(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) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 2
POL(evenodd(x1, x2)) = 1 + 2·x1 + 2·x2
POL(false) = 1
POL(not(x1)) = x1
POL(s(x1)) = x1
POL(true) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
evenodd(0, s(0)) → false
(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(s(x), s(0)) → evenodd(x, 0)
Q is empty.
(3) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(4) 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(s(x), s(0)) → evenodd(x, 0)
The set Q consists of the following terms:
not(true)
not(false)
evenodd(x0, 0)
evenodd(s(x0), s(0))
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(6) 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(s(x), s(0)) → evenodd(x, 0)
The set Q consists of the following terms:
not(true)
not(false)
evenodd(x0, 0)
evenodd(s(x0), s(0))
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(8) 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(s(x), s(0)) → evenodd(x, 0)
The set Q consists of the following terms:
not(true)
not(false)
evenodd(x0, 0)
evenodd(s(x0), s(0))
We have to consider all minimal (P,Q,R)-chains.
(9) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(10) 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))
R is empty.
The set Q consists of the following terms:
not(true)
not(false)
evenodd(x0, 0)
evenodd(s(x0), s(0))
We have to consider all minimal (P,Q,R)-chains.
(11) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
not(true)
not(false)
evenodd(x0, 0)
evenodd(s(x0), s(0))
(12) 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))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- EVENODD(x, 0) → EVENODD(x, s(0))
The graph contains the following edges 1 >= 1
- EVENODD(s(x), s(0)) → EVENODD(x, 0)
The graph contains the following edges 1 > 1, 2 > 2
(14) YES