(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
Lexicographic path order with status [LPO].
Quasi-Precedence:
s1 > 0
Status:
s1: [1]
0: []
The following usable rules [FROCOS05] were oriented:
none
(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