Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 QDP
↳7 QDPOrderProof (⇔)
↳8 QDP
↳9 DependencyGraphProof (⇔)
↳10 TRUE

(0) Obligation:

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

not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)

Q is empty.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

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

not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)

The set Q consists of the following terms:

not(true)
not(false)
evenodd(x0, 0)
evenodd(0, s(0))
evenodd(s(x0), s(0))

(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:

EVENODD(x, 0) → NOT(evenodd(x, s(0)))
EVENODD(x, 0) → EVENODD(x, s(0))
EVENODD(s(x), s(0)) → EVENODD(x, 0)

The TRS R consists of the following rules:

not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)

The set Q consists of the following terms:

not(true)
not(false)
evenodd(x0, 0)
evenodd(0, s(0))
evenodd(s(x0), s(0))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(6) Obligation:

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

EVENODD(s(x), s(0)) → EVENODD(x, 0)
EVENODD(x, 0) → EVENODD(x, s(0))

The TRS R consists of the following rules:

not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)

The set Q consists of the following terms:

not(true)
not(false)
evenodd(x0, 0)
evenodd(0, s(0))
evenodd(s(x0), s(0))

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


EVENODD(s(x), s(0)) → EVENODD(x, 0)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
EVENODD(x1, x2)  =  x1
s(x1)  =  s(x1)
0  =  0
not(x1)  =  not
true  =  true
false  =  false
evenodd(x1, x2)  =  x2

Lexicographic path order with status [LPO].
Quasi-Precedence:
[0, not, false] > s1
[0, not, false] > true

Status:
true: []
not: []
false: []
s1: [1]
0: []


The following usable rules [FROCOS05] were oriented:

not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)

(8) Obligation:

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

EVENODD(x, 0) → EVENODD(x, s(0))

The TRS R consists of the following rules:

not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)

The set Q consists of the following terms:

not(true)
not(false)
evenodd(x0, 0)
evenodd(0, s(0))
evenodd(s(x0), s(0))

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(10) TRUE