1. 2011-03-01 ISS

2. (Presburger )

3. IIJ
Ticket #4242 (hackage)
Data.Map operations can produce invalid trees
Description
With 6.13.20100801:
Prelude Data.Map> valid $ deleteMin $
deleteMin $ fromList [ (i, ()) | i <-
[0,2,5,1,6,4,8,9,7,11,10,3] ]
False

4. Tree

5. ( 10,000 )

6. template = <<EOF
non_neg := { [a,b,c,d,e] :
a >= 0 &&
b >= 0 &&
c >= 0 &&
d >= 0 &&
e >= 0};
double0 := { [a,b,c,d,e] :
(!((gamma * (1 + e) <= 20 * (1 + (c + d + 1)) &&
(delta * (a + 1) >= 10* (b + 1) &&
delta * (b + 1) >= 10* (a + 1) &&
!
((delta * (a + 1) >= 10* (b + 1 + 1) &&
delta * (b + 1 + 1) >= 10* (a + 1))) &&
c + d + 1 + e + 1 = b + 1 &&
delta * (c + 1) >= 10* (d + 1) &&
delta * (d + 1) >= 10* (c + 1) &&
delta * (c + d + 1 + 1) >= 10* (e + 1) &&

7. : Presburger
(0, 1, +, <)
: Omega, Prestaf, LASH,
Mona
(0, 1, +, ×, <)

10. (in Coq)

11. Data.Map
Coq
IIJ-II( IIJ )
Haskell Data.Map ( )
SLIB, MIT-Scheme
Journal of Functional Programming

12. ( )

13. :
:
A B
A B
A A
A A

14. :
:
A A
A A

16. :
1. Waitfree ML
2.
ML
3.

17. •
•
•
•