Created
March 21, 2010 11:16
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Binary search division
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ultraDiv ∷ Integer → Integer → (Integer, Integer) | |
-- pre: 1 <= d <= n | |
-- post: a*d + b == n | |
-- && (A) x > d . (a*x > n) | |
-- where | |
-- (a, b) = ultraDiv n d | |
ultraDiv n d = bDiv n d 1 n | |
where | |
-- pre: same as ultraDiv | |
-- && d * lower <= n && d * upper > n | |
-- post: same as ultraDiv | |
-- invariant: lower <= middle < upper | |
-- && (A) x in [0, lower] . (x * d <= n) | |
-- && (A) x in (upper..] . (x * d > n) | |
-- variant: upper-lower | |
bDiv n d lower upper | |
| lower ≡ upper-1 = (lower, n - d*lower) | |
| middle*d > n = bDiv n d lower middle | |
| otherwise = bDiv n d middle upper | |
where | |
middle = (upper + lower) `div` 2 | |
ultraDiv2 ∷ Integer → Integer → (Integer, Integer) | |
-- pre: 1 <= d <= n | |
-- post: a*d + b == n | |
-- && (A) x > d . (a * x > n) | |
-- where | |
-- (a, b) = ultraDiv2 n d | |
ultraDiv2 n d = bDiv n d 1 n | |
where | |
-- pre: same as ultraDiv2 | |
-- && d * lower <= n && d * (lower+i) >= n | |
-- post: same as ultraDiv2 | |
-- invariant: lower <= lower+i | |
-- && (A) x in [0, lower] . (x * d <= n) | |
-- && (A) x in [lower+i..] . (x * d >= n) | |
-- variant: i | |
bDiv n d lower i | |
| i ≡ 0 = (lower, n - d*lower) | |
| (lower+i)*d ≤ n = bDiv n d (lower+i) (i `div` 2) | |
| otherwise = bDiv n d lower (i `div` 2) |
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