Termination w.r.t. Q proof of /tmp/tmp2hwJkF/ExIntrod_Zan97

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

    ↳13 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 PisEmptyProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 QDPOrderProof (⇔)

    ↳23 QDP

    ↳24 PisEmptyProof (⇔)

    ↳25 TRUE

  ↳26 QDP

    ↳27 QDPOrderProof (⇔)

    ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 PisEmptyProof (⇔)

    ↳32 TRUE

  ↳33 QDP

    ↳34 QDPOrderProof (⇔)

    ↳35 QDP

    ↳36 QDPOrderProof (⇔)

    ↳37 QDP

    ↳38 PisEmptyProof (⇔)

    ↳39 TRUE

  ↳40 QDP

    ↳41 QDPOrderProof (⇔)

    ↳42 QDP

    ↳43 QDPOrderProof (⇔)

    ↳44 QDP

    ↳45 QDPOrderProof (⇔)

    ↳46 QDP

    ↳47 QDPOrderProof (⇔)

    ↳48 QDP

    ↳49 QDPOrderProof (⇔)

    ↳50 QDP

    ↳51 PisEmptyProof (⇔)

    ↳52 TRUE

  ↳53 QDP

    ↳54 QDPOrderProof (⇔)

    ↳55 QDP

    ↳56 QDPOrderProof (⇔)

    ↳57 QDP

    ↳58 PisEmptyProof (⇔)

    ↳59 TRUE

  ↳60 QDP

(0) Obligation:

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

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

ACTIVE(fact(X)) → MARK(if(zero(X), s(0), prod(X, fact(p(X)))))
ACTIVE(fact(X)) → IF(zero(X), s(0), prod(X, fact(p(X))))
ACTIVE(fact(X)) → ZERO(X)
ACTIVE(fact(X)) → S(0)
ACTIVE(fact(X)) → PROD(X, fact(p(X)))
ACTIVE(fact(X)) → FACT(p(X))
ACTIVE(fact(X)) → P(X)
ACTIVE(add(0, X)) → MARK(X)
ACTIVE(add(s(X), Y)) → MARK(s(add(X, Y)))
ACTIVE(add(s(X), Y)) → S(add(X, Y))
ACTIVE(add(s(X), Y)) → ADD(X, Y)
ACTIVE(prod(0, X)) → MARK(0)
ACTIVE(prod(s(X), Y)) → MARK(add(Y, prod(X, Y)))
ACTIVE(prod(s(X), Y)) → ADD(Y, prod(X, Y))
ACTIVE(prod(s(X), Y)) → PROD(X, Y)
ACTIVE(if(true, X, Y)) → MARK(X)
ACTIVE(if(false, X, Y)) → MARK(Y)
ACTIVE(zero(0)) → MARK(true)
ACTIVE(zero(s(X))) → MARK(false)
ACTIVE(p(s(X))) → MARK(X)
MARK(fact(X)) → ACTIVE(fact(mark(X)))
MARK(fact(X)) → FACT(mark(X))
MARK(fact(X)) → MARK(X)
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))
MARK(if(X1, X2, X3)) → IF(mark(X1), X2, X3)
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(zero(X)) → ACTIVE(zero(mark(X)))
MARK(zero(X)) → ZERO(mark(X))
MARK(zero(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(0) → ACTIVE(0)
MARK(prod(X1, X2)) → ACTIVE(prod(mark(X1), mark(X2)))
MARK(prod(X1, X2)) → PROD(mark(X1), mark(X2))
MARK(prod(X1, X2)) → MARK(X1)
MARK(prod(X1, X2)) → MARK(X2)
MARK(p(X)) → ACTIVE(p(mark(X)))
MARK(p(X)) → P(mark(X))
MARK(p(X)) → MARK(X)
MARK(add(X1, X2)) → ACTIVE(add(mark(X1), mark(X2)))
MARK(add(X1, X2)) → ADD(mark(X1), mark(X2))
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)
MARK(true) → ACTIVE(true)
MARK(false) → ACTIVE(false)
FACT(mark(X)) → FACT(X)
FACT(active(X)) → FACT(X)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)
ZERO(mark(X)) → ZERO(X)
ZERO(active(X)) → ZERO(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
PROD(mark(X1), X2) → PROD(X1, X2)
PROD(X1, mark(X2)) → PROD(X1, X2)
PROD(active(X1), X2) → PROD(X1, X2)
PROD(X1, active(X2)) → PROD(X1, X2)
P(mark(X)) → P(X)
P(active(X)) → P(X)
ADD(mark(X1), X2) → ADD(X1, X2)
ADD(X1, mark(X2)) → ADD(X1, X2)
ADD(active(X1), X2) → ADD(X1, X2)
ADD(X1, active(X2)) → ADD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 8 SCCs with 23 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ADD(X1, mark(X2)) → ADD(X1, X2)
ADD(mark(X1), X2) → ADD(X1, X2)
ADD(active(X1), X2) → ADD(X1, X2)
ADD(X1, active(X2)) → ADD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ADD(X1, mark(X2)) → ADD(X1, X2)
ADD(X1, active(X2)) → ADD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ADD(x1, x2) = x2
mark(x1) = mark(x1)
active(x1) = active(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

ADD(mark(X1), X2) → ADD(X1, X2)
ADD(active(X1), X2) → ADD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ADD(mark(X1), X2) → ADD(X1, X2)
ADD(active(X1), X2) → ADD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ADD(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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

P(active(X)) → P(X)
P(mark(X)) → P(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P(active(X)) → P(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
P(x1) = x1
active(x1) = active(x1)
mark(x1) = x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

P(mark(X)) → P(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P(mark(X)) → P(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > P₁

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(17) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(18) TRUE

(19) Obligation:

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

PROD(X1, mark(X2)) → PROD(X1, X2)
PROD(mark(X1), X2) → PROD(X1, X2)
PROD(active(X1), X2) → PROD(X1, X2)
PROD(X1, active(X2)) → PROD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROD(X1, mark(X2)) → PROD(X1, X2)
PROD(X1, active(X2)) → PROD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROD(x1, x2) = x2
mark(x1) = mark(x1)
active(x1) = active(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

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

PROD(mark(X1), X2) → PROD(X1, X2)
PROD(active(X1), X2) → PROD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROD(mark(X1), X2) → PROD(X1, X2)
PROD(active(X1), X2) → PROD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROD(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(24) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(25) TRUE

(26) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = x1
active(x1) = active(x1)
mark(x1) = x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(28) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > S₁

The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(31) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(32) TRUE

(33) Obligation:

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

ZERO(active(X)) → ZERO(X)
ZERO(mark(X)) → ZERO(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ZERO(active(X)) → ZERO(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZERO(x1) = x1
active(x1) = active(x1)
mark(x1) = x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(35) Obligation:

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

ZERO(mark(X)) → ZERO(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ZERO(mark(X)) → ZERO(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > ZERO₁

The following usable rules [FROCOS05] were oriented: none

(37) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(38) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(39) TRUE

(40) Obligation:

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

IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3) = x3
mark(x1) = mark(x1)
active(x1) = active(x1)

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(42) Obligation:

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

IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

IF(X1, mark(X2), X3) → IF(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3) = IF(x2)
mark(x1) = mark(x1)
active(x1) = x1

Recursive Path Order [RPO].
Precedence:

mark₁ > IF₁

The following usable rules [FROCOS05] were oriented: none

(44) Obligation:

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

IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

IF(X1, active(X2), X3) → IF(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3) = IF(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)

Recursive Path Order [RPO].
Precedence:

mark₁ > IF₁
active₁ > IF₁

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

IF(active(X1), X2, X3) → IF(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3) = IF(x1, x2, x3)
mark(x1) = x1
active(x1) = active(x1)

Recursive Path Order [RPO].
Precedence:

active₁ > IF₃

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

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

IF(mark(X1), X2, X3) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

IF(mark(X1), X2, X3) → IF(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3) = IF(x1)
mark(x1) = mark(x1)

Recursive Path Order [RPO].
Precedence:

mark₁ > IF₁

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(51) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(52) TRUE

(53) Obligation:

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

FACT(active(X)) → FACT(X)
FACT(mark(X)) → FACT(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FACT(active(X)) → FACT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FACT(x1) = x1
active(x1) = active(x1)
mark(x1) = x1

Recursive Path Order [RPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

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

FACT(mark(X)) → FACT(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FACT(mark(X)) → FACT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:

mark₁ > FACT₁

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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

(58) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(59) TRUE

(60) Obligation:

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

MARK(fact(X)) → ACTIVE(fact(mark(X)))
ACTIVE(fact(X)) → MARK(if(zero(X), s(0), prod(X, fact(p(X)))))
MARK(fact(X)) → MARK(X)
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))
ACTIVE(add(0, X)) → MARK(X)
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(zero(X)) → ACTIVE(zero(mark(X)))
ACTIVE(add(s(X), Y)) → MARK(s(add(X, Y)))
MARK(zero(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(prod(s(X), Y)) → MARK(add(Y, prod(X, Y)))
MARK(s(X)) → MARK(X)
MARK(prod(X1, X2)) → ACTIVE(prod(mark(X1), mark(X2)))
ACTIVE(if(true, X, Y)) → MARK(X)
MARK(prod(X1, X2)) → MARK(X1)
MARK(prod(X1, X2)) → MARK(X2)
MARK(p(X)) → ACTIVE(p(mark(X)))
ACTIVE(if(false, X, Y)) → MARK(Y)
MARK(p(X)) → MARK(X)
MARK(add(X1, X2)) → ACTIVE(add(mark(X1), mark(X2)))
ACTIVE(p(s(X))) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
mark(fact(X)) → active(fact(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(zero(X)) → active(zero(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2)))
mark(p(X)) → active(p(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(true) → active(true)
mark(false) → active(false)
fact(mark(X)) → fact(X)
fact(active(X)) → fact(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
zero(mark(X)) → zero(X)
zero(active(X)) → zero(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
prod(mark(X1), X2) → prod(X1, X2)
prod(X1, mark(X2)) → prod(X1, X2)
prod(active(X1), X2) → prod(X1, X2)
prod(X1, active(X2)) → prod(X1, X2)
p(mark(X)) → p(X)
p(active(X)) → p(X)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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