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c = totalcost(bs))
∧
(subbudget(b
1
(bf)) =
B
∧
subbudget(b
2
(bf))
=
B
∧
cost(b
1
(bf))
>
cost(b
2
(bf))
⇒
∧
bs = subbudget(b
2
(bf))
∧
c = cost(b
2
(bf)))
(subbudget(b
1
(bf))
=
B
∧
subbudget(b
2
(bf)) =
B
∧
cost(b
1
(bf))
≤
cost(b
2
(bf))
⇒
bs = subbudget(b
1
(bf))
∧
c = cost(b
1
(bf)))
∧
(subbudget(b
1
(bf))
=
B
∧
subbudget(b
2
(bf)) =
B
∧
cost(b
1
(bf))
>
cost(b
2
(bf))
⇒
bs = scale(subbudget(b
2
(bf)), cost(b
2
(bf))/cost(b
1
(bf)))
∧
c = totalcost(bs))
∧
(subbudget(b
1
(bf))
=
B
∧
subbudget(b
2
(bf))
=
B
∧
bs = meet(subbudget(b
1
(bf)), subbudget(b
2
(bf)))
∧
c = totalcost(bs))) ]
end,
join: B
×
B
→
B
join(b
1
,b
2
)
≡
if b
1
=
B
∨
b
2
=
B
then
B
else
[ bf
→
mk_CF(c,bs)
|
bf:BF, c:C, bs:B
•
(bf
∈
dom b
2
⇒
bs = join(subbudget(b
1
(bf)), subbudget(b
2
(bf)))
dom b
1
∧
bf
∈
∧
c = totalcost(subbudget(b
1
(bf))),
+ totalcost(subbudget(b
2
(bf))))
∧
(bf
∈
dom b
2
⇒
bs = subbudget(b
1
(bf))
dom b
1
∧
bf
∈
∧
c = cost(b
1
(bf)))
∧
(bf
∈
dom b
2
⇒
bs = subbudget(b
2
(bf))
dom b
1
∧
bf
∈
∧
c = cost(b
2
(bf))) ]
end
Let
bf
ij..k
range over values of type BF, and let
c
ij..k
range over values of type
C. Budgets then have a sub-structure of the following general form (here in one
of many schematic unfoldings):
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