Termination Proof using AProVE

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.

     ↳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.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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 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.
Used Argument Filtering System:

G(x₁) -> G(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 7

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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 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.
Used Argument Filtering System:

H(x₁) -> H(x₁)
ok(x₁) -> ok(x₁)
mark(x₁) -> mark(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 8

             ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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 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.
Used Argument Filtering System:

H(x₁) -> H(x₁)
mark(x₁) -> mark(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 8

             ↳AFS

...

               →DP Problem 9

                 ↳Dependency Graph

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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(mark(X)) -> F(X)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(ok(x₁)) = x₁

POL(F(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

F(x₁) -> F(x₁)
mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 10

             ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

Dependency Pair:

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))

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ok(x₁)) = 1 + x₁

POL(F(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

F(x₁) -> F(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 10

             ↳AFS

...

               →DP Problem 11

                 ↳Dependency Graph

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →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 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.
Used Argument Filtering System:

ACTIVE(x₁) -> ACTIVE(x₁)
h(x₁) -> h(x₁)
f(x₁) -> f(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 12

             ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →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 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.
Used Argument Filtering System:

ACTIVE(x₁) -> ACTIVE(x₁)
f(x₁) -> f(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 12

             ↳AFS

...

               →DP Problem 13

                 ↳Dependency Graph

       →DP Problem 5

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳Argument Filtering and 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(f(X)) -> PROPER(X)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(g(x₁)) = x₁

POL(PROPER(x₁)) = x₁

POL(h(x₁)) = x₁

POL(f(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
f(x₁) -> f(x₁)
h(x₁) -> h(x₁)
g(x₁) -> g(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

           →DP Problem 14

             ↳Argument Filtering and Ordering

       →DP Problem 6

         ↳Nar

Dependency Pairs:

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

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
h(x₁) -> h(x₁)
g(x₁) -> g(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

           →DP Problem 14

             ↳AFS

...

               →DP Problem 15

                 ↳Argument Filtering and Ordering

       →DP Problem 6

         ↳Nar

Dependency Pair:

PROPER(g(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 AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(g(x₁)) = 1 + x₁

POL(PROPER(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
g(x₁) -> g(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

           →DP Problem 14

             ↳AFS

...

               →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 17

             ↳Nar

...

               →DP Problem 23

                 ↳Argument Filtering and 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 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(f(X')))) -> TOP(f(mark(g(h(f(X'))))))
TOP(ok(f(X''))) -> TOP(mark(g(h(f(X'')))))
TOP(ok(f(h(X')))) -> TOP(f(h(active(X'))))

The following usable rules w.r.t. to the 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)) = x₁

POL(h(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(TOP(x₁)) = 1 + x₁

POL(ok(x₁)) = 1 + x₁

POL(f(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

TOP(x₁) -> TOP(x₁)
ok(x₁) -> ok(x₁)
h(x₁) -> h(x₁)
f(x₁) -> f(x₁)
active(x₁) -> active(x₁)
mark(x₁) -> mark(x₁)
g(x₁) -> g(x₁)
proper(x₁) -> proper(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 17

             ↳Nar

...

               →DP Problem 24

                 ↳Argument Filtering and 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(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'))))

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(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'))))

The following usable rules w.r.t. to the AFS can be oriented:

proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
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(proper(x₁)) = x₁

POL(g(x₁)) = x₁

POL(h(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

POL(TOP(x₁)) = 1 + x₁

POL(ok(x₁)) = x₁

POL(f(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

TOP(x₁) -> TOP(x₁)
mark(x₁) -> mark(x₁)
f(x₁) -> f(x₁)
proper(x₁) -> proper(x₁)
g(x₁) -> g(x₁)
h(x₁) -> h(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 17

             ↳Nar

...

               →DP Problem 25

                 ↳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:
0:03 minutes

POL(mark(x₁))	= x₁
POL(H(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

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

POL(g(x₁))	= x₁
POL(PROPER(x₁))	= x₁
POL(h(x₁))	= x₁
POL(f(x₁))	= 1 + x₁