Term Rewriting System R:
[X]
af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

MARK(f(X)) -> AF(mark(X))
MARK(f(X)) -> MARK(X)
MARK(h(X)) -> MARK(X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation


Dependency Pairs:

MARK(h(X)) -> MARK(X)
MARK(f(X)) -> MARK(X)


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




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

MARK(f(X)) -> MARK(X)
two new Dependency Pairs are created:

MARK(f(f(X''))) -> MARK(f(X''))
MARK(f(h(X''))) -> MARK(h(X''))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

MARK(f(h(X''))) -> MARK(h(X''))
MARK(f(f(X''))) -> MARK(f(X''))
MARK(h(X)) -> MARK(X)


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




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

MARK(h(X)) -> MARK(X)
three new Dependency Pairs are created:

MARK(h(h(X''))) -> MARK(h(X''))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))

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:

MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(X''))) -> MARK(f(X''))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(X''))) -> MARK(h(X''))
MARK(f(h(X''))) -> MARK(h(X''))


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




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

MARK(f(f(X''))) -> MARK(f(X''))
two new Dependency Pairs are created:

MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))

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:

MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(X''))) -> MARK(h(X''))
MARK(f(h(X''))) -> MARK(h(X''))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




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

MARK(f(h(X''))) -> MARK(h(X''))
three new Dependency Pairs are created:

MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

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:

MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(X''))) -> MARK(h(X''))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




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

MARK(h(h(X''))) -> MARK(h(X''))
three new Dependency Pairs are created:

MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

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:

MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




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

MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
two new Dependency Pairs are created:

MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(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


Dependency Pairs:

MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




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

MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
three new Dependency Pairs are created:

MARK(h(f(h(h(X''''''))))) -> MARK(f(h(h(X''''''))))
MARK(h(f(h(f(f(X'''''''')))))) -> MARK(f(h(f(f(X'''''''')))))
MARK(h(f(h(f(h(X'''''''')))))) -> MARK(f(h(f(h(X'''''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 8
Polynomial Ordering


Dependency Pairs:

MARK(h(f(h(f(h(X'''''''')))))) -> MARK(f(h(f(h(X'''''''')))))
MARK(h(f(h(f(f(X'''''''')))))) -> MARK(f(h(f(f(X'''''''')))))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(h(X''''''))))) -> MARK(f(h(h(X''''''))))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

MARK(h(f(h(f(h(X'''''''')))))) -> MARK(f(h(f(h(X'''''''')))))
MARK(h(f(h(f(f(X'''''''')))))) -> MARK(f(h(f(f(X'''''''')))))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))
MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(h(X''''''))))) -> MARK(f(h(h(X''''''))))


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(MARK(x1))=  1 + x1  
  POL(h(x1))=  1 + x1  
  POL(f(x1))=  x1  

resulting in one new DP problem.



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


Dependency Pairs:

MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 10
Polynomial Ordering


Dependency Pair:

MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))


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(MARK(x1))=  1 + x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 11
Dependency Graph


Dependency Pair:


Rules:


af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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