


f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))


Strategy:

innermost




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

G(h(h(x'')), y'') -> G(h(x''), y'')
four new Dependency Pairs are created:

G(h(h(h(x''''))), y'''') -> G(h(h(x'''')), y'''')
G(h(h(h(h(x'''''''')))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(h(x'''''''''')))), y'''') -> G(h(h(h(x''''''''''))), y'''')
G(h(h(h(h(h(x''''''''''''))))), y'''') -> G(h(h(h(h(x'''''''''''')))), y'''')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 7
Forward Instantiation Transformation


Dependency Pairs:

G(h(h(h(h(h(x''''''''''''))))), y'''') -> G(h(h(h(h(x'''''''''''')))), y'''')
G(h(h(h(h(x'''''''''')))), y'''') -> G(h(h(h(x''''''''''))), y'''')
G(h(h(h(h(x'''''''')))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(x''''))), y'''') -> G(h(h(x'''')), y'''')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')
G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')


Rules:


f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))


Strategy:

innermost




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

G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')
seven new Dependency Pairs are created:

G(h(h(h(h(x'''''''')))), y''') -> G(h(h(h(x''''''''))), y''')
G(h(h(h(h(x'''''''''')))), y'') -> G(h(h(h(x''''''''''))), y'')
G(h(h(h(h(x'''''''')))), y'') -> G(h(h(h(x''''''''))), y'')
G(h(h(h(h(h(x''''''''''''))))), y'') -> G(h(h(h(h(x'''''''''''')))), y'')
G(h(h(h(h(x''''''')))), y'') -> G(h(h(h(x'''''''))), y'')
G(h(h(h(h(h(x''''''''''))))), y'') -> G(h(h(h(h(x'''''''''')))), y'')
G(h(h(h(h(h(h(x'''''''''''''')))))), y'') -> G(h(h(h(h(h(x''''''''''''''))))), y'')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 8
Argument Filtering and Ordering


Dependency Pairs:

G(h(h(h(h(h(h(x'''''''''''''')))))), y'') -> G(h(h(h(h(h(x''''''''''''''))))), y'')
G(h(h(h(h(h(x''''''''''))))), y'') -> G(h(h(h(h(x'''''''''')))), y'')
G(h(h(h(h(x''''''')))), y'') -> G(h(h(h(x'''''''))), y'')
G(h(h(h(h(h(x''''''''''''))))), y'') -> G(h(h(h(h(x'''''''''''')))), y'')
G(h(h(h(h(x'''''''')))), y'') -> G(h(h(h(x''''''''))), y'')
G(h(h(h(h(x'''''''''')))), y'') -> G(h(h(h(x''''''''''))), y'')
G(h(h(h(h(x'''''''')))), y''') -> G(h(h(h(x''''''''))), y''')
G(h(h(h(h(x'''''''''')))), y'''') -> G(h(h(h(x''''''''''))), y'''')
G(h(h(h(h(x'''''''')))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(x''''))), y'''') -> G(h(h(x'''')), y'''')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')
G(h(h(h(h(h(x''''''''''''))))), y'''') -> G(h(h(h(h(x'''''''''''')))), y'''')


Rules:


f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

G(h(h(h(h(h(h(x'''''''''''''')))))), y'') -> G(h(h(h(h(h(x''''''''''''''))))), y'')
G(h(h(h(h(h(x''''''''''))))), y'') -> G(h(h(h(h(x'''''''''')))), y'')
G(h(h(h(h(x''''''')))), y'') -> G(h(h(h(x'''''''))), y'')
G(h(h(h(h(h(x''''''''''''))))), y'') -> G(h(h(h(h(x'''''''''''')))), y'')
G(h(h(h(h(x'''''''')))), y'') -> G(h(h(h(x''''''''))), y'')
G(h(h(h(h(x'''''''''')))), y'') -> G(h(h(h(x''''''''''))), y'')
G(h(h(h(h(x'''''''')))), y''') -> G(h(h(h(x''''''''))), y''')
G(h(h(h(h(x'''''''''')))), y'''') -> G(h(h(h(x''''''''''))), y'''')
G(h(h(h(h(x'''''''')))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(x''''))), y'''') -> G(h(h(x'''')), y'''')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(h(h(h(x''''''''''''))))), y'''') -> G(h(h(h(h(x'''''''''''')))), y'''')


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(G(x1, x2))=  x1 + x2  
  POL(h(x1))=  1 + x1  
  POL(F(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
G(x1, x2) -> G(x1, x2)
h(x1) -> h(x1)


   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 9
Dependency Graph


Dependency Pairs:

G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')


Rules:


f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes