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

(0) Obligation:

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

a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)

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:

A__F(g(X), Y) → A__F(mark(X), f(g(X), Y))
A__F(g(X), Y) → MARK(X)
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)
MARK(g(X)) → MARK(X)

The TRS R consists of the following rules:

a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(g(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:
A__F(x0, x1, x2)  =  A__F(x0, x1)
MARK(x0, x1)  =  MARK(x0)

Tags:
A__F has argument tags [0,1,1] and root tag 0
MARK has argument tags [1,0] and root tag 0

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A__F(x₁, x₂)) = 1 + x₁ + x₂   
POL(MARK(x₁)) = 1 + x₁   
POL(a__f(x₁, x₂)) = x₁   
POL(f(x₁, x₂)) = x₁   
POL(g(x₁)) = 1 + x₁   
POL(mark(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
a__f(X1, X2) → f(X1, X2)

(4) Obligation:

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

A__F(g(X), Y) → A__F(mark(X), f(g(X), Y))
A__F(g(X), Y) → MARK(X)
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__F(g(X), Y) → MARK(X)
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__F(x0, x1, x2)  =  A__F(x0, x1)
MARK(x0, x1)  =  MARK(x0, x1)

Tags:
A__F has argument tags [2,0,0] and root tag 1
MARK has argument tags [1,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A__F(x₁, x₂)) = 1 + x₁   
POL(MARK(x₁)) = 0   
POL(a__f(x₁, x₂)) = 1 + x₁   
POL(f(x₁, x₂)) = 1 + x₁   
POL(g(x₁)) = x₁   
POL(mark(x₁)) = x₁   

The following usable rules [FROCOS05] were oriented:

mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
a__f(X1, X2) → f(X1, X2)

(6) Obligation:

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

A__F(g(X), Y) → A__F(mark(X), f(g(X), Y))

The TRS R consists of the following rules:

a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__F(g(X), Y) → A__F(mark(X), f(g(X), Y))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__F(x0, x1, x2)  =  A__F(x1, x2)

Tags:
A__F has argument tags [3,2,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A__F(x₁, x₂)) = x₂   
POL(a__f(x₁, x₂)) = x₁   
POL(f(x₁, x₂)) = x₁   
POL(g(x₁)) = 1 + x₁   
POL(mark(x₁)) = x₁   

The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

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

a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)

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

(9) PisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE