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, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(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)
proper(c) → ok(c)
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, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(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)
proper(c) → ok(c)
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: {3}
a: empty set
b: empty set
c: 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, b, X) → f(X, X, X)
c → a
c → b

The replacement map contains the following entries:

f: {3}
a: empty set
b: empty set
c: empty set

(3) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

c → a
c → b

Used ordering:
Polynomial interpretation [POLO]:

POL(a) = 1   
POL(b) = 1   
POL(c) = 2   
POL(f(x₁, x₂, x₃)) = x₃   

(4) Obligation:

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

f(a, b, X) → f(X, X, X)

The replacement map contains the following entries:

f: {3}
a: empty set
b: empty set

(5) CSRInnermostProof (EQUIVALENT transformation)

The CSR is orthogonal. By [CS_Inn] we can switch to innermost.

(6) Obligation:

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

f(a, b, X) → f(X, X, X)

The replacement map contains the following entries:

f: {3}
a: empty set
b: empty set

Innermost Strategy.

(7) CSDependencyPairsProof (EQUIVALENT transformation)

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

(8) Obligation:

Q-restricted context-sensitive dependency pair problem:
For all symbols f in {f, F} we have µ(f) = {3}.

The ordinary context-sensitive dependency pairs DP_o are:

F(a, b, X) → F(X, X, X)

The TRS R consists of the following rules:

f(a, b, X) → f(X, X, X)

The set Q consists of the following terms:

f(a, b, x0)

(9) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
The rules F(a, b, z0) → F(z0, z0, z0) and F(a, b, x0) → F(x0, x0, x0) form no chain, because ECap^µ(F(z0, z0, z0)) = F(z0, z0, z0) does not unify with F(a, b, x0).

(10) TRUE