Termination w.r.t. Q proof of TRS/TRCSRExIntrod_Zan97

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.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
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 [13,14,18] 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.