Term Rewriting System R:
[x]
w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

W(r(x)) -> W(x)
B(r(x)) -> B(x)
B(w(x)) -> W(b(x))
B(w(x)) -> B(x)

Furthermore, R contains three SCCs.


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


Dependency Pair:

W(r(x)) -> W(x)


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




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

W(r(x)) -> W(x)
one new Dependency Pair is created:

W(r(r(x''))) -> W(r(x''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 4
Forward Instantiation Transformation
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst


Dependency Pair:

W(r(r(x''))) -> W(r(x''))


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




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

W(r(r(x''))) -> W(r(x''))
one new Dependency Pair is created:

W(r(r(r(x'''')))) -> W(r(r(x'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 4
FwdInst
             ...
               →DP Problem 5
Argument Filtering and Ordering
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst


Dependency Pair:

W(r(r(r(x'''')))) -> W(r(r(x'''')))


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




The following dependency pair can be strictly oriented:

W(r(r(r(x'''')))) -> W(r(r(x'''')))


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
W(x1) -> W(x1)
r(x1) -> r(x1)


   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 4
FwdInst
             ...
               →DP Problem 6
Dependency Graph
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst


Dependency Pair:


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

B(r(x)) -> B(x)


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




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

B(r(x)) -> B(x)
one new Dependency Pair is created:

B(r(r(x''))) -> B(r(x''))

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
       →DP Problem 3
FwdInst


Dependency Pair:

B(r(r(x''))) -> B(r(x''))


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




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

B(r(r(x''))) -> B(r(x''))
one new Dependency Pair is created:

B(r(r(r(x'''')))) -> B(r(r(x'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 7
FwdInst
             ...
               →DP Problem 8
Argument Filtering and Ordering
       →DP Problem 3
FwdInst


Dependency Pair:

B(r(r(r(x'''')))) -> B(r(r(x'''')))


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




The following dependency pair can be strictly oriented:

B(r(r(r(x'''')))) -> B(r(r(x'''')))


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
B(x1) -> B(x1)
r(x1) -> r(x1)


   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 7
FwdInst
             ...
               →DP Problem 9
Dependency Graph
       →DP Problem 3
FwdInst


Dependency Pair:


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

B(w(x)) -> B(x)


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




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

B(w(x)) -> B(x)
one new Dependency Pair is created:

B(w(w(x''))) -> B(w(x''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
           →DP Problem 10
Forward Instantiation Transformation


Dependency Pair:

B(w(w(x''))) -> B(w(x''))


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




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

B(w(w(x''))) -> B(w(x''))
one new Dependency Pair is created:

B(w(w(w(x'''')))) -> B(w(w(x'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
           →DP Problem 10
FwdInst
             ...
               →DP Problem 11
Argument Filtering and Ordering


Dependency Pair:

B(w(w(w(x'''')))) -> B(w(w(x'''')))


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




The following dependency pair can be strictly oriented:

B(w(w(w(x'''')))) -> B(w(w(x'''')))


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

w(r(x)) -> r(w(x))


Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
B(x1) -> B(x1)
w(x1) -> w(x1)
r(x1) -> x1


   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
           →DP Problem 10
FwdInst
             ...
               →DP Problem 12
Dependency Graph


Dependency Pair:


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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