File size: 1,796 Bytes
e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | import Mathlib
namespace OAI
noncomputable section
namespace Problem348
abbrev BinaryMatrix (n : ℕ) := Fin n → Fin n → Bool
abbrev IncreasingMap (k n : ℕ) := {f : Fin k → Fin n // StrictMono f}
def orderedCopies {k n : ℕ} (H : BinaryMatrix k) (A : BinaryMatrix n) :
Finset (IncreasingMap k n × IncreasingMap k n) := by
classical
exact Finset.univ.filter (fun rc =>
∀ i j, A ((rc.1).val i) ((rc.2).val j) = H i j)
def copyCount {k n : ℕ} (H : BinaryMatrix k) (A : BinaryMatrix n) : ℕ :=
(orderedCopies H A).card
def HFree {k n : ℕ} (H : BinaryMatrix k) (A : BinaryMatrix n) : Prop :=
copyCount H A = 0
def hammingDistance {n : ℕ} (A B : BinaryMatrix n) : ℕ := by
classical
exact (Finset.univ.filter (fun rc : Fin n × Fin n =>
A rc.1 rc.2 ≠ B rc.1 rc.2)).card
def etaBit (a b : ℕ) : Bool :=
decide (((a / (2 ^ (5 - b))) % 2) = 1)
def anchor64 (u v : Fin 64) : Bool :=
if 32 ≤ u.val ∧ 59 ≤ v.val then
etaBit (u.val - 32) (v.val - 58)
else
decide (u.val ≠ v.val)
def fixedH : BinaryMatrix 66 := fun u v =>
if hu : u.val < 64 then
if hv : v.val < 64 then
anchor64 ⟨u.val, hu⟩ ⟨v.val, hv⟩
else
decide ((v.val = 64 ∧ u.val = 0) ∨
(v.val = 65 ∧ u.val = 1))
else
if v.val < 64 then
decide ((u.val = 64 ∧ v.val = 0) ∨
(u.val = 65 ∧ v.val = 1))
else
decide (u.val = 65 ∨ v.val = 64)
def fixedMinEdits {n : ℕ} (A : BinaryMatrix n) : ℕ := by
classical
exact
(Finset.univ.image (fun B : BinaryMatrix n =>
if HFree fixedH B then hammingDistance A B else n * n + 1)).min'
(by simp)
def fixedDistance {n : ℕ} (A : BinaryMatrix n) : ℝ :=
(fixedMinEdits A : ℝ) / ((n : ℝ) ^ 2)
end Problem348
end
end OAI
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