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(X, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
g(ok(X)) → ok(g(X))
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(X, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
g(ok(X)) → ok(g(X))
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: empty set
g: {1}
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(X, g(X), Y) → f(Y, Y, Y)
g(b) → c
b → c

The replacement map contains the following entries:

f: empty set
g: {1}
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:

g(b) → c
b → c

Used ordering:
Polynomial interpretation [POLO]:

POL(b) = 2   
POL(c) = 1   
POL(f(x₁, x₂, x₃)) = 0   
POL(g(x₁)) = 2 + 2·x₁   

(4) Obligation:

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

f(X, g(X), Y) → f(Y, Y, Y)

The replacement map contains the following entries:

f: empty set
g: {1}

(5) CSDependencyPairsProof (EQUIVALENT transformation)

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

(6) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {g} are replacing on all positions.
The symbols in {f, F} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

F(X, g(X), Y) → F(Y, Y, Y)

The TRS R consists of the following rules:

f(X, g(X), Y) → f(Y, Y, Y)

Q is empty.

(7) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
The rules F(z0, g(z0), z1) → F(z1, z1, z1) and F(x0, g(x0), x1) → F(x1, x1, x1) form no chain, because ECap^µ(F(z1, z1, z1)) = F(z1, z1, z1) does not unify with F(x0, g(x0), x1).

(8) TRUE