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))

Innermost 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

         ↳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:

     ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 8

                 ↳Argument Filtering and 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 that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

G(x₁) -> G(x₁)
ok(x₁) -> ok(x₁)

     ↳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.

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 10

             ↳FwdInst

...

               →DP Problem 16

                 ↳Argument Filtering and 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(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''''''))))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
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

         ↳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 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.

     ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 18

             ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 18

             ↳FwdInst

...

               →DP Problem 19

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 18

             ↳FwdInst

...

               →DP Problem 20

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 18

             ↳FwdInst

...

               →DP Problem 21

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 18

             ↳FwdInst

...

               →DP Problem 22

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 18

             ↳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(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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 18

             ↳FwdInst

...

               →DP Problem 24

                 ↳Argument Filtering and 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(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''''''))))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
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

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 18

             ↳FwdInst

...

               →DP Problem 25

                 ↳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.

     ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

           →DP Problem 26

             ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

           →DP Problem 26

             ↳FwdInst

...

               →DP Problem 27

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

           →DP Problem 26

             ↳FwdInst

...

               →DP Problem 28

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

           →DP Problem 26

             ↳FwdInst

...

               →DP Problem 29

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

           →DP Problem 26

             ↳FwdInst

...

               →DP Problem 30

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

           →DP Problem 26

             ↳FwdInst

...

               →DP Problem 31

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

           →DP Problem 26

             ↳FwdInst

...

               →DP Problem 32

                 ↳Argument Filtering and 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(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''''''))))

The following usable rules for innermost can be oriented:

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

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

           →DP Problem 26

             ↳FwdInst

...

               →DP Problem 33

                 ↳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.

     ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 35

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 36

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 37

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 38

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 39

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 40

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 41

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 42

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 43

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 44

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 45

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 46

                 ↳Forward Instantiation Transformation

       →DP Problem 6

         ↳Nar

Dependency Pairs:

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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 47

                 ↳Forward Instantiation Transformation

       →DP Problem 6

         ↳Nar

Dependency Pairs:

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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 48

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 49

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 50

                 ↳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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳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(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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳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(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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 53

                 ↳Argument Filtering and 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(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'''''''')))))

The following usable rules for innermost can be oriented:

g(ok(X)) -> ok(g(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))

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳FwdInst

       →DP Problem 5

         ↳FwdInst

           →DP Problem 34

             ↳FwdInst

...

               →DP Problem 54

                 ↳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.

     ↳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:

     ↳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 55

             ↳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:

     ↳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 55

             ↳Nar

...

               →DP Problem 56

                 ↳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:

     ↳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 55

             ↳Nar

...

               →DP Problem 57

                 ↳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:

     ↳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 55

             ↳Nar

...

               →DP Problem 58

                 ↳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:

     ↳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 55

             ↳Nar

...

               →DP Problem 59

                 ↳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:

     ↳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 55

             ↳Nar

...

               →DP Problem 60

                 ↳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:

     ↳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 55

             ↳Nar

...

               →DP Problem 61

                 ↳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:

     ↳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 55

             ↳Nar

...

               →DP Problem 62

                 ↳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:

     ↳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 55

             ↳Nar

...

               →DP Problem 63

                 ↳Argument Filtering and 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 pairs can be strictly oriented:

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

The following usable rules for innermost 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))
active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(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))

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

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

     ↳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 55

             ↳Nar

...

               →DP Problem 64

                 ↳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))

Strategy:

innermost

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 for innermost can be oriented:

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

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

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

     ↳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 55

             ↳Nar

...

               →DP Problem 65

                 ↳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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:27 minutes