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

(0) Obligation:

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

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
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

and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The TRS R 2 is

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))

The signature Sigma is {cond}

(2) Obligation:

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

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond(true, x0)
and(x0, false)
and(false, x0)
and(true, true)
even(0)
even(s(0))
even(s(s(x0)))
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(y))
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:

COND(true, x) → COND(and(even(x), gr(x, 0)), p(x))
COND(true, x) → AND(even(x), gr(x, 0))
COND(true, x) → EVEN(x)
COND(true, x) → GR(x, 0)
COND(true, x) → P(x)
EVEN(s(s(x))) → EVEN(x)
GR(s(x), s(y)) → GR(x, y)

The TRS R consists of the following rules:

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond(true, x0)
and(x0, false)
and(false, x0)
and(true, true)
even(0)
even(s(0))
even(s(s(x0)))
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(y))
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 2 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) 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:

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond(true, x0)
and(x0, false)
and(false, x0)
and(true, true)
even(0)
even(s(0))
even(s(s(x0)))
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(y))
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.

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)
cond(x1, x2)  =  cond(x1)
true  =  true
and(x1, x2)  =  and
even(x1)  =  even
gr(x1, x2)  =  gr
0  =  0
p(x1)  =  x1
false  =  false
y  =  y

Recursive path order with status [RPO].
Quasi-Precedence:

s₁ > [true, and, even, gr, 0, false, y]
cond₁ > [true, and, even, gr, 0, false, y]

Status:

s₁: [1]
cond₁: multiset
true: multiset
and: multiset
even: multiset
gr: multiset
0: multiset
false: multiset
y: multiset

The following usable rules [FROCOS05] were oriented:

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

(9) Obligation:

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

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond(true, x0)
and(x0, false)
and(false, x0)
and(true, true)
even(0)
even(s(0))
even(s(s(x0)))
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(y))
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:

COND(true, x) → COND(and(even(x), gr(x, 0)), p(x))

The TRS R consists of the following rules:

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond(true, x0)
and(x0, false)
and(false, x0)
and(true, true)
even(0)
even(s(0))
even(s(s(x0)))
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(y))
p(0)
p(s(x0))

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