Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

a__f1(X) -> a__if3(mark1(X), c, f1(true))
a__if3(true, X, Y) -> mark1(X)
a__if3(false, X, Y) -> mark1(Y)
mark1(f1(X)) -> a__f1(mark1(X))
mark1(if3(X1, X2, X3)) -> a__if3(mark1(X1), mark1(X2), X3)
mark1(c) -> c
mark1(true) -> true
mark1(false) -> false
a__f1(X) -> f1(X)
a__if3(X1, X2, X3) -> if3(X1, X2, X3)

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

a__f1(X) -> a__if3(mark1(X), c, f1(true))
a__if3(true, X, Y) -> mark1(X)
a__if3(false, X, Y) -> mark1(Y)
mark1(f1(X)) -> a__f1(mark1(X))
mark1(if3(X1, X2, X3)) -> a__if3(mark1(X1), mark1(X2), X3)
mark1(c) -> c
mark1(true) -> true
mark1(false) -> false
a__f1(X) -> f1(X)
a__if3(X1, X2, X3) -> if3(X1, X2, X3)

Q is empty.

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

MARK1(if3(X1, X2, X3)) -> MARK1(X1)
A__F1(X) -> A__IF3(mark1(X), c, f1(true))
MARK1(f1(X)) -> A__F1(mark1(X))
A__IF3(false, X, Y) -> MARK1(Y)
A__F1(X) -> MARK1(X)
A__IF3(true, X, Y) -> MARK1(X)
MARK1(if3(X1, X2, X3)) -> MARK1(X2)
MARK1(if3(X1, X2, X3)) -> A__IF3(mark1(X1), mark1(X2), X3)
MARK1(f1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__f1(X) -> a__if3(mark1(X), c, f1(true))
a__if3(true, X, Y) -> mark1(X)
a__if3(false, X, Y) -> mark1(Y)
mark1(f1(X)) -> a__f1(mark1(X))
mark1(if3(X1, X2, X3)) -> a__if3(mark1(X1), mark1(X2), X3)
mark1(c) -> c
mark1(true) -> true
mark1(false) -> false
a__f1(X) -> f1(X)
a__if3(X1, X2, X3) -> if3(X1, X2, X3)

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

↳ QTRS
  ↳ DependencyPairsProof
QDP

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

MARK1(if3(X1, X2, X3)) -> MARK1(X1)
A__F1(X) -> A__IF3(mark1(X), c, f1(true))
MARK1(f1(X)) -> A__F1(mark1(X))
A__IF3(false, X, Y) -> MARK1(Y)
A__F1(X) -> MARK1(X)
A__IF3(true, X, Y) -> MARK1(X)
MARK1(if3(X1, X2, X3)) -> MARK1(X2)
MARK1(if3(X1, X2, X3)) -> A__IF3(mark1(X1), mark1(X2), X3)
MARK1(f1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__f1(X) -> a__if3(mark1(X), c, f1(true))
a__if3(true, X, Y) -> mark1(X)
a__if3(false, X, Y) -> mark1(Y)
mark1(f1(X)) -> a__f1(mark1(X))
mark1(if3(X1, X2, X3)) -> a__if3(mark1(X1), mark1(X2), X3)
mark1(c) -> c
mark1(true) -> true
mark1(false) -> false
a__f1(X) -> f1(X)
a__if3(X1, X2, X3) -> if3(X1, X2, X3)

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