Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
    ↳5 QDP
        ↳6 QDPOrderProof (⇔)
        ↳7 QDP
        ↳8 QDPOrderProof (⇔)
        ↳9 QDP
        ↳10 PisEmptyProof (⇔)
        ↳11 TRUE
    ↳12 QDP
        ↳13 QDPOrderProof (⇔)
        ↳14 QDP
        ↳15 QDPOrderProof (⇔)
        ↳16 QDP
        ↳17 PisEmptyProof (⇔)
        ↳18 TRUE
    ↳19 QDP
        ↳20 QDPOrderProof (⇔)
        ↳21 QDP
        ↳22 QDPOrderProof (⇔)
        ↳23 QDP
        ↳24 PisEmptyProof (⇔)
        ↳25 TRUE
    ↳26 QDP
        ↳27 QDPOrderProof (⇔)
        ↳28 QDP
        ↳29 QDPOrderProof (⇔)
        ↳30 QDP
        ↳31 QDPOrderProof (⇔)
        ↳32 QDP
        ↳33 PisEmptyProof (⇔)
        ↳34 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(f(X)) → MARK(g(h(f(X))))
ACTIVE(f(X)) → G(h(f(X)))
ACTIVE(f(X)) → H(f(X))
MARK(f(X)) → ACTIVE(f(mark(X)))
MARK(f(X)) → F(mark(X))
MARK(f(X)) → MARK(X)
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
MARK(h(X)) → H(mark(X))
MARK(h(X)) → MARK(X)
F(mark(X)) → F(X)
F(active(X)) → F(X)
G(mark(X)) → G(X)
G(active(X)) → G(X)
H(mark(X)) → H(X)
H(active(X)) → H(X)

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 4 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(active(X)) → H(X)
H(mark(X)) → H(X)

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


H(active(X)) → H(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
H(x1)  =  H(x1)

Tags:
H has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

H(mark(X)) → H(X)

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


H(mark(X)) → H(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
H(x1)  =  H(x1)

Tags:
H has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

G(active(X)) → G(X)
G(mark(X)) → G(X)

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


G(active(X)) → G(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
G(x1)  =  G(x1)

Tags:
G has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

G(mark(X)) → G(X)

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


G(mark(X)) → G(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
G(x1)  =  G(x1)

Tags:
G has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(18) TRUE

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(active(X)) → F(X)
F(mark(X)) → F(X)

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F(active(X)) → F(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(x1)  =  F(x1)

Tags:
F has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
active1: multiset


The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(mark(X)) → F(X)

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F(mark(X)) → F(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(x1)  =  F(x1)

Tags:
F has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(24) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(25) TRUE

(26) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(X)) → MARK(g(h(f(X))))
MARK(f(X)) → MARK(X)
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
MARK(h(X)) → MARK(X)

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(f(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x1)  =  MARK(x1)
ACTIVE(x1)  =  ACTIVE(x1)

Tags:
MARK has tags [1]
ACTIVE has tags [1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
f(x1)  =  f(x1)
mark(x1)  =  x1
g(x1)  =  x1
h(x1)  =  x1
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
f1: multiset


The following usable rules [FROCOS05] were oriented:

mark(f(X)) → active(f(mark(X)))
active(f(X)) → mark(g(h(f(X))))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(active(X)) → f(X)
f(mark(X)) → f(X)
h(active(X)) → h(X)
h(mark(X)) → h(X)
g(active(X)) → g(X)
g(mark(X)) → g(X)

(28) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(X)) → MARK(g(h(f(X))))
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
MARK(h(X)) → MARK(X)

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(h(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x1)  =  MARK(x1)
ACTIVE(x1)  =  ACTIVE(x1)

Tags:
MARK has tags [1]
ACTIVE has tags [1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
f(x1)  =  f
mark(x1)  =  x1
g(x1)  =  g
h(x1)  =  h(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[f, g, h1]

Status:
f: []
g: []
h1: [1]


The following usable rules [FROCOS05] were oriented:

mark(f(X)) → active(f(mark(X)))
active(f(X)) → mark(g(h(f(X))))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(active(X)) → f(X)
f(mark(X)) → f(X)
h(active(X)) → h(X)
h(mark(X)) → h(X)
g(active(X)) → g(X)
g(mark(X)) → g(X)

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(X)) → MARK(g(h(f(X))))
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))

The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(f(X)) → ACTIVE(f(mark(X)))
ACTIVE(f(X)) → MARK(g(h(f(X))))
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x1)  =  MARK(x1)
ACTIVE(x1)  =  ACTIVE(x1)

Tags:
MARK has tags [1]
ACTIVE has tags [0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
f(x1)  =  f
mark(x1)  =  mark(x1)
g(x1)  =  g
h(x1)  =  h
active(x1)  =  active

Recursive path order with status [RPO].
Quasi-Precedence:
[mark1, h] > active > f > g

Status:
f: multiset
mark1: multiset
g: []
h: []
active: multiset


The following usable rules [FROCOS05] were oriented:

f(active(X)) → f(X)
f(mark(X)) → f(X)
h(active(X)) → h(X)
h(mark(X)) → h(X)
g(active(X)) → g(X)
g(mark(X)) → g(X)

(32) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(f(X)) → mark(g(h(f(X))))
mark(f(X)) → active(f(mark(X)))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(33) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(34) TRUE