Term Rewriting System R:
[X]
active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(f(X)) -> G(h(f(X)))
ACTIVE(f(X)) -> H(f(X))
ACTIVE(f(X)) -> F(active(X))
ACTIVE(f(X)) -> ACTIVE(X)
ACTIVE(h(X)) -> H(active(X))
ACTIVE(h(X)) -> ACTIVE(X)
F(mark(X)) -> F(X)
F(ok(X)) -> F(X)
H(mark(X)) -> H(X)
H(ok(X)) -> H(X)
PROPER(f(X)) -> F(proper(X))
PROPER(f(X)) -> PROPER(X)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(X)
PROPER(h(X)) -> H(proper(X))
PROPER(h(X)) -> PROPER(X)
G(ok(X)) -> G(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains six SCCs.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:

G(ok(X)) -> G(X)


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

G(ok(X)) -> G(X)
one new Dependency Pair is created:

G(ok(ok(X''))) -> G(ok(X''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 7
Forward Instantiation Transformation
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:

G(ok(ok(X''))) -> G(ok(X''))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

G(ok(ok(X''))) -> G(ok(X''))
one new Dependency Pair is created:

G(ok(ok(ok(X'''')))) -> G(ok(ok(X'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 7
FwdInst
             ...
               →DP Problem 8
Polynomial Ordering
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:

G(ok(ok(ok(X'''')))) -> G(ok(ok(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(ok(ok(ok(X'''')))) -> G(ok(ok(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(G(x1))=  1 + x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 7
FwdInst
             ...
               →DP Problem 9
Dependency Graph
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

F(ok(X)) -> F(X)
F(mark(X)) -> F(X)


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

F(mark(X)) -> F(X)
two new Dependency Pairs are created:

F(mark(mark(X''))) -> F(mark(X''))
F(mark(ok(X''))) -> F(ok(X''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 10
Forward Instantiation Transformation
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

F(mark(ok(X''))) -> F(ok(X''))
F(mark(mark(X''))) -> F(mark(X''))
F(ok(X)) -> F(X)


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

F(ok(X)) -> F(X)
three new Dependency Pairs are created:

F(ok(ok(X''))) -> F(ok(X''))
F(ok(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(mark(ok(X'''')))) -> F(mark(ok(X'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 10
FwdInst
             ...
               →DP Problem 11
Forward Instantiation Transformation
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

F(ok(mark(ok(X'''')))) -> F(mark(ok(X'''')))
F(mark(mark(X''))) -> F(mark(X''))
F(ok(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(ok(X''))) -> F(ok(X''))
F(mark(ok(X''))) -> F(ok(X''))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

F(mark(mark(X''))) -> F(mark(X''))
two new Dependency Pairs are created:

F(mark(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(mark(mark(ok(X'''')))) -> F(mark(ok(X'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 10
FwdInst
             ...
               →DP Problem 12
Forward Instantiation Transformation
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

F(mark(mark(ok(X'''')))) -> F(mark(ok(X'''')))
F(mark(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(ok(X''))) -> F(ok(X''))
F(mark(ok(X''))) -> F(ok(X''))
F(ok(mark(ok(X'''')))) -> F(mark(ok(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

F(mark(ok(X''))) -> F(ok(X''))
three new Dependency Pairs are created:

F(mark(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(mark(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(mark(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 10
FwdInst
             ...
               →DP Problem 13
Forward Instantiation Transformation
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

F(mark(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))
F(mark(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(ok(mark(ok(X'''')))) -> F(mark(ok(X'''')))
F(mark(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(ok(X''))) -> F(ok(X''))
F(mark(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(mark(mark(ok(X'''')))) -> F(mark(ok(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

F(ok(ok(X''))) -> F(ok(X''))
three new Dependency Pairs are created:

F(ok(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(ok(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(ok(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 10
FwdInst
             ...
               →DP Problem 14
Forward Instantiation Transformation
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

F(ok(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))
F(mark(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(mark(mark(ok(X'''')))) -> F(mark(ok(X'''')))
F(mark(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(ok(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(mark(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(ok(mark(ok(X'''')))) -> F(mark(ok(X'''')))
F(mark(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

F(ok(mark(mark(X'''')))) -> F(mark(mark(X'''')))
two new Dependency Pairs are created:

F(ok(mark(mark(mark(X''''''))))) -> F(mark(mark(mark(X''''''))))
F(ok(mark(mark(ok(X''''''))))) -> F(mark(mark(ok(X''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 10
FwdInst
             ...
               →DP Problem 15
Forward Instantiation Transformation
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

F(mark(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))
F(ok(mark(mark(ok(X''''''))))) -> F(mark(mark(ok(X''''''))))
F(mark(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(mark(mark(ok(X'''')))) -> F(mark(ok(X'''')))
F(mark(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(mark(mark(mark(X''''''))))) -> F(mark(mark(mark(X''''''))))
F(ok(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(ok(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(mark(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(ok(mark(ok(X'''')))) -> F(mark(ok(X'''')))
F(ok(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

F(ok(mark(ok(X'''')))) -> F(mark(ok(X'''')))
three new Dependency Pairs are created:

F(ok(mark(ok(ok(X''''''))))) -> F(mark(ok(ok(X''''''))))
F(ok(mark(ok(mark(mark(X'''''''')))))) -> F(mark(ok(mark(mark(X'''''''')))))
F(ok(mark(ok(mark(ok(X'''''''')))))) -> F(mark(ok(mark(ok(X'''''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 10
FwdInst
             ...
               →DP Problem 16
Polynomial Ordering
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

F(ok(mark(ok(mark(ok(X'''''''')))))) -> F(mark(ok(mark(ok(X'''''''')))))
F(ok(mark(ok(mark(mark(X'''''''')))))) -> F(mark(ok(mark(mark(X'''''''')))))
F(ok(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))
F(ok(mark(mark(ok(X''''''))))) -> F(mark(mark(ok(X''''''))))
F(mark(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(mark(mark(ok(X'''')))) -> F(mark(ok(X'''')))
F(mark(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(ok(mark(mark(mark(X''''''))))) -> F(mark(mark(mark(X''''''))))
F(ok(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(ok(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(mark(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(ok(mark(ok(ok(X''''''))))) -> F(mark(ok(ok(X''''''))))
F(mark(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(ok(mark(ok(mark(ok(X'''''''')))))) -> F(mark(ok(mark(ok(X'''''''')))))
F(ok(mark(ok(mark(mark(X'''''''')))))) -> F(mark(ok(mark(mark(X'''''''')))))
F(ok(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))
F(ok(mark(mark(ok(X''''''))))) -> F(mark(mark(ok(X''''''))))
F(ok(mark(mark(mark(X''''''))))) -> F(mark(mark(mark(X''''''))))
F(ok(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(ok(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(ok(mark(ok(ok(X''''''))))) -> F(mark(ok(ok(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))=  x1  
  POL(ok(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 10
FwdInst
             ...
               →DP Problem 17
Dependency Graph
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

F(mark(ok(mark(mark(X''''''))))) -> F(ok(mark(mark(X''''''))))
F(mark(mark(ok(X'''')))) -> F(mark(ok(X'''')))
F(mark(mark(mark(X'''')))) -> F(mark(mark(X'''')))
F(mark(ok(ok(X'''')))) -> F(ok(ok(X'''')))
F(mark(ok(mark(ok(X''''''))))) -> F(ok(mark(ok(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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
FwdInst
             ...
               →DP Problem 18
Polynomial Ordering
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:

F(mark(mark(mark(X'''')))) -> F(mark(mark(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(mark(mark(mark(X'''')))) -> F(mark(mark(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 10
FwdInst
             ...
               →DP Problem 19
Dependency Graph
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

H(ok(X)) -> H(X)
H(mark(X)) -> H(X)


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

H(mark(X)) -> H(X)
two new Dependency Pairs are created:

H(mark(mark(X''))) -> H(mark(X''))
H(mark(ok(X''))) -> H(ok(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 20
Forward Instantiation Transformation
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

H(mark(ok(X''))) -> H(ok(X''))
H(mark(mark(X''))) -> H(mark(X''))
H(ok(X)) -> H(X)


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

H(ok(X)) -> H(X)
three new Dependency Pairs are created:

H(ok(ok(X''))) -> H(ok(X''))
H(ok(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(mark(ok(X'''')))) -> H(mark(ok(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 20
FwdInst
             ...
               →DP Problem 21
Forward Instantiation Transformation
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

H(ok(mark(ok(X'''')))) -> H(mark(ok(X'''')))
H(mark(mark(X''))) -> H(mark(X''))
H(ok(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(ok(X''))) -> H(ok(X''))
H(mark(ok(X''))) -> H(ok(X''))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

H(mark(mark(X''))) -> H(mark(X''))
two new Dependency Pairs are created:

H(mark(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(mark(mark(ok(X'''')))) -> H(mark(ok(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 20
FwdInst
             ...
               →DP Problem 22
Forward Instantiation Transformation
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

H(mark(mark(ok(X'''')))) -> H(mark(ok(X'''')))
H(mark(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(ok(X''))) -> H(ok(X''))
H(mark(ok(X''))) -> H(ok(X''))
H(ok(mark(ok(X'''')))) -> H(mark(ok(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

H(mark(ok(X''))) -> H(ok(X''))
three new Dependency Pairs are created:

H(mark(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(mark(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(mark(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(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 20
FwdInst
             ...
               →DP Problem 23
Forward Instantiation Transformation
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

H(mark(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(X''''''))))
H(mark(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(ok(mark(ok(X'''')))) -> H(mark(ok(X'''')))
H(mark(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(ok(X''))) -> H(ok(X''))
H(mark(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(mark(mark(ok(X'''')))) -> H(mark(ok(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

H(ok(ok(X''))) -> H(ok(X''))
three new Dependency Pairs are created:

H(ok(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(ok(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(ok(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(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 20
FwdInst
             ...
               →DP Problem 24
Forward Instantiation Transformation
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

H(ok(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(X''''''))))
H(mark(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(mark(mark(ok(X'''')))) -> H(mark(ok(X'''')))
H(mark(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(ok(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(mark(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(ok(mark(ok(X'''')))) -> H(mark(ok(X'''')))
H(mark(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

H(ok(mark(mark(X'''')))) -> H(mark(mark(X'''')))
two new Dependency Pairs are created:

H(ok(mark(mark(mark(X''''''))))) -> H(mark(mark(mark(X''''''))))
H(ok(mark(mark(ok(X''''''))))) -> H(mark(mark(ok(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 20
FwdInst
             ...
               →DP Problem 25
Forward Instantiation Transformation
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

H(mark(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(X''''''))))
H(ok(mark(mark(ok(X''''''))))) -> H(mark(mark(ok(X''''''))))
H(mark(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(mark(mark(ok(X'''')))) -> H(mark(ok(X'''')))
H(mark(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(mark(mark(mark(X''''''))))) -> H(mark(mark(mark(X''''''))))
H(ok(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(ok(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(mark(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(ok(mark(ok(X'''')))) -> H(mark(ok(X'''')))
H(ok(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

H(ok(mark(ok(X'''')))) -> H(mark(ok(X'''')))
three new Dependency Pairs are created:

H(ok(mark(ok(ok(X''''''))))) -> H(mark(ok(ok(X''''''))))
H(ok(mark(ok(mark(mark(X'''''''')))))) -> H(mark(ok(mark(mark(X'''''''')))))
H(ok(mark(ok(mark(ok(X'''''''')))))) -> H(mark(ok(mark(ok(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 20
FwdInst
             ...
               →DP Problem 26
Polynomial Ordering
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

H(ok(mark(ok(mark(ok(X'''''''')))))) -> H(mark(ok(mark(ok(X'''''''')))))
H(ok(mark(ok(mark(mark(X'''''''')))))) -> H(mark(ok(mark(mark(X'''''''')))))
H(ok(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(X''''''))))
H(ok(mark(mark(ok(X''''''))))) -> H(mark(mark(ok(X''''''))))
H(mark(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(mark(mark(ok(X'''')))) -> H(mark(ok(X'''')))
H(mark(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(ok(mark(mark(mark(X''''''))))) -> H(mark(mark(mark(X''''''))))
H(ok(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(ok(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(mark(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(ok(mark(ok(ok(X''''''))))) -> H(mark(ok(ok(X''''''))))
H(mark(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

H(ok(mark(ok(mark(ok(X'''''''')))))) -> H(mark(ok(mark(ok(X'''''''')))))
H(ok(mark(ok(mark(mark(X'''''''')))))) -> H(mark(ok(mark(mark(X'''''''')))))
H(ok(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(X''''''))))
H(ok(mark(mark(ok(X''''''))))) -> H(mark(mark(ok(X''''''))))
H(ok(mark(mark(mark(X''''''))))) -> H(mark(mark(mark(X''''''))))
H(ok(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(ok(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(ok(mark(ok(ok(X''''''))))) -> H(mark(ok(ok(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))=  x1  
  POL(H(x1))=  1 + x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
           →DP Problem 20
FwdInst
             ...
               →DP Problem 27
Dependency Graph
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

H(mark(ok(mark(mark(X''''''))))) -> H(ok(mark(mark(X''''''))))
H(mark(mark(ok(X'''')))) -> H(mark(ok(X'''')))
H(mark(mark(mark(X'''')))) -> H(mark(mark(X'''')))
H(mark(ok(ok(X'''')))) -> H(ok(ok(X'''')))
H(mark(ok(mark(ok(X''''''))))) -> H(ok(mark(ok(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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 3
FwdInst
           →DP Problem 20
FwdInst
             ...
               →DP Problem 28
Polynomial Ordering
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:

H(mark(mark(mark(X'''')))) -> H(mark(mark(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

H(mark(mark(mark(X'''')))) -> H(mark(mark(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  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
           →DP Problem 20
FwdInst
             ...
               →DP Problem 29
Dependency Graph
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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
FwdInst
       →DP Problem 4
Forward Instantiation Transformation
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

ACTIVE(h(X)) -> ACTIVE(X)
ACTIVE(f(X)) -> ACTIVE(X)


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

ACTIVE(f(f(X''))) -> ACTIVE(f(X''))
ACTIVE(f(h(X''))) -> ACTIVE(h(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 4
FwdInst
           →DP Problem 30
Forward Instantiation Transformation
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

ACTIVE(h(h(X''))) -> ACTIVE(h(X''))
ACTIVE(h(f(f(X'''')))) -> ACTIVE(f(f(X'''')))
ACTIVE(h(f(h(X'''')))) -> ACTIVE(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
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 30
FwdInst
             ...
               →DP Problem 31
Forward Instantiation Transformation
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

ACTIVE(f(f(f(X'''')))) -> ACTIVE(f(f(X'''')))
ACTIVE(f(f(h(X'''')))) -> ACTIVE(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
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 30
FwdInst
             ...
               →DP Problem 32
Forward Instantiation Transformation
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

ACTIVE(f(h(h(X'''')))) -> ACTIVE(h(h(X'''')))
ACTIVE(f(h(f(f(X''''''))))) -> ACTIVE(h(f(f(X''''''))))
ACTIVE(f(h(f(h(X''''''))))) -> ACTIVE(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 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 30
FwdInst
             ...
               →DP Problem 33
Forward Instantiation Transformation
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

ACTIVE(h(h(h(X'''')))) -> ACTIVE(h(h(X'''')))
ACTIVE(h(h(f(f(X''''''))))) -> ACTIVE(h(f(f(X''''''))))
ACTIVE(h(h(f(h(X''''''))))) -> ACTIVE(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 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 30
FwdInst
             ...
               →DP Problem 34
Forward Instantiation Transformation
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

ACTIVE(h(f(f(f(X''''''))))) -> ACTIVE(f(f(f(X''''''))))
ACTIVE(h(f(f(h(X''''''))))) -> ACTIVE(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 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 30
FwdInst
             ...
               →DP Problem 35
Forward Instantiation Transformation
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

ACTIVE(h(f(h(h(X''''''))))) -> ACTIVE(f(h(h(X''''''))))
ACTIVE(h(f(h(f(f(X'''''''')))))) -> ACTIVE(f(h(f(f(X'''''''')))))
ACTIVE(h(f(h(f(h(X'''''''')))))) -> ACTIVE(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 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 30
FwdInst
             ...
               →DP Problem 36
Polynomial Ordering
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  1 + x1  
  POL(h(x1))=  1 + x1  
  POL(mark(x1))=  0  
  POL(ok(x1))=  0  
  POL(f(x1))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 30
FwdInst
             ...
               →DP Problem 37
Dependency Graph
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 30
FwdInst
             ...
               →DP Problem 38
Polynomial Ordering
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:

ACTIVE(f(f(f(X'''')))) -> ACTIVE(f(f(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(f(f(f(X'''')))) -> ACTIVE(f(f(X'''')))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  1 + x1  
  POL(mark(x1))=  0  
  POL(ok(x1))=  0  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 30
FwdInst
             ...
               →DP Problem 39
Dependency Graph
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(X)) -> PROPER(X)
PROPER(g(X)) -> PROPER(X)
PROPER(f(X)) -> PROPER(X)


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(f(X)) -> PROPER(X)
three new Dependency Pairs are created:

PROPER(f(f(X''))) -> PROPER(f(X''))
PROPER(f(g(X''))) -> PROPER(g(X''))
PROPER(f(h(X''))) -> PROPER(h(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 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(f(h(X''))) -> PROPER(h(X''))
PROPER(f(g(X''))) -> PROPER(g(X''))
PROPER(f(f(X''))) -> PROPER(f(X''))
PROPER(g(X)) -> PROPER(X)
PROPER(h(X)) -> PROPER(X)


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(g(X)) -> PROPER(X)
five new Dependency Pairs are created:

PROPER(g(g(X''))) -> PROPER(g(X''))
PROPER(g(h(X''))) -> PROPER(h(X''))
PROPER(g(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(f(h(X'''')))) -> PROPER(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 41
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(g(h(X''))) -> PROPER(h(X''))
PROPER(g(g(X''))) -> PROPER(g(X''))
PROPER(f(g(X''))) -> PROPER(g(X''))
PROPER(f(f(X''))) -> PROPER(f(X''))
PROPER(h(X)) -> PROPER(X)
PROPER(f(h(X''))) -> PROPER(h(X''))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(h(X)) -> PROPER(X)
nine new Dependency Pairs are created:

PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 42
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(h(X''))) -> PROPER(h(X''))
PROPER(g(g(X''))) -> PROPER(g(X''))
PROPER(f(g(X''))) -> PROPER(g(X''))
PROPER(f(f(X''))) -> PROPER(f(X''))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(f(h(X''))) -> PROPER(h(X''))
PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(h(X'''')))) -> PROPER(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 43
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(h(X''))) -> PROPER(h(X''))
PROPER(g(g(X''))) -> PROPER(g(X''))
PROPER(f(g(X''))) -> PROPER(g(X''))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(f(h(X''))) -> PROPER(h(X''))
PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(f(g(X''))) -> PROPER(g(X''))
five new Dependency Pairs are created:

PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 44
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(h(X''))) -> PROPER(h(X''))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(g(h(X''))) -> PROPER(h(X''))
PROPER(g(g(X''))) -> PROPER(g(X''))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(f(h(X''))) -> PROPER(h(X''))
nine new Dependency Pairs are created:

PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 45
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(h(X''))) -> PROPER(h(X''))
PROPER(g(g(X''))) -> PROPER(g(X''))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(g(g(X''))) -> PROPER(g(X''))
five new Dependency Pairs are created:

PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 46
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(h(X''))) -> PROPER(h(X''))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(g(h(X''))) -> PROPER(h(X''))
nine new Dependency Pairs are created:

PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 47
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(g(f(f(X'''')))) -> PROPER(f(f(X'''')))
three new Dependency Pairs are created:

PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(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 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 48
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(g(f(g(X'''')))) -> PROPER(f(g(X'''')))
five new Dependency Pairs are created:

PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 49
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(g(f(h(X'''')))) -> PROPER(f(h(X'''')))
nine new Dependency Pairs are created:

PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 50
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(h(h(X''))) -> PROPER(h(X''))
nine new Dependency Pairs are created:

PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 51
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(h(f(f(h(X''''''))))) -> PROPER(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 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 52
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(h(f(g(X'''')))) -> PROPER(f(g(X'''')))
five new Dependency Pairs are created:

PROPER(h(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(h(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(h(f(g(f(h(X'''''''')))))) -> PROPER(f(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 53
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(h(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(h(f(h(X'''')))) -> PROPER(f(h(X'''')))
nine new Dependency Pairs are created:

PROPER(h(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(h(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(h(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(h(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(h(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(h(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 54
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(h(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(h(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(h(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(h(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
five new Dependency Pairs are created:

PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(h(g(g(f(f(X'''''''')))))) -> PROPER(g(g(f(f(X'''''''')))))
PROPER(h(g(g(f(g(X'''''''')))))) -> PROPER(g(g(f(g(X'''''''')))))
PROPER(h(g(g(f(h(X'''''''')))))) -> PROPER(g(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 55
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(g(g(f(h(X'''''''')))))) -> PROPER(g(g(f(h(X'''''''')))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(g(g(f(g(X'''''''')))))) -> PROPER(g(g(f(g(X'''''''')))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(f(f(X'''''''')))))) -> PROPER(g(g(f(f(X'''''''')))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(h(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(h(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(h(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(h(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
nine new Dependency Pairs are created:

PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(h(g(h(f(f(X'''''''')))))) -> PROPER(g(h(f(f(X'''''''')))))
PROPER(h(g(h(f(g(X'''''''')))))) -> PROPER(g(h(f(g(X'''''''')))))
PROPER(h(g(h(f(h(X'''''''')))))) -> PROPER(g(h(f(h(X'''''''')))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(h(g(h(g(f(f(X''''''''''))))))) -> PROPER(g(h(g(f(f(X''''''''''))))))
PROPER(h(g(h(g(f(g(X''''''''''))))))) -> PROPER(g(h(g(f(g(X''''''''''))))))
PROPER(h(g(h(g(f(h(X''''''''''))))))) -> PROPER(g(h(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 56
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(g(h(g(f(h(X''''''''''))))))) -> PROPER(g(h(g(f(h(X''''''''''))))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(h(g(f(g(X''''''''''))))))) -> PROPER(g(h(g(f(g(X''''''''''))))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(h(g(f(f(X''''''''''))))))) -> PROPER(g(h(g(f(f(X''''''''''))))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(h(g(h(f(h(X'''''''')))))) -> PROPER(g(h(f(h(X'''''''')))))
PROPER(h(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(g(h(f(g(X'''''''')))))) -> PROPER(g(h(f(g(X'''''''')))))
PROPER(h(g(h(f(f(X'''''''')))))) -> PROPER(g(h(f(f(X'''''''')))))
PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(g(g(f(h(X'''''''')))))) -> PROPER(g(g(f(h(X'''''''')))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(g(g(f(g(X'''''''')))))) -> PROPER(g(g(f(g(X'''''''')))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(f(f(X'''''''')))))) -> PROPER(g(g(f(f(X'''''''')))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(h(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(h(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(h(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

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

PROPER(h(g(f(f(f(X'''''''')))))) -> PROPER(g(f(f(f(X'''''''')))))
PROPER(h(g(f(f(g(X'''''''')))))) -> PROPER(g(f(f(g(X'''''''')))))
PROPER(h(g(f(f(h(X'''''''')))))) -> PROPER(g(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 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 57
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(g(h(g(f(h(X''''''''''))))))) -> PROPER(g(h(g(f(h(X''''''''''))))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(h(g(f(g(X''''''''''))))))) -> PROPER(g(h(g(f(g(X''''''''''))))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(h(g(f(f(X''''''''''))))))) -> PROPER(g(h(g(f(f(X''''''''''))))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(g(f(f(h(X'''''''')))))) -> PROPER(g(f(f(h(X'''''''')))))
PROPER(h(g(f(f(g(X'''''''')))))) -> PROPER(g(f(f(g(X'''''''')))))
PROPER(h(g(f(f(f(X'''''''')))))) -> PROPER(g(f(f(f(X'''''''')))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(h(g(h(f(h(X'''''''')))))) -> PROPER(g(h(f(h(X'''''''')))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(g(h(f(g(X'''''''')))))) -> PROPER(g(h(f(g(X'''''''')))))
PROPER(h(g(h(f(f(X'''''''')))))) -> PROPER(g(h(f(f(X'''''''')))))
PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(g(g(f(h(X'''''''')))))) -> PROPER(g(g(f(h(X'''''''')))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(g(g(f(g(X'''''''')))))) -> PROPER(g(g(f(g(X'''''''')))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(f(f(X'''''''')))))) -> PROPER(g(g(f(f(X'''''''')))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(h(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(h(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(h(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(h(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(h(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
five new Dependency Pairs are created:

PROPER(h(g(f(g(g(X'''''''')))))) -> PROPER(g(f(g(g(X'''''''')))))
PROPER(h(g(f(g(h(X'''''''')))))) -> PROPER(g(f(g(h(X'''''''')))))
PROPER(h(g(f(g(f(f(X''''''''''))))))) -> PROPER(g(f(g(f(f(X''''''''''))))))
PROPER(h(g(f(g(f(g(X''''''''''))))))) -> PROPER(g(f(g(f(g(X''''''''''))))))
PROPER(h(g(f(g(f(h(X''''''''''))))))) -> PROPER(g(f(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 58
Forward Instantiation Transformation
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(g(h(g(f(h(X''''''''''))))))) -> PROPER(g(h(g(f(h(X''''''''''))))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(h(g(f(g(X''''''''''))))))) -> PROPER(g(h(g(f(g(X''''''''''))))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(h(g(f(f(X''''''''''))))))) -> PROPER(g(h(g(f(f(X''''''''''))))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(h(g(f(g(f(h(X''''''''''))))))) -> PROPER(g(f(g(f(h(X''''''''''))))))
PROPER(h(g(f(g(f(g(X''''''''''))))))) -> PROPER(g(f(g(f(g(X''''''''''))))))
PROPER(h(g(f(g(f(f(X''''''''''))))))) -> PROPER(g(f(g(f(f(X''''''''''))))))
PROPER(h(g(f(g(h(X'''''''')))))) -> PROPER(g(f(g(h(X'''''''')))))
PROPER(h(g(f(g(g(X'''''''')))))) -> PROPER(g(f(g(g(X'''''''')))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(g(f(f(h(X'''''''')))))) -> PROPER(g(f(f(h(X'''''''')))))
PROPER(h(g(f(f(g(X'''''''')))))) -> PROPER(g(f(f(g(X'''''''')))))
PROPER(h(g(f(f(f(X'''''''')))))) -> PROPER(g(f(f(f(X'''''''')))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(h(g(h(f(h(X'''''''')))))) -> PROPER(g(h(f(h(X'''''''')))))
PROPER(h(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(g(h(f(g(X'''''''')))))) -> PROPER(g(h(f(g(X'''''''')))))
PROPER(h(g(h(f(f(X'''''''')))))) -> PROPER(g(h(f(f(X'''''''')))))
PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(g(g(f(h(X'''''''')))))) -> PROPER(g(g(f(h(X'''''''')))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(g(g(f(g(X'''''''')))))) -> PROPER(g(g(f(g(X'''''''')))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(f(f(X'''''''')))))) -> PROPER(g(g(f(f(X'''''''')))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(h(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(h(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(h(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

PROPER(h(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
nine new Dependency Pairs are created:

PROPER(h(g(f(h(h(X'''''''')))))) -> PROPER(g(f(h(h(X'''''''')))))
PROPER(h(g(f(h(f(f(X''''''''''))))))) -> PROPER(g(f(h(f(f(X''''''''''))))))
PROPER(h(g(f(h(f(g(X''''''''''))))))) -> PROPER(g(f(h(f(g(X''''''''''))))))
PROPER(h(g(f(h(f(h(X''''''''''))))))) -> PROPER(g(f(h(f(h(X''''''''''))))))
PROPER(h(g(f(h(g(g(X''''''''''))))))) -> PROPER(g(f(h(g(g(X''''''''''))))))
PROPER(h(g(f(h(g(h(X''''''''''))))))) -> PROPER(g(f(h(g(h(X''''''''''))))))
PROPER(h(g(f(h(g(f(f(X'''''''''''')))))))) -> PROPER(g(f(h(g(f(f(X'''''''''''')))))))
PROPER(h(g(f(h(g(f(g(X'''''''''''')))))))) -> PROPER(g(f(h(g(f(g(X'''''''''''')))))))
PROPER(h(g(f(h(g(f(h(X'''''''''''')))))))) -> PROPER(g(f(h(g(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 59
Polynomial Ordering
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(g(h(g(f(h(X''''''''''))))))) -> PROPER(g(h(g(f(h(X''''''''''))))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(h(g(f(g(X''''''''''))))))) -> PROPER(g(h(g(f(g(X''''''''''))))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(h(g(f(f(X''''''''''))))))) -> PROPER(g(h(g(f(f(X''''''''''))))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(h(g(f(h(g(f(h(X'''''''''''')))))))) -> PROPER(g(f(h(g(f(h(X'''''''''''')))))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(g(f(h(g(f(g(X'''''''''''')))))))) -> PROPER(g(f(h(g(f(g(X'''''''''''')))))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(g(f(h(g(f(f(X'''''''''''')))))))) -> PROPER(g(f(h(g(f(f(X'''''''''''')))))))
PROPER(h(g(f(h(g(h(X''''''''''))))))) -> PROPER(g(f(h(g(h(X''''''''''))))))
PROPER(h(g(f(h(g(g(X''''''''''))))))) -> PROPER(g(f(h(g(g(X''''''''''))))))
PROPER(h(g(f(h(f(h(X''''''''''))))))) -> PROPER(g(f(h(f(h(X''''''''''))))))
PROPER(h(g(f(h(f(g(X''''''''''))))))) -> PROPER(g(f(h(f(g(X''''''''''))))))
PROPER(h(g(f(h(f(f(X''''''''''))))))) -> PROPER(g(f(h(f(f(X''''''''''))))))
PROPER(h(g(f(h(h(X'''''''')))))) -> PROPER(g(f(h(h(X'''''''')))))
PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(h(g(f(g(f(h(X''''''''''))))))) -> PROPER(g(f(g(f(h(X''''''''''))))))
PROPER(h(g(f(g(f(g(X''''''''''))))))) -> PROPER(g(f(g(f(g(X''''''''''))))))
PROPER(h(g(f(g(f(f(X''''''''''))))))) -> PROPER(g(f(g(f(f(X''''''''''))))))
PROPER(h(g(f(g(h(X'''''''')))))) -> PROPER(g(f(g(h(X'''''''')))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(g(f(f(h(X'''''''')))))) -> PROPER(g(f(f(h(X'''''''')))))
PROPER(h(g(f(f(g(X'''''''')))))) -> PROPER(g(f(f(g(X'''''''')))))
PROPER(h(g(f(f(f(X'''''''')))))) -> PROPER(g(f(f(f(X'''''''')))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(h(g(h(f(h(X'''''''')))))) -> PROPER(g(h(f(h(X'''''''')))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(g(h(f(g(X'''''''')))))) -> PROPER(g(h(f(g(X'''''''')))))
PROPER(h(g(h(f(f(X'''''''')))))) -> PROPER(g(h(f(f(X'''''''')))))
PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(g(g(f(h(X'''''''')))))) -> PROPER(g(g(f(h(X'''''''')))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(g(g(f(g(X'''''''')))))) -> PROPER(g(g(f(g(X'''''''')))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(f(f(X'''''''')))))) -> PROPER(g(g(f(f(X'''''''')))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(h(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(h(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(h(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(h(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(h(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(g(f(g(g(X'''''''')))))) -> PROPER(g(f(g(g(X'''''''')))))
PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PROPER(f(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(f(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(f(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(f(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(f(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(f(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(f(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(f(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(f(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(f(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(f(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(f(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(f(f(h(X'''')))) -> PROPER(f(h(X'''')))
PROPER(f(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(f(f(g(X'''')))) -> PROPER(f(g(X'''')))
PROPER(f(f(f(X'''')))) -> PROPER(f(f(X'''')))
PROPER(f(g(g(X'''')))) -> PROPER(g(g(X'''')))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  x1  
  POL(PROPER(x1))=  1 + x1  
  POL(h(x1))=  x1  
  POL(mark(x1))=  0  
  POL(ok(x1))=  0  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 60
Dependency Graph
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(g(h(g(f(h(X'''''''')))))) -> PROPER(h(g(f(h(X'''''''')))))
PROPER(h(g(h(g(f(h(X''''''''''))))))) -> PROPER(g(h(g(f(h(X''''''''''))))))
PROPER(g(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))
PROPER(h(g(h(g(f(g(X''''''''''))))))) -> PROPER(g(h(g(f(g(X''''''''''))))))
PROPER(g(h(g(f(f(X'''''''')))))) -> PROPER(h(g(f(f(X'''''''')))))
PROPER(h(g(h(g(f(f(X''''''''''))))))) -> PROPER(g(h(g(f(f(X''''''''''))))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(g(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(h(g(f(h(g(f(h(X'''''''''''')))))))) -> PROPER(g(f(h(g(f(h(X'''''''''''')))))))
PROPER(g(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(g(f(h(g(f(g(X'''''''''''')))))))) -> PROPER(g(f(h(g(f(g(X'''''''''''')))))))
PROPER(g(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(g(f(h(g(f(f(X'''''''''''')))))))) -> PROPER(g(f(h(g(f(f(X'''''''''''')))))))
PROPER(h(g(f(h(g(h(X''''''''''))))))) -> PROPER(g(f(h(g(h(X''''''''''))))))
PROPER(h(g(f(h(g(g(X''''''''''))))))) -> PROPER(g(f(h(g(g(X''''''''''))))))
PROPER(h(g(f(h(f(h(X''''''''''))))))) -> PROPER(g(f(h(f(h(X''''''''''))))))
PROPER(h(g(f(h(f(g(X''''''''''))))))) -> PROPER(g(f(h(f(g(X''''''''''))))))
PROPER(h(g(f(h(f(f(X''''''''''))))))) -> PROPER(g(f(h(f(f(X''''''''''))))))
PROPER(h(g(f(h(h(X'''''''')))))) -> PROPER(g(f(h(h(X'''''''')))))
PROPER(h(f(h(g(f(h(X''''''''''))))))) -> PROPER(f(h(g(f(h(X''''''''''))))))
PROPER(h(g(f(g(f(h(X''''''''''))))))) -> PROPER(g(f(g(f(h(X''''''''''))))))
PROPER(h(g(f(g(f(g(X''''''''''))))))) -> PROPER(g(f(g(f(g(X''''''''''))))))
PROPER(h(g(f(g(f(f(X''''''''''))))))) -> PROPER(g(f(g(f(f(X''''''''''))))))
PROPER(h(g(f(g(h(X'''''''')))))) -> PROPER(g(f(g(h(X'''''''')))))
PROPER(h(f(h(g(f(g(X''''''''''))))))) -> PROPER(f(h(g(f(g(X''''''''''))))))
PROPER(h(g(f(f(h(X'''''''')))))) -> PROPER(g(f(f(h(X'''''''')))))
PROPER(h(g(f(f(g(X'''''''')))))) -> PROPER(g(f(f(g(X'''''''')))))
PROPER(h(g(f(f(f(X'''''''')))))) -> PROPER(g(f(f(f(X'''''''')))))
PROPER(h(f(h(g(f(f(X''''''''''))))))) -> PROPER(f(h(g(f(f(X''''''''''))))))
PROPER(h(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(h(f(h(X''''''))))) -> PROPER(h(f(h(X''''''))))
PROPER(h(g(h(f(h(X'''''''')))))) -> PROPER(g(h(f(h(X'''''''')))))
PROPER(g(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(g(h(f(g(X'''''''')))))) -> PROPER(g(h(f(g(X'''''''')))))
PROPER(h(g(h(f(f(X'''''''')))))) -> PROPER(g(h(f(f(X'''''''')))))
PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(g(f(h(g(h(X'''''''')))))) -> PROPER(f(h(g(h(X'''''''')))))
PROPER(g(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(g(g(f(h(X''''''))))) -> PROPER(g(f(h(X''''''))))
PROPER(h(g(g(f(h(X'''''''')))))) -> PROPER(g(g(f(h(X'''''''')))))
PROPER(g(g(f(g(X''''''))))) -> PROPER(g(f(g(X''''''))))
PROPER(h(g(g(f(g(X'''''''')))))) -> PROPER(g(g(f(g(X'''''''')))))
PROPER(g(g(f(f(X''''''))))) -> PROPER(g(f(f(X''''''))))
PROPER(h(g(g(f(f(X'''''''')))))) -> PROPER(g(g(f(f(X'''''''')))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(h(f(h(g(g(X'''''''')))))) -> PROPER(f(h(g(g(X'''''''')))))
PROPER(h(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(h(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(h(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(g(f(h(f(h(X'''''''')))))) -> PROPER(f(h(f(h(X'''''''')))))
PROPER(h(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(f(h(f(g(X'''''''')))))) -> PROPER(f(h(f(g(X'''''''')))))
PROPER(g(f(h(f(f(X'''''''')))))) -> PROPER(f(h(f(f(X'''''''')))))
PROPER(g(f(h(h(X''''''))))) -> PROPER(f(h(h(X''''''))))
PROPER(g(f(g(f(h(X'''''''')))))) -> PROPER(f(g(f(h(X'''''''')))))
PROPER(g(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(g(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(f(g(X'''''''')))))) -> PROPER(f(g(f(g(X'''''''')))))
PROPER(g(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(g(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(f(g(f(f(X'''''''')))))) -> PROPER(f(g(f(f(X'''''''')))))
PROPER(h(f(g(h(X''''''))))) -> PROPER(f(g(h(X''''''))))
PROPER(h(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(h(f(g(X''''''))))) -> PROPER(h(f(g(X''''''))))
PROPER(h(f(f(h(X''''''))))) -> PROPER(f(f(h(X''''''))))
PROPER(h(f(f(g(X''''''))))) -> PROPER(f(f(g(X''''''))))
PROPER(g(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(f(f(f(X''''''))))) -> PROPER(f(f(f(X''''''))))
PROPER(h(h(f(f(X''''''))))) -> PROPER(h(f(f(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(f(g(g(X''''''))))) -> PROPER(f(g(g(X''''''))))
PROPER(h(g(f(g(g(X'''''''')))))) -> PROPER(g(f(g(g(X'''''''')))))
PROPER(h(h(g(f(g(X'''''''')))))) -> PROPER(h(g(f(g(X'''''''')))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 61
Polynomial Ordering
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

g(ok(X)) -> ok(g(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  1 + x1  
  POL(PROPER(x1))=  1 + x1  
  POL(h(x1))=  x1  
  POL(mark(x1))=  0  
  POL(ok(x1))=  0  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 62
Dependency Graph
       →DP Problem 6
Nar


Dependency Pairs:

PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 63
Polynomial Ordering
       →DP Problem 6
Nar


Dependency Pair:

PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  1 + x1  
  POL(h(x1))=  1 + x1  
  POL(mark(x1))=  0  
  POL(ok(x1))=  0  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
           →DP Problem 40
FwdInst
             ...
               →DP Problem 64
Dependency Graph
       →DP Problem 6
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(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
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Narrowing Transformation


Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(mark(X)) -> TOP(proper(X))
three new Dependency Pairs are created:

TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(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 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Narrowing Transformation


Dependency Pairs:

TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(ok(X)) -> TOP(active(X))
three new Dependency Pairs are created:

TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(ok(f(X''))) -> TOP(f(active(X'')))
TOP(ok(h(X''))) -> TOP(h(active(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 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Nar
             ...
               →DP Problem 66
Narrowing Transformation


Dependency Pairs:

TOP(ok(h(X''))) -> TOP(h(active(X'')))
TOP(ok(f(X''))) -> TOP(f(active(X'')))
TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(mark(f(X''))) -> TOP(f(proper(X'')))
three new Dependency Pairs are created:

TOP(mark(f(f(X')))) -> TOP(f(f(proper(X'))))
TOP(mark(f(g(X')))) -> TOP(f(g(proper(X'))))
TOP(mark(f(h(X')))) -> TOP(f(h(proper(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 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Nar
             ...
               →DP Problem 67
Narrowing Transformation


Dependency Pairs:

TOP(mark(f(h(X')))) -> TOP(f(h(proper(X'))))
TOP(mark(f(g(X')))) -> TOP(f(g(proper(X'))))
TOP(ok(f(X''))) -> TOP(f(active(X'')))
TOP(mark(f(f(X')))) -> TOP(f(f(proper(X'))))
TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(ok(h(X''))) -> TOP(h(active(X'')))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(mark(g(X''))) -> TOP(g(proper(X'')))
three new Dependency Pairs are created:

TOP(mark(g(f(X')))) -> TOP(g(f(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(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 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Nar
             ...
               →DP Problem 68
Narrowing Transformation


Dependency Pairs:

TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(mark(g(f(X')))) -> TOP(g(f(proper(X'))))
TOP(mark(f(g(X')))) -> TOP(f(g(proper(X'))))
TOP(ok(h(X''))) -> TOP(h(active(X'')))
TOP(ok(f(X''))) -> TOP(f(active(X'')))
TOP(mark(f(f(X')))) -> TOP(f(f(proper(X'))))
TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(mark(f(h(X')))) -> TOP(f(h(proper(X'))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(mark(h(X''))) -> TOP(h(proper(X'')))
three new Dependency Pairs are created:

TOP(mark(h(f(X')))) -> TOP(h(f(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(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 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Nar
             ...
               →DP Problem 69
Narrowing Transformation


Dependency Pairs:

TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(h(f(X')))) -> TOP(h(f(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(mark(g(f(X')))) -> TOP(g(f(proper(X'))))
TOP(mark(f(h(X')))) -> TOP(f(h(proper(X'))))
TOP(mark(f(g(X')))) -> TOP(f(g(proper(X'))))
TOP(ok(h(X''))) -> TOP(h(active(X'')))
TOP(ok(f(X''))) -> TOP(f(active(X'')))
TOP(mark(f(f(X')))) -> TOP(f(f(proper(X'))))
TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(ok(f(X''))) -> TOP(f(active(X'')))
three new Dependency Pairs are created:

TOP(ok(f(f(X')))) -> TOP(f(mark(g(h(f(X'))))))
TOP(ok(f(f(X')))) -> TOP(f(f(active(X'))))
TOP(ok(f(h(X')))) -> TOP(f(h(active(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 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Nar
             ...
               →DP Problem 70
Rewriting Transformation


Dependency Pairs:

TOP(ok(f(h(X')))) -> TOP(f(h(active(X'))))
TOP(ok(f(f(X')))) -> TOP(f(f(active(X'))))
TOP(ok(f(f(X')))) -> TOP(f(mark(g(h(f(X'))))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(h(f(X')))) -> TOP(h(f(proper(X'))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(mark(g(f(X')))) -> TOP(g(f(proper(X'))))
TOP(mark(f(h(X')))) -> TOP(f(h(proper(X'))))
TOP(mark(f(g(X')))) -> TOP(f(g(proper(X'))))
TOP(ok(h(X''))) -> TOP(h(active(X'')))
TOP(mark(f(f(X')))) -> TOP(f(f(proper(X'))))
TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(ok(f(f(X')))) -> TOP(f(mark(g(h(f(X'))))))
one new Dependency Pair is created:

TOP(ok(f(f(X')))) -> TOP(mark(f(g(h(f(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 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Nar
             ...
               →DP Problem 71
Narrowing Transformation


Dependency Pairs:

TOP(ok(f(f(X')))) -> TOP(mark(f(g(h(f(X'))))))
TOP(ok(f(f(X')))) -> TOP(f(f(active(X'))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(h(f(X')))) -> TOP(h(f(proper(X'))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(mark(g(f(X')))) -> TOP(g(f(proper(X'))))
TOP(mark(f(h(X')))) -> TOP(f(h(proper(X'))))
TOP(mark(f(g(X')))) -> TOP(f(g(proper(X'))))
TOP(ok(h(X''))) -> TOP(h(active(X'')))
TOP(mark(f(f(X')))) -> TOP(f(f(proper(X'))))
TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(ok(f(h(X')))) -> TOP(f(h(active(X'))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(ok(h(X''))) -> TOP(h(active(X'')))
three new Dependency Pairs are created:

TOP(ok(h(f(X')))) -> TOP(h(mark(g(h(f(X'))))))
TOP(ok(h(f(X')))) -> TOP(h(f(active(X'))))
TOP(ok(h(h(X')))) -> TOP(h(h(active(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 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Nar
             ...
               →DP Problem 72
Rewriting Transformation


Dependency Pairs:

TOP(ok(h(h(X')))) -> TOP(h(h(active(X'))))
TOP(ok(h(f(X')))) -> TOP(h(f(active(X'))))
TOP(ok(h(f(X')))) -> TOP(h(mark(g(h(f(X'))))))
TOP(ok(f(h(X')))) -> TOP(f(h(active(X'))))
TOP(ok(f(f(X')))) -> TOP(f(f(active(X'))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(h(f(X')))) -> TOP(h(f(proper(X'))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(mark(g(f(X')))) -> TOP(g(f(proper(X'))))
TOP(mark(f(h(X')))) -> TOP(f(h(proper(X'))))
TOP(mark(f(g(X')))) -> TOP(f(g(proper(X'))))
TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(mark(f(f(X')))) -> TOP(f(f(proper(X'))))
TOP(ok(f(f(X')))) -> TOP(mark(f(g(h(f(X'))))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(ok(h(f(X')))) -> TOP(h(mark(g(h(f(X'))))))
one new Dependency Pair is created:

TOP(ok(h(f(X')))) -> TOP(mark(h(g(h(f(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 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Nar
             ...
               →DP Problem 73
Polynomial Ordering


Dependency Pairs:

TOP(ok(h(f(X')))) -> TOP(mark(h(g(h(f(X'))))))
TOP(ok(h(f(X')))) -> TOP(h(f(active(X'))))
TOP(ok(f(f(X')))) -> TOP(mark(f(g(h(f(X'))))))
TOP(ok(f(h(X')))) -> TOP(f(h(active(X'))))
TOP(ok(f(f(X')))) -> TOP(f(f(active(X'))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(h(f(X')))) -> TOP(h(f(proper(X'))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(mark(g(f(X')))) -> TOP(g(f(proper(X'))))
TOP(mark(f(h(X')))) -> TOP(f(h(proper(X'))))
TOP(mark(f(g(X')))) -> TOP(f(g(proper(X'))))
TOP(mark(f(f(X')))) -> TOP(f(f(proper(X'))))
TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(ok(h(h(X')))) -> TOP(h(h(active(X'))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(active(x1))=  0  
  POL(proper(x1))=  0  
  POL(g(x1))=  0  
  POL(h(x1))=  0  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(TOP(x1))=  x1  
  POL(f(x1))=  1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar
           →DP Problem 65
Nar
             ...
               →DP Problem 74
Polynomial Ordering


Dependency Pairs:

TOP(ok(h(f(X')))) -> TOP(mark(h(g(h(f(X'))))))
TOP(ok(h(f(X')))) -> TOP(h(f(active(X'))))
TOP(ok(f(f(X')))) -> TOP(mark(f(g(h(f(X'))))))
TOP(ok(f(h(X')))) -> TOP(f(h(active(X'))))
TOP(ok(f(f(X')))) -> TOP(f(f(active(X'))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(h(f(X')))) -> TOP(h(f(proper(X'))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(mark(g(f(X')))) -> TOP(g(f(proper(X'))))
TOP(mark(f(h(X')))) -> TOP(f(h(proper(X'))))
TOP(mark(f(g(X')))) -> TOP(f(g(proper(X'))))
TOP(mark(f(f(X')))) -> TOP(f(f(proper(X'))))
TOP(ok(h(h(X')))) -> TOP(h(h(active(X'))))


Rules:


active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))