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__fact1(X) -> a__if3(a__zero1(mark1(X)), s1(0), prod2(X, fact1(p1(X))))
a__add2(0, X) -> mark1(X)
a__add2(s1(X), Y) -> s1(a__add2(mark1(X), mark1(Y)))
a__prod2(0, X) -> 0
a__prod2(s1(X), Y) -> a__add2(mark1(Y), a__prod2(mark1(X), mark1(Y)))
a__if3(true, X, Y) -> mark1(X)
a__if3(false, X, Y) -> mark1(Y)
a__zero1(0) -> true
a__zero1(s1(X)) -> false
a__p1(s1(X)) -> mark1(X)
mark1(fact1(X)) -> a__fact1(mark1(X))
mark1(if3(X1, X2, X3)) -> a__if3(mark1(X1), X2, X3)
mark1(zero1(X)) -> a__zero1(mark1(X))
mark1(prod2(X1, X2)) -> a__prod2(mark1(X1), mark1(X2))
mark1(p1(X)) -> a__p1(mark1(X))
mark1(add2(X1, X2)) -> a__add2(mark1(X1), mark1(X2))
mark1(s1(X)) -> s1(mark1(X))
mark1(0) -> 0
mark1(true) -> true
mark1(false) -> false
a__fact1(X) -> fact1(X)
a__if3(X1, X2, X3) -> if3(X1, X2, X3)
a__zero1(X) -> zero1(X)
a__prod2(X1, X2) -> prod2(X1, X2)
a__p1(X) -> p1(X)
a__add2(X1, X2) -> add2(X1, X2)

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

a__fact1(X) -> a__if3(a__zero1(mark1(X)), s1(0), prod2(X, fact1(p1(X))))
a__add2(0, X) -> mark1(X)
a__add2(s1(X), Y) -> s1(a__add2(mark1(X), mark1(Y)))
a__prod2(0, X) -> 0
a__prod2(s1(X), Y) -> a__add2(mark1(Y), a__prod2(mark1(X), mark1(Y)))
a__if3(true, X, Y) -> mark1(X)
a__if3(false, X, Y) -> mark1(Y)
a__zero1(0) -> true
a__zero1(s1(X)) -> false
a__p1(s1(X)) -> mark1(X)
mark1(fact1(X)) -> a__fact1(mark1(X))
mark1(if3(X1, X2, X3)) -> a__if3(mark1(X1), X2, X3)
mark1(zero1(X)) -> a__zero1(mark1(X))
mark1(prod2(X1, X2)) -> a__prod2(mark1(X1), mark1(X2))
mark1(p1(X)) -> a__p1(mark1(X))
mark1(add2(X1, X2)) -> a__add2(mark1(X1), mark1(X2))
mark1(s1(X)) -> s1(mark1(X))
mark1(0) -> 0
mark1(true) -> true
mark1(false) -> false
a__fact1(X) -> fact1(X)
a__if3(X1, X2, X3) -> if3(X1, X2, X3)
a__zero1(X) -> zero1(X)
a__prod2(X1, X2) -> prod2(X1, X2)
a__p1(X) -> p1(X)
a__add2(X1, X2) -> add2(X1, X2)

Q is empty.

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

MARK1(prod2(X1, X2)) -> MARK1(X2)
MARK1(prod2(X1, X2)) -> MARK1(X1)
MARK1(fact1(X)) -> MARK1(X)
MARK1(add2(X1, X2)) -> MARK1(X2)
A__IF3(false, X, Y) -> MARK1(Y)
A__FACT1(X) -> A__IF3(a__zero1(mark1(X)), s1(0), prod2(X, fact1(p1(X))))
A__ADD2(0, X) -> MARK1(X)
MARK1(if3(X1, X2, X3)) -> MARK1(X1)
A__FACT1(X) -> A__ZERO1(mark1(X))
A__ADD2(s1(X), Y) -> A__ADD2(mark1(X), mark1(Y))
A__IF3(true, X, Y) -> MARK1(X)
A__FACT1(X) -> MARK1(X)
A__P1(s1(X)) -> MARK1(X)
A__PROD2(s1(X), Y) -> A__PROD2(mark1(X), mark1(Y))
MARK1(add2(X1, X2)) -> A__ADD2(mark1(X1), mark1(X2))
MARK1(p1(X)) -> MARK1(X)
MARK1(p1(X)) -> A__P1(mark1(X))
A__ADD2(s1(X), Y) -> MARK1(X)
A__PROD2(s1(X), Y) -> MARK1(X)
MARK1(zero1(X)) -> MARK1(X)
MARK1(s1(X)) -> MARK1(X)
MARK1(add2(X1, X2)) -> MARK1(X1)
A__ADD2(s1(X), Y) -> MARK1(Y)
A__PROD2(s1(X), Y) -> MARK1(Y)
MARK1(prod2(X1, X2)) -> A__PROD2(mark1(X1), mark1(X2))
A__PROD2(s1(X), Y) -> A__ADD2(mark1(Y), a__prod2(mark1(X), mark1(Y)))
MARK1(if3(X1, X2, X3)) -> A__IF3(mark1(X1), X2, X3)
MARK1(zero1(X)) -> A__ZERO1(mark1(X))
MARK1(fact1(X)) -> A__FACT1(mark1(X))

The TRS R consists of the following rules:

a__fact1(X) -> a__if3(a__zero1(mark1(X)), s1(0), prod2(X, fact1(p1(X))))
a__add2(0, X) -> mark1(X)
a__add2(s1(X), Y) -> s1(a__add2(mark1(X), mark1(Y)))
a__prod2(0, X) -> 0
a__prod2(s1(X), Y) -> a__add2(mark1(Y), a__prod2(mark1(X), mark1(Y)))
a__if3(true, X, Y) -> mark1(X)
a__if3(false, X, Y) -> mark1(Y)
a__zero1(0) -> true
a__zero1(s1(X)) -> false
a__p1(s1(X)) -> mark1(X)
mark1(fact1(X)) -> a__fact1(mark1(X))
mark1(if3(X1, X2, X3)) -> a__if3(mark1(X1), X2, X3)
mark1(zero1(X)) -> a__zero1(mark1(X))
mark1(prod2(X1, X2)) -> a__prod2(mark1(X1), mark1(X2))
mark1(p1(X)) -> a__p1(mark1(X))
mark1(add2(X1, X2)) -> a__add2(mark1(X1), mark1(X2))
mark1(s1(X)) -> s1(mark1(X))
mark1(0) -> 0
mark1(true) -> true
mark1(false) -> false
a__fact1(X) -> fact1(X)
a__if3(X1, X2, X3) -> if3(X1, X2, X3)
a__zero1(X) -> zero1(X)
a__prod2(X1, X2) -> prod2(X1, X2)
a__p1(X) -> p1(X)
a__add2(X1, X2) -> add2(X1, X2)

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

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

MARK1(prod2(X1, X2)) -> MARK1(X2)
MARK1(prod2(X1, X2)) -> MARK1(X1)
MARK1(fact1(X)) -> MARK1(X)
MARK1(add2(X1, X2)) -> MARK1(X2)
A__IF3(false, X, Y) -> MARK1(Y)
A__FACT1(X) -> A__IF3(a__zero1(mark1(X)), s1(0), prod2(X, fact1(p1(X))))
A__ADD2(0, X) -> MARK1(X)
MARK1(if3(X1, X2, X3)) -> MARK1(X1)
A__FACT1(X) -> A__ZERO1(mark1(X))
A__ADD2(s1(X), Y) -> A__ADD2(mark1(X), mark1(Y))
A__IF3(true, X, Y) -> MARK1(X)
A__FACT1(X) -> MARK1(X)
A__P1(s1(X)) -> MARK1(X)
A__PROD2(s1(X), Y) -> A__PROD2(mark1(X), mark1(Y))
MARK1(add2(X1, X2)) -> A__ADD2(mark1(X1), mark1(X2))
MARK1(p1(X)) -> MARK1(X)
MARK1(p1(X)) -> A__P1(mark1(X))
A__ADD2(s1(X), Y) -> MARK1(X)
A__PROD2(s1(X), Y) -> MARK1(X)
MARK1(zero1(X)) -> MARK1(X)
MARK1(s1(X)) -> MARK1(X)
MARK1(add2(X1, X2)) -> MARK1(X1)
A__ADD2(s1(X), Y) -> MARK1(Y)
A__PROD2(s1(X), Y) -> MARK1(Y)
MARK1(prod2(X1, X2)) -> A__PROD2(mark1(X1), mark1(X2))
A__PROD2(s1(X), Y) -> A__ADD2(mark1(Y), a__prod2(mark1(X), mark1(Y)))
MARK1(if3(X1, X2, X3)) -> A__IF3(mark1(X1), X2, X3)
MARK1(zero1(X)) -> A__ZERO1(mark1(X))
MARK1(fact1(X)) -> A__FACT1(mark1(X))

The TRS R consists of the following rules:

a__fact1(X) -> a__if3(a__zero1(mark1(X)), s1(0), prod2(X, fact1(p1(X))))
a__add2(0, X) -> mark1(X)
a__add2(s1(X), Y) -> s1(a__add2(mark1(X), mark1(Y)))
a__prod2(0, X) -> 0
a__prod2(s1(X), Y) -> a__add2(mark1(Y), a__prod2(mark1(X), mark1(Y)))
a__if3(true, X, Y) -> mark1(X)
a__if3(false, X, Y) -> mark1(Y)
a__zero1(0) -> true
a__zero1(s1(X)) -> false
a__p1(s1(X)) -> mark1(X)
mark1(fact1(X)) -> a__fact1(mark1(X))
mark1(if3(X1, X2, X3)) -> a__if3(mark1(X1), X2, X3)
mark1(zero1(X)) -> a__zero1(mark1(X))
mark1(prod2(X1, X2)) -> a__prod2(mark1(X1), mark1(X2))
mark1(p1(X)) -> a__p1(mark1(X))
mark1(add2(X1, X2)) -> a__add2(mark1(X1), mark1(X2))
mark1(s1(X)) -> s1(mark1(X))
mark1(0) -> 0
mark1(true) -> true
mark1(false) -> false
a__fact1(X) -> fact1(X)
a__if3(X1, X2, X3) -> if3(X1, X2, X3)
a__zero1(X) -> zero1(X)
a__prod2(X1, X2) -> prod2(X1, X2)
a__p1(X) -> p1(X)
a__add2(X1, X2) -> add2(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 1 SCC with 2 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP

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

MARK1(prod2(X1, X2)) -> MARK1(X2)
MARK1(prod2(X1, X2)) -> MARK1(X1)
MARK1(fact1(X)) -> MARK1(X)
MARK1(add2(X1, X2)) -> MARK1(X2)
A__IF3(false, X, Y) -> MARK1(Y)
A__FACT1(X) -> A__IF3(a__zero1(mark1(X)), s1(0), prod2(X, fact1(p1(X))))
A__ADD2(0, X) -> MARK1(X)
MARK1(if3(X1, X2, X3)) -> MARK1(X1)
A__ADD2(s1(X), Y) -> A__ADD2(mark1(X), mark1(Y))
A__FACT1(X) -> MARK1(X)
A__IF3(true, X, Y) -> MARK1(X)
A__P1(s1(X)) -> MARK1(X)
A__PROD2(s1(X), Y) -> A__PROD2(mark1(X), mark1(Y))
MARK1(add2(X1, X2)) -> A__ADD2(mark1(X1), mark1(X2))
MARK1(p1(X)) -> MARK1(X)
MARK1(p1(X)) -> A__P1(mark1(X))
A__ADD2(s1(X), Y) -> MARK1(X)
A__PROD2(s1(X), Y) -> MARK1(X)
MARK1(zero1(X)) -> MARK1(X)
MARK1(add2(X1, X2)) -> MARK1(X1)
MARK1(s1(X)) -> MARK1(X)
A__ADD2(s1(X), Y) -> MARK1(Y)
A__PROD2(s1(X), Y) -> MARK1(Y)
MARK1(prod2(X1, X2)) -> A__PROD2(mark1(X1), mark1(X2))
A__PROD2(s1(X), Y) -> A__ADD2(mark1(Y), a__prod2(mark1(X), mark1(Y)))
MARK1(if3(X1, X2, X3)) -> A__IF3(mark1(X1), X2, X3)
MARK1(fact1(X)) -> A__FACT1(mark1(X))

The TRS R consists of the following rules:

a__fact1(X) -> a__if3(a__zero1(mark1(X)), s1(0), prod2(X, fact1(p1(X))))
a__add2(0, X) -> mark1(X)
a__add2(s1(X), Y) -> s1(a__add2(mark1(X), mark1(Y)))
a__prod2(0, X) -> 0
a__prod2(s1(X), Y) -> a__add2(mark1(Y), a__prod2(mark1(X), mark1(Y)))
a__if3(true, X, Y) -> mark1(X)
a__if3(false, X, Y) -> mark1(Y)
a__zero1(0) -> true
a__zero1(s1(X)) -> false
a__p1(s1(X)) -> mark1(X)
mark1(fact1(X)) -> a__fact1(mark1(X))
mark1(if3(X1, X2, X3)) -> a__if3(mark1(X1), X2, X3)
mark1(zero1(X)) -> a__zero1(mark1(X))
mark1(prod2(X1, X2)) -> a__prod2(mark1(X1), mark1(X2))
mark1(p1(X)) -> a__p1(mark1(X))
mark1(add2(X1, X2)) -> a__add2(mark1(X1), mark1(X2))
mark1(s1(X)) -> s1(mark1(X))
mark1(0) -> 0
mark1(true) -> true
mark1(false) -> false
a__fact1(X) -> fact1(X)
a__if3(X1, X2, X3) -> if3(X1, X2, X3)
a__zero1(X) -> zero1(X)
a__prod2(X1, X2) -> prod2(X1, X2)
a__p1(X) -> p1(X)
a__add2(X1, X2) -> add2(X1, X2)

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