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:

f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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:

F(a, f(b, x)) → F(a, f(a, f(a, x)))
F(a, f(b, x)) → F(a, f(a, x))
F(a, f(b, x)) → F(a, x)
F(b, f(a, x)) → F(b, f(b, f(b, x)))
F(b, f(a, x)) → F(b, f(b, x))
F(b, f(a, x)) → F(b, x)

The TRS R consists of the following rules:

f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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 2 SCCs.

(4) Complex Obligation (AND)

(5) Obligation:

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

F(b, f(a, x)) → F(b, f(b, x))
F(b, f(a, x)) → F(b, f(b, f(b, x)))
F(b, f(a, x)) → F(b, x)

The TRS R consists of the following rules:

f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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.

F(b, f(a, x)) → F(b, f(b, x))
F(b, f(a, x)) → F(b, 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, x2)  =  F(x1, x2)

Tags:
F has tags [0,1]

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

POL(a) = 1   
POL(b) = 0   
POL(f(x₁, x₂)) = 1 + x₁ + x₂   

The following usable rules [FROCOS05] were oriented:

f(b, f(a, x)) → f(b, f(b, f(b, x)))

(7) Obligation:

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

F(b, f(a, x)) → F(b, f(b, f(b, x)))

The TRS R consists of the following rules:

f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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.

F(b, f(a, x)) → F(b, f(b, f(b, 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, x2)  =  F(x2)

Tags:
F has tags [1,1]

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

POL(a) = 1   
POL(b) = 0   
POL(f(x₁, x₂)) = x₁ + x₂   

The following usable rules [FROCOS05] were oriented:

f(b, f(a, x)) → f(b, f(b, f(b, x)))

(9) Obligation:

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

f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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:

F(a, f(b, x)) → F(a, f(a, x))
F(a, f(b, x)) → F(a, f(a, f(a, x)))
F(a, f(b, x)) → F(a, x)

The TRS R consists of the following rules:

f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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.

F(a, f(b, x)) → F(a, f(a, x))
F(a, f(b, x)) → F(a, 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, x2)  =  F(x1, x2)

Tags:
F has tags [0,1]

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

POL(a) = 0   
POL(b) = 1   
POL(f(x₁, x₂)) = 1 + x₁ + x₂   

The following usable rules [FROCOS05] were oriented:

f(a, f(b, x)) → f(a, f(a, f(a, x)))

(14) Obligation:

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

F(a, f(b, x)) → F(a, f(a, f(a, x)))

The TRS R consists of the following rules:

f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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.

F(a, f(b, x)) → F(a, f(a, f(a, 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, x2)  =  F(x2)

Tags:
F has tags [1,1]

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

POL(a) = 0   
POL(b) = 1   
POL(f(x₁, x₂)) = x₁ + x₂   

The following usable rules [FROCOS05] were oriented:

f(a, f(b, x)) → f(a, f(a, f(a, x)))

(16) Obligation:

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

f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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