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 QDPOrderProof (⇔)
    ↳11 QDP
    ↳12 PisEmptyProof (⇔)
    ↳13 TRUE
  ↳14 QDP
    ↳15 QDPOrderProof (⇔)
    ↳16 QDP
    ↳17 QDPOrderProof (⇔)
    ↳18 QDP
    ↳19 PisEmptyProof (⇔)
    ↳20 TRUE
  ↳21 QDP
    ↳22 QDPOrderProof (⇔)
    ↳23 QDP
    ↳24 QDPOrderProof (⇔)
    ↳25 QDP
    ↳26 QDPOrderProof (⇔)
    ↳27 QDP
    ↳28 PisEmptyProof (⇔)
    ↳29 TRUE
  ↳30 QDP
    ↳31 QDPOrderProof (⇔)
    ↳32 QDP
    ↳33 QDPOrderProof (⇔)
    ↳34 QDP
    ↳35 PisEmptyProof (⇔)
    ↳36 TRUE
  ↳37 QDP
    ↳38 QDPOrderProof (⇔)
    ↳39 QDP
    ↳40 QDPOrderProof (⇔)
    ↳41 QDP
    ↳42 PisEmptyProof (⇔)
    ↳43 TRUE
  ↳44 QDP
    ↳45 QDPOrderProof (⇔)
    ↳46 QDP
    ↳47 QDPOrderProof (⇔)
    ↳48 QDP
    ↳49 PisEmptyProof (⇔)
    ↳50 TRUE
  ↳51 QDP
    ↳52 QDPOrderProof (⇔)
    ↳53 QDP
    ↳54 QDPOrderProof (⇔)
    ↳55 QDP
    ↳56 PisEmptyProof (⇔)
    ↳57 TRUE
  ↳58 QDP
    ↳59 QDPOrderProof (⇔)
    ↳60 QDP
    ↳61 QDPOrderProof (⇔)
    ↳62 QDP
    ↳63 QDPOrderProof (⇔)
    ↳64 QDP
    ↳65 PisEmptyProof (⇔)
    ↳66 TRUE
  ↳67 QDP
    ↳68 QDPOrderProof (⇔)
    ↳69 QDP
    ↳70 QDPOrderProof (⇔)
    ↳71 QDP
    ↳72 QDPOrderProof (⇔)
    ↳73 QDP
    ↳74 QDPOrderProof (⇔)
    ↳75 QDP
    ↳76 QDPOrderProof (⇔)
    ↳77 QDP
    ↳78 PisEmptyProof (⇔)
    ↳79 TRUE
  ↳80 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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 10 SCCs with 28 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(ok(X1), ok(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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lexicographic Path Order [LPO].
Precedence:
mark1 > ADD1


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

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(ok(X1), ok(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)  =  ADD(x2)
mark(x1)  =  mark
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:
ok1 > [ADD1, mark]


The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

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

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

(10) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(11) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

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

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

Lexicographic Path Order [LPO].
Precedence:
[P1, ok1]


The following usable rules [FROCOS05] were oriented: none

(16) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(17) 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: Combined order from the following AFS and order.
P(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(18) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

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

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(X1, mark(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)  =  PROD(x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
mark1 > PROD1


The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

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

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

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

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROD(ok(X1), ok(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)  =  PROD(x2)
mark(x1)  =  mark
ok(x1)  =  ok(x1)

Lexicographic Path Order [LPO].
Precedence:
ok1 > [PROD1, mark]


The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

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

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

(26) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(27) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

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

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

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
[S1, ok1]


The following usable rules [FROCOS05] were oriented: none

(32) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(33) 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: Combined order from the following AFS and order.
S(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(34) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) Obligation:

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

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

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
[ZERO1, ok1]


The following usable rules [FROCOS05] were oriented: none

(39) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(40) 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: Combined order from the following AFS and order.
ZERO(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(41) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(42) PisEmptyProof (EQUIVALENT transformation)

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

(43) TRUE

(44) Obligation:

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

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

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(ok(X1), ok(X2), ok(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(x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
ok1 > 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)

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

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(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(48) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(49) PisEmptyProof (EQUIVALENT transformation)

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

(50) TRUE

(51) Obligation:

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

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

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
[FACT1, ok1]


The following usable rules [FROCOS05] were oriented: none

(53) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(mark(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
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(55) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(56) PisEmptyProof (EQUIVALENT transformation)

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

(57) TRUE

(58) Obligation:

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

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

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

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(if(X1, X2, X3)) → PROPER(X1)
PROPER(if(X1, X2, X3)) → PROPER(X2)
PROPER(if(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → PROPER(X)
PROPER(prod(X1, X2)) → PROPER(X1)
PROPER(prod(X1, X2)) → PROPER(X2)
PROPER(p(X)) → PROPER(X)
PROPER(add(X1, X2)) → PROPER(X1)
PROPER(add(X1, X2)) → PROPER(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
if(x1, x2, x3)  =  if(x1, x2, x3)
fact(x1)  =  x1
zero(x1)  =  x1
s(x1)  =  s(x1)
prod(x1, x2)  =  prod(x1, x2)
p(x1)  =  p(x1)
add(x1, x2)  =  add(x1, x2)

Lexicographic Path Order [LPO].
Precedence:
s1 > PROPER1
prod2 > PROPER1
add2 > PROPER1


The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

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

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

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(fact(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
fact(x1)  =  fact(x1)
zero(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
[PROPER1, fact1]


The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

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

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

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

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(zero(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
zero(x1)  =  zero(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(64) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(65) PisEmptyProof (EQUIVALENT transformation)

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

(66) TRUE

(67) Obligation:

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

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

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

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(prod(X1, X2)) → ACTIVE(X1)
ACTIVE(prod(X1, X2)) → ACTIVE(X2)
ACTIVE(add(X1, X2)) → ACTIVE(X1)
ACTIVE(add(X1, X2)) → ACTIVE(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
if(x1, x2, x3)  =  x1
fact(x1)  =  x1
zero(x1)  =  x1
s(x1)  =  x1
prod(x1, x2)  =  prod(x1, x2)
p(x1)  =  x1
add(x1, x2)  =  add(x1, x2)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(69) Obligation:

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

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

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

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
if(x1, x2, x3)  =  if(x1)
fact(x1)  =  x1
zero(x1)  =  x1
s(x1)  =  x1
p(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(71) Obligation:

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

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

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(zero(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
fact(x1)  =  x1
zero(x1)  =  zero(x1)
s(x1)  =  s(x1)
p(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

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

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

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(fact(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
fact(x1)  =  fact(x1)
p(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
[ACTIVE1, fact1]


The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

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

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

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(p(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
p(x1)  =  p(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(77) 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)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(78) PisEmptyProof (EQUIVALENT transformation)

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

(79) TRUE

(80) Obligation:

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

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

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