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))

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
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →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))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(G(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 7
             ↳Dependency Graph
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →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))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →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))





The following dependency pair can be strictly oriented:

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


There are no usable rules 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))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 8
             ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar


Dependency Pair:

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))





The following dependency pair can be strictly oriented:

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


There are no usable rules 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))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 8
             ↳Polo
             ...
               →DP Problem 9
                 ↳Dependency Graph
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →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))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polynomial Ordering
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →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))





The following dependency pair can be strictly oriented:

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


There are no usable rules 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))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 10
             ↳Polynomial Ordering
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar


Dependency Pair:

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))





The following dependency pair can be strictly oriented:

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


There are no usable rules 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))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 10
             ↳Polo
             ...
               →DP Problem 11
                 ↳Dependency Graph
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →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))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polynomial Ordering
       →DP Problem 5
         ↳Polo
       →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))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
           →DP Problem 12
             ↳Polynomial Ordering
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar


Dependency Pair:

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))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
           →DP Problem 12
             ↳Polo
             ...
               →DP Problem 13
                 ↳Dependency Graph
       →DP Problem 5
         ↳Polo
       →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))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polynomial Ordering
       →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))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 14
             ↳Polynomial Ordering
       →DP Problem 6
         ↳Nar


Dependency Pairs:

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))





The following dependency pair can be strictly oriented:

PROPER(g(X)) -> PROPER(X)


There are no usable rules 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(PROPER(x1))=  x1  
  POL(f(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 14
             ↳Polo
             ...
               →DP Problem 15
                 ↳Polynomial Ordering
       →DP Problem 6
         ↳Nar


Dependency Pair:

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))





The following dependency pair can be strictly oriented:

PROPER(f(X)) -> PROPER(X)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 14
             ↳Polo
             ...
               →DP Problem 16
                 ↳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))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →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))





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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳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))





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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 18
                 ↳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))





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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 19
                 ↳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))





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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 20
                 ↳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))





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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 21
                 ↳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))





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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 22
                 ↳Narrowing 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))





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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 23
                 ↳Polynomial Ordering


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





The following dependency pair can be strictly oriented:

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


Additionally, the following usable rules 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))
g(ok(X)) -> ok(g(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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 24
                 ↳Polynomial Ordering


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





The following dependency pairs can be strictly oriented:

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'))))


Additionally, the following usable rules 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))
g(ok(X)) -> ok(g(X))


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 25
                 ↳Polynomial Ordering


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





The following dependency pair can be strictly oriented:

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


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

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


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 26
                 ↳Polynomial Ordering


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(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(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(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))





The following dependency pairs can be strictly oriented:

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(f(X')))) -> TOP(f(f(active(X'))))
TOP(ok(f(h(X')))) -> TOP(f(h(active(X'))))


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

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


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 27
                 ↳Polynomial Ordering


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(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'))))


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))





The following dependency pair can be strictly oriented:

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


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

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


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 28
                 ↳Polynomial Ordering


Dependency Pairs:

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


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))





The following dependency pairs can be strictly oriented:

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


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

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


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Nar
           →DP Problem 17
             ↳Nar
             ...
               →DP Problem 29
                 ↳Dependency Graph


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))





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:05 minutes