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

0 QTRS
↳1 AAECC Innermost (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
  ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 PisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 QDP

(0) Obligation:

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

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is

neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

The TRS R 2 is

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))

The signature Sigma is {cond2, cond1}

(2) Obligation:

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

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0)
cond2(true, x0)
cond2(false, x0)
neq(0, 0)
neq(0, s(x0))
neq(s(x0), 0)
neq(s(x0), s(y))
even(0)
even(s(0))
even(s(s(x0)))
div2(0)
div2(s(0))
div2(s(s(x0)))
p(0)
p(s(x0))

(3) DependencyPairsProof (EQUIVALENT transformation)

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

(4) Obligation:

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

COND1(true, x) → COND2(even(x), x)
COND1(true, x) → EVEN(x)
COND2(true, x) → COND1(neq(x, 0), div2(x))
COND2(true, x) → NEQ(x, 0)
COND2(true, x) → DIV2(x)
COND2(false, x) → COND1(neq(x, 0), p(x))
COND2(false, x) → NEQ(x, 0)
COND2(false, x) → P(x)
NEQ(s(x), s(y)) → NEQ(x, y)
EVEN(s(s(x))) → EVEN(x)
DIV2(s(s(x))) → DIV2(x)

The TRS R consists of the following rules:

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0)
cond2(true, x0)
cond2(false, x0)
neq(0, 0)
neq(0, s(x0))
neq(s(x0), 0)
neq(s(x0), s(y))
even(0)
even(s(0))
even(s(s(x0)))
div2(0)
div2(s(0))
div2(s(s(x0)))
p(0)
p(s(x0))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

DIV2(s(s(x))) → DIV2(x)

The TRS R consists of the following rules:

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0)
cond2(true, x0)
cond2(false, x0)
neq(0, 0)
neq(0, s(x0))
neq(s(x0), 0)
neq(s(x0), s(y))
even(0)
even(s(0))
even(s(s(x0)))
div2(0)
div2(s(0))
div2(s(s(x0)))
p(0)
p(s(x0))

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.


DIV2(s(s(x))) → DIV2(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DIV2(x1)  =  x1
s(x1)  =  s(x1)
cond1(x1, x2)  =  x2
true  =  true
cond2(x1, x2)  =  x2
even(x1)  =  even(x1)
neq(x1, x2)  =  x1
0  =  0
div2(x1)  =  x1
false  =  false
p(x1)  =  x1
y  =  y

Lexicographic path order with status [LPO].
Precedence:
s1 > 0 > true > even1 > false
s1 > y

Status:
y: []
even1: [1]
true: []
false: []
s1: [1]
0: []

The following usable rules [FROCOS05] were oriented:

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

(9) Obligation:

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

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0)
cond2(true, x0)
cond2(false, x0)
neq(0, 0)
neq(0, s(x0))
neq(s(x0), 0)
neq(s(x0), s(y))
even(0)
even(s(0))
even(s(s(x0)))
div2(0)
div2(s(0))
div2(s(s(x0)))
p(0)
p(s(x0))

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:

EVEN(s(s(x))) → EVEN(x)

The TRS R consists of the following rules:

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0)
cond2(true, x0)
cond2(false, x0)
neq(0, 0)
neq(0, s(x0))
neq(s(x0), 0)
neq(s(x0), s(y))
even(0)
even(s(0))
even(s(s(x0)))
div2(0)
div2(s(0))
div2(s(s(x0)))
p(0)
p(s(x0))

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.


EVEN(s(s(x))) → EVEN(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
EVEN(x1)  =  x1
s(x1)  =  s(x1)
cond1(x1, x2)  =  x2
true  =  true
cond2(x1, x2)  =  x2
even(x1)  =  even(x1)
neq(x1, x2)  =  x1
0  =  0
div2(x1)  =  x1
false  =  false
p(x1)  =  x1
y  =  y

Lexicographic path order with status [LPO].
Precedence:
s1 > 0 > true > even1 > false
s1 > y

Status:
y: []
even1: [1]
true: []
false: []
s1: [1]
0: []

The following usable rules [FROCOS05] were oriented:

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

(14) Obligation:

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

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0)
cond2(true, x0)
cond2(false, x0)
neq(0, 0)
neq(0, s(x0))
neq(s(x0), 0)
neq(s(x0), s(y))
even(0)
even(s(0))
even(s(s(x0)))
div2(0)
div2(s(0))
div2(s(s(x0)))
p(0)
p(s(x0))

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

(15) PisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE

(17) Obligation:

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

COND2(true, x) → COND1(neq(x, 0), div2(x))
COND1(true, x) → COND2(even(x), x)
COND2(false, x) → COND1(neq(x, 0), p(x))

The TRS R consists of the following rules:

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0)
cond2(true, x0)
cond2(false, x0)
neq(0, 0)
neq(0, s(x0))
neq(s(x0), 0)
neq(s(x0), s(y))
even(0)
even(s(0))
even(s(s(x0)))
div2(0)
div2(s(0))
div2(s(s(x0)))
p(0)
p(s(x0))

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