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:

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
active(f(X1, X2, X3)) → f(X1, active(X2), X3)
f(X1, mark(X2), X3) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) QTRSToCSRProof (EQUIVALENT transformation)

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

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
active(f(X1, X2, X3)) → f(X1, active(X2), X3)
f(X1, mark(X2), X3) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top, active: active, mark: mark, ok: ok, proper: proper
The replacement map contains the following entries:

f: {2}
a: empty set
b: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).

(2) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

f(a, X, X) → f(X, b, b)
b → a

The replacement map contains the following entries:

f: {2}
a: empty set
b: empty set

(3) CSDependencyPairsProof (EQUIVALENT transformation)

Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.

(4) Obligation:

Q-restricted context-sensitive dependency pair problem:
For all symbols f in {f, F} we have µ(f) = {2}.

The ordinary context-sensitive dependency pairs DP_o are:

F(a, X, X) → F(X, b, b)
F(a, X, X) → B

The TRS R consists of the following rules:

f(a, X, X) → f(X, b, b)
b → a

Q is empty.

(5) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 1 less node.

(6) Obligation:

Q-restricted context-sensitive dependency pair problem:
For all symbols f in {f, F} we have µ(f) = {2}.

The TRS P consists of the following rules:

F(a, X, X) → F(X, b, b)

The TRS R consists of the following rules:

f(a, X, X) → f(X, b, b)
b → a

Q is empty.

(7) QCSUsableRulesProof (EQUIVALENT transformation)

The following rules are not useable [DA_EMMES] and can be deleted:

f(a, x0, x0) → f(x0, b, b)

(8) Obligation:

Q-restricted context-sensitive dependency pair problem:
For all symbols f in {F} we have µ(f) = {2}.

The TRS P consists of the following rules:

F(a, X, X) → F(X, b, b)

The TRS R consists of the following rules:

b → a

Q is empty.

(9) QCSDPInstantiationProcessor (EQUIVALENT transformation)

Using the Context-Sensitive Instantiation[LPAR08,DA_EMMES] Processor
the pair F(a, X, X) → F(X, b, b)
was transformed to the following new pairs:

F(a, b, b) → F(b, b, b)

(10) Obligation:

Q-restricted context-sensitive dependency pair problem:

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

b → a

Q is empty.

(11) PIsEmptyProof (EQUIVALENT transformation)

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

(12) TRUE