(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) 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:
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)
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 1 SCC with 1 less node.
(4) 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)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
0 = 0
s(x1) = s(x1)
From the DPs we obtained the following set of size-change graphs:
- EVENODD(x, 0) → EVENODD(x, s(0)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1
- EVENODD(s(x), s(0)) → EVENODD(x, 0) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 > 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(6) TRUE